In mean density these stars display a range of , while the range in surface gravity is , illustrating the significant fact that the mean density varies far more widely than the surface gravity. The latter quantity is the important one in determining the pressure that may be assumed to exist in the reversing layer. If the masses of the very luminous stars of low mean density, such as W Crucis, exceed the solar mass, as they most probably do, the hypothetical radii are increased, and the range in surface gravity becomes even smaller than before.

The data for stars of known mean density and radius permit the estimation of the range in surface gravity, and hence of the range in pressure, encountered in the reversing layer. In the absence of knowledge of the appropriate optical depth, however, the actual pressure cannot be deduced from such considerations, and recourse must be made to more indirect methods. The present view is based upon a number of considerations, none of which would alone be of great weight. All of the conclusions, taken together, however, indicate that the upper limit of the pressure for the region in which the Fraunhofer lines originate is of the order of .

Attention was first called to the probability of an extremely low pressure in the reversing layer by R. H. Fowler and Milne,[62] in advancing the form of ionization theory which is to be analyzed in later chapters. The conclusion that the pressure in the reversing layer is exceedingly low was a direct outcome of their discussion, and they mentioned that the results from other methods converged in the same direction.

Russell and Stewart,[63] in a specific discussion of the pressures at the surface of the sun, have established beyond question, and on quite other grounds, that the pressure for the solar reversing layer is indeed of the order suggested by Fowler and Milne. The value need then no longer be regarded as a result of the Fowler-Milne theory, and may be used without redundancy in deriving a stellar temperature scale from that theory.

METHODS OF ESTIMATING REVERSING LAYER PRESSURES

Russell and Stewart examined the evidence for reversing layer pressures derived from the following sources: (a) Shifts of spectral lines due to pressure, (b) Sharpness of lines, (c) Widths of lines, (d) Flash spectrum, (e) Equilibrium of outer layers, (g) Ionization and chemical equilibrium in the solar atmosphere. In addition to these we have (f) the observed limit of the Balmer series in the hotter stars, where the hydrogen lines are at or near their maximum. These sources of evidence will now be briefly discussed.

(a) Shifts of Spectral Lines.—It was at one time supposed that displacements of spectral lines, corresponding to pressures of several atmospheres, could be found in stellar spectra. More recent work,[64] however, has shown conclusively that the pressure shifts that occur are so small that it is impossible to estimate a pressure from them with any approach to accuracy. The estimated pressures are of the same order as their probable errors. This being so, the most that can be expected of the method based upon pressure effects is a demonstration of whether or no the pressure exceeds 0.1 atmosphere, and this question has now been satisfactorily answered in the negative.

(b) Sharpness of lines.—The occurrence, as sharp distinct lines in the spectra of the stellar atmosphere, of lines that are diffuse in the laboratory at atmospheric pressure, and only become sharp when the pressure is very much reduced, indicates that the pressure in the reversing layer must be extremely low. The mere existence of distinct hydrogen lines points to a pressure of less than half an atmosphere, as was shown by Evershed,[65] and the lines 4111, 4097, 3912 of chromium,[66] 3421, 3183 of barium,[67] and 4355, 4108, 3972 of calcium,[68] which are sharp and distinct in the solar spectrum, but which only lose their diffuseness in the laboratory under vacuum conditions, indicates pressures probably far lower than 0.1 atmospheres. The lines of doubly ionized nitrogen, which are seen as sharp clear absorption lines in the early stars and the cooler stars,[69] are also somewhat hazy under even the finest laboratory conditions,[70] and probably arise in regions of very low pressure in the stellar atmosphere.

(c) Widths of lines.—The width of an absorption line, produced by “Rayleigh Scattering” close to resonance conditions, is given by Stewart[71] as where is the observed width of the line, the wave-length expressed in the same units, and the number of molecules per square centimeter column in the line of sight.

It is unfortunate that the widths of Fraunhofer lines are hard to measure and difficult to interpret. Results obtained from objective prism spectra will probably differ from those derived with the aid of a slit spectrograph, and moreover, in estimating a line with wings it is hard to judge what should be regarded as the “true” line width. Russell and Stewart[72] estimate for the lines in the solar spectrum. Then, on the assumption that the reversing layer has a thickness of a hundred kilometers, the partial pressure of neutral sodium in the reversing layer, as derived by Russell and Stewart from the formula just quoted, is of the order . At the solar temperature, 5600°, about 99 per cent of the sodium present is in the ionized condition,[73] and thus the total partial pressure of sodium atoms may be of the order . If it be assumed that sodium constitutes about 5 per cent of the total material present, the total pressure thus derived is of the order .

The lines are of course the ultimate lines of neutral sodium. It will be shown[74] in Chapter IX that the partial electron pressure in the region from which ultimate lines originate is probably between at maximum. When 99 per cent of the atoms are ionized, the pressure rises by a factor of about 100, and the corresponding partial electron pressure becomes between . As the total pressure is probably, at the solar temperature, about twice the partial electron pressure, the total pressure should be nearer to .

