# This code is a part of XMM: Generate and Analyse (XGA), a module designed for the XMM Cluster Survey (XCS).
# Last modified by David J Turner (david.turner@sussex.ac.uk) 26/03/2021, 16:58. Copyright (c) David J Turner
from typing import Union, List
import numpy as np
from astropy.units import Quantity, Unit, UnitConversionError, kpc, deg
from .base import BaseModel1D
from ..utils import r500, r200, r2500
[docs]class KingProfile1D(BaseModel1D):
"""
An XGA model implementation of the King profile, describing an isothermal sphere. This describes a
radial density profile and assumes spherical symmetry.
"""
def __init__(self, x_unit: Union[str, Unit] = 'kpc', y_unit: Union[str, Unit] = Unit('Msun/Mpc^3'),
cust_start_pars: List[Quantity] = None):
"""
The init of a subclass of the XGA BaseModel1D class, describing a basic model for galaxy cluster gas
density, the king profile.
: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.
"""
# 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('Msun/Mpc^3'), Unit('1/cm^3')]
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_core_starts = [Quantity(100, 'kpc'), Quantity(0.2, 'deg'), Quantity(0.05, r200), Quantity(0.1, r500),
Quantity(0.5, r2500)]
# TODO MAKE THE NEW START PARAMETERS MORE SENSIBLE
norm_starts = [Quantity(1e+13, 'Msun/Mpc^3'), Quantity(1e-3, '1/cm^3')]
start_pars = [Quantity(1, ''), r_core_starts[xu_ind], norm_starts[yu_ind]]
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)
r_core_priors = [{'prior': Quantity([0, 2000], 'kpc'), 'type': 'uniform'},
{'prior': Quantity([0, 1], 'deg'), 'type': 'uniform'},
{'prior': Quantity([0, 1], r200), 'type': 'uniform'},
{'prior': Quantity([0, 1], r500), 'type': 'uniform'},
{'prior': Quantity([0, 1], r2500), 'type': 'uniform'}]
norm_priors = [{'prior': Quantity([1e+12, 1e+16], 'Msun/Mpc^3'), 'type': 'uniform'},
{'prior': Quantity([0, 10], '1/cm^3'), 'type': 'uniform'}]
priors = [{'prior': Quantity([0, 3]), 'type': 'uniform'}, r_core_priors[xu_ind], norm_priors[yu_ind]]
nice_pars = [r"$\beta$", r"R$_{\rm{core}}$", "S$_{0}$"]
info_dict = {'author': 'placeholder', 'year': 'placeholder', 'reference': 'placeholder',
'general': 'The un-projected version of the beta profile, suitable for a simple fit\n'
' to 3D density distributions. Describes a simple isothermal sphere.'}
super().__init__(x_unit, y_unit, start_pars, priors, 'king', 'King Profile', nice_pars, 'Gas Density',
info_dict)
[docs] @staticmethod
def model(x: Quantity, beta: Quantity, r_core: Quantity, norm: Quantity) -> Quantity:
"""
The model function for the king profile.
:param Quantity x: The radii to calculate y values for.
:param Quantity beta: The beta slope parameter of the model.
:param Quantity r_core: The core radius.
:param Quantity norm: The normalisation of the model.
:return: The y values corresponding to the input x values.
:rtype: Quantity
"""
return norm * ((1 + (x / r_core)**2)**(-3 * beta))
[docs] def derivative(self, x: Quantity, dx: Quantity = Quantity(0, ''), use_par_dist: bool = False) -> Quantity:
"""
Calculates the gradient of the king profile at a given point, overriding the numerical method implemented
in the BaseModel1D class, as this simple model has an easily derivable first derivative.
: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
"""
x = x[..., None]
if not use_par_dist:
beta, r_core, norm = self._model_pars
else:
beta, r_core, norm = self.par_dists
return (-6*beta*norm*x/np.power(r_core, 2))*np.power((1+np.power(x/r_core, 2)), (-3*beta) - 1)
[docs]class SimpleVikhlininDensity1D(BaseModel1D):
"""
An XGA model implementation of a simplified version of Vikhlinin's full density model. Used relatively recently
in https://doi.org/10.1051/0004-6361/201833325 by Ghirardini et al., a simplified form of Vikhlinin's full
density model, which can be found in https://doi.org/10.1086/500288.
