  Hot and Warm Jupiters (HJs&WJs) are gas-giant planets orbiting their host
stars at short orbital periods, posing a challenge to their efficient in-situ
formation. Therefore, most of the HJs&WJs are thought to have migrated from an
initially farther-out birth locations. Current migration models, i.e
disc-migration (gas-dissipation driven) and eccentric-migration (tidal
evolution driven), fail to produce the occurrence rate and orbital properties
of HJs&WJs. Here we study the role of the thermal evolution and its coupling to
tidal evolution. We use the AMUSE, numerical environment, and MESA, planetary
evolution modeling, to model in detail the coupled internal and orbital
evolution of gas-giants during their eccentric-migration. In a companion paper,
we use a simple semi-analytic model, validated by our numerical model, and run
a population-synthesis study. We consider the initially inflated radii of
gas-giants (expected following their formation), as well study the effects of
the potential slowed contraction and even re-inflation of gas-giants (due to
tidal and radiative heating) on the eccentric-migration. Tidal forces that
drive eccentric-migration are highly sensitive to the planetary structure and
radius. Consequently, we find that this form of inflated eccentric-migration
operates on significantly (up to an order of magnitude) shorter timescales than
previously studied eccentric-migration models. Thereby, inflated
eccentric-migration gives rise to more rapid formation of HJs&WJs, higher
occurrence rates of WJs, and higher rates of tidal disruptions, compared with
previous eccentric migration models which consider constant ~Jupiter radii for
HJ&WJ progenitors. Coupled thermal-dynamical evolution of eccentric gas-giants
can therefore play a key-role in their evolution.
