Package 'celestial'

Collection of Common Astronomical Conversion Routines and Functions

Description

Various functions for converting between commonly used coordinate systems in astronomy and making cosmological calculations.

Details

Package:	celestial
Type:	Package
Version:	1.5.6
Date:	2023-11-23
License:	GPL-3
Depends:	R (>= 3.00), RANN, NISTunits, pracma
Suggest:	plotrix, igraph

There are a number of functions included, but the most useful for astronomy conversions are the decimal degrees to DMS/HMS formats used at many telescopes: deg2dms, deg2hms, dms2deg, hms2deg. It also contains functions for various cosmological calculations (i.e. distance, volume and age for different cosmologies and redshifts).

Author(s)

Aaron Robotham

Maintainer: Aaron Robotham <[email protected]>

---

Transforms 3D cartesian coordinates to spherical coordinates

Description

Transforms 3D cartesian coordinates to spherical coordinates. The user can choose to return the spherical coordinates in degrees or radians.

Usage

car2sph(x, y, z, deg = TRUE)

Arguments

x	x values, can also contain a matrix of x, y and z (in that order).
y	y values.
z	z values
deg	Should degrees be returned (default) or radians.

Details

This is a low level function that is used for plot transformations.

Value

A data.frame is returned containing the columns long (longitude), lat (latitude) and radius.

Author(s)

Aaron Robotham

See Also

sph2car

Examples

print(car2sph(x=1,y=1,z=0,deg=TRUE))

---

Cosmological distance calculations

Description

These functions allow comoving, angular size and luminosity distances to be calculated for a given redshift, it can also return look back time. They use curvature correctly, calculated internally using the relation OmegaM+OmegaL+OmegaR+OmegaK=1, but by default they assume a flat Universe where only OmegaM needs to be specified and OmegaR=0 (so no radiation pressure at any epoch).

Usage

cosdist(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1, wprime = 0,
age=FALSE, ref, error=FALSE)
cosdistz(z=1)
cosdistzeff(zref = 1, zem = 2)
cosdista(z=1)
cosdistCoDist(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistLumDist(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistAngDist(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistAngDist12(z1=1, z2=2, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0,
w0 = -1, wprime = 0, ref)
cosdistCoDistTran(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistCoDist12ang(z1=1, z2=2, ang=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, w0 = -1, wprime = 0, inunit='deg', ref) 
cosdistLumDist12ang(z1=1, z2=2, ang=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, w0 = -1, wprime = 0, inunit='deg', ref)
cosdistAngDist12ang(z1=1, z2=2, ang=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, w0 = -1, wprime = 0, inunit='deg', ref)
cosdistzem12ang(z1=1, z2=2, ang=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, w0 = -1, wprime = 0, inunit='deg', ref)
cosdistzeff12ang(z1=1, z2=2, ang=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, w0 = -1, wprime = 0, inunit='deg', ref)
cosdistDistMod(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistAngScale(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistAngSize(z=1, Size=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1,
wprime=0, Dim=1, Dist='Co',  outunit='deg', ref)
cosdistAngArea(z=1, Size=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1,
wprime=0, Dim=2, Dist='Co',  outunit='deg2', ref)
cosdistCoVol(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistHubTime(H0=100)
cosdistUniAgeNow(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistUniAgeAtz(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistTravelTime(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1,
wprime = 0, ref)
cosdistRelError(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1, wprime = 0,
ref)
cosdistCrit(z_lens=1, z_source=2, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0,
w0 = -1, wprime = 0, ref)

Arguments

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 as defined at z=0 (default is 0.3).
OmegaL	Omega Lambda as defined at z=0 (default is for a flat Universe with OmegaL = 1-OmegaM-OmegaR = 0.7).
OmegaR	Omega Radiation as defined at z=0 (default is 0, but OmegaM/3400 is typical).
w0	The value of dark energy equation of state as defined at z=0. See cosgrow for more details.
wprime	The evolution term that governs how the dark energy equation of state evolves with redshift. See cosgrow for more details.
age	Flag for cosdist function to return age or not- this slows calculation, so is by default turned off.
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. This overrides any other settings for H0, OmegaM and OmegaL. If OmegaR is missing from the reference set then it is inherited from the function input (0 by default). See cosref for details.
error	Flag for cosdist to calculate the relative error for distance/age values.
z1	Redshift for object 1, where z1 must be > -1 (can be a vector) and less than z2.
z2	Redshift for object 2, where z2 must be > -1 (can be a vector) and greater than z1.
zref	Redshift for the reference object, i.e. the object that we caste as the observer of another object at zem.
zem	Redshift for the emitting object, i.e. the object that we caste as being observed by another object at zref.
z_lens	Redshift, where z_lens must be > -1 (can be a vector) and z_lens < z_source.
z_source	Redshift, where z_source must be > -1 (can be a vector) and z_lens < z_source.
ang	The observed angular separation between object 1 and object 2 in degrees.
Size	The 1D size of the object (i.e. diameter or total length) in Mpc. Either comoving or angular, as specified by Dist. For cosdistAngArea this is always taken to be the diameter of either the projected or 3D object. The Size specified should be assuming the same cosmology as provided, so be careful with your H0!
Dim	Specifies whether the object being considered is 1D (a line) 2D (e.g. a face on galaxy) or 3D (e.g. dark matter halo). This makes a very small modification to the geometry used (tan of 1D/2D and sin for 3D), but is only noticeable for large structures at low redshifts.
Dist	Determines the distance type of the specified Size, i.e. angular / physical distances (Ang) or with respect to comoving distances (Co).
inunit	The units of angular coordinate provided for ang. Allowed options are deg for degrees, amin for arc minutes, asec for arc seconds, and rad for radians.
outunit	For cosdistAngSize units of angular coordinate output. Allowed options are deg for degress (default), amin for arc minutes, asec for arc seconds, and rad for radians. For cosdistAngArea units of angular area output. Allowed options are deg2 for square degrees (default), amin2 for square arc minutes, asec2 for square arc seconds and rad2 or sr for steradians.

Details

Functions are largely based on D. W. Hogg et al. 1999 and Wright et al. 2006.

Negative value of z> -1 are allowed, which produces future predictions based on present day cosmology.

cosdistAngDist12 is only available for OmegaK>=0.

Value

cosdist

Returns a data.frame (even if only 1 redshift if requested) with the following columns:

z	Requested redshift
a	Universe expansion factor, as given by a=1/(1+z)
CoDist	Line-of-sight (i.e. radial) comoving distance in units of Mpc
LumDist	Luminosity distance in units of Mpc
AngDist	Angular diameter distance in units of Mpc
CoDistTran	Transverse comoving distance in units of Mpc
DistMod	The distance modulus used where AbsMag = ApMag - DistMod, and DistMod = 5log10(LumDist)+25 in units of mag
AngScale	Physical projected scale of an object at z in units of kpc/arcsec
CoVol	Comoving volume of Universe within z in units of Gpc^3

If age=TRUE is set then additional age-related information is calculated for each z as extra columns:

HubTime	Approximate Hubble age of the Universe in units of Gyrs
UniAgeNow	Age of the Universe now in units of Gyrs
UniAgeAtz	Age of the Universe at the specified redshift (z) in units of Gyrs
TravelTime	Light travel time from the specified redshift (AKA look back time) in units of Gyrs

If error=TRUE is set then the relative error for distance/age values is calculated for each z as an extra column:

RelError	Relative error of the distance/age integrals (this is the main source of error in the calculations)

The outputs of the standalone functions are:

cosdistz	Returns the input redshift (only included for clarity).
cosdistzeff	Returns the apparent redshift that the object at zref will observe the object at zem for the Universe age that zref is observed to have now. This is given by $(1+zem)/(1+zref)$.
cosdista	Returns the Universe expansion factor, as given by $a=1/(1+z)$.
cosdistCoDist	Returns the line-of-sight (i.e. radial) comoving distance in units of Mpc. For a flat Universe (OmegaK=0) this is exactly the samething as the transverse comoving distance, and by extension it is also the proper motion distance.
cosdistLumDist	Returns the luminosity distance in units of Mpc.
cosdistAngDist	Returns the angular diameter distance in units of Mpc.
cosdistAngDist12	Returns the radial angular diameter distance separation in units of Mpc between objects at z1 and z2 that have small angular separations on sky.
cosdistCoDistTran	Returns the transverse comoving distance in units of Mpc. This is equivilant to the proper motion distance for all values of Universe curvature (OmegaK !=0), and is the same thing as the line-of-sight comoving distance for a flat Universe (OmegaK=0).
cosdistCoDist12ang	Returns the total comoving distance in units of Mpc between objects at z1 and z2 with a separation ang. This works for curved cosmologies (i.e. OmegaK!=0) and for large radial and tangential separations. For small separations at a certain value of z for both objects the result is very similar to cosdistCoDistTran(z)*sin(ang*pi/180). This function was mostly extracted from Eqn 3.19 in Peacock (1999).
cosdistLumDist12ang	Returns the total luminosity distance in units of Mpc between objects at z1 and z2 with a separation ang. This is equal to cosdistCoDist12ang*(1+zeff), where zeff is the apparent redshift that the object at z1 will observe the object at z2 for the Universe age that z1 is observed to have now. See cosdistCoDist12ang for details.
cosdistAngDist12ang	Returns the total angular diameter distance in units of Mpc between objects at z1 and z2 with a separation ang. This is equal to cosdistCoDist12ang/(1+zeff), where zeff is the apparent redshift that the object at z1 will observe the object at z2 for the Universe age that z1 is observed to have now. See cosdistCoDist12ang for details.
cosdistzem12ang	Returns the apparent redshift that the object at z1 would observe the object at z2 to be for our current Universe age. See cosdistCoDist12ang for details.
cosdistzeff12ang	Returns the apparent redshift that the object at z1 would observe the object at z2 to be for the Universe age that z1 is observed to have now. See cosdistCoDist12ang for details.
cosdistDistMod	Returns the distance modulus used where AbsMag = ApMag - DistMod, and DistMod = 5log10(LumDist)+25 in units of mag.
cosdistAngScale	Returns the physical projected scale of an object at z in units of kpc/arcsec.
cosdistAngSize	Returns the angular size (length or diameter) of an object (by default in degrees).
cosdistAngArea	Returns the angular area of an object (by default degrees^2), taking the specified Size to be the diameter.
cosdistCoVol	Returns the comoving volume of Universe within z in units of Gpc^3.
cosdistHubTime	Returns the approximate Hubble age of the Universe in units of Gyrs.
cosdistUniAgeNow	Returns the age of the Universe now in units of Gyrs.
cosdistUniAgeAtz	Returns the age of the Universe at the specified redshift (z) in units of Gyrs.
cosdistTravelTime	Returns the light travel time from the specified redshift (AKA look back time) in units of Gyrs.
cosdistRelError	Returns the relative error of the distance/age integrals (this is the main source of error in the calculations).
cosdistCrit	Returns the critical surface mass density, SigmaC (see also cosNFW).

