# 7.4.3 Underlying orbital models

Four kinds of orbital solutions are considered here, labelled as SB1, SB1C, TrendSB1 (FirstDegreeTrendSB1, SecondDegreeTrendSB1) and StochasticSB1. They are described below. The models will be fitted through a least-squares procedure, the objective function being the ${\chi}^{2}$ except otherwise stated. The errors on the parameters are extracted in the classical way from the variance-covariance matrix.

## Orbital models of type SB1

The general eccentric Keplerian model is, for the spectroscopic channel, expressed in terms of the Campbell coefficients (see Equation 7.49 and the appendices in Halbwachs et al. 2023). The model radial velocity is given by

$$\mathrm{RV}(t)=\gamma +K\left[\mathrm{cos}(v(t)+\omega )+e\mathrm{cos}(\omega )\right]$$ | (7.30) |

where $v(t)$ is the true anomaly which is deduced from the eccentric anomaly $E$ by

$$\mathrm{tan}\frac{v}{2}=\sqrt{\frac{1+e}{1-e}}\mathrm{tan}\frac{E}{2}$$ | (7.31) |

which in turn is expressed as a function of the mean anomaly $M$

$$M=E-e\mathrm{sin}(E)=\frac{2\pi}{P}(t-{T}_{0})$$ | (7.32) |

where ${T}_{0}$ is the time of passage at periastron. The $\omega $ is the longitude of periastron measured in the sense of the orbit with its origin at the ascending node. In this model, the eccentricity $e$ enters as a very non-linear parameter. The parameters to determine are $P$, $\gamma $, $K$, $e$, $\omega $, ${T}_{0}$ (respectively, the period, the centre-of-mass velocity, the semi-amplitude, the eccentricity, the longitude of periastron and the time of passage at periastron).

The first step of the computation is devoted to the determination of the period. Some significance tests are made on the very existence of the period. If the period is not significant, the model is fitting noise and is considered as being invalid. This is a statistical decision.

## Orbital models of type SB1C

Another particular kind of orbit is the circular one, i.e. $e=0$. In this case, the model is much more simple and is expressed by the following equation

$$\mathrm{RV}(t)=\gamma +K[\mathrm{cos}(\frac{2\pi}{P}(t-{T}_{0})]$$ | (7.33) |

where here ${T}_{0}$ is the time of maximum velocity and the $\omega $ is fixed at zero corresponding to the maximum velocity. The free parameters to determine are restricted to $P$, $\gamma $, $K$, ${T}_{0}$. The model is linear except, of course, for the determination of the period. The significance level of the period is here also determined.

## Models of type TrendSB1, non_linear_spectro

It happens that the radial velocity evolution as a function of time just exhibits a trend. This could be a transient behaviour just observed by chance or may correspond to a small piece of slow motion. It could also corresponds to an orbital motion with a period well longer than the span of time of the observations by the satellite. This span of time is 34 months for the Gaia DR3. However, it could be slightly shorter for some objects because it combines with the gaps in the scanning law. The present trend modelling is restricted to one-degree and two-degree polynomials with time as the independent variable. The formalism used for the fit is the same as for Hipparcos. The coefficients given in the nss_non_linear_spectro table are the coefficients of the polynomials with respect to the Gaia DR3 reference epoch. The non_linear_spectro solution is compared to the SB1 orbital solutions using a ${\chi}^{2}$ test. The goodness of fit residuals are compared and the best solution is kept. It should be made clear that this comparison could be an approximation.

## Models of type StochasticSB1

When neither the fit of a non_linear_spectro solution nor the fit of an orbital solution corresponding to a significant period are possible, the solution is classified as “StochasticSB1” and not published in the NSS tables, retaining the median RV in the main catalogue instead. These are objects that have very little chance of being associated with a mere orbital or trend motion. Some badly sampled orbits and high degree multiple systems could fall into this category, perhaps momentarily. Some of them will certainly benefit from an increase of the number of transits. These last results are not part of the Gaia DR3 release. However, the solutions were transmitted to the solution combiner, Section 7.7, and thousands could be rescued, thanks to the combination with an astrometric orbit.

## The significance level of the period

To search for the period, a Fourier method is used. In power spectra, the peaks in the periodogram are quite numerous. At a fixed frequency and under the null-hypothesis of a process generating white-noise, the height of the peaks is distributed as a decreasing exponential. A test must be made to decide if a candidate frequency is significant by computing the significance level which is the probability under the null-hypothesis of a white-noise process to observe a peak at least as high. Since the algorithm is considering the highest peak, what is called here “significance level” (SL) is the probability under the null-hypothesis of white-noise to observe the highest peak at least as high. If this significance level is zero, the peak could not be considered as generated by the noise and thus a high suspicion is permitted that the outstanding peak is due to a deterministic periodic signal. In the case of even sampling of the time-series, such a significance level can be easily computed. However, in the case of irregular sampling and particularly in the case of the Gaia sampling, the situation is much more complex. To test for the significance level of the period, the procedure relies on the use of Monte-Carlo simulations performed in the conditions of the Gaia sampling. The first dependency of the statistical distribution is on the number of data points in the time series. Therefore, such simulations were done for each of the possibilities on this number. Keeping this number constant, it is easy to notice through the simulations that the statistical distribution of the highest peak corresponding to different configurations of the time sampling gives very similar statistical distributions. This is particularly true for the interesting zone where the significance level is lower than 0.2. Therefore various Monte-Carlo simulations were performed and the value of the height of the highest peak associated with various fixed significance levels has been computed. From these simulations, look-up tables are build that are used by the algorithms. Details on the process are given in Gosset et al. (2022).

The probability (conf_spectro_period) given in the NSS nss_two_body_orbit table is defined as 1–SL. It should be made clear that the significance level SL of the period mentioned here is unrelated to the significance of the spectroscopic solution, which has been defined as the signal-to-noise of the semi_amplitude_primary $K$. In the case of the spectroscopic processing, the so-called primary is associated with the sole star which is dominantly participating to the spectrum definition.