A parameter-free model for temperature and pressure profiles in luminous, stable stars

Abstract

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.