Author(s)

Aaron Robotham

References

Based on the equations in:

Davis T.M. & Lineweaver, Charles H., 2004, PASA, 21, 97

Hogg D.W., 1999, arXiv, 9905116

Liske J., 2000, MNRAS, 319, 557L

Peacock J.A., 1999, Cosmological Physics, Cambridge University Press

Wright E.L., 2006, PASP, 118, 1711

See Also

cosvol, cosmap, cosgrow, cosref, cosNFW

Examples

## Not run: 
cosdist(0.3,70,age=TRUE)
cosdist(0.3,70,age=TRUE,ref='Planck')
cosdistz(0.3)
cosdista(0.3)
cosdistCoDist(0.3,70)
cosdistLumDist(0.3,70)
cosdistAngDist(0.3,70)
cosdistAngDist12(0.3,0.5,70)
cosdistCoDistTran(0.3,70)
cosdistCoDist12ang(0,2,10)
cosdistDistMod(0.3,70)
cosdistAngScale(0.3,70)
cosdistAngSize(0.3,1,70)
cosdistCoVol(0.3,70)
cosdistHubTime(70)
cosdistUniAgeNow(0.3,70)
cosdistUniAgeAtz(0.3,70)
cosdistTravelTime(0.3,70)
cosdistRelError(0.3)
cosdistCrit(0.3,0.5,70)
cosdistzeff(1,2)
cosdistzem12ang(1,2)
cosdistzeff12ang(1,2)

#A check of the comoving separation between objects function:

cosdistCoDistTran(2,OmegaM = 0.3, OmegaL=1)*sin(pi/180)
cosdistCoDist12ang(2,2,ang=1,OmegaM=0.3,OmegaL=1)

#Very close, however cosdistCoDist12ang lets us go further:

cosdistCoDist12ang(1,2,ang=10,OmegaM=0.3,OmegaL=1)
cosdistCoDist12ang(2,2,ang=180,OmegaM=0.3,OmegaL=1)

#The second number should be be the same as:

cosdistCoDist(2,OmegaM=0.3,OmegaL=1)*2

#Example 1 by John Peacock for EDS Universe (answer should be nearly 3):

cosdistzem12ang(3,4,56.4,H0=100,OmegaM=1,OmegaL=0)

#Example 2 by John Peacock for EDS Universe (answer should be nearly 2995 Mpc/h):

cosdistCoDist12ang(3,4,56.4,H0=100,OmegaM=1,OmegaL=0)

#Example 3 by John Peacock for Milne Universe (answer should be nearly 5294 Mpc/h):

cosdistCoDist12ang(3,4,56,H0=100,OmegaM=0,OmegaL=0)

#Example 4 by John Peacock for Milne Universe (answer should be nearly 4.846):

cosdistzeff12ang(3,4,56,H0=100,OmegaM=0,OmegaL=0)

#Example 5 by John Peacock for Milne Universe (answer should be nearly 364 Mpc/h):

cosdistAngDist12ang(3,4,56,H0=100,OmegaM=0,OmegaL=0)

#Nice plot of distance estimates:

redshifts=seq(0,3,by=0.01)
plot(redshifts, cosdistCoDist(redshifts, ref='planck'), type='l', col='darkgreen',
xlab='Redshift / z', ylab='Distance / Mpc')
lines(redshifts, cosdistLumDist(redshifts, ref='planck'), col='red')
lines(redshifts, cosdistAngDist(redshifts, ref='planck'), col='blue')
legend('topleft', legend=c('Comoving Distance', 'Luminosity Distance', 'Angular Diameter Distance'),
col=c('darkgreen', 'red', 'blue'),lty=1)

plot(redshifts, cosdistTravelTime(redshifts, ref='planck'), type='l',
xlab='Redshift / z', ylab='Light travel time / Yrs')

#Actual time example (Figure 1 of Davis & Lineweaver 2004)
zseq=10^seq(-2,6,len=1e3)-1
dists=cosdistCoDist(zseq, ref='737')*0.00326
times=cosdistTravelTime(zseq, ref='737')
plot(dists, times, type='l', xlab='Comoving Distance / Glyr',
ylab='Time / Gyr')
abline(v=0, h=0, lty=1)
abline(h=c(min(times), max(times)), lty=2)
abline(v=c(min(dists), max(dists)), lty=2)

#Conformal time example (Figure 1 of Davis & Lineweaver 2004):
#Mpc to Glyr conversion is 0.00326

zseq=10^seq(-2,6,len=1e3)-1
dists=cosdistCoDist(zseq, ref='737')*0.00326
plot(dists, dists, type='l',
xlab='Comoving Distance / Glyr', ylab='Conformal Time / Gyr')
abline(v=0, h=0, lty=1)
abline(h=c(min(dists), max(dists)), lty=2)
abline(v=c(min(dists), max(dists)), lty=2)

## End(Not run)

---

Cosmological growth and evolution calculations

Description

These functions allow various properties of the expansion of the Universe to be calculated: e.g. OmegaM/OmegaL/OmegaR/OmegaK for ay redshift, growth rate and growth factor, sigma8, and RhoCrit and RhoMean. They use curvature correctly, calculated internally using the relation OmegaM+OmegaL+OmegaR+OmegaK=1, but by default they assume a flat Universe where only OmegaM needs to be specified and OmegaR=0 (so no radiation pressure at any epoch).

Usage

cosgrow(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0 = -1, wprime = 0,
  Sigma8=0.8, fSigma8=FALSE, Dist='Co',
  Mass='Msun', ref)
  cosgrowz(z = 1)
  cosgrowa(z = 1)
  cosgrowH(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0,
  ref)
  cosgrowCoVel(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1,
  wprime=0, ref)
  cosgrowPecVel(z=1, zob=1)
  cosgrowOmegaM(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0, ref)
  cosgrowOmegaL(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0, ref)
  cosgrowOmegaR(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0, ref)
  cosgrowOmegaK(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0, ref)
  cosgrowDecelq(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0, ref)
  cosgrowEoSwDE(z=1, w0=-1, wprime=0)
  cosgrowRhoDE(z=1,w0=-1, wprime=0, rhoDE=1)
  cosgrowFactor(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0, ref)
  cosgrowRate(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0,
  Sigma8=0.8, fSigma8=FALSE, ref)
  cosgrowSigma8(z=1, OmegaM=0.3,OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0,
  Sigma8=0.8, ref)
  cosgrowFactorApprox(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0,
  ref)
  cosgrowRateApprox(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0,
  Sigma8=0.8, fSigma8=FALSE, ref)
  cosgrowSigma8Approx(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1, wprime=0,
  Sigma8=0.8, ref)
  cosgrowRhoCrit(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1,
  wprime=0, Dist='Co', Mass='Msun', ref)
  cosgrowRhoMean(z=1, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, w0=-1,
  wprime=0, Dist='Co', Mass='Msun', ref)
  cosgrowDeltaVir(z=1, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0, ref)

Arguments

z	Cosmological redshift, where z must be > -1 (can be a vector).
zob	Observed redshift, where z must be > -1 (can be a vector). Essentially a combination of the cosmological redshift z and the peculiar velocity of the objects with respect to this.
H0	Hubble constant as defined at z=0 (default is H0=100 (km/s)/Mpc).
OmegaM	Omega Matter as defined at z=0 (default is 0.3).
OmegaL	Omega Lambda as defined at z=0 (default is for a flat Universe with OmegaL = 1-OmegaM-OmegaR = 0.7).
OmegaR	Omega Radiation as defined at z=0 (default is 0, but ~OmegaM/3400 is typical).
w0	The value of dark energy equation of state as defined at z=0.
wprime	The evolution term that governs how the dark energy equation of state evolves with redshift.
rhoDE	The z=0 reference energy density for dark energy.
Sigma8	The value of Sigma8 to use if fsigma8=TRUE (by default this is a reasonable 0.8 for simplicity).
fSigma8	Logical to express whether the growth rate of structure calculated by cosgrow, cosgrowRate or cosgrowRateApprox is given as f*Sigma8 (TRUE) or simply f (FALSE). This is useful for redshift space distortion comparisons (RSD), since RSD strictly measures f*Sigma8.
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), with respect to comoving distances (Co) or with respect to physical metres (m).
Mass	Determines the mass type, i.e. whether Rho critical energy or mean mass densities are calculated with respect to solar masses (Msun) or with respect to kilograms (kg).
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. If ref=137 or ref=737 no specific Sigma8 is assumed, instead Sigma8 is set to whatever the input value is set to (by default this is 0.8).

