def nfw(m, z, dm=0, ref_in='200c', ref_out='500c',
c=1., err=1e-6, scaling='duffy08', full_output=False):
"""
Convert the mass of a cluster from one overdensity radius to another
assuming an NFW profile.
Parameters
----------
m : float
mass, in units of solar mass
z : float
redshift
dm : float (optional)
mass uncertainty
ref_in : {'2500c', '500c', '200c', '180c', '100c',
'500a', '200a', '100a'} (default '200c')
overdensity at which the input mass is measured.
The last letter indicates whether the overdensity
is with respect to the critical ('c') or average
('a') density of the Universe at redshift z.
ref_out : {'2500c', '500c', '200c', '180c', '100c',
'500a', '200a', '100a'} (default '500c')
overdensity at which the output mass is measured.
c : float (default 1)
either a fixed concentration or a correction factor
to the Duffy et al. relation (useful, e.g., for
estimating uncertainties due to the c-M relation).
See parameter duffy.
err : float (default 1e-6)
allowed difference for convergence
scaling : {'duffy08', 'dutton14'} (optional)
If given, use the corresponding concentration
relation with a correction factor *c*. If False,
the concentration is fixed to the value of *c*.
Only possible if ref_in is either '200c' or '200a'.
full_output : bool (default False)
If True, also return the concentration used.
Returns
-------
m_out : float
The mass at the output overdensity. If dm>0 then an
uncertainty on this mass is also returned (m_out is
then a tuple of length 2).
c : float (optional)
concentration. This is returned if full_output is
set to True.
"""
#if ref_in != '200c':
#return read_nfw(m, z, dm, ref_in=ref_in, ref_out=ref_out)
if ref_in not in ('200a', '200c'):
duffy = False
if not numpy.iterable(m):
m = numpy.array([m])
if not numpy.iterable(dm):
dm = numpy.array([dm])
# iteratively calculate output mass
def _mass(c, mx, scale, err):
mx_out = 0
mass = mx
while abs(mx_out/mass - 1) > err:
mx_out = mass
r_out = (3 * mx_out / (4 * numpy.pi * rho_out)) ** (1./3.)
x = r_out / scale
mass = mx * (numpy.log(1 + x) - x / (1 + x)) / \
(numpy.log(1 + c) - c / (1 + c))
return mass
# density contrasts
rho_in = int(ref_in[:-1]) * cosmology.density(z, ref=ref_in[-1])
rho_out = int(ref_out[:-1]) * cosmology.density(z, ref=ref_out[-1])
# radii
r_in = conversions.rsph(m, z, ref=ref_in, unit='Mpc')
if scaling in ('duffy08', 'dutton14'):
c = c * scalings.cM(m, z, ref=ref_in, scaling=scaling)
scale = r_in / c
# mass and uncertainty (if defined)
m_out = numpy.array([_mass(ci, mi, rs, err)
for ci, mi, rs in izip(c, m, scale)])
if numpy.any(dm > 0):
dm_hi = numpy.array([_mass(ci, mi+dmi, rs, err)
for ci, mi, dmi, rs in izip(c, m, dm, scale)])
dm_lo = numpy.array([_mass(ci, mi-dmi, rs, err)
for ci, mi, dmi, rs in izip(c, m, dm, scale)])
m_out = (m_out, (dm_hi+dm_lo)/2)
if full_output:
return m_out, c
return m_out
def upp(r, m500, z, runit='Mpc', unit='astro',
self_similar=True, profile='sph'):
"""
The Universal Pressure Profile (UPP) from Arnaud et al. (2010)
Parameters
----------
r : float
Radius at which to estimate the pressure
m500 : float
Cluster mass within r500
z : float
Cluster redshift
runit : {'Mpc', 'kpc', 'r500'} (default 'Mpc')
Whether R is in absolute units ('Mpc', 'kpc') or in
units relative to r500 ('r500'). In the first case,
r500 will be estimated from m500 assuming a
spherical cluster.
unit : {'astro', 'cgs', 'mks'} (default 'astro')
Pressure units. Default value is astro, which
returns the pressure in units of Msun·Mpc⁻¹·s⁻². In
cgs, the units are erg·cm⁻³ = g·cm⁻¹·s⁻² while in
mks they are kg·m⁻¹·s⁻².
self_similar : bool (default True)
Whether to use the "self-similar" scaling or the
"universal" scaling.
profile : {'sph', 'cyl'} (default 'sph')
Whether the used profile is spherical or
cylindrical, i.e., integrated along the line of
sight.
