Internal Equilibrium Conditions

Mathematically the conditions for the internal equilibrium of a star can be expressed as four differential equations governing the distribution of mass, gas pressure and energy production and transport in the star. These equations will now be derived.

Hydrostatic Equilibrium. The force of gravity pulls the stellar material towards the centre. It is resisted by the pressure force due to the thermal motions of the gas molecules. The first equilibrium condition is that these forces be in equilibrium.

Consider a cylindrical volume element at the distance r from the centre of the star (Fig. 10.1). Its volume is d V = d A dr, where d A is its base area and dr its height; its mass is dm = p dA dr, where p = p(r) is the gas density at the radius r. If the mass inside radius r is Mr, the gravitational force on the volume element will be d Fg = -

GMr dm

Gravitational Pressure
Fig. 10.1. In hydrostatic equilibrium the sum of the gravitational and pressure force acting on a volume element is zero

Since the pressure decreases outwards, dP will be negative and the force d Fp positive. The equilibrium condition is that the total force acting on the volume element should be zero, i. e.

or where G is the gravitational constant. The minus sign in this expression means that the force is directed towards the centre of the star. If the pressure at the lower surface of the volume element is P and at its upper surface P + d P, the net force of pressure acting on the element is d Fp = P dA - (P + dP)dA

dr"

GMr p

This is the equation of hydrostatic equilibrium.

Mass Distribution. The second equation gives the mass contained within a given radius. Consider a spherical shell of thickness dr at the distance r from the centre (Fig. 10.2). Its mass is

Hannu Karttunen et al. (Eds.), Stellar Structure.

In: Hannu Karttunen et al. (Eds.), Fundamental Astronomy, 5th Edition. pp. 229-242 (2007) DOI: 11685739_10 © Springer-Verlag Berlin Heidelberg 2007

Magnetic Momentum Spherical Shell

Fig. 10.3. The energy flowing out of a spherical shell is the sum of the energy flowing into it and the energy generated within the shell

Mass Loss Rate Sphere Density
Fig. 10.2. The mass of a thin spherical shell is the product of its volume and its density

giving the mass continuity equation d Mr

"dT

Energy Production. The third equilibrium condition expresses the conservation of energy, requiring that any energy produced in the star has to be carried to the surface and radiated away. We again consider a spherical shell of thickness dr and mass dMr at the radius r (Fig. 10.3). Let Lr be the energy flux, i.e. the amount of energy passing through the surface r per unit time. If s is the energy production coefficient, i. e. the amount of energy released in the star per unit time and mass, then dLr = Lr+dr — Lr = sdMr = 4nr2 ps dr . Thus the energy conservation equation is dLr dr

The rate at which energy is produced depends on the distance to the centre. Essentially all of the energy radiated by the star is produced in the hot and dense core. In the outer layers the energy production is negligible and Lr is almost constant.

Fig. 10.3. The energy flowing out of a spherical shell is the sum of the energy flowing into it and the energy generated within the shell

The Temperature Gradient. The fourth equilibrium equation gives the temperature change as a function of the radius, i.e. the temperature gradient dT/dr. The form of the equation depends on how the energy is transported: by conduction, convection or radiation.

In the interiors of normal stars conduction is very inefficient, since the electrons carrying the energy can only travel a short distance without colliding with other particles. Conduction only becomes important in compact stars, white dwarfs and neutron stars, where the mean free path of photons is extremely short, but that of some electrons can be relatively large. In normal stars conductive energy transport can be neglected.

In radiative energy transport, photons emitted in hotter parts of the star are absorbed in cooler regions, which they heat. The star is said to be in radiative equilibrium, when the energy released in the stellar interior is carried outwards entirely by radiation.

The radiative temperature gradient is related to the energy flux Lr according to dr dr

where a = 4a/c = 7.564 x 10-16 J m-3 K-4 is the radiation constant, c the speed of light, and p the density. The mass absorption coefficient k gives the amount of absorption per unit mass. Its value depends on the temperature, density and chemical composition.

In order to derive (10.4), we consider the equation of radiative transfer (5.44). In terms of the variables used in the present chapter, it may be written cos 0—— = -KvP Iv + ]v . dr

In this equation kv is replaced with a suitable mean value k. The equation is then multiplied with cos 0 and integrated over all directions and frequencies. On the left hand side, Iv may be approximated with the Planck function Bv. The frequency integral may then be evaluated by means of (5.16). On the right-hand side, the first term can be expressed in terms of the flux density according to (4.2) and the integral over directions of the second gives zero, since ]v does not depend on 0. One thus obtains

4n d jac

the gas moving with the bubble obeys the adiabatic equation of state

where P is the pressure of the gas and y , the adiabatic exponent

is the ratio of the specific heats in constant pressure and constant volume. This ratio of the specific heats depends on the ionization of the gas, and can be computed when the temperature, density and chemical composition are known.

Taking the derivative of (10.5) we get the expression for the convective temperature gradient dT _ 1 \ T dP dr = V - y) P dr

Finally, using the relation Lr

between the flux density Fr and the energy flux Lr, one obtains (10.4).

The derivative dT/dr is negative, since the temperature increases inwards. Clearly there has to be a temperature gradient, if energy is to be transported by radiation: otherwise the radiation field would be the same in all directions and the net flux Fr would vanish.

If the radiative transfer of energy becomes inefficient, the absolute value of the radiative temperature gradient becomes very large. In that case motions are set up in the gas, which carry the energy outwards more efficiently than the radiation. In these convective motions, hot gas rises upwards into cooler layers, where it loses its energy and sinks again. The rising and sinking gas elements also mix the stellar material, and the composition of the convective parts of a star becomes homogeneous. Radiation and conduction, on the other hand, do not mix the material, since they move only energy, not gas.

In order to derive the temperature gradient for the convective case, consider a rising bubble. Assume that

In the practical computation of stellar structure, one uses either (10.4) or (10.7), depending on which equation gives a less steep temperature gradient. In the outermost layers of stars heat exchange with the surroundings must be taken into account, and (10.7) is no longer a good approximation. An often used method for calculating the convective temperature gradient in that case is the mixing-length theory. The theory of convection is a difficult and still imperfectly understood problem, which is beyond the scope of this presentation.

The convective motions set in when the radiative temperature gradient becomes larger in absolute value than the adiabatic gradient, i. e. if either the radiative gradient becomes steep or if the convective gradient becomes small. From (10.4) it can be seen that a steep radiative gradient is expected, if either the energy flux density or the mass absorption coefficient becomes large. The convective gradient may become small, if the adiabatic exponent approaches 1.

Boundary Conditions. In order to obtain a well-posed problem, some boundary conditions have to be prescribed for the preceding differential equations:

- There are no sources of energy or mass at the centre inside the radius r = 0; thus M0 = 0 and L0 = 0.

- The total mass within the radius R of the star is fixed,

- The temperature and pressure at the stellar surface have some determinate values, TR and PR. These will be very small compared to those in the interior, and thus it is usually sufficient to take TR = 0 and Pr = 0.

In addition to these boundary conditions one needs an expression for the pressure, which is given by the equation of state as well as expressions for the mass absorption coefficient and the energy generation rate, which will be considered later. The solution of the basic differential equations give the mass, temperature, density and energy flux as functions of the radius. The stellar radius and luminosity can then be calculated and compared with the observations.

The properties of a stellar equilibrium model are essentially determined once the mass and the chemical composition have been given. This result is known as the Vogt-Russell theorem.

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