def vikhlinin_temperature_profile(r="r", T_0="T_0", a="a", b="b", c="c",
r_t="r_t", T_min="T_min",
r_cool="r_cool", a_cool="a_cool"):
"""
A temperature profile for galaxy clusters from
Vikhlinin, A., Kravtsov, A., Forman, W., et al.
2006, ApJ, 640, 691.
Parameters
----------
r : string
The symbol for the radius variable.
T_0 : string
The symbol for the scale temperature of the profile.
a : string
The symbol for the inner logarithmic slope.
b : string
The symbol for the width of the transition region.
c : string
The symbol for the outer logarithmic slope.
r_t : string
The symbol for the scale radius.
T_min : string
The symbol for the minimum temperature.
r_cool : string
The symbol for the cooling radius.
a_cool : string
The symbol for the logarithmic slope in the cooling region.
"""
r, T_0, a, b, c, r_t, T_min, r_cool, a_cool = \
symbols((r, T_0, a, b, c, r_t, T_min, r_cool, a_cool))
x = (r/r_cool)**a_cool
t = (r/r_t)**(-a)/((1.+(r/r_t)**b)**(c/b))
profile = T_0*t*(x+T_min/T_0)/(x+1)
return Formula1D(profile, r, [T_0, a, b, c, r_t, T_min, r_cool, a_cool])
def AM06_density_profile(r="r", rho_0="rho_0", a="a", a_c="a_c", c="c",
alpha="alpha", beta="beta"):
"""
The density profile for galaxy clusters suggested by
Ascasibar, Y., & Markevitch, M. 2006, ApJ, 650, 102.
Works best in concert with the ``AM06_temperature_profile``.
Parameters
----------
r : string
The symbol for the radius variable.
rho_0 : string
The symbol for the scale density of the profile.
a : string
The symbol for the scale radius.
a_c : string
The symbol for the scale radius of the cool core.
c : string
The symbol for the scale of the temperature drop of the cool
core.
alpha : string
The symbol for the first slope parameter.
beta : string
The symbol for the second slope parameter.
"""
r, rho_0, a, a_c, c, alpha, beta = symbols((r, rho_0, a, a_c, c,
alpha, beta))
profile = rho_0*(1+r/a_c)*(1+r/a_c/c)**alpha*(1+r/a)**beta
return Formula1D(profile, r, [rho_0, a, a_c, c, alpha, beta])
def linear(x="x", a="a", b="b"):
"""
A linear formula.
Parameters
----------
x : string
The symbol for the independent variable.
a : string
The symbol for the slope of the line.
b : string
The symbol for the intercept.
"""
x, a, b = symbols((x, a, b))
formula = a * x + b
return Formula1D(formula, x, [a, b])
def hernquist_mass_profile(r="r", M_0="M_0", a="a"):
"""
A Hernquist mass profile (Hernquist, L. 1990,
ApJ, 356, 359).
Parameters
----------
r : string
The symbol for the radius variable.
M_0 : string
The symbol for the total mass of the profile.
a : string
The symbol for the scale radius.
"""
r, M_0, a = symbols((r, M_0, a))
profile = M_0*r**2/(r+a)**2
return Formula1D(profile, r, [M_0, a])
def hernquist_density_profile(r="r", M_0="M_0", a="a"):
"""
A Hernquist density profile (Hernquist, L. 1990,
ApJ, 356, 359).
Parameters
----------
r : string
The symbol for the radius variable.
M_0 : string
The symbol for the total mass of the profile.
a : string
The symbol for the scale radius.
"""
r, M_0, a = symbols((r, M_0, a))
profile = M_0/(2*pi*a**3)/((r/a)*(1+r/a)**3)
return Formula1D(profile, r, [M_0, a])
def NFW_density_profile(r="r", rho_s="rho_s", r_s="r_s"):
"""
An NFW density profile (Navarro, J.F., Frenk, C.S.,
& White, S.D.M. 1996, ApJ, 462, 563).
