lenstronomy.LensModel.Profiles package¶
Submodules¶
lenstronomy.LensModel.Profiles.arc_perturbations module¶
- class ArcPerturbations[source]¶
Bases:
LensProfileBase
Uses radial and tangential fourier modes within a specific range in both directions to perturb a lensing potential.
- function(x, y, coeff, d_r, d_phi, center_x, center_y)[source]¶
- Parameters:
x – x-coordinate
y – y-coordinate
coeff – float, amplitude of basis
d_r – period of radial sinusoidal in units of angle
d_phi – period of tangential sinusoidal in radian
center_x – center of rotation for tangential basis
center_y – center of rotation for tangential basis
- Returns:
- derivatives(x, y, coeff, d_r, d_phi, center_x, center_y)[source]¶
- Parameters:
x – x-coordinate
y – y-coordinate
coeff – float, amplitude of basis
d_r – period of radial sinusoidal in units of angle
d_phi – period of tangential sinusoidal in radian
center_x – center of rotation for tangential basis
center_y – center of rotation for tangential basis
- Returns:
f_x, f_y
- hessian(x, y, coeff, d_r, d_phi, center_x, center_y)[source]¶
- Parameters:
x – x-coordinate
y – y-coordinate
coeff – float, amplitude of basis
d_r – period of radial sinusoidal in units of angle
d_phi – period of tangential sinusoidal in radian
center_x – center of rotation for tangential basis
center_y – center of rotation for tangential basis
- Returns:
f_xx, f_yy, f_xy
lenstronomy.LensModel.Profiles.base_profile module¶
- class LensProfileBase(*args, **kwargs)[source]¶
Bases:
object
This class acts as the base class of all lens model functions and indicates raise statements and default outputs if these functions are not defined in the specific lens model class.
To implement a new lens model profile you should: 1. make a new python file in this folder 2. create a class inheriting this class; YourModel(LensProfileBase) 3. write new definitions following the same input and output conventions as this base class
function(x, y, <other parameters>) derivatives(x, y, <other parameters>) hessian(x, y, <other parameters>)
- set the variables for sampling the new profile
param_names = [“param1”, “param2”, …] lower_limit_default = {“param1”: value, “param2: value, …} upper_limit_default = {“param1”: value, “param2: value, …}
give the new profile a meaningful name and add it in the LensModel.profile_list_base class
write test functions in the test/test_LensModel/test_Profiles folder with a new file with test_<profile name>.py
With that, you should be good to go and import and use it for any purpose. Further definitions in the class are optional and only used for certain applications (such as kinematics)
- function(*args, **kwargs)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- derivatives(*args, **kwargs)[source]¶
Deflection angles.
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- hessian(*args, **kwargs)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- density_lens(*args, **kwargs)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity. (optional definition)
\[\kappa(x, y) = \int_{-\infty}^{\infty} \rho(x, y, z) dz\]- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- mass_3d_lens(*args, **kwargs)[source]¶
Mass enclosed a 3d sphere or radius r given a lens parameterization with angular units The input parameter are identical as for the derivatives definition. (optional definition)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- mass_2d_lens(*args, **kwargs)[source]¶
Two-dimensional enclosed mass at radius r (optional definition)
\[M_{2d}(R) = \int_{0}^{R} \rho_{2d}(r) 2\pi r dr\]with \(\rho_{2d}(r)\) is the density_2d_lens() definition
The mass definition is such that:
\[\alpha = mass_2d / r / \pi\]with alpha is the deflection angle
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- set_static(**kwargs)[source]¶
Pre-computes certain computations that do only relate to the lens model parameters and not to the specific position where to evaluate the lens model.
- Parameters:
kwargs – lens model parameters
- Returns:
no return, for certain lens model some private self variables are initiated
lenstronomy.LensModel.Profiles.chameleon module¶
- class Chameleon(static=False)[source]¶
Bases:
LensProfileBase
Class of the Chameleon model (See Suyu+2014) an elliptical truncated double isothermal profile.
- param_names = ['alpha_1', 'w_c', 'w_t', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'alpha_1': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.8, 'e2': -0.8, 'w_c': 0, 'w_t': 0}¶
- upper_limit_default = {'alpha_1': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.8, 'e2': 0.8, 'w_c': 100, 'w_t': 100}¶
- function(x, y, alpha_1, w_c, w_t, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – ra-coordinate
y – dec-coordinate
alpha_1 – deflection angle at 1 (arcseconds) from the center
w_c – see Suyu+2014
w_t – see Suyu+2014
e1 – ellipticity parameter
e2 – ellipticity parameter
center_x – ra center
center_y – dec center
- Returns:
lensing potential
- derivatives(x, y, alpha_1, w_c, w_t, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – ra-coordinate
y – dec-coordinate
alpha_1 – deflection angle at 1 (arcseconds) from the center
w_c – see Suyu+2014
w_t – see Suyu+2014
e1 – ellipticity parameter
e2 – ellipticity parameter
center_x – ra center
center_y – dec center
- Returns:
deflection angles (RA, DEC)
- hessian(x, y, alpha_1, w_c, w_t, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – ra-coordinate
y – dec-coordinate
alpha_1 – deflection angle at 1 (arcseconds) from the center
w_c – see Suyu+2014
w_t – see Suyu+2014
e1 – ellipticity parameter
e2 – ellipticity parameter
center_x – ra center
center_y – dec center
- Returns:
second derivatives of the lensing potential (Hessian: f_xx, f_xy, f_yx, f_yy)
- density_lens(r, alpha_1, w_c, w_t, e1=0, e2=0, center_x=0, center_y=0)[source]¶
Spherical average density as a function of 3d radius.
- Parameters:
r – 3d radius
alpha_1 – deflection angle at 1 (arcseconds) from the center
w_c – see Suyu+2014
w_t – see Suyu+2014
e1 – ellipticity parameter
e2 – ellipticity parameter
center_x – ra center
center_y – dec center
- Returns:
matter density at 3d radius r
- mass_3d_lens(r, alpha_1, w_c, w_t, e1=0, e2=0, center_x=0, center_y=0)[source]¶
Mass enclosed 3d radius.
- Parameters:
r – 3d radius
alpha_1 – deflection angle at 1 (arcseconds) from the center
w_c – see Suyu+2014
w_t – see Suyu+2014
e1 – ellipticity parameter
e2 – ellipticity parameter
center_x – ra center
center_y – dec center
- Returns:
mass enclosed 3d radius r
- param_convert(alpha_1, w_c, w_t, e1, e2)[source]¶
Convert the parameter alpha_1 (deflection angle one arcsecond from the center) into the “Einstein radius” scale parameter of the two NIE profiles.
- Parameters:
alpha_1 – deflection angle at 1 (arcseconds) from the center
w_c – see Suyu+2014
w_t – see Suyu+2014
e1 – eccentricity modulus
e2 – eccentricity modulus
- Returns:
- class DoubleChameleon[source]¶
Bases:
LensProfileBase
Class of the Chameleon model (See Suyu+2014) an elliptical truncated double isothermal profile.
- param_names = ['alpha_1', 'ratio', 'w_c1', 'w_t1', 'e11', 'e21', 'w_c2', 'w_t2', 'e12', 'e22', 'center_x', 'center_y']¶
- lower_limit_default = {'alpha_1': 0, 'center_x': -100, 'center_y': -100, 'e11': -0.8, 'e12': -0.8, 'e21': -0.8, 'e22': -0.8, 'ratio': 0, 'w_c1': 0, 'w_c2': 0, 'w_t1': 0, 'w_t2': 0}¶
- upper_limit_default = {'alpha_1': 100, 'center_x': 100, 'center_y': 100, 'e11': 0.8, 'e12': 0.8, 'e21': 0.8, 'e22': 0.8, 'ratio': 100, 'w_c1': 100, 'w_c2': 100, 'w_t1': 100, 'w_t2': 100}¶
- function(x, y, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
- Parameters:
x – ra-coordinate
y – dec-coordinate
alpha_1 – deflection angle at 1 (arcseconds) from the center
ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
w_c1 – Suyu+2014 for first profile
w_t1 – Suyu+2014 for first profile
e11 – ellipticity parameter for first profile
e21 – ellipticity parameter for first profile
w_c2 – Suyu+2014 for second profile
w_t2 – Suyu+2014 for second profile
e12 – ellipticity parameter for second profile
e22 – ellipticity parameter for second profile
center_x – ra center
center_y – dec center
- Returns:
lensing potential
- derivatives(x, y, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
- Parameters:
x – ra-coordinate
y – dec-coordinate
alpha_1 – deflection angle at 1 (arcseconds) from the center
ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
w_c1 – Suyu+2014 for first profile
w_t1 – Suyu+2014 for first profile
e11 – ellipticity parameter for first profile
e21 – ellipticity parameter for first profile
w_c2 – Suyu+2014 for second profile
w_t2 – Suyu+2014 for second profile
e12 – ellipticity parameter for second profile
e22 – ellipticity parameter for second profile^V
center_x – ra center
center_y – dec center
- Returns:
deflection angles (RA, DEC)
- hessian(x, y, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
- Parameters:
x – ra-coordinate
y – dec-coordinate
alpha_1 – deflection angle at 1 (arcseconds) from the center
ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
w_c1 – Suyu+2014 for first profile
w_t1 – Suyu+2014 for first profile
e11 – ellipticity parameter for first profile
e21 – ellipticity parameter for first profile
w_c2 – Suyu+2014 for second profile
w_t2 – Suyu+2014 for second profile
e12 – ellipticity parameter for second profile
e22 – ellipticity parameter for second profile
center_x – ra center
center_y – dec center
- Returns:
second derivatives of the lensing potential (Hessian: f_xx, f_yy, f_xy)
- density_lens(r, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
- Parameters:
r – 3d radius
alpha_1 – deflection angle at 1 (arcseconds) from the center
ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
w_c1 – Suyu+2014 for first profile
w_t1 – Suyu+2014 for first profile
e11 – ellipticity parameter for first profile
e21 – ellipticity parameter for first profile
w_c2 – Suyu+2014 for second profile
w_t2 – Suyu+2014 for second profile
e12 – ellipticity parameter for second profile
e22 – ellipticity parameter for second profile
center_x – ra center
center_y – dec center
- Returns:
3d density at radius r
- mass_3d_lens(r, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
- Parameters:
r – 3d radius
alpha_1 – deflection angle at 1 (arcseconds) from the center
ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
w_c1 – Suyu+2014 for first profile
w_t1 – Suyu+2014 for first profile
e11 – ellipticity parameter for first profile
e21 – ellipticity parameter for first profile
w_c2 – Suyu+2014 for second profile
w_t2 – Suyu+2014 for second profile
e12 – ellipticity parameter for second profile
e22 – ellipticity parameter for second profile
center_x – ra center
center_y – dec center
- Returns:
mass enclosed 3d radius
- set_static(alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
Pre-computes certain computations that do only relate to the lens model parameters and not to the specific position where to evaluate the lens model.
- Parameters:
kwargs – lens model parameters
- Returns:
no return, for certain lens model some private self variables are initiated
- class TripleChameleon[source]¶
Bases:
LensProfileBase
Class of the Chameleon model (See Suyu+2014) an elliptical truncated double isothermal profile.
- param_names = ['alpha_1', 'ratio12', 'ratio13', 'w_c1', 'w_t1', 'e11', 'e21', 'w_c2', 'w_t2', 'e12', 'e22', 'w_c3', 'w_t3', 'e13', 'e23', 'center_x', 'center_y']¶
- lower_limit_default = {'alpha_1': 0, 'center_x': -100, 'center_y': -100, 'e11': -0.8, 'e12': -0.8, 'e13': -0.8, 'e21': -0.8, 'e22': -0.8, 'e23': -0.8, 'ratio12': 0, 'ratio13': 0, 'w_c1': 0, 'w_c2': 0, 'w_c3': 0, 'w_t1': 0, 'w_t2': 0, 'w_t3': 0}¶
- upper_limit_default = {'alpha_1': 100, 'center_x': 100, 'center_y': 100, 'e11': 0.8, 'e12': 0.8, 'e13': 0.8, 'e21': 0.8, 'e22': 0.8, 'e23': 0.8, 'ratio12': 100, 'ratio13': 100, 'w_c1': 100, 'w_c2': 100, 'w_c3': 100, 'w_t1': 100, 'w_t2': 100, 'w_t3': 100}¶
- function(x, y, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]¶
- Parameters:
alpha_1 –
ratio12 – ratio of first to second amplitude
ratio13 – ratio of first to third amplitude
w_c1 –
w_t1 –
e11 –
e21 –
w_c2 –
w_t2 –
e12 –
e22 –
center_x –
center_y –
- Returns:
- derivatives(x, y, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]¶
- Parameters:
alpha_1 –
ratio12 – ratio of first to second amplitude
ratio13 – ratio of first to third amplidute
w_c1 –
w_t1 –
e11 –
e21 –
w_c2 –
w_t2 –
e12 –
e22 –
center_x –
center_y –
- Returns:
- hessian(x, y, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]¶
- Parameters:
alpha_1 –
ratio12 – ratio of first to second amplitude
ratio13 – ratio of first to third amplidute
w_c1 –
w_t1 –
e11 –
e21 –
w_c2 –
w_t2 –
e12 –
e22 –
center_x –
center_y –
- Returns:
- density_lens(r, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]¶
- Parameters:
r – 3d radius
alpha_1 –
ratio12 – ratio of first to second amplitude
ratio13 – ratio of first to third amplitude
w_c1 –
w_t1 –
e11 –
e21 –
w_c2 –
w_t2 –
e12 –
e22 –
center_x –
center_y –
- Returns:
density at radius r (spherical average)
- mass_3d_lens(r, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]¶
- Parameters:
r – 3d radius
alpha_1 –
ratio12 – ratio of first to second amplitude
ratio13 – ratio of first to third amplitude
w_c1 –
w_t1 –
e11 –
e21 –
w_c2 –
w_t2 –
e12 –
e22 –
center_x –
center_y –
- Returns:
mass enclosed 3d radius
- set_static(alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]¶
Pre-computes certain computations that do only relate to the lens model parameters and not to the specific position where to evaluate the lens model.
- Parameters:
kwargs – lens model parameters
- Returns:
no return, for certain lens model some private self variables are initiated
- class DoubleChameleonPointMass[source]¶
Bases:
LensProfileBase
Class of the Chameleon model (See Suyu+2014) an elliptical truncated double isothermal profile.
- param_names = ['alpha_1', 'ratio_chameleon', 'ratio_pointmass', 'w_c1', 'w_t1', 'e11', 'e21', 'w_c2', 'w_t2', 'e12', 'e22', 'center_x', 'center_y']¶
- lower_limit_default = {'alpha_1': 0, 'center_x': -100, 'center_y': -100, 'e11': -0.8, 'e12': -0.8, 'e21': -0.8, 'e22': -0.8, 'ratio_chameleon': 0, 'ratio_pointmass': 0, 'w_c1': 0, 'w_c2': 0, 'w_t1': 0, 'w_t2': 0}¶
- upper_limit_default = {'alpha_1': 100, 'center_x': 100, 'center_y': 100, 'e11': 0.8, 'e12': 0.8, 'e21': 0.8, 'e22': 0.8, 'ratio_chameleon': 100, 'ratio_pointmass': 100, 'w_c1': 100, 'w_c2': 100, 'w_t1': 100, 'w_t2': 100}¶
- function(x, y, alpha_1, ratio_pointmass, ratio_chameleon, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
#TODO chose better parameterization for combining point mass and Chameleon profiles
- Parameters:
x – ra-coordinate
y – dec-coordinate
alpha_1 – deflection angle at 1 (arcseconds) from the center
ratio_pointmass – ratio of point source Einstein radius to combined Chameleon deflection angle at r=1
ratio_chameleon – ratio in deflection angles at r=1 for the two Chameleon profiles
w_c1 – Suyu+2014 for first profile
w_t1 – Suyu+2014 for first profile
e11 – ellipticity parameter for first profile
e21 – ellipticity parameter for first profile
w_c2 – Suyu+2014 for second profile
w_t2 – Suyu+2014 for second profile
e12 – ellipticity parameter for second profile
e22 – ellipticity parameter for second profile
center_x – ra center
center_y – dec center
- Returns:
lensing potential
- derivatives(x, y, alpha_1, ratio_pointmass, ratio_chameleon, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
- Parameters:
x –
y –
alpha_1 –
ratio_pointmass – ratio of point source Einstein radius to combined Chameleon deflection angle at r=1
ratio_chameleon – ratio in deflection angles at r=1 for the two Chameleon profiles
w_c1 – Suyu+2014 for first profile
w_t1 – Suyu+2014 for first profile
e11 – ellipticity parameter for first profile
e21 – ellipticity parameter for first profile
w_c2 – Suyu+2014 for second profile
w_t2 – Suyu+2014 for second profile
e12 – ellipticity parameter for second profile
e22 – ellipticity parameter for second profile
center_x – ra center
center_y – dec center
- Returns:
- hessian(x, y, alpha_1, ratio_pointmass, ratio_chameleon, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]¶
- Parameters:
x –
y –
alpha_1 –
ratio_pointmass – ratio of point source Einstein radius to combined Chameleon deflection angle at r=1
ratio_chameleon – ratio in deflection angles at r=1 for the two Chameleon profiles
w_c1 – Suyu+2014 for first profile
w_t1 – Suyu+2014 for first profile
e11 – ellipticity parameter for first profile
e21 – ellipticity parameter for first profile
w_c2 – Suyu+2014 for second profile
w_t2 – Suyu+2014 for second profile
e12 – ellipticity parameter for second profile
e22 – ellipticity parameter for second profile
center_x – ra center
center_y – dec center
- Returns:
lenstronomy.LensModel.Profiles.cnfw module¶
- class CNFW[source]¶
Bases:
LensProfileBase
this class computes the lensing quantities of a cored NFW profile: rho = rho0 * (r + r_core)^-1 * (r + rs)^-2 alpha_Rs is the normalization equivalent to the deflection angle at rs in the absence of a core
- model_name = 'CNFW'¶
- param_names = ['Rs', 'alpha_Rs', 'r_core', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'r_core': 0}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'r_core': 100}¶
- function(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]¶
- Parameters:
x – angular position
y – angular position
Rs – angular turn over point
alpha_Rs – deflection at Rs (in the absence of a core
r_core – core radius
center_x – center of halo
center_y – center of halo
- Returns:
- derivatives(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]¶
Deflection angles.
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- hessian(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy.
- density(R, Rs, rho0, r_core)[source]¶
Three dimensional truncated NFW profile.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (central core density)
- Returns:
rho(R) density
- density_lens(R, Rs, alpha_Rs, r_core)[source]¶
Computes the density at 3d radius r given lens model parameterization.
The integral in the LOS projection of this quantity results in the convergence quantity.
- density_2d(x, y, Rs, rho0, r_core, center_x=0, center_y=0)[source]¶
Projected two dimenstional NFW profile (kappa*Sigma_crit)
- Parameters:
x (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
- Returns:
Epsilon(R) projected density at radius R
- mass_3d(R, Rs, rho0, r_core)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
R –
Rs –
rho0 –
r_core –
- Returns:
- mass_3d_lens(R, Rs, alpha_Rs, r_core)[source]¶
Mass enclosed a 3d sphere or radius r given a lens parameterization with angular units.
- Returns:
- alpha_r(R, Rs, rho0, r_core)[source]¶
Deflection angel of NFW profile along the radial direction.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
- Returns:
Epsilon(R) projected density at radius R
- cnfw_gamma(R, Rs, rho0, r_core, ax_x, ax_y)[source]¶
Shear gamma of NFW profile (times Sigma_crit) along the projection to coordinate ‘axis’.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
- Returns:
Epsilon(R) projected density at radius R
lenstronomy.LensModel.Profiles.cnfw_ellipse module¶
- class CNFW_ELLIPSE[source]¶
Bases:
LensProfileBase
This class contains functions concerning the NFW profile.
relation are: R_200 = c * Rs
- param_names = ['Rs', 'alpha_Rs', 'r_core', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'r_core': 0}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'r_core': 100}¶
- function(x, y, Rs, alpha_Rs, r_core, e1, e2, center_x=0, center_y=0)[source]¶
Returns double integral of NFW profile.
