# 6.3.4 Line Spread Function (LSF)

The RVS LSF model is the same as that used in the astrometry (see Section 3.3.5). The LSF $L(u)$ is modelled as the linear combination of a 1D mean profile ${H}_{0}(u)$ of fixed amplitude and a set of $N$ 1D Basis Functions (BF) ${H}_{n}(u)$ with amplitudes ${h}_{n}$, where

$$L(u)={H}_{0}(u)+\sum _{n=1}^{N}{h}_{n}{H}_{n}(u).$$ | (6.2) |

The BFs were derived from a large set of theoretical LSFs (see Section 5.3.5). ${H}_{n}(u)$ are fixed and ${h}_{n}$ are calibrated by simultaneously fitting them to data. It is the trended ${h}_{n}$ which are used to reconstruct the time-dependent LSF in Equation 6.2 (see Figure 6.4).

LSF-AL ${h}_{n}$ are fitted by comparing the entire observed spectrum to high-resolution ground-based spectra of the same star, convolved with the LSF-AL ${H}_{n}(u)$, hence the single waveband in Table 6.6. There are many more 2D windows than suitable high-resolution ground-based spectra, which permits the LSF-AC ${h}_{n}$ to be fitted as a function of wavelength in 12 different 2-nm wavebands.

Property | AL | AC |

CaU duration (hr) | 1 | 0.5 |

Input data | Calibrated spectra | 2D windows |

Data amount | 1-6 spectra | $\sim $10 2D windows |

Data manipulation | Spectra Doppler-shifted to rest | Average AC profile |

Data aggregation | No | 1 spline per CaU |

# BFs ($N$) | 8 | 5 |

# wavebands | 1 | 12 |

Model | High-res. spectra $\u229b$ with AL BFs | AC BFs only |

Sun phase | No | Yes |

CaU trending | Median or linear | Spline |

## LSF-AL

Gaia DR2 used two different LSF-AL. One was an on-ground LSF-AL per CCD and field-of-view configuration, derived from the aforementioned theoretical LSFs. These were used for the data acquired after the first decontamination at OBMT 1317. The other LSF-AL was derived from in-flight EPSL data, in order to take account of the degradation of the RVS resolution before the OBMT 1317 decontamination. Both of these LSF-AL are described in more detail in Sartoretti et al. (2018, Section 5.4).

Therefore, Gaia DR2 had two different LSF-AL trend epochs, but the LSF-AL in each configuration was constant with time within both epochs. Gaia DR3 improves on this by aggregating spectra over one-hour Calibration Units (CaUs) and deriving ${h}_{n}$ in every configuration and every CaU. Each configuration and CaU has its LSF-AL constructed using Equation 6.2. If it passes a quality assessment, the ${h}_{n}$ value is included in the trend, which is either a constant (median) or linear fit. It is these trended ${h}_{n}$ values that are used to reconstruct the LSF-AL to be convolved with the synthetic spectra and thus to generate the templates. These templates are then compared to observed spectra for wavelength calibration (see Section 6.3.3) and to derive radial velocities (see Section 6.4.8).

## LSF-AC

The DR3 RVS pipeline derives the RVS LSF-AC for the first time. It is needed as input to deblend overlapping spectra, to model the AC flux distribution, and for the estimation of the flux loss out of the window, which is taken into account in the estimation of grvs_mag (see Section 6.4.5).

2D windows allow AC fluxes to be summed in the AL direction to derive an AC profile for each waveband. A spline is fit to each AC profile. The AC peak position is found and used to recentre to a standard grid with the AC peak position at the centre.

The size of the windows used to obtain the data is 10 AC pixels, which only measures the core of the LSF-AC. A wing profile is extrapolated by adding a series of 15 equi-spaced data points on a straight line, starting at the end points of the aforementioned spline and ending at zero at $\pm $20 pixels (the extent of the theoretical LSFs). The extended AC profiles are then normalised such that the area under the curve is 1.0. A spline is fit to the extended AC profile and resampled to a standard grid.

The AC splines are averaged over all 2D windows for each configuration, waveband and CaU combination. The averaged AC spline is then fit to the LSF-AC ${H}_{n}(u)$ to derive the CaU ${h}_{n}$, rejecting any outliers. The CaU ${h}_{n}$ are trended using splines.

## AC peak position

Gaia DR3 also derives the RVS AC peak position of the LSF-AC for the first time. Like the LSF-AC, the AC peak position is also needed as input to deblend overlapping spectra to model the position of the AC flux distribution and for the estimation of the flux loss out of the window, which is taken into account in the estimation of grvs_mag (see Section 6.4.5). The AC peak position is found when calibrating the LSF-AC (see Section 6.3.4). It is fit as a function of field angles, as follows:

$$A{C}_{\text{CCD}}=\sum _{m=0}^{2}\sum _{n=0}^{2}{P}_{mn}{\eta}^{m}{\zeta}^{n}+A\mathrm{sin}(\theta )+B\mathrm{cos}(\theta )+C\mathrm{sin}(2\theta )+D\mathrm{cos}(2\theta ),$$ | (6.3) |

where $\eta $ and $\zeta $ are the field angles field-of-view coordinates of the calibration stars at the fiducial time of the CCD, $\theta $ is the Sun phase at the time of the observation. Calculating $A{C}_{\text{CCD}}$ during the EPSL does not require the coefficients $A$, $B$, $C$ and $D$ because the EPSL has a constant precession rate. The NSL has a precession period, which the coefficients $A$, $B$, $C$ and $D$ model. The CaU duration is 30 hours, which means 2D windows are aggregated for 30 hours, collecting 600$-$700 windows per CaU per configuration, with which to derive $A{C}_{\text{CCD}}$.

Each ${P}_{mn}$ is trended over its CaU values with a median, with the exception of ${P}_{00}$, which is modelled with a quadratic. The coefficients $A$, $B$, $C$ and $D$ are trended with a sinusoid.