8.3.4 Error model for Astrometry
Author(s): Thierry Pauwels, Aldo Dell’Oro, Federica Spoto, Paolo Tanga
Systematic and random uncertainties
Each transit can contain up to 9 positions and, when giving the uncertainties, there are complicated correlations between the errors of different CCDs of the same transit. The ideal situation would be to have, for each transit, a complete 9 $\times $ 9 covariance matrix, where each element of the covariance matrix would be a 2 $\times $ 2 covariance matrix of Right Ascension and Declination. However, the task to determine the precise correlations of the various contributions to the error budget is very complicated and difficult. Therefore, we adopted a simplified approach and separated the uncertainty into a systematic and a random part. The systematic part is the part of the uncertainty that is the same for all positions of the same transit; the random part is the part of the uncertainty that is statistically independent from one CCD to another. This means that we assumed that the complete covariance matrix W of the transit could be written as:
$$\bm{W}=\left(\begin{array}{cccc}\hfill {\bm{W}}_{1}\hfill & \hfill \mathbf{\U0001d7ce}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \mathbf{\U0001d7ce}\hfill \\ \hfill \mathbf{\U0001d7ce}\hfill & \hfill {\bm{W}}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \mathbf{\U0001d7ce}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill \mathbf{\U0001d7ce}\hfill & \hfill \mathbf{\U0001d7ce}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\bm{W}}_{9}\hfill \end{array}\right)+\left(\begin{array}{cccc}\hfill {\bm{W}}_{\text{s}}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill \\ \hfill {\bm{W}}_{\text{s}}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {\bm{W}}_{\text{s}}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill \end{array}\right),$$  (8.10) 
where ${\bm{W}}_{n}$ is the covariance matrix of Right Ascension (RA or $\alpha $) and Declination (DEC or $\delta $) of the AF$n$ position, which we call the random uncertainty of the AF$n$ position, ${\bm{W}}_{\text{s}}$ is a constant covariance matrix throughout the transit, which we will call the systematic uncertainty of the transit, and $\mathbf{\U0001d7ce}$ is a $2\times 2$ matrix of zeros. Note that although we call this uncertainty ‘systematic’, ‘systematic’ is to be understood ‘inside a transit’, and we assume that there is no correlation from one transit to another transit.
The systematic uncertainties are given in the archive in the table sso_observation in the fields ra_error_systematic, dec_error_systematic, and ra_dec_correlation_systematic. The random uncertainties are given in the same table in the fields ra_error_random, dec_error_random, and ra_dec_correlation_random.
Uncertainty from the centroiding
The uncertainty from centroiding was received from signal processing in pixels in an ALAC coordinate system and converted to Right Ascension and Declination. The errors in AL and AC are normally uncorrelated, but after rotation to an $(\alpha ,\delta )$ coordinate system, the errors of $\alpha $ and $\delta $ become highly correlated.
In AL, uncertainties are small and show the extreme precision of Gaia. For SSOs fainter than $\sim $16 mag, in AL, this will be the dominating contribution to the random uncertainty. These uncertainties are purely random without any contribution to the systematic uncertainty of the transit.
In AC, for objects fainter than magnitude 13, the binning policy was to bin all pixels in AC to a single sample, and our only information was that the object was inside this sample. Therefore, the position was given as the centre of the sample and the uncertainty was given as the dispersion of a rectangular distribution over the sample. Moreover, the errors from one CCD to another were treated as uncorrelated. For AC, this was in fact an oversimplification. The transmitted window was defined in SM based on an onboard centroiding. This means that the SSO was normally in one of the central two pixels of the window. This window was propagated to the AF CCDs so as to keep the object in the centre pixels in the case of a star. The proper motion of the SSO will cause it to drift away from the centre. In the first CCDs (AF1, AF2, …), the SSO will still be rather close to the central pixel, so that the given uncertainties were overestimated. But, for the subsequent CCDs, this was no longer the case. For the last CCD, the error will be underestimated if the object was about to leave the transmitted window along the AC border. However, no attempt has been made to determine if the object was really leaving the transmitted window nor whether this was along the AL or AC border. The drift away from the window centre is systematic and causes the errors in reality to be correlated. Nevertheless, we think the simplified approach was justified, given the fact that the uncertainties are large, and, consequently, weights will be small whenever used in, e.g., orbit computation. A more correct model would hardly contribute anything to the orbit computation. And even with a more correct error model, we would still be left with the problem that errors in AC are highly nonGaussian.
