Base objects

class glow.lenses.PsiGeneral(p_phys={}, p_prec={})

Base class for a generic non-axisymmetric lens.

Parameters:

p_physdict, optional

Physical parameters of the lens.

p_precdict, optional

Precision parameters of the lens.

Attributes:

p_physdict

Default parameters updated with the input.

p_precdict

Default parameters updated with the input.

isAxisymbool, default=False

Allows to easily differentiate a general lens from the special axisymmetric case (instance of PsiAxisym).

asymp_indexfloat or None, default=None

asymp_amplitudefloat or None, default=None

If the asymptotic behaviour of the lens is

\[\psi'\sim\frac{A_\psi}{r^\gamma}\ ,\quad r\to\infty\]

the asymptotic index and amplitude are defined as

\[\begin{split}\begin{align} \gamma_\text{asymp} &= (\gamma+1)/2\\ A_\text{asymp} &= A_\psi/2^{\gamma_\text{asymp}} \end{align}\end{split}\]

Methods

check_general_input()	Check the input upon initialization.
check_input()	(subclass defined, optional) Check the input of the lens implementation.
ddphi_Fermat_ddx1(x1, x2)	Return only d11 from ddphi_Fermat_vec().
ddphi_Fermat_ddx2(x1, x2)	Return only d22 from ddphi_Fermat_vec().
ddphi_Fermat_dx1dx2(x1, x2)	Return only d12 from ddphi_Fermat_vec().
ddphi_Fermat_vec(x1, x2)	Hessian of the Fermat potential.
ddpsi_ddx1(x1, x2)	Return only d11 from ddpsi_vec().
ddpsi_ddx2(x1, x2)	Return only d22 from ddpsi_vec().
ddpsi_dx1dx2(x1, x2)	Return only d12 from ddpsi_vec().
ddpsi_vec(x1, x2)	(subclass defined) Hessian of the lensing potential.
default_general_params()	Fill the default parameters.
default_params()	(subclass defined) Initialize the default parameters.
display_info()	Print internal information in human-readable form.
dphi_Fermat_dx1(x1, x2, y)	Return only d1 from dphi_Fermat_vec().
dphi_Fermat_dx2(x1, x2, y)	Return only d2 from dphi_Fermat_vec().
dphi_Fermat_vec(x1, x2, y)	Gradient of the Fermat potential.
dpsi_dx1(x1, x2)	Return only d1 from dpsi_vec().
dpsi_dx2(x1, x2)	Return only d2 from dpsi_vec().
dpsi_vec(x1, x2)	(subclass defined) Gradient of the lensing potential.
phi_Fermat(x1, x2, y)	Fermat potential.
psi(x1, x2)	(subclass defined) Lensing potential.
shear(x1, x2)	Magnification matrix.

default_general_params()

Fill the default parameters.

Update the parameters common for all lenses (none in this case). Then call default_params() (subclass defined).

Returns:

p_physdict

Default physical parameters.

p_precdict

Default precision parameters.

check_general_input()

Check the input upon initialization.

It first calls check_input() (subclass defined) to perform any checks that the user desires. It then checks that the input only updates existing keys in p_phys and p_prec, otherwise throws a warning.

The idea is that we do not use a wrong name that is then ignored.

Warns:

LensWarning

Warning

If the subclass will add a new parameter without an entry in default_params(), it must be manually added to self.p_phys_default_keys or self.p_prec_default_keys in check_input() with self.p_phys_default_keys.add(new_key).

display_info()

Print internal information in human-readable form.

phi_Fermat(x1, x2, y)

Fermat potential.

Defined without setting the minimum time-delay to zero

\[\phi_F(x_1, x_2, y) = \frac{1}{2}(x_1^2+x_2^2+y^2) -x_1y -\psi(x_1, x_2)\]

Parameters:

x1float or array

Coordinate x1 (x1 is defined aligned with y).

x2float or array

Coordinate x2.

yfloat or array

Impact parameter.

Returns:

phifloat or array

dphi_Fermat_vec(x1, x2, y)

Gradient of the Fermat potential.

Parameters:

x1float or array

Coordinate x1 (x1 is defined aligned with y).

x2float or array

Coordinate x2.

yfloat or array

Impact parameter.

