V isib ility

In the Appendix we will discuss magnitude types, § A.l, and the detection of galaxies, § A.2, and give a demonstration of visibility theory, § A.2.1. These sections give a broader view of the topic and useful demonstrations of the problems. In this section we will lay down the basic equations of visibility theory, used to calculate the volume over which a galaxy can be seen. This framework makes it possible to incorporate many different limits at once.

To start with let us imagine what happens to our measurements when a galaxy of constant intrinsic luminosity and size is viewed at different redshifts. As the galaxy is moved further away it appears fainter, eventually approaching the faint magnitude limit. It also appears to be smaller, approaching the minimum diameter limit. In a static. Euclidean Universe, with no background light, it is very easy to calculate the maximum distance to which the galaxy can be seen to with these two limits.

d i = - M -25)

dg = 206.3 Mpc (2.25)

dmin d>max ~ rnin{d\ , dg)

where M and D are the absolute magnitude and intrinsic size of the galaxy (in kpc) and mfaint

Chapter 2: A Set of Useful Tools For Producing and Using the BBD. 36 of 206.3 comes from converting the angular size to radians (2.063 x 10^) and converting kpc to Mpc,

However, there are extra complications such as the optical background, which has a similar surface brightness as the centres of disk galaxies. This causes the light loss problems discussed at length in § A.l, which are shown to be profile dependent. Also since the Universe is expanding there are important cosmological effects, such as surface brightness dimming (see Eqn. 2.5) and the effect of the K-correction (see Eqn. 2.6) and evolution. Thus the same galaxy will differ in the amount of light loss and ratio of apparent diameter to scale size if it is viewed at a different redshift.

This means that both di and dg are surface brightness dependent. They are functions of the surface brightness profile of the galaxy, which depends on morphological type and redshift, and also the detection isophote.

2.4.1 A Thorough Example; Exponential Profiles.

Below is the full visibility treatment of face-on disk galaxies with isophotal magnitudes. The equations below are reproduced from Disney & Phillipps (1983) and Phillipps, Davies and Disney (1990). These are used to calculate the volume over which a galaxy of absolute magni­ tude M , and central surface brightness po can be seen. The theory determines the maximum distance to which a galaxy can be seen, using two constraints: the apparent magnitude that the galaxy would have, and the apparent size that the galaxy would have. The first constraint sets a limit on the luminosity distance to the galaxy, which is the distance that a galaxy is at when it becomes too faint to be seen.

dl = [fipiim - Po)]^ içfp.2{mum-M-20-{KYe){z))] (2.26) where f{pum — Po) is the fraction of light above the isophotal detection threshold and is profile dependent, see Eqn 2.16. Using /? = 1 and k = 1.678, the fraction of light seen is:

/(Miim - Ho ) = l - [ i + 0.41n(10)(/iH„ - mo)] (2.27)

where pum is modified according to Eqn 2.9 to take into account cosmological dimming and the (K+e) correction.

Thus the maximum distance has a surface brightness dependence. The luminosity distance is a function of redshift and cosmological parameters, calculated from the proper distance, see

§ 2.2.

The maximum distance can be found numerically by, for instance, a Newton- Raphson iteration using the fact that

cZi(;z)-(^i,(2:) = 0 (2.28)

Chapter 2: A Set of Useful Tools For Producing and Using the BBD. 37 The second constraint, the size limit is found by a similar method. The size limit is:

d2 = Ggitiiim - Mo) Mpc (2.29)

where C is a profile dependent constant. g{pnm—Po) is related to the isophotal limit by Eqn 2.18 and is equivalent to the isophotal radius in scale lengths, see Eqn 2.17 for an exponential disk.

6iim is the minimum apparent diameter. For a spiral disk with an exponential profile:

C = (2.30)

In this case dg is an angular-diameter distance, not a luminosity distance. In the same way as for the magnitude limit, the maximum redshift can be found numerically by solving Eqn 2.31.

dg(zg) - dA{z2) = 0 (2.31)

Once the redshifts zi and zg, which are the solutions of Eqn 2.28 and Eqn 2.31 , have been found, the maximum redshift is the minimum of zi and zg.

Zmax — min(zi,zg) (2.32)

The equations can also be used to find the minimum redshift.

Zmin —m a x ( z 3 , Z 4 ) (2.33)

where zg is calculated in the same way as zi, using the bright magnitude limit rather than the faint magnitude limit and Z4 is calculated in the same way as zg, using the maximum apparent

diameter limit rather than the minimum apparent diameter limit.

Additional cuts for minimum or maximum redshift can be imposed at this stage. Many surveys have a minimum redshift cut to avoid problems with peculiar velocities dwarfing the Hubble flow velocity. Cuts are made at high redshift to avoid the worst photometric problems or evolutionary problems. Also cuts are made to produce luminosity functions or BBDs within a redshift range to test evolution.

The visibility V{M,po) is calculated using Eqn 2.4 and these limits. It represents the vol­ ume over which a spiral disk galaxy of absolute magnitude M and central surface brightness po

can be observed. The central surface brightness can be converted to effective surface brightness using Eqn 2.24.

As well as isophotal magnitude and diameter limits, there are similar limits used for total magnitudes, half-light diameters and apparent surface brightness limits. These are particularly useful if the magnitudes have been corrected to total magnitudes. In these cases, the isophotal limit only affects the completeness of the survey.

Chapter 2: A Set of Useful Tools For Producing and Using the BBD. 38 The total magnitude limits are simple to calculate. For total magnitudes f{po — pum) in Eqn 2.26 is equal to 1. Thus;

dl = jyipc (2.34)

Similarly, the half-light diameter does not depend on the isophote, only on the central surface brightness. Thus Eqn 2.29 becomes:

dg = 1.678 C 1Q[0 2(A0-M)1 gg)

where g{po — pum), the number of scale lengths at the isophote is replaced by 1.678, the number of scale lengths at the half-light radius, see Eqn 2.14. di and dg can be related to the luminosity and angular-diameter distances as before and solved to get z (see Eqns 2.2 - 2.31).

In Chapters 4 and 5 we have also used apparent surface brightness limits pe^^, to avoid regions of low completeness. These are different from the isophotal limit. An isophotal limit is the surface brightness level that isophotal magnitudes and areas are measured at, whereas an apparent surface brightness limit is a simple cut removing some detected objects. The redshift can be found by solving Eqn 2.36 for z.

101ogjQ(l + z) + {K + e)(z) 3- Pe — p ^ ^ = 0 (2.36)