- derivatives(x, y, Rs, alpha_Rs, r_core, e1, e2, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function (integral of NFW)
- hessian(x, y, Rs, alpha_Rs, r_core, e1, e2, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy.
lenstronomy.LensModel.Profiles.const_mag module¶
- class ConstMag(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class implements the macromodel potential of Diego et al.
<https://www.aanda.org/articles/aa/pdf/2019/07/aa35490-19.pdf>`_ Convergence and shear are computed according to Diego2018
- param_names = ['center_x', 'center_y', 'mu_r', 'mu_t', 'parity', 'phi_G']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'mu_r': 1, 'mu_t': 1000, 'parity': -1, 'phi_G': 0.0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'mu_r': 1, 'mu_t': 1000, 'parity': 1, 'phi_G': 3.141592653589793}¶
- function(x, y, mu_r, mu_t, parity, phi_G, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coord (in angles)
y – y-coord (in angles)
mu_r – radial magnification
mu_t – tangential magnification
parity – parity side of the macromodel. Either +1 (positive parity) or -1 (negative parity)
phi_G – shear orientation angle (relative to the x-axis)
- Returns:
lensing potential
- derivatives(x, y, mu_r, mu_t, parity, phi_G, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coord (in angles)
y – y-coord (in angles)
mu_r – radial magnification
mu_t – tangential magnification
parity – parity of the side of the macromodel. Either +1 (positive parity) or -1 (negative parity)
phi_G – shear orientation angle (relative to the x-axis)
- Returns:
deflection angle (in angles)
- hessian(x, y, mu_r, mu_t, parity, phi_G, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coord (in angles)
y – y-coord (in angles)
mu_r – radial magnification
mu_t – tangential magnification
parity – parity of the side of the macromodel. Either +1 (positive parity) or -1 (negative parity)
phi_G – shear orientation angle (relative to the x-axis)
- Returns:
hessian matrix (in angles)
lenstronomy.LensModel.Profiles.constant_shift module¶
- class Shift(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Lens model with a constant shift of the deflection field.
- param_names = ['alpha_x', 'alpha_y']¶
- lower_limit_default = {'alpha_x': -1000, 'alpha_y': -1000}¶
- upper_limit_default = {'alpha_x': 1000, 'alpha_y': 1000}¶
- function(x, y, alpha_x, alpha_y)[source]¶
- Parameters:
x – coordinate in image plane (angle)
y – coordinate in image plane (angle)
alpha_x – shift in x-direction (angle)
alpha_y – shift in y-direction (angle)
- Returns:
lensing potential
lenstronomy.LensModel.Profiles.convergence module¶
- class Convergence(*args, **kwargs)[source]¶
Bases:
LensProfileBase
A single mass sheet (external convergence)
- model_name = 'CONVERGENCE'¶
- param_names = ['kappa', 'ra_0', 'dec_0']¶
- lower_limit_default = {'dec_0': -100, 'kappa': -10, 'ra_0': -100}¶
- upper_limit_default = {'dec_0': 100, 'kappa': 10, 'ra_0': 100}¶
- function(x, y, kappa, ra_0=0, dec_0=0)[source]¶
Lensing potential.
- Parameters:
x – x-coordinate
y – y-coordinate
kappa – (external) convergence
- Returns:
lensing potential
- derivatives(x, y, kappa, ra_0=0, dec_0=0)[source]¶
Deflection angle.
- Parameters:
x – x-coordinate
y – y-coordinate
kappa – (external) convergence
- Returns:
deflection angles (first order derivatives)
- hessian(x, y, kappa, ra_0=0, dec_0=0)[source]¶
Hessian matrix.
- Parameters:
x – x-coordinate
y – y-coordinate
kappa – external convergence
ra_0 – zero point of polynomial expansion (no deflection added)
dec_0 – zero point of polynomial expansion (no deflection added)
- Returns:
second order derivatives f_xx, f_xy, f_yx, f_yy
lenstronomy.LensModel.Profiles.coreBurkert module¶
- class CoreBurkert(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Lensing properties of a modified Burkert profile with variable core size normalized by rho0, the central core density.
- param_names = ['Rs', 'alpha_Rs', 'r_core', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 1, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'r_core': 0.5}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 100, 'center_x': 100, 'center_y': 100, 'r_core': 50}¶
- function(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]¶
- Parameters:
x – angular position
y – angular position
Rs – angular turn over point
alpha_Rs – deflection angle at Rs
center_x – center of halo
center_y – center of halo
- Returns:
- derivatives(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]¶
Deflection angles :param x: x coordinate :param y: y coordinate :param Rs: scale radius :param alpha_Rs: deflection angle at Rs :param r_core: core radius :param center_x:
- Parameters:
center_y –
- Returns:
- hessian(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]¶
- Parameters:
x – x coordinate
y – y coordinate
Rs – scale radius
alpha_Rs – deflection angle at Rs
r_core – core radius
center_x –
center_y –
- Returns:
- mass_2d(R, Rs, rho0, r_core)[source]¶
Analytic solution of the projection integral (convergence)
- Parameters:
R – projected distance
Rs – scale radius
rho0 – central core density
r_core – core radius
- coreBurkAlpha(R, Rs, rho0, r_core, ax_x, ax_y)[source]¶
Deflection angle.
- Parameters:
R –
Rs –
rho0 –
r_core –
ax_x –
ax_y –
- Returns:
- density(R, Rs, rho0, r_core)[source]¶
Three dimensional cored Burkert profile.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – characteristic density
- Returns:
rho(R) density
- density_2d(x, y, Rs, rho0, r_core, center_x=0, center_y=0)[source]¶
Projected two dimenstional core Burkert profile (kappa*Sigma_crit)
- Parameters:
x – x coordinate
y – y coordinate
Rs – scale radius
rho0 – central core density
r_core – core radius
- mass_3d(R, Rs, rho0, r_core)[source]¶
- Parameters:
R – projected distance
Rs – scale radius
rho0 – central core density
r_core – core radius
lenstronomy.LensModel.Profiles.cored_density module¶
- class CoredDensity(*args, **kwargs)[source]¶
Bases:
LensProfileBase
class for a uniform cored density dropping steep in the outskirts This profile is e.g. featured in Blum et al. 2020 https://arxiv.org/abs/2001.07182v1
- ..math::
rho_c(r) = frac{2}{pi} Sigma_{c} R_c^3 left(R_c^2 + r^2 right)^{-2}
with the convergence profile as
- ..math::
kappa_c(theta) = left(1 + frac{theta^2}{theta_c^2} right)^{-3/2}.
An approximate mass-sheet degeneracy can then be written as
- ..math::
kappa_{lambda_c}(theta) = lambda_c kappa(theta) + (1-lambda_c) kappa_c(theta).
- param_names = ['sigma0', 'r_core', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'r_core': 0, 'sigma0': -1}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'r_core': 100, 'sigma0': 10}¶
- function(x, y, sigma0, r_core, center_x=0, center_y=0)[source]¶
Potential of cored density profile.
- Parameters:
x – x-coordinate in angular units
y – y-coordinate in angular units
sigma0 – convergence in the core
r_core – core radius
center_x – center of the profile
center_y – center of the profile
- Returns:
lensing potential at (x, y)
- derivatives(x, y, sigma0, r_core, center_x=0, center_y=0)[source]¶
Deflection angle of cored density profile.
- Parameters:
x – x-coordinate in angular units
y – y-coordinate in angular units
sigma0 – convergence in the core
r_core – core radius
center_x – center of the profile
center_y – center of the profile
- Returns:
alpha_x, alpha_y at (x, y)
- hessian(x, y, sigma0, r_core, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in angular units
y – y-coordinate in angular units
sigma0 – convergence in the core
r_core – core radius
center_x – center of the profile
center_y – center of the profile
- Returns:
Hessian df/dxdx, df/dxdy, df/dydx, df/dydy at position (x, y)
- static alpha_r(r, sigma0, r_core)[source]¶
Radial deflection angle of the cored density profile.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
deflection angle
- static d_alpha_dr(r, sigma0, r_core)[source]¶
Radial derivatives of the radial deflection angle.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
dalpha/dr
- static kappa_r(r, sigma0, r_core)[source]¶
Convergence of the cored density profile. This routine is also for testing.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
convergence at r
- static density(r, sigma0, r_core)[source]¶
Rho(r) = 2/pi * Sigma_crit R_c**3 * (R_c**2 + r**2)**(-2)
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
density at radius r
- density_lens(r, sigma0, r_core)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
desnity at radius r
- density_2d(x, y, sigma0, r_core, center_x=0, center_y=0)[source]¶
Projected density at projected radius r.
- Parameters:
x – x-coordinate in angular units
y – y-coordinate in angular units
sigma0 – convergence in the core
r_core – core radius
center_x – center of the profile
center_y – center of the profile
- Returns:
projected density
- mass_2d(r, sigma0, r_core)[source]¶
Mass enclosed in cylinder of radius r.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
mass enclosed in cylinder of radius r
lenstronomy.LensModel.Profiles.cored_density_2 module¶
- class CoredDensity2(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Class for a uniform cored density dropping steep in the outskirts credits for suggesting this profile goes to Kfir Blum.
\[\rho(r) = 2/\pi * \Sigma_{\rm crit} R_c^2 * (R_c^2 + r^2)^{-3/2}\]This profile drops like an NFW profile as math:rho(r)^{-3}.
- model_name = 'CORED_DENSITY_2'¶
- param_names = ['sigma0', 'r_core', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'r_core': 0, 'sigma0': -1}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'r_core': 100, 'sigma0': 10}¶
- function(x, y, sigma0, r_core, center_x=0, center_y=0)[source]¶
Potential of cored density profile.
- Parameters:
x – x-coordinate in angular units
y – y-coordinate in angular units
sigma0 – convergence in the core
r_core – core radius
center_x – center of the profile
center_y – center of the profile
- Returns:
lensing potential at (x, y)
- derivatives(x, y, sigma0, r_core, center_x=0, center_y=0)[source]¶
Deflection angle of cored density profile.
- Parameters:
x – x-coordinate in angular units
y – y-coordinate in angular units
sigma0 – convergence in the core
r_core – core radius
center_x – center of the profile
center_y – center of the profile
- Returns:
alpha_x, alpha_y at (x, y)
- hessian(x, y, sigma0, r_core, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in angular units
y – y-coordinate in angular units
sigma0 – convergence in the core
r_core – core radius
center_x – center of the profile
center_y – center of the profile
- Returns:
Hessian df/dxdx, df/dxdy, df/dydx, df/dydy at position (x, y)
- static alpha_r(r, sigma0, r_core)[source]¶
Radial deflection angle of the cored density profile.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
deflection angle
- static d_alpha_dr(r, sigma0, r_core)[source]¶
Radial derivatives of the radial deflection angle.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
dalpha/dr
- static kappa_r(r, sigma0, r_core)[source]¶
Convergence of the cored density profile. This routine is also for testing.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
convergence at r
- static density(r, sigma0, r_core)[source]¶
Rho(r) = 2/pi * Sigma_crit R_c**3 * (R_c**2 + r**2)**(-3/2)
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
density at radius r
- density_lens(r, sigma0, r_core)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
density at radius r
- density_2d(x, y, sigma0, r_core, center_x=0, center_y=0)[source]¶
Projected density at projected radius r.
- Parameters:
x – x-coordinate in angular units
y – y-coordinate in angular units
sigma0 – convergence in the core
r_core – core radius
center_x – center of the profile
center_y – center of the profile
- Returns:
projected density
- static mass_2d(r, sigma0, r_core)[source]¶
Mass enclosed in cylinder of radius r.
- Parameters:
r – radius (angular scale)
sigma0 – convergence in the core
r_core – core radius
- Returns:
mass enclosed in cylinder of radius r
lenstronomy.LensModel.Profiles.cored_density_exp module¶
- class CoredDensityExp(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class contains functions concerning an exponential cored density profile, namely.
- ..math::
rho(r) = rho_0 exp(- (theta / theta_c)^2)
- param_names = ['kappa_0', 'theta_c', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'kappa_0': 0, 'theta_c': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'kappa_0': 10, 'theta_c': 100}¶
- static rhotilde(kappa_0, theta_c)[source]¶
Computes the central density in angular units :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :return: central density in 1/arcsec.
- function(x, y, kappa_0, theta_c, center_x=0, center_y=0)[source]¶
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
lensing potential (in arcsec^2)
- static alpha_radial(r, kappa_0, theta_c)[source]¶
Returns the radial part of the deflection angle :param r: angular position (normally in units of arc seconds) :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :return: radial deflection angle.
- derivatives(x, y, kappa_0, theta_c, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function (lensing potential), which are the deflection angles.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
deflection angle in x, deflection angle in y
- hessian(x, y, kappa_0, theta_c, center_x=0, center_y=0)[source]¶
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
- density(R, kappa_0, theta_c)[source]¶
Three dimensional density profile in angular units (rho0_physical = rho0_angular Sigma_crit / D_lens)
- Parameters:
R – projected angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
- Returns:
rho(R) density
- density_lens(r, kappa_0, theta_c)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
- Returns:
density rho(r)
- static kappa_r(R, kappa_0, theta_c)[source]¶
Convergence of the cored density profile. This routine is also for testing.
- Parameters:
R – radius (angular scale)
kappa_0 – convergence in the core
theta_c – core radius
- Returns:
convergence at r
- density_2d(x, y, kappa_0, theta_c, center_x=0, center_y=0)[source]¶
Projected two dimensional ULDM profile (convergence * Sigma_crit), but given our units convention for rho0, it is basically the convergence.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
- Returns:
Epsilon(R) projected density at radius R
- static mass_3d(R, kappa_0, theta_c)[source]¶
Mass enclosed a 3d sphere or radius r :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param R: radius in arcseconds :return: mass of soliton in angular units.
lenstronomy.LensModel.Profiles.cored_density_mst module¶
- class CoredDensityMST(profile_type='CORED_DENSITY')[source]¶
Bases:
LensProfileBase
Approximate mass-sheet transform of a density core.
This routine takes the parameters of the density core and subtracts a mass-sheet that approximates the cored profile in it’s center to counter-act (in approximation) this model. This allows for better sampling of the mass-sheet transformed quantities that do not have strong covariances. The subtraction of the mass-sheet is done such that the sampler returns the real central convergence of the original model (but be careful, the output of quantities like the Einstein angle of the main deflector are still the not-scaled one). Attention!!! The interpretation of the result is that the mass sheet as ‘CONVERGENCE’ that is present needs to be subtracted in post- processing.
- param_names = ['lambda_approx', 'r_core', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'lambda_approx': -1, 'r_core': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'lambda_approx': 10, 'r_core': 100}¶
- function(x, y, lambda_approx, r_core, center_x=0, center_y=0)[source]¶
Lensing potential of approximate mass-sheet correction.
- Parameters:
x – x-coordinate
y – y-coordinate
lambda_approx – approximate mass sheet transform
r_core – core radius of the cored density profile
center_x – x-center of the profile
center_y – y-center of the profile
- Returns:
lensing potential correction
- derivatives(x, y, lambda_approx, r_core, center_x=0, center_y=0)[source]¶
Deflection angles of approximate mass-sheet correction.
- Parameters:
x – x-coordinate
y – y-coordinate
lambda_approx – approximate mass sheet transform
r_core – core radius of the cored density profile
center_x – x-center of the profile
center_y – y-center of the profile
- Returns:
alpha_x, alpha_y
- hessian(x, y, lambda_approx, r_core, center_x=0, center_y=0)[source]¶
Hessian terms of approximate mass-sheet correction.
- Parameters:
x – x-coordinate
y – y-coordinate
lambda_approx – approximate mass sheet transform
r_core – core radius of the cored density profile
center_x – x-center of the profile
center_y – y-center of the profile
- Returns:
df/dxx, df/dxy, df/dyx, df/dyy
lenstronomy.LensModel.Profiles.cored_steep_ellipsoid module¶
- class CSE(axis='product_avg')[source]¶
Bases:
LensProfileBase
Cored steep ellipsoid (CSE) :param axis: ‘major’ or ‘product_avg’ ; whether to evaluate corresponding to r= major axis or r= sqrt(ab) source: Keeton and Kochanek (1998) Oguri 2021: https://arxiv.org/pdf/2106.11464.pdf
\[\kappa(u;s) = \frac{A}{2(s^2 + \xi^2)^{3/2}}\]with
\[\xi(x, y) = \sqrt{x^2 + \frac{y^2}{q^2}}\]- param_names = ['A', 's', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'A': -1000, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 's': 0}¶
- upper_limit_default = {'A': 1000, 'center_x': -100, 'center_y': -100, 'e1': 0.5, 'e2': 0.5, 's': 10000}¶
- function(x, y, a, s, e1, e2, center_x, center_y)[source]¶
- Parameters:
x – coordinate in image plane (angle)
y – coordinate in image plane (angle)
a – lensing strength
s – core radius
e1 – eccentricity
e2 – eccentricity
center_x – center of profile
center_y – center of profile
- Returns:
lensing potential
- derivatives(x, y, a, s, e1, e2, center_x, center_y)[source]¶
- Parameters:
x – coordinate in image plane (angle)
y – coordinate in image plane (angle)
a – lensing strength
s – core radius
e1 – eccentricity
e2 – eccentricity
center_x – center of profile
center_y – center of profile
- Returns:
deflection in x- and y-direction
- hessian(x, y, a, s, e1, e2, center_x, center_y)[source]¶
- Parameters:
x – coordinate in image plane (angle)
y – coordinate in image plane (angle)
a – lensing strength
s – core radius
e1 – eccentricity
e2 – eccentricity
center_x – center of profile
center_y – center of profile
- Returns:
hessian elements f_xx, f_xy, f_yx, f_yy
- class CSEMajorAxis(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Cored steep ellipsoid (CSE) along the major axis source: Keeton and Kochanek (1998) Oguri 2021: https://arxiv.org/pdf/2106.11464.pdf
\[\kappa(u;s) = \frac{A}{2(s^2 + \xi^2)^{3/2}}\]with
\[\xi(x, y) = \sqrt{x^2 + \frac{y^2}{q^2}}\]- param_names = ['A', 's', 'q', 'center_x', 'center_y']¶
- lower_limit_default = {'A': -1000, 'center_x': -100, 'center_y': -100, 'q': 0.001, 's': 0}¶
- upper_limit_default = {'A': 1000, 'center_x': -100, 'center_y': -100, 'e2': 0.5, 'q': 0.99999, 's': 10000}¶
- function(x, y, a, s, q)[source]¶
- Parameters:
x – coordinate in image plane (angle)
y – coordinate in image plane (angle)
a – lensing strength
s – core radius
q – axis ratio
- Returns:
lensing potential
- class CSEMajorAxisSet[source]¶
Bases:
LensProfileBase
A set of CSE profiles along a joint center and axis.
- function(x, y, a_list, s_list, q)[source]¶
- Parameters:
x – coordinate in image plane (angle)
y – coordinate in image plane (angle)
a_list – list of lensing strength
s_list – list of core radius
q – axis ratio
- Returns:
lensing potential
- class CSEProductAvg[source]¶
Bases:
LensProfileBase
Cored steep ellipsoid (CSE) evaluated at the product-averaged radius sqrt(ab), such that mass is not changed when increasing ellipticity.
