cosNFW {celestial}

Navarro Frenk and White profile

Description

Density and total mass values for Navaro Frenk and White (NFW) profiles

Usage

cosNFW(Rad=0, Rho0=2.412e15, Rs=0.03253)
cosNFWmass_c(Rho0=2.412e15, Rs=0.03253, c=5, Munit = 1, Lunit = 1e+06)
cosNFWmass_Rmax(Rho0=2.412e15, Rs=0.03253, Rmax=0.16265, Munit = 1, Lunit = 1e+06)
cosNFWvcirc(Rad = 0.16264, Mvir = 1e+12, c = 5, f = Inf, z = 0, H0 = 100, OmegaM = 0.3,
OmegaL = 1 - OmegaM - OmegaR, OmegaR = 0, Rho = "crit", Dist = "Co", DeltaVir = 200,
Munit = 1, Lunit = 1e+06, Vunit = 1000, ref)
cosNFWvesc(Rad = 0.16264, Mvir = 1e+12, c = 5, f = Inf, z = 0, H0 = 100, OmegaM = 0.3,
OmegaL = 1 - OmegaM - OmegaR, OmegaR = 0, Rho = "crit", Dist = "Co", DeltaVir = 200,
Munit = 1, Lunit = 1e+06, Vunit = 1000, ref)
cosNFWsigma(Rad=0.03253, Rs=0.03253, c=5, z = 0, H0 = 100, OmegaM = 0.3,
OmegaL = 1-OmegaM-OmegaR, OmegaR=0, Rho = "crit", DeltaVir = 200, Munit = 1,
Lunit = 1e+06, Vunit = 1000, ref)
cosNFWsigma_mean(Rad=0.03253, Rs=0.03253, c=5, z = 0, H0 = 100, OmegaM = 0.3,
OmegaL = 1-OmegaM-OmegaR, OmegaR=0, Rho = "crit", DeltaVir = 200, Munit = 1,
Lunit = 1e+06, Vunit = 1000, ref)
cosNFWgamma(Rad=0.03253, Rs=0.03253, c=5, SigmaC=1, z = 0, H0 = 100,
OmegaM = 0.3, OmegaL = 1-OmegaM-OmegaR, OmegaR=0, Rho = "crit", DeltaVir = 200,
Munit = 1, Lunit = 1e+06, Vunit = 1000, ref)
cosNFWduffym2c(M=2e12, z = 0, H0 = 100, OmegaM = 0.3, OmegaL = 1-OmegaM-OmegaR,
OmegaR=0, Rho = "crit", A=6.71, B=-0.091, C=-0.44, Munit = 1, ref)

Arguments

Mvir

Mass within virial radius in units of 'Munit'.

Rad

Radius at which to calculate output in units of 'Lunit'. Either this is a 3D radius (cosNFW) or a projected 2D radius (cosNFWsigma/cosNFWsigma_mean).

Rho0

The normalising factor.

Rs

The NFW profile scale radius, where Rs=Rmax/c, in units of 'Munit'.

c

The NFW profile concentration parameter, where c=Rmax/Rs.

f

The NFW profile truncation radius in units of Rmax.

Rmax

The NFW profile Rmax parameter, where Rmax=Rs*c, in units of 'Lunit'.

SigmaC

The critical surface mass density (when SigmaC=1 we compute the excess surface density / ESD). See cosdistCrit for general computation given source and lens redshifts.

M

The halo mass required for computing the Duffy (2008) mass to concentration conversion in units of 'Munit'. Here the halo mass required for input is the 200 times overdense with respect to critical variation.

z

Cosmological redshift, where z must be > -1 (can be a vector).

H0

Hubble constant as defined at z=0 (default is H0=100 (km/s)/Mpc).

OmegaM

Omega matter (default is 0.3).

OmegaL

Omega Lambda (default is for a flat Universe with OmegaL = 1-OmegaM = 0.7).

