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Stellar structure

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Structure of stars

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Stellar structure models describe the internal structure of astar in detail and make predictions about theluminosity, thecolor and thefuture evolution of the star. Different classes and ages of stars have different internal structures, reflecting theirelemental makeup and energy transport mechanisms.

Heat transport

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For energy transport refer toRadiative transfer.

The different transport mechanisms of high-mass, intermediate-mass and low-mass stars

Different layers of the stars transport heat up and outwards in different ways, primarilyconvection andradiative transfer, butthermal conduction is important inwhite dwarfs.

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 anadiabatic process. In this case, the rising parcel isbuoyant and continues to rise if it is warmer than the surrounding gas; if the rising parcel is cooler than the surrounding gas, it will fall back to its original height.^[1] In regions with a low temperature gradient and a low enoughopacity to allow energy transport via radiation, radiation is the dominant mode of energy transport.

The internal structure of amain sequence star depends upon the mass of the star.

In stars with masses of 0.3–1.5solar masses (M_☉), including the Sun, hydrogen-to-helium fusion occurs primarily viaproton–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 radiativecores with convective envelopes in the outer portion of the star.

In massive stars (greater than about 1.5M_☉), the core temperature is above about 1.8×10⁷K, sohydrogen-to-helium fusion occurs primarily via theCNO 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 coreconvective. In the outer portion of the star, the temperature gradient is shallower but the temperature is high enough that the hydrogen is nearly fullyionized, so the star remains transparent toultraviolet radiation. Thus, massive stars have aradiative 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

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Temperature profile in the Sun

Mass inside a given radius in the Sun

Density profile in the Sun

Pressure profile in the Sun

The simplest commonly usedmodel of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is in asteady state and that it isspherically symmetric. It contains four basicfirst-order differential equations: two represent howmatter andpressure vary with radius; two represent howtemperature andluminosity vary with radius.^[4]

In forming thestellar structure equations (exploiting the assumed spherical symmetry), one considers the matterdensity $\rho (r)$ , temperature $T(r)$ , total pressure (matter plus radiation) $P(r)$ , luminosity $l(r)$ , and energy generation rate per unit mass $\epsilon (r)$ in a spherical shell of a thickness ${\mbox{d}}r$ at a distance $r {\displaystyle r}$ from the center of the star. The star is assumed to be inlocal thermodynamic equilibrium (LTE) so the temperature is identical for matter andphotons. Although LTE does not strictly hold because the temperature a given shell "sees" below itself is always hotter than the temperature above, this approximation is normally excellent because the photonmean free path, $\lambda$ , is much smaller than the length over which the temperature varies considerably, i.e. $\lambda \ll T/|\nabla T|$ .

First is a statement ofhydrostatic equilibrium: the outward force due to thepressure gradient within the star is exactly balanced by the inward force due togravity. This is sometimes referred to as stellar equilibrium.

{{\mbox{d}}P \over {\mbox{d}}r}=-{Gm\rho  \over r^{2}}

where $m(r)$ is the cumulative mass inside the shell at $r {\displaystyle r}$ andG is thegravitational constant. The cumulative mass increases with radius according to themass continuity equation:

{{\mbox{d}}m \over {\mbox{d}}r}=4\pi r^{2}\rho .

Integrating the mass continuity equation from the star center ( $r=0$ ) to the radius of the star ( $r=R$ ) yields the total mass of the star.

Considering the energy leaving the spherical shell yields theenergy equation:

{{\mbox{d}}l \over {\mbox{d}}r}=4\pi r^{2}\rho (\epsilon -\epsilon _{\nu })

where $\epsilon _{\nu }$ is the luminosity produced in the form ofneutrinos (which usually escape the star without interacting with ordinary matter) per unit mass. Outside the core of the star, where nuclear reactions occur, no energy is generated, so the luminosity is constant.

The energy transport equation takes differing forms depending upon the mode of energy transport. For conductive energy transport (appropriate for awhite dwarf), the energy equation is

{{\mbox{d}}T \over {\mbox{d}}r}=-{1 \over k}{l \over 4\pi r^{2}},

wherek is thethermal conductivity.

In the case of radiative energy transport, appropriate for the inner portion of a solar massmain sequence star and the outer envelope of a massive main sequence star,

{{\mbox{d}}T \over {\mbox{d}}r}=-{3\kappa \rho l \over 64\pi r^{2}\sigma T^{3}},

where $\kappa$ is theopacity of the matter, $\sigma$ is theStefan–Boltzmann constant, and theBoltzmann constant is set to one.

