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Astron. Astrophys. 348, 364-370 (1999)

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3. The "superhump" mass ratio as a calibrator

To calibrate the Linewidth/K vs. mass ratio in SU UMa stars we will use results of the tidal resonance model. Osaki (1985) derived an analytical expression for the precession rate of the eccentric Keplerian orbit of matter at the disk's outer edge under the influence of the secondary's perturbing gravitational force:


where [FORMULA], [FORMULA], [FORMULA] and a are the precession and orbital frequencies, the disk outer radius and the binary separation, respectively.

If the superhump frequency ([FORMULA]) reflects the displacement between the orbital and precession frequency, then:


Replacing the precession frequency in Eq. 5, defining the observable [FORMULA] and assuming a disk radius equal to the 3:1 tidal resonance radius, i.e. about 0.46a (Whitehurst 1988a, Osaki 1989, Lubow 1991) we roughly obtain:


Due to the success of the tidal resonance model in reproducing the superhumps seen in SU UMa stars (e.g. Osaki 1985, Whitehurst 1988a,b, Hirose & Osaki 1990, Hirose et al. 1991, Lubow 1991, Whitehurst & King 1991, Lubow 1992, Hirose & Osaki 1993, Murray 1998), Eq. 7 seems to be a good tool for estimating the mass ratio in non eclipsing SU UMa stars. A test for Eq. 7 can be made with the data of the 4 eclipsing SU UMa stars for which independent mass ratios are available. The result of this comparison, given in Table 1, indicates that the model reproduces well the observed mass ratio, within the observational uncertainties.


Table 1. Comparison of observed ([FORMULA]) and predicted ([FORMULA], from Eq. 7) mass ratios. The [FORMULA] parameter is from Patterson (1998) and references therein.

However, Eq. 7 was derived for one orbiting particle assuming gravity as the main driving force for precession, whereas the real phenomenon involves the collective motion of many particles probably influenced by pressure forces and viscosity (Murray 1998). Murray's main result is that [FORMULA] is not only a function of the mass ratio (as previous studies suggested) but also a function of the gas pressure and viscosity. For example, [FORMULA] increases by 15% when the gas pressure is incremented by a factor 5 in one of his simulations. In general, the [FORMULA] changes found by Murray are of the same order of magnitude as that observed during superoutbursts of SU UMa stars. However, they can also be explained uniquely as changes in disk radius through Eq. 5. This was done by Patterson et al. (1993) in order to explain the [FORMULA] d/d commonly observed in SU UMa stars. Therefore, in our current stage of knowledge, we cannot discriminate between a shrinking disk or viscosity/pressure changes as causes for the [FORMULA] changes.

As a working hypothesis we will assume that Eq. 7 is a first order approach to the mass ratio of SU UMa stars. We will call the mass ratio so derived the "superhump" mass ratio ([FORMULA]). We estimate an intrinsic uncertainty [FORMULA] [FORMULA] 0.017 [FORMULA], obtained by propagating errors in Eq. 7 and assuming a typical [FORMULA] variation of 15% through superoutburst.

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© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999