import numpy as np
from typing import Tuple
import autoarray as aa
from autogalaxy.profiles.mass.total.power_law import PowerLaw
def psi_from(grid, axis_ratio, core_radius):
"""
Returns the $\Psi$ term in expressions for the calculation of the deflection of an elliptical isothermal mass
distribution. This is used in the `Isothermal` and `Chameleon` `MassProfile`'s.
The expression for Psi is:
$\Psi = \sqrt(q^2(s^2 + x^2) + y^2)$
Parameters
----------
grid
The (y,x) coordinates of the grid, in an arrays of shape (total_coordinates, 2)
axis_ratio
Ratio of profiles ellipse's minor and major axes (b/a)
core_radius
The radius of the inner core
Returns
-------
float
The value of the Psi term.
"""
return np.sqrt(
np.add(
np.multiply(
axis_ratio**2.0, np.add(np.square(grid[:, 1]), core_radius**2.0)
),
np.square(grid[:, 0]),
)
)
[docs]class Isothermal(PowerLaw):
def __init__(
self,
centre: Tuple[float, float] = (0.0, 0.0),
ell_comps: Tuple[float, float] = (0.0, 0.0),
einstein_radius: float = 1.0,
):
"""
Represents an elliptical isothermal density distribution, which is equivalent to the elliptical power-law
density distribution for the value slope = 2.0.
Parameters
----------
centre
The (y,x) arc-second coordinates of the profile centre.
ell_comps
The first and second ellipticity components of the elliptical coordinate system.
einstein_radius
The arc-second Einstein radius.
"""
super().__init__(
centre=centre,
ell_comps=ell_comps,
einstein_radius=einstein_radius,
slope=2.0,
)
@property
def axis_ratio(self):
axis_ratio = super().axis_ratio
return min(axis_ratio, 0.99999)
[docs] @aa.grid_dec.grid_2d_to_vector_yx
@aa.grid_dec.grid_2d_to_structure
@aa.grid_dec.transform
@aa.grid_dec.relocate_to_radial_minimum
def deflections_yx_2d_from(self, grid: aa.type.Grid2DLike):
"""
Calculate the deflection angles on a grid of (y,x) arc-second coordinates.
For coordinates (0.0, 0.0) the analytic calculation of the deflection angle gives a NaN. Therefore,
coordinates at (0.0, 0.0) are shifted slightly to (1.0e-8, 1.0e-8).
Parameters
----------
grid
The grid of (y,x) arc-second coordinates the deflection angles are computed on.
"""
factor = (
2.0
* self.einstein_radius_rescaled
* self.axis_ratio
/ np.sqrt(1 - self.axis_ratio**2)
)
psi = psi_from(grid=grid, axis_ratio=self.axis_ratio, core_radius=0.0)
deflection_y = np.arctanh(
np.divide(np.multiply(np.sqrt(1 - self.axis_ratio**2), grid[:, 0]), psi)
)
deflection_x = np.arctan(
np.divide(np.multiply(np.sqrt(1 - self.axis_ratio**2), grid[:, 1]), psi)
)
return self.rotated_grid_from_reference_frame_from(
grid=np.multiply(factor, np.vstack((deflection_y, deflection_x)).T)
)
[docs] @aa.grid_dec.grid_2d_to_structure
@aa.grid_dec.transform
@aa.grid_dec.relocate_to_radial_minimum
def shear_yx_2d_from(self, grid: aa.type.Grid2DLike):
"""
Calculate the (gamma_y, gamma_x) shear vector field on a grid of (y,x) arc-second coordinates.
The result is returned as a `ShearYX2D` dats structure, which has shape [total_shear_vectors, 2], where
entries for [:,0] are the gamma_2 values and entries for [:,1] are the gamma_1 values.
Note therefore that this convention means the FIRST entries in the array are the gamma_2 values and the SECOND
entries are the gamma_1 values.
Parameters
----------
grid
The grid of (y,x) arc-second coordinates the deflection angles are computed on.
"""
convergence = self.convergence_2d_from(grid=grid)
gamma_2 = (
-2
* convergence
* np.divide(grid[:, 1] * grid[:, 0], grid[:, 1] ** 2 + grid[:, 0] ** 2)
)
gamma_1 = -convergence * np.divide(
grid[:, 1] ** 2 - grid[:, 0] ** 2, grid[:, 1] ** 2 + grid[:, 0] ** 2
)
shear_field = self.rotated_grid_from_reference_frame_from(
grid=np.vstack((gamma_2, gamma_1)).T, angle=self.angle * 2
)
return aa.VectorYX2DIrregular(values=shear_field, grid=grid)
[docs]class IsothermalSph(Isothermal):
def __init__(
self, centre: Tuple[float, float] = (0.0, 0.0), einstein_radius: float = 1.0
):
"""
Represents a spherical isothermal density distribution, which is equivalent to the spherical power-law
density distribution for the value slope: float = 2.0
Parameters
----------
centre
The (y,x) arc-second coordinates of the profile centre.
einstein_radius
The arc-second Einstein radius.
"""
super().__init__(
centre=centre, ell_comps=(0.0, 0.0), einstein_radius=einstein_radius
)
@property
def axis_ratio(self):
return 1.0
[docs] @aa.grid_dec.grid_2d_to_structure
@aa.grid_dec.transform
@aa.grid_dec.relocate_to_radial_minimum
def potential_2d_from(self, grid: aa.type.Grid2DLike):
"""
Calculate the potential on a grid of (y,x) arc-second coordinates.
Parameters
----------
grid
The grid of (y,x) arc-second coordinates the deflection angles are computed on.
"""
eta = self.elliptical_radii_grid_from(grid)
return 2.0 * self.einstein_radius_rescaled * eta
[docs] @aa.grid_dec.grid_2d_to_vector_yx
@aa.grid_dec.grid_2d_to_structure
@aa.grid_dec.transform
@aa.grid_dec.relocate_to_radial_minimum
def deflections_yx_2d_from(self, grid: aa.type.Grid2DLike):
"""
Calculate the deflection angles on a grid of (y,x) arc-second coordinates.
Parameters
----------
grid
The grid of (y,x) arc-second coordinates the deflection angles are computed on.
"""
return self._cartesian_grid_via_radial_from(
grid=grid,
radius=np.full(grid.shape[0], 2.0 * self.einstein_radius_rescaled),
)