The case of convective energy transport does not have a known rigorous mathematical formulation, and involvesturbulence in the gas. Convective energy transport is usually modeled usingmixing length theory. This treats the gas in the star as containing discrete elements which roughly retain the temperature, density, and pressure of their surroundings but move through the star as far as a characteristic length, called themixing length.^[5] For amonatomic ideal gas, when the convection isadiabatic, meaning that the convective gas bubbles don't exchange heat with their surroundings, mixing length theory yields

{{\mbox{d}}T \over {\mbox{d}}r}=\left(1-{1 \over \gamma }\right){T \over P}{{\mbox{d}}P \over {\mbox{d}}r},

where $\gamma =c_{p}/c_{v}$ is theadiabatic index, the ratio ofspecific heats in the gas. (For a fully ionizedideal gas, $\gamma =5/3$ .) When the convection is not adiabatic, the true temperature gradient is not given by this equation. For example, in the Sun the convection at the base of the convection zone, near the core, is adiabatic but that near the surface is not. The mixing length theory contains two free parameters which must be set to make the model fit observations, so it is aphenomenological theory rather than a rigorous mathematical formulation.^[6]

Also required are theequations 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 structurecodes (meaning computer programs calculating the model's variables) either interpolate in a density-temperature grid to obtain the opacity needed, or use afitting 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 fromnuclear physics experiments, usingreaction 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 ofboundary conditions, a solution of these equations completely describes the behavior of the star. Typical boundary conditions set the values of the observable parameters appropriately at the surface ( $r=R$ ) and center ( $r=0$ ) of the star: $P(R)=0$ , meaning the pressure at the surface of the star is zero; $m(0)=0$ , there is no mass inside the center of the star, as required if the mass density remainsfinite; $m(R)=M$ , the total mass of the star is the star's mass; and $T(R)=T_{eff}$ , the temperature at the surface is theeffective temperature of the star.

Although nowadays stellar evolution models describe the main features ofcolor–magnitude diagrams, important improvements have to be made in order to remove uncertainties which are linked to the limited knowledge of transport phenomena. The most difficult challenge remains the numerical treatment of turbulence.^{[citation needed]} Some research teams are developing simplified modelling of turbulence in 3D calculations.

Rapid evolution

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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]

References

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^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) andHansen, Kawaler & Trimble (2004, §7.1)
^Hansen, Kawaler & Trimble (2004, §5.1)
^^a ^bOstlie, Dale A. and Carrol, Bradley W.,An introduction to Modern Stellar Astrophysics, Addison-Wesley (2007)
^Iglesias, C. A.; Rogers, F. J. (June 1996), "Updated Opal Opacities",Astrophysical Journal,464: 943–+,Bibcode:1996ApJ...464..943I,doi:10.1086/177381.
^Rauscher, T.; Heger, A.; Hoffman, R. D.; Woosley, S. E. (September 2002), "Nucleosynthesis in Massive Stars with Improved Nuclear and Stellar Physics",The Astrophysical Journal,576 (1):323–348,arXiv:astro-ph/0112478,Bibcode:2002ApJ...576..323R,doi:10.1086/341728.
^Moya, A.; Garrido, R. (August 2008), "Granada oscillation code (GraCo)",Astrophysics and Space Science,316 (1–4):129–133,arXiv:0711.2590,Bibcode:2008Ap&SS.316..129M,doi:10.1007/s10509-007-9694-2,S2CID 16150778.
^Mueller, E. (July 1986), "Nuclear-reaction networks and stellar evolution codes – The coupling of composition changes and energy release in explosive nuclear burning",Astronomy and Astrophysics,162 (1–2):103–108,Bibcode:1986A&A...162..103M.

Sources

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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,arXiv:astro-ph/9610099,Bibcode:1997ApJ...484..329K,doi:10.1086/304333,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

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opacity code retrieved November 2009
TheYellow 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)
TheBaSTI database of stellar evolution tracks
Stellar atmospheres: A contribution to the observational study of high temperature in the reversing layers of stars, (1925) by Cecilia Payne-Gaposchkin, Cambridge: The Observatory.

Stars

List

Formation

Evolution

Classification

Early Late Main sequence O B A F G K M Subdwarf O B WR OB Subgiant Giant Blue Red Yellow Bright giant Supergiant Blue Red Yellow Hypergiant Yellow Carbon S CN CH White dwarf Chemically peculiar Am Ap/Bp CEMP HgMn He-weak Barium Lambda Boötis Lead Technetium Be Shell B[e] Helium Extreme Blue straggler
Remnants	Compact star Parker's star White dwarf Helium planet Neutron Radio-quiet Pulsar Binary X-ray Magnetar Stellar black hole X-ray binary Burster SGR
Hypothetical	Blue dwarf Black dwarf Exotic Boson Electroweak Strange Preon Planck Dark Dark-energy Quark Q Black hole star Black Hawking Quasi-star Gravastar Thorne–Żytkow object Iron Blitzar White hole