Same as CSEMajorAxis but evaluated at r=sqrt(q)*r_original
Keeton and Kochanek (1998) Oguri 2021: https://arxiv.org/pdf/2106.11464.pdf
\[\kappa(u;s) = \frac{A}{2(s^2 + \xi^2)^{3/2}}\]with
\[\xi(x, y) = \sqrt{qx^2 + \frac{y^2}{q}}\]- param_names = ['A', 's', 'q', 'center_x', 'center_y']¶
- lower_limit_default = {'A': -1000, 'center_x': -100, 'center_y': -100, 'q': 0.001, 's': 0}¶
- upper_limit_default = {'A': 1000, 'center_x': -100, 'center_y': -100, 'e2': 0.5, 'q': 0.99999, 's': 10000}¶
- function(x, y, a, s, q)[source]¶
- Parameters:
x – coordinate in image plane (angle)
y – coordinate in image plane (angle)
a – lensing strength
s – core radius
q – axis ratio
- Returns:
lensing potential
- class CSEProductAvgSet[source]¶
Bases:
LensProfileBase
A set of CSE profiles along a joint center and axis.
- function(x, y, a_list, s_list, q)[source]¶
- Parameters:
x – coordinate in image plane (angle)
y – coordinate in image plane (angle)
a_list – list of lensing strength
s_list – list of core radius
q – axis ratio
- Returns:
lensing potential
lenstronomy.LensModel.Profiles.curved_arc_const module¶
- class CurvedArcConstMST[source]¶
Bases:
LensProfileBase
Lens model that describes a section of a highly magnified deflector region. The parameterization is chosen to describe local observables efficient.
Observables are: - curvature radius (basically bending relative to the center of the profile) - radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope) - tangential stretch (plus sign). Infinity means at critical curve - direction of curvature - position of arc
Requirements: - Should work with other perturbative models without breaking its meaning (say when adding additional shear terms) - Must best reflect the observables in lensing - minimal covariances between the parameters, intuitive parameterization.
- param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'direction', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'radial_stretch': -5, 'tangential_stretch': -100}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'radial_stretch': 5, 'tangential_stretch': 100}¶
- function(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
ATTENTION: there may not be a global lensing potential!
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- derivatives(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- hessian(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- class CurvedArcConst(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Curved arc lensing with orientation of curvature perpendicular to the x-axis with unity radial stretch.
- param_names = ['tangential_stretch', 'curvature', 'direction', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'tangential_stretch': -100}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'tangential_stretch': 100}¶
- function(x, y, tangential_stretch, curvature, direction, center_x, center_y)[source]¶
ATTENTION: there may not be a global lensing potential!
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- derivatives(x, y, tangential_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- hessian(x, y, tangential_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
lenstronomy.LensModel.Profiles.curved_arc_sis_mst module¶
- class CurvedArcSISMST[source]¶
Bases:
LensProfileBase
Lens model that describes a section of a highly magnified deflector region. The parameterization is chosen to describe local observables efficient.
Observables are: - curvature radius (basically bending relative to the center of the profile) - radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope) - tangential stretch (plus sign). Infinity means at critical curve - direction of curvature - position of arc
Requirements: - Should work with other perturbative models without breaking its meaning (say when adding additional shear terms) - Must best reflect the observables in lensing - minimal covariances between the parameters, intuitive parameterization.
- param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'direction', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'radial_stretch': -5, 'tangential_stretch': -100}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'radial_stretch': 5, 'tangential_stretch': 100}¶
- static stretch2sis_mst(tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
parameters in terms of a spherical SIS + MST resulting in the same observables
- static sis_mst2stretch(theta_E, kappa_ext, center_x_sis, center_y_sis, center_x, center_y)[source]¶
Turn Singular power-law lens model into stretch parameterization at position (center_x, center_y) This is the inverse function of stretch2spp()
- Parameters:
theta_E – Einstein radius of SIS profile
kappa_ext – external convergence (MST factor 1 - kappa_ext)
center_x_sis – center of SPP model
center_y_sis – center of SPP model
center_x – center of curved model definition
center_y – center of curved model definition
- Returns:
tangential_stretch, radial_stretch, curvature, direction
- Returns:
- function(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
ATTENTION: there may not be a global lensing potential!
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- derivatives(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- hessian(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
lenstronomy.LensModel.Profiles.curved_arc_spp module¶
- class CurvedArcSPP[source]¶
Bases:
LensProfileBase
Lens model that describes a section of a highly magnified deflector region. The parameterization is chosen to describe local observables efficient.
Observables are: - curvature radius (basically bending relative to the center of the profile) - radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope) - tangential stretch (plus sign). Infinity means at critical curve - direction of curvature - position of arc
Requirements: - Should work with other perturbative models without breaking its meaning (say when adding additional shear terms) - Must best reflect the observables in lensing - minimal covariances between the parameters, intuitive parameterization.
- param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'direction', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'radial_stretch': -5, 'tangential_stretch': -100}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'radial_stretch': 5, 'tangential_stretch': 100}¶
- static stretch2spp(tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
parameters in terms of a spherical power-law profile resulting in the same observables
- static spp2stretch(theta_E, gamma, center_x_spp, center_y_spp, center_x, center_y)[source]¶
Turn Singular power-law lens model into stretch parameterization at position (center_x, center_y) This is the inverse function of stretch2spp()
- Parameters:
theta_E – Einstein radius of SPP model
gamma – power-law slope
center_x_spp – center of SPP model
center_y_spp – center of SPP model
center_x – center of curved model definition
center_y – center of curved model definition
- Returns:
tangential_stretch, radial_stretch, curvature, direction
- function(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
ATTENTION: there may not be a global lensing potential!
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- derivatives(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- hessian(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
lenstronomy.LensModel.Profiles.curved_arc_spt module¶
- class CurvedArcSPT[source]¶
Bases:
LensProfileBase
Curved arc model based on SIS+MST with an additional non-linear shear distortions applied on the source coordinates around the center.
This profile is effectively a Source Position Transform of a curved arc and a shear distortion.
- param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'direction', 'gamma1', 'gamma2', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'gamma1': -0.5, 'gamma2': -0.5, 'radial_stretch': -5, 'tangential_stretch': -100}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'gamma1': 0.5, 'gamma2': 0.5, 'radial_stretch': 5, 'tangential_stretch': 100}¶
- function(x, y, tangential_stretch, radial_stretch, curvature, direction, gamma1, gamma2, center_x, center_y)[source]¶
ATTENTION: there may not be a global lensing potential!
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
gamma1 – non-linear reduced shear distortion in the source plane
gamma2 – non-linear reduced shear distortion in the source plane
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- derivatives(x, y, tangential_stretch, radial_stretch, curvature, direction, gamma1, gamma2, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
gamma1 – non-linear reduced shear distortion in the source plane
gamma2 – non-linear reduced shear distortion in the source plane
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- hessian(x, y, tangential_stretch, radial_stretch, curvature, direction, gamma1, gamma2, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
gamma1 – non-linear reduced shear distortion in the source plane
gamma2 – non-linear reduced shear distortion in the source plane
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
lenstronomy.LensModel.Profiles.curved_arc_tan_diff module¶
- class CurvedArcTanDiff[source]¶
Bases:
LensProfileBase
Curved arc model with an additional non-zero tangential stretch differential in tangential direction component.
Observables are: - curvature radius (basically bending relative to the center of the profile) - radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope) - tangential stretch (plus sign). Infinity means at critical curve - direction of curvature - position of arc
Requirements: - Should work with other perturbative models without breaking its meaning (say when adding additional shear terms) - Must best reflect the observables in lensing - minimal covariances between the parameters, intuitive parameterization.
- param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'dtan_dtan', 'direction', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'dtan_dtan': -10, 'radial_stretch': -5, 'tangential_stretch': -100}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'dtan_dtan': 10, 'radial_stretch': 5, 'tangential_stretch': 100}¶
- static stretch2sie_mst(tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)[source]¶
- Parameters:
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
dtan_dtan – d(tangential_stretch) / d(tangential direction) / tangential stretch
direction – float, angle in radian
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
parameters in terms of a spherical SIS + MST resulting in the same observables
- function(x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)[source]¶
ATTENTION: there may not be a global lensing potential!
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
dtan_dtan – d(tangential_stretch) / d(tangential direction) / tangential stretch
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- derivatives(x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
dtan_dtan – d(tangential_stretch) / d(tangential direction) / tangential stretch
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
- hessian(x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)[source]¶
- Parameters:
x –
y –
tangential_stretch – float, stretch of intrinsic source in tangential direction
radial_stretch – float, stretch of intrinsic source in radial direction
curvature – 1/curvature radius
direction – float, angle in radian
dtan_dtan – d(tangential_stretch) / d(tangential direction) / tangential stretch
center_x – center of source in image plane
center_y – center of source in image plane
- Returns:
lenstronomy.LensModel.Profiles.dipole module¶
- class Dipole(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Class for dipole response of two massive bodies (experimental)
- param_names = ['com_x', 'com_y', 'phi_dipole', 'coupling']¶
- lower_limit_default = {'com_x': -100, 'com_y': -100, 'coupling': -10, 'phi_dipole': -10}¶
- upper_limit_default = {'com_x': 100, 'com_y': 100, 'coupling': 10, 'phi_dipole': 10}¶
- function(x, y, com_x, com_y, phi_dipole, coupling)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
lenstronomy.LensModel.Profiles.elliptical_density_slice module¶
- class ElliSLICE(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class computes the lensing quantities for an elliptical slice of constant density. Based on Schramm 1994 https://ui.adsabs.harvard.edu/abs/1994A%26A…284…44S/abstract
Computes the lensing quantities of an elliptical slice with semi major axis ‘a’ and semi minor axis ‘b’, centered on ‘center_x’ and ‘center_y’, oriented with an angle ‘psi’ in radian, and with constant surface mass density ‘sigma_0’. In other words, this lens model is characterized by the surface mass density :
- ..math::
- kappa(x,y) = left{
- begin{array}{ll}
sigma_0 & mbox{if } frac{x_{rot}^2}{a^2} + frac{y_{rot}^2}{b^2} leq 1 0 & mbox{else}
end{array}
right}.
with
- ..math::
x_{rot} = x_c cos psi + y_c sin psi y_{rot} = - x_c sin psi + y_c cos psi x_c = x - center_x y_c = y - center_y
- param_names = ['a', 'b', 'psi', 'sigma_0', 'center_x', 'center_y']¶
- lower_limit_default = {'a': 0.0, 'b': 0.0, 'center_x': -100.0, 'center_y': -100.0, 'psi': -1.5707963267948966}¶
- upper_limit_default = {'a': 100.0, 'b': 100.0, 'center_x': 100.0, 'center_y': 100.0, 'psi': 1.5707963267948966}¶
- function(x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0)[source]¶
Lensing potential.
- Parameters:
a – float, semi-major axis, must be positive
b – float, semi-minor axis, must be positive
psi – float, orientation in radian
sigma_0 – float, surface mass density, must be positive
center_x – float, center on the x axis
center_y – float, center on the y axis
- derivatives(x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0)[source]¶
Lensing deflection angle.
- Parameters:
a – float, semi-major axis, must be positive
b – float, semi-minor axis, must be positive
psi – float, orientation in radian
sigma_0 – float, surface mass density, must be positive
center_x – float, center on the x axis
center_y – float, center on the y axis
- hessian(x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0)[source]¶
Lensing second derivatives.
- Parameters:
a – float, semi-major axis, must be positive
b – float, semi-minor axis, must be positive
psi – float, orientation in radian
sigma_0 – float, surface mass density, must be positive
center_x – float, center on the x axis
center_y – float, center on the y axis
- alpha_in(x, y, kwargs_slice)[source]¶
Deflection angle for (x,y) inside the elliptical slice.
- Parameters:
kwargs_slice – dict, dictionary with the slice definition (a,b,psi,sigma_0)
- alpha_ext(x, y, kwargs_slice)[source]¶
Deflection angle for (x,y) outside the elliptical slice.
- Parameters:
kwargs_slice – dict, dictionary with the slice definition (a,b,psi,sigma_0)
lenstronomy.LensModel.Profiles.epl module¶
- class EPL[source]¶
Bases:
LensProfileBase
Elliptical Power Law mass profile.
\[\kappa(x, y) = \frac{3-\gamma}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q}} \right)^{\gamma-1}\]with \(\theta_{E}\) is the (circularized) Einstein radius, \(\gamma\) is the negative power-law slope of the 3D mass distributions, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate system aligned with the major and minor axis of the lens.
In terms of eccentricities, this profile is defined as
\[\kappa(r) = \frac{3-\gamma}{2} \left(\frac{\theta'_{E}}{r \sqrt{1 - e*\cos(2*\phi)}} \right)^{\gamma-1}\]with \(\epsilon\) is the ellipticity defined as
\[\epsilon = \frac{1-q^2}{1+q^2}\]And an Einstein radius \(\theta'_{\rm E}\) related to the definition used is
\[\left(\frac{\theta'_{\rm E}}{\theta_{\rm E}}\right)^{2} = \frac{2q}{1+q^2}.\]The mathematical form of the calculation is presented by Tessore & Metcalf (2015), https://arxiv.org/abs/1507.01819. The current implementation is using hyperbolic functions. The paper presents an iterative calculation scheme, converging in few iterations to high precision and accuracy.
A (faster) implementation of the same model using numba is accessible as ‘EPL_NUMBA’ with the iterative calculation scheme. An alternative implementation of the same model using a fortran code FASTELL is implemented as ‘PEMD’ profile.
- param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 1.5, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 2.5, 'theta_E': 100}¶
- param_conv(theta_E, gamma, e1, e2)[source]¶
Converts parameters as defined in this class to the parameters used in the EPLMajorAxis() class.
- Parameters:
theta_E – Einstein radius as defined in the profile class
gamma – negative power-law slope
e1 – eccentricity modulus
e2 – eccentricity modulus
- Returns:
b, t, q, phi_G
- set_static(theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
theta_E – Einstein radius
gamma – power law slope
e1 – eccentricity component
e2 – eccentricity component
center_x – profile center
center_y – profile center
- Returns:
self variables set
- function(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
gamma – power law slope
e1 – eccentricity component
e2 – eccentricity component
center_x – profile center
center_y – profile center
- Returns:
lensing potential
- derivatives(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
gamma – power law slope
e1 – eccentricity component
e2 – eccentricity component
center_x – profile center
center_y – profile center
- Returns:
alpha_x, alpha_y
- hessian(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
gamma – power law slope
e1 – eccentricity component
e2 – eccentricity component
center_x – profile center
center_y – profile center
- Returns:
f_xx, f_xy, f_yx, f_yy
- mass_3d_lens(r, theta_E, gamma, e1=None, e2=None)[source]¶
Computes the spherical power-law mass enclosed (with SPP routine)
- Parameters:
r – radius within the mass is computed
theta_E – Einstein radius
gamma – power-law slope
e1 – eccentricity component (not used)
e2 – eccentricity component (not used)
- Returns:
mass enclosed a 3D radius r.
- density_lens(r, theta_E, gamma, e1=None, e2=None)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – radius within the mass is computed
theta_E – Einstein radius
gamma – power-law slope
e1 – eccentricity component (not used)
e2 – eccentricity component (not used)
- Returns:
mass enclosed a 3D radius r
- class EPLMajorAxis[source]¶
Bases:
LensProfileBase
This class contains the function and the derivatives of the elliptical power law.
\[\kappa = (2-t)/2 * \left[\frac{b}{\sqrt{q^2 x^2 + y^2}}\right]^t\]where with \(t = \gamma - 1\) (from EPL class) being the projected power-law slope of the convergence profile, critical radius b, axis ratio q.
Tessore & Metcalf (2015), https://arxiv.org/abs/1507.01819
- param_names = ['b', 't', 'q', 'center_x', 'center_y']¶
- function(x, y, b, t, q)[source]¶
Returns the lensing potential.
- Parameters:
x – x-coordinate in image plane relative to center (major axis)
y – y-coordinate in image plane relative to center (minor axis)
b – critical radius
t – projected power-law slope
q – axis ratio
- Returns:
lensing potential
- derivatives(x, y, b, t, q)[source]¶
Returns the deflection angles.
- Parameters:
x – x-coordinate in image plane relative to center (major axis)
y – y-coordinate in image plane relative to center (minor axis)
b – critical radius
t – projected power-law slope
q – axis ratio
- Returns:
f_x, f_y
- hessian(x, y, b, t, q)[source]¶
Hessian matrix of the lensing potential.
- Parameters:
x – x-coordinate in image plane relative to center (major axis)
y – y-coordinate in image plane relative to center (minor axis)
b – critical radius
t – projected power-law slope
q – axis ratio
- Returns:
f_xx, f_yy, f_xy
- class EPLQPhi[source]¶
Bases:
LensProfileBase
Class to model a EPL sampling over q and phi instead of e1 and e2.
- param_names = ['theta_E', 'gamma', 'q', 'phi', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'gamma': 1.5, 'phi': -3.141592653589793, 'q': 0, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'gamma': 2.5, 'phi': 3.141592653589793, 'q': 1, 'theta_E': 100}¶
- function(x, y, theta_E, gamma, q, phi, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
gamma – power law slope
q – axis ratio
phi – position angle
center_x – profile center
center_y – profile center
- Returns:
lensing potential
- derivatives(x, y, theta_E, gamma, q, phi, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
gamma – power law slope
q – axis ratio
phi – position angle
center_x – profile center
center_y – profile center
- Returns:
alpha_x, alpha_y
- hessian(x, y, theta_E, gamma, q, phi, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
gamma – power law slope
q – axis ratio
phi – position angle
center_x – profile center
center_y – profile center
- Returns:
f_xx, f_xy, f_yx, f_yy
- mass_3d_lens(r, theta_E, gamma, q=None, phi=None)[source]¶
Computes the spherical power-law mass enclosed (with SPP routine).
- Parameters:
r – radius within the mass is computed
theta_E – Einstein radius
gamma – power-law slope
q – axis ratio (not used)
phi – position angle (not used)
- Returns:
mass enclosed a 3D radius r.
- density_lens(r, theta_E, gamma, q=None, phi=None)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – radius within the mass is computed
theta_E – Einstein radius
gamma – power-law slope
q – axis ratio (not used)
phi – position angle (not used)
- Returns:
mass enclosed a 3D radius r
lenstronomy.LensModel.Profiles.epl_numba module¶
- class EPL_numba[source]¶
Bases:
LensProfileBase
” Elliptical Power Law mass profile - computation accelerated with numba
\[\kappa(x, y) = \frac{3-\gamma}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q}} \right)^{\gamma-1}\]with \(\theta_{E}\) is the (circularized) Einstein radius, \(\gamma\) is the negative power-law slope of the 3D mass distributions, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate system aligned with the major and minor axis of the lens.
In terms of eccentricities, this profile is defined as
\[\kappa(r) = \frac{3-\gamma}{2} \left(\frac{\theta'_{E}}{r \sqrt{1 − e*\cos(2*\phi)}} \right)^{\gamma-1}\]with \(\epsilon\) is the ellipticity defined as
\[\epsilon = \frac{1-q^2}{1+q^2}\]And an Einstein radius \(\theta'_{\rm E}\) related to the definition used is
\[\left(\frac{\theta'_{\rm E}}{\theta_{\rm E}}\right)^{2} = \frac{2q}{1+q^2}.\]The mathematical form of the calculation is presented by Tessore & Metcalf (2015), https://arxiv.org/abs/1507.01819. The current implementation is using hyperbolic functions. The paper presents an iterative calculation scheme, converging in few iterations to high precision and accuracy.
A (slower) implementation of the same model using hyperbolic functions without the iterative calculation is accessible as ‘EPL’ not requiring numba.
- param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 1.5, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 2.5, 'theta_E': 100}¶
- static function(x, y, theta_E, gamma, e1, e2, center_x=0.0, center_y=0.0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
lensing potential
- static derivatives(x, y, theta_E, gamma, e1, e2, center_x=0.0, center_y=0.0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
deflection angles alpha_x, alpha_y
- static hessian(x, y, theta_E, gamma, e1, e2, center_x=0.0, center_y=0.0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
Hessian components f_xx, f_yy, f_xy
lenstronomy.LensModel.Profiles.flexion module¶
- class Flexion(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Class for flexion.