Parameters
----------
r : string
The symbol for the radius variable.
rho_s : string
The symbol for the scale density of the profile.
r_s : string
The symbol for the scale radius.
"""
r, rho_s, r_s = symbols((r, rho_s, r_s))
profile = rho_s/((r/r_s)*(1+r/r_s)**2)
return Formula1D(profile, r, [rho_s, r_s])
def power_law(x="x", K="K", x_scale="x_scale", alpha="alpha"):
"""
A power-law formula.
Parameters
----------
x : string
The symbol for the independent variable.
K : string
The symbol for the normalization.
x_scale : string
The symbol for the scale value.
alpha : string
The symbol for the power-law index.
"""
x, K, x_scale, alpha = symbols((x, K, x_scale, alpha))
formula = K * (x / x_scale)**alpha
return Formula1D(formula, x, [x_scale, K, alpha])
def exponential(x="x", A="A", x_0="x_0", x_s="x_s"):
"""
An exponential formula.
Parameters
----------
x : string
The symbol for the independent variable.
A : string
The symbol for the normalization.
x_0 : string
The symbol for the y-intercept.
x_s : string
The symbol for the scale parameter.
"""
x, A, x_0, x_s = symbols((x, A, x_0, x_s))
formula = A * exp((x - x_0) / x_s)
return Formula1D(formula, x, [A, x_0, x_s])
def sNFW_mass_profile(r="r", M="M", a="a"):
"""
A "super-NFW" mass profile (Lilley, E. J.,
Wyn Evans, N., & Sanders, J.L. 2018, MNRAS).
Parameters
----------
r : string
The symbol for the radius variable.
M : string
The symbol for the total mass of the profile.
a : string
The symbol for the scale radius.
"""
r, M, a = symbols((r, M, a))
x = r/a
profile = M*(1-(2+3*x)/(2*(1+x)**(3/2)))
return Formula1D(profile, r, [M, a])
def sNFW_density_profile(r="r", M="M", a="a"):
"""
A "super-NFW" density profile (Lilley, E. J.,
Wyn Evans, N., & Sanders, J.L. 2018, MNRAS).
Parameters
----------
r : string
The symbol for the radius variable.
M : string
The symbol for the total mass of the profile.
a : string
The symbol for the scale radius.
"""
r, M, a = symbols((r, M, a))
x = r/a
profile = 3*M/(16*pi*a**3)/(x*(1+x)**(5/2))
return Formula1D(profile, r, [M, a])
def NFW_mass_profile(r="r", rho_s="rho_s", r_s="r_s"):
"""
An NFW mass profile (Navarro, J.F., Frenk, C.S.,
& White, S.D.M. 1996, ApJ, 462, 563).
Parameters
----------
r : string
The symbol for the radius variable.
rho_s : string
The symbol for the scale density of the profile.
r_s : string
The symbol for the scale radius.
"""
r, rho_s, r_s = symbols((r, rho_s, r_s))
x = r/r_s
profile = 4*pi*rho_s*r_s**3*(log(1+x)-x/(1+x))
return Formula1D(profile, r, [rho_s, r_s])
def tNFW_density_profile(r="r", rho_s="rho_s", r_s="r_s", r_t="r_t"):
"""
A truncated NFW density profile ().
Parameters
----------
r : string
The symbol for the radius variable.
rho_s : string
The symbol for the scale density of the profile.
r_s : string
The symbol for the scale radius.
r_t : string
The symbol for the truncation radius.
"""
r, rho_s, r_s, r_t = symbols((r, rho_s, r_s, r_t))
profile = rho_s/((r/r_s)*(1+r/r_s)**2)
profile /= (1+(r/r_t)**2)
return Formula1D(profile, r, [rho_s, r_s, r_t])
def beta_model_profile(r="r", rho_c="rho_c", r_c="r_c", beta="beta"):
"""
A beta-model density profile (like for galaxy clusters,
Cavaliere A., Fusco-Femiano R., 1976, A&A, 49, 137).
Parameters
----------
r : string
The symbol for the radius variable.
rho_c : string
The symbol for the scale density.
r_c : string
The symbol for the core radius.
beta : string
The symbol for the beta parameter.
