Image Source.

An relevant paper showed up on the astro-ph preprint server this morning, “A Bayesian Kepler Periodogram Detects a Second Planet in HD 208487” by P. Ç. Gregory of the University of British Columbia.

Gregory employs a technique known as the parallel tempering Markov Chain Monte Carlo algorithm to argue that the HD 208487 data set contains two planets. The first planet (which was previously announced by Tinney et al. 2005, and confirmed by Butler et al. 2006) has a period of 130 days and a minimum mass 37% that of Jupiter. The second planet in Gregory’s model lies out at a period of 908 days, and has 46% of Jupiter’s mass.

Interestingly, the console does not recover Gregory’s parameters precisely, but it does find a fit that’s extremely similar. (I just uploaded the fit to the systemic back-end.) The radial velocity reflex curve looks like this:

wheras the planetary configuration (at the moment when the first radial velocity data point was obtained on Aug. 8th, 1998) looks like this:

It’s interesting to look at the fits for the latest HD 208487 dataset that have been submitted by participants in the systemic collaboration. At the moment, there are six different fits:

On September 4th, mikevald submitted a 2-planet fit that is a close analog of the one published by Gregory. In the last several days, dstew and andy have also turned in fits that have essentially the same configuration as obtained by Gregory. That’s definitely cool.

In addition to the five fits that look like the Gregory configuration, with the outer planet at a period of P~1000 days, there’s also a completely different take on the system that was submitted this morning by Olweg. In the Olweg fit, the second planet lies interior to the known planet, and has a period of only 29 days. The chi-square is less than one, indicating a slight degree of overfitting. When overfitting occurs, it can easily be remedied by a slight random perturbation of the parameters. It’s very interesting that this fit was completely missed by the Bayesian Kepler Periodogram, so I thought I would have a closer look at Olweg’s model system.

The Olweg radial velocity curve is radically different from the Gregory fit:

The 28.68 day inner planet has a mass of 0.16 Jupiter masses (about 50 Earth masses) and travels on an orbit of modest (e=0.18) eccentricity. There’s a fair amount of planet-planet interaction in this system over the time scale of the radial velocity observations. By the time the fit reaches the end of the data set, there’s a noticeable difference between the keplerian model fit and a self consistent (integrated) model fit:

The system is stable, however, when I did a short test integration of 100,000 years. The secular interaction between the two planets causes the two orbits to execute a complicated dance over a timescale of several thousand years, with the periastron angle of the inner planet orbit mostly librating around an anti-aligned configuration.

As I’ve remarked in an earlier post, we’re currently in the progress of upgrading the downloadable console so that it will be capable of computing estimates of the uncertainties in the orbital elements of a fit. A good way to generate uncertainties in this context is to use the so-called bootstrap method. In the bootstrap, one re-draws the original radial velocity data set with replacement, thus producing alternate realizations of the data in which a fraction of the points appear more than once, and in which a fraction do not appear at all. One then fits to these new datasets, thereby building up distributions for each orbital element. (For more detail, see this paper, which describes in detail how this procedure was applied to the radial velocity data set for HD 209458.) When I run a self-consistent bootstrap analysis based on the Olweg fit, I get the following mean values and standard deviations for the parameters:

This fit is thus quite well constrained, and is a completely viable competing model for describing the hd208487 planetary system. I think the situation here really underscores the value of the systemic collaboration. Many radial velocity data-sets can be fitted by completely different models that offer equally robust fits to the data, while simultaneously maintaining small uncertainties on their bootstrap-estimated parameters.

So how do we know which HD 208487 system (if either) is correct? I’m hoping that the Monte Carlo simulation that will make up phase II of the systemic project will give a great deal of insight into when a particular orbital model can be deemed secure.