A Moving Target—Revising the Mass of Proxima Centauri c^*

Informed by the Damasso et al. (2020) radial velocity evidence for a suspected second companion, Proxima Centauri  c, we reanalyzed astrometry originally secured with Hubble Space Telescope Fine Guidance Sensor 3 (FGS 3 Benedict et al. 1999), and detected an astrometric signature suggesting a mass, ${ \mathcal M }$_c = 18 ± 5 ${{ \mathcal M }}_{\oplus }$ (Benedict & McArthur 2020). The Damasso et al. (2020) results are listed as RV and the Benedict & McArthur (2020) results as FGS+RV in Table 1. Within a month of our publication additional astrometric information became available (Gratton et al. 2020). This note reports the results of incorporating that new astrometry into our modeling.

Table 1.  Orbital Elements for Proxima  c

Parameter	RVa	(FGS+RV)b	SPHEREc	(FGS+RV+SPHERE)d
P (days)	1900 ${}_{-82}^{+96}$	1902 ± 24	1896 ± 34	1928 ± 20
P (yr)	⋯	5.21 ± 0.07	5.19 ± 0.09	5.28 ± 0.06
T0e	55892 ± 100	55891 ± 24	56262 ± 32	56202 ± 21
ωA (deg)	0.3 ± 0.06	0 ± 2	⋯	173 ± 4
ωc(deg)	⋯	⋯	−2 ± 5	−4 ± 4
Ω (deg)	⋯	298 ± 20	330 ± 2	331 ± 1
i (deg)	⋯	18 ± 4	135 ± 2	133 ± 1
f	0	0.02 ± 0.0	0.06 ± 0.02	0.04 ± 0.01
KA (m s−1)	1.2 ± 0.4	1.2 ± 0.1	⋯	1.1 ± 0.2
α (mas)	⋯	0.5 ± 0.1	⋯	0.20 ± 0.03
a ('')	⋯	⋯	1.04 ± 0.02	1.06 ± 0.01
ϖ (mas)	⋯	767.02 ± 0.11	⋯	767.14 ± 0.14
μR.A.(mas yr−1)	⋯	−3779.23 ± 0.13	⋯	−3779.06 ± 0.06
μDecl. (mas yr−1)	763.64 ± 0.52	⋯	763.63 ± 0.05
${ \mathcal M }$A (${{ \mathcal M }}_{\odot }\,$)	⋯	0.11 ± 0.02	⋯	0.094 ± 0.003
${ \mathcal M }$c (${{ \mathcal M }}_{\oplus }$)	5.8 ± 1.9g	18 ± 5	⋯	7 ± 1
Gaia
ϖ (mas)	⋯	768.5 ± 0.2	⋯	⋯
μR.A. (mas yr−1)	⋯	3781.3 ± 0.1	⋯	⋯
μDecl. (mas yr−1)	⋯	769.8 ± 0.2	⋯

Notes.

^aFrom Damasso et al. (2020). ^bFrom Benedict & McArthur (2020). ^cFrom our analysis of the (ρ, θ) from Gratton et al. (2020), Table 2. ^dThis paper. ^eJD-2400000.0. ^fValues without stated errors are assumed. ^gMinimum mass, ${ \mathcal M }$ sin i .

Gratton et al. (2020) utilized the high contrast imaging system SPHERE  to detect directly the companion Proxima Centauri  c, producing a total of nine measurements of position angle and separation over a span of 3.3 years. These are listed in their table 2, along with estimated signal-to-noise ratios (S/N). With that data and the FGS astrometry of the primary, Proxima Centauri  A, we have an opportunity to treat the Proxima Centauri  A-c system as a binary star, to estimate the masses of both primary and secondary by applying the modeling described at length in Benedict et al. (2016). Note that the mean epochs for these two sets of astrometric data differ by a quarter century.

We first determined an orbit from the SPHERE  relative position angles (θ) and separations (ρ) by themselves (Gratton et al. 2020, Table 2). We decomposed the θ , ρ into x and y along R.A. and decl. respectively. To achieve near unity reduced χ² (χ²/degrees of freedom) for the orbital fitting, we assigned errors to x and y, ${\sigma }_{{\rm{x}},{\rm{y}}}=\displaystyle \frac{4\times \mathrm{res}}{{\rm{S}}/{\rm{N}}}$, where res = 25 mas (millisecond of arc) is the estimated resolution, thus assigning an rms error of 51 mas along each axis. The resulting orbital parameters are listed as SPHERE in Table 1. This orbit exhibits an rms residual of 15 mas.

We then model the FGS measurements of Proxima  A and the SPHERE measurements of Proxima  c as a two component binary (Equations (3)–(6), and (9), Benedict et al. 2016), again including the Damasso et al. (2020) RV orbital elements as priors introduced as data with errors. Our modeling derives a mass fraction, thus mass estimates for both the primary, Proxima  A, and secondary, Proxima  c. We list the final orbital elements and masses under FGS+RV+SPHERE in Table 1. FGS residuals to the Proxima  A orbit have an rms of 1.2 mas. The SPHERE residuals to the Proxima  c orbit have an rms of 20 mas.

The inclination from FGS + RV only was i = 18° ± 4°. Including the SPHERE measurements yields i = 133° ± 1°. The difference in Proxima  A FGS rms astrometric residuals with and without the SPHERE data is less than 0.05 mas. The SPHERE measurements obviously dominate the orbit modeling with a/σ_a = 119, compared to α/σ_α = 6. FGS astrometry with per observation precision of 1 mas is unable to conclusively argue for or against the higher inclination. Benedict & McArthur (2020) adopted ${ \mathcal M }$_A = 0.11 ${{ \mathcal M }}_{\odot }\,$ (Benedict et al. 1999) which, along with the lower inclination, yielded ${ \mathcal M }$_c = 18 ± 5 ${{ \mathcal M }}_{\oplus }$. Our modeling, including SPHERE, finds ${ \mathcal M }$_c = 6 ± 1 ${{ \mathcal M }}_{\oplus }$, and for Proxima  A, ${ \mathcal M }$_A = 0.094 ± 0.003 ${{ \mathcal M }}_{\odot }$. While each set of observations has deficiencies in precision (Benedict et al. 1999; Damasso et al. 2020) and even interpretation (Gratton et al. 2020), they together, along with the combined Hipparcos+Gaia results from Kervella et al. (2020), argue for the reality of Proxima  c. Additional high-contrast imaging of Proxima  c and a vastly improved perturbation orbit of Proxima  A from Gaia, along with additional future RV (Damasso & Del Sordo 2020) observations should more effectively characterize this closest, non-solar planetary system.

G.F.B. and B.E.M. thank McDonald Observatory for Research Fellowship appointments. See Benedict et al. (2017); Benedict & Harrison (2017) for many additional acknowledgments.