# Stellar structure

﻿
Stellar structure

Stars of different mass and age have varying internal structures. Stellar structure models describe the internal structure of a star in detail and make detailed predictions about the luminosity, the color and the future evolution of the star.

Energy transport

Different layers of the stars transport heat up and outwards in different ways, primarily convection and radiative transfer, but thermal conduction is important in white 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 an adiabatic process. In this case, the rising parcel is buoyant and continues to rise if it is warmer than the surrounding gas; if the rising particle is cooler than the surrounding gas, it will fall back to its original height. [harvtxt|Hansen|Kawaler|Trimble|2004|loc=§5.1.1] In regions with a low temperature gradient and a low enough opacity to allow energy transport via radiation, radiation is the dominant mode of energy transport.

The internal structure of a main sequence star depends upon the mass of the star.

In solar mass stars (0.3&ndash;1.5 solar masses), 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 solar masses), the core temperature is above about $1.8 imes 10^7$ 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. [harvtxt |Hansen|Kawaler|Trimble|2004|loc=Tbl. 1.1] 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. Giants are also fully convective. [harvtxt |Hansen|Kawaler|Trimble|2004|loc=§2.2.1]

Equations of stellar structure

The simplest commonly used model of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is in a steady state and that it is spherically symmetric. It contains four basic first-order differential equations: two represent how matter and pressure vary with radius; two represent how temperature and luminosity vary with radius. [This discussion follows those of, e. g., harvtxt |Zeilik|Gregory|1998|loc=&sect;16-1&ndash;16-2 and harvtxt |Hansen|Kawaler|Trimble|2004|loc=&sect;7.1.]

In forming the stellar structure equations (exploiting the assumed spherical symmetry), one considers the matter density $ho\left(r\right)$, temperature $T\left(r\right)$, total pressure (matter plus radiation) $P\left(r\right)$, luminosity $l\left(r\right)$, and energy generation rate per unit mass $epsilon\left(r\right)$ in a spherical shell of a thickness $mbox\left\{d\right\}r$ at a distance $r$ from the center of the star. The star is assumed to be in local thermodynamic equilibrium (LTE) so the temperature is identical for matter and photons. 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 photon mean free path, $lambda$, is much smaller than the length over which the temperature varies considerably, i. e. $lambda ll T/| abla T|$.

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.:$\left\{mbox\left\{d\right\} P over mbox\left\{d\right\} r\right\} = - \left\{ G m ho over r^2 \right\}$,where $m\left(r\right)$ is the cumulative mass inside the shell at $r$ and "G" is the gravitational constant. The cumulative mass increases with radius according to the "mass continuity equation:":$\left\{mbox \left\{d\right\} m over mbox\left\{d\right\} r\right\} = 4 pi r^2 ho .$

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 the "energy equation:":$\left\{mbox\left\{d\right\} l over mbox\left\{d\right\} r\right\} = 4 pi r^2 ho \left( epsilon - epsilon_ u \right)$,where $epsilon_ u$ is the luminosity produced in the form of neutrinos (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 luminosity transport (appropriate for a white dwarf), the energy equation is:$\left\{mbox\left\{d\right\} T over mbox\left\{d\right\} r\right\} = - \left\{1 over k\right\} \left\{ l over 4 pi r^2 \right\},$where "k" is the thermal conductivity.

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,:$\left\{mbox\left\{d\right\} T over mbox\left\{d\right\} r\right\} = - \left\{3 kappa ho l over 64 pi r^2 sigma T^3\right\},$where $kappa$ is the opacity of the matter, $sigma$ is the Stefan-Boltzmann constant, and the Boltzmann constant is set to one.

The case of convective luminosity transport (appropriate for non-radiative portions of main sequence stars and all of giants and low mass stars) does not have a known rigorous mathematical formulation. Convective energy transport is usually modeled using mixing length theory. Mixing length theory 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 the "mixing length". [harvtxt|Hansen|Kawaler|Trimble|2004|loc=&sect;5.1] For a monatomic ideal gas, mixing length theory yields:$\left\{mbox\left\{d\right\} T over mbox\left\{d\right\} r\right\} = left\left(1 - \left\{1 over gamma\right\} ight\right) \left\{T over P \right\} \left\{ mbox\left\{d\right\} P over mbox\left\{d\right\} r\right\},$where $gamma = c_p / c_v$ is the adiabatic index, the ratio of specific heats in the gas. (For a fully ionized ideal gas, $gamma = 5/3$.)

Also required is the equation of state, relating the pressure to other local variables appropriate for the material, such as temperature, density, chemical composition, etc. Relevant equations of state may have to include the perfect gas law, radiation pressure, pressure due to degenerate electrons, etc.

Combined with a set of boundary 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: $P\left(R\right) = 0$, meaning the pressure at the surface of the star is zero; $m\left(0\right) = 0$, there is no mass inside the center of the star, as required if the mass density remains finite; $m\left(R\right) = M$, the total mass of the star is the star's mass; and $T\left(R\right) = T_\left\{eff\right\}$, the temperature at the surface is the effective temperature of the star.

