5.8 Expected Telemetry Rate

As it was mentioned at various times, the amount of data one would ideally like the GAIA satellite to transmit to the ground is far larger than the available telemetry rate. Any observing proposal must therefore provide an estimate of the necessary effort in terms of telemetry rate which is implied by the suggested observations.

Generally speaking, we would like to observe galaxies in all sky areas that trigger the detection and from a suitable surrounding area, and to send the corresponding data to the ground. Such a surrounding area could be defined as composed by all the square areas of a given side that are adjacent to the areas where an excess surface brightness has been detected, as it is shown in Figure 5.3, where a side of 2 arcsec was chosen for illustrative purposes.

Figure 5.3: Galaxy Data Transmission. The solid squares indicate the sky areas where an excess surface brightness has been detected, while the dashed line delimits the sky area from which galaxy data are trasmitted to the ground. Right arrows indicate the satellite scan direction. The GAIA Astro Airy Disk is also shown to illustrate the high resolution achievable in galaxy observations.

The outlined observation strategy appears to satisfactorily cover the galaxy regions whose surface brightness is just below the detection limit, thus allowing to study the galaxy morphology in greater detail. In the following, however, in order to estimate the telemetry rate required for galaxy observations, a few simplifying assumptions will be made. It is assumed that data are transmitted from circular areas centered on the galaxy center, and that the radius $r_t$ of this areas can be written as

$$r_t=r_e+\Delta\,r~,$$ (5.7)

where $\Delta\,r$ is a positive constant. The overall sky area $\Omega_t$ within a radius $r_t$ for all galaxies brighter than $I$ could then be used as a rough estimate of the overall sky area from which data should be transmitted to the ground in order to observe all galaxies down to this magnitude. Neglecting the possible superposition on the sky between different galaxies, which is correct as far as the sky area in consideration is reasonably small, $\Omega_t$ can be written as

$$\begin{split}\Omega_t(I)&=\Omega_{sky}\int_{-\infty}^I\pi\,r_... ...{d}I'+ \pi\,(\Delta\,r)^2\,\,\Omega_{sky}\,N_c(I)~, \end{split}$$ (5.8)

Table 5.7: Fraction of sky $\Omega _t/\Omega _{sky}$ within a radius $r_t=r_e+\Delta\,r$ for all galaxies brighter than $I$ for some values of $I$ and $\Delta\,r$. $\Omega _t/\Omega _{sky}$ in $10^{-6}$ sky, $\Delta\,r$ in arcsec and $I$ in magnitudes. Values calculated via Newton integration (see Section 9.7 in [Press et al. 1996]) of Equation 5.8.

$\frac{\Omega_t}{\Omega_{sky}}$		$\Delta\,r$
		2	4	6	8	10
$I$	10	5.9559699	6.4369017	6.9178336	7.3987655	7.8796973
	11	13.015628	14.507042	15.998455	17.489869	18.981282
	12	28.234900	32.617360	36.999820	41.382281	45.764741
	13	62.711973	74.930181	87.148388	99.366598	111.58480
	14	147.90940	180.29082	212.67225	245.05368	277.43510
	15	379.93125	461.71573	543.50020	625.28469	707.06916
	16	1059.6399	1257.0964	1454.5528	1652.0093	1849.4657
	17	3118.8339	3576.2198	4033.6057	4490.9917	4948.3775
	18	9354.6744	10375.459	11396.243	12417.028	13437.812
	19	27814.818	30020.274	32225.728	34431.184	36636.639
	20	80550.210	85187.870	89825.529	94463.189	99100.849

Numerical values of $\Omega _t/\Omega _{sky}$ for different values of $I$ and $\Delta\,r$ are given in Table 5.7. From these it can be concluded that, with $\Delta\,r=6$ arcsec, all galaxies brighter than the detection limit $I_{det}=17$ would cover about $0.4\%$ of the sky. The telemetry rate required to cover such a sky area can be derived as function of the adopted sample size under the assumptions that each sample value is coded into 16 bits and that a loss-less compression factor of 16/5 can be applied before transmission. The assumption of such a relatively high compression factor is deemed realistic, since most sample values will be low, thus allowing a very efficient compression.

Under these assumptions, the required telemetry rate after compression $TR$ for galaxy observations can be estimated as

$$TR=\sin\phi_{lim}\,B\,\frac{\Omega_t}{\Omega_{sky}}\,\frac{v_s\,h_{fov}}{A_{sam}}\,BPS~,$$ (5.9)

where $\phi_{lim}$ is the chosen lower limit in Galactic latitude for galaxy observation ( $\sin \phi_{lim}$ is the fraction of sky where the absolute value of the Galactic latitude $b$ is greater than $\phi$), $B$ the number of photometric bands in which observations will be carried out, $v_s$ the scan velocity of the satellite, $h_{fov}$ the height of the field of view, $A_{sam}$ the sample size and $BPS$ indicated the number of bits per sample after compression. Values of $TR$ are given in Table 5.8 for the presently assumed values of the relevant parameters.

Table 5.8: Telemetry rate after compression required for galaxy observations as function of the sample size. Values calculated from Equation 5.9, where $\phi_{lim}=15~\textrm{deg}$, $B=5$, $\Omega_t/\Omega_{sky}=0.004$ and 5 bits/sample after compression have been assumed. Sample size expressed in pixels, telemetry rate in kbits/s.

$6\times8$	$6\times4$	$6\times2$	$1\times8$
108	217	433	650

As mentioned in Section 2.6, a total telemetry rate of about 1 Mbit/s after compression is presently foreseen for GAIA. From Table 5.8, it appears that the observation of galaxies, carried out with $6\times8$ or $6\times4$ pixels/sample, as suggested in Section 5.7, would require a significant, but probably not unreasonable, part of the total telemetry. On the other hand, observing the assumed fraction of the sky with a smaller sample size would require a prohibitively high telemetry rate.