# Standard solar model
The **standard solar model** (SSM) is a calibrated, one-dimensional mathematical description of the Sun's interior that evolves a 1 M☉ star from the pre-main sequence to the present solar age, matching the observed solar luminosity, radius, and surface heavy-element-to-hydrogen ratio without using any helioseismological constraints in its definition[1](https://www.sns.ias.edu/~jnb/Papers/Preprints/BP2000/paper.pdf). It serves as the reference against which solar neutrino fluxes, oscillation frequencies, and models of other Sun-like stars are judged[2](https://inspirehep.net/files/357ed3f24028e54acfbeb65938618b37).
| Key fact | Detail |
|---|---|
| Calibration targets | Present-day luminosity, radius, and surface Z/X; the mixing-length parameter and initial helium and metal mass fractions are adjusted to meet them[3](https://iopscience.iop.org/article/10.3847/1538-4357/835/2/202) |
| Adopted age | 4.57 Gyr, the solar system age[4](https://ar5iv.labs.arxiv.org/html/0910.3690) |
| Sound-speed agreement | Better than 0.1% r.m.s. in the solar interior for the BP2000 model; peak residuals of 0.3% (GS98) versus about 1% (AGSS09) just below the convective envelope[1](https://www.sns.ias.edu/~jnb/Papers/Preprints/BP2000/paper.pdf) • [4](https://ar5iv.labs.arxiv.org/html/0910.3690) |
| Neutrino fluxes (GS98) | pp 5.97×10¹⁰, ⁷Be 5.07×10⁹, ⁸B 5.94×10⁶ cm⁻² s⁻¹[5](https://ar5iv.labs.arxiv.org/html/0811.2424) |
| Abundance problem | Low-Z (AGSS09) models miss the convective-zone boundary and surface helium by 11σ and 5σ[6](https://iopscience.iop.org/article/10.1088/0004-637X/705/2/L123) |
| CNO neutrinos | Borexino measured a CNO flux of 6.7 (+1.2/−0.8)×10⁸ cm⁻² s⁻¹, agreeing with high-metallicity models[7](https://ar5iv.labs.arxiv.org/html/2307.14636) |
| Nominal constants | IAU 2015 Resolution B3 fixed L☉ = 3.828×10²⁶ W, R☉ = 6.957×10⁸ m, T_eff = 5772 K as exact conversion factors[8](https://www.iau.org/common/Uploaded%20files/IAUGA2015-Resolution-B3-recommended-nominal-conversion.pdf) |
## Historical development
The lineage begins with Jonathan Lane, who in 1869 first wrote down the equations describing a gas sphere in hydrostatic equilibrium, constructing the first physical model of the solar interior[9](http://www.ap.smu.ca/~guenther/evolution/solar_model_history.html). Erika Böhm-Vitense refined the mixing-length theory by adopting a mixing length proportional to the pressure scale height in work published in 1953 and 1958[9](http://www.ap.smu.ca/~guenther/evolution/solar_model_history.html).
## Physical assumptions and governing equations
In the SSM, energy is transported by radiation in the interior and by convection in the outer layers. Thermal convection is the dominant energy-transport mechanism in the outer one-third of the Sun by radius, where it also drives the global fluid circulations and surface magnetic fields[10](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-122414-034534). Convection is treated under the mixing-length theory, with the mixing length as the adjustable parameter[11](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html).
**Microphysics.** Microscopic diffusion, which depends on gravitational settling and radiative acceleration built on opacity spectra, substantially improved the quality of the SSM; seismic data show that the tachocline, the transition region between the radiative interior and the convective envelope, has an extension of about 5% of the solar radius[2](https://inspirehep.net/files/357ed3f24028e54acfbeb65938618b37). Standard models include no convective overshooting[11](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html).
**Physics deliberately left out.** Convective overshoot is excluded from the standard framework[11](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html). Helioseismology has confirmed the presence of dynamical processes not included in the SSM framework that partly limit its predictions, for example of the present neutrino fluxes[2](https://inspirehep.net/files/357ed3f24028e54acfbeb65938618b37). Seismic inferences also suggest that convective velocities are substantially smaller than those predicted by theory and simulations, challenging models of internal differential rotation that rely on convective stresses[10](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-122414-034534).
## Calibration and key parameters
A calibration evolves a 1 M☉ model from the pre-main sequence to the solar system age of 4.57 Gyr and forces it to match the present-day solar radius, luminosity, and photospheric Z/X[4](https://ar5iv.labs.arxiv.org/html/0910.3690). The three adjusted quantities are the mixing-length parameter and the initial helium and metal mass fractions[3](https://iopscience.iop.org/article/10.3847/1538-4357/835/2/202).