- param_names = ['g1', 'g2', 'g3', 'g4', 'ra_0', 'dec_0']¶
- lower_limit_default = {'dec_0': -100, 'g1': -0.1, 'g2': -0.1, 'g3': -0.1, 'g4': -0.1, 'ra_0': -100}¶
- upper_limit_default = {'dec_0': 100, 'g1': 0.1, 'g2': 0.1, 'g3': 0.1, 'g4': 0.1, 'ra_0': 100}¶
- function(x, y, g1, g2, g3, g4, ra_0=0, dec_0=0)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
lenstronomy.LensModel.Profiles.flexionfg module¶
- class Flexionfg[source]¶
Bases:
LensProfileBase
Flexion consist of basis F flexion and G flexion (F1,F2,G1,G2), see formulas 2.54, 2.55 in Massimo Meneghetti 2017 - “Introduction to Gravitational Lensing”.
- param_names = ['F1', 'F2', 'G1', 'G2', 'ra_0', 'dec_0']¶
- lower_limit_default = {'F1': -0.1, 'F2': -0.1, 'G1': -0.1, 'G2': -0.1, 'dec_0': -100, 'ra_0': -100}¶
- upper_limit_default = {'F1': 0.1, 'F2': 0.1, 'G1': 0.1, 'G2': 0.1, 'dec_0': 100, 'ra_0': 100}¶
- function(x, y, F1, F2, G1, G2, ra_0=0, dec_0=0)[source]¶
Lensing potential.
- Parameters:
x – x-coordinate
y – y-coordinate
F1 – F1 flexion, derivative of kappa in x direction
F2 – F2 flexion, derivative of kappa in y direction
G1 – G1 flexion
G2 – G2 flexion
ra_0 – center x-coordinate
dec_0 – center y-coordinate
- Returns:
lensing potential
- derivatives(x, y, F1, F2, G1, G2, ra_0=0, dec_0=0)[source]¶
Deflection angle.
- Parameters:
x – x-coordinate
y – y-coordinate
F1 – F1 flexion, derivative of kappa in x direction
F2 – F2 flexion, derivative of kappa in y direction
G1 – G1 flexion
G2 – G2 flexion
ra_0 – center x-coordinate
dec_0 – center x-coordinate
- Returns:
deflection angle.
- hessian(x, y, F1, F2, G1, G2, ra_0=0, dec_0=0)[source]¶
Hessian matrix.
- Parameters:
x – x-coordinate
y – y-coordinate
F1 – F1 flexion, derivative of kappa in x direction
F2 – F2 flexion, derivative of kappa in y direction
G1 – G1 flexion
G2 – G2 flexion
ra_0 – center x-coordinate
dec_0 – center y-coordinate
- Returns:
second order derivatives f_xx, f_yy, f_xy
- static transform_fg(F1, F2, G1, G2)[source]¶
Basis transform from (F1,F2,G1,G2) to (g1,g2,g3,g4).
- Parameters:
F1 – F1 flexion, derivative of kappa in x direction
F2 – F2 flexion, derivative of kappa in y direction
G1 – G1 flexion
G2 – G2 flexion
- Returns:
g1,g2,g3,g4 (phi_xxx, phi_xxy, phi_xyy, phi_yyy)
lenstronomy.LensModel.Profiles.gauss_decomposition module¶
This module contains the class to compute lensing properties of any elliptical profile using Shajib (2019)’s Gauss decomposition.
- class GaussianEllipseKappaSet(use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Bases:
LensProfileBase
This class computes the lensing properties of a set of concentric elliptical Gaussian convergences.
- param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma': 0}¶
- upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma': 100}¶
- __init__(use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
- Parameters:
use_scipy_wofz (
bool
) – To initiateclass GaussianEllipseKappa
. IfTrue
, Gaussian lensing will usescipy.special.wofz
function. SetFalse
for lower precision, but faster speed.min_ellipticity (
float
) – To be passed toclass GaussianEllipseKappa
. Minimum ellipticity for Gaussian elliptical lensing calculation. For lower ellipticity than min_ellipticity the equations for the spherical case will be used.
- function(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Compute the potential function for a set of concentric elliptical Gaussian convergence profiles.
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinateamp (
numpy.array
withdtype=float
) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)sigma (
numpy.array
withdtype=float
) – Standard deviation of Gaussiane1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordianate of centroid
- Returns:
Potential for elliptical Gaussian convergence
- Return type:
float
, ornumpy.array
withshape = x.shape
- derivatives(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Compute the derivatives of function angles \(\partial f/\partial x\), \(\partial f/\partial y\) at \(x,\ y\) for a set of concentric elliptic Gaussian convergence profiles.
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinateamp (
numpy.array
withdtype=float
) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)sigma (
numpy.array
withdtype=float
) – Standard deviation of Gaussiane1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordianate of centroid
- Returns:
Deflection angle \(\partial f/\partial x\), \(\partial f/\partial y\) for elliptical Gaussian convergence
- Return type:
tuple
(float, float)
or(numpy.array, numpy.array)
with eachnumpy
array’s shape equal tox.shape
- hessian(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Compute Hessian matrix of function \(\partial^2f/\partial x^2\), \(\partial^2 f/\partial y^2\), \(\partial^2 f/\partial x\partial y\) for a set of concentric elliptic Gaussian convergence profiles.
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinateamp (
numpy.array
withdtype=float
) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)sigma (
numpy.array
withdtype=float
) – Standard deviation of Gaussiane1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordianate of centroid
- Returns:
Hessian \(\partial^2f/\partial x^2\), \(\partial^2/\partial x\partial y\), \(\partial^2/\partial y\partial x\), \(\partial^2 f/\partial y^2\) for elliptical Gaussian convergence.
- Return type:
tuple
(float, float, float)
, or(numpy.array, numpy.array, numpy.array)
with eachnumpy
array’s shape equal tox.shape
- density_2d(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Compute the density of a set of concentric elliptical Gaussian convergence profiles \(\sum A/(2\pi \sigma^2) \exp(-( x^2+y^2/q^2)/2\sigma^2)\).
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinateamp (
numpy.array
withdtype=float
) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)sigma (
numpy.array
withdtype=float
) – Standard deviation of Gaussiane1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordianate of centroid
- Returns:
Density \(\kappa\) for elliptical Gaussian convergence
- Return type:
float
, ornumpy.array
with shape equal tox.shape
- class GaussDecompositionAbstract(n_sigma=15, sigma_start_mult=0.02, sigma_end_mult=15.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Bases:
object
This abstract class sets up a template for computing lensing properties of an elliptical convergence through Shajib (2019)’s Gauss decomposition.
- __init__(n_sigma=15, sigma_start_mult=0.02, sigma_end_mult=15.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Set up settings for the Gaussian decomposition. For more details about the decomposition parameters, see Shajib (2019).
- Parameters:
n_sigma (
int
) – Number of Gaussian componentssigma_start_mult (
float
) – Lower range of logarithmically spaced sigmassigma_end_mult (
float
) – Upper range of logarithmically spaced sigmasprecision (
int
) – Numerical precision of Gaussian decompositionuse_scipy_wofz (
bool
) – To be passed toclass GaussianEllipseKappa
. IfTrue
, Gaussian lensing will usescipy.special.wofz
function. SetFalse
for lower precision, but faster speed.min_ellipticity (
float
) – To be passed toclass GaussianEllipseKappa
. Minimum ellipticity for Gaussian elliptical lensing calculation. For lower ellipticity than min_ellipticity the equations for the spherical case will be used.
- gauss_decompose(**kwargs)[source]¶
Compute the amplitudes and sigmas of Gaussian components using the integral transform with Gaussian kernel from Shajib (2019). The returned values are in the convention of eq. (2.13).
- Parameters:
kwargs – Keyword arguments to send to
func
- Returns:
Amplitudes and standard deviations of the Gaussian components
- Return type:
tuple
(numpy.array, numpy.array)
- abstract get_scale(**kwargs)[source]¶
Abstract method to identify the keyword argument for the scale size among the profile parameters of the child class’ convergence profile.
- Parameters:
kwargs – Keyword arguments
- Returns:
Scale size
- Return type:
float
- abstract get_kappa_1d(y, **kwargs)[source]¶
Abstract method to compute the spherical Sersic profile at y. The concrete method has to defined by the child class.
- Parameters:
y (
float
ornumpy.array
) – y coordinatekwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile
- function(x, y, e1=0.0, e2=0.0, center_x=0.0, center_y=0.0, **kwargs)[source]¶
Compute the deflection potential of a Gauss-decomposed elliptic convergence.
- Parameters:
x (
float
) – x coordinatey (
float
) – y coordinatee1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordinate of centroidkwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile
- Returns:
Deflection potential
- Return type:
float
- derivatives(x, y, e1=0.0, e2=0.0, center_x=0.0, center_y=0.0, **kwargs)[source]¶
Compute the derivatives of the deflection potential \(\partial f/\partial x\), \(\partial f/\partial y\) for a Gauss-decomposed elliptic convergence.
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinatee1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordinate of centroidkwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile
- Returns:
Derivatives of deflection potential
- Return type:
tuple
(type(x), type(x))
- hessian(x, y, e1=0.0, e2=0.0, center_x=0.0, center_y=0.0, **kwargs)[source]¶
Compute the Hessian of the deflection potential \(\partial^2f/\partial x^2\), \(\partial^2 f/ \partial y^2\), \(\partial^2 f/\partial x\partial y\) of a Gauss-decomposed elliptic Sersic convergence.
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinatee1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordinate of centroidkwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile
- Returns:
Hessian of deflection potential
- Return type:
tuple
(type(x), type(x), type(x))
- density_2d(x, y, e1=0.0, e2=0.0, center_x=0.0, center_y=0.0, **kwargs)[source]¶
Compute the convergence profile for Gauss-decomposed elliptic Sersic profile.
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinatee1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordinate of centroidkwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile in the child class.
- Returns:
Convergence profile
- Return type:
type(x)
- class SersicEllipseGaussDec(n_sigma=15, sigma_start_mult=0.02, sigma_end_mult=15.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Bases:
GaussDecompositionAbstract
This class computes the lensing properties of an elliptical Sersic profile using the Shajib (2019)’s Gauss decomposition method.
- param_names = ['k_eff', 'R_sersic', 'n_sersic', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'R_sersic': 0.0, 'center_x': -100.0, 'center_y': -100.0, 'e1': -0.5, 'e2': -0.5, 'k_eff': 0.0, 'n_sersic': 0.5}¶
- upper_limit_default = {'R_sersic': 100.0, 'center_x': 100.0, 'center_y': 100.0, 'e1': 0.5, 'e2': 0.5, 'k_eff': 100.0, 'n_sersic': 8.0}¶
- get_kappa_1d(y, **kwargs)[source]¶
Compute the spherical Sersic profile at y.
- Parameters:
y (
float
) – y coordinatekwargs – Keyword arguments
- Keyword Arguments:
n_sersic (
float
) – Sersic indexR_sersic (
float
) – Sersic scale radiusk_eff (
float
) – Sersic convergence at R_sersic
- Returns:
Sersic function at y
- Return type:
type(y)
- class NFWEllipseGaussDec(n_sigma=15, sigma_start_mult=0.005, sigma_end_mult=50.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Bases:
GaussDecompositionAbstract
This class computes the lensing properties of an elliptical, projected NFW profile using Shajib (2019)’s Gauss decomposition method.
- param_names = ['Rs', 'alpha_Rs', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5}¶
- __init__(n_sigma=15, sigma_start_mult=0.005, sigma_end_mult=50.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Set up settings for the Gaussian decomposition. For more details about the decomposition parameters, see Shajib (2019).
- Parameters:
n_sigma (
int
) – Number of Gaussian componentssigma_start_mult (
float
) – Lower range of logarithmically spaced sigmassigma_end_mult (
float
) – Upper range of logarithmically spaced sigmasprecision (
int
) – Numerical precision of Gaussian decompositionuse_scipy_wofz (
bool
) – To be passed toclass GaussianEllipseKappa
. IfTrue
, Gaussian lensing will usescipy.special.wofz
function. SetFalse
for lower precision, but faster speed.min_ellipticity (
float
) – To be passed toclass GaussianEllipseKappa
. Minimum ellipticity for Gaussian elliptical lensing calculation. For lower ellipticity than min_ellipticity the equations for the spherical case will be used.
- class GaussDecompositionAbstract3D(n_sigma=15, sigma_start_mult=0.02, sigma_end_mult=15.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Bases:
GaussDecompositionAbstract
This abstract class sets up a template for computing lensing properties of a convergence from 3D spherical profile through Shajib (2019)’s Gauss decomposition.
- gauss_decompose(**kwargs)[source]¶
Compute the amplitudes and sigmas of Gaussian components using the integral transform with Gaussian kernel from Shajib (2019). The returned values are in the convention of eq. (2.13).
- Parameters:
kwargs – Keyword arguments to send to
func
- Returns:
Amplitudes and standard deviations of the Gaussian components
- Return type:
tuple
(numpy.array, numpy.array)
- class CTNFWGaussDec(n_sigma=15, sigma_start_mult=0.01, sigma_end_mult=20.0, precision=10, use_scipy_wofz=True)[source]¶
Bases:
GaussDecompositionAbstract3D
This class computes the lensing properties of an projection from a spherical cored-truncated NFW profile using Shajib (2019)’s Gauss decomposition method.
- param_names = ['r_s', 'r_core', 'r_trunc', 'a', 'rho_s', 'center_xcenter_y']¶
- lower_limit_default = {'a': 0.0, 'center_x': -100, 'center_y': -100, 'r_core': 0, 'r_s': 0, 'r_trunc': 0, 'rho_s': 0}¶
- upper_limit_default = {'a': 10.0, 'center_x': 100, 'center_y': 100, 'r_core': 100, 'r_s': 100, 'r_trunc': 100, 'rho_s': 1000}¶
- __init__(n_sigma=15, sigma_start_mult=0.01, sigma_end_mult=20.0, precision=10, use_scipy_wofz=True)[source]¶
Set up settings for the Gaussian decomposition. For more details about the decomposition parameters, see Shajib (2019).
- Parameters:
n_sigma (
int
) – Number of Gaussian componentssigma_start_mult (
float
) – Lower range of logarithmically spaced sigmassigma_end_mult (
float
) – Upper range of logarithmically spaced sigmasprecision (
int
) – Numerical precision of Gaussian decompositionuse_scipy_wofz (
bool
) – To be passed toclass GaussianEllipseKappa
. IfTrue
, Gaussian lensing will usescipy.special.wofz
function. SetFalse
for lower precision, but faster speed.
- get_kappa_1d(y, **kwargs)[source]¶
Compute the spherical cored-truncated NFW profile at y.
- Parameters:
y (
float
) – y coordinatekwargs – Keyword arguments
- Keyword Arguments:
r_s (
float
) – Scale radiusr_trunc (
float
) – Truncation radiusr_core (
float
) – Core radiusrho_s (
float
) – Density normalizationa (
float
) – Core regularization parameter
- Returns:
projected NFW profile at y
- Return type:
type(y)
- get_scale(**kwargs)[source]¶
Identify the scale size from the keyword arguments.
- Parameters:
kwargs – Keyword arguments
- Keyword Arguments:
r_s (
float
) – Scale radiusr_trunc (
float
) – Truncation radiusr_core (
float
) – Core radiusrho_s (
float
) – Density normalizationa (
float
) – Core regularization parameter
- Returns:
NFW scale radius
- Return type:
float
lenstronomy.LensModel.Profiles.gaussian_ellipse_kappa module¶
This module defines class GaussianEllipseKappa
to compute the lensing properties
of an elliptical Gaussian profile with ellipticity in the convergence using the formulae
from Shajib (2019).
- class GaussianEllipseKappa(use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Bases:
LensProfileBase
This class contains functions to evaluate the derivative and hessian matrix of the deflection potential for an elliptical Gaussian convergence.
The formulae are from Shajib (2019).
- param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma': 0}¶
- upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma': 100}¶
- __init__(use_scipy_wofz=True, min_ellipticity=1e-05)[source]¶
Setup which method to use the Faddeeva function and the ellipticity limit for spherical approximation.
- Parameters:
use_scipy_wofz (
bool
) – IfTrue
, usescipy.special.wofz
.min_ellipticity (
float
) – Minimum allowed ellipticity. Forq > 1 - min_ellipticity
, values for spherical case will be returned.
- function(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Compute the potential function for elliptical Gaussian convergence.
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinateamp (
float
) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)sigma (
float
) – Standard deviation of Gaussiane1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordianate of centroid
- Returns:
Potential for elliptical Gaussian convergence
- Return type:
float
, ornumpy.array
with shape equal tox.shape
- derivatives(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Compute the derivatives of function angles \(\partial f/\partial x\), \(\partial f/\partial y\) at \(x,\ y\).
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinateamp (
float
) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)sigma (
float
) – Standard deviation of Gaussiane1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordianate of centroid
- Returns:
Deflection angle \(\partial f/\partial x\), \(\partial f/\partial y\) for elliptical Gaussian convergence.
- Return type:
tuple
(float, float)
or(numpy.array, numpy.array)
with eachnumpy.array
’s shape equal tox.shape
.
- hessian(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Compute Hessian matrix of function \(\partial^2f/\partial x^2\), \(\partial^2 f/\partial y^2\), \(\partial^2/\partial x\partial y\).
- Parameters:
x (
float
ornumpy.array
) – x coordinatey (
float
ornumpy.array
) – y coordinateamp (
float
) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)sigma (
float
) – Standard deviation of Gaussiane1 (
float
) – Ellipticity parameter 1e2 (
float
) – Ellipticity parameter 2center_x (
float
) – x coordinate of centroidcenter_y (
float
) – y coordianate of centroid
- Returns:
Hessian \(A/(2 \pi \sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\) for elliptical Gaussian convergence.
- Return type:
tuple
(float, float, float)
, or(numpy.array, numpy.array, numpy.array)
with eachnumpy.array
’s shape equal tox.shape
.
- density_2d(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Compute the density of elliptical Gaussian \(A/(2 \pi \sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\).
- Parameters:
x (
float
ornumpy.array
) – x coordinate.y (
float
ornumpy.array
) – y coordinate.amp (
float
) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)sigma (
float
) – Standard deviation of Gaussian.e1 (
float
) – Ellipticity parameter 1.e2 (
float
) – Ellipticity parameter 2.center_x (
float
) – x coordinate of centroid.center_y (
float
) – y coordianate of centroid.
- Returns:
Density \(\kappa\) for elliptical Gaussian convergence.
- Return type:
float
, ornumpy.array
with shape =x.shape
.
- static sgn(z)[source]¶
Compute the sign function \(\mathrm{sgn}(z)\) factor for deflection as sugggested by Bray (1984). For current implementation, returning 1 is sufficient.
- Parameters:
z (
complex
) – Complex variable \(z = x + \mathrm{i}y\)- Returns:
\(\mathrm{sgn}(z)\)
- Return type:
float
- sigma_function(x, y, q)[source]¶
Compute the function \(\varsigma (z; q)\) from equation (4.12) of Shajib (2019).
- Parameters:
x (
float
ornumpy.array
) – Real part of complex variable, \(x = \mathrm{Re}(z)\)y (
float
ornumpy.array
) – Imaginary part of complex variable, \(y = \mathrm{Im}(z)\)q (
float
) – Axis ratio
- Returns:
real and imaginary part of \(\varsigma(z; q)\) function
- Return type:
tuple
(type(x), type(x))
lenstronomy.LensModel.Profiles.gaussian_ellipse_potential module¶
- class GaussianEllipsePotential[source]¶
Bases:
LensProfileBase
This class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix with ellipticity in the convergence.
the calculation follows Glenn van de Ven et al. 2009
- param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma': 0}¶
- upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma': 100}¶
- derivatives(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function.
- hessian(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
lenstronomy.LensModel.Profiles.gaussian_kappa module¶
- class GaussianKappa[source]¶
Bases:
LensProfileBase
This class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix.
