Transit Timing of the White Dwarf-Cold Jupiter System WD 1856+534

We present new transit timing measurements for the white dwarf-cold Jupiter system WD 1856+534, extending the baseline of observations from 311 epochs to 1498 epochs. The planet is unlikely to have survived the host star's red-giant phase at its present location and is likely too small for common-envelope evolution to take place. As such, a plausible explanation for the short semimajor axis is that the exoplanet started out on a much larger orbit and then spiraled inward through high-eccentricity tidal migration (HETM). A past transit-timing analysis found tentative evidence for orbital growth, which could have been interpreted as a residual effect of HETM, but we find the data are consistent with a constant-period model after adding 18 new transit measurements. We use the estimated period derivative $\dot{P} = 0.04\pm0.43$ ms yr$^{-1}$ to place a lower limit on the planetary tidal quality factor of $Q_p'\gtrsim 3.1 \times 10^6$, consistent with that of Jupiter in our own Solar System. We also test for the presence of companion planets in the system, which could have excited WD 1856 b onto an eccentric orbit via the Kozai-Lidov process, and ultimately rule out the presence of an additional planet with a mass greater than $4.1\,M_J$ and a period shorter than 1500 days. We find no evidence for nonzero eccentricity, with an upper limit of $e\lesssim10^{-2}$. If the planet indeed reached its current orbit through HETM, the low present-day eccentricity indicates that the migration process has now ceased, and any further orbital evolution will be governed solely by weak planetary tides.