"""
h = cosmology.h
h70 = h / 0.7
if self_similar:
alpha_p = 0
Po = 8.130 / h70**1.5
c500 = 1.156
gamma = 0.3292
alpha = 1.0620
beta = 5.4807
else:
alpha_p = 0.12
Po = 8.403 / h70**1.5
c500 = 1.177
gamma = 0.3081
alpha = 1.0510
beta = 5.4905
def alpha_prime(x):
""" Eq. 8 """
if self_similar:
return 0
f = (2*x) ** 3
f /= (1+f)
return 0.10 - f * (alpha_p+0.10)
def I(x):
""" Eq. 24 """
f = lambda u: p(u, c500) * u**2
return 3 * integrate.quad(f, 0, x)[0]
def J(x):
""" Eq. 27 """
f = lambda u: p(u) * numpy.sqrt(u**2 - x**2) * u
return I(5) - 3*integrate.quad(f, x, 5)[0]
def P(x, m500, z):
""" UPP itself, Eq. 13 """
exp = alpha_p + alpha_prime(x, alpha_p, self_similar)
return p(x) * P500(m500, z) * (m500/3e14) ** exp
def P500(m500, z, unit='astro'):
""" Eq. 5 """
# in units of keV·cm⁻³
pp = 1.65e-3 * cosmology.E(z)**(8./3.) * \
(m500 / 3e14)**(2./3.) * h70**2
# in units of erg·cm⁻³ = g·cm⁻¹·s⁻²
pp *= units.keV
if unit == 'cgs':
return pp
# in units of kg·m⁻¹·s⁻²
if unit == 'mks':
return pp / units.kg * units.m
# in units of Msun·Mpc⁻¹·s⁻²
if unit == 'astro':
return pp / units.Msun * units.Mpc
return pp
def p(x, c500):
""" Eq. 11 """
cx = c500 * x
exp = (beta-gamma) / alpha
return Po / (cx**gamma * (1+cx**alpha)**exp)
def Ysph(R, r500, m500, z):
f = lambda r: P(r/r500, m500, z) * r**2
i = 4 * numpy.pi * integrate.quad(f, 0, R)[0]
return i * constants.Msun * constants.sigmaT / \
(constants.me * constants.c**2)
def Ycyl(R, r500, m500, z):
""" Eq. 15 """
rb = 5 * r500
f = lambda x, r: 4 * numpy.pi * r * P(x/r500, m500, z) * \
x / numpy.sqrt(x**2-r**2)
i = integrate.dblquad(f, 0, R, lambda r: r, lambda r: Rb)[0]
return i * constants.Msun * constants.sigmaT / \
(constants.me * constants.c**2)
# the calculation itself
if runit == 'r500':
x = r
else:
r500 = conversions.rsph(m500, z, ref='500c', unit=runit)
x = r / r500
if profile == 'sph':
Ax = 2.925e-5 * I(x) / h70
y = Ax * (m500/3e14)**1.78
elif profile == 'cyl':
Bx = 2.925e-5 * J(x) / h70
y = Bx * (m500/3e14)**1.78
return y
def sigma_profile(sigma, aperture, r200, z=0, dsigma=0, orbits="iso", concentration="duffy08", err=1e-3):
"""
A correction for incomplete coverage based on the theoretical velocity
dispersion profile of Mamon, Biviano & Murante (2010). The profiles
were kindly provided by Gary Mamon
Returns
-------
s200 : float
the corrected velocity dispersion at r200
aperture : float
the final estimate of the radial coverage, based on the
new r200
r200 : float
the final estimate of r200
"""
import conversions
import scalings
import scipy
from scipy import integrate, interpolate
def fraction(r0, c=4, orbits="iso"):
## these values kindly provided by Gary Mamon, are in a file in the
## folder where this module is located
## here, s = sigma_LOS / sqrt[GM(r_s)/r_s]. However, we are only
## interested in a ratio.
# r / r_s
x = [
0.000,
0.100,
0.126,
0.158,
0.200,
0.251,
0.316,
0.398,
0.501,
0.631,
0.794,
1.000,
1.259,
1.585,
1.995,
2.512,
3.162,
3.981,
5.012,
6.310,
7.943,
10.000,
]
# s for isotropic orbits
if orbits == "iso":
s = [
0.000,
0.625,
0.638,
0.650,
0.661,
0.670,
0.677,
0.682,
0.684,
0.682,
0.678,
0.669,
0.658,
0.642,
0.624,
0.603,
0.579,
0.554,
0.527,
0.499,
0.471,
0.442,
]
# s for progressively more radial orbits, beta=r/2/(r+r_s)
elif orbits == "radial":
s = [
0.000,
0.700,
0.714,
0.727,
0.737,
0.745,
0.750,
0.751,
0.748,
0.740,
0.728,
0.712,
0.691,
0.667,
0.640,
0.611,
0.581,
0.549,
0.518,
0.486,
0.455,
0.425,
]
# in units of r200
r = scipy.array(x) / c
profile = interpolate.interp1d(r, s)
integrand = lambda R: profile(R) * R
# these are integrated velocity dispersions
# sigma_ap = 2 * integrate.romberg(integrand, 0.1, r0) / r0**2
# sigma_r200 = 2 * integrate.romberg(integrand, 0.1, 1) # / 1**2
sigma_ap = profile(r0)
sigma_r200 = profile(1)
return sigma_r200 / sigma_ap
if type(concentration) == float:
c = concentration
elif concentration in ("dolag04", "duffy08"):
c = scalings.csigma(sigma, z, dsigma, scaling=concentration)[0]
else:
msg = "Value for argument concentration not valid, see help page"
raise ValueError(msg)
s1 = sigma # velocity dispersion at r200 -- changing with changing r200
r0 = aperture * r200 # initial radial coverage -- fixed!
r1 = [r200] # r200 -- should go changing
r2 = aperture * r200 # radial coverage
x = []
while abs(r1[-1] - r2) / r1[-1] > err:
r2 = r1[-1]
# x = fraction(r0/r1, c, orbits=orbits)
x.append(fraction(r0 / r1[-1], c, orbits=orbits))
s1 = x[-1] * sigma
m1 = scalings.sigma(s1, z)
# r1 = conversions.rsph(m1, z)
r1.append(conversions.rsph(m1, z))
ap = r0 / r1[-1]
if scipy.isnan(ap):
print x
print r1
print c, sigma, s1, r0, r2, ap
print ""
# exit()
r200new = r1[-1]
return s1, ap, r200new