Returns:

d1float or array

Derivative of the Fermat potential wrt x1.

d2float or array

Derivative of the Fermat potential wrt x2.

ddphi_Fermat_vec(x1, x2)

Hessian of the Fermat potential.

Parameters:

x1float or array

Coordinate x1.

x2float or array

Coordinate x2.

Returns:

d11float or array

2nd derivative of the Fermat potential wrt x1 twice.

d12float or array

2nd derivative of the Fermat potential wrt x1 and x2.

d22float or array

2nd derivative of the Fermat potential wrt x2 twice.

dphi_Fermat_dx1(x1, x2, y)

Return only d1 from dphi_Fermat_vec().

dphi_Fermat_dx2(x1, x2, y)

Return only d2 from dphi_Fermat_vec().

ddphi_Fermat_ddx1(x1, x2)

Return only d11 from ddphi_Fermat_vec().

ddphi_Fermat_ddx2(x1, x2)

Return only d22 from ddphi_Fermat_vec().

ddphi_Fermat_dx1dx2(x1, x2)

Return only d12 from ddphi_Fermat_vec().

dpsi_dx1(x1, x2)

Return only d1 from dpsi_vec().

dpsi_dx2(x1, x2)

Return only d2 from dpsi_vec().

ddpsi_ddx1(x1, x2)

Return only d11 from ddpsi_vec().

ddpsi_ddx2(x1, x2)

Return only d22 from ddpsi_vec().

ddpsi_dx1dx2(x1, x2)

Return only d12 from ddpsi_vec().

shear(x1, x2)

Magnification matrix.

The Jacobian matrix for the lens mapping is [1]

\[A_{ij} = \frac{\partial y_i}{\partial x_j} = \partial_i\partial_j\phi_F = \delta_{ij} - \partial_i\partial_j\psi\]

It can be written as

\[\begin{split}A = \begin{pmatrix}1-\kappa-\gamma_1 & -\gamma_2\\ -\gamma_2 & 1-\kappa+\gamma_1 \end{pmatrix}\end{split}\]

The convergence, \(\kappa\), and shear, \(\gamma\), can be expressed in terms of the second derivatives of the lensing potential

\[\begin{split}\begin{align} \kappa &= (\psi_{11} + \psi_{22})/2\\ \gamma_1 &= (\psi_{11} - \psi_{22})/2\\ \gamma_2 &= \psi_{12}\\ \gamma &= \sqrt{\gamma_1^2 + \gamma_2^2} \end{align}\end{split}\]

The eigenvalues of \(A\) are

\[\begin{split}\begin{align} \lambda_1 &= 1 - \kappa - \gamma\\ \lambda_2 &= 1 - \kappa + \gamma\\ \text{tr}\,A &= \lambda_1+\lambda_2 = 2(1-\kappa)\\ \det A &= \lambda_1\lambda_2 = (1-\kappa)^2-\gamma^2 \end{align}\end{split}\]

Finally, the magnification, \(\mu\), is defined as

\[\mu = \frac{1}{|\det A|}\]

Parameters:

x1, x2float or array

Coordinates.

Returns:

ddict

Dictionary with keys:

• gamma1 (float or array) – \(\gamma_1\)

• gamma2 (float or array) – \(\gamma_2\)

• kappa (float or array) – \(\kappa\)

• gamma (float or array) – \(\gamma\)

• lambda1 (float or array) – \(\lambda_1\)

• lambda2 (float or array) – \(\lambda_2\)

• detA (float or array) – \(\det A\)

• trA (float or array) – \(\text{tr}\,A\)

• mag (float or array) – \(\mu\)

References

[1]

P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses, (1992).

default_params()

(subclass defined) Initialize the default parameters.

Returns:

p_physdict

Default physical parameters.

p_precdict

Default precision parameters.

check_input()

(subclass defined, optional) Check the input of the lens implementation.

psi(x1, x2)

(subclass defined) Lensing potential.

Parameters:

x1, x2float or array

Coordinates.

Returns:

psifloat or array or None, default=None

dpsi_vec(x1, x2)

(subclass defined) Gradient of the lensing potential.

Parameters:

x1, x2float or array

Coordinates.

Returns:

d1float or array or None, default=None

Derivative of the lensing potential wrt x1.

d2float or array or None, default=None

Derivative of the lensing potential wrt x2.

ddpsi_vec(x1, x2)

(subclass defined) Hessian of the lensing potential.

