# This code is a part of X-ray: Generate and Analyse (XGA), a module designed for the XMM Cluster Survey (XCS).
# Last modified by David J Turner (turne540@msu.edu) 08/06/2023, 22:40. Copyright (c) The Contributors
from typing import Union, List
import numpy as np
from astropy.constants import k_B
from astropy.units import Quantity, Unit, UnitConversionError, kpc, deg
from .base import BaseModel1D
from ..utils import r500, r200, r2500
[docs]class SimpleVikhlininTemperature1D(BaseModel1D):
"""
An XGA model implementation of the simplified version of Vikhlinin's temperature model. This is for the
description of 3D temperature profiles of galaxy clusters.
:param Unit/str x_unit: The unit of the x-axis of this model, kpc for instance. May be passed as a string
representation or an astropy unit object.
:param Unit/str y_unit: The unit of the output of this model, keV for instance. May be passed as a string
representation or an astropy unit object.
:param List[Quantity] cust_start_pars: The start values of the model parameters for any fitting function that
used start values. The units are checked against default start values.
"""
def __init__(self, x_unit: Union[str, Unit] = 'kpc', y_unit: Union[str, Unit] = Unit('keV'),
cust_start_pars: List[Quantity] = None):
"""
The init of a subclass of the XGA BaseModel1D class, describing a simple version of the galaxy cluster
temperature profile model created by Vikhlinin et al.
"""
# If a string representation of a unit was passed then we make it an astropy unit
if isinstance(x_unit, str):
x_unit = Unit(x_unit)
if isinstance(y_unit, str):
y_unit = Unit(y_unit)
poss_y_units = [Unit('keV'), Unit('K')]
y_convertible = [u.is_equivalent(y_unit) for u in poss_y_units]
if not any(y_convertible):
allowed = ", ".join([u.to_string() for u in poss_y_units])
raise UnitConversionError("{p} is not convertible to any of the allowed units; "
"{a}".format(p=y_unit.to_string(), a=allowed))
else:
yu_ind = y_convertible.index(True)
poss_x_units = [kpc, deg, r200, r500, r2500]
x_convertible = [u.is_equivalent(x_unit) for u in poss_x_units]
if not any(x_convertible):
allowed = ", ".join([u.to_string() for u in poss_x_units])
raise UnitConversionError("{p} is not convertible to any of the allowed units; "
"{a}".format(p=x_unit.to_string(), a=allowed))
else:
xu_ind = x_convertible.index(True)
r_cool_starts = [Quantity(50, 'kpc'), Quantity(0.01, 'deg'), Quantity(0.05, r200), Quantity(0.1, r500),
Quantity(0.5, r2500)]
r_tran_starts = [Quantity(200, 'kpc'), Quantity(0.015, 'deg'), Quantity(0.2, r200), Quantity(0.4, r500),
Quantity(0.7, r2500)]
t_min_starts = [Quantity(3, 'keV'), (Quantity(3, 'keV')/k_B).to('K')]
t_zero_starts = [Quantity(6, 'keV'), (Quantity(6, 'keV')/k_B).to('K')]
start_pars = [r_cool_starts[xu_ind], Quantity(1, ''), t_min_starts[yu_ind], t_zero_starts[yu_ind],
r_tran_starts[xu_ind], Quantity(1, '')]
if cust_start_pars is not None:
# If the custom start parameters can run this gauntlet without tripping an error then we're all good
# This method also returns the custom start pars converted to exactly the same units as the default
start_pars = self.compare_units(cust_start_pars, start_pars)
rc_priors = [{'prior': Quantity([10, 500], 'kpc'), 'type': 'uniform'},
{'prior': Quantity([0.0, 0.032951243], 'deg'), 'type': 'uniform'},
{'prior': Quantity([0, 0.5], r200), 'type': 'uniform'},
{'prior': Quantity([0, 0.3], r500), 'type': 'uniform'},
