A More Precise Mass for GJ 1214 b and the Frequency of Multi-Planet Systems Around Mid-M Dwarfs

We present an intensive effort to refine the mass and orbit of the enveloped terrestrial planet GJ 1214 b using 165 radial velocity (RV) measurements taken with the HARPS spectrograph over a period of ten years. We conduct a joint analysis of the RVs with archival Spitzer/IRAC transits and measure a planetary mass and radius of $8.17\pm 0.43 M_{\oplus}$ and $2.742^{+0.050}_{-0.053} R_{\oplus}$. Assuming GJ 1214 b is an Earth-like core surrounded by a H/He envelope, we measure an envelope mass fraction of $X_{\rm env}= 5.24^{+0.30}_{-0.29}$%. GJ 1214 b remains a prime target for secondary eclipse observations of an enveloped terrestrial, the scheduling of which benefits from our tight constraint on the orbital eccentricity of $<0.063$ at 95% confidence, which narrows the secondary eclipse window to 2.8 hours. By combining GJ 1214 with other mid-M dwarf transiting systems with intensive RV follow-up, we calculate the frequency of mid-M dwarf planetary systems with multiple small planets and find that $90^{+5}_{-21}$% of mid-M dwarfs with a known planet with mass $\in [1,10] M_{\oplus}$ and orbital period $\in [0.5,50]$ days, will host at least one additional planet. We rule out additional planets around GJ 1214 down to $3 M_{\oplus}$ within 10 days such that GJ 1214 is a single-planet system within these limits, a result that has a $44^{+9}_{-5}$% probability given the prevalence of multi-planet systems around mid-M dwarfs. We also investigate mid-M dwarf RV systems and show that the probability that all reported RV planet candidates are real planets is $<12$% at 99% confidence, although this statistical argument is unable to identify the probable false positives.