Although nowadays stellar evolution models describes the main features of color magnitude diagrams, important improvements have to be made in order to remove uncertainties which are linked to our limited knowledge of transport phenomena. The most difficult challenge remains the numerical treatment of turbulence. Some research teams are developing simplified modelling of turbulence in 3D calculations.

ee also

*Polytrope

References

General references

*citation| title=Stellar Structure and Evolution | first1=R. | last1=Kippenhahn | first2=A. | last2=Weigert | publisher=Springer-Verlag | year=1990
*citation|last=Hansen | last2=Kawaler | last3=Trimble | first=Carl J. | first2=Steven D. | first3=Virginia | publisher=Springer | edition=2nd | year=2004 | title=Stellar Interiors | isbn=0387200894
*citation | last1=Kennedy | first1=Dallas C. | last2=Bludman | first2=Sidney A. | title=Variational Principles for Stellar Structure | year=1997 | journal=Astrophysical Journal | volume=484 | pages=329 | id=arxiv|astro-ph|9610099 | doi=10.1086/304333
*citation | first1=Achim | last1=Weiss | first2=Wolfgang | last2=Hillebrandt | first3=Hans-Christoph | last3=Thomas | first4=H. | last4=Ritter | title=Cox and Giuli's Principles of Stellar Structure | publisher=Cambridge Scientific Publishers | year=2004
*citation | last=Zeilik | first=Michael A. | last2=Gregory | first2=Stephan A. | title=Introductory Astronomy & Astrophysics | edition=4th | year=1998 | publisher=Saunders College Publishing | isbn=0030062284

* [http://www-pat.llnl.gov/Research/OPAL OPAL opacity code]
* The [http://astro.ensc-rennes.fr/index.php?pw=ycesam Yellow CESAM code] , stellar evolution and structure FORTRAN source code
* [http://theory.kitp.ucsb.edu/%7Epaxton/EZ-intro.html EZ to Evolve ZAMS Stars] a FORTRAN 90 software derived from Eggleton's Stellar Evolution Code, a web-based interface can be found [http://shayol.bartol.udel.edu/~rhdt/ezweb here] .
* [http://obswww.unige.ch/~mowlavi/evol/stev_database.html Geneva Grids of Stellar Evolution Models] (some of them including rotational induced mixing)
* The [http://www.oa-teramo.inaf.it/BASTI BaSTI] database of stellar evolution tracks

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• stellar structure — žvaigždžių sandara statusas T sritis fizika atitikmenys: angl. stellar structure vok. Sternaufbau, m; Sternstruktur, f rus. строение звёзд, n pranc. constitution des étoiles, f; structure des étoiles, f …   Fizikos terminų žodynas

• Stellar evolution — Life cycle of a Sun like star Stellar evolution is the process by which a star undergoes a sequence of radical changes during its lifetime. Depending on the mass of the star, this lifetime ranges from only a few million years (for the most… …   Wikipedia

• Stellar birthline — The stellar birthline is a predicted line on the Hertzsprung Russell diagram that relates the colour and luminosity of protostars at the end of the initial phase of accretion onto them. [cite journal | author = Stahler, S.W. | year = 1983 |… …   Wikipedia

• structure des étoiles — žvaigždžių sandara statusas T sritis fizika atitikmenys: angl. stellar structure vok. Sternaufbau, m; Sternstruktur, f rus. строение звёзд, n pranc. constitution des étoiles, f; structure des étoiles, f …   Fizikos terminų žodynas

• Stellar kinematics — is the study of the movement of stars without needing to understand how they acquired their motion. This differs from stellar dynamics, which takes into account gravitational effects. The motion of a star relative to the Sun can provide useful… …   Wikipedia

• Stellar wind bubble — is the astronomical term usually used to describe a cavity light years across filled with hot gas blown into the interstellar medium by the high velocity (several thousand km/s) stellar wind from a single massive star of type O or B. Weaker… …   Wikipedia

• Stellar classification — In astronomy, stellar classification is a classification of stars based on their spectral characteristics. The spectral class of a star is a designated class of a star describing the ionization of its chromosphere, what atomic excitations are… …   Wikipedia

• Stellar association — A stellar association is a very loose star cluster, looser than both open clusters and globular clusters. Stellar associations will normally contain from 10 to 100 or more stars. The stars share a common origin, but have become gravitationally… …   Wikipedia

• Stellar magnetic field — A stellar magnetic field is a magnetic field generated by the motion of conductive plasma inside a main sequence (hydrogen burning) star. This motion is created through convection, which is a form of energy transport involving the physical… …   Wikipedia

• Stellar populations — ▪ Table Stellar populations Population I disk population Population II extreme Population I older Population I intermediate Population II halo Population II members gas A type stars stars of galactic nucleus high velocity stars with z velocities… …   Universalium