Two decades of optical timing of the shortest-period binary star system HM Cancri

The shortest-period binary star system known to date, RX J0806.3+1527 (HM Cancri), has now been observed in the optical for more than two decades. Although it is thought to be a double degenerate binary undergoing mass transfer, an early surprise was that its orbital frequency, $f_0$, is currently increasing as the result of gravitational wave radiation. This is unusual since it was expected that the mass donor was degenerate and would expand on mass loss, leading to a decreasing $f_0$. We exploit two decades of high-speed photometry to precisely quantify the trajectory of HM Cancri, allowing us to find that $\ddot f_0$ is negative, where $\ddot f_0~=~(-5.38\pm2.10)\times10^{-27}$ Hz s$^{-2}$. Coupled with our positive frequency derivative, we show that mass transfer is counteracting gravitational-wave dominated orbital decay and that HM Cancri will turn around within $2100\pm800\,$yrs from now. We present Hubble Space Telescope ultra-violet spectra which display Lyman-$α$ absorption, indicative of the presence of hydrogen accreted from the donor star. We use these pieces of information to explore a grid of permitted donor and accretor masses with the Modules for Experiments in Stellar Astrophysics suite, finding models in good accordance with many of the observed properties for a cool and initially hydrogen-rich extremely-low-mass white dwarf ($\approx0.17\,$M$_\odot$) coupled with a high accretor mass white dwarf ($\approx 1.0\,$M$_\odot$). Our measurements and models affirm that HM~Cancri is still one of the brightest verification binaries for the Laser Interferometer Space Antenna spacecraft.