OmegaR

Omega Radiation (default is 0, but OmegaM/3400 is typical).

Rho

Set whether the critical energy density is used (crit) or the mean mass density (mean).

Dist

Determines the distance type, i.e. whether the Rho critical energy or mean mass densities are calculated with respect to angular / physical distances (Ang) or with respect to comoving distances (Co). In effect this means Rvir values are either angular / physical (Ang) or comoving (Co). It does not affect Mvir <-> Sigma conversions, but does affect Mvir <-> Rvir and Rvir <-> Sigma.

DeltaVir

Desired overdensity of the halo with respect to Rho.

Munit

Base mass unit in multiples of Msun.

Lunit

Base length unit in multiples of parsecs.

Vunit

Base velocity unit in multiples of m/s.

A

Parameter used for Duffy mass to concentration relation.

B

Parameter used for Duffy mass to concentration relation.

C

Parameter used for Duffy mass to concentration relation.

ref

The name of a reference cosmology to use, one of 137 / 737 / Planck / Planck13 / Planck15 / Planck18 / WMAP / WMAP9 / WMAP7 / WMAP5 / WMAP3 / WMAP1 / Millennium / GiggleZ. Planck = Planck18 and WMAP = WMAP9. The usage is case insensitive, so wmap9 is an allowed input. See cosref for details. This overrides any other settings for H0/ OmegaM and OmegaL.

Details

These functions calculate various aspects of the NFW profile.

Value

cosNFW

Returns the instantaneous NFW profile density.

cosNFWmass_c

Returns the total mass given Rs and c in Msun/h.

cosNFWmass_Rmax

Returns the total mass given Rs and Rmax in Msun/h.

cosNFWvcirc

Returns the circular Keplarian orbit velocity for a given radius assuming an NFW halo potential.

cosNFWvesc

Returns the minimum escape (or unbinding) velocity for a given radius assuming an NFW halo potential.

cosNFWsigma

Returns the line-of-sight surface mass density at Rad (Eqn. 11 of Wright & Brainerd, 2000).

cosNFWsigma_mean

Returns the means surface mass density within Rad (Eqn. 13 of Wright & Brainerd, 2000).

cosNFWgamma

Returns the radial dependence of the weak lensing shear (Eqn. 12 of Wright & Brainerd, 2000).

cosNFWduffym2c

Returns the Duffy et al (2008) predicted concentration for a given halo mass.

Author(s)

Aaron Robotham

References

Duffy A.R., et al., 2008, MNRAS, 390L

Navarro J.F., Frenk C.S., White Simon D.M., 1996, ApJ, 462

Wright C.O. & Brainerd T.G., 2000, ApJ, 534

See Also

cosvol, cosmap, cosdist, cosgrow, coshalo

Examples

#What difference do we see if we use the rad_mean200 radius rather than rad_crit200

rad_crit200=coshaloMvirToRvir(1e12,Lunit=1e6)
rad_mean200=coshaloMvirToRvir(1e12,Lunit=1e6,Rho='mean')
cosNFWmass_Rmax(Rmax=rad_crit200) #By construction we should get ~10^12 Msun/h
cosNFWmass_Rmax(Rmax=rad_mean200) #For the same profile this is a factor 1.31 larger

#Shear checks:

plot(10^seq(-2,2,by=0.1), cosNFWgamma(10^seq(-2,2,by=0.1),Rs=0.2,c=10), type='l',
log='xy', xlab='R/Rs', ylab='ESD')
legend('topright', legend=c('Rs=0.2','c=10'))

