  Planetary rotation rates and obliquities provide information regarding the
history of planet formation, but have not yet been measured for evolved
extrasolar planets. Here we investigate the theoretical and observational
perspective of the Rossiter-McLauglin effect during secondary eclipse (RMse)
ingress and egress for transiting exoplanets. Near secondary eclipse, when the
planet passes behind the parent star, the star sequentially obscures light from
the approaching and receding parts of the rotating planetary surface. The
temporal block of light emerging from the approaching (blue-shifted) or
receding (red-shifted) parts of the planet causes a temporal distortion in the
planet's spectral line profiles resulting in an anomaly in the planet's radial
velocity curve. We demonstrate that the shape and the ratio of the
ingress-to-egress radial velocity amplitudes depends on the planetary
rotational rate, axial tilt and impact factor (i.e. sky-projected planet
spin-orbital alignment). In addition, line asymmetries originating from
different layers in the atmosphere of the planet could provide information
regarding zonal atmospheric winds and constraints on the hot spot shape for
giant irradiated exoplanets. The effect is expected to be most-pronounced at
near-infrared wavelengths, where the planet-to-star contrasts are large. We
create synthetic near-infrared, high-dispersion spectroscopic data and
demonstrate how the sky-projected spin axis orientation and equatorial velocity
of the planet can be estimated. We conclude that the RMse effect could be a
powerful method to measure exoplanet spins.
