Today's guest post is by Ellen Price, a Senior astrophysics major at Caltech.
Professor Johnson pitched me this project idea just after I took his Introduction to Astronomy class (Ay20) in 2012. At that time, I was a sophomore with very little research experience, I knew absolutely nothing about exoplanets. In fact, I had pretty recently considered dropping my astrophysics major entirely. I felt like I had a lot of catching up to do, but classes were a lot more enjoyable when I felt like they mattered in the context of my research. Prof. Johnson’s Ay117 (Statistics and Data Analysis for Astronomers) class, for example, was immeasurably important for me – I learned scientific programming in Python and Bayesian statistics for the first time.
I attended the Exolab group meetings and started to pick up exoplanet jargon and, eventually, I started to absorb the science, too. Prof. Johnson warned me up front that this wasn’t going to be a “packaged” project for an undergrad, and it wasn’t! By the summer following my sophomore year I was still working on it, which is when Dr. Leslie Rogers (Caltech) and Dr.Rebekah Dawson (Berkeley) got involved, bringing some new ideas and a lot of expertise. Then, the project took an interesting turn and we put it on hold temporarily so I could develop a different project, since we needed that result to move forward. Now, as a senior at Caltech, the paper is finally finished, accepted to ApJ, and posted on the arXiv. If doing research as an undergrad has taught me anything, it’s that this is really what I want to do with my life and my career.
How Low Can You Go? The Photoeccentric Effect for Planets of Various Sizes
My project involves measuring the properties of exoplanet orbits. When browsing through published exoplanet characterization papers, it can be easy to take for granted the process by which a planet’s parameters were obtained. I have certainly been guilty of scrolling straight to the nice, typeset table near the end of the paper that lists all the orbital parameters and taking those numbers at face value. But some parameters are a lot harder to measure than others.
Take orbital eccentricity and the related parameter, argument of periastron (an angle that describes the way the planet’s elliptical orbit is “turned” with respect to the viewer) as examples of parameters that seem notoriously difficult to measure. Fixing eccentricity to be zero – that is, assuming a circular orbit – reduces the parameter space we have to explore by two dimensions, which means faster convergence for the fitting procedure. Yet eccentricity is such an interesting parameter! It has implications for dynamical mechanisms (like the Kozai mechanism) and for planet habitability. It’s worth finding a better way to measure it that will work even when the data is noisy or the planet is small.
Kipping et al. (2011) describe a way in which orbital eccentricity can be constrained in multi-planet systems. Dawson & Johnson (2012) go further to show that the eccentricity of a single planet can be constrained from a light curve if we have an estimate of the stellar density from some other source, even when that estimate is rough; they coin the term “photoeccentric effect” to describe the way a planet’s eccentricity is encoded in its light curve. The key to all this comes down to the way a planet’s speed depends on both e and ω. Kepler’s second law states that a planet sweeps out equal areas in equal times (a consequence of conservation of angular momentum), so it moves faster at periapse (when it is closest to the star) than at apoapse (when it is farthest from the star). If we assume that the orbit is circular when we fit it, we would be forced to conclude a different stellar density to account for this difference in speed. Combined with an estimate of the “true” stellar density (from, say, asteroseismology), we can constrain eccentricity after fitting the transit, the computationally favorable approach.
All that already existed in the literature when I started this project. We wanted to know the limits of the photoeccentric effect with respect to the signal-to-noise ratio of the transit. Unlike “most” results in astronomy, this did not reduce to a simple power law relation. The stellar density estimate we derive from fitting the transit depends on many individual parameters, and the uncertainty on that measurement – which ultimately determines our uncertainty in the eccentricity measurement – also depends on those parameters. We were able to form some general conclusions about the photoeccentric effect in the low signal-to-noise regime, however.
- The value of the eccentricity determines how well you can measure it. We are more likely to observe transiting planets if their orbits are highly eccentric (this is not the same as saying that planets are likely to have highly eccentric orbits!). When we take this into account in our calculations, we find that measuring high eccentricities is favored by this method. If the light curve says the eccentricity is low, but our prior assumptions say it is high, the probabilities “fight” with each other, and we get a wide probability distribution; if the light curve and our assumptions say the eccentricity is high, the probability distribution will be narrow at a high value. But…
- If the transit is very noisy, you are likely to measure the wrong eccentricity but with misleadingly high precision. If the light curve is very noisy, we cannot constrain any of the model parameters very well; it is uninformative. In those cases, we essentially fall back on our prior assumptions, which bias us sharply towards higher eccentricity whether or not the eccentricity was really high. This should not be seen as a problem with the method, but rather a limitation – if the signal-to-noise is very low, the photoeccentric effect should not be used to constrain eccentricity. Which leads us to...
- There are critical values of the planet-to-star radius ratio, $R_/R_\star$, for a given noise level below which the photoeccentric effect constrains eccentricity poorly. By simulating probability distributions of transit parameters and combining them together into a probability distribution for stellar density, we were able to predict the “best-case” uncertainty in eccentricity that this method could yield for representative planet parameters. As we would expect, as the noise level decreases, we can use this method to constrain the eccentricities of increasingly smaller planets.