Details

The above functions are heavily based on the equations in Hamilton A.J.S., 2001, MNRAS 322 419 and Lahav O., et al., 1991, MNRAS, 251, 136.

Negative value of z> -1 are allowed, which produces future predictions based on present day cosmology.

The approximation routines are generally accurate to sub 1 percent, and since they do not involve numerical integration they are substantially faster when computing large grids of numbers, i.e. they are recommended for plots, since the accuracy is sub the line width.

Value

cosgrow

Returns a data.frame (even if only 1 redshift if requested) with the following parameters evaluated at the respective redshift/s:

z	Requested redshift
a	Universe expansion factor, as given by a=1/(1+z)
H	Hubble expansion rate in units of (km/s)/Mpc
CoVel	Cosmological recession velocity in units of km/s
OmegaM	Omega Matter
OmegaL	Omega Lambda
OmegaR	Omega Radiation
OmegaK	Omega K(c)urvature
Decelq	Traditional deceleration parameter q
Factor	Exact growth factor (g, see cosgrowFactor below for details)
Rate	Exact growth rate (f or f*Sigma8, see cosgrowRate below for details)
Sigma8	Power spectrum fluctuation amplitude on the scale 8 Mpc/z
RhoCrit	Critical energy density of the Universe at z, where $\rho_{crit}=(3.H(z)^2)/(8.\pi.G)$, in units of $M_{\odot}/Mpc^3$
RhoMean	Mean mass density of the Universe at z, where $\rho_{mean}=\rho_{crit}.\Omega_{M}(z)$, in units of $M_{\odot}/Mpc^3$

The outputs of the standalone functions are:

cosgrowz	Returns the input redshift (only included for clarity).
cosgrowa	Returns the Universe expansion factor, as given by a=1/(1+z).
cosgrowH	Returns the value of the Hubble expansion rate at z, in units of km/s/Mpc.
cosgrowCoVel	Returns the value of the cosmological recession velocity of the object in units of km/s.
cosgrowPecVel	Returns the value of the peculiar velocity of the object in units of km/s.
cosgrowOmegaM	Returns the value of Omega Matter at z.
cosgrowOmegaL	Returns the value of Omega Lambda at z.
cosgrowOmegaR	Returns the value of Omega Radiation at z.
cosgrowOmegaK	Returns the value of Omega K(c)urvature at z.
cosgrowDecelq	Returns the traditional deceleration parameter q, given by q=OmegaM/2+Omegar-OmegaL.
cosgrowEoSwDE	Returns w for the dark energy equation of state, where $P=w.\rho_{DE}.c^2$ and $w=w_0+2.w'.(1-1/(1+z))$, as described in Wright (2006).
cosgrowRhoDE	Returns the energy density for dark energy, given by $\rho_{DE}.e^{-6.w'.(1-1/(1 + z))}/(1 + z)^{-(3 + 3.w_0 + 6.w')}$, as described in Wright (2006)..
cosgrowFactor	Returns the exact value of the growth factor (typically referred to as 'g' in the astronomy literature), at z. This is defined such that it equals 1 at z=Inf and is less than 1 at lower z.
cosgrowRate	Returns either the true (typically referred to as 'f' in the astronomy literature) or RSD type (f*Sigma8) value of the growth rate of structure, at z. This is defined such that it equals 1 at z=Inf and is less than 1 at lower z.
cosgrowSigma8	Returns the power spectrum fluctuation amplitude on the scale 8 Mpc/z at z, and is unitless.
cosgrowFactorApprox	Returns the approximate value of the growth factor (typically referred to as 'g' in the astronomy literature), at z. This is defined such that it equals 1 at z=Inf and is less than 1 at lower z.
cosgrowRateApprox	Returns either the approximate true (typically referred to as 'f' in the astronomy literature) or approximate RSD type (f*Sigma8) value of the growth rate of structure, at z. This is defined such that it equals 1 at z=Inf and is less than 1 at lower z.
cosgrowSigma8Approx	Returns the approximate power spectrum fluctuation amplitude on the scale 8 Mpc/z at z, and is unitless.
cosgrowRhoCrit	Returns the critical energy density of the Universe at z, where $\rho_{crit}=(3.H(z)^2)/(8.\pi.G)$, in units of $M_{\odot}/Mpc^3$.
cosgrowRhoMean	Returns the mean mass density of the Universe at z, where $\rho_{mean}=\rho_{crit}.\Omega_{M}(z)$, in units of $M_{\odot}/Mpc^3$.
cosgrowDeltaVir	Returns the delta-critical virial radius overdensity criterion for a range of flat Universes with varying OmegaM. Taken from Eqn. 6 of Bryan & Norman (1998).

Note

The difference between RhoCrit and RhoMean at z=0 is simply RhoMean=RhoCrit*OmegaM. Corrected for 1/(1+z)^3 RhoMean stays constant with redshift (as it should- to first order we do not gain or lose mass within a comoving volume).

The growth rate and growth factor does not make use of OmegaR in the cosgrow function, hence OmegaR cannot be provided in the individual functions. This is because correctly accounting for the effect of radiation pressure before the surface of last scattering (z~1100) on the growth rate of structure is highly complex, and beyond the scope of this package. In the case of cosgrow, even if OmegaR is specified it is, in effect, set to zero when making growth factor and rate calculations.

The evolution of the dark matter equation of state (w) is parameterised as described in Wright (2006).

It is important to remember that H is in physical units for both the numerator and denominator (i.e. 'proper' at a given redshift, so the units are km/s / pMpc). To ask the question "is the Universe accelerating?" is to really ask "is the expansion factor accelerating?". This requires the denominator to be in comoving units (so rescaling for proper distances today) and measuring the differential with time or redshift. You will only find an accelerating Universe when dividing H(z)/(1+z)! See the examples to see how we can find this location, and that it is consistent with the start of acceleration calculated from the decleration parameter (q) directly.

Author(s)

Aaron Robotham

References

Based on the equations in:

Bryan & Norman, 1998, ApJ, 495, 80

Davis T.M. & Lineweaver, Charles H., 2004, PASA, 21, 97

Davis T.M. & Scrimgeour M.I., 2014, MNRAS, 442, 1117

Hamilton A.J.S., 2001, MNRAS 322 419

Lahav O., et al., 1991, MNRAS, 251, 136

Peacock J.A., 1999, Cosmological Physics, Cambridge University Press

Wright E.L., 2006, PASP, 118, 1711

See Also

cosvol, cosmap, cosdist, cosref, coshalo

Examples

cosgrow(0.3)
  cosgrow(0.3,ref='Planck')
  cosgrowz(0.3)
  cosgrowa(0.3)
  cosgrowH(0.3)
  cosgrowCoVel(0.3)
  cosgrowPecVel(0.3,0.31)
  cosgrowOmegaM(0.3)
  cosgrowOmegaL(0.3)
  cosgrowOmegaK(0.3)
  sum(cosgrowOmegaM(0.3)+cosgrowOmegaL(0.3)+cosgrowOmegaK(0.3)) #Still 1.
  cosgrowDecelq(0.3)
  cosgrowEoSwDE(0.3)
  cosgrowFactor(0.3)
  cosgrowFactorApprox(0.3) #Approximation better than 1% for reasonable cosmologies.
  cosgrowRate(0.3)
  cosgrowRateApprox(0.3) #Approximation better than 1% for reasonable cosmologies.
  cosgrowRhoCrit(0.3)
  cosgrowRhoMean(0.3)
  cosgrowRhoMean(0)-cosgrowRhoMean(2,Dist='Ang')/(1+2)^3 #Mass is conserved in co-vol
  cosgrowRhoMean(0)-cosgrowRhoMean(10,Dist='Co') #Mass is conserved in co-vol
  
  # Various recessional velocities (see Figure 2 of Davis & Lineweaver 2004):
  
  plot(10^seq(-1,4,by=0.01), cosgrowCoVel(10^seq(-1,4,by=0.01), ref='planck')
  /299792.458, type='l', log='x', xlab='z', ylab='Cosmological Recession Velocity / c')
  lines(10^seq(-1,4,by=0.01), cosgrowPecVel(0,10^seq(-1,4,by=0.01))/299792.458, col='red')
  lines(10^seq(-1,4,by=0.01), 10^seq(-1,4,by=0.01), col='blue')
  abline(h=1,v=1.5,lty=2)
  legend('topleft', legend=c('GR', 'SR', 'Approx (cz)', 'Superluminal'), lty=c(1,1,1,2),
  col=c('black','red','blue','black'))
  