Since 2015, reported solar quantities have been anchored to the IAU nominal constants: a nominal solar radius of 6.957×10⁸ m, nominal luminosity of 3.828×10²⁶ W, and nominal effective temperature of 5772 K. These are defined as exact conversion factors close to the current best estimates, not as the true solar properties[8](https://www.iau.org/common/Uploaded%20files/IAUGA2015-Resolution-B3-recommended-nominal-conversion.pdf).
## Helioseismology tests
Helioseismology provides the sharpest test of the SSM because the model is defined without seismic constraints. The BP2000 model agreed with helioseismic sound-speed inversions to better than 0.1% r.m.s. averaged over the solar interior[1](https://www.sns.ias.edu/~jnb/Papers/Preprints/BP2000/paper.pdf), and by the mid-1990s calibrated models already showed good overall agreement with the inferred solar structure[12](https://link.springer.com/article/10.1007/s41116-020-00028-3). The remaining discrepancy between the helioseismic radial sound-speed profile and the SSM prediction is smaller than 2%, concentrated in regions likely related to radiative-opacity uncertainties or an anomalous internal composition[13](https://ar5iv.labs.arxiv.org/html/1701.03926).
**Where the residuals sit.** High-Z models agree well with solar data overall but show a systematic problem right below the bottom of the convective envelope, linked to the steepness of the molecular-weight and temperature gradients there; low-Z models disagree much more generally[3](https://iopscience.iop.org/article/10.3847/1538-4357/835/2/202). A global comparison of the B16 models with solar observables yields 2.7σ for the high-Z (GS98) model and 4.7σ for the low-Z (AGSS09) model; excluding the sound-speed points in the narrow region just below the convective envelope improves these to 0.9σ and 3.0σ[3](https://iopscience.iop.org/article/10.3847/1538-4357/835/2/202).
One seismic observable remains unexplained by the high- and low-metallicity standard models considered: neither reproduces the observed gravity-mode period spacing, with disagreement exceeding 100σ. The family of standard models does narrow the expected period spacing to between about 2150 and 2190 s[11](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html).
## Solar neutrinos as a probe
The SSM predicts internal observables such as neutrino fluxes and oscillation frequencies, and its reliability has been tested by neutrino detections and helioseismology[2](https://inspirehep.net/files/357ed3f24028e54acfbeb65938618b37). The historical tension was severe: the Bahcall–Ulrich SSM predicted a chlorine (³⁷Cl) capture rate of 7.9(1±0.33) SNU (solar neutrino unit, a neutrino capture rate measure), an uncertainty of about ±2.6 SNU, while Raymond Davis's experiment observed 2.0 ± 0.3 SNU, the original solar neutrino problem[14](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.60.297). Bahcall's 1992 review quoted 8.0 ± 3.0 SNU for chlorine-37 and 132 (+21/−17) SNU for gallium-71, with errors spanning the total theoretical range including 3σ input-parameter uncertainties[15](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.64.885).
**Predicted fluxes.** The BPS08 calibration gives, for the GS98 composition, pp 5.97×10¹⁰, ⁷Be 5.07×10⁹, and ⁸B 5.94×10⁶ cm⁻² s⁻¹; the corresponding AGS05 values are 6.04×10¹⁰, 4.55×10⁹, and 4.72×10⁶[5](https://ar5iv.labs.arxiv.org/html/0811.2424). Predicted total capture rates are 8.46 (+0.87/−0.88) SNU for chlorine and 127.9 (+8.1/−8.2) SNU for gallium under GS98, versus 6.86 and 120.5 SNU under AGS05[5](https://ar5iv.labs.arxiv.org/html/0811.2424). Uncertainty budgets differ by flux: solar composition dominates the CNO flux uncertainties (13% for ¹³N, 12% for ¹⁵O, 17% for ¹⁷F), while for ⁸B the dominant terms are opacity (6.8%), diffusion (4.2%), and the S₁₇ cross section (3.8%)[5](https://ar5iv.labs.arxiv.org/html/0811.2424).
## The solar abundance problem
In 2005, new three-dimensional atmosphere simulations and better atomic data led Asplund and colleagues to a downward revision of the most abundant heavy elements, C, N, O, and Ne, by about 30%, lowering the solar metallicity relative to Grevesse & Sauval (1998)[11](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html). SSMs built on the revised abundances no longer agree with helioseismology: the base of the convective envelope becomes too shallow and the surface helium abundance too low[11](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html). Quantitatively, the AGSS09 model's convective-zone boundary radius and surface helium abundance remain 11σ and 5σ away from the helioseismic values[6](https://iopscience.iop.org/article/10.1088/0004-637X/705/2/L123), and the sound-speed peak below the convective zone is about 1% for AGSS09, an improvement over AGS05 but far above GS98, whose overall sound-speed agreement is a factor of 4 better[6](https://iopscience.iop.org/article/10.1088/0004-637X/705/2/L123) • [4](https://ar5iv.labs.arxiv.org/html/0910.3690). The discrepancy, affecting the convective envelope, the sound-speed profile, and the convection-zone depth, was originally called the Convective Zone Problem and later renamed the Solar Abundance Problem[16](https://arxiv.org/html/2608.23155).