{'prior': Quantity([0, 1], r2500), 'type': 'uniform'}]
rt_priors = [{'prior': Quantity([100, 500], 'kpc'), 'type': 'uniform'},
{'prior': Quantity([0.001, 0.032951243], 'deg'), 'type': 'uniform'},
{'prior': Quantity([0.1, 0.5], r200), 'type': 'uniform'},
{'prior': Quantity([0.07, 0.3], r500), 'type': 'uniform'},
{'prior': Quantity([0.2, 1], r2500), 'type': 'uniform'}]
t0_priors = [{'prior': Quantity([0.5, 15], 'keV'), 'type': 'uniform'},
{'prior': (Quantity([0.5, 15], 'keV')/k_B).to('K'), 'type': 'uniform'}]
tm_priors = [{'prior': Quantity([0.1, 6], 'keV'), 'type': 'uniform'},
{'prior': (Quantity([0.1, 6], 'keV') / k_B).to('K'), 'type': 'uniform'}]
priors = [rc_priors[xu_ind], {'prior': Quantity([0, 5]), 'type': 'uniform'}, tm_priors[yu_ind],
t0_priors[yu_ind], rt_priors[xu_ind], {'prior': Quantity([0, 5]), 'type': 'uniform'}]
nice_pars = [r"R$_{\rm{cool}}$", r"a$_{\rm{cool}}$", r"T$_{\rm{min}}$", "T$_{0}$", r"R$_{\rm{T}}$", "c"]
info_dict = {'author': 'Ghirardini et al.', 'year': 2019,
'reference': 'https://doi.org/10.1051/0004-6361/201833325',
'general': "A simplified, 'functional', form of Vikhlinin's temperature model.\n"
" This model has 6 free parameters rather than the 9 free parameters\n"
" of the original"}
super().__init__(x_unit, y_unit, start_pars, priors, 'simple_vikhlinin_temp',
'Simplified Vikhlinin Profile', nice_pars, 'Gas Temperature', info_dict)
[docs] @staticmethod
def model(x: Quantity, r_cool: Quantity, a_cool: Quantity, t_min: Quantity, t_zero: Quantity, r_tran: Quantity,
c_power: Quantity) -> Quantity:
"""
The model function for the simplified Vikhlinin temperature profile.
:param Quantity x: The radii to calculate y values for.
:param Quantity r_cool: Parameter describing the radius of the cooler core region.
:param Quantity a_cool: Power law parameter for the cooler core region.
:param Quantity t_min: A minimum temperature parameter for the model.
:param Quantity t_zero: A normalising temperature parameter for the model.
:param Quantity r_tran: The radius of the transition region of this broken power law model.
:param Quantity c_power: The power law index for the part of the model which describes the outer region of
the cluster.
:return: The y values corresponding to the input x values.
:rtype: Quantity
"""
try:
cool_expr = ((t_min / t_zero) + (x / r_cool)**a_cool) / (1 + (x / r_cool)**a_cool)
out_expr = 1 / ((1 + (x / r_tran)**2)**(c_power / 2))
result = t_zero * cool_expr * out_expr
except ZeroDivisionError:
result = np.NaN
return result
[docs] def derivative(self, x: Quantity, dx: Quantity = Quantity(0, ''), use_par_dist: bool = False) -> Quantity:
"""
Calculates the gradient of the simple Vikhlinin temperature profile at a given point, overriding the
numerical method implemented in the BaseModel1D class.
:param Quantity x: The point(s) at which the slope of the model should be measured.
:param Quantity dx: This makes no difference here, as this is an analytical derivative. It has
been left in so that the inputs for this method don't vary between models.
:param bool use_par_dist: Should the parameter distributions be used to calculate a derivative
distribution; this can only be used if a fit has been performed using the model instance.
Default is False, in which case the current parameters will be used to calculate a single value.
:return: The calculated slope of the model at the supplied x position(s).