  # Comparison of the various energy densities that make up the Universe for Planck 2013:
  
  plot(cosdistUniAgeAtz(10^seq(-3,4.9,by=0.1), ref='Planck')*1e9,
  cosgrowRhoCrit(10^seq(-3,4.9,by=0.1), ref='Planck', Dist='m', Mass='kg')*
  cosgrowOmegaR(10^seq(-3,4.9,by=0.1), ref='Planck'), type='l',log='xy',
  xlab='Years since Universe formed', ylab=expression('Energy Density'*(kg/m^3)))
  
  lines(cosdistUniAgeAtz(10^seq(-3,4.9,by=0.1), ref='Planck')*1e9,
  cosgrowRhoCrit(10^seq(-3,4.9,by=0.1), ref='Planck', Dist='m', Mass='kg')*
  cosgrowOmegaM(10^seq(-3,4.9,by=0.1), ref='Planck'), col='red')
  
  lines(cosdistUniAgeAtz(10^seq(-3,4.9,by=0.1), ref='Planck')*1e9,
  cosgrowRhoCrit(10^seq(-3,4.9,by=0.1), ref='Planck', Dist='m', Mass='kg')*
  cosgrowOmegaL(10^seq(-3,4.9,by=0.1), ref='Planck'), col='blue')

  abline(v=cosdistUniAgeAtz(0.33,ref='Planck')*1e9,lty=2) # Matter = Vacuum
  abline(v=cosdistUniAgeAtz(3391,ref='Planck')*1e9,lty=2) # Matter = Radiation
  
  legend('topright', legend=c('Radiation Energy Density', 'Matter Energy Density',
  'Vacuum Energy Density'), lty=1, col=c('black','red','blue'))
  
  # Where's the acceleration?
  plot(seq(0,2,len=1e3),cosgrowH(seq(0,2,len=1e3)),type='l',xlab='z',
  ylab='H(z) / km/s / pMpc')
  # There it is!
  plot(seq(0,2,len=1e3),cosgrowH(seq(0,2,len=1e3))/(1+seq(0,2,len=1e3)),
  type='l',xlab='z',ylab='H(z) / km/s / cMpc')
  #When does it start accelerating?
  accel.loc=which.min(abs(cosgrowDecelq(seq(0,2,len=1e3))))
  abline(v=seq(0,2,len=1e3)[accel.loc],lty=2)

---

Virial halo conversion functions

Description

All 6 Virial parameter conversion functions. Each can map precisely to the other as a one paramter function.

Usage

coshaloMvirToSigma(Mvir=1e+12, z=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, Rho='crit', Dist='Co', DeltaVir=200, Munit=1, Lunit=1e6, Vunit=1e3, Dim=3, ref)
coshaloSigmaToMvir(Sigma=230, z=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, Rho='crit', Dist='Co', DeltaVir=200, Munit=1, Lunit=1e6, Vunit=1e3, Dim=3, ref)
coshaloMvirToRvir(Mvir=1e12, z=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, Rho='crit', Dist='Co', DeltaVir=200, Munit=1, Lunit=1e6, Vunit=1e3, Dim=3, ref)
coshaloRvirToMvir(Rvir=162.635, z=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, Rho='crit', Dist='Co', DeltaVir=200, Munit=1, Lunit=1e6, Vunit=1e3, Dim=3, ref)
coshaloSigmaToRvir(Sigma=230, z=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, Rho='crit', Dist='Co', DeltaVir=200, Munit=1, Lunit=1e6, Vunit=1e3, Dim=3, ref)
coshaloRvirToSigma(Rvir=162.635, z=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, Rho='crit', Dist='Co', DeltaVir=200, Munit=1, Lunit=1e6, Vunit=1e3, Dim=3, ref)
coshaloSigmaToTvir(Sigma=230, Vunit=1e3, Tunit='K', Type='halo', Dim=3)

Arguments

Mvir	Mass within virial radius in units of 'Munit'.
Sigma	Velocity dispersion (3D) within virial radius in units of 'Vunit'. For coshaloSigmaToTvir the Sigma input should be the virial Sigma which can be found by setting DeltaVir='get' in the the other coshalo functions.
Rvir	Virial radius (3D) in units of 'Lunit'.
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 as defined at z=0 (default is 0.3).
OmegaL	Omega Lambda as defined at z=0 (default is for a flat Universe with OmegaL = 1-OmegaM = 0.7).
OmegaR	Omega Radiation as defined at z=0 (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. If set to 'get' it will estimate the required DeltaVir for a virial collapse using the cosgrowDeltaVir function.
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.
Type	Specify the 'halo' or 'gas' temperature to be computed.
Tunit	Specify the output temperature to be Kelvin ('K'), 'eV' or 'keV'.
Dim	The dimensional type for the halo (either the 2 or 3). 3 (default) means quantities are intrinsic 3D values. 2 means quantities are for projected systems (i.e. radius and velocity dispersion are compressed). From comparisons to simulations (so NFW, c~5 halos) Rvir[proj]=Rvir[3D]/1.37 and Sigma[proj]=Sigma[3D]/sqrt(3). The former has dependence on the halo profile (so is approximate), whereas the latter is a dimensionality argument that should hold for any virialised system. Note that for projected systems Sigma is measured along one dimension: the line-of-site.
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 allow for various analytic conversions between the 3 major properties related to virial radius: the mass, velocity dispresion and size. The default properties calculate properties for 1e12 Msun halos and assume masses in Msun, velocities in km/s and distances in Kpc.

Value

coshaloMvirToSigma	Outputs approximate velocity dispersion (in units of Vunit) given mass (this is exactly the escape velocity at Rvir).
coshaloSigmaToMvir	Outputs mass (in units of Munit) given velocity dispersion.
coshaloMvirToRvir	Outputs radius (in units of Lunit) given mass.
coshaloRvirToMvir	Outputs mass (in units of Munit) given radius.
coshaloSigmaToRvir	Outputs radius (in units of Lunit) given velocity dispersion.
coshaloRvirToSigma	Outputs approximate velocity dispersion (in units of Vunit) given radius (this is exactly the escape velocity at Rvir).
coshaloSigmaToTvir	Output temperture (in units of Tunit) given velocity dispersion. Based on Eqns. 3/7/8/9 of Bryan & Norman (1998).

Author(s)

Aaron Robotham, Chris Power

References

coshaloSigmaToTvir based on the equations in:

Bryan & Norman, 1998, ApJ, 495, 80

See Also

cosvol, cosmap, cosdist, cosgrow, cosNFW

Examples

coshaloMvirToSigma(1e13) # Velocity in km/s
coshaloMvirToSigma(1e13, Vunit=1) # Velocity in m/s
coshaloSigmaToMvir(coshaloMvirToSigma(1e13, Vunit=1),Vunit=1)
coshaloMvirToRvir(1e13) #Radius in kpc
coshaloSigmaToRvir(coshaloMvirToSigma(1e13, Vunit=1),Vunit=1)

#Some sanity checks

rho_crit200=cosgrowRhoCrit(z=0)*200 #200 times rho critical at z=0
rho_mean200=cosgrowRhoMean(z=0)*200 #200 times rho mean at z=0
#For a 10^12 Msun/h halo, the radius in Mpc/h where the contained density equals rho_crit*200
rad_crit200=(1e12/rho_crit200*3/4/pi)^(1/3)
coshaloMvirToRvir(1e12,Lunit=1e6)-rad_crit200 # ~0 as expected
#For a 10^12 Msun/h halo, the radius in Mpc/h where the contained density equals rho_crit*200
rad_mean200=(1e12/rho_mean200*3/4/pi)^(1/3) # ~0 as expected
coshaloMvirToRvir(1e12,Lunit=1e6,Rho='mean')-rad_mean200

---

Cosmological Mapping Functions

Description

Functions for mapping from one arbitrary cosmological parameter to another. This includes the provision of a generic interpolation function and another exact value lookup.

Usage

cosmapval(val=50, cosparam="CoVol", H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0,
w0=-1, wprime=0, Sigma8=0.8, fSigma8=FALSE, zrange=c(-0.99,100), res=100, iter=8,
out='cos', degen='lo', ref)
cosmapfunc(cosparamx="CoVol", cosparamy="z", H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR,
OmegaR=0, w0=-1, wprime=0, Sigma8=0.8, fSigma8=FALSE, zrange=c(0,20), step='z', res=100,
degen='lo', ref)