In AC, for objects brighter than magnitude 13, 2D windows were downlinked to earth, and a 2D centroid fitting was possible. In that case, the uncertainty in the AC direction is comparable to (but still a bit larger than) the uncertainty in the AL direction.
Uncertainties given in the $(\alpha ,\delta )$ coordinate system may appear to be large due to the large uncertainty in AC, which, after rotation, has a contribution to the uncertainty in both Right Ascension and Declination. However, taking into account the correlation, the user can recover the precise AL component of the position. Whereas uncertainties in Right Ascension and Declination are typically of the order of 500 $\mathrm{mas}$, the real uncertainty in AL is often smaller than 1 $\mathrm{mas}$.
Excess noise
In Gaia DR2, we found that positional errors for bright objects were much larger than the computed uncertainties. Therefore, SSOs brighter than magnitude 10 were removed from the catalogue. In Gaia DR3, for SSOs, a different policy was adopted. This socalled excess noise was modelled by comparing the postfit residuals of the positions to their expected distribution from the computed centroiding uncertainty and added to the error budget. This excess noise is only dependent on the preliminary magnitude and, for DR3, the following formula was derived and adopted:
$$\{\begin{array}{cc}{\u03f5}_{\mathrm{ex},\mathrm{AL}}=0.72\text{mas}\times \mathrm{exp}[0.63578({m}_{G}10)],\hfill & \text{in AL},\hfill \\ {\u03f5}_{\mathrm{ex},\mathrm{AC}}=1.5\text{mas}\times \mathrm{exp}[0.32673({m}_{G}10)],\hfill & \text{in AC},\hfill \end{array}$$  (8.11) 
where, again, ${m}_{G}$ is the preliminary magnitude.
The contribution from the excess noise is only to the random uncertainty. There is no contribution to the systematic uncertainty. The source of this noise is unknown. Possible explanations might be the photocentric shift or the extended angular size of the object.
Uncertainty from the attitude
The attitude of the satellite gives a contribution to both the random and the systematic uncertainty. Here again, the model that we used was a simplification of the reality. The errors in the OGA3 (OGA for OnGround Attitude from AGIS) attitude have typical frequencies of the order of 0.2 Hz. This means that the error from the attitude between two adjacent CCDs, i.e., with epochs 4.5 s apart, will be highly correlated, whereas positions from CCDs that are far from each other, such as AF1 and AF9, will hardly be correlated. A detailed model of this is, however, hard to set up. Therefore, we adopted the same approximation as in Equation 8.10. The typical random and systematic uncertainties from the OGA3 attitude were determined by analysing, on one hand, the positions of bright stars with welldetermined proper motions and parallaxes, and, on the other hand, the positions of SSOs. In both cases, we concentrated on the magnitude range where the centroiding errors are negligible compared to the errors from the attitude. Table 8.5 lists the uncertainties that were deduced from an analysis of AGIS 3.2, the version of AGIS to process the SSOs. These values were adopted for the uncertainty contribution from the attitude for SSOs.
AL  AC  
Contribution to the systematic uncertainty  0.051 $\mathrm{mas}$  2.3 $\mathrm{mas}$ 
Contribution to the random uncertainty  0.20 $\mathrm{mas}$  1.37 $\mathrm{mas}$ 
Note that when one computes the motion on the sky of an SSO, the error due to the attitude will also cause an error on the computed motion of the SSO. This error has, however, not been modelled for AGIS 3.2.
The above mentioned values are those for which the attitude behaved well. There are known epochs for which the attitude behaved less well. If the attitude was known to be too poor, the position was rejected (see Section 8.3.3). However, in the case of poorer attitude not bad enough to reject the position, the systematic uncertainty in AL of the attitude was adapted. 666 epoch intervals were identified with such increased systematic uncertainty in AL. The largest value of the increased systematic uncertainty in AL is 0.166 $\mathrm{mas}$. Larger values caused the transit to be rejected. 11.7 % of the transits have such increased systematic uncertainties in AL.
Uncertainty from nonapplied corrections
In Gaia DR2, a basic processing had been applied with the intention to apply highprecision corrections in subsequent versions of the catalogue. Without the highprecision corrections, the positions were affected with additional errors, which were included in the error model. However, in Gaia DR3, the positions are:

•
defined to be those of the photocentre and not the centre of mass;

•
defined to be those not corrected for the effect of relativistic light bending;

•
derived using the most complete model for aberration correction.
Hence, in Gaia DR3, there is no more contribution to the error budget from the socalled nonapplied corrections.
Combined uncertainty
The three contributions to the uncertainty were supposed to come from statistically independent errors. Therefore the total uncertainty was computed as
$${\sigma}_{\mathrm{combined}}=\sqrt{{\sigma}_{\mathrm{centroid}}^{2}+{\sigma}_{\mathrm{excessNoise}}^{2}+{\sigma}_{\mathrm{attitude}}^{2}}.$$  (8.12) 