June 14, 191 7J 



NATURE 



309 



the mass and mean density are given, and also 

 the molecular weight and ratio of specific heats (y) 

 of the material, we can find at once the tempera- 

 ture at any internal point. Let us take a star of 

 mass \\ times that of the sun and of mean density 

 o"oo2 gm. /cm.3; for illustration, the average 

 molecular weight will be taken as 54 {e.g. iron 

 vapour dissociated into atoms at the high tem- 

 perature). For y we shall take 4, but any pos- 

 sible change in y makes comparatively little 

 difference in the results, so far as we require them. 

 For this star the calculated temperature at the 

 centre is 150,000,000°; half-way from the centre 

 to the boundary it is 42,000,000°. But the tem- 

 perature of which we have some observational 

 knowledge is not given immediately by these 

 calculations; according- to observation, the "effec- 

 tive temperature " of a star of this density would 

 probably be about 6500°. This term does not 

 refer to the temperature at any particular point, 

 but measures the total outflow of heat per unit 

 surface. Now, the outflow of heat evidently de- 

 pends on two conditions — the temperature gra- 

 dient (more strictly the gradient of T*), and the 

 transparency of the material ; therefore, the tem- 

 perature-distribution being calculated as already 

 explained, we can deduce the transparency neces- 

 sary to give the observed effective temperature of 

 6500°. The result is startling. We find the 

 material must be so absorbent that a thickness of 

 one-hundredth of a millimetre (at atmospheric 

 density) would be almost perfectly opaque. There 

 is little doubt that such opacity is imf>ossible. 

 Conversely, if we adopt any reasonable absorp- 

 tion coefficient, the effective temperature would 

 have to be above 100,000°, which is decisively 

 contradicted by observation. 



A way out of this discrepancy is found if we 

 take into account the effect of the pressure of 

 radiation. Fortunately, this effect can be calcu- 

 lated rigorously without introducing any addi- 

 tional assumption or hyp>othesis. Suppose that a 

 beam of radiation carrying energy E falls on a 

 sheet of material which absorbs feE and transmits 

 (i - fe)E. It is known from the theory of electro- 

 magnetic waves that radiant energy E carries a 

 forward-momentum E/c, where c is the velocity 

 of light; similarly, the emergent beam carries 

 momentum (i - fe)E/c. The difference feE/c can- 

 not be lost, and must evidently remain in the 

 absorbing material. The material thus gains 

 momentum, or, in other w^ords, experiences a 

 pressure. The amount of the pressure feE/c in- 

 volves the coefficient of absorption k, of which 

 we have no immediate observational knowledge ; 

 but it is the same coefficient which has already 

 entered into the calculations of the opacity of the 

 material, so that the introduction of radiation- 

 pressure into the theory brings in no additional 

 unknowns or arbitrary quantities. 



The radiation-pressure is thus proportional to 

 fe, and to the approximately known outflow of 

 energy. The preposterous value of fe already 

 found would, if adopted, lead to a pressure far 

 exceeding gravity, so that the star would be 



NO. 2485, VOL. 99] 



blown to pieces. But the radiation-pressure 

 modifies the internal distribution of pressure and 

 temperature; it supports some of the weight of 

 the outer layers of the star, and consequently a 

 lower temperature will suffice to maintain the 

 given density. The smaller temperature- 

 gradient causes less tendency to outflow of heat, 

 and there is accordingly no need for so high an 

 opacity to oppose it. By calculation we find that 

 for a star of mass 1*5 times the sun, and mole- 

 cular weight 54, radiation-pressure will counter- 

 balance i9/20ths of gravity ; somew^hat un- 

 expectedly, this fraction depends neither on the 

 density of the star (so long as it is a perfect gas) 

 nor on the effective temperature, but it alters a 

 little with the mass of the star. The pressures 

 and temperatures are then reduced throughout in 

 the ratio 1/20; for the star already considered, 

 the corrected value of the central temperature is 

 7,000,000°. Assuming an effective temp>erature 

 of 6500°, we can now calculate the new value 

 of fe; it amounts to 30 C.G.S. units, i.e. 

 1/30 gm. per sq. cm. section will reduce 

 the radiation passing through it in the ratio i/e. 

 It is of considerable interest to note that this is 

 of the same order of magnitude as the absorption 

 of X-rays by solid material ; for at the high tem- 

 peratures here concerned the radiation would be 

 of very short wave-length and of the nature <rf 

 soft X-rays. 



The approximate balance between radiation- 

 pressure and gravity leads to an important rela- 

 tion between stellar temperatures and densities. 

 It is easy to put this relation in a more rigorous 

 form ; but it will suffice here to express the condi- 

 tion as radiation-pressure = gravity. If T is the 

 effective temperature of the star, and g the value 

 of gravity at the surface, the outflow of radiation 

 (per unit area) varies as T*, and the condition is 



We shall assume that fe is the same for all stars. 

 Now g depends on the mass and mean density in 

 the ratio M pf. Hence 



TocM^y. 



The range of mass in different stars is trifling 

 compared with the great range of density. Thus 

 the leading result is that the elective temperature 

 of a giant star is proportional to the sixth-root of 

 the density. To test this, w^e take the densities 

 given by Russell * for the different types, and, 

 assuming that stars of the solar type (G) have the 

 sun's effective temperature (6000'^), we calculate 

 by the sixth-root law the temperatures of the 

 other types. 



DcDsity Effective 



Type (©=1) temperature 



A ... xV • -• 10,800' 



G ... 3V-0 ••• 6,000" 



K ... oi ... 4,250= 



^I ••• isicKT - 2,950' 



The calculated numbers in the last column agree 

 almost exactly with the temperatures usually 



* Loc. cit , pp. 282-83. 



