Atmospheric Escape From a Hot Jupiter: Energy-Limited Loss Rates at Densities Far Below the Bound

This paper computes where exobase region sits relative to the density bound that arises in Einstein-Cartan gravity when the spin of fermionic matter sources torsion. The bound on the neutron branch is 9.72e+57 kilograms per cubic metre, and on the electron branch 2.87e+51 kilograms per cubic metre. The primary system examined, exobase region, has a density of 1.00e-09 kilograms per cubic metre, which is a fraction 1.03e-67 of the neutron-branch bound — a margin of 67.0 orders of magnitude. The fractional correction that the bound makes to the effective density, and hence to structure quantities built from it, is that same ratio to leading order. Further systems examined here are exobase region at 1.00e-09 kilograms per cubic metre; upper atmosphere at XUV absorption at 1.00e-07 kilograms per cubic metre; photosphere at 1.00e-03 kilograms per cubic metre. The conclusion is stated plainly: at these densities the correction is present in the equations and absent from every digit that observation or modelling can reach, and this paper quantifies that absence system by system rather than asserting it. The study is computed from stated assumptions; it is not a measurement.