2. The theoretical linewidth/K vs mass ratio relation
The linewidth/K vs mass ratio relation is based on the assumption that the emission lines are broadened by Doppler broadening in a Keplerian disk. Since the linewidth and the radial velocity semiamplitude K of the accreting primary have the same dependence on the inclination angle, the ratio R between these quantities is independent of i. If the accretion disk radius is a constant fraction of the Roche lobe radius of the primary then R should scale with the mass ratio.
The deduction by Warner (1973) assumes binary synchronus rotation, circular orbits and that particles ejected from the inner Lagrangian point at thermal velocities, conserve their angular momentum about the primary and eventually take up a circular orbit about the primary at a radius . Using the Kepler's third law for circular orbits along with the relation between K and the orbital period, he found:
where vsini is the projected rotational velocity of the disk at the radius , deduced from the width of the emission lines at some intensity level and is the distance from the center of the primary to the inner Lagrangian point in units of the semi-major axis of the orbit. Expressing this distance in units of the binary separation as , Shafter (1983) obtained:
Using the Roche approximation (Kopal 1959), an expression for can be obtained:
which is accurate to 1% for 0.1
10, as deduced from the tables of
Plavec and Kratchovil (1964). From the above arguments, we should
expect a relationship:
where is a calibration parameter depending on the line intensity level where vsini is measured. The above relation has been used but almost always restricted to dwarf novae above the period gap (Warner 1973, Piotrowski 1975, Robinson 1976, Shafter 1983 and Jurcevic et al. 1994). For example, Jurcevic et al. (1994, hereafter J94) used = FWHM (the full width at half maximum of the emission line) and found = 2.00 0.02 calibrating a sample of 12 dwarf novae, including only 2 SU UMa stars, viz. HT Cas and WZ Sge. Looking in their Fig. 5, it is evident that their relationship is relatively well established for above-the-gap dwarf novae, but a large data gap is seen in the domain of SU UMa stars (the low q, upper right corner of their Fig. 5). In fact, J94 extrapolated the relationship to the domain of SU UMa stars basically using the data of WZ Sge. They claim that removing this star from their sample changes only by 5%. We think that this extrapolation deserves an empirical confirmation, moreover considering that the gas dynamics in the disks of dwarf novae above and below the period gap could be substantially different.
© European Southern Observatory (ESO) 1999
Online publication: July 26, 1999