- param_names = ['amp', 'sigma', 'center_x', 'center_y']¶
- lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'sigma': 0}¶
- upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'sigma': 100}¶
- derivatives(x, y, amp, sigma, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function.
- hessian(x, y, amp, sigma, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
- density_2d(x, y, amp, sigma, center_x=0, center_y=0)[source]¶
- Parameters:
x –
y –
amp –
sigma –
center_x –
center_y –
- Returns:
lenstronomy.LensModel.Profiles.gaussian_potential module¶
- class Gaussian(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix.
- param_names = ['amp', 'sigma_x', 'sigma_y', 'center_x', 'center_y']¶
- lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'sigma': 0}¶
- upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'sigma': 100}¶
lenstronomy.LensModel.Profiles.general_nfw module¶
lenstronomy.LensModel.Profiles.hernquist module¶
- class Hernquist(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Class to compute the Hernquist 1990 model, which is in 3d: rho(r) = rho0 / (r/Rs * (1 + (r/Rs))**3)
in lensing terms, the normalization parameter ‘sigma0’ is defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
Examples for converting angular to physical mass units¶
>>> from lenstronomy.Cosmo.lens_cosmo import LensCosmo >>> from astropy.cosmology import FlatLambdaCDM >>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3, Ob0=0.05) >>> lens_cosmo = LensCosmo(z_lens=0.5, z_source=1.5, cosmo=cosmo)
Here we compute the angular scale of Rs on the sky (in arc seconds) and the deflection the normalization sigma0 from the total stellar mass in M_sol and Rs in [Mpc]:
>>> sigma0, rs_angle = lens_cosmo.hernquist_phys2angular(mass=10**11, rs=0.02)
And here we perform the inverse calculation given Rs_angle and alpha_Rs to return the physical halo properties.
>>> m_tot, rs = lens_cosmo.hernquist_angular2phys(sigma0=sigma0 rs_angle=rs_angle)
The lens model calculation uses angular units as arguments! So to execute a deflection angle calculation one uses
>>> from lenstronomy.LensModel.Profiles.hernquist import Hernquist >>> hernquist = Hernquist() >>> alpha_x, alpha_y = hernquist.derivatives(x=1, y=1, Rs=rs_angle, sigma0=sigma0, center_x=0, center_y=0)
- param_names = ['sigma0', 'Rs', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'center_x': -100, 'center_y': -100, 'sigma0': 0}¶
- upper_limit_default = {'Rs': 100, 'center_x': 100, 'center_y': 100, 'sigma0': 100}¶
- static density(r, rho0, Rs)[source]¶
Computes the 3-d density.
- Parameters:
r – 3-d radius
rho0 – density normalization
Rs – Hernquist radius
- Returns:
density at radius r
- density_lens(r, sigma0, Rs)[source]¶
Density as a function of 3d radius in lensing parameters This function converts the lensing definition sigma0 into the 3d density.
- Parameters:
r – 3d radius
sigma0 – rho0 * Rs (units of projected density)
Rs – Hernquist radius
- Returns:
enclosed mass in 3d
- density_2d(x, y, rho0, Rs, center_x=0, center_y=0)[source]¶
Projected density along the line of sight at coordinate (x, y)
- Parameters:
x – x-coordinate
y – y-coordinate
rho0 – density normalization
Rs – Hernquist radius
center_x – x-center of the profile
center_y – y-center of the profile
- Returns:
projected density
- static mass_3d(r, rho0, Rs)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r – 3-d radius within the mass is integrated (same distance units as density definition)
rho0 – density normalization
Rs – Hernquist radius
- Returns:
enclosed mass
- mass_3d_lens(r, sigma0, Rs)[source]¶
Mass enclosed a 3d sphere or radius r for lens parameterisation This function converts the lensing definition sigma0 into the 3d density.
- Parameters:
r – radius
sigma0 – rho0 * Rs (units of projected density)
Rs – Hernquist radius
- Returns:
enclosed mass in 3d
- mass_2d(r, rho0, Rs)[source]¶
Mass enclosed projected 2d sphere of radius r.
- Parameters:
r – projected radius
rho0 – density normalization
Rs – Hernquist radius
- Returns:
mass enclosed 2d projected radius
- mass_2d_lens(r, sigma0, Rs)[source]¶
Mass enclosed projected 2d sphere of radius r Same as mass_2d but with input normalization in units of projected density.
- Parameters:
r – projected radius
sigma0 – rho0 * Rs (units of projected density)
Rs – Hernquist radius
- Returns:
mass enclosed 2d projected radius
- static mass_tot(rho0, Rs)[source]¶
Total mass within the profile.
- Parameters:
rho0 – density normalization
Rs – Hernquist radius
- Returns:
total mass within profile
- function(x, y, sigma0, Rs, center_x=0, center_y=0)[source]¶
Lensing potential.
- Parameters:
x – x-coordinate position (units of angle)
y – y-coordinate position (units of angle)
sigma0 – normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
Rs – Hernquist radius in units of angle
center_x – x-center of the profile (units of angle)
center_y – y-center of the profile (units of angle)
- Returns:
lensing potential at (x,y)
- derivatives(x, y, sigma0, Rs, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate position (units of angle)
y – y-coordinate position (units of angle)
sigma0 – normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
Rs – Hernquist radius in units of angle
center_x – x-center of the profile (units of angle)
center_y – y-center of the profile (units of angle)
- Returns:
derivative of function (deflection angles in x- and y-direction)
- hessian(x, y, sigma0, Rs, center_x=0, center_y=0)[source]¶
Hessian terms of the function.
- Parameters:
x – x-coordinate position (units of angle)
y – y-coordinate position (units of angle)
sigma0 – normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
Rs – Hernquist radius in units of angle
center_x – x-center of the profile (units of angle)
center_y – y-center of the profile (units of angle)
- Returns:
df/dxdx, df/dxdy, df/dydx, df/dydy
- static rho2sigma(rho0, Rs)[source]¶
Converts 3d density into 2d projected density parameter.
- Parameters:
rho0 – 3d density normalization of Hernquist model
Rs – Hernquist radius
- Returns:
sigma0 defined quantity in projected units
- static sigma2rho(sigma0, Rs)[source]¶
Converts projected density parameter (in units of deflection) into 3d density parameter.
- Parameters:
sigma0 – density defined quantity in projected units
Rs – Hernquist radius
- Returns:
rho0 the 3d density normalization of Hernquist model
- grav_pot(x, y, rho0, Rs, center_x=0, center_y=0)[source]¶
#TODO decide whether these functions are needed or not
gravitational potential (modulo 4 pi G and rho0 in appropriate units) :param x: x-coordinate position (units of angle) :param y: y-coordinate position (units of angle) :param rho0: density normalization parameter of Hernquist profile :param Rs: Hernquist radius in units of angle :param center_x: x-center of the profile (units of angle) :param center_y: y-center of the profile (units of angle) :return: gravitational potential at projected radius
lenstronomy.LensModel.Profiles.hernquist_ellipse module¶
- class Hernquist_Ellipse[source]¶
Bases:
LensProfileBase
This class contains functions for the elliptical Hernquist profile.
Ellipticity is defined in the potential.
- param_names = ['sigma0', 'Rs', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma0': 0}¶
- upper_limit_default = {'Rs': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma0': 100}¶
- function(x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns double integral of NFW profile.
- derivatives(x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function (integral of NFW)
- hessian(x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
- density(r, rho0, Rs, e1=0, e2=0)[source]¶
Computes the 3-d density.
- Parameters:
r – 3-d radius
rho0 – density normalization
Rs – Hernquist radius
- Returns:
density at radius r
- density_lens(r, sigma0, Rs, e1=0, e2=0)[source]¶
Density as a function of 3d radius in lensing parameters This function converts the lensing definition sigma0 into the 3d density.
- Parameters:
r – 3d radius
sigma0 – rho0 * Rs (units of projected density)
Rs – Hernquist radius
- Returns:
enclosed mass in 3d
- density_2d(x, y, rho0, Rs, e1=0, e2=0, center_x=0, center_y=0)[source]¶
Projected density along the line of sight at coordinate (x, y)
- Parameters:
x – x-coordinate
y – y-coordinate
rho0 – density normalization
Rs – Hernquist radius
center_x – x-center of the profile
center_y – y-center of the profile
- Returns:
projected density
- mass_2d_lens(r, sigma0, Rs, e1=0, e2=0)[source]¶
Mass enclosed projected 2d sphere of radius r Same as mass_2d but with input normalization in units of projected density.
- Parameters:
r – projected radius
sigma0 – rho0 * Rs (units of projected density)
Rs – Hernquist radius
- Returns:
mass enclosed 2d projected radius
- mass_2d(r, rho0, Rs, e1=0, e2=0)[source]¶
Mass enclosed projected 2d sphere of radius r.
- Parameters:
r – projected radius
rho0 – density normalization
Rs – Hernquist radius
- Returns:
mass enclosed 2d projected radius
- mass_3d(r, rho0, Rs, e1=0, e2=0)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r – 3-d radius within the mass is integrated (same distance units as density definition)
rho0 – density normalization
Rs – Hernquist radius
- Returns:
enclosed mass
- mass_3d_lens(r, sigma0, Rs, e1=0, e2=0)[source]¶
Mass enclosed a 3d sphere or radius r in lensing parameterization.
- Parameters:
r – 3-d radius within the mass is integrated (same distance units as density definition)
sigma0 – rho0 * Rs (units of projected density)
Rs – Hernquist radius
- Returns:
enclosed mass
lenstronomy.LensModel.Profiles.hernquist_ellipse_cse module¶
- class HernquistEllipseCSE[source]¶
Bases:
Hernquist_Ellipse
This class contains functions for the elliptical Hernquist profile.
Ellipticity is defined in the convergence. Approximation with CSE profile introduced by Oguri 2021: https://arxiv.org/pdf/2106.11464.pdf
- param_names = ['sigma0', 'Rs', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma0': 0}¶
- upper_limit_default = {'Rs': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma0': 100}¶
- function(x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns double integral of NFW profile.
lenstronomy.LensModel.Profiles.hessian module¶
- class Hessian(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Class for constant Hessian distortion (second order) The input is in the same convention as the LensModel.hessian() output.
- param_names = ['f_xx', 'f_yy', 'f_xy', 'f_yx', 'ra_0', 'dec_0']¶
- lower_limit_default = {'dec_0': -100, 'f_xx': -100, 'f_xy': -100, 'f_yx': -100, 'f_yy': -100, 'ra_0': -100}¶
- upper_limit_default = {'dec_0': 100, 'f_xx': 100, 'f_xy': 100, 'f_yx': 100, 'f_yy': 100, 'ra_0': 100}¶
- function(x, y, f_xx, f_yy, f_xy, f_yx, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
f_xx – dalpha_x/dx
f_yy – dalpha_y/dy
f_xy – dalpha_x/dy
f_yx – dalpha_y/dx
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
lensing potential
- derivatives(x, y, f_xx, f_yy, f_xy, f_yx, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
f_xx – dalpha_x/dx
f_yy – dalpha_y/dy
f_xy – dalpha_x/dy
f_yx – dalpha_y/dx
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
deflection angles
- hessian(x, y, f_xx, f_yy, f_xy, f_yx, ra_0=0, dec_0=0)[source]¶
Hessian. Attention: If f_xy != f_yx then this function is not accurate!
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
f_xx – dalpha_x/dx
f_yy – dalpha_y/dy
f_xy – dalpha_x/dy
f_yx – dalpha_y/dx
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
f_xx, f_yy, f_xy
lenstronomy.LensModel.Profiles.interpol module¶
- class Interpol(grid=False, min_grid_number=100, kwargs_spline=None)[source]¶
Bases:
LensProfileBase
Class which uses an interpolation of a lens model and its first and second order derivatives.
See also the tests in lenstronomy.test.test_LensModel.test_Profiles.test_interpol.py for example use cases as checks against known analytic models.
The deflection angle is in the same convention as the one in the LensModel module, meaning that: source position = image position - deflection angle
- param_names = ['grid_interp_x', 'grid_interp_y', 'f_', 'f_x', 'f_y', 'f_xx', 'f_yy', 'f_xy']¶
- lower_limit_default = {}¶
- upper_limit_default = {}¶
- __init__(grid=False, min_grid_number=100, kwargs_spline=None)[source]¶
- Parameters:
grid – bool, if True, computes the calculation on a grid
min_grid_number – minimum numbers of positions to compute the interpolation on a grid, otherwise in a loop
kwargs_spline – keyword arguments for the scipy.interpolate.RectBivariateSpline() interpolation (optional) if =None, a default linear interpolation is chosen.
- function(x, y, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]¶
- Parameters:
x – x-coordinate (angular position), float or numpy array
y – y-coordinate (angular position), float or numpy array
grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
- Returns:
potential at interpolated positions (x, y)
- derivatives(x, y, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]¶
Returns df/dx and df/dy of the function.
- Parameters:
x – x-coordinate (angular position), float or numpy array
y – y-coordinate (angular position), float or numpy array
grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
- Returns:
f_x, f_y at interpolated positions (x, y)
- hessian(x, y, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
- Parameters:
x – x-coordinate (angular position), float or numpy array
y – y-coordinate (angular position), float or numpy array
grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
- Returns:
f_xx, f_xy, f_yx, f_yy at interpolated positions (x, y)
- class InterpolScaled(grid=True, min_grid_number=100, kwargs_spline=None)[source]¶
Bases:
LensProfileBase
Class for handling an interpolated lensing map and has the freedom to scale its lensing effect.
Applications are e.g. mass to light ratio.
- param_names = ['scale_factor', 'grid_interp_x', 'grid_interp_y', 'f_', 'f_x', 'f_y', 'f_xx', 'f_yy', 'f_xy']¶
- lower_limit_default = {'scale_factor': 0}¶
- upper_limit_default = {'scale_factor': 100}¶
- __init__(grid=True, min_grid_number=100, kwargs_spline=None)[source]¶
- Parameters:
grid – bool, if True, computes the calculation on a grid
min_grid_number – minimum numbers of positions to compute the interpolation on a grid
kwargs_spline – keyword arguments for the scipy.interpolate.RectBivariateSpline() interpolation (optional) if =None, a default linear interpolation is chosen.
- function(x, y, scale_factor=1, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]¶
- Parameters:
x – x-coordinate (angular position), float or numpy array
y – y-coordinate (angular position), float or numpy array
scale_factor – float, overall scaling of the lens model relative to the input interpolation grid
grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
- Returns:
potential at interpolated positions (x, y)
- derivatives(x, y, scale_factor=1, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]¶
- Parameters:
x – x-coordinate (angular position), float or numpy array
y – y-coordinate (angular position), float or numpy array
scale_factor – float, overall scaling of the lens model relative to the input interpolation grid
grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
- Returns:
deflection angles in x- and y-direction at position (x, y)
- hessian(x, y, scale_factor=1, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]¶
- Parameters:
x – x-coordinate (angular position), float or numpy array
y – y-coordinate (angular position), float or numpy array
scale_factor – float, overall scaling of the lens model relative to the input interpolation grid
grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
- Returns:
second derivatives of the lensing potential f_xx, f_yy, f_xy at position (x, y)
lenstronomy.LensModel.Profiles.multi_gaussian_kappa module¶
- class MultiGaussianKappa[source]¶
Bases:
LensProfileBase
- param_names = ['amp', 'sigma', 'center_x', 'center_y']¶
- lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'sigma': 0}¶
- upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'sigma': 100}¶
- function(x, y, amp, sigma, center_x=0, center_y=0, scale_factor=1)[source]¶
- Parameters:
x –
y –
amp –
sigma –
center_x –
center_y –
- Returns:
- derivatives(x, y, amp, sigma, center_x=0, center_y=0, scale_factor=1)[source]¶
- Parameters:
x –
y –
amp –
sigma –
center_x –
center_y –
- Returns:
- hessian(x, y, amp, sigma, center_x=0, center_y=0, scale_factor=1)[source]¶
- Parameters:
x –
y –
amp –
sigma –
center_x –
center_y –
- Returns:
- class MultiGaussianKappaEllipse[source]¶
Bases:
LensProfileBase
- param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma': 0}¶
- upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma': 100}¶
- function(x, y, amp, sigma, e1, e2, center_x=0, center_y=0, scale_factor=1)[source]¶
- Parameters:
x –
y –
amp –
sigma –
center_x –
center_y –
- Returns:
- derivatives(x, y, amp, sigma, e1, e2, center_x=0, center_y=0, scale_factor=1)[source]¶
- Parameters:
x –
y –
amp –
sigma –
center_x –
center_y –
- Returns:
- hessian(x, y, amp, sigma, e1, e2, center_x=0, center_y=0, scale_factor=1)[source]¶
- Parameters:
x –
y –
amp –
sigma –
center_x –
center_y –
- Returns:
lenstronomy.LensModel.Profiles.multipole module¶
- class Multipole(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class contains a multipole contribution (for 1 component with m>=2) This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf Equation B12 m : int, multipole order, m>=2 a_m : float, multipole strength phi_m : float, multipole orientation in radian
- param_names = ['m', 'a_m', 'phi_m', 'center_x', 'center_y']¶
- lower_limit_default = {'a_m': 0, 'center_x': -100, 'center_y': -100, 'm': 2, 'phi_m': -3.141592653589793}¶
- upper_limit_default = {'a_m': 100, 'center_x': 100, 'center_y': 100, 'm': 100, 'phi_m': 3.141592653589793}¶
- function(x, y, m, a_m, phi_m, center_x=0, center_y=0)[source]¶
Lensing potential of multipole contribution (for 1 component with m>=2) This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf
- Parameters:
x – x-coordinate to evaluate function
y – y-coordinate to evaluate function
m – int, multipole order, m>=2
a_m – float, multipole strength
phi_m – float, multipole orientation in radian
center_x – x-position
center_y – y-position
- Returns:
lensing potential
- derivatives(x, y, m, a_m, phi_m, center_x=0, center_y=0)[source]¶
Deflection of a multipole contribution (for 1 component with m>=2) This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf Equation B12
- Parameters:
x – x-coordinate to evaluate function
y – y-coordinate to evaluate function
m – int, multipole order, m>=2
a_m – float, multipole strength
phi_m – float, multipole orientation in radian
center_x – x-position
center_y – y-position
- Returns:
deflection angles alpha_x, alpha_y
- hessian(x, y, m, a_m, phi_m, center_x=0, center_y=0)[source]¶
Hessian of a multipole contribution (for 1 component with m>=2) This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf
- Parameters:
x – x-coordinate to evaluate function
y – y-coordinate to evaluate function
m – int, multipole order, m>=2
a_m – float, multipole strength
phi_m – float, multipole orientation in radian
center_x – x-position
center_y – y-position
- Returns:
f_xx, f_xy, f_yx, f_yy
lenstronomy.LensModel.Profiles.nfw module¶
- class NFW(interpol=False, num_interp_X=1000, max_interp_X=10)[source]¶
Bases:
LensProfileBase
This class contains functions concerning the NFW profile.
relation are: R_200 = c * Rs The definition of ‘Rs’ is in angular (arc second) units and the normalization is put in with regard to a deflection angle at ‘Rs’ - ‘alpha_Rs’. To convert a physical mass and concentration definition into those lensing quantities for a specific redshift configuration and cosmological model, you can find routines in lenstronomy.Cosmo.lens_cosmo.py
Examples for converting angular to physical mass units¶
>>> from lenstronomy.Cosmo.lens_cosmo import LensCosmo >>> from astropy.cosmology import FlatLambdaCDM >>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3, Ob0=0.05) >>> lens_cosmo = LensCosmo(z_lens=0.5, z_source=1.5, cosmo=cosmo)
Here we compute the angular scale of Rs on the sky (in arc seconds) and the deflection angle at Rs (in arc seconds):
>>> Rs_angle, alpha_Rs = lens_cosmo.nfw_physical2angle(M=10**13, c=6)
And here we perform the inverse calculation given Rs_angle and alpha_Rs to return the physical halo properties.