#How do critical, mean 200 and 500 masses evolve with redshift? Let's see:

zseq=10^seq(-2, 1, by=0.1)
con=seq(2, 20, by=0.01)
concol=rainbow(length(con), start=0, end=5/6)
rad_crit200=coshaloMvirToRvir(1, z=zseq, Rho='crit', DeltaVir=200, ref='Planck15')
rad_crit500=coshaloMvirToRvir(1, z=zseq, Rho='crit', DeltaVir=500, ref='Planck15')
rad_mean200=coshaloMvirToRvir(1, z=zseq, Rho='mean', DeltaVir=200, ref='Planck15')
rad_mean500=coshaloMvirToRvir(1, z=zseq, Rho='mean', DeltaVir=500, ref='Planck15')
rad_vir=coshaloMvirToRvir(1, z=zseq, Rho='crit', DeltaVir='get', ref='Planck15')

plot(1, 1, type='n', xlim=c(0.01,10), ylim=c(0.8,1.55), xlab='Redshift',
ylab='M200c / M500c', log='x')
for(i in 1:length(con)){
lines(zseq, cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_crit200)/
cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_crit500), col=concol[i])
}

plot(1, 1, type='n', xlim=c(0.01,10), ylim=c(0.8,1.55), xlab='Redshift',
ylab='M200m / M500m', log='x')
for(i in 1:length(con)){
lines(zseq, cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_mean200)/
cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_mean500), col=concol[i])
}

plot(1, 1, type='n', xlim=c(0.01,10), ylim=c(0.8,1.55), xlab='Redshift',
ylab='M200m / M200c',log='x')
for(i in 1:length(con)){
lines(zseq, cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_mean200)/
cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_crit200), col=concol[i])
}

plot(1, 1, type='n', xlim=c(0.01,10), ylim=c(0.8,1.55), xlab='Redshift',
ylab='M500m / M500c', log='x')
for(i in 1:length(con)){
lines(zseq, cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_mean500)/
cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_crit500), col=concol[i])
}

plot(1, 1, type='n', xlim=c(0.01,10), ylim=c(0.8,1.55), xlab='Redshift',
ylab='Mvir / M200c',log='x')
for(i in 1:length(con)){
lines(zseq, cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_vir)/
cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_crit200), col=concol[i])
}

plot(1, 1, type='n', xlim=c(0.01,10), ylim=c(0.8,1.55), xlab='Redshift',
ylab='Mvir / M200m',log='x')
for(i in 1:length(con)){
lines(zseq, cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_vir)/
cosNFWmass_Rmax(Rho0=1, Rs=rad_crit200[1]/con[i], Rmax=rad_mean200), col=concol[i])
}

plot(zseq, rad_crit200/rad_crit500, type='l', xlim=c(0.01,10), ylim=c(0.8,1.55),
xlab='Redshift', ylab='R200 / R500', log='x')

plot(zseq, rad_mean200/rad_crit200, type='l', xlim=c(0.01,10), ylim=c(0.8,1.55), 
xlab='Redshift', ylab='Rm / Rc', log='x')

plot(zseq, rad_vir/rad_crit200, type='l', xlim=c(0.01,10), ylim=c(0.8,1.55), 
xlab='Redshift', ylab='Rvir / R200c', log='x')

plot(zseq, rad_vir/rad_mean200, type='l', xlim=c(0.01,10), ylim=c(0.8,1.55), 
xlab='Redshift', ylab='Rvir / R200c', log='x')

#R200m and R200c go either side of Rvir, so by cosmic conspiracy the mean is nearly flat:

plot(zseq, 2*rad_vir/(rad_mean200+rad_crit200), type='l', xlim=c(0.01,10),
ylim=c(0.8,1.55), xlab='Redshift', ylab='2Rvir / (R200c+R200m)', log='x')

#To check Vcirc and Vesc for a 10^12 Msun halo:

plot(0:400, cosNFWvcirc(0:400,f=1,Lunit=1e3), type='l', lty=1, xlab='R / kpc',
ylab='V / km/s', ylim=c(0,500))
lines(0:400, cosNFWvesc(0:400,f=1,Lunit=1e3), lty=2)
legend('topright', legend=c('Vel-Circ','Vel-Escape'), lty=c(1,2))
abline(v=coshaloMvirToRvir(Lunit=1e3), lty=3)