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 _ { eff } \sim 50 \textrm { K } , at which a Neptune emits can be used to estimate the mass of its atmosphere , M _ { atm } .
Explicitly , we find the relation T _ { eff } \propto M _ { 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 .