We assembled the historical light curve of the BL Lac object Mkn 421
and searched for its possible periodicity using the Jurkevich
method. Our results indicate that this object is very active and
probably has two periodic activities. One
period is of 
years and the other is of 
years which, if real, superposes on the former. The former
period has a higher confidence. The period of 23 years is about half
the time interval between the well observed outbursts in 1934
and 1982. If the period is real, outbursts probably occurred
between 1953 and 1968, where unfortunately, no published data are available.
We must remember, however, that there is some noise on the curve
(Fig. 2 (click here)) and that the total observation range spans only
four times the period of 23 years. More
observations are required to assess the reality of this period.
The period of one year and multiple found in Mkn 421, in ON 231 (Liu et al. 1995) and in 3C 120 (Jurkevich et al. 1971) are spurious results due to the existence of a long time-scale period and a cycle of one year in the astronomical optical observations.
Regarding
the 23 years period, we tentatively provide below a theoretical explanation.
Sillanpää et al. (1988a) suggest a binary black hole model to explain
the quasi-periodic behaviour found in BL Lac object OJ 287. However, there
are several difficulties with the binary model: observed periodicity is not
exact, the period in OJ 287 corresponding to the minimum of brightness is quite
doubtful, observed burst structures are very broad, the system is short-lived
due to gravitational radiation and dynamically unstable due to the interaction
between secondary black hole and disk. Periodicity has probably been found in
many BL Lac objects (Liu 1996) and therefore is not probably at the
origin of binary black holes. The fact that the duration of a burst is around
half the quasi-period can be interpreted in terms of thermal instabilities
in a slim accretion disk in AGNs. Some simulations have shown that slim disks
can indeed be subject to limit-cycle type oscillations, as in the case of dwarf
novae although with a different oscillation behavior (Taam & Lin 1984;
Lasota
& Pelat 1991; Honma et al. 1991). The basic characteristics
of the thermal limit cycles depend strongly on the viscosity parameter
, central black hole mass 
,
accretion rate 
and generalized stress tensor parameter 
(cf. Wallinder et al. 1992). However, the time duration of the bursts
is almost independent of both 
and 
, and may be written
empirically as
when 
and 
where 
, 
being the Eddington accretion rate and
the accretion efficiency (Honma et al. 1991).
The time interval between
subsequent bursts depends strongly on 
, but weakly on 
.
As both the origin and the properties of the presumed viscosity in
accretion disks are unknown at present, its hydro-magnetic origin is
one of the options. (Horiuchi & Kato (1990) suggested that 
may hold if the escape rate of the magnetic field is low. With
these values of the parameters, the thermal limit cycle time 
(period) should be of the order of 2
, i.e.
For Mkn 421, if we adopt the typical values of 
, 
and 
and search the central black hole
mass M to get a period of 22.4 years, we find an estimated mass of
. This mass is reasonable if the parent
galaxies of BL Lac objects are FR I radio galaxies.
Acknowledgements
We are grateful to the referee, Dr. D. Alloin, for her helpful comments.