The uncertainties in calculating the value of the pulsation constant Q for Delta Scuti stars are examined through calculations of error propagation for Mbol, logg and Teff determined from uvbyBeta photometry. With the assumption of a normal distribution of errors, the empirically derived uncertainties in the application of the calibrations indicate an uncertainty in the Q value of +-18%. Systematic errors are also discussed. This relatively large uncertainty needs to be taken into consideration if pulsation mode identifications are based on pulsational Q values.
The comparison of observed periods with theoretical Q values is one of the methods to identify specific radial and nonradial pulsation modes. The best-known Q values are the values of 0.033 and 0.026 days for the radial fundamental and first overtone modes, respectively. The correctness of these mode identifications often depends on the accuracy of the calculation of the observed Q values from the observed periods. The calculated Q value contains a fairly large error caused mainly by the uncertainties in determining the luminosity, gravity and effective temperature of the stars. These uncertainties need to be determined.
The Q value is defined by , where P is the period of pulsation and p and po, are the values of the mean density of the star and the Sun, respectively. Since the mean density is not observed directly, we will express the mean density in terms of other stellar parameters.
The mean density can be expressed as , where M denotes the mass of the star and R the stellar radius. Since the static gravity, ,
We can eliminate the radius R by introducing the luminosity , where Teff represents the effective temperature. Hence:
The solar values, including the bolometric correction, now need to be substituted. Since there exist different conventions concerning the bolometric correction, a small systematic error will be introduced if the same convention is not adopted for both the Sun and the pulsating star.
We will now adopt the following solar values (Allen 1976): Mv,o = 4 83, Mbol,o = 4 75, (B.C.o = -0.08), Teff,o = 5770 K, logg o = 4.44. Substituting these value in the equation 3 gives:
So far no assumption of the actual values of the mass have been made. Also, the correct application of the equation requires that the variability of logg, Mbol and Teff through the pulsational cycle be considered.
The most commonly used determination of Mbol, logg, and Tcff relies on measured uvbyBeta indices and their calibrations. The logg and Teff values can be determined from one of many available model-atmosphere grids (e.g. calibrations by Philip and Relya 1979), while for Mv the empirical absolute magnitude calibrations for A and F stars by Crawford (1975, 1979) are available. Potential problems for some stars concerning the hidden mass term are examined in Section 4.
How do the observational uncertainties in measuring the uvbyBeta indices affect the derived Q value? The two relevant observed quantities are ,B and cl. (The effects of interstellar reddening on the cl index can be corrected by computing the (b- y)O index with the stan- dard calibrations of Crawford.) Let us consider a typical position of a Delta Scuti star in the Hertzsprung-Russell Diagram at 7750 K and log g = 3.75. The empirical calibrations of uvbyBeta photometry by Crawford and the model-atmosphere calibrations by Breger (1974) provide the following linear relations at this position in the Hertzsprung-Russell Diagram:
if the small effects of the bolometric corrections are ignored for the purposes of this calculation. (Should the bolometric correction actually be assumed to be zero, the error in the calculated Q value would lie between +-2%.)
The observational uncertainties in measuring Beta and c1 can be assumed to be independent of each other. Consequently,
What are the uncertainties in c1 and Beta Crawford (1975, 1979) determined that Mv could be predicted from uvbyBeta to an accuracy of ~0.3 mag. This uncertainty includes photometric errors and 'cosmic scatter' (rotational-velocity effects etc.). Since Mv depends on the measured values of Beta and c1, we can express the uncertainty of +-0.3 mag in terms of these two parameters. For independent observational uncertainties in Beta and c1 and the linear expression of Mv we can derive:
A reasonable error in the Beta index of Sigma(Beta)=+-0.01 mag leads to Sigma(c1) = +-0.029. Note that our approach is treating problems of the astrophysical calibrations of c1 (such as rotational effects and cosmic scatter) as observational errors. This appears reasonable since the error calculation is not concerned with the actual source of error (apart from a normal distribution of the errors). Substituting these values into the equation for Q/P leads to
This implies an uncertainty in the derived Q value of 18%. For Q = 0.033 days the Q value would be uncertain by +-0.006 days! The linearizations of the photometric calibrations used in the previous calculation apply only to 7750 K and logg = 3.75. However, for other parts in the Lower Instability Strip similar results for the uncertainty of the Q value are found.
It appears difficult to escape this large uncertainty of 18%. It might therefore even be difficult to distinguish between the radial fundamental and the first overtone to a certainty of two standard deviations. This does, of course, not imply that pulsation modes cannot be identified on the basis of Q values, but does suggest a certain amount of caution if only the period lengths are avilable as a means of pulsation mode identification.
The widely used method of using narrowband photometry to calculate Q values contains two hidden assumptions which are both related to the stellar mass. The absolute magnitude calibrations of Crawford (1975, 1979) rely on measured atmospheric parameters such as the size of the Balmer discontinuity (cl index). Such an absolute magnitude calibration is valid only if the star to be calibrated has the same size (mass) as that the calibrating stars. This argument does, of course, apply to all photometric or spectroscopic absolute magnitude calibrations.
The first assumption is the internal consistency of the two different calibrations used (the model-atmosphere calibration of photometry to obtain logg and Teff, and the empirical absolute magnitude calibration). Mbol and (logg, Teff) are not really independent parameters. To some extent logg and Mbol can be considered as equivalent quantities both relying on the measured size of the Balmer discontinuity.
Even for normal Population I stars it might be useful to check that the adopted absolute magnitude and (logg, Teff ) calibrations are equivalent to each other by computing the size of the mass through the easily derived formula
and to check whether or not the derived mass is reasonable for such a Population I star. This numerically correct mass needs to be derived even if the star does not have a normal mass! The calculation checks only the consistency of the two calibrations used. We refer to an older discussion by Breger and Kuhi (1970), which presented an explanation for unusually small photometric masses for A/F stars in the literature.
Some pulsating A/F stars, such as the Population II star SX Phe, have masses which are approximately a factor of two smaller than those of normal Population I stars. For these stars the photometric calibrations are not valid for two reasons. The first reason appears obvious: the standard photometric calibrations for stars with normal metal-abundance are not valid for stars with weak metal lines. The second reason may be even more important: if the mass of the star is not the same as the mass of the stars used for the calibration for Mv at the same position in the Hertzsprung-Russell Diagram, the derived Mv values will be incorrect by a factor given in equation 13. The photometric Mv values are, astrophysically speaking, not really absolute magnitude calibrations (the distance and size of the star are not known), but gravity calibrations.
What would be the size of the systematic error in the calculated Q value for a low-mass star? Let us consider a star with a mass of half the mass of a normal Population I star of the same gravity and temperature. The absolute magnitude predicted from the normal (high-mass) photometric or spectroscopic calibrations would be too bright by 0.753 mag (= -2.5 log 0.5). The derived Q value would therefore also be incorrect. For a radial pulsator with Q = 0.033 days, the value of Q = 0.028 days would be calculated. This systematic error in Q would be 0.005 days!
A final comment concerns possible systematic errors in the Q value correlated with the rotational velocity of the star. Crawford (1979) noticed that photometric distance moduli of the stars in the Pleiades cluster show a trend with the measured rotational velocity, v sin i, presumably caused by the effects of v sin i on c1. The effect on his calibration was a 'cosmic scatter' of +- of 0.2 mag. This uncertainty in the calibrations of c1 was included in our calculation and should not cause further uncertainties in the Q value. However, the effect is systematic with rotational velocity. We speculate that this could be the reason for known disagreements in some sharp-lined Delta Scuti stars between the mode identification and the calculated Q value.