H.-E. Fröhlich

A Bayesian search for stellar activity cycles

Der RS CVn Veränderliche HK Lac weist zwei Aktivitätszyklen auf. Ein bereits bekannter Zyklus kann jetzt auf 13,37 +/- 0,08 Jahre (90%-Intervall) eingeschränkt werden, ein weiterer auf 9,48 +/- 0,13 Jahre. Der 6,7-Jahre-Zyklus entpuppt sich als bloße Oberschwingung.

The RS CVn binary HK Lac exhibits two activity cycles, which establishes firmly the multi-periodicity of dynamo action in these overactive stars. We improve the previously published cycle period to 13.37 +/- 0.08 years and present strong evidence of an additional cycle with 9.48 +/- 0.13 years. The already known 6.7-years cycle turns out to be a mere overtone of the dominating 13.4-years cycle.

Minor decadal brightness variations in very active stars may reveal cyclical behaviour.

If there is more than one fundamental mode indicated in the data, the question arises whether such a multi-periodicity, which would be of considerable interest, is - in view of the noise - really required by the data or not. A Bayesian time series analysis allows one to compare quantitatively a set of hypotheses: (1) no periodicity at all, (2) one fundamental mode with overtones, (3) two modes... Moreover, inspecting the marginal distribution of a period parameter, a mean value as well as a confidence region can be assigned to it.

Starting with the simplest model, the zero hypothesis, with only two free parameters - an offset and a linear trend -, we have successively refined the analysis by introducing at first one and then two sinusoidal frequencies with unknown frequencies, amplitudes and phases. Overtones have been allowed too in order to match any non-sinusoidality. The most ambitious model considered here is described by a total of twelve free parameters. When we have to leave off this sequence of increasingly complex fitting functions?

In a Bayesian's view each model or hypothesis can be assigned a value, its strength which measures its reliability. In mathematical terms it is the likelihood integrated over parameter space, with the weight function being the prior density distribution of all parameters proper. Finding the average likelihood is computationally much more demanding than just searching for the most probable set of parameters, their modal values. Because the weight distribution is normalized, there is an inherent statistical penalty for inspecting too large a number of free parameters. In the case of an over-ambitious model the gain in goodness of fit is more than compensated for by an over-inflated parameter space! Of course, choices as the dimension and the extent of the parameter space must be made beforehand.

The Bayesian approach has been applied to a long-term photometry of the active RS CVn binary HK Lac. Over a time span of 48 years 4766 brightness estimates have been collected, photographic (from Sonneberg Sky-patrol plates) as well as photoelectric ones (from Automatic Photoelectric Telescopes). The most ambitious model, that with two fundamental modes, the first one comprising two overtones, survived the evaluation!

Although the amplitudes are integrated away analytically the numerical integration over the remaining six-dimensions required the combined computing power of AIP's 128-nodes PC cluster "Sanssouci".


Figure: B light curve of HK Lac. The green line connects seasonal averages, the red one shows median values. Bars indicate the standard error of the mean. Over-plotted are the photoelectric measurements.