The Equation of State
- PDF / 1,115,968 Bytes
- 39 Pages / 439.37 x 666.142 pts Page_size
- 65 Downloads / 210 Views
The Equation of State∗
From the density of about 10−23 g cm−3 in the interstellar medium to a density of about 1015 g cm−3 in neutron stars the difference amounts to a factor 1038 . Over such different conditions, the physical state of the matter differs a lot. The physical state of a medium is described by an equation of state, which is a relation P = P(, T ) between the pressure P, density and temperature T for a given chemical composition. It is given either by an analytical expression, like the law of perfect gas, or by a table of numerical values. This equation plays an essential role in stellar evolution. An overview of the different physical states of matter is given in Fig. 7.8. The study of the thermodynamic coefficients is generally not the beloved subject of students. However, a star is a thermodynamic machine, thus the thermodynamic parameters, such as the specific heats, the adiabatic exponents Γi , etc., determine most of the stellar properties. For example, they tell us how a star is heating when it contracts, whether the nuclear reactions are stable or not, when a stellar core collapses to make a supernova, etc.
7.1 Excitation and Ionization of Gases The excitation and ionization of the stellar medium influences the matter properties, such as the mean molecular weight μ , the specific heats CP and CV as well as other thermodynamic properties. Partial ionization also increases the number of possible atomic transitions and thus the matter opacity.
7.1.1 Excitation Let us consider a system of N atoms with energy states En , En . We want to know the numbers Nn , Nn of atoms with energy states n, n . . . as a function of T . These numbers determine the gas properties and the relative intensities of spectral lines. We ∗ This
chapter may form the matter of a basic introductory course.
A. Maeder, Physics, Formation and Evolution of Rotating Stars, Astronomy and Astrophysics Library, DOI 10.1007/978-3-540-76949-1 7, c Springer-Verlag Berlin Heidelberg 2009
137
138
7
The Equation of State
consider a situation of equilibrium for non-degenerate particles, i.e., for relatively low densities. The number of atoms with an electron in the level n is according to Boltzmann statistics (Appendix C.5) Nn = gn eψ e−En /(kT ) ,
(7.1)
where gn is the statistical weight of level n. The ratio of the numbers of electrons in levels n and n is Nn g = n e−(En −En )/(kT ) . Nn gn
(7.2)
The total number is N = ∑ Nn with n
gn Nn = e− En, 1 /(kT ) N1 g1 and N = N1 ∑ n
Nn N1 = N1 g1
En, 1 = En − E1 ,
(7.3)
∑ gn e−En, 1 /(kT ) .
(7.4)
n
u(T )
u(T ) is the partition function, it can be regarded as the statistical weight of the whole system of atoms. The fraction of atoms in state n is Nn Nn N1 gn − (En −E1 )/(kT ) = = e . N N1 N u(T )
(7.5)
Function u(T ) tends toward infinity because n → ∞ and the differences En, 1 are finite. However, the interactions between ions lower the continuum level and limit the summation in (7.4). Thus, one can often take only the first terms in u(T ). Quantiti
Data Loading...