>>> rho0, Rs, c, r200, M200 = lens_cosmo.nfw_angle2physical(Rs_angle=Rs_angle, alpha_Rs=alpha_Rs)
The lens model calculation uses angular units as arguments! So to execute a deflection angle calculation one uses
>>> from lenstronomy.LensModel.Profiles.nfw import NFW >>> nfw = NFW() >>> alpha_x, alpha_y = nfw.derivatives(x=1, y=1, Rs=Rs_angle, alpha_Rs=alpha_Rs, center_x=0, center_y=0)
- profile_name = 'NFW'¶
- param_names = ['Rs', 'alpha_Rs', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100}¶
- __init__(interpol=False, num_interp_X=1000, max_interp_X=10)[source]¶
- Parameters:
interpol – bool, if True, interpolates the functions F(), g() and h()
num_interp_X – int (only considered if interpol=True), number of interpolation elements in units of r/r_s
max_interp_X – float (only considered if interpol=True), maximum r/r_s value to be interpolated (returning zeros outside)
- function(x, y, Rs, alpha_Rs, center_x=0, center_y=0)[source]¶
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
lensing potential
- derivatives(x, y, Rs, alpha_Rs, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function (integral of NFW), which are the deflection angles.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
deflection angle in x, deflection angle in y
- hessian(x, y, Rs, alpha_Rs, center_x=0, center_y=0)[source]¶
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
- static density(R, Rs, rho0)[source]¶
Three-dimensional NFW profile.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
- Returns:
rho(R) density
- density_lens(r, Rs, alpha_Rs)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – 3d radios
Rs – turn-over radius of NFW profile
alpha_Rs – deflection at Rs
- Returns:
density rho(r)
- density_2d(x, y, Rs, rho0, center_x=0, center_y=0)[source]¶
Projected two-dimensional NFW profile (kappa)
- Parameters:
x – x-coordinate
y – y-coordinate
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
center_x – x-centroid position
center_y – y-centroid position
- Returns:
Epsilon(R) projected density at radius R
- mass_3d(r, Rs, rho0)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r – 3d radius
Rs – scale radius
rho0 – density normalization (characteristic density)
- Returns:
M(<r)
- mass_3d_lens(r, Rs, alpha_Rs)[source]¶
Mass enclosed a 3d sphere or radius r. This function takes as input the lensing parameterization.
- Parameters:
r – 3d radius
Rs – scale radius
alpha_Rs – deflection (angular units) at projected Rs
- Returns:
M(<r)
- mass_2d(R, Rs, rho0)[source]¶
Mass enclosed a 2d cylinder or projected radius R.
- Parameters:
R – projected radius
Rs – scale radius
rho0 – density normalization (characteristic density)
- Returns:
mass in cylinder.
- mass_2d_lens(R, Rs, alpha_Rs)[source]¶
- Parameters:
R – projected radius
Rs – scale radius
alpha_Rs – deflection (angular units) at projected Rs
- Returns:
mass enclosed 2d cylinder <R
- nfw_potential(R, Rs, rho0)[source]¶
Lensing potential of NFW profile (Sigma_crit D_OL**2)
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
- Returns:
Epsilon(R) projected density at radius R
- nfw_alpha(R, Rs, rho0, ax_x, ax_y)[source]¶
Deflection angle of NFW profile (times Sigma_crit D_OL) along the projection to coordinate ‘axis’.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
ax_x (same as R) – projection to either x- or y-axis
ax_y (same as R) – projection to either x- or y-axis
- Returns:
Epsilon(R) projected density at radius R
- nfw_gamma(R, Rs, rho0, ax_x, ax_y)[source]¶
Shear gamma of NFW profile (times Sigma_crit) along the projection to coordinate ‘axis’.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
ax_x (same as R) – projection to either x- or y-axis
ax_y (same as R) – projection to either x- or y-axis
- Returns:
Epsilon(R) projected density at radius R
lenstronomy.LensModel.Profiles.nfw_ellipse module¶
- class NFW_ELLIPSE(interpol=False, num_interp_X=1000, max_interp_X=10)[source]¶
Bases:
LensProfileBase
This class contains functions concerning the NFW profile with an ellipticity defined in the potential parameterization of alpha_Rs and Rs is the same as for the spherical NFW profile.
from Glose & Kneib: https://cds.cern.ch/record/529584/files/0112138.pdf
relation are: R_200 = c * Rs
- profile_name = 'NFW_ELLIPSE'¶
- param_names = ['Rs', 'alpha_Rs', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5}¶
- __init__(interpol=False, num_interp_X=1000, max_interp_X=10)[source]¶
- Parameters:
interpol – bool, if True, interpolates the functions F(), g() and h()
num_interp_X – int (only considered if interpol=True), number of interpolation elements in units of r/r_s
max_interp_X – float (only considered if interpol=True), maximum r/r_s value to be interpolated (returning zeros outside)
- function(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns elliptically distorted NFW lensing potential.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
lensing potential
- derivatives(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function, calculated as an elliptically distorted deflection angle of the spherical NFW profile.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
deflection in x-direction, deflection in y-direction
- hessian(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy the calculation is performed as a numerical differential from the deflection field. Analytical relations are possible.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
- mass_3d_lens(r, Rs, alpha_Rs, e1=1, e2=0)[source]¶
- Parameters:
r – radius (in angular units)
Rs –
alpha_Rs –
e1 –
e2 –
- Returns:
- density_lens(r, Rs, alpha_Rs, e1=1, e2=0)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – 3d radios
Rs – turn-over radius of NFW profile
alpha_Rs – deflection at Rs
- Returns:
density rho(r)
lenstronomy.LensModel.Profiles.nfw_ellipse_cse module¶
- class NFW_ELLIPSE_CSE(high_accuracy=True)[source]¶
Bases:
NFW_ELLIPSE
this class contains functions concerning the NFW profile with an ellipticity defined in the convergence parameterization of alpha_Rs and Rs is the same as for the spherical NFW profile Approximation with CSE profile introduced by Oguri 2021: https://arxiv.org/pdf/2106.11464.pdf Match to NFW using CSEs is approximate: kappa matches to ~1-2%
relation are: R_200 = c * Rs
- profile_name = 'NFW_ELLIPSE_CSE'¶
- param_names = ['Rs', 'alpha_Rs', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5}¶
- __init__(high_accuracy=True)[source]¶
- Parameters:
high_accuracy (boolean) – if True uses a more accurate larger set of CSE profiles (see Oguri 2021)
- function(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns elliptically distorted NFW lensing potential.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
lensing potential
- derivatives(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function, calculated as an elliptically distorted deflection angle of the spherical NFW profile.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
deflection in x-direction, deflection in y-direction
- hessian(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy the calculation is performed as a numerical differential from the deflection field. Analytical relations are possible.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
lenstronomy.LensModel.Profiles.nfw_mass_concentration module¶
This module contains a class to compute the Navarro-Frank-White function in mass/kappa space.
- class NFWMC(z_lens, z_source, cosmo=None, static=False)[source]¶
Bases:
LensProfileBase
this class contains functions parameterises the NFW profile with log10 M200 and the concentration rs/r200 relation are: R_200 = c * Rs
ATTENTION: the parameterization is cosmology and redshift dependent! The cosmology to connect mass and deflection relations is fixed to default H0=70km/s Omega_m=0.3 flat LCDM. It is recommended to keep a given cosmology definition in the lens modeling as the observable reduced deflection angles are sensitive in this parameterization. If you do not want to impose a mass-concentration relation, it is recommended to use the default NFW lensing profile parameterized in reduced deflection angles.
- param_names = ['logM', 'concentration', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'concentration': 0.01, 'logM': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'concentration': 1000, 'logM': 16}¶
- __init__(z_lens, z_source, cosmo=None, static=False)[source]¶
- Parameters:
z_lens – redshift of lens
z_source – redshift of source
cosmo – astropy cosmology instance
static – boolean, if True, only operates with fixed parameter values
- set_static(logM, concentration, *args, **kwargs)[source]¶
- Parameters:
logM – log10(M200)
concentration – halo concentration c = r_200 / r_s
- Returns:
- function(x, y, logM, concentration, center_x=0, center_y=0)[source]¶
- Parameters:
x – angular position
y – angular position
Rs – angular turn over point
alpha_Rs – deflection at Rs
center_x – center of halo
center_y – center of halo
- Returns:
lenstronomy.LensModel.Profiles.nfw_vir_trunc module¶
- class NFWVirTrunc(z_lens, z_source, cosmo=None)[source]¶
Bases:
LensProfileBase
this class contains functions concerning the NFW profile that is sharply truncated at the virial radius https://arxiv.org/pdf/astro-ph/0304034.pdf
relation are: R_200 = c * Rs
lenstronomy.LensModel.Profiles.nie module¶
- class NIE[source]¶
Bases:
LensProfileBase
Non-singular isothermal ellipsoid (NIE)
\[\kappa = \theta_E/2 \left[s^2_{scale} + qx^2 + y^2/q]−1/2\]- param_names = ['theta_E', 'e1', 'e2', 's_scale', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 's_scale': 0, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 's_scale': 100, 'theta_E': 10}¶
- function(x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale
center_x – profile center
center_y – profile center
- Returns:
lensing potential
- derivatives(x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale
center_x – profile center
center_y – profile center
- Returns:
alpha_x, alpha_y
- hessian(x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale
center_x – profile center
center_y – profile center
- Returns:
f_xx, f_xy, f_yx, f_yy
- density_lens(r, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
3d mass density at 3d radius r. This function assumes spherical symmetry/ignoring the eccentricity.
- Parameters:
r – 3d radius
theta_E – Einstein radius
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale
center_x – profile center
center_y – profile center
- Returns:
3d mass density at 3d radius r
- mass_3d_lens(r, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
Mass enclosed a 3d radius r. This function assumes spherical symmetry/ignoring the eccentricity.
- Parameters:
r – 3d radius
theta_E – Einstein radius
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale
center_x – profile center
center_y – profile center
- Returns:
3d mass density at 3d radius r
- class NIEMajorAxis(diff=1e-10)[source]¶
Bases:
LensProfileBase
This class contains the function and the derivatives of the non-singular isothermal ellipse. See Keeton and Kochanek 1998, https://arxiv.org/pdf/astro-ph/9705194.pdf
\[\kappa = b * (q2(s2 + x2) + y2)^{−1/2}`\]- param_names = ['b', 's', 'q', 'center_x', 'center_y']¶
- function(x, y, b, s, q)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
lenstronomy.LensModel.Profiles.nie_potential module¶
- class NIE_POTENTIAL[source]¶
Bases:
LensProfileBase
This class implements the elliptical potential of Eq.
(67) of LECTURES ON GRAVITATIONAL LENSING and Eq. (1) of Blandford & Kochanek 1987, mapped to Eq. (8) of Barnaka1998 to find the ellipticity bounds
- param_names = ['center_x', 'center_y', 'theta_E', 'theta_c', 'e1', 'e2']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': 0, 'e2': 0, 'theta_E': 0, 'theta_c': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.2, 'e2': 0.2, 'theta_E': 10, 'theta_c': 10}¶
- set_static(theta_E, theta_c, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate in image plane
y – y-coordinate in image plane
theta_E – Einstein radius
theta_c – core radius
e1 – eccentricity component
e2 – eccentricity component
center_x – profile center
center_y – profile center
- Returns:
self variables set
- function(x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coord (in angles)
y – y-coord (in angles)
theta_E – Einstein radius (in angles)
theta_c – core radius (in angles)
e1 – eccentricity component, x direction(dimensionless)
e2 – eccentricity component, y direction (dimensionless)
- Returns:
lensing potential
- derivatives(x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coord (in angles)
y – y-coord (in angles)
theta_E – Einstein radius (in angles)
theta_c – core radius (in angles)
e1 – eccentricity component, x direction(dimensionless)
e2 – eccentricity component, y direction (dimensionless)
- Returns:
deflection angle (in angles)
- hessian(x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coord (in angles)
y – y-coord (in angles)
theta_E – Einstein radius (in angles)
theta_c – core radius (in angles)
e1 – eccentricity component, x direction(dimensionless)
e2 – eccentricity component, y direction (dimensionless)
- Returns:
hessian matrix (in angles)
- class NIEPotentialMajorAxis(diff=1e-10)[source]¶
Bases:
LensProfileBase
This class implements the elliptical potential of Eq.
(67) of LECTURES ON GRAVITATIONAL LENSING and Eq. (1) of Blandford & Kochanek 1987, mapped to Eq. (8) of Barnaka1998 to find the ellipticity bounds
- param_names = ['theta_E', 'theta_c', 'eps', 'center_x', 'center_y']¶
lenstronomy.LensModel.Profiles.numerical_deflections module¶
- class TabulatedDeflections(custom_class)[source]¶
Bases:
LensProfileBase
A user-defined class that returns deflection angles given a set of observed coordinates on the sky (x, y).
This class has similar functionality as INTERPOL, with the difference being that the interpolation for this class is done prior to class creation. When used with routines in the lenstronomy.Sampling, this class effectively acts as a fixed lens model with no keyword arguments.
- profile_name = 'TABULATED_DEFLECTIONS'¶
- param_names = []¶
- lower_limit_default = {}¶
- upper_limit_default = {}¶
- __init__(custom_class)[source]¶
- Parameters:
custom_class – a user-defined class that has a __call___ method that returns deflection angles
Code example:
>>> custom_class = CustomLensingClass() >>> alpha_x, alpha_y = custom_class(x, y, **kwargs)
or equivalently:
>>> from lenstronomy.LensModel.lens_model import LensModel >>> lens_model_list = ['NumericalAlpha'] >>> lens_model = LensModel(lens_model_list, numerical_alpha_class=custom_class) >>>> alpha_x, alpha_y = lens_model.alpha(x, y, **kwargs)
- function(x, y, center_x=0, center_y=0, **kwargs)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- derivatives(x, y, center_x=0, center_y=0, **kwargs)[source]¶
- Parameters:
x – x coordinate [arcsec]
y – x coordinate [arcsec]
center_x – deflector x center [arcsec]
center_y – deflector y center [arcsec]
kwargs – keyword arguments for the custom profile
- Returns:
- hessian(x, y, center_x=0, center_y=0, **kwargs)[source]¶
Returns the components of the hessian matrix :param x: x coordinate [arcsec] :param y: y coordinate [arcsec] :param center_x: the deflector x coordinate :param center_y: the deflector y coordinate :param kwargs: keyword arguments for the profile :return: the derivatives of the deflection angles that make up the hessian matrix.
lenstronomy.LensModel.Profiles.p_jaffe module¶
- class PJaffe[source]¶
Bases:
LensProfileBase
class to compute the DUAL PSEUDO ISOTHERMAL MASS DISTRIBUTION based on Eliasdottir (2007) https://arxiv.org/pdf/0710.5636.pdf Appendix A
Module name: ‘PJAFFE’;
An alternative name is dPIED (in the elliptical scenario)
This profile is for the spherical case. For an elliptical version, use “PJAFFE_ELLIPSE” (ellipticitly in the potential) # TODO: add/revise name once ellipticity in the mass is available
The 3D density distribution is
\[\rho(r) = \frac{\rho_0}{(1+r^2/Ra^2)(1+r^2/Rs^2)}\]with \(Rs > Ra\).
The projected density is
\[\Sigma(R) = \Sigma_0 \frac{Ra Rs}{Rs-Ra}\left(\frac{1}{\sqrt{Ra^2+R^2}} - \frac{1}{\sqrt{Rs^2+R^2}} \right)\]with
\[\Sigma_0 = \pi \rho_0 \frac{Ra Rs}{Rs + Ra}\]In the lensing parameterization,
\[\sigma_0 = \frac{\Sigma_0}{\Sigma_{\rm crit}}\]- param_names = ['sigma0', 'Ra', 'Rs', 'center_x', 'center_y']¶
- lower_limit_default = {'Ra': 0, 'Rs': 0, 'center_x': -100, 'center_y': -100, 'sigma0': 0}¶
- upper_limit_default = {'Ra': 100, 'Rs': 100, 'center_x': 100, 'center_y': 100, 'sigma0': 10}¶
- density(r, rho0, Ra, Rs)[source]¶
Computes the density.
- Parameters:
r – radial distance from the center (in 3D)
rho0 – density normalization (see class documentation above)
Ra – core radius
Rs – transition radius from logarithmic slope -2 to -4
- Returns:
density at r
- density_2d(x, y, rho0, Ra, Rs, center_x=0, center_y=0)[source]¶
Projected density.
- Parameters:
x – projected coordinate on the sky
y – projected coordinate on the sky
rho0 – density normalization (see class documentation above)
Ra – core radius
Rs – transition radius from logarithmic slope -2 to -4
center_x – center of profile
center_y – center of profile
- Returns:
projected density
- mass_3d(r, rho0, Ra, Rs)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r – radial distance from the center (in 3D)
rho0 – density normalization (see class documentation above)
Ra – core radius
Rs – transition radius from logarithmic slope -2 to -4
- Returns:
M(<r)
- mass_3d_lens(r, sigma0, Ra, Rs)[source]¶
Mass enclosed a 3d sphere or radius r given a lens parameterization with angular units.
- Parameters:
r – radial distance from the center (in 3D)
sigma0 – density normalization (see class documentation above)
Ra – core radius
Rs – transition radius from logarithmic slope -2 to -4
- Returns:
M(<r) in angular units (modulo critical mass density)
- mass_2d(r, rho0, Ra, Rs)[source]¶
Mass enclosed projected 2d sphere of radius r.
- Parameters:
r – radial distance from the center in projection
rho0 – density normalization (see class documentation above)
Ra – core radius
Rs – transition radius from logarithmic slope -2 to -4
- Returns:
Sigma(<r)
- mass_tot(rho0, Ra, Rs)[source]¶
Total mass within the profile.
- Parameters:
rho0 – density normalization (see class documentation above)
Ra – core radius
Rs – transition radius from logarithmic slope -2 to -4
- Returns:
total mass
- grav_pot(r, rho0, Ra, Rs)[source]¶
Gravitational potential (modulo 4 pi G and rho0 in appropriate units)
- Parameters:
r – radial distance from the center (in 3D)
rho0 – density normalization (see class documentation above)
Ra – core radius
Rs – transition radius from logarithmic slope -2 to -4
- Returns:
gravitational potential (modulo 4 pi G and rho0 in appropriate units)
- function(x, y, sigma0, Ra, Rs, center_x=0, center_y=0)[source]¶
Lensing potential.
- Parameters:
x – projected coordinate on the sky
y – projected coordinate on the sky
sigma0 – sigma0/sigma_crit (see class documentation above)
Ra – core radius (see class documentation above)
Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
center_x – center of profile
center_y – center of profile
- Returns:
lensing potential
- derivatives(x, y, sigma0, Ra, Rs, center_x=0, center_y=0)[source]¶
Deflection angles.
- Parameters:
x – projected coordinate on the sky
y – projected coordinate on the sky
sigma0 – sigma0/sigma_crit (see class documentation above)
Ra – core radius (see class documentation above)
Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
center_x – center of profile
center_y – center of profile
- Returns:
f_x, f_y
- hessian(x, y, sigma0, Ra, Rs, center_x=0, center_y=0)[source]¶
Hessian of lensing potential.
- Parameters:
x – projected coordinate on the sky
y – projected coordinate on the sky
sigma0 – sigma0/sigma_crit (see class documentation above)
Ra – core radius (see class documentation above)
Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
center_x – center of profile
center_y – center of profile
- Returns:
f_xx, f_xy, f_yx, f_yy
- rho2sigma(rho0, Ra, Rs)[source]¶
Converts 3d density into 2d projected density parameter, Equation A4 in Eliasdottir (2007)
- Parameters:
rho0 – density normalization
Ra – core radius (see class documentation above)
Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
- Returns:
projected density normalization
lenstronomy.LensModel.Profiles.p_jaffe_ellipse module¶
- class PJaffe_Ellipse[source]¶
Bases:
LensProfileBase
class to compute the DUAL PSEUDO ISOTHERMAL ELLIPTICAL MASS DISTRIBUTION based on Eliasdottir (2007) https://arxiv.org/pdf/0710.5636.pdf Appendix A with the ellipticity implemented in the potential
Module name: ‘PJAFFE_ELLIPSE’;
An alternative name is dPIED.
