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Astron. Astrophys. 361, 952-958 (2000)

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5. Discussion

There are four possible explanations for the mass-transfer behaviour of the secondary in AM Her. (i) The spottedness of the L1 region is not untypical.

If the secondaries in CV's have unusually large spot covering fractions over their entire surfaces and very dark spots, one would expect their flux radii to be quite small, i.e. the optically visible surfaces would be smaller than their geometric areas. The low-mass stellar models of Baraffe & Chabrier (1996) show that, for M dwarf masses in the range [FORMULA], the luminosity goes as [FORMULA]. The Roche geometry demands that the mean density of the stars at a given [FORMULA] be (roughly) constant, i.e. that [FORMULA]. Thus, the temperature of an M dwarf CV secondary at a given [FORMULA] goes as [FORMULA]. The effective temperature of a spotted star with photospheric temperature T and an area filling factor f of its totally dark spots is [FORMULA], so the temperature in this case is proportional to [FORMULA]. Imagine that we have two systems at the same orbital period and with secondaries with the same spectral class (i.e. T) but one is spotted and the other is not: the spottedness of the spotted star must then be

[EQUATION]

i.e. the mass and hence the radius of the spotted star must be smaller than the unspotted one with the same spectral type (e.g. for [FORMULA], [FORMULA], [FORMULA]). This effect is increased if the secondaries are transfering material (Kolb & Baraffe 2000) since this causes them to have larger radii at constant mass and hence smaller radii, masses, and later spectral types at constant [FORMULA]. Deviations of this kind have been seen by Beuermann et al. (1998): CV's above and below the gap tend to be of later spectral type than the equivalent (solar abundance) main-sequence stars of the same mean density. While the systems above the orbital period gap are expected to be out of thermal equilibrium, the stars below the gap should be very near the main-sequence (unless they have evolved to brown dwarfs: Kolb & Baraffe 1999). It is true that the CV secondaries below the gap are individually "indistinguishable from ZAMS stars" (Beuermann et al. 1998), but there is a clear trend: on the average they are over a 1/2 spectral type cooler than expected (about 5-10% in temperature). If this effect is due to massive spottedness, we would predict that the stars are undermassive by the factor

[EQUATION]

or about 60% for [FORMULA].

(ii) The spottedness of the L1 region is untypical.

If the spottedness of the L1 region is unusual, there must be some mechanism which preferentially produces magnetic flux around the L1-point and/or which forces spot groups which appear at higher latitudes to wander down towards the L1-point. Granzer et al. (2000) have calculated the dynamics of flux tubes within the convective envelope of a rapidly rotating [FORMULA] star and find that spots are most likely to occur at relatively high latitudes. Thus, in order to explain the wide range of mass-transfer rates in AM Her, we need a means of bringing the already emergent flux down to the L1-point: we will present a description of such a process elsewhere (Hessman & Gänsicke 2000, in preparation).

(iii) The spots are not very___ dark.

In this case, the secondaries need not be particularly spotted if the magnetic fields are not too strong. However, if the magnetic fields are small, then little depression of the stellar surface occurs and we would not expect to see dramatic changes in the mass-transfer rates. The temperature difference and magnetic field strength can be crudely estimated by shifting the atmospheric structure of an M4 giant ([FORMULA] near the L1-point is considerably reduced from that of a dwarf) down in radius until the density equal to that of the high-state stream ([FORMULA]) has been reduced by a factor of 20. Using the NextGen models of Hauschildt et al. (1999), we estimate magnetic fields of [FORMULA] G and corresponding temperature drops by roughly [FORMULA] K: these are exactly the fields and dark spots expected from a rapidly rotating and magnetically active M star.

(iv) The secondaries are only moderately spotted.

