2018 D2 | An unseen companion in Auriga

During the Apache Point Observatory Galactic Evolution Experiment (APOGEE), astronomers searched archival spectra of "single" stars for radial velocity (RV) variations. Among about 200 stars with the largest accelerations, they looked for periodic brightness variations in ASAS-SN data. The star 2M05215658+4359220 showed both a large RV variation and photometric variability. Assume its RV variation is caused by an unseen companion, and ignore proper motion.

2.1 Table 3 gives three APOGEE RV measurements.

(D2.1.1) Use a simple linear model to estimate the maximum apparent acceleration $a_{\max} = \Delta{\rm RV}/\Delta t$ (km s⁻¹ day⁻¹). (6)

(D2.1.2) Estimate the mass of the unseen companion. (9)

2.2 Follow-up spectra from the 1.5 m Tillinghast Reflector Echelle Spectrograph (TRES) give the RVs in Table 4.

(D2.2.1) Plot RV against time and draw a suitable sinusoid through the data. Estimate the orbital period $P_{\rm orb}$ and the RV semi-amplitude $K$. (14)

(D2.2.2) If the orbit is circular, find the minimum orbital radius $r_{\rm orb}$ of the star, in $R_\odot$ and au. (4)

(D2.2.3) The mass function is

where 1 is the visible star, 2 the companion, and $i$ the inclination. Calculate it in $M_\odot$. (7)

2.3 Table 5 gives the stellar parameters. The light curve period equals the orbital period, so assume $P_{\rm rot} = P_{\rm orb}\equiv P$ and $i_{\rm rot} = i_{\rm orb}\equiv i$.

(D2.3.1) Calculate the luminosity $L_1$ ($L_\odot$), radius $R_1$ ($R_\odot$), $\sin i$, and mass $M_1$ ($M_\odot$) of the visible star, with uncertainties. (16)

(D2.3.2) What type of star is it: blue giant, yellow main sequence star, red giant, red main sequence star, or white dwarf? (4)

(D2.3.3) From the mass function, plot $M_2$ (vertical) against $M_1$ for the most probable $\sin i$ and for $\sin i\pm\Delta\sin i$. (10)

(D2.3.4) Shade the region $[M_1-\Delta M_1, M_1+\Delta M_1]$ and draw dashed lines at the maximum masses of a white dwarf and a neutron star. What is the possible mass of the companion, and what could it be? (5)