Stars/Sun/Standard models

The Standard Solar Model (SSM) is the currently popular model for the Sun. There are many models of the Sun’s current state, evolution and interior.

Theoretical solar models
A zero-age solar, or protosolar, model composition is estimated using [carbonaceous Ivuna, Tanzania] CI chondrites.

For "some elements condensation temperatures are not known well, if at all. These elements include several alkalis (Rb, Cs), halogens (F, Cl, Br, I), and trace elements (Bi, In, Hg, Pb, Sn, and Tl)."

"Carbon and oxygen are two abundant elements governing much of the chemistry of the other, less abundant elements. A lower absolute oxygen abundance will lower condensation temperatures of O-bearing compounds. In addition to the absolute O abundance, the C/O ratio influences condensation temperatures. The C/O ratio from the determination by Allende Prieto et al. (2001, 2002) is 0.5, which is slightly higher than the C/O ratio of 0.49 found by Grevesse & Sauval (1998), and clearly higher than 0.42 from Anders & Grevesse (1989). An increase in the C/O ratio toward unity lowers the condensation temperatures of oxides and silicates, and the initial oxide and silicate condensates are replaced by C-bearing compounds (e.g., Larimer 1975; Larimer & Bartholomay 1979; Lodders & Fegley 1993; Krot et al. 2000). Changes in abundances of other elements such as sulfur or phosphorus also mean that their condensation temperatures will change."

"When compared to the photosphere, meteorites are depleted in noble gases and H, C, N, and O, which readily form gaseous compounds, and enriched in elements (e.g., Li) that are processed in the Sun. Of the 83 naturally occurring elements, there are 56 for which a comparison of photospheric and CI chondrite abundances can be done. Excluded from the comparison are elements lighter than fluorine, the noble gases, and elements for which no or only very unreliable photospheric abundance determinations exist [...]. Of the 56 elements for which the comparison can be done, the relative abundances of 31 elements in the photosphere and in CI chondrites agree within 10%; increasing the comparison to within 15% yields agreement for 41 elements. Therefore, the usually more precise analyses of CI chondrites can be used to refine photospheric abundances."

Models "of the Sun’s evolution and interior show that currently observed photospheric abundances (relative to hydrogen) must be lower than those of the proto-Sun because helium and other heavy elements have settled toward the Sun’s interior since the time of the Sun’s formation some 4.55 Gyr ago."

Photospheric "abundances relative to hydrogen are not representative of the solar system, and only the protosolar (i.e., unfractionated with respect to hydrogen) abundances represent the" "solar system elemental abundances."

The "abundances for a few elements (e.g., F, Cl, Rb, Sn, Sb) are those determined almost 30 years ago and that for some elements (As, Se, Br, Te, I, Cs, Ta, Re, Hg, Bi) no photospheric determinations exist because there are no observable lines in the solar spectrum. The abundances of the noble gases He, Ne, Ar, Kr, and Xe cannot be derived from the photospheric spectrum. Ne and Ar abundances can be derived from coronal sources such as solar wind (SW), solar flares, or solar energetic particles (SEP). The He abundance is derived indirectly from results of helioseismology."

"No solar abundance data are available for Kr and Xe".

"The abundances of F, Cl, and Tl are derived from sunspot spectra (Hall & Noyes 1969, 1972; Lambert, Mallia, & Smith 1972)."

Chondritic "meteorites in their present form are not unaltered equilibrium condensates from the solar nebula. Chondrites have experienced mineralogical alterations by thermal metamorphism, and in the case of CI and CM chondrites, also by aqueous alteration on their parent bodies. However, the chondrite parent bodies themselves must have accumulated from nebular condensates. Assuming that the parent body alterations occurred in a closed system, the chemical composition then still remains that of the nebular condensates accreted by the chondritic meteorite parent bodies. However, some individual phases, such as calcium-aluminum-rich inclusions in chondrites, may represent more or less unaltered condensates (see § 3.2)."

"Many of the recent solar models (e.g., Christensen-Dalsgaard 1998; Boothroyd & Sackmann 2003) are calibrated to the observed Z/X ratio from the abundance table by Grevesse & Noels (1993), mainly because opacity tables are available for this composition."

"The abundance determinations of Li in the Sun and in CI chondrites are relatively certain and show that the photospheric Li abundance is about 150 times lower than the value preserved in meteorites."

Protons
The following fusion reaction produces neutrinos and accompanying gamma-rays of the energy indicated:


 * $$\mathrm{_1^1H} + \mathrm{_1^1H} \rightarrow \mathrm{_{1}^{2}D} + e^+ + \nu_e + \gamma (0.42 MeV). $$

Observation of gamma rays of this energy likely indicate this reaction is occurring nearby.

