HAT-P-13b is a Jupiter-mass transiting exoplanet that has settled onto a
stable, short-period, and mildly eccentric orbit as a consequence of the action
of tidal dissipation and perturbations from a second, highly eccentric, outer
companion. Due to the special orbital configuration of the HAT-P-13 system, the
magnitude of HAT-P-13b's eccentricity ($e_b$) is in part dictated by its Love
number ($k_{2_b}$), which is in turn a proxy for the degree of central mass
concentration in its interior. Thus, the measurement of $e_b$ constrains
$k_{2_b}$ and allows us to place otherwise elusive constraints on the mass of
HAT-P-13b's core ($M_{\rm{core,b}}$). In this study we derive new constraints
on the value of $e_b$ by observing two secondary eclipses of HAT-P-13b with the
Infrared Array Camera on board the $\textit{Spitzer Space Telescope}$. We fit
the measured secondary eclipse times simultaneously with radial velocity
measurements and find that $e_b = 0.00700 \pm 0.00100$. We then use
octupole-order secular perturbation theory to find the corresponding $k_{2_b} =
0.31^{+0.08}_{-0.05}$. Applying structural evolution models, we then find, with
68\% confidence, that $M_{\rm{core,b}}$ is less than 25 Earth masses
($M_{\oplus}$). The most likely value of $M_{\rm{core,b}} = 11 M_{\oplus}$,
which is similar to the core mass theoretically required for runaway gas
accretion. This is the tightest constraint to date on the core mass of a hot
Jupiter. Additionally, we find that the measured secondary eclipse depths,
which are in the 3.6 $\mu$m and 4.5 $\mu$m bands, best match atmospheric model
predictions with a dayside temperature inversion and relatively efficient
day-night circulation.
