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The historic, classical thermodynamic model of star interiors neglects luminosity (𝐿), and consequently predicts ultrahigh central solar temperatures (𝑇 ~ 15 × 106 K). Modern models yield similar 𝑇 profiles mostly because local thermal equilibrium and multiple free parameters are used. Instead, long-term stability of stars signifies disequilibrium where energy generated equals energy emitted. We assume that heat is generated in a shell defining the core and use Fourier’s model, which describes diffusion of heat, including via radiation, to predict the 𝑇 profile. Under steady-state, power 𝐿 transmitted through each shell is constant above the zone of energy generation. Hence, 𝐿 is independent of spherical radius (𝑠), so the Stefan-Boltzmann law dictates 𝑇(𝑠), and material properties are irrelevant. Temperature is constant in the core and proportional to 𝐿¼𝑠−½ above. A point source core sets the upper limit on 𝑇(𝑠), giving 𝑇average = (6/5)𝑇surface. Core size or convecting regions little affect our results. We also construct a parameter-free model for interior pressure (𝑃) and density (ρ) by inserting our 𝑇(𝑠) formula into an ideal gas law (𝑃/ρ 𝛼 𝑇) while using the equation for hydrostatic gravitational compression. We find 𝑃 𝛼 𝑠−3, ρ 𝛼 𝑠−5/2, and ρaverage = 6 × ρsurface. Another result, 𝐿 𝛼 mass3.3, agrees with accepted empirical rules for main sequence stars, and validates our model. The total solar mass already “burned” suggests that fusion occurs near 𝑠surf/400 where 𝑃 ~ 0.5 × 1012 Pa, in agreement with H-bomb pressure estimates. Implications are discussed. 

Abstract. An inverse method is devised to probe Earth's thermalstate without assuming its mineralogy. This constrains thermal conductivity(κ) in the lower mantle (LM) by combining seismologic models of bulkmodulus (B) and pressure (P) vs. depth (z) with a new result, ∂ln(κ) / ∂P ∼ 7.33/BT, and available high temperature (T) data onκ for lengths exceeding millimeters. Considering large samples accounts forthe recently revealed dependence of heat transport properties onlength scale. Applying separation of variables to seismologic ∂B/∂P vs. depth isolates changes with T. The resulting LM dT / dz dependson ∂2B/∂P2 and ∂B/∂T, whichvary little among dense phases. Because seismic ∂B/∂P isdiscontinuous and model dependent ∼ 200 km above the core,unlike the LM, our results are extrapolated through this tiny layer (D′′).Flux and power are calculated from dT / dz for cases of high (oxide) and low(silicate) κ. Geotherm calculations are independent of κ,and thus of LM mineralogy, but require specifying a reference temperature atsome depth: a wide range is considered. Limitations on deep melting are usedto ascertain which of our geotherm, flux, and power curves best representEarth's interior. Except for an oxide composition with miniscule ∂2B/∂P2, the LM heats the core, causing it to melt. Deepheating is attributed to cyclical stresses from > 1000 km dailyand monthly fluctuations of the barycenter inside the LM. 

Available data on insulating, semiconducting, and metallic solids verify our new model that incorporates steady-state heat flow into a macroscopic, thermodynamic description of solids, with agreement being best for isotropic examples. Our model is based on: (1) mass and energy conservation; (2) Fourier’s law; (3) Stefan–Boltzmann’s law; and (4) rigidity, which is a large, yet heretofore neglected, energy reservoir with no counterpart in gases. To account for rigidity while neglecting dissipation, we consider the ideal, limiting case of a perfectly frictionless elastic solid (PFES) which does not generate heat from stress. Its equation-of-state is independent of the energetics, as in the historic model. We show that pressure-volume work (PdV) in a PFES arises from internal interatomic forces, which are linked to Young’s modulus (Ξ) and a constant (n) accounting for cation coordination. Steady-state conditions are adiabatic since heat content (Q) is constant. Because average temperature is also constant and the thermal gradient is fixed in space, conditions are simultaneously isothermal: Under these dual restrictions, thermal transport properties do not enter into our analysis. We find that adiabatic and isothermal bulk moduli (B) are equal. Moreover, Q/V depends on temperature only. Distinguishing deformation from volume changes elucidates how solids thermally expand. These findings lead to simple descriptions of the two specific heats in solids: ∂ln(cP)/∂P = −1/B; cP = nΞ times thermal expansivity divided by density; cP = cVnΞ/B. Implications of our validated formulae are briefly covered.