"""
def __init__(self, x_unit: Union[str, Unit] = 'kpc', y_unit: Union[str, Unit] = Unit('Msun/Mpc^3'),
cust_start_pars: List[Quantity] = None):
"""
The init of a subclass of the XGA BaseModel1D class, describing a simplified version of Vikhlinin et al.'s
model for the gas density profile of a galaxy cluster.
: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.
"""
# 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('Msun/Mpc^3'), Unit('1/cm^3')]
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_core_starts = [Quantity(100, 'kpc'), Quantity(0.2, 'deg'), Quantity(0.05, r200), Quantity(0.1, r500),
Quantity(0.5, r2500)]
r_s_starts = [Quantity(300, 'kpc'), Quantity(0.35, 'deg'), Quantity(0.15, r200), Quantity(0.3, r500),
Quantity(1.0, r2500)]
norm_starts = [Quantity(1e+13, 'Msun/Mpc^3'), Quantity(1e-3, '1/cm^3')]
start_pars = [Quantity(1, ''), r_core_starts[xu_ind], Quantity(1, ''), r_s_starts[xu_ind], Quantity(2, ''),
norm_starts[yu_ind]]
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)
r_core_priors = [{'prior': Quantity([0, 2000], 'kpc'), 'type': 'uniform'},
{'prior': Quantity([0, 1], 'deg'), 'type': 'uniform'},
{'prior': Quantity([0, 1], r200), 'type': 'uniform'},
{'prior': Quantity([0, 1], r500), 'type': 'uniform'},
{'prior': Quantity([0, 1], r2500), 'type': 'uniform'}]
norm_priors = [{'prior': Quantity([1e+12, 1e+16], 'Msun/Mpc^3'), 'type': 'uniform'},
{'prior': Quantity([0, 10], '1/cm^3'), 'type': 'uniform'}]
priors = [{'prior': Quantity([0, 3]), 'type': 'uniform'}, r_core_priors[xu_ind],
{'prior': Quantity([0, 3]), 'type': 'uniform'}, r_core_priors[xu_ind],
{'prior': Quantity([0, 5]), 'type': 'uniform'}, norm_priors[yu_ind]]
nice_pars = [r"$\beta$", r"R$_{\rm{core}}$", r"$\alpha$", r"R$_{\rm{s}}$", r"$\epsilon$", r"N$_{0}$"]
info_dict = {'author': 'Ghirardini et al.', 'year': 2019,
'reference': 'https://doi.org/10.1051/0004-6361/201833325',
'general': "A simplified form of Vikhlinin's full density model, a type of broken\n"
" power law that deals well with most galaxy cluster density profile."}
super().__init__(x_unit, y_unit, start_pars, priors, 'simple_vikhlinin_dens', 'Simplified Vikhlinin Profile',
nice_pars, 'Gas Density', info_dict)
[docs] @staticmethod
def model(x: Quantity, beta: Quantity, r_core: Quantity, alpha: Quantity, r_s: Quantity, epsilon: Quantity,
norm: Quantity) -> Quantity:
"""
The model function for the simplified Vikhlinin density profile.
:param Quantity x: The radii to calculate y values for.
:param Quantity beta: The beta parameter of the model.
:param Quantity r_core: The core radius of the model.
:param Quantity alpha: The alpha parameter of the model.
:param Quantity r_s: The radius near where a change of slope by epsilon occurs.
:param Quantity epsilon: The epsilon parameter of the model.
:param Quantity norm: The overall normalisation of the model.
:return: The y values corresponding to the input x values.