Arguments

val	The value/s to be mapped from parameter cosparamx to parameter cosparamy (this can be a vector or a single number).
cosparam	Cosmological parameter, must be one of: z, a, CoDist, LumDist, CoDistTran, DistMod, AngScale, CoVol, UniAgeAtz, TravelTime (see cosdist help for further description of these) H, CoVel, OmegaM, OmegaL, OmegaK, Factor, Rate, RhoCrit (see cosgrow help for further description of these). Note that AngDist and AngSize are not an option for cosparam due to degenerate redshift solutions causing problems with the approxfun mapping.
cosparamx	Cosmological parameter, must be one of: z, a, CoDist, LumDist, CoDistTran, DistMod, AngScale, CoVol, UniAgeAtz, TravelTime (see cosdist help for further description of these) H, CoVel, OmegaM, OmegaL, OmegaK, Factor, Rate, RhoCrit (see cosgrow help for further description of these). Note that AngDist and AngSize are not an option for cosparamx due to degenerate redshift solutions causing problems with the approxfun mapping.
cosparamy	Cosmological parameter, must be one of: z, a, CoDist, LumDist, AngDist, CoDistTran, DistMod, AngScale, CoVol, UniAgeAtz, TravelTime (see cosdist help for further description of these) H, CoVel, OmegaM, OmegaL, OmegaK, Factor, Rate, RhoCrit (see cosgrow help for further description of these).
H0	Hubble constant as defined at z=0 (default is H0=100 (km/s)/Mpc).
OmegaM	Omega Matter as defined at z=0 (default is 0.3).
OmegaL	Omega Lambda as defined at z=0 (default is for a flat Universe with OmegaL = 1-OmegaM = 0.7).
OmegaR	Omega Radiation as defined at z=0 (default is 0, but OmegaM/3400 is typical).
w0	The value of dark energy equation of state as defined at z=0. See cosgrow for more details.
wprime	The evolution term that governs how the dark energy equation of state evolves with redshift. See cosgrow for more details.
Sigma8	The value of Sigma8 as defined at z=0 to use if fsigma8=TRUE (by default this is a reasonable 0.8 for simplicity).
fSigma8	Logical to express whether the growth rate of structure calculated by cosgrow, cosgrowRate or cosgrowRateApprox is given as f*Sigma8 (TRUE) or simply f (FALSE). This is useful for redshift space distortion comparisons (RSD), since RSD strictly measures f*Sigma8.
zrange	Lower and upper z limits that the approxfun mapping is generated over (increase range if default is not sufficient, and decrease if it is wasteful, i.e. the possible redshift window is known to be quite narrow).
step	The type of stepping used. Allowed values are 'z' (uniform stepping in z), 'logz' (uniform stepping in log10(1+z) and expansion factor 'a' (uniform stepping in a=1/(1+z)). Default is z. For mappings using time (UniAgeNow, UniAgeAtz, TravelTime) or comoving quantities (CoDist, CoDistTran, CoVol) or distance modulus (DistMod) 'a' or 'logz' map the numeric range more uniformly. This is because a and log10(1+z) are approximately linear in light travel time (positive and negative correlation respectively), and typically they have better behaviour than stepping uniformly in z directly.
res	The resolution of steps. Larger numbers will be more accurate, but will be slower to compute.
iter	The number of iterations to make when calculating the exact location of a given cosmological parameter when using cosmapval.
out	Either out='cos', in which case the output is a data.frame containing the output of cosdist and cosgrow for the specified input vaue, or out='z', in which case the output is a vector of the corresponding redshift (z) values.
degen	In cases where solutions are degenerate (multiple y solutions for a single x), this specifies whether to calculate the lower y solution (degen='lo'), or the higher y solutions (degen='hi').
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. If ref=137 or ref=737 no specific Sigma8 is assumed, instead Sigma8 is set to whatever the input value is set to (by default this is 0.8).

Details

The default zrange and res should be sufficient for most reasonable cosmologies if the approximate redshift location of the region to be mapped is entirely unknown.

Predictions into the future are possible if val is set to negative (distance and volume parameters) or below their present day value (age and growth parameters). However, many potential values are outside of the asymptotic limits, e.g. using the default 737 cosmology H is tending to 83.666, i.e. it will fail if you request H=83 but work if your resuest H=84.

The default res and iter for cosmapval is appropriate for most mappings with -0.99 < z < 100 using a fiducial 737 cosmology. If this proves insufficient (this should be obvious from error column) then increase both of these. Overall accuracy goes as res^iter.

Value

If out='cos', cosmapval contains the concatenation of the cosdist (with age=TRUE and error=TRUE) and cosgrow functions for parameter 'cosparam' at value 'val'. The 'z' and 'a' columns are only included once (from the output of cosdist). See cosdist and cosgrow for information on the cosdist and cosgrow outputs. If out='z', then cosmapval merely returns the corresponding redshifts.

The cosmapval output (when out='cos') includes an additional final column named 'MapError' which gives the approximate relative error of the values returned compared to the desired lookup location. Smaller is obviously better, but at the cost of computational time.

cosmapfunc uses base R approxfun to map cosparamx onto cosparamy between zrange[1] and zrange[2] in uniform steps of expansion factor (a=1/(1+z)). cosmofunc returns the output function created by approxfun.

Author(s)

Aaron Robotham

References

Based on the equations in:

Hogg D.W., 1999, arXiv, 9905116

Wright E.L., 2006, PASP, 118, 1711

See Also

cosdist, cosvol, cosgrow

Examples

## Not run: 
tempfunc=cosmapfunc('CoVol', 'UniAgeAtz')
tempfunc(50)

cosmapval(50:60, 'CoVol')

#A future prediction:

cosmapval(59, 'H', H0=70)

## End(Not run)

---

Cosmology parameter data included in celestial package.

Description

cosref: Cosmology H0 / OmegaM / OmegaL / OmegaR (via OmegaM/zeq) and Sigma8 parameters taken from Planck (13/15/18), WMAP (1/3/5/9), Millennium Simulation and GiggleZ. Not all of these exist for each source, so NA values are used in these cases.

For Planck we use the second column of the main cosmology table, which does no use external data.

Usage

data(cosref)

Details

The included data is a table of the following cosmological parameters:

Ref	H0	OmegaM	OmegaL	OmegaR	Sigma8
737	70.0	0.300	0.700	NA	NA
137	100.0	0.300	0.700	NA	NA
Planck	68.4	0.301	0.699	8.985075e-05	0.793
Planck18	68.4	0.301	0.699	8.985075e-05	0.793
Planck15	67.8	0.308	0.692	9.150327e-05	0.815
Planck13	67.3	0.315	0.685	9.289295e-05	0.829
WMAP	69.7	0.288	0.712	8.780488e-05	0.817
WMAP9	69.7	0.288	0.712	8.780488e-05	0.817
WMAP7	70.4	0.275	0.725	8.569648e-05	0.816
WMAP5	70.5	0.274	0.726	8.45679e-05	0.812
WMAP3	70.4	0.268	0.732	NA	0.776
WMAP1	72.0	0.290	0.710	NA	0.900
Millennium	73.0	0.250	0.750	NA	0.900
GiggleZ	70.5	0.273	0.727	NA	0.812

Author(s)

Aaron Robotham

References

Name	Full Reference	arXiv Refence
737	Simplified concordance cosomology	NA
137	Simplified concordance cosomology	NA
Planck 18	Planck Collaboration, 2018, arXiv, 1807.06209	arxiv:1807.06209
Planck 15	Planck Collaboration, 2015, A&A, 594, 13	arxiv:1502.01589
Planck 13	Planck Collaboration, 2014, A&A, 571, 16	arXiv:1303.5076v3
WMAP9	Hinshaw G., et al., 2013, ApJS, 208, 19	arXiv:1212.5226v3
WMAP7	Komatsu E., et al., 2010, ApJS, 192, 18	arXiv:1001.4538v3
WMAP5	Komatsu E., et al., 2009, ApJS, 180, 306	arXiv:0803.0547v2
WMAP3	Spergel D. N., et al., 2007, ApJS, 170, 377	arXiv:astro-ph/0603449v2
WMAP1	Spergel D. N., et al., 2003, ApJS, 148, 175	arXiv:astro-ph/0302209v3
Millennium	Springel V., et al., 2005, Nature, 435, 629	arXiv:astro-ph/0504097v2
GiggleZ	Poole G. B., et al., 2015, MNRAS, 449, 1454	arXiv:1407.0390v1

See Also

cosvol, cosmap, cosdist, cosgrow

Examples

data(cosref)
cosref[cosref[,'Ref']=='Planck',]

---

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 as defined at z=0 (default is 0.3).
OmegaL	Omega Lambda as defined at z=0 (default is for a flat Universe with OmegaL = 1-OmegaM = 0.7).
OmegaR	Omega Radiation as defined at z=0 (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)

---

Orbital functions

Description

A variety of obital analysis functions. These are useful for setting up initial conditions for merging systems etc.

Usage

cosorbVisViva(mass=1e12, Rad=162.635, SemiMajRad=162.635, Munit=1, Lunit=1e3, Vunit=1)
cosorbFreeFall(M1=1e12, M2=1, Rad=162.635, Munit=1, Lunit=1e3, Vunit=1, Tunit=1e9)
cosorbRocheRad(M1=1e12, M2=1e10, Size=35.03865, Rfac=2.44)
cosorbRocheSize(M1=1e12, M2=1e10, Rad=396.8294, Rfac=2.44)
  
potential_part(part, eval = NULL, mass = 1, soft = 1, Munit = 1, Lunit = 1000,
  Vunit = 1) 
kinetic_part(vel, mass=NULL, Vunit = 1)

Arguments

mass	Mass in units of 'Munit'.
M1	Mass of primary body in units of 'Munit'.
M2	Mass of secondary body in units of 'Munit'.
Rad	Separation between bodies in units of 'Lunit' (for cosorbRocheSize this is in arbitrary units).
part	Cartesian particle positions [x,y,z] in units of 'Lunit'.
eval	Cartesian evalution positions [x,y,z] in units of 'Lunit'. If this is not provided then the potentials are evaluated at the positions of part.
vel	Cartesian particle velocities [vx,vy,vz] in units of 'Vunit'.
soft	Numeric softening in units of 'Lunit'.
SemiMajRad	The semi major radius of the orbit (a > 0 for ellipses, a = Rad for circles, 1/a = 0 for parabolas, and a < 0 for hyperbolas).
Size	The size radius of the secondary object. Inside of this radius the object is bound to the secondary, outside of this radius the object is stripped by the primary.
Rfac	The Roche factor. Approximately taken to be 2.44, but in reality it varies depending on the shape of the potentials etc.
Munit	Base mass unit in multiples of Msun.
Lunit	Base length unit in multiples of parsecs.
Vunit	Base velocity unit in multiples of km/s.
Tunit	Base time unit in multiples of years.