The 3D density distribution is
\[\rho(r) = \frac{\rho_0}{(1+r^2/Ra^2)(1+r^2/Rs^2)}\]with \(Rs > Ra\).
The projected density is
\[\Sigma(R) = \Sigma_0 \frac{Ra Rs}{Rs-Ra}\left(\frac{1}{\sqrt{Ra^2+R^2}} - \frac{1}{\sqrt{Rs^2+R^2}} \right)\]with
\[\Sigma_0 = \pi \rho_0 \frac{Ra Rs}{Rs + Ra}\]In the lensing parameterization,
\[\sigma_0 = \frac{\Sigma_0}{\Sigma_{\rm crit}}\]- param_names = ['sigma0', 'Ra', 'Rs', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'Ra': 0, 'Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma0': 0}¶
- upper_limit_default = {'Ra': 100, 'Rs': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma0': 10}¶
- function(x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns double integral of NFW profile.
- derivatives(x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function (integral of NFW)
lenstronomy.LensModel.Profiles.pemd module¶
- class PEMD(suppress_fastell=False)[source]¶
Bases:
LensProfileBase
Class for power law ellipse mass density profile (PEMD). This class effectively calls the class SPEMD_SMOOTH with a fixed and very small central smoothing scale to perform the numerical integral using the FASTELL code by Renan Barkana. An alternative implementation of the same model using pure python with analytical functions is probided as ‘EPL’ profile.
\[\kappa(x, y) = \frac{3-\gamma}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q}} \right)^{\gamma-1}\]with \(\theta_{E}\) is the (circularized) Einstein radius, \(\gamma\) is the negative power-law slope of the 3D mass distributions, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate system aligned with the major and minor axis of the lens.
In terms of eccentricities, this profile is defined as
\[\kappa(r) = \frac{3-\gamma}{2} \left(\frac{\theta'_{E}}{r \sqrt{1 − e*\cos(2*\phi)}} \right)^{\gamma-1}\]with \(\epsilon\) is the ellipticity defined as
\[\epsilon = \frac{1-q^2}{1+q^2}\]And an Einstein radius \(\theta'_{\rm E}\) related to the definition used is
\[\left(\frac{\theta'_{\rm E}}{\theta_{\rm E}}\right)^{2} = \frac{2q}{1+q^2}.\]- param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 1.5, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 2.5, 'theta_E': 100}¶
- __init__(suppress_fastell=False)[source]¶
- Parameters:
suppress_fastell – bool, if True, does not raise if fastell4py is not installed
- function(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
lensing potential
- derivatives(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
deflection angles alpha_x, alpha_y
- hessian(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
Hessian components f_xx, f_xy, f_yx, f_yy
- mass_3d_lens(r, theta_E, gamma, e1=None, e2=None)[source]¶
Computes the spherical power-law mass enclosed (with SPP routine).
- Parameters:
r – radius within the mass is computed
theta_E – Einstein radius
gamma – power-law slope
e1 – eccentricity component (not used)
e2 – eccentricity component (not used)
- Returns:
mass enclosed a 3D radius r
- density_lens(r, theta_E, gamma, e1=None, e2=None)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – radius within the mass is computed
theta_E – Einstein radius
gamma – power-law slope
e1 – eccentricity component (not used)
e2 – eccentricity component (not used)
- Returns:
mass enclosed a 3D radius r
lenstronomy.LensModel.Profiles.point_mass module¶
- class PointMass[source]¶
Bases:
LensProfileBase
Class to compute the physical deflection angle of a point mass, given as an Einstein radius.
- param_names = ['theta_E', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'theta_E': 100}¶
- function(x, y, theta_E, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coord (in angles)
y – y-coord (in angles)
theta_E – Einstein radius (in angles)
- Returns:
lensing potential
lenstronomy.LensModel.Profiles.sersic module¶
- class Sersic(smoothing=1e-05, sersic_major_axis=False)[source]¶
Bases:
SersicUtil
,LensProfileBase
this class contains functions to evaluate a Sersic mass profile: https://arxiv.org/pdf/astro-ph/0311559.pdf
\[\kappa(R) = \kappa_{\rm eff} \exp \left[ -b_n (R/R_{\rm Sersic})^{\frac{1}{n}}\right]\]with \(b_{n}\approx 1.999n-0.327\)
Examples¶
Example for converting physical mass units into convergence units used in the definition of this profile.
We first define an AstroPy cosmology instance and a LensCosmo class instance with a lens and source redshift.
>>> from lenstronomy.Cosmo.lens_cosmo import LensCosmo >>> from astropy.cosmology import FlatLambdaCDM >>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3, Ob0=0.05) >>> lens_cosmo = LensCosmo(z_lens=0.5, z_source=1.5, cosmo=cosmo)
We define the half-light radius R_sersic (arc seconds on the sky) and Sersic index n_sersic
>>> R_sersic = 2 >>> n_sersic = 4
Here we compute k_eff, the convergence at the half-light radius R_sersic for a stellar mass in Msun
>>> k_eff = lens_cosmo.sersic_m_star2k_eff(m_star=10**11.5, R_sersic=R_sersic, n_sersic=n_sersic)
And here we perform the inverse calculation given k_eff to return the physical stellar mass.
>>> m_star = lens_cosmo.sersic_k_eff2m_star(k_eff=k_eff, R_sersic=R_sersic, n_sersic=n_sersic)
The lens model calculation uses angular units as arguments! So to execute a deflection angle calculation one uses
>>> from lenstronomy.LensModel.Profiles.sersic import Sersic >>> sersic = Sersic() >>> alpha_x, alpha_y = sersic.derivatives(x=1, y=1, k_eff=k_eff, R_sersic=R_sersic, center_x=0, center_y=0)
- param_names = ['k_eff', 'R_sersic', 'n_sersic', 'center_x', 'center_y']¶
- lower_limit_default = {'R_sersic': 0, 'center_x': -100, 'center_y': -100, 'k_eff': 0, 'n_sersic': 0.5}¶
- upper_limit_default = {'R_sersic': 100, 'center_x': 100, 'center_y': 100, 'k_eff': 10, 'n_sersic': 8}¶
- function(x, y, n_sersic, R_sersic, k_eff, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate
y – y-coordinate
n_sersic – Sersic index
R_sersic – half light radius
k_eff – convergence at half light radius
center_x – x-center
center_y – y-center
- Returns:
lenstronomy.LensModel.Profiles.sersic_ellipse_kappa module¶
- class SersicEllipseKappa[source]¶
Bases:
LensProfileBase
This class contains the function and the derivatives of an elliptical sersic profile with the ellipticity introduced in the convergence (not the potential).
This requires the use of numerical integrals (Keeton 2004)
- param_names = ['k_eff', 'R_sersic', 'n_sersic', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'R_sersic': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'k_eff': 0, 'n_sersic': 0.5}¶
- upper_limit_default = {'R_sersic': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'k_eff': 10, 'n_sersic': 8}¶
- function(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- derivatives(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]¶
Deflection angles.
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
lenstronomy.LensModel.Profiles.sersic_ellipse_potential module¶
- class SersicEllipse[source]¶
Bases:
LensProfileBase
this class contains functions to evaluate a Sersic mass profile: https://arxiv.org/pdf/astro-ph/0311559.pdf
- param_names = ['k_eff', 'R_sersic', 'n_sersic', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'R_sersic': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'k_eff': 0, 'n_sersic': 0.5}¶
- upper_limit_default = {'R_sersic': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'k_eff': 10, 'n_sersic': 8}¶
- function(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]¶
Returns Gaussian.
lenstronomy.LensModel.Profiles.sersic_utils module¶
- class SersicUtil(smoothing=1e-05, sersic_major_axis=False)[source]¶
Bases:
object
- __init__(smoothing=1e-05, sersic_major_axis=False)[source]¶
- Parameters:
smoothing – smoothing scale of the innermost part of the profile (for numerical reasons)
sersic_major_axis – boolean; if True, defines the half-light radius of the Sersic light profile along the semi-major axis (which is the Galfit convention) if False, uses the product average of semi-major and semi-minor axis as the convention (default definition for all light profiles in lenstronomy other than the Sersic profile)
- k_bn(n, Re)[source]¶
Returns normalisation of the sersic profile such that Re is the half light radius given n_sersic slope.
- static b_n(n)[source]¶
B(n) computation. This is the approximation of the exact solution to the relation, 2*incomplete_gamma_function(2n; b_n) = Gamma_function(2*n).
- Parameters:
n – the sersic index
- Returns:
b(n)
- get_distance_from_center(x, y, e1, e2, center_x, center_y)[source]¶
Get the distance from the center of Sersic, accounting for orientation and axis ratio :param x:
- Parameters:
y –
e1 – eccentricity
e2 – eccentricity
center_x – center x of sersic
center_y – center y of sersic
- alpha_abs(x, y, n_sersic, r_eff, k_eff, center_x=0, center_y=0)[source]¶
- Parameters:
x –
y –
n_sersic –
r_eff –
k_eff –
center_x –
center_y –
- Returns:
- d_alpha_dr(x, y, n_sersic, r_eff, k_eff, center_x=0, center_y=0)[source]¶
- Parameters:
x –
y –
n_sersic –
r_eff –
k_eff –
center_x –
center_y –
- Returns:
- density(x, y, n_sersic, r_eff, k_eff, center_x=0, center_y=0)[source]¶
De-projection of the Sersic profile based on Prugniel & Simien (1997) :return:
- total_flux(amp, R_sersic, n_sersic, e1=0, e2=0, **kwargs)[source]¶
Computes analytical integral to compute total flux of the Sersic profile.
- Parameters:
amp – amplitude parameter in Sersic function (surface brightness at R_sersic
R_sersic – half-light radius in semi-major axis
n_sersic – Sersic index
e1 – eccentricity
e2 – eccentricity
- Returns:
Analytic integral of the total flux of the Sersic profile
lenstronomy.LensModel.Profiles.shapelet_pot_cartesian module¶
- class CartShapelets(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class contains the function and the derivatives of the cartesian shapelets.
- param_names = ['coeffs', 'beta', 'center_x', 'center_y']¶
- lower_limit_default = {'beta': 0, 'center_x': -100, 'center_y': -100, 'coeffs': [0]}¶
- upper_limit_default = {'beta': 100, 'center_x': 100, 'center_y': 100, 'coeffs': [100]}¶
- function(x, y, coeffs, beta, center_x=0, center_y=0)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- derivatives(x, y, coeffs, beta, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function.
- hessian(x, y, coeffs, beta, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
- H_n(n, x)[source]¶
Constructs the Hermite polynomial of order n at position x (dimensionless)
- Parameters:
n – The n’the basis function.
x – 1-dim position (dimensionless)
- Returns:
array– H_n(x).
- Raises:
AttributeError, KeyError
lenstronomy.LensModel.Profiles.shapelet_pot_polar module¶
- class PolarShapelets[source]¶
Bases:
LensProfileBase
This class contains the function and the derivatives of the Singular Isothermal Sphere.
- param_names = ['coeffs', 'beta', 'center_x', 'center_y']¶
- lower_limit_default = {'beta': 0, 'center_x': -100, 'center_y': -100, 'coeffs': [0]}¶
- upper_limit_default = {'beta': 100, 'center_x': 100, 'center_y': 100, 'coeffs': [100]}¶
- function(x, y, coeffs, beta, center_x=0, center_y=0)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
lenstronomy.LensModel.Profiles.shear module¶
- class Shear(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Class for external shear gamma1, gamma2 expression.
- param_names = ['gamma1', 'gamma2', 'ra_0', 'dec_0']¶
- lower_limit_default = {'dec_0': -100, 'gamma1': -0.5, 'gamma2': -0.5, 'ra_0': -100}¶
- upper_limit_default = {'dec_0': 100, 'gamma1': 0.5, 'gamma2': 0.5, 'ra_0': 100}¶
- function(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
gamma1 – shear component
gamma2 – shear component
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
lensing potential
- derivatives(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
gamma1 – shear component
gamma2 – shear component
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
deflection angles
- hessian(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
gamma1 – shear component
gamma2 – shear component
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
f_xx, f_xy, f_yx, f_yy
- class ShearGammaPsi[source]¶
Bases:
LensProfileBase
class to model a shear field with shear strength and direction. The translation ot the cartesian shear distortions is as follow:
\[\gamma_1 = \gamma_{ext} \cos(2 \phi_{ext}) \gamma_2 = \gamma_{ext} \sin(2 \phi_{ext})\]- param_names = ['gamma_ext', 'psi_ext', 'ra_0', 'dec_0']¶
- lower_limit_default = {'dec_0': -100, 'gamma_ext': 0, 'psi_ext': -3.141592653589793, 'ra_0': -100}¶
- upper_limit_default = {'dec_0': 100, 'gamma_ext': 1, 'psi_ext': 3.141592653589793, 'ra_0': 100}¶
- static function(x, y, gamma_ext, psi_ext, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
gamma_ext – shear strength
psi_ext – shear angle (radian)
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
- class ShearReduced[source]¶
Bases:
LensProfileBase
Reduced shear distortions \(\gamma' = \gamma / (1-\kappa)\). This distortion keeps the magnification as unity and, thus, does not change the size of apparent objects. To keep the magnification at unity, it requires.
\[(1-\kappa)^2) - \gamma_1^2 - \gamma_2^ = 1\]Thus, for given pair of reduced shear \((\gamma'_1, \gamma'_2)\), an additional convergence term is calculated and added to the lensing distortions.
- param_names = ['gamma1', 'gamma2', 'ra_0', 'dec_0']¶
- lower_limit_default = {'dec_0': -100, 'gamma1': -0.5, 'gamma2': -0.5, 'ra_0': -100}¶
- upper_limit_default = {'dec_0': 100, 'gamma1': 0.5, 'gamma2': 0.5, 'ra_0': 100}¶
- function(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
gamma1 – shear component
gamma2 – shear component
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
lensing potential
- derivatives(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
gamma1 – shear component
gamma2 – shear component
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
deflection angles
- hessian(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y0-coordinate (angle)
gamma1 – shear component
gamma2 – shear component
ra_0 – x/ra position where shear deflection is 0
dec_0 – y/dec position where shear deflection is 0
- Returns:
f_xx, f_xy, f_yx, f_yy
lenstronomy.LensModel.Profiles.sie module¶
- class SIE(NIE=True)[source]¶
Bases:
LensProfileBase
Class for singular isothermal ellipsoid (SIS with ellipticity)
\[\kappa(x, y) = \frac{1}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q}} \right)\]with \(\theta_{E}\) is the (circularized) Einstein radius, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate system aligned with the major and minor axis of the lens.
In terms of eccentricities, this profile is defined as
\[\kappa(r) = \frac{1}{2} \left(\frac{\theta'_{E}}{r \sqrt{1 − e*\cos(2*\phi)}} \right)\]with \(\epsilon\) is the ellipticity defined as
\[\epsilon = \frac{1-q^2}{1+q^2}\]And an Einstein radius \(\theta'_{\rm E}\) related to the definition used is
\[\left(\frac{\theta'_{\rm E}}{\theta_{\rm E}}\right)^{2} = \frac{2q}{1+q^2}.\]- param_names = ['theta_E', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'theta_E': 100}¶
- __init__(NIE=True)[source]¶
- Parameters:
NIE – bool, if True, is using the NIE analytic model. Otherwise it uses PEMD with gamma=2 from fastell4py
- function(x, y, theta_E, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angular coordinates)
y – y-coordinate (angular coordinates)
theta_E – Einstein radius
e1 – eccentricity
e2 – eccentricity
center_x – centroid
center_y – centroid
- Returns:
- derivatives(x, y, theta_E, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angular coordinates)
y – y-coordinate (angular coordinates)
theta_E – Einstein radius
e1 – eccentricity
e2 – eccentricity
center_x – centroid
center_y – centroid
- Returns:
- hessian(x, y, theta_E, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angular coordinates)
y – y-coordinate (angular coordinates)
theta_E – Einstein radius
e1 – eccentricity
e2 – eccentricity
center_x – centroid
center_y – centroid
- Returns:
- static theta2rho(theta_E)[source]¶
Converts projected density parameter (in units of deflection) into 3d density parameter.
- Parameters:
theta_E –
- Returns:
- static mass_3d(r, rho0, e1=0, e2=0)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r – radius in angular units
rho0 – density at angle=1
- Returns:
mass in angular units
- mass_3d_lens(r, theta_E, e1=0, e2=0)[source]¶
Mass enclosed a 3d sphere or radius r given a lens parameterization with angular units.
- Parameters:
r – radius in angular units
theta_E – Einstein radius
- Returns:
mass in angular units
- mass_2d(r, rho0, e1=0, e2=0)[source]¶
Mass enclosed projected 2d sphere of radius r.
- Parameters:
r –
rho0 –
e1 –
e2 –
- Returns:
- grav_pot(x, y, rho0, e1=0, e2=0, center_x=0, center_y=0)[source]¶
Gravitational potential (modulo 4 pi G and rho0 in appropriate units)
- Parameters:
x –
y –
rho0 –
e1 –
e2 –
center_x –
center_y –
- Returns:
- density_lens(r, theta_E, e1=0, e2=0)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – radius in angles
theta_E – Einstein radius
e1 – eccentricity component
e2 – eccentricity component
- Returns:
density
lenstronomy.LensModel.Profiles.sis module¶
- class SIS(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class contains the function and the derivatives of the Singular Isothermal Sphere.
\[\kappa(x, y) = \frac{1}{2} \left(\frac{\theta_{E}}{\sqrt{x^2 + y^2}} \right)\]with \(\theta_{E}\) is the Einstein radius,
- param_names = ['theta_E', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'theta_E': 100}¶
- function(x, y, theta_E, center_x=0, center_y=0)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- derivatives(x, y, theta_E, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function.
- hessian(x, y, theta_E, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
- static rho2theta(rho0)[source]¶
Converts 3d density into 2d projected density parameter :param rho0:
- Returns:
- static theta2rho(theta_E)[source]¶
Converts projected density parameter (in units of deflection) into 3d density parameter :param theta_E: Einstein radius :return:
- static mass_3d(r, rho0)[source]¶
Mass enclosed a 3d sphere or radius r :param r: radius in angular units :param rho0: density at angle=1 :return: mass in angular units.
- mass_3d_lens(r, theta_E)[source]¶
Mass enclosed a 3d sphere or radius r given a lens parameterization with angular units.
- Parameters:
r – radius in angular units
theta_E – Einstein radius
- Returns:
mass in angular units
- static mass_2d(r, rho0)[source]¶
Mass enclosed projected 2d sphere of radius r :param r:
- Parameters:
rho0 –
- Returns:
- mass_2d_lens(r, theta_E)[source]¶
- Parameters:
r – radius
theta_E – Einstein radius
- Returns:
mass within a radius in projection
- grav_pot(x, y, rho0, center_x=0, center_y=0)[source]¶
Gravitational potential (modulo 4 pi G and rho0 in appropriate units) :param x:
- Parameters:
y –
rho0 –
center_x –
center_y –
- Returns:
- static density(r, rho0)[source]¶
Computes the density :param r: radius in angles :param rho0: density at angle=1 :return: density at r.
lenstronomy.LensModel.Profiles.sis_truncate module¶
- class SIS_truncate(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class contains the function and the derivatives of the Singular Isothermal Sphere.
