3 Angular Size

Since galaxies are not sharp-edged objects, their angular size can be variously defined (see [Mihalas and Binney(1981)] and [Binney and Merrifield(1998)]). As far as studies of surface brightness distribution are concerned, however, the parameter most widely used to characterize the size of a galaxy is its effective radius. This can be roughly defined as the radius encircling half of the light emitted by the galaxy, but in practice its measurement is usually performed through a more complicated process.

Surface photometry of galaxies (see e.g. [Jedrzejewski(1987)] for E galaxies and [Kent(1985)] for D galaxies) is usually analysed by fitting ellipses to the isophotes and by plotting their surface brightness versus their radius, which is defined as the geometric mean of the ellipse's semi-axes $a$ and $b$, i.e. $r=\sqrt{ab}$. The resulting plot is then called the surface brightness radial profile of the galaxy. In this context, the effective radius of the galaxy is defined as the radius of the isophote encircling half of the light emitted by the galaxy, also called the effective isophote. The effective radius and the effective surface brightness, the latter being the surface brightness of the effective isophote, are usually indicated with $r_e$ and $\mu_e$, respectively.

Until the launch of HST, accurate measurements of the small angular sizes of faint galaxies were made virtually impossible by the phenomenon of seeing. The Medium Deep Survey ([Ratnatunga et al.(1999)]), the first survey project to be carried out with HST superb instrumentation, has recently brought to an end this long-standing lack of meaningful data, while [Im et al.(1995)] have demonstrated the potential of angular size measurements to discriminate between currently competing cosmological models.

[Casertano et al.(1995)] have obtained effective radii for about 10,000 galaxies from Wide Field and Planetary Camera (WF/PC) parallel observations of random fields in the $I$ band. As shown in their Figure 6, the observed angular size distribution as function of $I$ magnitude shows a large scatter about the median value, mainly due to the intrinsic scatter in linear size and redshift distribution. The same figure also shows that the observed relation between the median effective radius and $I$ magnitude is well-fit by the theoretically predicted relation for galaxies of constant central surface brightness $\mu_0=19.3$ mag/arcsec$^2$and absolute magnitude $M_I=-20.5$ in the context of a mild luminosity evolution scenario. This latter relation asymptotically approaches the linear relation in $\log\,r_e$ vs. $I$ that is measured in local samples of bright spiral galaxies following Freeman's law, and was therefore taken as a description of the relation between the galaxy effective radius and magnitude in our model. Least-square polynomial fit showed that its accurate description required a fourth-degree polynomial, which is represented in Figure 3, together with the Euclidean extrapolation to faint magnitudes of the local linear relation.

Figure 3: Median galaxy effective radius in the $I$ band. The straight line represents the Euclidean extrapolation to faint magnitudes of the result valid for local samples of spiral galaxies following Freeman's law, while the curve represents the best fourth-degree polynomial fit to the theoretically predicted relation that best fits the observations at faint magnitudes. Note the pronounced divergence of the two curves for $I\gtrsim19$. From [Casertano et al.(1995)].

The relation that in the following will be used to express $r_e$ as function of $I$ is therefore

$$r_e(I)= (a_r+b_r\,I+c_r\,I^2+d_r\,I^3+e_r\,I^4)~~~\mathrm{[arcsec]}~.$$ (3)

The values of the five parameters contained in Equation 3 are given in Table 3.

Table: Parameters of $\log\,r_e$ vs. $I$ least-square fourth-degree polynomial best-fit. $r_e$ expressed in arcsec.

$a_r$	$b_r$	$c_r$	$d_r$	$e_r$
3.57702 $\,\cdot\,10^{0}$	-2.12805 $\,\cdot\,10^{-1}$	5.34616 $\,\cdot\,10^{-3}$	-4.62001 $\,\cdot\,10^{-4}$	1.28947 $\,\cdot\,10^{-5}$

Note that it is assumed not only that $r_e$ depends on $I$ only, but also that the same relation holds for all galaxies, irrespective of their morphological types. From a statistical point of view, however, and as far as Equation 3 holds on average, the net effect on the total number of detected galaxies should be negligible. Effective radii calculated from Equation 3, are given in Table 4. A rough estimation of the typical surface brightness of the central regions of galaxies can be given by the average surface brightness inside the effective radius $<\!\mu\!>_e$. Under our assumptions, and from the definition of effective radius, this quantity is equal for Es and Ds and can be written as

$$\begin{split}<\!\mu\!>_e&=-2.5\,\log\,\left(\frac{F/2}{\pi\,r... ...I))+I_{\mathrm{[mag]}} ~~~\mathrm{[mag/arcsec^2]~.} \end{split}$$ (4)

Values of $<\!\mu\!>_e$ are given in Table 5. To characterize the total fraction of the sky occupied by galaxies brighter than a given magnitude, one can define $\Omega_e$ as the total solid angle that lies within the effective radius of all galaxies brighter than a given magnitude. Since superposition of different galaxies on the same sky regions is negligible, at least in the magnitude range we are considering, $\Omega_e$ can simply be written as

$$% latex2html id marker 2758 \begin{split}\Omega_e(I)&=\frac{... ...me\,3}+2\,e_r\,I^{\prime\,4}\right]\,\mathrm{d}I'~, \end{split}$$ (5)

where $\Omega_{sky}$ is the solid angle spanned by the whole sky and the factor $60^4$ was introduced to take into account the fact that the coefficients giving $N$ and $r_e$ as function of $I$ in Equations 1 and 3 are expressed in different angular units. Values of $\Omega_e/\Omega_{sky}$, calculated through Romberg integration of Equation 5, are given in Table 4.

Table 4: Number and size of galaxies according to our statistical model. Modelled differential and cumulative number counts, effective radius and fraction of sky inside the effective radius at different $I$.

$I$	$N$	$N_c$	$r_e$	$\Omega_e/\Omega_{sky}$
mag	deg$^{-2}$ mag$^{-1}$	deg$^{-2}$	arcsec	$10^{-6}$ sky
10	0.008183	0.005151	44.72	5.350
11	0.03777	0.02458	28.64	10.93
12	0.1654	0.1114	18.29	21.15
13	0.6867	0.4791	11.68	38.88
14	2.705	1.958	7.468	68.08
15	10.10	7.596	4.801	114.0
16	35.79	27.99	3.115	183.6
17	120.3	98.00	2.049	285.9
18	383.3	325.9	1.374	433.3
19	1159	1030	0.9446	644.0
20	3322	3093	0.6707	946.5
21	9032	8826	0.4954	1389
22	23290	23940	0.3838	2060
23	56970	61760	0.3147	3132
24	132200	151500	0.2758	4969