**Which scale fits better?** On helioseismology, GS98 clearly fits better[4](https://ar5iv.labs.arxiv.org/html/0910.3690). A 2024 analysis offers a warning: inversions of helioseismic data, performed directly, point to a low metal content in the Sun's convective envelope, which matches what spectroscopic studies using full 3D models have found.[17](https://www.aanda.org/articles/aa/full_html/2024/06/aa48312-23/aa48312-23.html)
## By the numbers
- Calibrated age: 4.57 Gyr[4](https://ar5iv.labs.arxiv.org/html/0910.3690)
- IAU nominal luminosity parameter: 3.828×10²⁶ W[8](https://www.iau.org/common/Uploaded%20files/IAUGA2015-Resolution-B3-recommended-nominal-conversion.pdf)
- Sound-speed agreement: better than 0.1% r.m.s. interior-wide (BP2000)[1](https://www.sns.ias.edu/~jnb/Papers/Preprints/BP2000/paper.pdf); peak residual 0.3% (GS98) versus 1% (AGSS09)[4](https://ar5iv.labs.arxiv.org/html/0910.3690)
- Global model–data comparison: 2.7σ (GS98) and 4.7σ (AGSS09), falling to 0.9σ and 3.0σ when the sub-envelope sound-speed region is excluded[3](https://iopscience.iop.org/article/10.3847/1538-4357/835/2/202)
- ⁸B flux: 5.94×10⁶ cm⁻² s⁻¹ (GS98)[5](https://ar5iv.labs.arxiv.org/html/0811.2424)
- Gravity-mode period spacing predicted range: 2150–2190 s[11](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html)
## Recent developments
**Borexino's final CNO results.** The collaboration's final CNO measurement gives an interaction rate of 6.7 (+1.2/−0.8) counts per day per 100 tonnes and a CNO neutrino flux at Earth of 6.7 (+1.2/−0.8)×10⁸ cm⁻² s⁻¹, in agreement with high-metallicity standard solar models; the no-CNO hypothesis is rejected at about 8σ[7](https://ar5iv.labs.arxiv.org/html/2307.14636). The collaboration's own publication page for the improved Phase-III measurement reports a slightly different figure, a rate of 6.7 (+2.0/−0.8) cpd/100 tonnes excluding the absence of CNO at about 7σ, with a corresponding flux of 6.6 (+2.0/−0.9)×10⁸ cm⁻² s⁻¹[18](https://borex.lngs.infn.it/papers/articles/improved-measurement-of-cno-solar-neutrinos-and-its-implications-for-the-ssm/).
**Neutrinos as abundance observables.** Combining the CNO result with the ⁸B flux yields a solar C+N abundance of N_CN = 5.81 (+1.22/−0.94)×10⁻⁴, in agreement with high-metallicity models[7](https://ar5iv.labs.arxiv.org/html/2307.14636); the Phase-III analysis gives N_CN = 5.78 (+1.86/−1.00)×10⁻⁴, showing a roughly 2σ tension with low-metallicity photospheric measurements and disfavoring the low-metallicity model B16-AGSS09met at 3.1σ relative to B16-GS98[18](https://borex.lngs.infn.it/papers/articles/improved-measurement-of-cno-solar-neutrinos-and-its-implications-for-the-ssm/). A conference summary notes the C+N abundance agrees well with the high-metallicity compilations MB22 and GS98 while showing a moderate ~2σ tension with AGSS09met, C11, and AAG21[19](https://inspirehep.net/files/05019b34676ab0247b726b21b3dc190a). A 2025 review of solar system abundances identifies neutrino fluxes from the pp chain and CNO cycles, as measured by Borexino, as far more advanced testable observables for solar elemental abundances[20](https://link.springer.com/content/pdf/10.1007/s11214-025-01146-w.pdf).
**Opacity tables and the limits of raising Z.** Post-2023 work compares solar models computed with the OPAL, OP, and OPLIB opacity tables; detailed comparisons are still required to determine the origin of the differences between them[17](https://www.aanda.org/articles/aa/full_html/2024/06/aa48312-23/aa48312-23.html). The same 2024 study concludes that raising the solar metallicity does not remove the degeneracies in solar modeling, so standard solar models cannot be used as an argument for a high-metallicity composition; removing the discrepancies requires modifying the temperature gradient just below the base of the convective zone, which can be simulated by a localized opacity increase of a few percent[17](https://www.aanda.org/articles/aa/full_html/2024/06/aa48312-23/aa48312-23.html).