- param_names = ['theta_E', 'r_trunc', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'r_trunc': 0, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'r_trunc': 100, 'theta_E': 100}¶
- function(x, y, theta_E, r_trunc, center_x=0, center_y=0)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
lenstronomy.LensModel.Profiles.spemd module¶
- class SPEMD(suppress_fastell=False)[source]¶
Bases:
LensProfileBase
Class for smooth power law ellipse mass density profile (SPEMD). This class effectively performs the FASTELL calculations by Renan Barkana. The parameters are changed and represent a spherically averaged Einstein radius an a logarithmic 3D mass profile slope.
The SPEMD mass profile is defined as follow:
\[\kappa(x, y) = \frac{3-\gamma}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q + s^2}} \right)^{\gamma-1}\]with \(\theta_{E}\) is the (circularized) Einstein radius, \(\gamma\) is the negative power-law slope of the 3D mass distributions, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate system aligned with the major and minor axis of the lens.
the FASTELL definitions are as follows:
The parameters are position \((x1,x2)\), overall factor (\(b\)), power (gam), axis ratio (arat) which is <=1, core radius squared (\(s2\)), and the output potential (\(\phi\)). The projected mass density distribution, in units of the critical density, is
\[\kappa(x1,x2)=b_{fastell} \left[u2+s2\right]^{-gam},\]with \(u2=\left[x1^2+x2^2/(arat^2)\right]\).
The conversion from lenstronomy definitions of this class to FASTELL are:
\[q_{fastell} \equiv q_{lenstronomy}\]\[gam \equiv (\gamma-1)/2\]\[b_{fastell} \equiv (3-\gamma)/2. * \left(\theta_{E}^2 / q\right)^{gam}\]\[s2_{fastell} = s_{lenstronomy}^2 * q\]- param_names = ['theta_E', 'gamma', 'e1', 'e2', 's_scale', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 0, 's_scale': 0, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 100, 's_scale': 100, 'theta_E': 100}¶
- function(x, y, theta_E, gamma, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale in the center of the profile (angle)
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
lensing potential
- derivatives(x, y, theta_E, gamma, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale in the center of the profile
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
deflection angles alpha_x, alpha_y
- hessian(x, y, theta_E, gamma, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale in the center of the profile
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
Hessian components f_xx, f_xy, f_yx, f_yy
- param_transform(x, y, theta_E, gamma, e1, e2, s_scale, center_x=0, center_y=0)[source]¶
Transforms parameters in the format of fastell4py.
- Parameters:
x – x-coordinate (angle)
y – y-coordinate (angle)
theta_E – Einstein radius (angle), pay attention to specific definition!
gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
e1 – eccentricity component
e2 – eccentricity component
s_scale – smoothing scale in the center of the profile
center_x – x-position of lens center
center_y – y-position of lens center
- Returns:
x-rotated, y-rotated, q_fastell, gam, s2, q, phi_G
- static convert_params(theta_E, gamma, q, s_scale)[source]¶
Converts parameter definitions into quantities used by the FASTELL fortran library.
- Parameters:
theta_E – Einstein radius
gamma – 3D power-law slope of mass profile
q – axis ratio minor/major
s_scale – float, smoothing scale in the core
- Returns:
pre-factors to SPEMP profile for FASTELL
lenstronomy.LensModel.Profiles.spep module¶
- class SPEP[source]¶
Bases:
LensProfileBase
Class for Softened power-law elliptical potential (SPEP)
- param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 0, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 100, 'theta_E': 100}¶
- function(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
- Parameters:
x (array of size (n)) – set of x-coordinates
theta_E (float.) – Einstein radius of lense
gamma (<2 float) – power law slope of mass profifle
e1 (-1<e1<1) – eccentricity
e2 (-1<e1<1) – eccentricity
- Returns:
function
- Raises:
AttributeError, KeyError
- derivatives(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
Deflection angles.
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- hessian(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- mass_3d_lens(r, theta_E, gamma, e1=None, e2=None)[source]¶
Computes the spherical power-law mass enclosed (with SPP routine)
- Parameters:
r – radius within the mass is computed
theta_E – Einstein radius
gamma – power-law slope
e1 – eccentricity component (not used)
e2 – eccentricity component (not used)
- Returns:
mass enclosed a 3D radius r
- density_lens(r, theta_E, gamma, e1=None, e2=None)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – radius within the mass is computed
theta_E – Einstein radius
gamma – power-law slope
e1 – eccentricity component (not used)
e2 – eccentricity component (not used)
- Returns:
mass enclosed a 3D radius r
lenstronomy.LensModel.Profiles.splcore module¶
- class SPLCORE(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This lens profile corresponds to a spherical power law (SPL) mass distribution with logarithmic slope gamma and a 3D core radius r_core.
\[\rho\left(r, \rho_0, r_c, \gamma\right) = \rho_0 \frac{{r_c}^\gamma}{\left(r^2 + r_c^2\right)^{\frac{\gamma}{2}}}\]The difference between this and EPL is that this model contains a core radius, is circular, and is also defined for gamma=3.
With respect to SPEMD, this model is different in that it is also defined for gamma = 3, is circular, and is defined in terms of a physical density parameter rho0, or the central density at r=0 divided by the critical density for lensing such that rho0 has units 1/arcsec.
This class is defined for all gamma > 1
- param_names = ['sigma0', 'center_x', 'center_y', 'r_core', 'gamma']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'gamma': 1.000001, 'r_core': 1e-06, 'sigma0': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'gamma': 5.0, 'r_core': 100, 'sigma0': 1000000000000.0}¶
- function(x, y, sigma0, r_core, gamma, center_x=0, center_y=0)[source]¶
Lensing potential (only needed for specific calculations, such as time delays)
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- derivatives(x, y, sigma0, r_core, gamma, center_x=0, center_y=0)[source]¶
- Parameters:
x – projected x position at which to evaluate function [arcsec]
y – projected y position at which to evaluate function [arcsec]
sigma0 – convergence at r = 0
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
center_x – x coordinate center of lens model [arcsec]
center_y – y coordinate center of lens model [arcsec]
- Returns:
deflection angle alpha in x and y directions
- hessian(x, y, sigma0, r_core, gamma, center_x=0, center_y=0)[source]¶
- Parameters:
x – projected x position at which to evaluate function [arcsec]
y – projected y position at which to evaluate function [arcsec]
sigma0 – convergence at r = 0
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
center_x – x coordinate center of lens model [arcsec]
center_y – y coordinate center of lens model [arcsec]
- Returns:
hessian elements
alpha_(x/y) = alpha_r * cos/sin(x/y / r)
- alpha(r, sigma0, r_core, gamma)[source]¶
Returns the deflection angle at r.
- Parameters:
r – radius [arcsec]
sigma0 – convergence at r=0
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
- Returns:
deflection angle at r
- static density(r, rho0, r_core, gamma)[source]¶
Returns the 3D density at r.
- Parameters:
r – radius [arcsec]
rho0 – convergence at r=0
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
- Returns:
density at r
- density_lens(r, sigma0, r_core, gamma)[source]¶
Returns the 3D density at r.
- Parameters:
r – radius [arcsec]
sigma0 – convergence at r=0
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
- Returns:
density at r
- density_2d(x, y, rho0, r_core, gamma)[source]¶
Returns the convergence at radius r.
- Parameters:
x – x position [arcsec]
y – y position [arcsec]
rho0 – convergence at r=0
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
- Returns:
convergence at r
- mass_3d(r, rho0, r_core, gamma)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r – radius [arcsec]
rho0 – density at r = 0 in units [rho_0_physical / sigma_crit] (which should be equal to [arcsec]) where rho_0_physical is a physical density normalization and sigma_crit is the critical density for lensing
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
- Returns:
mass inside radius r
- mass_3d_lens(r, sigma0, r_core, gamma)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r – radius [arcsec]
sigma0 – convergence at r = 0
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
- Returns:
mass inside radius r
- mass_2d(r, rho0, r_core, gamma)[source]¶
Mass enclosed projected 2d disk of radius r.
- Parameters:
r – radius [arcsec]
rho0 – density at r = 0 in units [rho_0_physical / sigma_crit] (which should be equal to [1/arcsec]) where rho_0_physical is a physical density normalization and sigma_crit is the critical density for lensing
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
- Returns:
projected mass inside disk of radius r
- mass_2d_lens(r, sigma0, r_core, gamma)[source]¶
Mass enclosed projected 2d disk of radius r.
- Parameters:
r – radius [arcsec]
sigma0 – convergence at r = 0 where rho_0_physical is a physical density normalization and sigma_crit is the critical density for lensing
r_core – core radius [arcsec]
gamma – logarithmic slope at r -> infinity
- Returns:
projected mass inside disk of radius r
lenstronomy.LensModel.Profiles.spp module¶
- class SPP(*args, **kwargs)[source]¶
Bases:
LensProfileBase
Class for circular power-law mass distribution.
- param_names = ['theta_E', 'gamma', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'gamma': 1.5, 'theta_E': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'gamma': 2.5, 'theta_E': 100}¶
- function(x, y, theta_E, gamma, center_x=0, center_y=0)[source]¶
- Parameters:
x (array of size (n)) – set of x-coordinates
y (array of size (n)) – set of y-coordinates
theta_E (float.) – Einstein radius of lens
gamma (<2 float) – power law slope of mass profile
- Returns:
function
- Raises:
AttributeError, KeyError
- derivatives(x, y, theta_E, gamma, center_x=0.0, center_y=0.0)[source]¶
Deflection angles.
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- hessian(x, y, theta_E, gamma, center_x=0.0, center_y=0.0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2.
- Parameters:
kwargs – keywords of the profile
- Returns:
raise as definition is not defined
- static rho2theta(rho0, gamma)[source]¶
Converts 3d density into 2d projected density parameter.
- Parameters:
rho0 –
gamma –
- Returns:
- static theta2rho(theta_E, gamma)[source]¶
Converts projected density parameter (in units of deflection) into 3d density parameter.
- Parameters:
theta_E –
gamma –
- Returns:
- static mass_3d(r, rho0, gamma)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r –
rho0 –
gamma –
- Returns:
- mass_2d(r, rho0, gamma)[source]¶
Mass enclosed projected 2d sphere of radius r.
- Parameters:
r –
rho0 –
gamma –
- Returns:
- mass_2d_lens(r, theta_E, gamma)[source]¶
- Parameters:
r – projected radius
theta_E – Einstein radius
gamma – power-law slope
- Returns:
2d projected radius enclosed
- grav_pot(x, y, rho0, gamma, center_x=0, center_y=0)[source]¶
Gravitational potential (modulo 4 pi G and rho0 in appropriate units)
- Parameters:
x –
y –
rho0 –
gamma –
center_x –
center_y –
- Returns:
- static density(r, rho0, gamma)[source]¶
Computes the density.
- Parameters:
r –
rho0 –
gamma –
- Returns:
lenstronomy.LensModel.Profiles.tnfw module¶
- class TNFW[source]¶
Bases:
LensProfileBase
This class contains functions concerning the truncated NFW profile with a truncation function (r_trunc^2)*(r^2+r_trunc^2)
density equation is:
\[\rho(r) = \frac{r_\text{trunc}^2}{r^2+r_\text{trunc}^2}\frac{\rho_0(\alpha_{R_s})}{r/R_s(1+r/R_s)^2}\]relation are: R_200 = c * Rs
- profile_name = 'TNFW'¶
- param_names = ['Rs', 'alpha_Rs', 'r_trunc', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'r_trunc': 0}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'r_trunc': 100}¶
- function(x, y, Rs, alpha_Rs, r_trunc, center_x=0, center_y=0)[source]¶
- Parameters:
x – angular position
y – angular position
Rs – angular turn over point
alpha_Rs – deflection at Rs
r_trunc – truncation radius
center_x – center of halo
center_y – center of halo
- Returns:
lensing potential
- derivatives(x, y, Rs, alpha_Rs, r_trunc, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function (integral of TNFW), which are the deflection angles.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
r_trunc – truncation radius (angular units)
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
deflection angle in x, deflection angle in y
- hessian(x, y, Rs, alpha_Rs, r_trunc, center_x=0, center_y=0)[source]¶
Returns d^2f/dx^2, d^2f/dxdy, d^2f/dydx, d^2f/dy^2 of the TNFW potential f.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
r_trunc – truncation radius (angular units)
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy
- static density(r, Rs, rho0, r_trunc)[source]¶
Three dimensional truncated NFW profile.
- Parameters:
r (float/numpy array) – radius of interest
Rs (float > 0) – scale radius
r_trunc (float > 0) – truncation radius (angular units)
- Returns:
rho(r) density
- density_2d(x, y, Rs, rho0, r_trunc, center_x=0, center_y=0)[source]¶
Projected two dimensional NFW profile (kappa*Sigma_crit)
- Parameters:
R (float/numpy array) – projected radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
r_trunc (float > 0) – truncation radius (angular units)
- Returns:
Epsilon(R) projected density at radius R
- mass_3d(r, Rs, rho0, r_trunc)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
r – 3d radius
Rs – scale radius
rho0 – density normalization (characteristic density)
r_trunc – truncation radius (angular units)
- Returns:
M(<r)
- nfw_potential(R, Rs, rho0, r_trunc)[source]¶
Lensing potential of truncated NFW profile.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
r_trunc (float > 0) – truncation radius (angular units)
- Returns:
lensing potential
- nfw_alpha(R, Rs, rho0, r_trunc, ax_x, ax_y)[source]¶
Deflection angle of NFW profile along the projection to coordinate axis.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
r_trunc (float > 0) – truncation radius (angular units)
axis (same as R) – projection to either x- or y-axis
- Returns:
- nfw_gamma(R, Rs, rho0, r_trunc, ax_x, ax_y)[source]¶
Shear gamma of NFW profile (times Sigma_crit) along the projection to coordinate ‘axis’.
- Parameters:
R (float/numpy array) – radius of interest
Rs (float) – scale radius
rho0 (float) – density normalization (characteristic density)
r_trunc (float > 0) – truncation radius (angular units)
axis (same as R) – projection to either x- or y-axis
- Returns:
- mass_2d(R, Rs, rho0, r_trunc)[source]¶
Analytic solution of the projection integral (convergence)
- Parameters:
R – projected radius
Rs – scale radius
rho0 – density normalization (characteristic density)
r_trunc – truncation radius (angular units)
- Returns:
mass enclosed 2d projected cylinder
lenstronomy.LensModel.Profiles.tnfw_ellipse module¶
- class TNFW_ELLIPSE[source]¶
Bases:
LensProfileBase
This class contains functions concerning the truncated NFW profile with an ellipticity defined in the potential parameterization of alpha_Rs, Rs and r_trunc is the same as for the spherical NFW profile.
from Glose & Kneib: https://cds.cern.ch/record/529584/files/0112138.pdf
relation are: R_200 = c * Rs
- profile_name = 'TNFW_ELLIPSE'¶
- param_names = ['Rs', 'alpha_Rs', 'r_trunc', 'e1', 'e2', 'center_x', 'center_y']¶
- lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'r_trunc': 0}¶
- upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'r_trunc': 100}¶
- function(x, y, Rs, alpha_Rs, r_trunc, e1, e2, center_x=0, center_y=0)[source]¶
Returns elliptically distorted NFW lensing potential.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
r_trunc – truncation radius
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
lensing potential
- derivatives(x, y, Rs, alpha_Rs, r_trunc, e1, e2, center_x=0, center_y=0)[source]¶
Returns df/dx and df/dy of the function, calculated as an elliptically distorted deflection angle of the spherical NFW profile.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
r_trunc – truncation radius
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
deflection in x-direction, deflection in y-direction
- hessian(x, y, Rs, alpha_Rs, r_trunc, e1, e2, center_x=0, center_y=0)[source]¶
Returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy the calculation is performed as a numerical differential from the deflection field. Analytical relations are possible.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
Rs – turn over point in the slope of the NFW profile in angular unit
alpha_Rs – deflection (angular units) at projected Rs
r_trunc – truncation radius
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
- mass_3d_lens(r, Rs, alpha_Rs, r_trunc, e1=1, e2=0)[source]¶
- Parameters:
r – radius (in angular units)
Rs – turn-over radius of NFW profile
alpha_Rs – deflection at Rs
r_trunc – truncation radius
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
- Returns:
- density_lens(r, Rs, alpha_Rs, r_trunc, e1=1, e2=0)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – 3d radios
Rs – turn-over radius of NFW profile
alpha_Rs – deflection at Rs
r_trunc – truncation radius
e1 – eccentricity component in x-direction
e2 – eccentricity component in y-direction
- Returns:
density rho(r)
lenstronomy.LensModel.Profiles.uldm module¶
- class Uldm(*args, **kwargs)[source]¶
Bases:
LensProfileBase
This class contains functions concerning the ULDM soliton density profile, whose good approximation is (see for example https://arxiv.org/pdf/1406.6586.pdf )
\[\rho = \rho_0 (1 + a(\theta/\theta_c)^2)^{-\beta}\]where \(\theta_c\) is the core radius, corresponding to the radius where the density drops by half its central value, :math: beta is the slope (called just slope in the parameters of this model), :math: rho_0 = kappa_0 Sigma_c/D_lens, and :math: a is a parameter, dependent on :math: beta, chosen such that :math: theta_c indeed corresponds to the radius where the density drops by half (simple math gives :math: a = 0.5^{-1/beta} - 1 ). For an ULDM soliton profile without contributions to background potential, it turns out that :math: beta = 8, a = 0.091. We allow :math: beta to be different from 8 to model solitons which feel the influence of background potential (see 2105.10873) The profile has, as parameters:
kappa_0: central convergence
theta_c: core radius (in arcseconds)
slope: exponent entering the profile, default value is 8
- param_names = ['kappa_0', 'theta_c', 'slope', 'center_x', 'center_y']¶
- lower_limit_default = {'center_x': -100, 'center_y': -100, 'kappa_0': 0, 'slope': 3.5, 'theta_c': 0}¶
- upper_limit_default = {'center_x': 100, 'center_y': 100, 'kappa_0': 1.0, 'slope': 10, 'theta_c': 100}¶
- static rhotilde(kappa_0, theta_c, slope=8)[source]¶
Computes the central density in angular units.
- Parameters:
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
- Returns:
central density in 1/arcsec
- function(x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8)[source]¶
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
lensing potential (in arcsec^2)
- static alpha_radial(r, kappa_0, theta_c, slope=8)[source]¶
Returns the radial part of the deflection angle.
- Parameters:
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
r – radius where the deflection angle is computed
- Returns:
radial deflection angle
- derivatives(x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8)[source]¶
Returns df/dx and df/dy of the function (lensing potential), which are the deflection angles.
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
deflection angle in x, deflection angle in y
- hessian(x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8)[source]¶
- Parameters:
x – angular position (normally in units of arc seconds)
y – angular position (normally in units of arc seconds)
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
center_x – center of halo (in angular units)
center_y – center of halo (in angular units)
- Returns:
Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
- density(R, kappa_0, theta_c, slope=8)[source]¶
Three dimensional ULDM profile in angular units (rho0_physical = rho0_angular Sigma_crit / D_lens)
- Parameters:
R – radius of interest
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
- Returns:
rho(R) density in angular units
- density_lens(r, kappa_0, theta_c, slope=8)[source]¶
Computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.
- Parameters:
r – 3d radius
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
- Returns:
density rho(r)
- static kappa_r(R, kappa_0, theta_c, slope=8)[source]¶
Convergence of the cored density profile. This routine is also for testing.
- Parameters:
R – radius (angular scale)
kappa_0 – convergence in the core
theta_c – core radius
slope – exponent entering the profile
- Returns:
convergence at r
- density_2d(x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8)[source]¶
Projected two dimensional ULDM profile (convergence * Sigma_crit), but given our units convention for rho0, it is basically the convergence.
- Parameters:
x – x-coordinate
y – y-coordinate
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
- Returns:
Epsilon(R) projected density at radius R
- mass_3d(R, kappa_0, theta_c, slope=8)[source]¶
Mass enclosed a 3d sphere or radius r.
- Parameters:
R – radius in arcseconds
kappa_0 – central convergence of profile
theta_c – core radius (in arcsec)
slope – exponent entering the profile
- Returns:
mass of soliton in angular units