4. Density evolution
The resemblance of morphological and photometric characteristics of suspected tidal tails of the HDFs galaxies with local objects allows us to use them to measure possible change with z of volume density of galaxies with tails (and, therefore, the rate of close encounters leading to the formation of extended tails).
The co-moving volume element in solid angle and redshift interval dz is
where - photometric distance (Eq. (1)), and (1 + z) for (e.g. Peebles 1993). The increase of the space density of galaxies with tidal tails we take in standard power-law form:
where - local volume density of such galaxies. By integrating Eqs. (2) and (3) we can find the expected number of objects within solid angle and in required range of z.
4.1. Local density of galaxies with tidal tails
We suppose that at the current epoch interactions and mergers accompanied by tail formation are almost entirely between bound pairs of galaxies (e.g. Toomre 1977). So we adopt that frequency of tidal tails among single objects (mergers) and in groups, is significantly lower than in pairs.
According to Karachentsev (1987), the relative frequency of galaxies with tails among the members of binary systems is 94/974=0.100.01. The fraction of paired galaxies in the local universe is not well determined. Various strategies give results between 5% and 15%. For instance, local pairing fraction is 7%1% according to Burkey et al. (1994), 6%-10% (Keel & van Soest 1992), 14%2% (Lawrence et al. 1989). The most intensive studies lead to 12%2% (Karachentsev 1987) and 10% (Xu & Sulentic 1991, Soares et al. 1995). Moreover, Xu & Sulentic (1991) found that the fraction of pairs is approximately constant (10%) over the luminosity range (see also Soares et al. 1995). Thus, we can adopt the value of 10%5% as a reasonable estimate of the local fraction of binary galaxies. Therefore, the relative fraction of galaxies with tidal tails at is 0.10.1=0.010.005.
To find total density of galaxies in the nearby part of the universe (), we considered the galaxy luminosity function (LF) according to Marzke et al. (1998). The adopted Schechter function parameters of the LF are: =-20.05, =5.410-3 Mpc-3 and =-1.12 (75 km/s/Mpc). By integrating LF from to -21.1 (the range of absolute luminosities of galaxies with tails in the HDF-N and HDF-S), we found that total volume density of galaxies is equal to 0.026 Mpc-3. Thus, 0.010.026=(2.61.3)10-4 Mpc-3.
The total angular area within which we searched tailed galaxies in two HDFs is 10.4 arcmin2 or 8.810-7 sr.
4.2. Exponent m from tidal structures
Varying exponent m, we can estimate the expected number of galaxies with tidal features in the HDFs. In Fig. 3 we present the results of calculations for two redshift intervals: 0.5-1.5 (total sample) and 0.5-1.0 (adopted cosmology is , or and =75 km/s/Mpc). As one can see, the total sample (25 objects) leads to . But this value must be considered as a low limit only due to strong underestimation of tidal tails at (Sect. 3). For the galaxies with (N=14) we obtain . Assuming Poisson error of N (=3.7), we have . Adding 50% uncertainty in the local space density , we have obtained a final estimation of m as . (Let us note also that two potential sources of errors - underestimation of value and omission of tailed galaxies in the HDFs - bias value of m in opposite directions and partially compensate each other.)
The value of m depends on the adopted cosmological model. For , we have (z=0.5-1.0). In our calculations for the model with a cosmological constant and zero spatial curvature (=0.3, =0.7, =1) we used the analytical approximation of the luminosity distance according to Pen (1999). In the framework of that model we have obtained .
To obtain more realistic error estimation, we must take into account the possible luminosity evolution of galaxies with redshift. Unfortunately, luminosity and surface brightness evolution of peculiar and interacting galaxies is poorly constrained at present (e.g. Roche et al. 1998). Moreover, Simard et al. (1999) claim that an apparent systematic increase in disk mean surface brightness to for bright () spiral galaxies is due to selection effects. Nevertheless, assuming that interacting galaxies undergo luminosity evolution between and 1, we estimated that the value of m must be decreased by : for and for .
© European Southern Observatory (ESO) 2000
Online publication: December 8, 1999