"""
r, rho_c, r_c, beta = symbols((r, rho_c, r_c, beta))
profile = rho_c*((1+(r/r_c)**2)**(-3*beta/2))
return Formula1D(profile, r, [r_c, rho_c, beta])
def gaussian(x="x", A="A", mu="mu", sigma="sigma"):
"""
A Gaussian formula.
Parameters
----------
x : string
The symbol for the independent variable.
A : string
The symbol for the normalization.
mu : string
The symbol for the mean.
sigma : string
The symbol for the standard deviation.
"""
x, A, mu, sigma = symbols((x, A, mu, sigma))
params = [A, mu, sigma]
formula = A * exp(-Rational(1, 2) *
((x - mu) / sigma)**2) / (sigma * sqrt(2 * pi))
return Formula1D(formula, x, params)
def einasto_density_profile(r="r", rho_0="rho_0", h="h", alpha="alpha"):
"""
A density profile where the logarithmic slope is a
power-law. The form here is that given in Equation 5 of
Retana-Montenegro et al. 2012, A&A, 540, A70.
Parameters
----------
r : string
The symbol for the radius variable.
rho_0 : string
The symbol for the core density.
h : string
The symbol for the scale radius.
alpha : string
The symbol for the power-law index.
"""
r, rho_0, h, alpha = symbols((r, rho_0, h, alpha))
x = r/h
profile = rho_0*exp(-x**alpha)
return Formula1D(profile, r, [rho_0, h, alpha])
def vikhlinin_density_profile(r="r", rho_0="rho_0", r_c="r_c",
r_s="r_s", alpha="alpha", beta="beta",
epsilon="epsilon", gamma=None):
"""
A modified beta-model density profile for galaxy
clusters from Vikhlinin, A., Kravtsov, A., Forman, W.,
et al. 2006, ApJ, 640, 691.
Parameters
----------
r : string
The symbol for the radius variable.
rho_0 : string
The symbol for the scale density of the profile.
r_c : string
The symbol for the core radius.
r_s : string
The symbol for the scale radius.
alpha : string
The symbol for the inner logarithmic slope parameter.
beta : string
The symbol for the middle logarithmic slope parameter.
epsilon : string
The symbol for the outer logarithmic slope parameter.
gamma : string
The symbol for controlling the width of the outer
transition. If None, it will be gamma = 3 by default.
"""
r, rho_0, r_c, r_s, alpha, beta, epsilon = \
symbols((r, rho_0, r_c, r_s, alpha, beta, epsilon))
params = [rho_0, r_c, r_s, alpha, beta, epsilon]
if gamma is None:
gamma = 3
else:
gamma = symbols(gamma)
params += [gamma]
profile = rho_0*(r/r_c)**(-alpha/2) * \
(1+(r/r_c)**2)**(-3*beta/2+alpha/4) * \
(1+(r/r_s)**gamma)**(-epsilon/gamma/2)
return Formula1D(profile, r, params)
def baseline_entropy_profile(r="r", K_0="K_0", K_200="K_200", r_200="r_200",
alpha="alpha"):
"""
The baseline entropy profile for galaxy clusters (Voit, G.M.,
Kay, S.T., & Bryan, G.L. 2005, MNRAS, 364, 909).
Parameters
----------
r : string
The symbol for the radius variable.
K_0 : string
The symbol for the central entropy floor.
K_200 : string
The symbol for the entropy at the radius *r_200*.
r_200 : string
The symbol for the virial radius of the profile.
alpha : string
The symbol for the logarithmic slope of the profile.
"""
r, K_0, K_200, r_200, alpha = symbols((r, K_0, K_200, r_200, alpha))
profile = K_0 + K_200*(r/r_200)**alpha
return Formula1D(profile, r, [K_0, K_200, r_200, alpha])
def AM06_temperature_profile(r="r", T_0="T_0", a="a", a_c="a_c", c="c"):
"""
The temperature profile for galaxy clusters suggested by
Ascasibar, Y., & Markevitch, M. 2006, ApJ, 650, 102.