Details

These functions allow for various analytic conversions between the 3 major properties related to virial radius: the mass, velocity dispresion and size. The default properties calculate properties for 1e12 Msun halos and assume masses in Msun, velocities in km/s and distances in Kpc.

Value

cosorbVisViva function gives the required velocity in units of Vunit to create the specified orbit.

cosorbFreeFall function gives the free fall time to static initial velocity separated bodies.

cosorbRocheRad function gives the orbital radius at which the secondary will become stripped within a specified bound radius.

cosorbRocheSize function gives the limiting bound radius of the secondary for a given system.

potential_part functions gives the total gravitational potential either at the positions of particles provided (part) or at the requested evaluation positions (eval).

kinetic_part functions gives the total kinetic energy of particles provided (vel)

Author(s)

Aaron Robotham, Chris Power

See Also

cosvol, cosmap, cosdist, cosgrow

Examples

cosorbVisViva(M=1e15, Rad=1, Lunit=1e6)
  cosorbFreeFall(M1=1e15, M2=1, Rad=1, Lunit=1e6)
  cosorbRocheRad(M1=1e12, M2=1e12, Size=162.635, Rfac=2.44)
  cosorbRocheSize(M1=1e12, M2=1e12, Rad=396.8294, Rfac=2.44)

---

Driver & Robotham (2010) cosmic variance calculator

Description

The main cosmic variance calculator function taken from Driver & Robotham (2010). cosvarcar is an interface to the Cartesian coordinate version, whilst cosvarsph is a utility interface to give approximate cosmic variance for astronomy survey regions (usually defined by RA, Dec and redshift limits).

Usage

cosvarcar(aside = 50, bside = 50, cside = 50, regions = 1)
cosvarsph(long = c(129, 141), lat = c(-2, 3), zmax = 1, zmin = 0, regions = 1,
inunit='deg', sep=":")
cosvararea(area=60, zmax=1, zmin=0, regions=1, inunit='deg2')

Arguments

aside	The aside (shortest projected side) of the Cartesian box, must be defined using 737 cosmology.
bside	The bside (longest projects side) of the Cartesian box, must be defined using 737 cosmology.
cside	The cside (radial side) of the Cartesian box, must be defined using 737 cosmology.
regions	How many well separated regions of this size will there be? The geometry provided is just for a single region, i.e. we reduce the single region CV by 1/sqrt(regions).
long	Upper and lower longitude (RA) limits of interest in units of inunit. If of length 1 then the number specified is assumed to be the upper limit and the lower limit is set to 0.
lat	Upper and lower latitude (Dec) limits of interest in units of inunit. If of length 1 then the number specified is assumed to be the upper limit and the lower limit is set to 0.
zmax	Maximum redshift of comoving cone.
zmin	Minimum redshift of comoving cone.
inunit	cosvarsph The units of angular coordinate provided for long and lat (see skyarea). cosvararea The units of angular area provided (see cosvol).
sep	When inunit='sex', sep defines the type of separator used for the HMS and DMS strings (i.e. H:M:S and D:M:S would be sep=':', which is the default). See hms2deg and dms2deg for more details.
area	Sky area in units of innunit (default is square degrees)

Details

These functions use the empircally motivated cosmic variance percentage formula provided in Driver & Robotham (2010) Eqn 4.

cosvarsph is a 'best effort' approximation of the comoving box subtended by the specified spherical coordinates using the following conversions:

CoDistLow = cosdistCoDist(z=zmin,H0=70,OmegaM=0.3)

CoDistHigh = cosdistCoDist(z=zmax,H0=70,OmegaM=0.3)

cside=CoDistHigh-CoDistLow

area=skyarea(long = long, lat = lat, inunit = inunit, outunit='deg2')[1]

volume=cosvol(area=area, zmax = zmax, zmin=zmin, H0 = 70, OmegaM = 0.3, inunit='deg2')[1]

aside=cos(mean(lat)*pi/180)*(abs(diff(long))/360)*(CoDistLow+cside/2)

bside=(abs(diff(long))/180)*(CoDistLow+cside/2)

scale=sqrt(volume*1e9/(aside*bside*cside))

aside=aside*scale

bside=bside*scale

return(cosvarcar(aside=aside, bside=bside, cside=cside, subsets=subsets))

cosvararea is a simplifed version of cosvarsph, where the assumption is that aside=bside (so the aspect ratio on the sky is 1:1).

Value

The output is the approximate percentage cosmic (or sample) variance that is expected for the volume specified.

Note

Many people get upset at the term 'cosmic variance' and prefer 'sample variance'. Whilst I am sympathetic to the argument, more astronomers are familiar with the former term.

These cosmic variance estimates are defined using SDSS at z~0.1, caveats abound at higher redshifts, but these numbers should serve as a reasonably conservative (i.e. pessimistic) upper limit.

Author(s)

Aaron Robotham and Simon Driver

References

Driver S.P. & Robotham A.S.G., 2010, MNRAS, 407, 2131

See Also

cosvol, skyarea

Examples

#Approximate CV of the GAMA equatorial regions:
cosvarsph(long=12, lat=5, zmax=0.5)*1/sqrt(3)
#Or using the GAMA sexigesimal coordinates (should be the same):
cosvarsph(long = c('11:36:0','12:24:0'), lat = c('-2:0:0','3:0:0'), zmax=0.5,
inunit='sex')*1/sqrt(3)
#Approximate CV of the SDSS:
cosvarsph(long=150, lat=100, zmax=0.3)

---

Cosmological volume calculator

Description

Given the sky area, two redshifts and the cosmology, this function calculates the comoving volume.

Usage

cosvol(area=60, zmax=1, zmin=0, H0=100, OmegaM=0.3, OmegaL=1-OmegaM-OmegaR, OmegaR=0,
w0=-1, wprime=0, inunit = "deg2", ref)

Arguments

area	Sky area in units of innunit (default is square degrees)
zmax	Maximum cosmological redshift of comoving cone.
zmin	Minimum cosmological redshift of comoving cone.
H0	Hubble constant as defined at z=0 (default is H0=100 (km/s)/Mpc)
OmegaM	Omega Matter as defined at z=0 (default is 0.3).
OmegaL	Omega Lambda as defined at z=0 (default is for a flat Universe with OmegaL = 1-OmegaM-OmegaR = 0.7).
OmegaR	Omega Radiation as defined at z=0 (default is 0, but OmegaM/3400 is typical).
w0	The value of dark energy equation of state as defined at z=0. See cosgrow for more details.
wprime	The evolution term that governs how the dark energy equation of state evolves with redshift. See cosgrow for more details.
inunit	The units of angular area provided. Allowed options are deg2 for square degrees, amin2 for square arc minutes, asec2 for square arc seconds and rad2 or sr for steradians.
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.

Value

A 3 element vector. The first element (voltot) specifies the comoving volume of the requested cone segment in Gpc^3, the second element (volmeanz) specifies the mean redshift when mass is uniformly distributed in the volume, the third element (volmedz) specifies the median redshift when mass is uniformly distributed in the volume.

Author(s)

Aaron Robotham

References

Based on the equations in:

Hogg D.W., 1999, arXiv, 9905116

Wright E.L., 2006, PASP, 118, 1711

See Also

cosdist,skyarea, cosmap, cosgrow

Examples

#Approximate volume of the GAMA survey (area given in skyarea example, zmax is approx
#limit of main galaxy sample):
TotalGAMAvol=cosvol(293.82,0.6)[1]
print(paste('The GAMA survey volume is ~',round(TotalGAMAvol,2),'Gpc^3'))

#Approximate volume of SDSS (area given for DR7, zmax is approx limit of main galaxy sample):
TotalSDSSvol=cosvol(8423,0.3)[1]
print(paste('The SDSS survey volume is ~',round(TotalSDSSvol,2),'Gpc^3'))

#Change of reference cosmology
cosvol(293.82,0.6,ref='Planck')

---

Convert decimal degrees to dms format.

Description

Convert decimal degrees to dms (degrees, minutes, seconds) format. This is probably most useful for declination conversion, since dms is fairly standard method of presenting declination coordinates. The decimal degrees=d+m/60+s/3600. Degrees should range from -90 to +90.

Usage

deg2dms(deg, type='mat', sep=':', digits=2)

Arguments

deg	The decimal degrees you are converting. All deg values should be -90<=deg<=90
type	The output type desired. If 'mat' then the output is a 3 column data.frame where column 1 is the degree, column 2 is the minutes and column 3 is the seconds. If 'cat' then the output is a single vector of strings where the separator is defined by the 'sep' argument.
sep	Defines the type of separator used when type='cat'. Any value other than 'DMS' and 'dms' is used for all separations, so the default ':' would produce an output like 3:34:45.5. If set to 'dms' or 'DMS' then the output is of the format 3d34m45.5s and 3D34M45.5s resepctively.
digits	The digits to print for angular seconds. See formatC for details on how digits is parsed.