:rtype: Quantity
"""
# This makes sure that the input radius, or radii, are being used properly. For a single x value, or a set of
# x values, we use [..., None] to ensure that (if the user has decided to use the parameter distribution) we
# get M distributions of the derivative. In the case where the user has passed a distribution of radii, really
# representative of a single radius but with uncertainty, then this does not trigger, and rather than an N
# by N (where N is the number of samples in each posterior) array, you get an M x N array
if x.isscalar or (not x.isscalar and x.ndim == 1):
x = x[..., None]
if not use_par_dist:
r_c, a, t_m, t_0, r_t, c = self._model_pars
else:
r_c, a, t_m, t_0, r_t, c = self.par_dists
p1 = (((x/r_t)**2)+1)**(-c/2)*((a*-(t_m-t_0))*((x**2)+(r_t**2))*((x/r_c)**a)
- c*x**2*(((x/r_c)**a)+1)*(t_m+(t_0*((x/r_c)**a))))
p2 = x*(x**2+r_t**2)*(((x/r_c)**a) + 1)**2
return p1/p2
[docs]class VikhlininTemperature1D(BaseModel1D):
"""
An XGA model implementation of the full version of Vikhlinin's temperature model. This is for the
description of 3D temperature profiles of galaxy clusters.
:param Unit/str x_unit: The unit of the x-axis of this model, kpc for instance. May be passed as a string
representation or an astropy unit object.
:param Unit/str y_unit: The unit of the output of this model, keV for instance. May be passed as a string
representation or an astropy unit object.
:param List[Quantity] cust_start_pars: The start values of the model parameters for any fitting function that
used start values. The units are checked against default start values.
"""
def __init__(self, x_unit: Union[str, Unit] = 'kpc', y_unit: Union[str, Unit] = Unit('keV'),
cust_start_pars: List[Quantity] = None):
"""
The init of a subclass of the XGA BaseModel1D class, describing the full version of the galaxy cluster
temperature profile model created by Vikhlinin et al.
"""
# If a string representation of a unit was passed then we make it an astropy unit
if isinstance(x_unit, str):
x_unit = Unit(x_unit)
if isinstance(y_unit, str):
y_unit = Unit(y_unit)
poss_y_units = [Unit('keV'), Unit('K')]
y_convertible = [u.is_equivalent(y_unit) for u in poss_y_units]
if not any(y_convertible):
allowed = ", ".join([u.to_string() for u in poss_y_units])
raise UnitConversionError("{p} is not convertible to any of the allowed units; "
"{a}".format(p=y_unit.to_string(), a=allowed))
else:
yu_ind = y_convertible.index(True)
poss_x_units = [kpc, deg, r200, r500, r2500]
x_convertible = [u.is_equivalent(x_unit) for u in poss_x_units]
if not any(x_convertible):
allowed = ", ".join([u.to_string() for u in poss_x_units])
raise UnitConversionError("{p} is not convertible to any of the allowed units; "
"{a}".format(p=x_unit.to_string(), a=allowed))
else:
xu_ind = x_convertible.index(True)
r_cool_starts = [Quantity(50, 'kpc'), Quantity(0.01, 'deg'), Quantity(0.05, r200), Quantity(0.1, r500),
Quantity(0.5, r2500)]
r_tran_starts = [Quantity(200, 'kpc'), Quantity(0.015, 'deg'), Quantity(0.2, r200), Quantity(0.4, r500),
Quantity(0.7, r2500)]
t_min_starts = [Quantity(3, 'keV'), (Quantity(3, 'keV') / k_B).to('K')]
t_zero_starts = [Quantity(6, 'keV'), (Quantity(6, 'keV') / k_B).to('K')]
start_pars = [r_cool_starts[xu_ind], Quantity(1, ''), t_min_starts[yu_ind], t_zero_starts[yu_ind],
r_tran_starts[xu_ind], Quantity(1, ''), Quantity(1, ''), Quantity(1, '')]
if cust_start_pars is not None:
# If the custom start parameters can run this gauntlet without tripping an error then we're all good
# This method also returns the custom start pars converted to exactly the same units as the default
start_pars = self.compare_units(cust_start_pars, start_pars)