Parameters:

x1, x2float or array

Coordinate x1.

Returns:

d11float or array or None, default=None

2nd derivative of the lensing potential wrt x1 twice.

d12float or array or None, default=None

2nd derivative of the lensing potential wrt x1 and x2.

d22float or array or None, default=None

2nd derivative of the lensing potential wrt x2 twice.

class glow.lenses.PsiAxisym(p_phys={}, p_prec={})

Bases: PsiGeneral

Base class for an axisymmetric lens.

Parameters:

p_physdict, optional

Physical parameters of the lens.

p_precdict, optional

Precision parameters of the lens.

Attributes:

isAxisymbool, default=True

Allows to easily differentiate this class from PsiGeneral.

Methods

ddpsi_ddx(x)	(subclass defined) Second derivative of the lensing potential.
ddpsi_vec(x1, x2)	Wrapper for ddpsi_ddx().
dpsi_dx(x)	(subclass defined) First derivative of the lensing potential.
dpsi_vec(x1, x2)	Wrapper for dpsi_dx().
psi(x1, x2)	Wrapper for psi_x().
psi_x(x)	(subclass defined) Lensing potential.
to_file(fname, xmin, xmax, Nx[, extension])	Evaluate the lensing potential and its derivatives on a grid and store it.

psi(x1, x2)

Wrapper for psi_x().

This method allows to use axisymmetric lenses with algorithms designed only for generic lenses. It defines the general lensing potential from its symmetric version as

\[\psi(x_1, x_2) = \psi\left(x=\sqrt{x_1^2+x_2^2}\right)\]

Parameters:

x1, x2float or array

Coordinates.

Returns:

psifloat or array

Lensing potential at (x1, x2).

dpsi_vec(x1, x2)

Wrapper for dpsi_dx().

The derivatives are evaluated as

\[\begin{split}\begin{align} \partial_1\psi &= \psi'\frac{x_1}{x}\\ \partial_2\psi &= \psi'\frac{x_2}{x} \end{align}\end{split}\]

Parameters:

x1, x2float or array

Coordinates.

Returns:

d1float or array

\(\partial_1\psi\)

d2float or array

\(\partial_2\psi\)

ddpsi_vec(x1, x2)

Wrapper for ddpsi_ddx().

The derivatives are evaluated as

\[\begin{split}\begin{align} \partial_{11}\psi &= (1-r_1^2)\psi'/x + r_1^2\psi''\\ \partial_{12}\psi &= r_1r_2(\psi'' - \psi'/x)\\ \partial_{22}\psi &= (1-r_2^2)\psi'/x + r_2^2\psi'' \end{align}\end{split}\]

with \(r_{1,2}=x_{1,2}/x\).

Parameters:

x1, x2float or array

Coordinates x1.

Returns:

d11float or array

\(\partial_{11}\psi\)

d12float or array

\(\partial_{12}\psi\)

d22float or array

\(\partial_{22}\psi\)

to_file(fname, xmin, xmax, Nx, extension='_lens.dat')

Evaluate the lensing potential and its derivatives on a grid and store it.

This method is called when a lens that has not been implemented in C is used in the C part of the code. The lens is then evaluated from xmin to xmax in a logarithmic grid with Nx points, stored and later read and interpolated from C. The files are stored in:

• fname + '_psi' + extension

• fname + '_dpsi' + extension

• fname + '_ddpsi' + extension

Parameters:

fnamestr

Root of the file name.

xminfloat

Minimum radius in the grid.

xmaxfloat

Maximum radius in the grid.

Nxfloat

Number of grid points.

extensionstr, optional

Extension of the file.

Notes

This method is intended for internal use, but can be used to save the lens to disk.

psi_x(x)

(subclass defined) Lensing potential.

Parameters:

xfloat or array

Radius.

Returns:

psifloat or array or None, default=None

dpsi_dx(x)

(subclass defined) First derivative of the lensing potential.

Parameters:

xfloat or array

Radius.

Returns:

dpsifloat or array or None, default=None

Derivative of the lensing potential wrt x.

ddpsi_ddx(x)

(subclass defined) Second derivative of the lensing potential.

Parameters:

xfloat or array

Radius.

Returns:

ddpsifloat or array or None, default=None

Second derivative of the lensing potential wrt x.