The total pressure derived in Chapter IX is the pressure corresponding to the median frequency of the sodium atoms that send out light to the exterior—it may be regarded as the average pressure for the visible sodium. The total pressure derived from the line width, on the other hand, is the pressure at the bottom of the layer of visible sodium, and might therefore be expected slightly to exceed the average pressure for the visible sodium atoms. The difference encountered, partial electron pressure, the total pressure should be nearer to for the average pressure, and partial electron pressure, the total pressure should be nearer to for the total absorption pressure, is in the direction that would be anticipated, although it is larger than might have been expected. Neither value is, however, of very high accuracy, and probably the agreement can be regarded as quite satisfactory.

If the same formula be applied to the hydrogen lines, which may have a width[75] of the order of 5Å, high values for the partial pressure of hydrogen are obtained. The behavior of hydrogen in the spectra of the cooler stars,[76] and the abnormally high abundance[77] derived for it in Chapter XIII, suggest that here, again, a definite abnormality of the behavior of hydrogen is involved.

(d) Flash Spectrum.—It was pointed out by Russell and Stewart[78] that the density in the region that gives the flash spectrum must be exceedingly low. If this were not the case, the intensity of the scattered sunlight would be great enough, as compared to the flash itself, to register on the plate as continuous background in the time required to photograph the flash. The pressure thus estimated, from the minimum amount of material required to give scattered sunlight strong enough to be registered, is less than .

(e) Radiative Equilibrium of the Outer Layers.—At the edge of a star, where radiation pressure and gravitation no longer balance, and in consequence the existence of temperature and pressure gradients, such as we observe in the reversing layer, becomes possible, the equations given by Eddington[79] for the equilibrium of the interior no longer hold. The outer layers fall off more steeply than the equations predict, and in consequence it is not possible to use the equations in deriving values for the pressure or density corresponding to a layer near the boundary at a given temperature. It is certain, however, that the density deduced from the equations will be far too high, and so the predicted density at a given temperature may be used to indicate that the pressures at the boundary of a giant star are indeed very low.

The following table is adapted from the one given by Eddington for the relation between distance, , from the center, density , and temperature , for a typical giant star of Class , effective temperature 6500°. The distance from the center is expressed in terms of the solar radius, the density in grams per cubic centimeter, and the temperature in absolute units. The last entry in the first column represents the total radius of the star.

0 0.1085 6,590,000° 4 0.0010 1,380,000°

1 0.0678 5,640,000 5 0.00015 730,000

2 0.0215 3,840,000 6 0.0000093 290,000

3 0.0050 2,370,000 6.9 0.000000 ......

At a depth where the temperature is 290,000°, ten times the temperature in the reversing layer of any known star, the density given is about . An atmosphere a hundred kilometers in thickness (the supposed approximate depth of the reversing layer) and of this density would contain only a hundred grams per square centimeter of surface. In order to bring the density into harmony with the densities derived for the reversing layer it is necessary to suppose that the value[80] of falls to 0.4 per cent of its value at 290,000° as the temperature falls, from 290,000° to 29,000°, to 10 per cent of its value. The fall of density displayed in the table appears to be rapid enough to warrant this supposition; and in any case, as was pointed out earlier, the actual fall is probably greater than the formula predicts. The general theory of stellar equilibrium is, then, consistent with very low pressures in the reversing layer. More than this cannot be said, as the formulae are not directly applicable.

(f) Observed Limit of the Balmer Series.—The earlier members of the Balmer series of hydrogen are produced by the transfer of electrons from 2-quantum orbits to 3-quantum orbits (), 4-quantum orbits (), and so forth. The later members of the series are associated with orbits of higher and higher quantum numbers. The major axis of the orbit varies as the square of the quantum number, and therefore a hydrogen atom which is producing, say, , is effectively much larger than one which is giving rise to . As was early suggested by Bohr,[81] the production of the higher members of the series must depend upon the possibility of existence of the corresponding outer orbits. As a preliminary assumption it appears probable that the existence of the larger orbits will depend on the proximity of neighboring atoms, and hence on the pressure.

The theoretical questions involved are very complex, and the present discussion is merely tentative. When the idea that the maximum number of lines that could be produced was a function of the pressure was first set forth, the available laboratory evidence appeared all to be in its favor. The maximum number of Balmer lines that had been produced in the vacuum tube was five, while it was well known that over twenty could be traced in absorption in some stellar atmospheres. Since that time, however, the work of R. W. Wood[82] has produced forty-seven lines of the Balmer absorption series of sodium in the laboratory at considerable pressures, and evidently the simple theory, relying on the mutual distances of the atoms to determine the number of lines that can be produced, cannot be applied in this case. The matter has been discussed by Franck,[83] who points out that the outermost effective orbit in the sodium atom that gives the forty-seventh line must embrace large numbers of other atoms. He suggests that collisions are chiefly responsible for the production of the absorption lines.

