55 Cancri – A tough nut to crack.
As soon as the new data sets for 55 Cancri from the Keck and Lick Observatories were made public last week, they were added to the downloadable systemic console and to the systemic backend. The newly released radial velocities can be combined with existing published data from both ELODIE and HET.
Just as we’d hoped, the systemic backend users got right down to brass tacks. As anyone who has gone up against 55 Cnc knows, it is the Gangkhar Puensum of radial velocity data sets. There are four telescopes, hundreds of velocities, a nearly twenty year baseline, and a 2.8 day inner periodicity. Keplerian models, furthermore, can’t provide fully definitive fits to the data. Planet-planet gravitational perturbations need to be taken into account to fully resolve the system.
Eugenio has specified a number of different incarnations of the data set. It’s generally thought that fits to partial data sets will be useful for building up to a final definitive fit. Here’s a snapshot of the current situation on the backend:
The “55cancriup_4datasets” aggregate contains all of the published data for all four telescopes. This is therefore the dataset that is most in need of being fully understood. The best fit so far has been provided by Mike Hall, who submitted on Nov. 9th. After I wrote to congratulate him, he replied,
Thanks Greg, [...] It actually slipped into place very easily. About 13-30 minutes of adding planets and polishing with simple Keplerian, then 25 iterations overnight with Hermite 4th Order.
The problem is that it seemed like I was getting sucked into a very deep chi^2 minimum, so getting alternative fits may be tricky!
Here’s a detail from his fit which illustrates the degree of difference between the Keplerian and the full dynamical model:
and here’s a thumbnail of the inner configuration of the system. It’s basically a self-consistent version of the best 5-Keplerian fit.
Mike’s fit has a reduced chi-square of 7.72. This would require a Gaussian stellar jitter of 6.53 m/s in order to drop the reduced chi-square to unity. Yet 55 Cancri is an old, inherently quiet star, and so I think it’s possible, even likely, that there is still a considerable improvement to be had. It’s just not clear how to make the breakthrough happen.
This situation is thus what we’ve been hoping for all along with the systemic collaboration: A world-famous star, a high-quality highly complex published data set, a tough unsolved computational problem, and the promise of a fascinating dynamical insight if the problem can be solved.
I’ll end with two comments posted by the frontline crew (Eric Diaz, Mike Hall, Petej, and Chris Thiessen) that I found quite striking. These are part of a very interesting discussion that’s going on right now inside the backend.
When something is this difficult to solve using the ordinary approaches, I start to look to improbable and difficult solutions. In the case of 55C, my hunch is that it’s a system where the integration is necessary, but not sufficient to build a correct solution. I think that the parameter space of solutions is so chaotic that the L-M minimization doesn’t explore it well, or that the inclination of the system is significant enough to skew the planet-to-planet interactions in the console, or both. Trojans or horseshoe orbits would fit these conditions. Perhaps other resonant or eccentric orbits would as well.
I think the high chi square results and flat periodograms after fitting the known planets also point to a 1:1 resonant solution or significant inclination. I just don’t think there’s enough K left to fit another significant planet unless it’s highly interactive with the others.
I’m going to keep working on this system in the hopes that we can find a solution (and because it’s really, really fun), but I suspect that a satisfactory answer won’t be found without a systematic search of the parameter space including inclination.
â€œNature is not stranger than we imagine but stranger than we can imagine.â€ Or words to that effect, I canâ€™t remember who said that but in all probability this system shall have more questions answered about it (or not as is often the case!) by direct imaging e.g. such as by the Terrestrial Planet Finder (TPF) mission to show what is really happening (if it is ever launched). The 55 Cancri system is listed as 63 on the top TPF 100 target stars.
In the meantime, we struggle onâ€¦ I donâ€™t think I can add anything else to what Eric and everyone else has said…