Evolution and Magnitudes of Candidate Planet Nine
  The recently renewed interest in a possible additional major body in the outer solar system prompted us to study the thermodynamic evolution of such an object. We assumed that it is a smaller version of Uranus and Neptune. We modeled the temporal evolution of the radius, temperature, intrinsic luminosity, and the blackbody spectrum of distant ice giant planets. The aim is also to provide estimates of the magnitudes in different bands to assess whether the object might be detectable. Simulations of the cooling and contraction were conducted for ice giants with masses of 5, 10, 20, and 50 Mearth that are located at 280, 700, and 1120 AU from the Sun. The core composition, the fraction of H/He, the efficiency of energy transport, and the initial luminosity were varied. The atmospheric opacity was set to 1, 50, and 100 times solar metallicity. We find for a nominal 10 Mearth planet at 700 AU at the current age of the solar system an effective temperature of 47 K, much higher than the equilibrium temperature of about 10 K, a radius of 3.7 Rearth, and an intrinsic luminosity of 0.006 Ljupiter. It has estimated apparent magnitudes of Johnson V, R, I, L, N, Q of 21.7, 21.4, 21.0, 20.1, 19.9, and 10.7, and WISE W1-W4 magnitudes of 20.1, 20.1, 18.6, and 10.2. The Q and W4 band and other observations longward of about 13 microns pick up the intrinsic flux. If candidate Planet 9 has a significant H/He layer and an efficient energy transport in the interior, then its luminosity is dominated by the intrinsic contribution, making it a self-luminous planet. At a likely position on its orbit near aphelion, we estimate for a mass of 5, 10, 20, and 50 Mearth a V magnitude from the reflected light of 24.3, 23.7, 23.3, and 22.6 and a Q magnitude from the intrinsic radiation of 14.6, 11.7, 9.2, and 5.8. The latter would probably have been detected by past surveys. 
Esther F. Linder and Christoph Mordasini
astro-ph.EP