Even though the simple theory is inapplicable to the laboratory conditions, it is not necessarily invalid in the stellar atmosphere, where conditions are far more simple, and where, in particular, the effects of collisions are negligible. There appears, moreover, to be a distinct observational correlation between the pressure and the number of observable hydrogen lines. The importance of the wave-length of the beginning of the continuous absorption, which lies just to the red of the last Balmer line observed, and extends toward the violet, was first indicated by Wright,[84] who recorded that the absorption head was farther to the red in Lyrae than in Cygni. This fact is obviously reflated to the difference in pressure in the atmospheres of the two stars, one of which is a normal star, while the other is a super-giant. The observational and theoretical importance of the question has also been discussed by Saha,[85] and by Nicholson.[86]

The observational data in the hands of the writers just quoted were very meagre, and the present writer and Miss Howe[87] have recently attempted to obtain information on the number of observed Balmer lines in a large number of stars, and to examine the correlation with absolute magnitude. A distinct correlation is found between the number of lines observed and the reduced proper motion, which is chosen as the best available criterion of absolute magnitude for the numerous stars involved (Class brighter than the fifth magnitude). It therefore appears that the pressure, and hence the proximity of the atoms, has some influence upon the possibility of the production of a line. The application of Bohr’s original suggestion is hence of considerable interest, and the resulting pressures may profitably be compared with the pressures otherwise derived for the reversing layer.

The maximum number of lines seen, while quite consistent for plates made with the same dispersion, is somewhat increased when the dispersion is made much greater. The number of lines seen in the spectra of various stars with strong hydrogen lines, made with a dispersion of about 40 mm. between and , varies between thirteen and twenty. The corresponding pressures, derived from Bohr’s estimate that a pressure of about 0.02 mm. would be required for the production of thirty-three Balmer lines, and on the assumption that the pressure varies as the sixth power of the quantum number, lie between and . These pressures are of course to be regarded as upper limits, for it is possible to miss several lines at the violet end of the series, and Wright, with larger dispersion, does indeed record twenty-four Balmer lines in Cygni; on the other hand it is not likely that the estimated number will exceed the actual number of lines.

The pressures in the reversing layer, as derived from the observed limit of the Balmer series, are then of the same order as the pressures derived by the other methods outlined above. This is of especial interest because the method, if applicable, is a direct one, and gives results for individual stars, whereas all the other methods, excepting the one based on pressure shifts, are essentially indirect.

(g) Ionization and Chemical Equilibrium.—The evidence adduced by Russell and Stewart[88] has been greatly amplified by Fowler and Milne,[89] and by the data bearing on their theory which were subsequently published by the writer[90] and by Menzel.[91] It is not intended to present the evidence from ionization theory here in support of the low pressures inferred by the other methods for the reversing layer. The pressure derived in the present chapter, and considered as independently established, will be used in Chapters VII ff. to derive a stellar temperature scale, for the reversing layer, from the line-intensity data presented.

SUMMARY

The following tabulation contains a synopsis of the reversing layer pressures derived by the methods that have been outlined.

The extreme tenuity of the stellar atmosphere appears to be unquestionably established by the data set forth above, and a maximum effective pressure of may therefore be assumed in a discussion of the spectra of reversing layers.

Pressure Shifts less than

Line Sharpness less than

Line Width

Flash Spectrum

Radiative Equilibrium order of

Limit of Balmer Series to (upper limit)

(Ionization to )

FOOTNOTES:

Notes in this chapter

Note 1 · M. N. R. A. S., 83, 403, 1923.Note 2 · Ap. J., 59, 197, 1924.Note 3 · St. John and Babcock, Ap. J., 60, 32, 1924.Note 4 · M. N. R. A. S., 82, 394, 1922.Note 5 · King, Ap. J., 41, 110, 1915.Note 6 · King, Ap. J., 48, 32, 1918.Note 7 · Saunders, quoted by Russell and Stewart, Ap. J., 59, 197, 1924.Note 8 · Payne, H. C. 256, 1924.Note 9 · A. Fowler, M. N. R. A. S., 80, 692, 1920.Note 10 · Stewart, Ap. J., in press.Note 11 · Ap. J., 59, 197, 1924.Note 12 · Chapter VI, p. 99.Note 13 · Chapter IX, p. 137.Note 14 · Shapley, H. B. 805, 1924.Note 15 · Chapter XIII, p. p. 188.Note 16 · Chapter V, p. 56.Note 17 · Ap. J., 59, 197, 1924.Note 18 · Eddington, Zeit. f. Phys., 7, 371, 1921.Note 19 · Russell and Stewart (loc. cit.) show that there are about 0.4 grams of matter above the photosphere per square centimeter of surface.Note 20 · Phil. Mag., 26, 9, 1913.Note 21 · Phil. Mag., 37, 456, 1919.Note 22 · Zeit. f. Phys., 1, 1, 1920.Note 23 · Wright, Nature, 109, 810, 1920.Note 24 · Saha, Nature, 114, 155, 1924.Note 25 · Nicholson, M. N. R. A. S., 85, 253, 1925.Note 26 · Unpublished.Note 27 · Ap. J., 59, 197, 1924.Note 28 · M. N. R. A. S., 83, 403, 1923; 84, 499, 1924.Note 29 · H. C. 256, 1924.Note 30 · H. C. 258, 1924.