Orbital period modulation and quadrupole moment changes in magnetically active close binaries

We discuss the main characteristics of the orbital period modulation in close binaries with late-type components. We focus on the various physical scenarios proposed to explain this phenomenon and, in particular, Hall’s (1989) suggestion that it may be connected with magnetic activity. Starting from the work of Applegate (1992) and Lanza et al. (1998a), we develop an integral approach to evaluate the gravitational quadrupole moment of an active star and its variations, which we consider to be an important driver of the observed orbital period changes. The method applies the tensor virial theorem after Chandrasekhar (1961) and directly relates the variation of the quadrupole moment with the changes of kinetic and magnetic energy of the stellar hydromagnetic dynamo. Particular effort has been applied in minimizing the number of free parameters entering the problem. A sample of 46 close binaries with period changes of alternate signs has been studied by our method. The amplitude of the quadrupole moment change appears to decrease with increasing angular velocity, implying that the time-variable part of the kinetic energy of rotation varies as δT /T ∝ Ω−0.93±0.10, with a correlation coefficient of 0.83. The length of the cycle of the orbital period modulation seems to be correlated with the angular velocity as Pmod ∝ Ω−0.36±0.10, but with a smaller correlation coefficient of 0.62. These results support the suggestion that a distributed non-linear dynamo is at work in the convective envelopes of very active stars and that it strongly affects the differential rotation. We also discuss the energy budget of the process responsible for the quadrupole moment variation and find that, on average, only ∼ 10% of the energy required to maintain the differential rotation may be lost by dissipation in the turbulent convective envelope during a cycle of the orbital period change. The problems of the magnetic field geometry and stability and the relationship between the length of the activity cycle, as determined by the change of the area of the starspots and the orbital period modulation, respectively, are also addressed.