  A change in the mass of the Galaxy with time will leave its imprint on the
motions of the stars, with stars having radially outward (mass loss) or inward
(mass accretion) bulk motions. Here we test the feasibility of using the mean
radial motion of stars in the stellar halo to constrain the rate of change of
mass in the Galaxy, for example, due to decay of dark matter into invisible
dark sector particles or more conservatively from the settling of baryons. In
the current $\Lambda$CDM paradigm of structure formation, the stellar halo is
formed by accretion of satellites onto the host galaxy. Over time, as the
satellites disrupt and phase mix, the mean radial motion $\langle V_{R}
\rangle$ of the stellar halo is eventually expected to be close to zero. But
most halos have substructures due to incomplete mixing of specific accretion
events and this can lead to nonzero $\langle V_{R} \rangle$ in them. Using
simulations, we measure the mean radial motion, $\langle V_{R} \rangle$, of
stars in 13 $\Lambda$CDM stellar halos lying in a spherical shell of radius 30
kpc. We find that for most halos, the shell motion is quite small, with 75\% of
halos having $\langle V_{R}\rangle \lesssim 1.2 \ {\rm km}s^{-1}$. When
substructures are removed by using a clustering algorithm, $\langle
V_{R}\rangle$ is reduced even further, with 75\% of halos having $\langle
V_{R}\rangle \lesssim 0.6 \ {\rm km}s^{-1}$. A value of $\langle V_{R}\rangle
\approx 0.6 \ {\rm km}s^{-1}$ can be attained corresponding to a galactic mass
loss rate of 2\% per Gyr. We show that this can place constraints on dark
matter decay parameters such as the decay lifetime and the kick velocity that
is imparted to the daughter particle. The advent of all-sky stellar surveys
involving millions to billions of stars is encouraging for detecting signatures
of dark matter decay.
