  K2-25b is a Neptune-sized exoplanet (3.45 Earth radii) that orbits its M4.5
host with a period of 3.48 days. Due to its membership in the Hyades Cluster,
the system has a known age (727 +/- 75 Myr). K2-25b's youth and its
similarities with Gl 436b suggested that K2-25b could be undergoing strong
atmospheric escape. We observed two transits of K2-25b at Lyman-alpha using
HST/STIS in order to search for escaping neutral hydrogen. We were unable to
detect an exospheric signature, but placed an upper limit of (R_p/R_s) < 0.56
at 95% confidence by fitting the light curve of the Lyman-alpha red-wing, or <
1.20 in the blue-wing. We reconstructed the intrinsic Lyman-alpha profile of
K2-25 to determine its Lyman-alpha flux, and analyzed XMM-Newton observations
to determined its X-ray flux. Based on the total X-ray and extreme ultraviolet
irradiation of the planet (8763 +/- 1049 erg/s/cm^2), we estimated the maximum
energy-limited mass loss rate of K2-25b to be 10.6 x 10^10 g/s (0.56 Earth
masses per 1 Gyr), five times larger than the similarly estimated mass loss
rate of Gl 436b (2.2 x 10^10 g/s). The photoionization time is about 3 hours,
significantly shorter than Gl 436b's 14 hours. A non-detection of a Lyman-alpha
transit could suggest K2-25b is not significantly losing its atmosphere, or
factors of the system are resulting in the mass loss being unobservable (e.g.,
atmosphere composition or the system's large high energy flux). Further
observations could provide more stringent constraints.