Value

A data.frame with the columns degrees, minutes and seconds if type='mat'. If type='cat' then a vector of strings with separators defined by the 'sep' argument.

Author(s)

Aaron Robotham

See Also

dms2deg

Examples

print(deg2dms(12.345))
print(deg2dms(12.345,type='cat',sep=':'))
print(deg2dms(12.345,type='cat',sep='dms'))
print(deg2dms(12.345,type='cat',sep='DMS'))

---

Convert decimal degrees to hms format.

Description

Convert decimal degrees to hms (hours, minutes, seconds) format. This is probably most useful for right-ascension (RA) conversion, since hms is fairly standard method of presenting RA coordinates. The decimal degrees=15*h+15*m/60+15*s/3600 (i.e. there are 24 hours in 360 degrees). Degrees should range from 0 to 360.

Usage

deg2hms(deg, type='mat', sep=':', digits=2)

Arguments

deg	The decimal degrees you are converting. All deg values should be 0<=d<=360.
type	The output type desired. If 'mat' then the output is a 3 column data.frame where column 1 is the degree, column 2 is the minutes and column 3 is the seconds. If 'cat' then the output is a single vector of strings where the separator is defined by the 'sep' argument.
sep	Defines the type of separator used when type='cat'. Any value other than 'DMS' and 'dms' is used for all separations, so the default ':' would produce an output like 3:34:45.5. If set to 'hms' or 'HMS' then the output is of the format 3h34m45.5s and 3H34M45.5s resepctively.
digits	The digits to print for angular seconds. See formatC for details on how digits is parsed.

Value

A data.frame with the columns degrees, minutes and seconds if type='mat'. If type='cat' then a vector of strings with separators defined by the 'sep' argument.

Author(s)

Aaron Robotham

See Also

hms2deg

Examples

deg2hms(123.456)
deg2hms(123.456,type='cat',sep=':')
deg2hms(123.456,type='cat',sep='hms')
deg2hms(123.456,type='cat',sep='HMS')

---

Convert DMS to degrees format.

Description

Convert DMS (degrees, minutes, seconds) to degrees format. This is probably most useful for declination conversion, since dms is fairly standard method of presenting declination coordinates. The decimal degrees=d+m/60+s/3600. Degrees should range from -90 to +90. Degrees and minutes should be integer and seconds can be decimal.

Usage

dms2deg(d,m,s,sign='d',sep=':')

Arguments

d	The integer number of degrees you are converting. If it is not integer then the floor of the number is taken. This can contain the sign of the declination when sign='d', but must be all positive if the sign argument is specified (this is required if d contains any 0s, see below). If sign is specified, all d values should be 0<=d<=90, otherwise d values should be -90<=d<=90.
m	The integer number of minutes you are converting. If it is not integer then the floor of the number is taken. All m values should be 0<=m<60.
s	The decimal number of seconds you are converting. All s values should be 0<=s<60.
sign	The sign of the declination. The default 'd' inherits the sign of the d argument. This is ambiguous when d is 0 since the sign of +/-0 is taken to be 0. If d contains any 0s, you must supply a vector of the same length as d with +ve or -ve values (e.g. +/- 1), the sign of these value will be taken as the sign for the declination.
sep	Defines the type of separator used when 'd' is a vector of strings. Any value other than 'DMS' and 'dms' is used for all separations, so the default ':' would be for an input like 3:34:45.5. If set to 'dms' or 'DMS' then the output is of the format 3d34m45.5s and 3D34M45.5s resepctively.

Value

A value of decimal degrees.

Author(s)

Aaron Robotham

See Also

deg2dms

Examples

print(dms2deg(70,45,19,-1))
print(dms2deg('-70:45:19'))
print(dms2deg('-70d45m19s',sep='dms'))
print(dms2deg(c('-70D45M19S','3D5M15S'),sep='DMS'))

---

Convert Equtorial to Galactic Coordinates

Description

Simple conversions accurate to about 0.1 arcseconds between J2000 Equatorial and MW Galactic.

Usage

eq2gal(RA, Dec, pole_RA = 192.859508, pole_Dec = 27.128336, eta = 32.932)
gal2eq(gal_long, gal_lat, pole_RA = 192.859508, pole_Dec = 27.128336, eta = 32.932)

Arguments

RA	Numeric vector; Right Ascension in degrees (J2000 system). For convenience this can also be a two column matrix / data.frame containing RA and Dec columns.
Dec	Numeric vector; Declination in degrees (J2000 system).
gal_long	Numeric vector; Galactic Longitude in degrees (J2000 system). For convenience this can also be a two column matrix / data.frame containing gal_long and gal_lat columns.
gal_lat	Numeric vector; Galactic Latitude in degrees (J2000 system).
pole_RA	Numeric scalar; Right Ascension of Galactic Pole in degrees (J2000 system).
pole_Dec	Numeric scalar; Declination of Galactic Pole in degrees (J2000 system).
eta	Numeric scalar; Tilt of the Earth's axis in degrees (J2000 system).

Details

Approximate simple conversions, accurate usully to 0.1 arcseconds

Value

eq2gal returns the Galactic longitude and latitude in degrees (two column output). gal2eq returns the Equatorial RA and Dec in degress (two column output).

Author(s)

Aaron Robotham

References

"Practical astronomy with your calculator" (Peter Duffett-Smith)

See Also

sph2car,car2sph

Examples

# Notice the conversion anbiguity at the poles

gal2eq(eq2gal(seq(0,360, len=19), seq(-90,90, len=19)))

---

Get Pixel Scale

Description

Given a FITSio of astro header, calculate the image pixel scale.

Usage

getpixscale(header, CD1_1 = 1, CD1_2 = 0, CD2_1 = 0, CD2_2 = 1)

Arguments

header	Full FITS header in table or vector format. Legal table format headers are provided by the read.fitshdr function or the hdr list output of read.fits in the astro package; the hdr output of readFITS in the FITSio package or the header output of magcutoutWCS. Missing header keywords are printed out and other header option arguments are used in these cases. See xy2radec.
CD1_1	FITS header CD1_1 for the Tan Gnomonic projection system. Change in RA-Tan in degrees along x-Axis.
CD1_2	FITS header CD1_2 for the Tan Gnomonic projection system. Change in RA-Tan in degrees along y-Axis.
CD2_1	FITS header CD2_1 for the Tan Gnomonic projection system. Change in Dec-Tan in degrees along x-Axis.
CD2_2	FITS header CD2_2 for the Tan Gnomonic projection system. Change in Dec-Tan in degrees along y-Axis.

Details

In most cases users will simply provide a valid header to find the WCS, but you can enter the CD values explicitly. Calculating the pixel scale from the latter is almost trivial, but the option is there for the curious/lazy.

Value

Numeric scalar; the image pixscale in asec/pixel (so typically a value of 0.1-0.5 for modern survey instruments).

Author(s)

Aaron Robotham

Examples

## Not run: 
#The answer should be almost exactly 0.2 asec/pixel:

#Using FITSio and ProFit packages
image = readFITS(system.file("extdata", 'KiDS/G266035fitim.fits', package="ProFit"))
getpixscale(image$hdr)
#Using astro package
image = read.fits(system.file("extdata", 'KiDS/G266035fitim.fits', package="ProFit"))
getpixscale(image$hdr[[1]])

## End(Not run)

---

Convert hms to degrees format.

Description

Convert hms (hours, minutes, seconds) to degrees format. This is probably most useful for right ascension (RA) conversion, since hms is fairly standard method of presenting RA coordinates. The decimal degrees=15*h+15*m/60+15*s/3600. Should range between 0 and 24 hours. Hours and minutes should be integer and seconds can be decimal.

Usage

hms2deg(h,m,s,sep=':')

Arguments

h	The integer number of hours you are converting. If it is not integer then the floor of the number is taken. All m values should be 0<=h<=24.
m	The integer number of minutes you are converting. If it is not integer then the floor of the number is taken. All m values should be 0<=m<60.
s	The decimal number of seconds you are converting. All s values should be 0<=s<60.
sep	Defines the type of separator used when 'h' is a vector of strings. Any value other than 'HMS' and 'hms' is used for all separations, so the default ':' would be for an input like 3:34:45.5. If set to 'hms' or 'HMS' then the output is of the format 3h34m45.5s and 3H34M45.5s resepctively.

Value

A value of decimal degrees.

Author(s)

Aaron Robotham

See Also

deg2hms

Examples

hms2deg(12,10,36)
hms2deg('12:10:36')
hms2deg('12h10m36s',sep='hms')
hms2deg(c('12H10M36S','3H4M10S'),sep='HMS')

---

IAU name creator.

Description

Creates IAU legal names for objects given coordinates, name and epoch.

Usage

IAUID(ra, dec, name = "GAMA", epoch = "J")

Arguments

ra	Right Ascension in decimal degrees.
dec	Declination in decimal degrees.
name	Name to be appended to IAU designation as a string.
epoch	Epoch, i.e. 'J' (default) or 'B'. Enter as a string.

Value

Text string that outputs an IAU legal name for an object.

Author(s)

Aaron Robotham

Examples

IAUID(123.45,67.89,'GAMA','J')

---

Planck's Law and Related Functions

Description

Functions related to Planck's Law of thermal radiation.