## Comparisons and extensions
**Dark-matter-accreting models.** Non-standard models in which the Sun captures dark matter can change the ⁸B neutrino flux by more than 15%, larger than the current SSM uncertainty, which allows such dark-matter particles to be excluded; in extreme capture scenarios an isothermal core forms, while milder capture only reduces the central temperature[13](https://ar5iv.labs.arxiv.org/html/1701.03926).
**Beyond spherical symmetry.** The SSM's one-dimensional, quasi-static framework excludes the dynamical processes helioseismology has confirmed in the real Sun, and these processes partly limit its predictions[2](https://inspirehep.net/files/357ed3f24028e54acfbeb65938618b37). The measured shortfall of convective velocities relative to simulations[10](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-122414-034534) and the unresolved gravity-mode period spacing[11](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html) mark where the standard model's simplifications show.
## References
1. [Bahcall, Pinsonneault & Basu (BP2000), Solar models: current epoch and composition uncertainties](https://www.sns.ias.edu/~jnb/Papers/Preprints/BP2000/paper.pdf)
2. [Turck-Chièze, The Standard Solar Model and beyond](https://inspirehep.net/files/357ed3f24028e54acfbeb65938618b37)
3. [Vinyoles et al. (2017), A New Generation of Standard Solar Models (B16), The Astrophysical Journal](https://iopscience.iop.org/article/10.3847/1538-4357/835/2/202)
4. [Serenelli et al. (2009/2011), New Results on Standard Solar Models](https://ar5iv.labs.arxiv.org/html/0910.3690)
5. [Serenelli et al. (BPS08), Solar neutrinos and the solar composition problem](https://ar5iv.labs.arxiv.org/html/0811.2424)
6. [Asplund et al. (AGSS09), New Solar Composition: The Problem with Solar Models Revisited, ApJ Letters](https://iopscience.iop.org/article/10.1088/0004-637X/705/2/L123)
7. [Borexino Collaboration, Final results of Borexino on CNO solar neutrinos](https://ar5iv.labs.arxiv.org/html/2307.14636)
8. [IAU 2015 Resolution B3: recommended nominal conversion constants for selected solar and planetary properties](https://www.iau.org/common/Uploaded%20files/IAUGA2015-Resolution-B3-recommended-nominal-conversion.pdf)
9. [History of the Solar Model, Saint Mary's University](http://www.ap.smu.ca/~guenther/evolution/solar_model_history.html)
10. [Seismic Sounding of Convection in the Sun, Annual Review of Fluid Mechanics](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-122414-034534)
11. [Buldgen et al. (2021), Standard solar models: Perspectives from updated solar neutrino fluxes and gravity-mode period spacing, Astronomy & Astrophysics](https://www.aanda.org/articles/aa/full_html/2021/07/aa40769-21/aa40769-21.html)
12. [Christensen-Dalsgaard et al., Solar structure and evolution, Living Reviews in Solar Physics](https://link.springer.com/article/10.1007/s41116-020-00028-3)
13. [The Sun as a probe of Fundamental Physics and Cosmology](https://ar5iv.labs.arxiv.org/html/1701.03926)
14. [Bahcall & Ulrich (1988), Solar models, neutrino experiments, and helioseismology, Reviews of Modern Physics](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.60.297)
15. [Bahcall (1992), Standard solar models and the solar neutrino problem, Reviews of Modern Physics](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.64.885)
16. [Solar chemical composition (review)](https://arxiv.org/html/2608.23155)
17. [Buldgen et al. (2024), In-depth analysis of solar models with high-metallicity abundances and updated opacity tables, Astronomy & Astrophysics](https://www.aanda.org/articles/aa/full_html/2024/06/aa48312-23/aa48312-23.html)
18. [Borexino, Improved Measurement of CNO Solar Neutrinos and Its Implications for the SSM](https://borex.lngs.infn.it/papers/articles/improved-measurement-of-cno-solar-neutrinos-and-its-implications-for-the-ssm/)
19. [Solar neutrinos from the CNO fusion cycle: Borexino discovery and implications for solar physics (conference proceedings)](https://inspirehep.net/files/05019b34676ab0247b726b21b3dc190a)
20. [Solar System Elemental Abundances from the Solar Photosphere and CI-Chondrites, Space Science Reviews (2025)](https://link.springer.com/content/pdf/10.1007/s11214-025-01146-w.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Astronomy › Stars and galaxies › Stellar astrophysics, structure, evolution, and variables › Stellar structure, atmospheres, and nucleosynthesis › Stellar interior models and energy transport*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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