:rtype: Quantity
"""
try:
# Calculates the ratio of the r_values to the r_core parameter
rc_rat = x / r_core
# Calculates the ratio of the r_values to the r_s parameter
rs_rat = x / r_s
first_term = rc_rat**(-alpha) / ((1+rc_rat**2)**((3 * beta) - (alpha / 2)))
second_term = 1 / ((1 + rs_rat**3)**(epsilon / 3))
result = norm * np.sqrt(first_term * second_term)
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 density 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
"""
x = x[..., None]
if not use_par_dist:
beta, r_core, alpha, r_s, epsilon, norm = self.model_pars
else:
beta, r_core, alpha, r_s, epsilon, norm = self.par_dists
# TODO DOUBLE CHECK THIS WHEN I'M LESS TIRED
first_term = -1*norm*np.sqrt(np.power(x/r_core, -alpha)*np.power((x**3/r_s**3) + 1, -epsilon/3)
*np.power((x**2/r_core**2) + 1, 0.5*(alpha-(6*beta))))
second_term = 1/(2*x*(x**2 + r_core**2)*(x**3 + r_s**3))
third_term = (x**3 + r_s**3)*(6*beta*x**2 + alpha*r_core**2) + x**3*epsilon*(x**2 + r_core**2)
return first_term*second_term*third_term
[docs]class VikhlininDensity1D(BaseModel1D):
"""
An XGA model implementation of Vikhlinin's full density model for galaxy cluster intra-cluster medium,
which can be found in https://doi.org/10.1086/500288. It is a radial profile, so an assumption
of spherical symmetry is baked in.
"""
def __init__(self, x_unit: Union[str, Unit] = 'kpc', y_unit: Union[str, Unit] = Unit('Msun/Mpc^3'),
cust_start_pars: List[Quantity] = None):
"""
The init of a subclass of the XGA BaseModel1D class, describing the full version of Vikhlinin et al.'s
model for the gas density profile of a galaxy cluster.
: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.
"""
# 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('Msun/Mpc^3'), Unit('1/cm^3')]
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_core_starts = [Quantity(100, 'kpc'), Quantity(0.2, 'deg'), Quantity(0.05, r200), Quantity(0.1, r500),
Quantity(0.5, r2500)]
r_s_starts = [Quantity(300, 'kpc'), Quantity(0.35, 'deg'), Quantity(0.15, r200), Quantity(0.3, r500),
Quantity(1.0, r2500)]
r_core_two_starts = [Quantity(50, 'kpc'), Quantity(0.1, 'deg'), Quantity(0.03, r200), Quantity(0.05, r500),
Quantity(0.25, r2500)]
norm_starts = [Quantity(1e+13, 'Msun/Mpc^3'), Quantity(1e-3, '1/cm^3')]
norm_two_starts = [Quantity(5e+12, 'Msun/Mpc^3'), Quantity(5e-4, '1/cm^3')]
start_pars = [Quantity(1, ''), r_core_starts[xu_ind], Quantity(1, ''), r_s_starts[xu_ind], Quantity(2, ''),
Quantity(3, ''), norm_starts[yu_ind], Quantity(1, ''), r_core_two_starts[xu_ind],
norm_two_starts[yu_ind]]
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)
r_core_priors = [{'prior': Quantity([0, 2000], 'kpc'), 'type': 'uniform'},
{'prior': Quantity([0, 1], 'deg'), 'type': 'uniform'},
{'prior': Quantity([0, 1], r200), 'type': 'uniform'},
{'prior': Quantity([0, 1], r500), 'type': 'uniform'},
{'prior': Quantity([0, 1], r2500), 'type': 'uniform'}]
norm_priors = [{'prior': Quantity([[1e+12, 1e+16]], 'Msun/Mpc^3'), 'type': 'uniform'},
{'prior': Quantity([0, 10], '1/cm^3'), 'type': 'uniform'}]
priors = [{'prior': Quantity([0, 3]), 'type': 'uniform'}, r_core_priors[xu_ind],
{'prior': Quantity([0, 3]), 'type': 'uniform'}, r_core_priors[xu_ind],
{'prior': Quantity([0, 5]), 'type': 'uniform'}, {'prior': Quantity([-5, 5]), 'type': 'uniform'},
norm_priors[yu_ind], {'prior': Quantity([0, 3]), 'type': 'uniform'}, r_core_priors[xu_ind],
norm_priors[yu_ind]]
nice_pars = [r"$\beta_{1}$", r"R$_{\rm{core,1}}$", r"$\alpha$", r"R$_{\rm{s}}$", r"$\epsilon$", r"$\gamma$",
r"N$_{01}$", r"$\beta_{2}$", r"R$_{\rm{core,2}}$", r"N$_{02}$"]
info_dict = {'author': 'Vikhlinin et al.', 'year': 2006,
'reference': 'https://doi.org/10.1086/500288',
'general': "The full model for cluster density profiles created by Vikhlinin et al.\n"
"This model has MANY free parameters which can be very hard to get constraints\n"
" on, and as such many people would use the simplified version which is implemented\n"
" as the SimpleVikhlininDensity1D class in XGA."}
super().__init__(x_unit, y_unit, start_pars, priors, 'vikhlinin_dens', 'Vikhlinin Profile',
nice_pars, 'Gas Density', info_dict)
[docs] @staticmethod
def model(x: Quantity, beta_one: Quantity, r_core_one: Quantity, alpha: Quantity, r_s: Quantity, epsilon: Quantity,
gamma: Quantity, norm_one: Quantity, beta_two: Quantity, r_core_two: Quantity, norm_two: Quantity):
"""
The model function for the full Vikhlinin density profile.
