The binary Yarkovsky effect on the primary asteroid with applications to singly synchronous binary asteroids

The binary Yarkovsky effect on the secondary asteroid (BYS) was recently discovered to influence binary asteroid systems by pushing the secondary asteroid toward a synchronous orbit on a short timescale. However, the binary Yarkovsky effect on the primary (BYP) remains less understood, partly due to non-linear effects from partial eclipses, but could have significant implications for singly synchronous binaries. In this work, we studied the BYP effect by numerical methods and estimated its induced orbital drifting rates for real binary asteroids. We find an empirical modified solution to estimate the effective BYP: the traditional BYP formula multiplied by $(r_s / r_p)^(\alpha -1 )$. We confirm that the BYP pushes the primary towards a synchronous orbit where its spin equals the mean motion. The parameter $\alpha$ is insensitive to the ratio of the spin rate to the mean motion and decreases slightly with increasing thermal inertia. For small binary systems with a typical thermal inertia of 200 tiu, $\alpha$ is approximately 1.7. The BYP is found to affect the mutual orbit of singly synchronous binaries with a timescale typically an order of magnitude longer than that of the BYS. Drift rates induced by the BYP for known small binary asteroids (primary radius < 1 km) range from -0.001 to -1 cm $yr^{-1}$. A comparative analysis with observed orbital drift rates shows agreement for pre-impact Didymos and 1996 FG$_3$ but discrepancies for 2001 SL$_9$ and 1999 KW$_4$, suggesting complex dynamics in these systems involving the BYP, the binary Yarkovsky-O'Keefe-Radzievskii-Paddack (BYORP) effect, and tides. The BYP is changing the mutual orbits of most discovered binary asteroids. We suggest that the BYP should be considered along with BYORP and tidal effects when studying binary systems' long-term dynamics.