"In ... the Cowan–Reines neutrino experiment, antineutrinos created in a nuclear reactor by beta decay reacted with protons producing neutrons and positrons:"



"The positron quickly finds an electron, and they annihilate each other. The two resulting gamma rays (γ) [511 keV each] are detectable. The neutron can be detected by its capture on an appropriate nucleus, releasing a gamma ray. The coincidence of both events – positron annihilation and neutron capture – gives a unique signature of an antineutrino interaction."

"It is fair to note, however, that almost all theories which invoke non-baryonic matter require some level of coincidence in order that the luminous and unseen mass contribute comparable densities (to within one or two powers often). For instance, in a neutrino-dominated universe, (mv/mproton) must be within a factor ~ 10 of nb/nγ. The only model that seems to evade this requirement is Witten’s (1984) idea that the quark-hadron phase transition may leave comparable amounts of material in ‘ordinary’ baryons and in ‘nuggets’ of exotic matter."

Neutrinos
"A star is considered to be at zero age (protostellar) when it is assumed to have a homogeneous composition and to be just beginning to derive most of its luminosity from nuclear reactions (so neglecting the period of contraction from a cloud of gas and dust). To obtain the SSM, a one solar mass stellar model at zero age is evolved numerically to the age of the Sun. The abundance of elements in the zero age solar model is estimated from primordial meteorites. Along with this abundance information, a reasonable guess at the zero-age luminosity (such as the present-day Sun's luminosity) is then converted by an iterative procedure into the correct value for the model, and the temperature, pressure and density throughout the model calculated by solving the equations of stellar structure numerically assuming the star to be in a steady state. The model is then evolved numerically up to the age of the Sun. Any discrepancy from the measured values of the Sun's luminosity, surface abundances, etc. can then be used to refine the model. For example, since the Sun formed, the helium and heavy elements have settled out of the photosphere by diffusion. As a result, the Solar photosphere now contains about 87% as much helium and heavy elements as the protostellar photosphere had; the protostellar Solar photosphere was 71.1% hydrogen, 27.4% helium, and 1.5% metals. A measure of heavy-element settling by diffusion is required for a more accurate model."

"Nuclear reactions in the core of the Sun change its composition, by converting hydrogen nuclei into helium nuclei by the proton-proton chain and (to a lesser extent in the Sun than in more massive stars) the CNO cycle. This decreases the mean molecular weight in the core of the Sun, which should lead to a decrease in pressure. This does not happen as instead the core contracts. By the Virial Theorem half of the gravitational potential energy released by this contraction goes towards raising the temperature of the core, and the other half is radiated away. By the ideal gas law this increase in temperature also increases the pressure and restores the balance of hydrostatic equilibrium. The luminosity of the Sun is increased by the temperature rise, increasing the rate of nuclear reactions. The outer layers expand to compensate for the increased temperature and pressure gradients, so the radius also increases. "

"Most of the neutrinos produced in the sun come from the first step of the pp chain but their energy is so low (<0.425 MeV) they are very difficult to detect. A rare side branch of the pp chain produces the "boron-8" neutrinos with a maximum energy of roughly 15 MeV, and these are the easiest neutrinos to detect.  A very rare interaction in the pp chain produces the "hep" neutrinos, the highest energy neutrinos predicted to be produced by our sun.  They are predicted to have a maximum energy of about 18 MeV."

"All of the interactions described above produce neutrinos with a spectrum of energies. The electron capture of 7Be produces neutrinos at either roughly 0.862 MeV (~90%) or 0.384 MeV (~10%). "

Sun
"The Standard Solar Model (SSM) refers to a mathematical treatment of the Sun as a spherical ball of gas (in varying states of ionisation, with the hydrogen in the deep interior being a completely ionised plasma). This model, technically the spherically symmetric quasi-static model of a star, has stellar structure described by several differential equations derived from basic physical principles. The model is constrained by boundary conditions, namely the luminosity, radius, age and composition of the Sun, which are well determined. The age of the Sun cannot be measured directly; one way to estimate it is from the age of the oldest meteorites, and models of the evolution of the solar system. The composition in the photosphere of the modern-day Sun, by mass, is 74.9% hydrogen and 23.8% helium. All heavier elements, called metals in astronomy, account for less than 2 percent of the mass. The SSM is used to test the validity of stellar evolution theory. In fact, the only way to determine the two free parameters of the stellar evolution model, the helium abundance and the mixing length parameter (used to model convection in the Sun), are to adjust the SSM to "fit" the observed Sun."

"The differential equations of stellar structure, such as the equation of hydrostatic equilibrium, are integrated numerically. The differential equations are approximated by difference equations. The star is imagined to be made up of spherically symmetric shells and the numerical integration carried out in finite steps making use of the equations of state, giving relationships for the pressure, the opacity and the energy generation rate in terms of the density, temperature and composition. "

Solar evolution
The current evolutionary model for the fusion-burning lifetime of the Sun, or a solar-mass star, is diagrammed at right.

Hypotheses

 * 1) A standard solar model encompassing electron-beam heating may be necessary.