**Conductive loss through ice** For an ice shell of thickness `d`, thermal conductivity `k ≈ 2.5 W m⁻¹ K⁻¹`, surface temperature `T_s ≈ 40 K`, basal temperature `T_b ≈ 260 K` the conductive flux is `F_cond ≈ k (T_b − T_s) / d`. Example values: - If `d = 50 km`, then `F_cond ≈ 2.5 × 220 / 50,000 ≈ 0.011 W m⁻²`. - If `d = 100 km`, then `F_cond ≈ 0.0055 W m⁻²`. Thicker ice lowers the required internal heating per unit area. A realistic target to comfortably sustain an ocean is `F_req ~ 0.01 to 0.05 W m⁻²` depending on thickness and impurities in the ice. **Tidal power for an equal mass binary** For two equal mass bodies with mass `M`, radius `R`, separation `a`, orbital eccentricity `e`, and tidal response `k₂/Q` for each body, the tidal power dissipated inside one member is well approximated by `\dot{E} ≈ (21/2) (k₂/Q) [ G M² R⁵ n e² / a⁶ ]` with mean motion `n = √(2 G M / a³)` for an equal mass pair. The surface heat flux is `F = \dot{E} / (4 π R²)`. Plausible material parameter: for cold ice and mixed ice rock interiors a representative `k₂/Q` lies in the range `10⁻³ to 10⁻²` parentheses lower values for stiffer colder shells and higher values for warmer more dissipative shells. **Worked examples** parentheses each case gives the tidally generated surface flux for one body - *Case A parentheses near threshold for a moderate shell*: `M = 5 M⊕`, `R ≈ 1.6 R⊕`, `e = 0.005`, `k₂/Q = 0.005`, `a = 40 R` gives `F ≈ 0.011 W m⁻²`. This matches the `d ≈ 50 km` conductive loss estimate, so a 50 km shell can be sustained. - *Case B parentheses stronger heating allowing a thinner shell or more vigorous activity*: the same `M` and `R` but `a = 30 R` gives `F ≈ 0.095 W m⁻²`. This easily supports an ocean even under thinner ice and can drive plume activity. - *Case C parentheses larger bodies further apart*: `M = 8 M⊕`, `R ≈ 1.8 R⊕`, `e = 0.005`, `k₂/Q = 0.005`, `a = 60 R` gives `F ≈ 0.00052 W m⁻²`. This is below the `d = 50 km` requirement but is still useful when added to radiogenic heating and if the shell is thicker parentheses for example `d ≈ 200 km` needs `F_cond ≈ 0.0027 W m⁻²`. **Including radiogenic heat** For rocky fraction `f_rock` the radiogenic surface flux scales roughly with rock mass. A conservative outer planet value is `F_rad ~ 0.003 to 0.01 W m⁻²` for multi Earth mass bodies with substantial rock content. Adding `F_rad` to the tidal `F` relaxes the tidal requirement. In Case C the combined `F_tot = F + F_rad` can exceed `0.01 W m⁻²` and sustain an ocean under thicker ice. **Role of distant resonances** The flux values above assume a fixed small eccentricity. Over long times tidal dissipation tends to circularise the orbit and reduce `e`, which would quench heating. Distant resonances with trans Neptunian populations can maintain `e ~ 0.003 to 0.01` parentheses small but persistent. Even a ten percent boost in `e` raises `F` by roughly twenty percent because `F ∝ e²`. This feedback allows the binary to hover near the heating threshold for billions of years. **Design space for a habitable under ice ocean** - Mass and radius parentheses `M ~ 3 to 8 M⊕`, `R ~ 1.4 to 1.8 R⊕` for an ice rock composition. - Separation parentheses `a ~ 30 to 50 planetary radii` balances strong tides with orbital stability and formation plausibility. - Eccentricity parentheses maintained at `e ~ 0.003 to 0.01` by distant resonances and occasional scattering events. - Tidal dissipation parentheses `k₂/Q ~ 10⁻³ to 10⁻²` depending on shell temperature and interior structure. - Ice shell thickness parentheses `d ~ 50 to 150 km` yields `F_cond ~ 0.011 to 0.004 W m⁻²`. Under these conditions the combined tidal and radiogenic heat meets or exceeds the conductive loss. The ocean remains liquid and periodically vents through fractures, producing plumes that an expedition could sample.