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
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
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.