A Scale-Dependent Dimensionality Model of Solar Structure: Modified Lane–Emden Solutions, Neutrino Fluxes, and Helioseismic Constraints

A Scale-Dependent Dimensionality Model of Solar Structure: Modified Lane–Emden Solutions, Neutrino Fluxes, and Helioseismic Constraints Christopher Merrill With computational collaboration by ChatGPT Abstract We explore a Solar model in which the effective spatial dimensionality, deff(r), varies weakly with scale, modifying the classical Lane–Emden equation. The perturbation induces a small radial dependence in the gravitational acceleration and alters the structural integrals governing nuclear reaction rates. By calibrating the modified solution to observed solar radius and mass, we compute neutrino fluxes, helioseismic sound-speed deviations, and perform a two-parameter (∆, λ) grid scan constrained by experiment. We find a best-fit region in which the reduced chi-square for the four major neutrino fluxes reaches χ2 red ≈0.16, with central sound-speed deviations remaining below 2 ×10−3, indicating consistency with helioseismic constraints. The best-fit parameter combination is ∆ ≈0.050 and λ≈0.147, corresponding to a weak dimensional perturbation that decays over a small fraction of the solar radius. 1 Introduction Standard Solar Models (SSMs) assume exact spherical symmetry in three spatial dimensions, implicitly fixing the gravitational acceleration and stellar structure equations. In this work, we consider a scenario in which the gravitational potential inherits a scale-dependent effective dimension: deff(ξ) = 3 + δ(ξ), δ(ξ) = ∆ e−λξ, where ∆ ≪1 and ξ=r/α is the usual dimensionless radial coordinate of the Lane–Emden formalism. Even extremely small departures from 3 can modify: •the hydrostatic equilibrium equation, •the Lane–Emden solution θ(ξ), •the resulting nuclear reaction kernels, 1 •predicted neutrino fluxes, •and the sound-speed profile cs(r). We rewrite the stellar structure equation so that the divergence term carries deff(r) instead of the fixed value 3, and investigate whether such a perturbation is consistent with solar data. 2 Methods 2.1 Modified Lane–Emden Equation A scale-dependent dimension modifies the divergence operator: 1 rdeff −1 d drrdeff −1dP dr=−GM(r)ρ(r) rdeff −1. In polytropic form with index n= 3: 1 ξ2+δ(ξ) d dξξ2+δ(ξ)dθ dξ=−θ3. When δ(ξ) = 0, the standard n= 3 Lane–Emden equation is recovered. The ODE is solved numerically using a two-step approach: 1. power-series expansion for ξ < 10−4, 2. direct numerical integration via solve ivp. 2.2 Solar Calibration For each model (standard or modified), the structural solution is matched to observed Solar mass and radius: M⊙= 4πα3ρcZξ1 0 ξ2θ3dξ, R⊙=αξ1. This uniquely determines (α, ρc) and yields the scaling of the central temperature: Tc∝(ρc)1/3. We report Tc(mod)/Tc(std) for each parameter set. 2.3 Neutrino Production Integrals For each neutrino branch iwith a temperature scaling βi, we compute: Ii=Zξ1 0 θ3+βiξ2dξ. The predicted flux ratio becomes Ri=I(mod) i I(std) i T(mod) c T(std) c!βi . We include pp, pep, 7Be, and 8B, with standard temperature exponents. 2 2.4 Helioseismic Sound-Speed Test For a polytropic model: c2 s(r)∝T(r)∝θ(ξ). We compute the fractional deviation: ∆cs cs =θmod −θstd θstd , sampled over 0.01 ≤r/R ≤0.98. 2.5 Parameter Scan We scan over: ∆∈[0.02,0.05], λ ∈[0.12,0.18], kdim ∈ {1,2,3}, with kdim governing how strongly deff enters the ODE. Each model produces four flux ratios compared to experiment: χ2=X i (Ri−Rexp i)2 σ2 i . The reduced chi-square is χ2 red =χ2/4. 3 Results 3.1 Lane–Emden Solutions The modified dimensionality shifts the surface zero from ξstd 1= 6.89685 →ξmod 1= 6.91627. 3.2 Dimensionality Profile The effective dimensionality profile is prescribed as deff(ξ)=3−∆ exp"−ξ λkdim #, so that the dimensional perturbation δ(ξ)≡deff(ξ)−3 is localized to the innermost core. For the best–fit parameters ∆ = 0.050, λ= 0.147, and kdim = 2, the central value is deff(0) = 3 −∆≃2.95, 3 a deviation of only ∼1.7% from strict three–dimensionality. The profile rises rapidly back toward deff ≈3byξ∼1 (corresponding to the inner few percent of the solar radius), so that the dimensional modification is confined to the nuclear–burning core and has negligible impact on the outer envelope. Figure 2 illustrates both deff(ξ) and δ(ξ) for this best–fit solution. The perturbation amplitude is small, but—as shown in later sections—large enough to modify the central temperature and the neutrino production kernels while remaining consistent with helioseismic constraints. 3.3 Helioseismic Sound-Speed Deviation For the best-fit region, max  ∆cs cs = 1.89 ×10−3. 3.4 Neutrino Flux Results For the fiducial point (∆, λ) = (0.045,0.160): Tmod c/Tstd c= 1.00387. Flux Rstruct Rtotal Rexp pp 0.9838 0.9991 1.020 ±0.100 pep 0.9876 0.9952 0.880 ±0.150 Be7 0.9741 1.0125 1.010 ±0.030 B8 0.9614 1.0386 1.040 ±0.070 Structural-only chi-square: χ2= 3.34, χ2 red = 0.83. With Tcscaling: χ2= 0.64, χ2 red = 0.16. 3.5 Parameter Scan and Best-Fit Region We find: (∆, λ)best ≈(0.050,0.147), χ2 red ≈0.160, Tmod c/Tstd c= 1.00373. 4 Conclusion We have shown that a weak, scale-dependent modification to the effective dimension of space can produce a consistent Solar model that: •preserves helioseismic sound-speed accuracy at the 10−3level, 4 •adjusts the structural integrals governing neutrino production, •matches all four major neutrino fluxes with χ2 red ≈0.16, •and yields a best-fit dimensional perturbation of order δ(0) ≈0.05 decaying over λ−1∼1 Lane–Emden units. The result is notable because a purely geometric modification—with no changes to nuclear physics—can improve the agreement between theory and experiment in a controlled, quantifiable way. Future work will explore more realistic (non-polytropic) Solar models, frequency-dependent helioseismic inversions, and the implications for other stars across the HR diagram. 5 Figure 1: Difference between modified and standard Lane–Emden solutions, ∆θ(ξ) = θmod −θstd. 6 Figure 2: Effective dimension profile deff (ξ) (upper curve) and perturbation δ(ξ) = deff −3 (lower curve) for the best–fit parameters (∆, λ) = (0.050,0.147) and kdim = 2. 7 Figure 3: Fractional sound-speed deviation ∆cs/csvs. radius. 8 Figure 4: Reduced chi-square χ2 red across the (∆, λ) grid. Contour is the 1–σregion. 9