Mathematics – Logic
Scientific paper
Jan 1931
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1931natur.127...16m&link_type=abstract
Nature, Volume 127, Issue 3192, pp. 16-27 (1931).
Mathematics
Logic
Scientific paper
PERHAPS the most striking general characteristic of the stars is that they can be divided into two groups of widely differing densities. In the first group, which comprises the majority of the known stars, the densities are of ?terrestrial? order of magnitude? that is to say, their mean densities are of the order of the known densities of gases, liquids, and solids. They range from one-millionth of that of water to ten or, in rare cases, perhaps fifty times that of water. In the second group the densities are of the order of 100,000 times that of water. Of the second group, the 'white dwarfs?? only a few examples are known, but they are all near-by stars, and it is generally agreed that they must be of very frequent occurrence in Nature, though difficult of discovery owing to their faintness. Whether stars exist of intermediate density remains for future observation. The possibility of the existence of matter in this dense state offers no difficulty. As pointed out by Eddington, we simply have to suppose the atoms ionised down to free electrons and bare nuclei. At these high densities the matter will form a degenerate gas, as first pointed out by R. H. Fowler. But this leaves entirely unsolved the question of why, under stellar conditions, matter sometimes takes up the 'normal? density and sometimes the high density. Owing to the probable great frequency of occurrence of dense stars, it might reasonably be asked of any theory of stellar constitution that it should account for dense stars in an unforced way. There are two main theories of stellar structure at the present moment. That of Sir James Jeans accounts for the existence of giants, dwarfs, and white dwarfs, but only at the cost of ad hoc hypotheses quite outside physics. It assumes stars to contain atoms of atomic weight higher than that observed on earth, and it assumes them to be relentlessly disappearing in the form of radiation ? it appeals to discontinuous changes of state consequent on successive ionisations, for which there is little warrant. I think it is true to say that the majority of astronomers do not accept this theory. The theory of Sir Arthur Eddington does not claim to account for the observed division of stars into dense stars and stars of ordinary density? nor does it establish the division of ordinary stars into giants and dwarfs. On the other hand, it claims to establish what is known as the mass-luminosity law from considerations of equilibrium only, that is, without introducing anything connected with the physics of the generation of energy. It claims to show that the observed fact that the brighter stars are the more massive can be deduced from the conditions expressing that the star is in a steady state, mechanically and thermally. It does this by making the hypothesis that the stars (giants and ordinary dwarfs) consist of perfect gas. Closer consideration of the actual formul used by the theory shows that it scarcely bears out the claims made for it by its originator. The 'formula for the luminosity? of a star makes the luminosity very nearly proportional to its effective temperature, and so the so-called proof of the mass-luminosity law involves a semi-empirical element, namely, an appeal to the observed effective temperatures of the stars, for the observed values of which the theory fails to account. Another difficulty encountered by the theory is that it makes the interiors of the more luminous (giant) stars cooler than those of the fainter stars, and it makes the interiors of both too cool for the temperature to have any appreciable influence on the rate of generation of energy, by stimu-lating, for example, the production of radioactive elements or the conversion of matter to radiation. The claim to establish the mass-luminosity law from mere equilibrium considerations cannot, however, be sustained for a moment. We may regard a star in a steady state as a system provided with an internal heating apparatus (the source of energy). It adjusts itself?state of aggregation, density distribution, temperature distribution?until the surface emission equals the internal generation of energy L. But provided the luminosity L is not too large (in order that the mass shall not burst under radiation pressure), it is clear that a given mass M can adjust itself to suit any arbitrary value of L. If, starting with one steady state, we then alter L (upwards or downwards) by altering the rate of supply of energy, the star will simply heat up or cool down until the surface emission is equal to the new volume of L?precisely like an electric fire. L and M are thus independent variables so far as steady-state considerations are concerned. The fact that L and M show a degree of correlation in Nature must be connected with facts of an altogether different order, namely, with the physics of energy-generation. It is essential to recognise the difference between the formal independence of L and M as regards steady-state considerations and the observed correlation of L with M in Nature. The observed mass-luminosity law must depend on the circumstance that in some way the more massive star contrives to provide itself with a stronger set of sources. The claim to establish the mass-luminosity law from equilibrium considerations only appears to me a philosophical blunder. Further, it is unphilosophical to assume the interior of a gas to be a perfect gas ? either knowledge of the interior is for ever unattainable or we should be able to infer it from the observable outer layers. When we dispense with the perfect gas hypothesis and at the same time recognise the independence of L and M as regards steady-state considerations, it is found that a rational analysis of stellar structure automatically accounts for the existence of dense stars without special hypothesis. Further, it shows, as common sense would lead us to expect, that the more luminous stars must have the hotter interiors. Here the temperatures are found to range up to 1010 degrees or higher, de pending on luminosity?a temperature sufficient to stimulate the conversion of matter into radiation. In addition, it shows that the central regions of stars must be very dense, ranging up to 107 grams cm.-3 or higher. Thus the difficulties met by earlier theories fall away as soon as the ground is cleared philosophically. The foregoing ideas suggest the following as the fundamental problems of stellar structure: (1) What are the configurations of equilibrium of a prescribed mass M as its luminosity L ranges from 0 upwards, M remaining constant ? (2) What is the effective temperature T, associated with a given pair (M, L) in a steady state ? (3) What is the value of L which will actually occur for the physical conditions disclosed by the answer to problem (1) ? We observe that the outer parts of a star are gaseous. Consequently we can solve the problem of the state of any actual star by integrating the equations of equilibrium from the boundary inwards ? we are entitled to assume the gas laws to go on holding until we find that the conditions are incompatible with them. We then change to a new equation of state, and carry on as before. We change our equation of state as often as may be necessary until we arrive at the centre. The answer to the first of the problems formulated above has been worked out, for certain types of source-distribution and opacity, by the method of inward integration. The results are sufficiently alike to be taken as affording insight into the nature of stellar structure in general, and are as follows. For a given mass M? of prescribed opacity, there exist two critical luminosities L1 and L0 (L1>L0) such that for L>L1 no configurations of equilibrium exist? for L1> L> L0 the density and temperature increase very rapidly as the centre is approached (T ocrÂ1 Â log------), so that in the centre there is a region of very high temperatures and densities where the gas laws are violated; for L = L0 a diffuse perfect gas configuration is possible? for L0>L>0 the only perfect gas configuration is a hollow shell provided with an internal, rigid supporting surface of spher
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