Title: Structure and Evolution of Internally Heated Hot Jupiters,
Abstract: Hot Jupiters receive strong stellar irradiation, producing equilibrium
temperatures of $1000 - 2500 \ \mathrm{Kelvin}$. Incoming irradiation directly
heats just their thin outer layer, down to pressures of $\sim 0.1 \
\mathrm{bars}$. In standard irradiated evolution models of hot Jupiters,
predicted transit radii are too small. Previous studies have shown that deeper
heating -- at a small fraction of the heating rate from irradiation -- can
explain observed radii. Here we present a suite of evolution models for HD
209458b where we systematically vary both the depth and intensity of internal
heating, without specifying the uncertain heating mechanism(s). Our models
start with a hot, high entropy planet whose radius decreases as the convective
interior cools. The applied heating suppresses this cooling. We find that very
shallow heating -- at pressures of $1 - 10 \ \mathrm{bars}$ -- does not
significantly suppress cooling, unless the total heating rate is $\gtrsim 10\%$
of the incident stellar power. Deeper heating, at $100 \ \mathrm{bars}$,
requires heating at only $1\%$ of the stellar irradiation to explain the
observed transit radius of $1.4 R_{\rm Jup}$ after 5 Gyr of cooling. In
general, more intense and deeper heating results in larger hot Jupiter radii.
Surprisingly, we find that heat deposited at $10^4 \ \mathrm{bars}$ -- which is
exterior to $\approx 99\%$ of the planet's mass -- suppresses planetary cooling
as effectively as heating at the center. In summary, we find that relatively
shallow heating is required to explain the radii of most hot Jupiters, provided
that this heat is applied early and persists throughout their evolution.