Usage

cosplanckLawRadFreq(nu,Temp=2.725)
cosplanckLawRadWave(lambda,Temp=2.725)
cosplanckLawEnFreq(nu,Temp=2.725)
cosplanckLawEnWave(lambda,Temp=2.725)
cosplanckLawRadFreqN(nu,Temp=2.725)
cosplanckLawRadWaveN(lambda,Temp=2.725)
cosplanckPeakFreq(Temp=2.725)
cosplanckPeakWave(Temp=2.725)
cosplanckSBLawRad(Temp=2.725)
cosplanckSBLawRad_sr(Temp=2.725)
cosplanckSBLawEn(Temp=2.725)
cosplanckLawRadPhotEnAv(Temp=2.725)
cosplanckLawRadPhotN(Temp=2.725)
cosplanckCMBTemp(z,Temp=2.725)

Arguments

nu	The frequency of radiation in Hertz (Hz).
lambda	The wavelength of radiation in metres (m).
Temp	The absolute temperature of the system in Kelvin (K).
z	Redshift, where z must be > -1 (can be a vector).

Details

The functions with Rad in the name are related the spectral radiance form of Planck's Law (typically designated I or B), whilst those with En are related to the spectral energy density form of Planck's Law (u), where $u=4\pi I/c$.

To calculate the number of photons in a mode we simply use $E=h\nu=h c / \lambda$.

Below h is the Planck constant, $k_B$ is the Boltzmann constant, c is the speed-of-light in a vacuum and $\sigma$ is the Stefan-Boltzmann constant.

cosplanckLawRadFreq is the spectral radiance per unit frequency version of Planck's Law, defined as:

$B_\nu(\nu,T) = I_\nu(\nu,T) = \frac{2 h \nu^3}{c^2} \frac{1}{e^{h \nu / k_B T}-1}$

cosplanckLawRadWave is the spectral radiance per unit wavelength version of Planck's Law, defined as:

$B_\lambda(\lambda,T) = I_\lambda(\lambda,T) = \frac{2 h c^2}{\lambda^5} \frac{1}{e^{h c / \lambda k_B T}-1}$

cosplanckLawRadFreqN is the number of photons per unit frequency, defined as:

$B_\nu(\nu,T) = I_\nu(\nu,T) = \frac{2 \nu^2}{c^2} \frac{1}{e^{h \nu / k_B T}-1}$

cosplanckLawRadWaveN is the number of photons per unit wavelength, defined as:

$B_\lambda(\lambda,T) = I_\lambda(\lambda,T) = \frac{2 c}{\lambda^4} \frac{1}{e^{h c / \lambda k_B T}-1}$

cosplanckLawEnFreq is the spectral energy density per unit frequency version of Planck's Law, defined as:

$u_\nu(\nu,T) = \frac{8 \pi h \nu^3}{c^3} \frac{1}{e^{h\nu/k_B T}-1}$

cosplanckLawEnWave is the spectral energy density per unit wavelength version of Planck's Law, defined as:

$u_\lambda(\lambda,T) = \frac{8 \pi h c}{\lambda^5} \frac{1}{e^{h c / \lambda k_B T}-1}$

cosplanckPeakFreq gives the location in frequency of the peak of $I_\nu(\nu,T)$, defined as:

$\nu_{peak} = 2.821 k_B T$

cosplanckPeakWave gives the location in wavelength of the peak of $I_\lambda(\lambda,T)$, defined as:

$\lambda_{peak} = 4.965 k_B T$

cosplanckSBLawRad gives the emissive power (or radiant exitance) version of the Stefan-Boltzmann Law, defined as:

$j^* = \sigma T^4$

cosplanckSBLawRad_sr gives the spectral radiance version of the Stefan-Boltzmann Law, defined as:

$L = \sigma T^4/\pi$

cosplanckSBLawEn gives the energy density version of the Stefan-Boltzmann Law, defined as:

$\epsilon = 4 \sigma T^4 / c$

Notice that $j^*$ and L merely differ by a factor of $\pi$, i.e. L is per steradian.

cosplanckLawRadPhotEnAv gives the average energy of the emitted black body photon, defined as:

$<E_{phot}> = 3.729282 \times 10^{-23} T$

cosplanckLawRadPhotN gives the total number of photons produced by black body per metre squared per second per steradian, defined as:

$N_{phot} = 1.5205 \times 10^{15} T^3 / \pi$

Various confidence building sanity checks of how to use these functions are given in the Examples below.

Value

Planck's Law in terms of spectral radiance:

cosplanckLawRadFreq	The power per steradian per metre squared per unit frequency for a black body (W.sr$^{-1}$.m$^{-2}$.Hz$^{-1}$).
cosplanckLawRadWave	The power per steradian per metre squared per unit wavelength for a black body (W.sr$^{-1}$.m$^{-2}$.m$^{-1}$).

Planck's Law in terms of spectral energy density:

cosplanckLawEnFreq	The energy per metre cubed per unit frequency for a black body (J.m$^{-3}$.Hz$^{-1}$).
cosplanckLawEnWave	The energy per metre cubed per unit wavelength for a black body (J.m$^{-3}$.m$^{-1}$).

Photon counts:

cosplanckLawRadFreqN	The number of photons per steradian per metre squared per second per unit frequency for a black body (photons.sr$^{-1}$.m$^{-2}$.s$^{-1}$.Hz$^{-1}$).
cosplanckLawRadWaveN	The number of photonsper steradian per metre squared per second per unit wavelength for a black body (photons.sr$^{-1}$.m$^{-2}$.s$^{-1}$.m$^{-1}$).

Peak locations (via Wien's displacement law):

cosplanckPeakFreq	The frequency location of the radiation peak for a black body as found in cosplanckLawRadFreq.
cosplanckPeakWave	The wavelength location of the radiation peak for a black body as found in cosplanckLawRadWave.

Stefan-Boltzmann Law:

cosplanckSBLawRad	Total energy radiated per metre squared per second across all wavelengths for a black body (W.m$^{-2}$). This is the emissive power version of the Stefan-Boltzmann Law.
cosplanckSBLawRad_sr	Total energy radiated per metre squared per second per steradian across all wavelengths for a black body (W.m$^{-2}$.sr$^{-1}$). This is the radiance version of the Stefan-Boltzmann Law.
cosplanckSBLawEn	Total energy per metre cubed across all wavelengths for a black body (J.m$^{-3}$). This is the energy density version of the Stefan-Boltzmann Law.

Photon properties:

cosplanckLawRadPhotEnAv	Average black body photon energy (J).
cosplanckLawRadPhotN	Total number of photons produced by black body per metre squared per second per steradian (m$^{-2}$.s$^{-1}$.sr$^{-1}$).

Cosmic Microwave Background:

cosplanckCMBTemp

The temperaure of the CMB at redshift z.

Author(s)

Aaron Robotham

References

Marr J.M., Wilkin F.P., 2012, AmJPh, 80, 399

See Also

cosgrow

Examples

#Classic example for different temperature stars:

waveseq=10^seq(-7,-5,by=0.01)
plot(waveseq, cosplanckLawRadWave(waveseq,5000),
log='x', type='l', xlab=expression(Wavelength / m),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*m^{-1}), col='blue')
lines(waveseq, cosplanckLawRadWave(waveseq,4000), col='green')
lines(waveseq, cosplanckLawRadWave(waveseq,3000), col='red')
legend('topright', legend=c('3000K','4000K','5000K'), col=c('red','green','blue'), lty=1)

#CMB now:

plot(10^seq(9,12,by=0.01), cosplanckLawRadFreq(10^seq(9,12,by=0.01)),
log='x', type='l', xlab=expression(Frequency / Hz),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*Hz^{-1}))
abline(v=cosplanckPeakFreq(),lty=2)

plot(10^seq(-4,-1,by=0.01), cosplanckLawRadWave(10^seq(-4,-1,by=0.01)),
log='x', type='l', xlab=expression(Wavelength / m),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*m^{-1}))
abline(v=cosplanckPeakWave(),lty=2)

#CMB at surface of last scattering:

TempLastScat=cosplanckCMBTemp(1100) #Note this is still much cooler than our Sun!

plot(10^seq(12,15,by=0.01), cosplanckLawRadFreq(10^seq(12,15,by=0.01),TempLastScat),
log='x', type='l', xlab=expression(Frequency / Hz),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*Hz^{-1}))
abline(v=cosplanckPeakFreq(TempLastScat),lty=2)

plot(10^seq(-7,-4,by=0.01), cosplanckLawRadWave(10^seq(-7,-4,by=0.01),TempLastScat),
log='x', type='l', xlab=expression(Wavelength / m),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*m^{-1}))
abline(v=cosplanckPeakWave(TempLastScat),lty=2)

#Exact number of photons produced by black body:

cosplanckLawRadPhotN()

#We can get pretty much the correct answer through direct integration, i.e.:

integrate(cosplanckLawRadFreqN,1e8,1e12)
integrate(cosplanckLawRadWaveN,1e-4,1e-1)

#Stefan-Boltzmann Law:

cosplanckSBLawRad_sr()

#We can get (almost, some rounding is off) the same answer by multiplying
#the total number of photons produced by a black body per metre squared per
#second per steradian by the average photon energy:

cosplanckLawRadPhotEnAv()*cosplanckLawRadPhotN()

---