  About ten percent of Sun-like ($1$-$2 M_\odot$) stars will engulf a $1$-$10
M_{\rm J}$ planet as they expand during the red giant branch (RGB) or
asymptotic giant branch (AGB) phase of their evolution. Once engulfed, these
planets experience a strong drag force in the star's convective envelope and
spiral inward, depositing energy and angular momentum. For these mass ratios,
the inspiral takes $\sim 10$-$10^{2}$ years ($\sim 10^{2}$-$10^{3}$ orbits);
the planet undergoes tidal disruption at a radius of $\sim R_\odot$. We use the
Modules for Experiments in Stellar Astrophysics (MESA) software instrument to
track the stellar response to the energy deposition while simultaneously
evolving the planetary orbit. For RGB stars, as well as AGB stars with $M_{\rm
p} \lesssim 5 M_{\rm J}$ planets, the star responds quasistatically but still
brightens measurably on a timescale of years. In addition, asteroseismic
indicators, such as the frequency spacing or rotational splitting, differ
before and after engulfment. For AGB stars, engulfment of a $M_{\rm p} \gtrsim
5 M_{\rm J}$ planet drives supersonic expansion of the envelope, causing a
bright, red, dusty eruption similar to a "luminous red nova." Based on the peak
luminosity, color, duration, and expected rate of these events, we suggest that
engulfment events on the AGB could be a significant fraction of low-luminosity
red novae in the Galaxy. We do not find conditions where the envelope is
ejected prior to the planet's tidal disruption, complicating the interpretation
of short-period giant planets orbiting white dwarfs as survivors of
common-envelope evolution.