If the global spottedness is not uniformly high, we need long-term large variations of the magnetic field strengths at the L1-point on the late M dwarfs of polars to produce long-term extended low- and high-states. The fully convective secondaries in most AM Her stars (i.e. below the period gap) are - according to standard dynamo theory - not supposed to show global solar-like magnetic cycles (e.g. Schüssler 1975; Küker & Rüdiger 1999) even though they are expected to be highly magnetic (e.g. like normal M dwarf flare stars). However, the M4 secondary in AM Her (Beuermann et al. 1998) may be just slightly above the mass-limit of [FORMULA] where the radiative core of a CV secondary disappears (Kolb & Baraffe 2000): the estimates for the primary mass and binary mass-ratio suggest [FORMULA]. Thus, AM Her's secondary may indeed show solar-type activity cycles which could show up as long-term variations in [FORMULA]. The period of solar-like cycles in late-type stars can be crudely expressed by the function (Ossendrijver 1997)

[EQUATION]

where [FORMULA] is the dynamo number and [FORMULA]. The dynamo number is proportional to the angular velocity difference in the over-shoot region between the radiative core and the convective envelope, the "[FORMULA]"-parameter of the "[FORMULA]" dynamo, and the cube of the core size, [FORMULA]. The cycle period and dynamo number of AM Her's small but rapidly rotating M-dwarf secondary would be considerably reduced by the small radius of the hypothetical radiative core ([FORMULA]) but this effect might be more than compensated by the rapid rotation of the envelope ([FORMULA]) if the radiative core is sufficiently slowly rotating. Indeed, there is a vague hint of a potential magnetic period in AM Her: the periods of quasi-steady accretion (best seen in Fig. 1) around JD 2444000, 2447000, and 2450000 might suggest a period of [FORMULA] days or [FORMULA]8 years, corresponding to a dynamo number [FORMULA], i.e. well above that of the sun. Given that the angular velocity difference across the over-shoot region is unlikely to be greater than [FORMULA], this implies an anomalously low value of [FORMULA].

A more likely explanation - certainly for most short period polars whose secondaries are undoubtably fully convective - is that the long-term mass-transfer variations are produced by the slow drift rather than oscillation of a global field produced by an [FORMULA]-dynamo (Rädler 1986). The numerical simulations by Küker & Rüdiger (1999) suggest that the very low Rossby-numbers of polar secondaries,

[EQUATION]

([FORMULA] is the convective turnover time) will produce a magnetic surface configuration resembling that of a dipole tilted into the orbital plane. The long-term high and low states would then be caused by the very slow drift of a quasi-dipolar spot group in and out of the L1-region. The moment of inertia of the highly magnetic white dwarf is so small, that the rotation of a large-scale field on the secondary can induce a rotation of the white dwarf as well, producing shifts in the phasing of the accretion phenomena. Fortunately, the synchronisation timescales of the few known (highly) asynchronous polars are of the order of hundreds of years (Mouchet et al. 1999) - much longer than the mass-transfer variations seen in AM Her and other polars to date - so the effect is probably negligible. This explanation for the mass-transfer variations predicts that all polars will show quasi-periodic mass-transfer cycles when observed for a long enough time - the observations of AM Her itself suggest decades.

Whatever the mechanism is which changes the amount of magnetic flux present at the L1-point, this result casts a very different light on the origin of short- and long-term mass-transfer variations in all types of CV's. While global radius variations due to, e.g., irradiation of the secondary (King et al. 1995) can change the maximum possible mass-transfer rates, this very long-term effect ([FORMULA]years) will not be easily discerned on the short-term if the observed rates are dominated by changes in the number, sizes, form, and filling factors of the spots on much shorter timescales (unless the mechanism simultaneously changes the spottedness).

Since there is no reason to assume that the mass-transfer variations seen in AM Her are not present in non-magnetic systems with similar periods, the same mechanism should be affecting the light curves of novalikes and dwarf novae as well: we have applied the mass-transfer history of AM Her to the latter in another paper (Schreiber et al. 2000).

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Online publication: October 10, 2000
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