5.6 Expected Number of Detected Galaxies

Introducing the data given in Sections 5.4 and 5.5 in Equation 5.5, an estimation of the signal-to-noise ratio $SNR$ produced by a given average surface brightness within an aperture of a given size can be given. In particular, we are interested in the range of angular sizes and average surface brightnesses covered by the inner regions of bright galaxies, which will obviously be the easiest to detect. These two quantities can be characterized by the galaxy effective radius and by the average surface brightness within the effective radius, which have been given, for galaxies of different total magnitudes, in Tables 4.4 and 4.5, respectively. Values of $SNR$ for apertures and average surface brightnesses in the relevant ranges are given in Table 5.3.

Table 5.3: Signal-to-Noise Ratio $SNR$ for Galaxy Detection, according to statistical formulae. $SNR$ is given for different values of aperture size and $I$-band average surface brightness within the aperture $<\!\mu\!>_{ap}$, according to Equation 5.5. Aperture size expressed in arcsec$^2$, $<\!\mu\!>_{ap}$ in mag/arcsec$^2$.

$SNR$		Aperture Size
		$4.0\!\times\!4.0$	$3.5\!\times\!3.5$	$3.0\!\times\!3.0$	$2.5\!\times\!2.5$	$2.0\!\times\!2.0$	$1.5\!\times\!1.5$	$1.0\!\times\!1.0$
$<\!\mu\!>_{ap}$	20.00	7.8341	6.9344	6.0047	5.0482	4.0683	3.0689	2.0544
	20.25	6.2371	5.5210	4.7811	4.0197	3.2395	2.4438	1.6360
	20.50	4.9633	4.3937	3.8050	3.1992	2.5783	1.9450	1.3021
	20.75	3.9483	3.4952	3.0270	2.5451	2.0512	1.5474	1.0360
	21.00	3.1398	2.7797	2.4074	2.0241	1.6314	1.2307	0.8240
	21.25	2.4964	2.2100	1.9141	1.6094	1.2972	0.9786	0.6552
	21.50	1.9844	1.7568	1.5216	1.2794	1.0312	0.7779	0.5208

These values can be used to estimate the number of galaxies that could be detected by GAIA ASM1 when adopting different aperture sizes. To ensure a safe detection, so as not to be swamped with undesirable data, a $SNR$ of about 4 is required, which is achieved, e.g., for $<\!\mu\!>_{ap}=20.5$ with a $3\times3$ arcsec$^2$ aperture. With such an aperture, most galaxies of $I=16$, which have $r_e=3.1$ and $<\!\mu\!>_e=20.5$, according to Tables 4.4 and 4.5, will be detected during most scans. On the other hand, fainter and smaller galaxies will only be detected some times, e.g. if the center of the galaxy happens to lie near the center of one of the apertures used for detection. Besides, since the average surface brightness within a radius smaller than the effective radius can be significantly larger than $<\!\mu\!>_e$, with a smaller aperture size one could detect an higher number of galaxies, but on the other hand such a choice could yield an higher number of false detections, resulting in loss of telemetry.

A more detailed understanding of the issue requires a more complete simulation taking into account the surface brightness radial profiles for E and D galaxies as they were modelled in Chapter 4 and the different possible positions of the galaxy center with respect to the aperture center. Numerically integrating the profiles of E and D galaxies of different magnitudes over square areas of different sizes and whose centers are randomly displaced from the galaxy center, one can obtain a clearer picture of which galaxies will be detected and with which probability. The percentages of detected galaxies obtained from these simulations are given as function of total magnitude and aperture size in Tables 5.4 and 5.5 for E and D galaxies, respectively, where a $SNR$ of 4 has been assumed to indicate a detection.

Table 5.4: Detection Probability of Elliptical Galaxies $P_{det,E}$ in the ASM1 as function of galaxy total $I$ magnitude and aperture size, according to numerical simulations. Detection probability expressed in percentage, aperture size in arcsec$^2$. See text for details.

$P_{det,E}$		Aperture Size
		$1.0\!\times\!1.0$	$1.5\!\times\!1.5$	$2.0\!\times\!2.0$	$2.5\!\times\!2.5$	$3.0\!\times\!3.0$	$3.5\!\times\!3.5$	$4.0\!\times\!4.0$
$I$	16.00	100	100	100	100	100	100	99
	16.25	100	100	100	98	97	95	91
	16.50	100	99	98	93	95	83	70
	16.75	100	94	90	84	76	58	46
	17.00	91	87	80	66	37	27	07
	17.25	72	65	52	25	00	00	00
	17.50	45	32	00	00	00	00	00
	17.75	07	00	00	00	00	00	00
	18.00	00	00	00	00	00	00	00

Table 5.5: Detection Probability of Disk Galaxies $P_{det,D}$ in the ASM1 as function of galaxy total $I$ magnitude and aperture size, according to numerical simulations. Detection probability expressed in percentage, aperture size in arcsec$^2$. See text for details.

