# 7.5.3 Underlying orbital models

Four kinds of orbital solutions are considered here and they are labelled SB2, SB2C, TrendSB2 and StochasticSB2. We describe them below.

## Orbital models of type SB2

The general eccentric Keplerian model is, for the spectroscopic channel, expressed in terms of the Campbell coefficients (see the appendices in Halbwachs et al. 2023). The model radial velocity is given by, for one of the component labelled 1,

$${\mathrm{RV}}_{1}(t)=\gamma +{K}_{1}\left[\mathrm{cos}(v(t)+\omega )+e\mathrm{cos}(\omega )\right]$$ | (7.36) |

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

$$\mathrm{tan}\frac{v}{2}=\sqrt{\frac{1+e}{1-e}}\mathrm{tan}\frac{E}{2}$$ | (7.37) |

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

$$M=E-e\mathrm{sin}(E)=\frac{2\pi}{P}(t-{T}_{0})$$ | (7.38) |

where ${T}_{0}$ is the time of passage at periastron. The $\omega $ is the longitude of periastron (for component 1) 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. Concerning the other component (2), the curve can be modelled by

$${\mathrm{RV}}_{2}(t)=\gamma -{K}_{2}\left[\mathrm{cos}(v(t)+\omega )+e\mathrm{cos}(\omega )\right].$$ | (7.39) |

The parameters to determine are $P$, $\gamma $, ${K}_{1}$, ${K}_{2}$, $e$, $\omega $, ${T}_{0}$ (respectively, the period, the centre-of-mass velocity, the semi-amplitude of component 1, the semi-amplitude of component 2, 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. It is interesting to note that in general

$$\frac{{\mathrm{RV}}_{1}-\gamma}{{\mathrm{RV}}_{2}-\gamma}=-\frac{{K}_{1}}{{K}_{2}}.$$ | (7.40) |

From this property, one deduces that in a ${\mathrm{RV}}_{2}$ versus ${\mathrm{RV}}_{1}$ diagram, the data should occupy a straight line whose slope is function of the mass ratio (noise effects excepted).

## Orbital models of type SB2C

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

$${\mathrm{RV}}_{1}(t)=\gamma +{K}_{1}\left[\mathrm{cos}(\frac{2\pi}{P}(t-{T}_{0}))\right]$$ | (7.41) |

$${\mathrm{RV}}_{2}(t)=\gamma -{K}_{2}\left[\mathrm{cos}(\frac{2\pi}{P}(t-{T}_{0}))\right]$$ | (7.42) |

where here ${T}_{0}$ is the time of maximum velocity of the first component and the $\omega $ is fixed at zero corresponding to the extremum velocities. The free parameters to determine are restricted to $P$, $\gamma $, ${K}_{1}$, ${K}_{2}$, ${T}_{0}$. The model is linear except, of course, for the determination of the period and for the constraint that ${\mathrm{RV}}_{1}$ and ${\mathrm{RV}}_{2}$ should be in antiphase. The significance of the period is here also determined.

## Models of type TrendSB2

The equivalent of the TrendSB1 (see Section 7.4.3) formalism can also be extended to the difference ${\mathrm{RV}}_{2}-{\mathrm{RV}}_{1}$. However, no SB2 object entered this category for Gaia DR3 and this model is not further detailed.

## Models of type StochasticSB2

When the fit of a TrendSB2 solution is not possible and the fit of an orbital solution corresponding to a significant period neither, the solution is classified as StochasticSB2. The RV is considered as constant but with an extra dispersion compared to the RV epoch uncertainties. This extra dispersion is estimated for the difference ${\mathrm{RV}}_{2}-{\mathrm{RV}}_{1}$, if possible.