  Recent analyses of the orbits of some Kuiper Belt objects hypothesize the
presence of an undiscovered Neptune-size planet at a very large separation from
the Sun. The energy budget of Neptunes on such distant orbits is dominated by
the internal heat released by their cooling rather than solar irradiation
(making them effectively "isolated"). The blackbody radiation that these
planets emit as they cool may provide the means for their detection. Here we
use an analytical toy model to study the cooling and radiation of isolated
Neptunes. This model can translate a detection (or a null detection) to a
constraint on the size and composition of the hypothesized "Planet Nine".
Specifically, the thick gas atmosphere of Neptune-like planets serves as an
insulating blanket which slows down their cooling. Therefore, a measurement of
the blackbody temperature, $T_{\rm eff}\sim 50\textrm{K}$, at which a Neptune
emits can be used to estimate the mass of its atmosphere, $M_{\rm atm}$.
Explicitly, we find the relation $T_{\rm eff}\propto M_{\rm atm}^{1/12}$.
Despite this weak relation, a measurement of the flux at the Wien tail can
constrain the atmospheric mass, at least to within a factor of a few, and
provide useful limits to possible formation scenarios of these planets.
Finally, we constrain the size and composition of Planet Nine by combining our
model with the null results of recent all-sky surveys.
