We describe a model of the fluid mechanics and thermodynamics of turbulent lava flows capable of thermally eroding sinuous rille channels on bodies without atmospheres. The model assumes Bingham plastic rheology for the lava and shows how the effects of radiant cooling and consequent crystallization change the rheology and control the point at which turbulence ceases. A correlation is found between magma volume eruption rate and the length and width of the eroded rille channel. Thus, simple measurements of rille length and width in images can provide reliable estimates of magma eruption rates. The model also predicts rille floor erosion rates, so that if rille depths are measured, eruption durations and erupted magma volumes can also be found. The model is applied in detail to six well-preserved lunar rilles and more generally to a published catalog of 214 lunar rilles. We find that rille-forming eruptions have magma volume eruption rates of a few times 104 m3 s−1, durations of up to 3 months, and erupted magma volumes up to ∼200 km3, consistent with theoretical predictions of basaltic magma ascent and eruption from deep mantle sources. The key requirement for rille formation, rather than mare lava flow deposit formation, is the turbulent eruption of a sufficiently large volume of low-viscosity magma at a sufficiently low eruption rate.