Assessing $TESS$'s Yield of Rocky Planets Around Nearby M Dwarfs

Terrestrial planets are easier to detect around M dwarfs than other types of stars, making them promising for next-generation atmospheric characterization studies. The $TESS$ mission has greatly increased the number of known M dwarf planets that we can use to perform population studies, allowing us to explore how the rocky planet occurrence rate varies with host radius, following in the footsteps of past work with $Kepler$ data. In this paper, we use simulations to assess $TESS$'s yield of small ($0.5\,R_\oplus \,<\,R_p\,<\,2\,R_\oplus$) planet candidates around nearby ($d\,<\,30$ pc) M dwarfs. We highlight the underappreciated fact that while $TESS$ was indeed expected to find a large number of planets around M dwarfs overall, it was not expected to have a high planetary yield for the latest M dwarfs. Furthermore, we find that $TESS$ has detected fewer planets around stars with $R_*\,<\,0.3 R_\odot$ than even was expected (11 observed vs. $24\,\pm\,5$ expected). We find evidence that the photometric noise of stars in the $TESS$ bandpass increases with decreasing radius for M dwarfs. However, this trend cannot explain the observed distribution of planets. Our main conclusions are: (1) the planet occurrence rate likely doesn't increase, and may decrease for the latest M dwarfs; and (2) there are at least 17, and potentially three times that number, transiting planets around nearby late M dwarfs that still won't be detected by the end of $TESS$'s 4th year.