  The Kepler mission revealed a class of planets known as ''super-puffs,'' with
masses only a few times larger than Earth's but radii larger than Neptune,
giving them very low mean densities. All three of the known planets orbiting
the young solar-type star Kepler 51 are super-puffs. The Kepler 51 system
thereby provides an opportunity for a comparative study of the structures and
atmospheres of this mysterious class of planets, which may provide clues about
their formation and evolution. We observed two transits each of Kepler 51b and
51d with the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope.
Combining new WFC3 transit times with re-analyzed Kepler data and updated
stellar parameters, we confirmed that all three planets have densities lower
than 0.1 g/cm$^{3}$. We measured the WFC3 transmission spectra to be
featureless between 1.15 and 1.63 $\mu$m, ruling out any variations greater
than 0.6 scale heights (assuming a H/He dominated atmosphere), thus showing no
significant water absorption features. We interpreted the flat spectra as the
result of a high-altitude aerosol layer (pressure $<$3 mbar) on each planet.
Adding this new result to the collection of flat spectra that have been
observed for other sub-Neptune planets, we find support for one of the two
hypotheses introduced by Crossfield and Kreidberg (2017), that planets with
cooler equilibrium temperatures have more high-altitude aerosols. We strongly
disfavor their other hypothesis that the H/He mass fraction drives the
appearance of large amplitude transmission features.
