Author(s): Dafydd W. Evans

The model used for calibrating the fluxes for Gaia DR2 is the same as that for Gaia DR1 (see Carrasco et al. 2016) except for the small-scale calibration in which for Gaia DR2 the calibrations are further divided by field-of-view. In the following, more details are given for the flux accumulations, i.e., the reference catalogue update.

As part of the iterations to establish the photometric reference system, accumulations are used to determine an average flux value (see Figure 5.4). The accumulations refer to weighted summations that are carried out for each source of their individual flux values. The three used in Gaia DR2 are ${\sum}_{k}{w}_{k}$, ${\sum}_{k}{I}_{sk}{w}_{k}$ and ${\sum}_{k}{I}_{sk}^{2}{w}_{k}$ where ${I}_{sk}$ is the individual flux measurement for observation $k$ and source $s$ and ${w}_{k}=1/{\sigma}_{k}^{2}$ is the weight used. ${\sigma}_{k}$ is the error on the flux quoted by the IPD, see Section 2.3.2). Note that the same notation as Carrasco et al. (2016) is used to avoid confusion.

As for Gaia DR1, inverse variance weighted mean values are used for the averages. Weighted means were chosen since the fluxes are heteroscedastic, i.e., the flux errors vary depending on the gate and window class configuration of the observation. The most significant configuration is gating since the effective exposure time of the observation is determined by this. In addition to this, sigma clipping has been introduced for the second data release. This is needed since the flux distribution can be non-Gaussian due to the presence of outliers. The sigma value for this filtering is determined from a robust estimate of the scatter of the observed fluxes. Median absolute deviation was used for this and a five sigma clipping was done. The reasons for measuring the value used for the sigma rather than using the formal errors from the IPD are that the errors can also be outlier in nature and that the source could be variable. In the latter case, too many observations would be filtered out if the formal errors were used in the sigma clipping.

As mentioned above, the weighting scheme used is the inverse variance ${w}_{k}$. Thus, after sigma clipping has been applied, the weighted mean flux for each source, $\overline{{I}_{s}}$, is given by

$$\overline{{I}_{s}}=\frac{{\sum}_{k}{I}_{sk}{w}_{k}}{{\sum}_{k}{w}_{k}}$$ | (5.3) |

Since the errors quoted for the flux quantities derived from the IPD do not fully account for modelling errors, they will be a slight underestimate of the observed error distribution. An example of such modelling errors is due to a simplified PSF/LSF being used in IPD. For Gaia DR2, the PSF/LSF did not account for dependencies on time nor colour. These dependencies will gradually be added in future processing cycles. To account for these modelling errors, the intrinsic scatter of the fluxes is used in the error calculation in the same way as used for the mean photometry of the Hipparcos catalogue ESA (1997):

$${\sigma}_{\overline{{I}_{s}}}=\sqrt{\frac{{\sum}_{k}{I}_{sk}^{2}{w}_{k}-{\overline{{I}_{s}}}^{2}{\sum}_{k}{w}_{k}}{{N}_{\mathrm{obs}}-1}}\cdot \frac{1}{\sqrt{{\sum}_{k}{w}_{k}}}$$ | (5.4) |

where ${N}_{\mathrm{obs}}$ is the number of CCD observations. The weighted mean flux and error given in Equation 5.3 and Equation 5.4 are the quantities used in the photometric calibrations as the reference fluxes.

Note that the above calculations are also carried out on the fluxes extracted from the BP/RP spectra: integrated ${G}_{\mathrm{BP}}$ and ${G}_{\mathrm{RP}}$ and the SSC fluxes (Section 5.1.3). As for Gaia DR1, these are treated as fluxes to be calibrated in the same way as the IPD-derived G-band fluxes and thus are accumulated in order to provide reference fluxes for these calibrations. These quantities are used as the colour information needed in the various calibrations described previously in Section 5.3.3 and Section 5.3.4, i.e., mean colour information is used rather than instantaneous values. In future processing cycles, sources detected as varying in colour will be calibrated using instantaneous colour values.

Other useful quantities can be derived from the accumulations that can help in detecting variables or determining whether the calibrations are at the photon noise level or not.

The reduced ${\chi}^{2}$, sometimes known as the unit weight variance, is given by

$$\frac{{\chi}^{2}}{\nu}=\frac{{\chi}^{2}}{{N}_{\mathrm{obs}}-1}=\frac{{\sum}_{k}{I}_{sk}^{2}{w}_{k}-{\overline{{I}_{s}}}^{2}{\sum}_{k}{w}_{k}}{{N}_{\mathrm{obs}}-1}$$ | (5.5) |

where $\nu $ is the degrees of freedom. The ${\chi}^{2}$ can be used to provide a quantity known as a P-value:

$$P=Q(\frac{\nu}{2},\frac{{\chi}^{2}}{2})$$ | (5.6) |

where Q is an incomplete gamma function (Press et al. 1993). P-values can be used to identify variable sources if the quoted errors are reliable and the calibrations have accounted for all systematic errors. For constant sources, the distribution of the P-values will be flat between 0 and 1, whereas for variable sources, it is strongly skewed towards 0. The advantage of P-values is that they are easier to interpret directly in comparison to the reduced ${\chi}^{2}$ and that the number of false positives from any variability detection limit is simply this limit multiplied by the number of sources. However, due to the underestimation of the individual flux errors from the IPD, the P-values for almost all sources is very close to 0 and thus are viewed as variable from this measure. It should be reiterated that, although no rescaling is carried out on the individual flux errors, the measurement of the error on the weighted mean flux, Equation 5.4, since it accounts for the scatter in the data for each source, is a realistic estimation of the error and is thus not underestimated.

Another way of estimating the additional scatter caused by variability is given by Equation 87 of van Leeuwen (1997) (page 349). This simplifies the variability as a sinusoidal variation and calculates an excess noise quantity:

$$A=\sqrt{\frac{2({\chi}^{2}-\nu )}{{\sum}_{k}{w}_{k}}}$$ | (5.7) |

When calculating this quantity, checks must be made that the numerator is not negative. If this is the case, then the source is unlikely to be variable and no excess noise is measurable.

For variable sources, it should be noted that the algorithm applied to generate the mean photometry, while quite robust for constant sources, is still not ideal. For a variable source the scatter of the measurements is larger, and this means that outliers may be not rejected because they are still within a few sigma from the median value. If these outliers are very faint, hence with correspondingly small uncertainties in flux, it may shift the weighted average estimate of the mean source photometry by several magnitudes. Users interested in photometric studies of variables should therefore exploit the epoch photometry provided in Gaia DR2 whenever possible.