Non-adiabatic stellar pulsation.

Abstract

The V-equation of Woltj er (I 936), which determines the non-adiabaticity, and hence the flux variations, associated with stellar pulsation, has been solved in a first-order theory under quite general conditions along the lines developed by M. Schwarzschild (1938). The manner of mechanical pulsation of the star here is that given by the solution of the adiabatic wave equation. In particular, the flux variation at the *Of papers presented at the Ninety-sixth Meeting of the American Astronomical Society, New York, N. Y., December 2629, 1956. point x may be written as H/H = ( H/H)ad + V, where V(x) = H is the non-adiabatic correction which must be applied to ( HH)ad, the flux variation computed on the basis of adiabatic oscillations. Here (p (x,x') is essentially a measure of the amount of internal thermal energy contained between shells of radii x' and x. The stability coefficient (Rosseland 1949) has also been expressed in terms of the above solution of the V-equation. The above theory is applicable to either radiative or convective transfer, or both. In order to determine the general conditions required for a pulsating star to exhibit the observed phase lag of roughly a quarter period in the emitted flux and the necessary instability for sustained pulsations, the above theory has been applied to a number of schematic variations of ( H/H)ad with . One of these corresponds to the case studied by Eddington (1927), and gives results in general agreement with his. Radiative red-giant models such as those of Schwarzschild and Li Hen (1949) do not yield the correct phase0 relationships, showing a phase lead of some 70 to 900 in the emitted flux. It appears that the only way in which H/H can have a large phase lag at the surface is for (6H/H)ad to experience an abrupt and rather large drop at some point near the surface. The value of the phase lag in the surface value of 6H/Hso obtained depends on the point at which the drop occurs, the heat-storage capacity of the material exterior to this point, and the details of the drop in ( HH)fld. Such an abrupt drop in ( HIJ)a will also cause the regions exterior to this point to exert a destabilizing influence on the pulsations. It appears, in fact, that for conditions likely to be expected in giant-star models, maximicrons instability will be attained for a phase lag in H/H at the surface of near a quarter period. An abrupt drop in the radiative part, at least, of ( fIH)a can result from (i) a sudden drop in , or (2) a sudden reversal in the sign of the temperature exponent in the opacity law. As noted by Whitney in a private communication and by Schatzman (1956), both of these effects will occur when fully ionized hydrogen is in the process of deionization, provided hydrogen is the principal constituent of the outer layers of the star. In addition, it appears to be possible to derive a theoretical period-luminosity relation which agrees reasonably well with observation, if fleff at the point at which (6H7H)ad drops is greater than about 3 or 4. The assumptions underlying this derivation are (I) that the point at which ( H/H)ad drops is determined by the degree of ionization of hydrogen at this point; (2) that the degree of ionization of hydrogen at this point has the same value in a sequence of homologous stars; and (3) that the phase lag in the surface value of 6H$H is a homology invariant. Recent studies by Schatzman (1956) of the convective-zone models of Mme. DumezilCurien (1954) show that ( H H)a , when both convective and radiative transfer are considered, does indeed drop rather abruptly near the surface of the star, attaining, in fact, a large negative value there. However, it remains to be seen whether this drop is of such character as to yield agreement with observation. Dumezil-Curien, Mme. P. 1954, Ann. Astroph. 17, 197. Eddington, A. S. 1927, M. N. 87, 539. Rosseland, S. 1949, The Pulsation Theory of Variable Stars (Oxford: Clarendon Press). Schatzman, E. 1956, Ann. Astroph. 19, 51. Schwarzschild, M. 1938, Z. Astroph. 15, 14. Schwarzschild, M. and Li Hen 1949, Al. N. 109, 631. Woltjer, J., Jr. 1936, B. A. N. 8, 17. Astronomy Department, Cornell University, Ithaca, N. V.