Estimating the Mass Escaping Rates of Radius-valley-spanning Planets in the TOI-431 System via X-Ray and Ultraviolet Evaporation

TOI-431 system has 3 close-in exoplanets, which gives an ideal lab to study gas escape. In this study, we measure the XUV luminosity for TOI-431 with XMM-Newton/EPIC-pn and OM data, then calculate the fluxes for the planets in the system. We find that, TOI-431 b's $\rm F_{XUV,b}=$$70286^{+12060}_{-2611}$$\rm \ erg\ cm^{-2}s^{-1}$ is 75 times of TOI-431 d $\rm F_{XUV,d}=$$935^{+160}_{-35}$$\rm \ erg\ cm^{-2}s^{-1}$. Adopting the energy limit method and hydrodynamic code $ATES$ with a set of He/H ratios, we obtain the mass-loss rates of $10^{10.51^{+0.07}_{-0.02}}$ g s$^{-1}$ for TOI-431 b, $10^{9.14^{+0.07}_{-0.02}}$ and $10^{9.84\sim 9.94}$ g s$^{-1}$ for TOI-431 d. We predict the $2.93\sim 7.91 \%$ H I Ly$\alpha$ and $0.19\sim 10.65\%$ He I triplet absorption depths for TOI-431 d, thus its gas escaping is detectable in principle. For both TOI-431 b and d, we select similar planets from the New Generation Planetary Population Synthesis (NGPPS) data. Then show that considering the mass-loss rates, TOI-431 b should be a naked solid planet, and TOI-431 d will likely maintain its gas envelope until the host star dies. According to the formation and evolution tracks, we find that TOI-431 b's potential birthplace (0.1-2 AU) should be inner than TOI-431 d (2-12 AU). Our results are consistent with the interpretation of the radius valley being caused by atmospheric escape. The intrinsic reason may be their birthplace, which will determine how close they can migrate to the host star, then lose mass and result in the Fulton gap.