Examples

Example 10.1 The Gravitational Acceleration at the Solar Surface

The expression for the gravitational acceleration is GMo g =

V 4nR3

1410 kg m-

This can be integrated from half the solar radius, r = R0/2, to the surface, where the pressure vanishes:

Using the solar mass M = 1.989 x 1030 kg and radius R = 6.96 x 108 m, one obtains g = 274ms-2 « 28gc , where g0 = 9.81 ms-2 is the gravitational acceleration at the surface of the Earth.

Example 10.2 The Average Density of the Sun The volume of a sphere with radius R is

thus the average density of the Sun is M 3 M

which gives

« 1 n 6.67 x 10-11 x 14102 x (6.96 x 108)2N/m2 « 1014 Pa.

This estimate is extremely rough, since the density increases strongly inwards.

Example 10.4 The Mean Molecular Weight of the Sun

In the outer layers of the Sun the initial chemical composition has not been changed by nuclear reactions. In this case one can use the values X = 0.71, Y = 0.27 and Z = 0.02. The mean molecular weight (10.10) is then

When the hydrogen is completely exhausted, X = 0 and Y = 0.98, and hence

Example 10.3 Pressure at Half the Solar Radius

The pressure can be estimated from the condition for the hydrostatic equilibrium (10.1). Suppose the density is constant and equal to the average density p. Then the mass within the radius r is

Mr = 3 npr3 , and the hydrostatic equation can be written GMrp 4nG p>2r d P

10.5 The Temperature of the Sun at

Using the density from Example 10.2 and the pressure from Example 10.3, the temperature can be estimated from the perfect gas law (10.8). Assuming the surface value for the mean molecular weight (Example 10.4), one obtains the temperature

Example 10.6 The Radiation Pressure in the Sun at Since the directions 0i are randomly distributed and r = R©/2

In the previous example we found that the temperature is T « 5 x 106 K. Thus the radiation pressure given by (10.11) is

and the distance from the starting point is r2 = x2 + y2

0A sin 0i\

The first term in square brackets can be written / N \

1 i = j independent,

This is about a thousand times smaller than the gas pressure estimated in Example 10.3. Thus it confirms that the use of the perfect gas law in Example 10.5 was correct.

Example 10.7 The Path of a Photon from the Centre of a Star to Its Surface

Radiative energy transport can be described as a random walk, where a photon is repeatedly absorbed and reemitted in a random direction. Let the step length of the walk (the mean free path) be d. Consider, for simplicity, the random walk in a plane. After one step the photon is absorbed at xi = d cos 0i , yi = d sin 0i , where 01 is an angle giving the direction of the step. After N steps the coordinates are

The same result applies for the second term in square brackets. Thus r2 = d2J2 (cos2 0i + sin2 0i ) = Nd2 After N steps the photon is at the distance r = from the starting point. Similarly, a drunkard taking a hundred one-metre steps at random will have wandered ten metres from his/her starting point. The same result applies in three dimensions.

The time taken by a photon to reach the surface from the centre depends on the mean free path d = 1/a = 1/Kp. The value of k at half the solar radius can be estimated from the values of density and temperature obtained in Example 10.2 and 10.5. The mass absorption coefficient in these conditions is found to be k = 10 m2/kg. (We shall not enter on how it is calculated.) The photon mean free path is then d = — « 10-4 m.

This should be a reasonable estimate in most of the solar interior. Since the solar radius r = 109 m, the number of steps needed to reach the surface will be N = (r/d)2 = 1026. The total path travelled by the photon is s = Nd = 1022 m, and the time taken is t = s/c = 106 years; a more careful calculation gives t = 107 years. Thus it takes 10 million years for the energy generated at the centre to radiate into space. Of

course the radiation that leaves the surface does not consist of the same gamma photons that were produced near the centre. The intervening scattering, emission and absorption processes have transformed the radiation into visible light (as can easily be seen). Telescopes Mastery

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