Since it is believed that a star's heat is maintained by libera tion of sub-atomic energy, another unknown condition enters into the problem, viz., the distribution of sub-atomic sources in the interior. This, however, has no very important effect on the main results of the investigation, and, by considering the extreme cases of a source wholly concentrated at the centre and a source evenly distributed through the mass, we can set limits to the uncertainty.
For the sun the central temperature is found to be of the order C. The mean temperature (averaged for equal mass) is 23,000,000°, and is less subject to uncertainty arising from ignorance of the distribution of the source of energy and other data. Moreover, all stars of the Main Series (see STELLAR Evo LUTION) have nearly the same internal temperature. The Main Series, which comprises the great majority of the stars, includes B type stars several hundred times more luminous than the sun, and red dwarfs, giving a hundredth or a thousandth of its light. It is remarkable that stars differing so widely in mass, in output of heat and light, and in surface temperature, should be so uni form in internal temperature. The giant stars have lower internal temperature, e.g., Capella, which is of the same spectral type as the sun but has only of its density, has a central temperature 10,000,000°.
The temperature gradient from the centre to the surface causes a flow of heat outwards, which is hindered by the opacity of the stellar material; hence, knowing the temperature distribution in the interior, and having observed the total outflow of heat from the star (i.e., its luminosity reduced to heat units) we can com pute the opacity. This astronomical opacity may be compared with the theoretical opacity for material of the density and tem perature concerned, as calculated from the modem theory of the atom. At present the results are not fully accordant, the astro nomical opacity being about 12 times the calculated opacity, i.e., the stars are about three magnitudes fainter than we should expect. The discrepancy is, however, the same for all stars, so that, if instead of attempting to predict theoretically the absolute bright ness, we use the law of variation of opacity with density and temperature to predict differences of brightness, the agreement is very good.
where M is now expressed in terms of the sun's mass, and p. is the average molecular weight in terms of the hydrogen atomic weight. Knowing L and M for any star we can calculate k; for example in Capella we find k= 120 C.G.S. units. This means that a screen containing 1 gram. per sq.cm. (equivalent to about 6 cm. of air) would stop about two-thirds of the radiation passing through it. At first sight this is a surprisingly high opacity. The old problem as to how the heat is brought up from the interior of a star to replace that radiated from the surface, has com pletely changed; we see rather that the star has to be constructed of highly opaque material in order to hold back the internal heat and permit it to come to the surface no faster than it does. The high opacity is, however, not so surprising when we realize that at ten million degrees the radiation in the interior consists of X-rays chiefly of wave-length 2 to 6 Angstroms, which are highly absorbed in a few millimetres of air. In fact the stellar opacity is less than the observed opacity in the laboratory—a fact ex plained by the high ionization of the stellar atoms, which throws a great part of their absorbing mechanism out of order.
According to modern theories of X-ray absorption, the co efficient of opacity should (approximately) be proportional to the density and inversely proportional to the power of the temperature (koo 01). By the use of this law we can eliminate k from the equation given above and obtain the relation between L and M. The density of the star nearly disappears from this re lation, so that the total radiation or absolute bolometric magni tude of a star is a function of the mass and molecular weight only, apart from a trivial correction dependent on the density, which can be calculated and applied when the spectral type is known. This predicted relation between L and M can be plotted as a graph ; it is called the mass-luminosity relation. It appears to be well confirmed by all the observational data available; but it must be remembered that accurately determined masses are rare, so that the test is not so complete as we should desire. The mass-luminosity relation cannot conveniently be expressed by an algebraic formula, but as a rough guide it may be stated that the heat outflow from a star varies as the third or fourth power of the mass in the most important part of the range.