rc_priors = [{'prior': Quantity([10, 500], 'kpc'), 'type': 'uniform'},
{'prior': Quantity([0.0, 0.032951243], 'deg'), 'type': 'uniform'},
{'prior': Quantity([0, 0.5], r200), 'type': 'uniform'},
{'prior': Quantity([0, 0.3], r500), 'type': 'uniform'},
{'prior': Quantity([0, 1], r2500), 'type': 'uniform'}]
rt_priors = [{'prior': Quantity([100, 500], 'kpc'), 'type': 'uniform'},
{'prior': Quantity([0.001, 0.032951243], 'deg'), 'type': 'uniform'},
{'prior': Quantity([0.1, 0.5], r200), 'type': 'uniform'},
{'prior': Quantity([0.07, 0.3], r500), 'type': 'uniform'},
{'prior': Quantity([0.2, 1], r2500), 'type': 'uniform'}]
t0_priors = [{'prior': Quantity([0.5, 15], 'keV'), 'type': 'uniform'},
{'prior': (Quantity([0.5, 15], 'keV') / k_B).to('K'), 'type': 'uniform'}]
tm_priors = [{'prior': Quantity([0.1, 6], 'keV'), 'type': 'uniform'},
{'prior': (Quantity([0.1, 6], 'keV') / k_B).to('K'), 'type': 'uniform'}]
priors = [rc_priors[xu_ind], {'prior': Quantity([0, 5]), 'type': 'uniform'}, tm_priors[yu_ind],
t0_priors[yu_ind], rt_priors[xu_ind], {'prior': Quantity([0, 5]), 'type': 'uniform'},
{'prior': Quantity([0, 5]), 'type': 'uniform'}, {'prior': Quantity([0, 5]), 'type': 'uniform'}]
nice_pars = [r"R$_{\rm{cool}}$", r"a$_{\rm{cool}}$", r"T$_{\rm{min}}$", "T$_{0}$", r"R$_{\rm{T}}$", "a", "b",
"c"]
info_dict = {'author': 'Vikhlinin et al.', 'year': 2005,
'reference': 'https://doi.org/10.1086/500288',
'general': "The full form of Vikhlinin's temperature model, describes a\n"
"cluster temperature profile from the core to the outskirts."}
super().__init__(x_unit, y_unit, start_pars, priors, 'vikhlinin_temp',
'Vikhlinin Profile', nice_pars, 'Gas Temperature', info_dict)
[docs] @staticmethod
def model(x: Quantity, r_cool: Quantity, a_cool: Quantity, t_min: Quantity, t_zero: Quantity, r_tran: Quantity,
a_power: Quantity, b_power: Quantity, c_power: Quantity) -> Quantity:
"""
The model function for the full Vikhlinin temperature profile.
:param Quantity x: The radii to calculate y values for.
:param float r_cool: Parameter describing the radius of the cooling region (I THINK - NOT CERTAIN YET).
:param float a_cool: Power law parameter for the cooling region (I THINK - NOT CERTAIN YET).
:param float t_min: A minimum temperature parameter for the model (I THINK - NOT CERTAIN YET).
:param float t_zero: A normalising temperature parameter for the model (I THINK - NOT CERTAIN YET).
:param float r_tran: The radius of the transition region of this broken power law model.
:param float a_power: The first power law index.
:param float b_power: The second power law index.
:param float c_power: the third power law index.
:return: The y values corresponding to the input x values.
:rtype: Quantity
"""
try:
power_rad_ratio = (x / r_cool)**a_cool
# The rest of the model expression
t_cool = (power_rad_ratio + (t_min / t_zero)) / (power_rad_ratio + 1)
# The ratio of the input radius (or radii) to the transition radius
rad_ratio = x / r_tran
t_outer = rad_ratio**(-a_power) / (1 + rad_ratio**b_power)**(c_power / b_power)
result = t_zero * t_cool * t_outer
except ZeroDivisionError:
result = np.NaN
return result
# So that things like fitting functions can be written generally to support different models
TEMP_MODELS = {"vikhlinin_temp": VikhlininTemperature1D, "simple_vikhlinin_temp": SimpleVikhlininTemperature1D}
TEMP_MODELS_PUB_NAMES = {n: m().publication_name for n, m in TEMP_MODELS.items()}
TEMP_MODELS_PAR_NAMES = {n: m().par_publication_names for n, m in TEMP_MODELS.items()}