  The orbits of many observed hot Jupiters are decaying rapidly due to tidal
interaction, eventually reaching the Roche limit. We analytically study the
ensuing coupled mass loss and orbital evolution during the Roche-lobe overflow
and find two possible scenarios. Planets with light cores $M_c\lesssim
6M_\oplus$ (assuming a nominal tidal dissipation factor $Q\sim 10^6$ for the
host star) are transformed into Neptune-mass gas planets, orbiting at a
separation (relative to the stellar radius) $a/R_\star\approx 3.5$. Planets
with heavier cores $M_c\gtrsim 6M_\oplus$ plunge rapidly until they are
destroyed at the stellar surface. Remnant gas-Neptunes, which are stable to
photo-evaporation, are absent from the observations, despite their unique
transit radius ($5-10R_\oplus$). This result suggests that $M_c\gtrsim
6M_\oplus$, providing a useful constraint on the poorly-known core mass that
may distinguish between different formation theories of gas giants.
Alternatively, if one assumes a prior of $M_c\approx 6 M_\oplus$ from the
core-accretion theory, our results suggest that $Q$ does not lie in the range
$10^6\lesssim Q\lesssim 10^7$.
