Photometric pulsation mode identifications for Delta Scuti stars rely to a large extent on the matching of one or more observed frequency ratios with specific known values.The most commonly used "magic ratios" for radial pulsation are near 3/4 and 4/5 for P1/Po and P2/P1, respectively.
Observations show that the large-amplitude Delta Scuti stars (dwarf cepheids) are radial pulsators. For stars with near-solar abundances, the most commonly observed ratio of P1/Po is 0.773 1) with very little variation. We note, however, that main-sequence stars cannot be found in this group.Among the low-amplitude variables, pulsation with purely radial modes seems to be rare. Nevertheless, a number of low-amplitude radial pulsators have been reported and for some the identification may indeed be correct. Whether most of these reported frequency determinations or radial mode identifications at these magic ratios can be confirmed for the low-amplitude pulsators, is an open question. The first problem therefore centers on the real range of the radial P1/P0 ratio as a function of luminosity and temperature inside the lower instability strip. As far as the mode identifications are concerned, the theoretical computations may provide helpful clues.
The >figure on the cover of the newsletter shows the results of Delta Scuti star models with solar abundances by Andreasen and Petersen (1988, Astr. Ap. 192 L4) Stellingwerf (1979, Ap. J. 227 935) and Fitch (1981, ApJ. 249 218). These models are, of course, not identical and the reader is referred to the papers for more details. We note that Andreasen and Petersen find the best fit with the observations by using enhanced opacities. The agreement of these models with observations is quite good. Also, the variation of the P1/Po ratio with luminosity and temperature inside the lower instability strip is 0.01 or less. lf this argument is correct, for normal stars the reported P1/Po matchings of ratios outside the 0.76 to 0.78 range would be in trouble. For P1/P2 a similar narrow range of period ratios is found.
But there exists another uncertainty and explanation for stars with two observed frequencies, practically any observed frequency ratio can be matched by nonradial pulsation because of the many different possible k, l, m and sin i values. The correctness of such nonradial identifications can be checked only if the star has three or more frequencies and if all the frequencies can be explained with a single set of stellar parameters.
The figure on the cover also shows the period ratios for nonradial modes of different k and l values, computed by Fitch. We note the relative closeness of these m = O values to the radial ratios. Once rotational splitting is introduced (-m <= l <= m), those possibilities also need to be considered.
We conclude that radial mode identifications based on observed period ratios are reliable only if the observed period ratios match the "magic numbers" exactly. Additional mode-identification methods such as phase differences and amplitude ratios (e.g. Balona, L. A. and Stobie, R. F., 1980, MNRAS 192, 625; Watson, R. D. 1988, Ap.Space Sci. 140, 255) can provide additional certainty about the correctness of such radialor nonradial identifications.Finally, the observed frequency ratio of 0.801 for VZ Cnc (Fitch, W. S. 1981, Ap.J. 249, 218; Cox, A. N., McNamara, B. J., and Ryan, W. 1984, Ap.J. 284, 250) can be explained well by the first and second radial overtones. However, Cox et al. have shown that with atmospheric helium depletion, the ratio between the radial first overtone and fundamental modes can increase to the observed value for VZ Cnc. Should this explanation really apply to VZ Cnc and other Delta Scuti stars, all the mode identifications based on frequency ratios must be reconsidered.