4. Starspot-induced mass-transfer variations
The mass-transfer rate from the secondary through the L1-point, , depends upon the geometric radius, , or cross-sectional area of the "nozzle" at the L1-point
where a is the binary separation, q is the mass-ratio , and (Meyer & Meyer-Hofmeister 1983), the mean mass-density at the sonic point of the flow, and , the sound speed of the gas, approximately given by the isothermal sound speed at the effective temperature of the secondary. Following the discussion in Lubow & Shu (1975), the size of the nozzle can be fairly accurately expressed in terms of the characteristic dimensionless size of the flow
where is the orbital frequency. Since , a typical value of is about 2000 km.
The mean mass-density is roughly given by the exponential fall-off of an isothermal atmosphere, , where is the radial amount that the secondary overflows its Roche volume, and is the isothermal density scale height given by , where is the effective gravity at the secondary's surface. The typical densities can be derived in systems with known orbital parameters and accretion rates. For example, AM Her has hr, , , K, and , resulting in cm, , km, and g cm -3.
We can simulate the spotted surface of AM Her's secondary star in an attempt to reproduce the statistical properties of the observed form of by assuming that the L1-point randomly samples the surface of a spotted region with uniform spot properties (i.e. spot depths and size distribution). The time-dependent mass-transfer curve can then be crudely thought of as the L1-point's random-walk through the simulated spotted landscape. Lacking a detailed model for the spatial and density structure of spots in late-type dwarfs, we will assume that all spots have circular cross-sections with different radii , that the radial increase in photospheric density away from the center of the spot is simply a Gauss function in r, and that no matter is lost from the very centers of the spots. In order to avoid a complex packing problem, we will also assume that individual spots can be arbitrarily superimposed spatially to produce spot groups whose depths are products of the individual spots. An example of such a spot simulation is shown in Fig. 6.
We further assume that the distribution of spot sizes is a power-law in spot radius. The properties of the spotted area are then solely determined by: (1) the spot filling factor
where is the mass-transfer rate in the absence of starspots and f goes from 0 (no spots whatsoever) to 1 (star totally covered with "dark" spots and no matter transferred); (2) the power-law index of the spot size distribution function; and (3,4) the extreme spot radii and .
The optimal parameter set was obtained by simulating many such spotted surfaces and then optimising the fit between the simulated and observed using a modified version of the Press et al. (1992) simplex algorithm. The nominal errors were computed using the procedure outlined in Zhang et al. (1987), in which the error is derived by holding each parameter constant at a slightly different value and optimising the other parameters: , , , . The resulting fit to the observed data is shown in Fig. 5 along with other simulations with 5- changes in the parameters (for which the other parameters were not optimised). Not unexpectedly, the fitted geometry parameters - especially f, and - are highly correlated: steeper spot size distributions (larger values of ) require slightly higher values of f and/or larger maximum radii .
The fitted power-law solution does a fairly good job of reproducing the overall shape of the observed (or rather ) curve: the reduced -value of the fit is 0.45. However, the simple model is not able to reproduce the small-scale structure responsible for the local peaks in (kinks in ), particularly those around caused by the middle-state "stand-still" phases (around 6000-7000 days in Fig. 3).
Fortunately, the filling factor - undoubtably the most important parameter describing the spottedness - is very well constrained: even totally random or fractal "spots" must have a filling factor not too far away from the observed value of (C=0.5)=0.75 and our assumption of very dark (i.e. deep) spots minimises the spot filling factor needed to explain the mass-transfer history. We are unlikely to have over-estimated the mean maximum mass-transfer rate (which goes into the definition of ) since there have been times during which the accretion luminosity was higher. If we have under-estimated the rate (we have no means of knowing whether there aren't always some spots in the L1-region), then again the spot filling factor is larger, not smaller.
The resulting simulated appearance of the spotted surface of AM Her near the L1-point (Fig. 6) shows how extremely spotted the secondary star in AM Her must be - at least at one special location - if the mean density of spots is roughly constant.
© European Southern Observatory (ESO) 2000
Online publication: October 10, 2000