Author(s): Thierry Pauwels, Aldo Dell’Oro, Federica Spoto, Paolo Tanga

Each transit can contain up to 9 positions, and when giving the uncertainties, there are complicated correlations between the errors of different CCDs for the same transit. To deal with all the possible correlations 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 the 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, as their practical exploitation would be.

Therefore, we adopted a simpler approach and separated the error into a systematic and a random part. The systematic part is the error component that repeats identically for all positions of the same transit; the random part is statistically independent from one CCD to another.

This choice translates into the assumption that the complete covariance matrix $W$ of the transit can be written as:

$$\bm{W}=\left(\begin{array}{cccc}\hfill {\bm{W}}_{1}\hfill & \hfill \mathbf{\U0001d7ce}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \mathbf{\U0001d7ce}\hfill \\ \hfill \mathbf{\U0001d7ce}\hfill & \hfill {\bm{W}}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \mathbf{\U0001d7ce}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill \mathbf{\U0001d7ce}\hfill & \hfill \mathbf{\U0001d7ce}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\bm{W}}_{9}\hfill \end{array}\right)+\left(\begin{array}{cccc}\hfill {\bm{W}}_{\text{s}}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill \\ \hfill {\bm{W}}_{\text{s}}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {\bm{W}}_{\text{s}}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\bm{W}}_{\text{s}}\hfill \end{array}\right)$$ | (4.8) |

where ${\bm{W}}_{n}$ is the covariance matrix of the right ascension and declination of the AF$n$ position, that we will call the random error of the AF$n$ position; ${\bm{W}}_{\text{s}}$ is a constant covariance matrix throughout the transit, which we will call the systematic error of the transit; and $\mathbf{\U0001d7ce}=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$.

The error from the centroiding is received from the IDU determination of the image parameters (IPD), expressed in pixels in an (AL, AC) 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 are highly correlated.

In AL, uncertainties are very small and correspond to the extreme precision of Gaia. These IPD uncertainties are considered to affect the random component without any contribution to the systematic error.

In AC, for objects fainter than magnitude 13, the binning policy is to bin all pixels in AC to a single sample, and the only information we have is that the object is inside this sample. Therefore the position is given as the centre of the sample and the uncertainty corresponds to the dispersion of a rectangular distribution over the sample. The errors from one CCD to another are treated as uncorrelated. This is in fact an oversimplification. The transmitted window is defined in SM based on an on-board centroiding. This means that the SSO is normally in one of the central two pixels of the window. This window is propagated to the AF CCDs and the object would remain in the centre pixels, if it were a star. The proper motion of an SSO, however, will produce a progressive drift away from the centre, the effect increasing linearly during the transit. In the first CCDs the SSO will still be rather close to the central pixel, so that the given uncertainties are overestimated. In the last CCDs the nominal error will be underestimated if the object tends to leave the window along the AC border of the transmitted window. However, no check has been done in Gaia DR2 to determine if an object is moving out of the window, along either the AL or AC border. The drift is a relevant effect and causes the errors to be correlated. Nevertheless, the adopted simplified approach is justified, given the fact that nominal errors turn out to be in any case large, and consequently the weights assigned to the AC position will be in any case very small. Even with a more correct error model, we would still be left with the problem that errors in AC are highly non-Gaussian.

In AC, for objects brighter than magnitude 13, 2-D windows are transmitted, and a two-dimensional centroid fitting is possible. In that case the uncertainty in AC direction is comparable to (but still larger than) the uncertainty in AL direction.

Uncertainties given in a (right ascension, declination) coordinate system may appear to be very 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 find back the very precise AL component of the position. While uncertainties in right ascension and declination will typically be of the order of 500 mas, the real uncertainty in AL is often smaller than 1 mas.

In order to facilitate this conversion the position angle of the scan is given for each position. It is contained in the position_angle_scan field and expressed using the usual astronomical convention of origin at equatorial North and positive values for rotations from North towards East.

All errors on the sky have the right ascension component multiplied by a factor $\mathrm{cos}(declination)$.

The attitude of the satellite gives a contribution to both the random and the systematic error. Here, again, the model that we present is a simplification of the reality. The errors in the OGA3 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, while positions from CCDs that are far from each other, such as AF1 and AF9, will have a lower correlation. A detailed model of this complex behaviour is very hard to set up.

Therefore, we determined the typical random and systematic error from the Oga3 attitude by analysing the positions of a large sample of bright stars with known proper motion and parallax, for which the centroiding errors are negligible compared to the errors from the attitude. Note that an error due to the attitude will also cause an error on velocity of the SSO, computed from its astrometry.

The following table lists the errors that we deduced from an analysis of AGIS 2.

AL | AC | |

Contribution to the systematic error | 0.32 mas | 1.2 mas |

Contribution to the error on the velocity | 0.03 mas/s | 0.1 mas/s |

Contribution to the random error | 0.5 mas | 2.1 mas |

For Gaia DR2, only a basic processing was performed. This means that a number of corrections that are necessary for the highest precision, were not applied.

Only rough estimates were made of the contribution to the error budget of effects including the discrepancy of light bending for a source at infinity with respect to a SSO (see Section 4.4.3), and the difference between the positions of the centre of mass and the photocentre. All these effects contribute only to the systematic error.