Works best in concert with the ``AM06_density_profile``.
Parameters
----------
r : string
The symbol for the radius variable.
T_0 : string
The symbol for the scale temperature of the profile.
a : string
The symbol for the scale radius.
a_c : string
The symbol for the scale radius of the cool core.
c : string
The symbol for the scale of the temperature drop of the cool
core.
"""
r, T_0, a, a_c, c = symbols((r, T_0, a, a_c, c))
profile = T_0/(1+r/a)*(c+r/a_c)/(1+r/a_c)
return Formula1D(profile, r, [T_0, a, a_c, c])
from formulas.base import Formula1D, Formula2D
from formulas.constants import FormulaConstant
from numpy.testing import assert_allclose
from sympy import Float
import yt.units as u
f_x = Formula1D("a*x**2+b","x",["a","b"])
g_x = Formula1D("p*cos(c*x)","x",["c","p"])
h_y = Formula1D("k*exp(d*y)","y",["k","d"])
j_o = Formula1D("2*v*o+z", "o", ["v","z"])
a = 50.*u.km/u.s**2
b = 30.*u.km
c = 10.*u.s**-1
d = 5.*u.s**-1
k = 10.*u.km
p = 20.*u.km
v = 3.0
z = 1.5
f_x.set_param_values(a=a, b=b)
g_x.set_param_values(p=p, c=c)
h_y.set_param_values(k=k, d=d)
j_o.set_param_values(v=v, z=z)
x = 60.*u.s
y = 15.*u.s
w = FormulaConstant("w", 2*u.km)
o = 3.4
def test_add():
n_x = f_x+g_x
assert n_x.ndim == 1
assert isinstance(n_x, Formula1D)
def maxwellian_velocity(v="v", A="A", sigma="sigma"):
v, A, sigma = symbols((v, A, sigma))
params = [A, sigma]
f = A * exp(-Rational(1, 2) * ((v / sigma)**2)) / (sigma * sqrt(2 * pi))
return Formula1D(f, v, params)
def maxwellian_speed(v="v", A="A", sigma="sigma"):
v, A, sigma = symbols((v, A, sigma))
params = [A, sigma]
f = A * exp(-Rational(1, 2) * ((v / sigma)**2)) / (sigma * sqrt(2 * pi))**3
return Formula1D(4 * pi * v * v * f, v, params)
from formulas.base import Formula1D, variable, Formula2D
from formulas.radial_profiles import NFW_density_profile
from formulas.general import gaussian
import yt.units as u
from numpy.testing import assert_allclose
import pytest
f_x = Formula1D("a*x**2+b", "x", ["a", "b"])
a = 50. * u.km / u.s**2
b = 30. * u.km
f_x.set_param_values(a=a, b=b)
g_x = gaussian(x="x", A="A_x", mu="mu_x", sigma="sigma_x")
g_y = gaussian(x="y", A="A_y", mu="mu_y", sigma="sigma_y")
g_z = gaussian(x="z", A="A_z", mu="mu_z", sigma="sigma_z")
A_x = A_y = A_z = 1.0
mu_x = mu_y = mu_z = 30 * u.s
sigma_x = sigma_y = sigma_z = 40 * u.s
g_x.set_param_values(A_x=A_x, mu_x=mu_x, sigma_x=sigma_x)
g_y.set_param_values(A_y=A_y, mu_y=mu_y, sigma_y=sigma_y)
g_z.set_param_values(A_z=A_z, mu_z=mu_z, sigma_z=sigma_z)
g_xy = g_x * g_y
g_xyz = g_xy * g_z
x = 60. * u.s
y = 25. * u.s
z = -10. * u.s
def test_unitless():
def cored_hernquist_density_profile(r="r", M_0="M_0", a="a", b="b"):
r, M_0, a, b = symbols((r, M_0, a, b))
profile = M_0*b/(2*pi*a**3)/((1+b*r/a)*(1+r/a)**3)
return Formula1D(profile, r, [M_0, a, b])