Prospects for detecting the Rossiter-McLaughlin effect of Earth-like planets: the test case of TRAPPIST-1b and c

The Rossiter-McLaughlin effect is the principal method of determining the sky-projected spin--orbit angle ($\beta$) of transiting planets. Taking the example of the recently discovered TRAPPIST-1 system, we explore how ultracool dwarfs facilitate the measurement of the spin--orbit angle for Earth-sized planets by creating an effect that can be an order of magnitude more ample than the Doppler reflex motion caused by the planet if the star is undergoing rapid rotation. In TRAPPIST-1's case we expect the semi-amplitudes of the Rossiter-McLaughlin effect to be $40-50$ m/s for the known transiting planets. Accounting for stellar jitter expected for ultracool dwarfs, instrumental noise, and assuming radial velocity precisions both demonstrated and anticipated for upcoming near-infrared spectrographs, we quantify the observational effort required to measure the planets' masses and spin--orbit angles. We conclude that if the planetary system is well-aligned then $\beta$ can be measured to a precision of $\lesssim 10^{\circ}$ if the spectrograph is stable at the level of 2 m/s. We also investigate the measure of $\Delta \beta$, the mutual inclination, when multiple transiting planets are present in the system. Lastly, we note that the rapid rotation rate of many late M-dwarfs will amplify the Rossiter-McLaughlin signal to the point where variations in the chromatic Rossiter-McLaughlin effect from atmospheric absorbers should be detectable.