$P_{det,D}$		Aperture Size
		$1.0\!\times\!1.0$	$1.5\!\times\!1.5$	$2.0\!\times\!2.0$	$2.5\!\times\!2.5$	$3.0\!\times\!3.0$	$3.5\!\times\!3.5$	$4.0\!\times\!4.0$
$I$	16.00	100	100	100	100	100	100	99
	16.25	100	99	99	97	97	94	91
	16.50	98	94	95	86	92	79	66
	16.75	81	86	82	76	63	55	46
	17.00	60	62	58	50	26	23	08
	17.25	32	27	20	02	00	00	00
	17.50	00	00	00	00	00	00	00

As expected, the number of detected galaxies increases steadily with the decrease in the aperture size both for Es and Ds. Below a certain aperture size, however, this estimation becomes rather uncertain due to the present poor knowledge of brightness profiles in the galaxy innermost regions. Besides, since some kind of median filtering will be required in order to discriminate between bright stars and galaxies, the measured signal will in fact be smaller than estimated, and this effect will become significant as the aperture size decreases. In any case, a lower limit to the aperture size must be set depending on the number of false detections that are deemed acceptable.

The thorough understanding of the problems connected with false detections requires the design, implementation and testing on real fields of a dedicated algorithm for galaxy detection, but this was beyond the scope of the present study. It is however believed that an aperture size of $2 \times 2$ arcsec$^2$ is large enough to be safely used in the following considerations. With such a choice, E and D galaxies of, e.g., $I=17$ are detected with a probability of about $80\%$ and $58\%$ respectively, whereas for $I\gtrsim17$ the detection probability quickly falls to zero. With an average number of scans of 75 per astrometric instrument (see Figure 2.4), should galaxies be observed for the whole mission in one Astro, an average of 60 and 45 scans would be obtained for Es and Ds, respectively, which, as we shall see in Chapter 7, are largely sufficient to reconstruct a high-resolution two-dimensional image. It can therefore be concluded that galaxies brighter than $I_{det}=17$ would be detected during the 60% of the scans with a $SNR=4$ in the ASM1 using an area of $2 \times 2$ arcsec$^2$ for detection. According to Table 4.4, there are about 4 million galaxies brighter than this limit on the whole sky. At low Galactic latitudes, though, galaxy detection becomes increasingly tricky due to the presence of Galactic nebulae and lots of stars. While it would be desirable to observe galaxies as well as Galactic nebulae down to very low Galactic latitudes, it is suspected that this could yield a large amount of false detections and thus loss of telemetry. Very conservatively, the galaxy detection could be carried out only when $\vert b\vert>15$, i.e. over 75% of the sky, thus leaving a total of 3 million observable galaxies. It must also be noted that the readnoise appears to dominate the total error budget, and that therefore the adoption of a larger sample size in the ASM1 could result in a much higher number of detected galaxies.

A typical, intrinsically bright, galaxy from the ``Third Reference Catalogue of Bright Galaxies'' ([de Vaucouleurs et al. 1991], RC3 in the following) has an absolute magnitude of $M_B=-19$, according to Figure 2 in [Impey and Bothun 1997], which using the average color index $B-I=2.0$ obtained for bright galaxies by [Prugniel and Héraudeau 1998] gives $M_I=-21$. Under this conservative assumption, for $I_{det}=17$ mag we obtain a distance modulus of $(m-M)_{det}=38$ mag and therefore a distance of $d_{det}=400$ Mpc or a redshift of $z_{det} \simeq 0.1$, using for the Hubble constant the value of $H_0=71$ Km/sMpc recently obtained by [Mould et al. 2000]. Clearly, since the RC3 only contains galaxies brighter than $B\simeq15.5$, i.e. on average $I\simeq13.5$, most detected galaxies will be intrinsically fainter, and therefore lie correspondingly farther, than assumed above, thus increasing the horizon of galaxy observations.

These numbers were derived under a number of assumptions, both optimistic and pessimistic. On the whole this gives a rather large uncertainty on the final numbers, but probably not more than a factor of two in each direction, i.e. from 1.5 to 6 million galaxies. Note, however, that following a different line of reasoning based on the analysis of several HST MDS fields, [Lindegren 2000] obtained a total number of 6 million detected galaxies, with an estimated uncertainty of about three in each direction. This remarkable agreement between estimations obtained with different methods confirms the reliability of the combination of statistical modelling and numerical siulation in mission planning.