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Weighing Milky Way Satellites with LISA

White dwarf stars are a well-established tool for studying Galactic stellar populations. Two white dwarfs in a tight binary system offer us an additional messenger - gravitational waves - for exploring the Milky Way and its immediate surroundings. Gravitational waves produced by double white dwarf (DWD) binaries can be detected by the future Laser Interferometer Space Antenna (LISA). Numerous and widespread DWDs have the potential to probe shapes, masses and formation histories of the stellar populations in the Galactic neighbourhood. In this work we outline a method for estimating the total stellar mass of Milky Way satellite galaxies based on the number of DWDs detected by LISA. To constrain the mass we perform a Bayesian inference using binary population synthesis models and considering the number of detected DWDs associated with the satellite and the measured distance to the satellite as the only inputs. Using a fiducial binary population synthesis model we find that for large satellites the stellar masses can be recovered to within 1) a factor two if the star formation history is known and 2) an order of magnitude when marginalising over different star formation history models. For smaller satellites we can place upper limits on their stellar mass. Gravitational wave observations can provide mass measurements for large satellites that are comparable, and in some cases more precise, than standard electromagnetic observations.