:param Quantity x: The radii to calculate y values for.
:param Quantity beta_one: The beta parameter of the model.
:param Quantity r_core_one: The core radius of the model.
:param Quantity alpha: The alpha parameter of the model.
:param Quantity r_s: The radius near where a change of slope by epsilon occurs.
:param Quantity epsilon: The epsilon parameter of the model.
:param Quantity gamma: Width of slope change transition region.
:param Quantity norm_one: The normalisation of the model first part of the model.
:param Quantity beta_two: The beta parameter slope of the small core part of the model.
:param Quantity r_core_two:The core radius of the small core part of the model.
:param Quantity norm_two: The normalisation of the additive, small core part of the model.
"""
try:
# Calculates the ratio of the r_values to the r_core_one parameter
rc1_rat = x / r_core_one
# Calculates the ratio of the r_values to the r_core_two parameter
rc2_rat = x / r_core_two
# Calculates the ratio of the r_values to the r_s parameter
rs_rat = x / r_s
first_term = rc1_rat**(-alpha) / ((1 + rc1_rat**2)**((3 * beta_one) - (alpha / 2)))
second_term = 1 / ((1 + rs_rat**gamma)**(epsilon / gamma))
additive_term = 1 / ((1 + rc2_rat**2)**(3 * beta_two))
except ZeroDivisionError:
return np.NaN
return np.sqrt((np.power(norm_one, 2) * first_term * second_term) + (np.power(norm_two, 2) * additive_term))
[docs] def derivative(self, x: Quantity, dx: Quantity = Quantity(0, ''), use_par_dist: bool = False) -> Quantity:
"""
Calculates the gradient of the full Vikhlinin density 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
"""
x = x[..., None]
if not use_par_dist:
b, rc, a, rs, e, g, n, b2, rc2, n2 = self.model_pars
else:
b, rc, a, rs, e, g, n, b2, rc2, n2 = self.par_dists
# Its horrible I know...
p1 = (-6*b2*(n2**2)*x*(((x/rc2)**2) + 1)**((-3*b2)-1)) / rc2**2
p2 = (-a*(n**2)*((x/rc)**(-a-1))*((((x/rc)**2)+1)**((a/2)-3*b))*((((x/rs)**g) + 1)**(-e/g)))/rc
p3 = (2*(n**2)*x*((a/2)-(3*b))*((x/rc)**(-a))*((((x/rc)**2)+1)**((a/2)-(3*b)-1))*((((x/rs)**g) + 1)**(-e/g)))/rc**2
p4 = -(n**2)*e*(x**(g-1))*(rs**(-g))*((x/rc)**(-a))*((((x/rc)**2)+1)**((a/2)-(3*b)))*((((x/rs)**g)+1)**(-e/g-1))
p5 = 2*np.sqrt((n2**2)*((((x/rc2)**2)+1)**(-3*b2)) + (n**2)*((x/rc)**(-a))*((((x/rc)**2)+1)**((a/2)-(3*b)))*((((x/rs)**g)+1)**(-e/g)))
return (p1 + p2 + p3 + p4) / p5
DENS_MODELS = {"simple_vikhlinin_dens": SimpleVikhlininDensity1D, 'king': KingProfile1D,
'vikhlinin_dens': VikhlininDensity1D}
DENS_MODELS_PAR_NAMES = {n: m().par_publication_names for n, m in DENS_MODELS.items()}
DENS_MODELS_PUB_NAMES = {n: m().publication_name for n, m in DENS_MODELS.items()}