10.6 Compact companions 10.6.2 Properties of the input data 10.6.4 Quality assessment and validation

10.6.3 Processing steps

A three-harmonic model is fitted to each light curve in the $G$ band:

G=A_{0}+\ \sum_{i=1}^{3}a_{i,\mathrm{c}}\cos\left[\frac{2\pi i}{P}(t-T_{0})% \right]+\ a_{i,\mathrm{s}}\sin\left[\frac{2\pi i}{P}(t-T_{0})\right],

(10.2)

with seven free parameters, $A_{0},a_{i,\mathrm{c}},a_{i,\mathrm{s}}$ , and $T_{0}$ derived from the condition $a_{2,\mathrm{s}}=0$ . $A_{0}$ reflects the mean magnitude of the model and is named model_mean_g. Three-harmonic models are derived also for time series in the $G_{\rm BP}$ and $G_{\rm RP}$ bands with at least 25 measurements (for fewer measurements, the model-related parameters are set to null or to an array of NaN). The semi-amplitude of each harmonic is defined as

A_{i}=\ \sqrt{a^{2}_{i,\mathrm{c}}+a^{2}_{i,\mathrm{s}}}\,\,\,\,\mbox{for~{}}i% =1,2,3,

(10.3)

and additional selection cuts based on semi-amplitudes in the $G$ band are performed on each system:

1.

$0.33<A_{2}/(\rm range\ in\ G)<0.6$ ,
2.

$A_{2}/A_{\mathrm{2,err}}>7$ ,
3.

$A_{1}/A_{\mathrm{1,err}}>3$ or $A_{3}/A_{\mathrm{3,err}}>3$ ,
4.

$A_{1}/A_{2}<1$ and $A_{3}/A_{2}<0.3$ .

The final stage is to identify ellipsoidal candidates that might have compact companions. This is done by using equation 1 of Gomel, Faigler, and Mazeh (2021a), which estimates the ellipsoidal leading amplitude $A_{2}$ as a function of the fill-out factor $f$ , the inclination $i$ , and the mass ratio $q$ :

A_{2}\approx\frac{1}{\overline{L}/L_{{}_{0}}}\alpha_{\mathrm{2}}\ f^{3}E^{3}(q% )\ q\ \sin^{2}i\ C(q,f)\,,

(10.4)

where $\overline{L}$ is the average luminosity of the star, $L_{{}_{0}}$ being the stellar brightness with no secondary at all, and $E(q)$ is the Eggleton (1983) approximation for the volume-averaged Roche-lobe radius in binary semi-major axis units. The ellipsoidal coefficient $\alpha_{2}$ depends on the linear limb- and gravity-darkening coefficients of the primary and is expected to be in the $1$ – $2$ range. The correction coefficient $C(q,f)$ starts at $1$ for $f=0$ (no correction), as expected, and rises monotonically as $f\to 1$ , obtaining a value of $\sim$ $1.5$ at $f\gtrsim 0.9$ (Gomel, Faigler, and Mazeh 2021b).

Assuming a fill-out factor $f=0.95$ , inclination of $90^{\circ}$ and a typical $\alpha_{2}$ of 1.3 for the $G$ -band (Claret 2019), the modified minimum mass ratio, mod_min_mass_ratio $\hat{q}_{\rm min}$ (Gomel, Faigler, and Mazeh 2021a) is solved for each ellipsoidal, within $10^{-9}<\hat{q}_{\rm min}<100$ , based on the observed second harmonic amplitude, $A_{2}$ , in the $G$ band. The uncertainty of $\hat{q}_{\rm min}$ is inherently large because of the uncertainty of $\alpha_{2}$ , which somewhat arbitrarily is adopted to be $0.1$ . Because the resulting distribution of $\hat{q}_{\rm min}$ is highly asymmetric, the following quantity is computed:

\mbox{{mod\_min\_mass\_ratio\_one\_sigma}}\equiv\hat{q}_{\rm min}-\sigma^{-}(% \hat{q}_{\rm min})\ ,

(10.5)

to represent the $15.9$ th percentile of its distribution, where $\sigma^{-}(\hat{q}_{\rm min})$ is the negative-side $1\sigma$ uncertainty of $\hat{q}_{\rm min}$ . In the same manner,

\mbox{{mod\_min\_mass\_ratio\_three\_sigma}}\equiv\hat{q}_{\rm min}-3\sigma^{-% }(\hat{q}_{\rm min})\

(10.6)

represents the $0.135$ th percentile of the $\hat{q}_{\rm min}$ distribution.

If no solution is found for Equation 10.4, then the minimum value of $\alpha_{2}$ for a solution of $\hat{q}_{\rm min}=100$ is calculated.

The final list of 6336 candidates is obtained by applying additional cuts to the results of the Compact Companion work package:

1.

$A_{2}/A_{\mathrm{2,err}}>10$ ,
2.

frequencygram peak $>12$ (in units of standard deviation of the frequencygram),
3.

$P<2.5$ day,
4.

$\hat{q}_{\rm min}>0.5$ .

gaia data release 3 documentation

10.6.3 Processing steps