import numpy as np
from scipy import special
from typing import Tuple
import autoarray as aa
from autogalaxy.profiles.mass.total.power_law_core import PowerLawCore
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class PowerLaw(PowerLawCore):
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,
slope: float = 2.0,
):
"""
Represents an elliptical power-law density distribution.
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.
slope
The density slope of the power-law (lower value -> shallower profile, higher value -> steeper profile).
"""
super().__init__(
centre=centre,
ell_comps=ell_comps,
einstein_radius=einstein_radius,
slope=slope,
core_radius=0.0,
)
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@aa.grid_dec.to_array
def potential_2d_from(self, grid: aa.type.Grid2DLike, **kwargs):
alpha = self.deflections_yx_2d_from(aa.Grid2DIrregular(grid), **kwargs)
alpha_x = alpha[:, 1]
alpha_y = alpha[:, 0]
x = grid[:, 1] - self.centre[1]
y = grid[:, 0] - self.centre[0]
return (x * alpha_x + y * alpha_y) / (3 - self.slope)
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@aa.grid_dec.to_vector_yx
@aa.grid_dec.transform
@aa.grid_dec.relocate_to_radial_minimum
def deflections_yx_2d_from(self, grid: aa.type.Grid2DLike, **kwargs):
"""
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).
This code is an adaption of Tessore & Metcalf 2015:
https://arxiv.org/abs/1507.01819
Parameters
----------
grid
The grid of (y,x) arc-second coordinates the deflection angles are computed on.
"""
slope = self.slope - 1.0
einstein_radius = (
2.0 / (self.axis_ratio**-0.5 + self.axis_ratio**0.5)
) * self.einstein_radius
factor = np.divide(1.0 - self.axis_ratio, 1.0 + self.axis_ratio)
b = np.multiply(einstein_radius, np.sqrt(self.axis_ratio))
angle = np.arctan2(
grid[:, 0], np.multiply(self.axis_ratio, grid[:, 1])
) # Note, this angle is not the position angle
R = np.sqrt(
np.add(np.multiply(self.axis_ratio**2, grid[:, 1] ** 2), grid[:, 0] ** 2)
)
z = np.add(
np.multiply(np.cos(angle), 1 + 0j), np.multiply(np.sin(angle), 0 + 1j)
)
complex_angle = (
2.0
* b
/ (1.0 + self.axis_ratio)
* (b / R) ** (slope - 1.0)
* z
* special.hyp2f1(1.0, 0.5 * slope, 2.0 - 0.5 * slope, -factor * z**2)
)
deflection_y = complex_angle.imag
deflection_x = complex_angle.real
rescale_factor = (self.ellipticity_rescale) ** (slope - 1)
deflection_y *= rescale_factor
deflection_x *= rescale_factor
return self.rotated_grid_from_reference_frame_from(
grid=np.vstack((deflection_y, deflection_x)).T
)
def convergence_func(self, grid_radius: float) -> float:
if grid_radius > 0.0:
return self.einstein_radius_rescaled * grid_radius ** (-(self.slope - 1))
return np.inf
@staticmethod
def potential_func(u, y, x, axis_ratio, slope, core_radius):
_eta_u = np.sqrt((u * ((x**2) + (y**2 / (1 - (1 - axis_ratio**2) * u)))))
return (
(_eta_u / u)
* ((3.0 - slope) * _eta_u) ** -1.0
* _eta_u ** (3.0 - slope)
/ ((1 - (1 - axis_ratio**2) * u) ** 0.5)
)
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class PowerLawSph(PowerLaw):
def __init__(
self,
centre: Tuple[float, float] = (0.0, 0.0),
einstein_radius: float = 1.0,
slope: float = 2.0,
):
"""
Represents a spherical power-law density distribution.
Parameters
----------
centre
The (y,x) arc-second coordinates of the profile centre.
einstein_radius
The arc-second Einstein radius.
slope
The density slope of the power-law (lower value -> shallower profile, higher value -> steeper profile).
"""
super().__init__(
centre=centre,
ell_comps=(0.0, 0.0),
einstein_radius=einstein_radius,
slope=slope,
)
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@aa.grid_dec.to_vector_yx
@aa.grid_dec.transform
@aa.grid_dec.relocate_to_radial_minimum
def deflections_yx_2d_from(self, grid: aa.type.Grid2DLike, **kwargs):
eta = self.radial_grid_from(grid=grid, **kwargs)
deflection_r = (
2.0
* self.einstein_radius_rescaled
* np.divide(
np.power(eta, (3.0 - self.slope)), np.multiply((3.0 - self.slope), eta)
)
)
return self._cartesian_grid_via_radial_from(grid=grid, radius=deflection_r)