## 4. Starspot-induced mass-transfer variationsThe mass-transfer rate from the secondary through the
L where 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 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
L 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 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
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
The resulting simulated appearance of the spotted surface of
AM Her near the L
© European Southern Observatory (ESO) 2000 Online publication: October 10, 2000 |