Stellar structure
Stellar structure models describe the internal structure of a
Energy transport
Different layers of the stars transport heat up and outwards in different ways, primarily
Convection is the dominant mode of energy transport when the temperature gradient is steep enough so that a given parcel of gas within the star will continue to rise if it rises slightly via an
The internal structure of a main sequence star depends upon the mass of the star.
In stars with masses of 0.3–1.5 solar masses (M☉), including the Sun, hydrogen-to-helium fusion occurs primarily via proton–proton chains, which do not establish a steep temperature gradient. Thus, radiation dominates in the inner portion of solar mass stars. The outer portion of solar mass stars is cool enough that hydrogen is neutral and thus opaque to ultraviolet photons, so convection dominates. Therefore, solar mass stars have radiative cores with convective envelopes in the outer portion of the star.
In massive stars (greater than about 1.5 M☉), the core temperature is above about 1.8×107 K, so hydrogen-to-helium fusion occurs primarily via the CNO cycle. In the CNO cycle, the energy generation rate scales as the temperature to the 15th power, whereas the rate scales as the temperature to the 4th power in the proton-proton chains.[2] Due to the strong temperature sensitivity of the CNO cycle, the temperature gradient in the inner portion of the star is steep enough to make the core convective. In the outer portion of the star, the temperature gradient is shallower but the temperature is high enough that the hydrogen is nearly fully ionized, so the star remains transparent to ultraviolet radiation. Thus, massive stars have a radiative envelope.
The lowest mass main sequence stars have no radiation zone; the dominant energy transport mechanism throughout the star is convection.[3]
Equations of stellar structure
The simplest commonly used
In forming the stellar structure equations (exploiting the assumed spherical symmetry), one considers the matter density , temperature , total pressure (matter plus radiation) , luminosity , and energy generation rate per unit mass in a spherical shell of a thickness at a distance from the center of the star. The star is assumed to be in
First is a statement of hydrostatic equilibrium: the outward force due to the pressure gradient within the star is exactly balanced by the inward force due to gravity. This is sometimes referred to as stellar equilibrium.
- ,
where is the cumulative mass inside the shell at and G is the gravitational constant. The cumulative mass increases with radius according to the mass continuity equation:
Integrating the mass continuity equation from the star center () to the radius of the star () yields the total mass of the star.
Considering the energy leaving the spherical shell yields the energy equation:
- ,
where is the luminosity produced in the form of
The energy transport equation takes differing forms depending upon the mode of energy transport. For conductive energy transport (appropriate for a white dwarf), the energy equation is
where k is the
In the case of radiative energy transport, appropriate for the inner portion of a solar mass main sequence star and the outer envelope of a massive main sequence star,
where is the opacity of the matter, is the
The case of convective energy transport does not have a known rigorous mathematical formulation, and involves
where is the
Also required are the equations of state, relating the pressure, opacity and energy generation rate to other local variables appropriate for the material, such as temperature, density, chemical composition, etc. Relevant equations of state for pressure may have to include the perfect gas law, radiation pressure, pressure due to degenerate electrons, etc. Opacity cannot be expressed exactly by a single formula. It is calculated for various compositions at specific densities and temperatures and presented in tabular form.[7] Stellar structure codes (meaning computer programs calculating the model's variables) either interpolate in a density-temperature grid to obtain the opacity needed, or use a fitting function based on the tabulated values. A similar situation occurs for accurate calculations of the pressure equation of state. Finally, the nuclear energy generation rate is computed from nuclear physics experiments, using reaction networks to compute reaction rates for each individual reaction step and equilibrium abundances for each isotope in the gas.[6][8]
Combined with a set of
Although nowadays stellar evolution models describe the main features of
Rapid evolution
The above simplified model is not adequate without modification in situations when the composition changes are sufficiently rapid. The equation of hydrostatic equilibrium may need to be modified by adding a radial acceleration term if the radius of the star is changing very quickly, for example if the star is radially pulsating.[9] Also, if the nuclear burning is not stable, or the star's core is rapidly collapsing, an entropy term must be added to the energy equation.[10]
See also
References
- ^ Hansen, Kawaler & Trimble (2004, §5.1.1)
- ^ Hansen, Kawaler & Trimble (2004, Tbl. 1.1)
- ^ Hansen, Kawaler & Trimble (2004, §2.2.1)
- ^ This discussion follows those of, e. g., Zeilik & Gregory (1998, §16-1–16-2) and Hansen, Kawaler & Trimble (2004, §7.1)
- ^ Hansen, Kawaler & Trimble (2004, §5.1)
- ^ a b Ostlie, Dale A. and Carrol, Bradley W., An introduction to Modern Stellar Astrophysics, Addison-Wesley (2007)
- doi:10.1086/177381.
- doi:10.1086/341728.
- S2CID 16150778.
- Bibcode:1986A&A...162..103M.
Sources
- Kippenhahn, R.; Weigert, A. (1990), Stellar Structure and Evolution, Springer-Verlag
- Hansen, Carl J.; Kawaler, Steven D.; Trimble, Virginia (2004), Stellar Interiors (2nd ed.), Springer, ISBN 0-387-20089-4
- Kennedy, Dallas C.; Bludman, Sidney A. (1997), "Variational Principles for Stellar Structure", Astrophysical Journal, 484 (1): 329–340, S2CID 16835178
- Weiss, Achim; Hillebrandt, Wolfgang; Thomas, Hans-Christoph; Ritter, H. (2004), Cox and Giuli's Principles of Stellar Structure, Cambridge Scientific Publishers, Bibcode:2004cgps.book.....W
- Zeilik, Michael A.; Gregory, Stephan A. (1998), Introductory Astronomy & Astrophysics (4th ed.), Saunders College Publishing, ISBN 0-03-006228-4
External links
- opacity code retrieved November 2009
- The Yellow CESAM code, stellar evolution and structure Fortran source code
- EZ to Evolve ZAMS Stars a FORTRAN 90 software derived from Eggleton's Stellar Evolution Code, a web-based interface can be found here [1].
- Geneva Grids of Stellar Evolution Models (some of them including rotational induced mixing)
- The BaSTI database of stellar evolution tracks