# 2.3.1 GOG Error models

Author(s): Eduard Masana

In GOG the recommendations from the various CUs have been included in order to obtain the most complete picture of Gaia performance as possible. The following section briefly describes the error models that have been used in GOG to help users to understand what is being used internally in GOG to generate the different outputs.

Note that in this version, the number of outliers, both in astrometry and photometry, is larger than expected. The origin of this is under investigation.

## Astrometric Data Models

To compute the astrometric combined parameters we have used the following simplified model following the recipe outlined in the Science Web Performance.

1. 1.

The end of mission precision on the astrometric parameters does not depend only on the error due to the location estimation with each CCD. There are calibration errors due to the CCD calibrations, the uncertainty of the attitude of the satellite and the uncertainty on the basic angle. Those are a function of magnitude and are detailed in Sect. 4.3 of de Bruijne et al. (2005). Then, for a source of a given magnitude we have calculated the field-of-view transit level, by doing:

 $\sigma_{cal}=\sigma_{cal_{CCDlevel}}/\sqrt{9}$
2. 2.

The same effective scan geometry factor is used for the whole sky.

3. 3.

Centroiding errors are computed using the Cramer Rao (CR) lower bound that requires the LSF derivative for each sample, the background, the readout noise and the source integrated signal. These values are used for the sigma parallax computation. We are adding the straylight effects to the background.

4. 4.

We are enabling the activation of gates (see Table 1.3).

The parallax error $\sigma_{\pi}$ is calculated following the expression

 $\sigma_{\pi}=m\cdot g_{\pi}\cdot\sqrt{\frac{\sigma_{\eta}^{2}}{N_{eff}}+\frac{% \sigma_{cal}^{2}}{N_{transit}}}$
• m is the contingency margin (1.2)

• $g_{\pi}=1.47/\sin{\xi}$ is a geometrical factor where $\xi$ is known as the solar aspect angle ($\xi=45^{\circ}$).

• $\sigma_{\eta}^{2}$ is the centroiding accuracy for a CCD according the Cramer-Rao algorithm (discrete form).

• $N_{eff}$ is the number of elementary CCD transits $(N_{strip}\times N_{transit})$

• $N_{transit}$ is the number of FoV transits.

• $N_{strip}$ is the number of CCDs in a row on the Gaia focal plane.

• $\sigma_{cal}$ is the field-of-view transit level calibration error.

We have assumed sky-averaged relations between the parallax error and the position and proper motion ones :

• $\sigma_{\alpha}=0.787\sigma_{\pi}$

• $\sigma_{\delta}=0.699\sigma_{\pi}$

• $\sigma_{\mu_{\alpha}}=0.556\sigma_{\pi}$

• $\sigma_{\mu_{\delta}}=0.496\sigma_{\pi}$

## Photometry Parameters

The photometric standard errors of the $G$, $G_{\rm BP}$ and $G_{\rm RP}$ bands are calculated following the recipe outlined in the Science Web Performance.

These errors include all known instrumental effects, including stray-light. These errors have a 20% margin for the $G_{\rm BP}$ and $G_{\rm RP}$ bands.

GOG uses the single CCD transit photometry error $\sigma_{p,j}$ for photometric band $j$, defined as:

 $\sigma_{p,G}[mag]=10^{-3}\cdot(0.04895\cdot z^{2}+1.8633\cdot z+0.0001985)^{1/2}$

to compute the photometric error in the G band.

where:

 $z=MAX[10^{0.4\cdot(12-15)},10^{0.4\cdot(G-15)}]$
 $\sigma_{p,BP/RP}[mag]=10^{-3}\cdot(10^{a_{BP/RP}}\cdot z^{2}+10^{b_{BP/RP}}% \cdot z+10^{c_{BP/RP}})^{1/2}$

to compute the photometric error in the BP/RP band.

where:

 $z=MAX[10^{0.4\cdot(11-15)},10^{0.4\cdot(G-15)}]$

and:

 $a_{BP}=-0.000562\cdot(V-I_{C})^{3}+0.044390\cdot(V-I_{C})^{2}+0.355123\cdot(V-% I_{C})+1.043270$
 $b_{BP}=-0.000400\cdot(V-I_{C})^{3}+0.018878\cdot(V-I_{C})^{2}+0.195768\cdot(V-% I_{C})+1.465592$
 $c_{BP}=+0.000262\cdot(V-I_{C})^{3}+0.060769\cdot(V-I_{C})^{2}-0.205807\cdot(V-% I_{C})-1.866968$
 $a_{RP}=-0.007597\cdot(V-I_{C})^{3}+0.114126\cdot(V-I_{C})^{2}-0.636628\cdot(V-% I_{C})+1.615927$
 $b_{RP}=-0.003803\cdot(V-I_{C})^{3}+0.057112\cdot(V-I_{C})^{2}-0.318499\cdot(V-% I_{C})+1.783906$
 $c_{RP}=-0.001923\cdot(V-I_{C})^{3}+0.027352\cdot(V-I_{C})^{2}-0.091569\cdot(V-% I_{C})-3.042268$

The sky-average end-of-mission median straylight can be estimated by the formula de Bruijne et al. (2005):

 $\sigma_{G,j}=m\cdot\sqrt{\frac{\sigma^{2}_{p,j}+\sigma_{cal}^{2}}{N_{transits}}}$
• m is the 20% contingency margin (m=1.2 for $G$ and 1.0 for BP/RP as the margin is already included in their parametrisation)

• $N_{transit}$ is the number of FoV transits.

• $\sigma^{2}_{p,j}$ is the photometric error defined above for the $j$ band.

• $\sigma_{cal}$ is the calibration error of 4 mmag.

## RVS Parameters

For the RVS epoch computation, CU6 tables based on the star physical parameter are used to obtain the $\sigma_{V_{r}}$ and the $\sigma_{vsini}$.

The sky-average end-of-mission error calculation includes all known instrumental effects, including stray-light. This calculation is based on the Science Web Performance.

The radial velocity performance formal error is given by:

 $\sigma_{Vrad}[kms^{-1}]=1+be^{a(V-12.7)}$

where a and b are the constants, defined in the Table 2.9 for different spectral types and V denotes the Johnson V magnitude.

This performance is valid for $G_{RVS}$ up to 16 mag, with:

 $V-G_{RVS}=0.0119+1.2092(V-I_{c})-0.0188(V-I_{c})^{2}-0.0005(V-I_{c})^{3}$

## Astrophysical Parameters

In the current version of GOG, the errors of the astrophysical parameters (effective temperature, $logg$, radius, mass, luminosity, $\alpha$ elements and extinction) are set to zero, as their models was not available at the time of the simulation.

## Binary systems treatment

Binary and multiple objects are processed through GOG. It decides if the systems are resolved or not based on the minimum separation $\rho$ (in mas) which depends on the magnitude difference in the $G$ band $\Delta_{G}$ following this formulae derived from simulations : $\log(\rho/0.71)={2.24+0.56\log{\sigma_{F}}+0.076\Delta_{G}+0.018\Delta_{G}^{2}}$ with $\sigma_{F}=\sqrt{10^{2(0.209G-4.899)}+0.027^{2}}$.