The cascading numbers of exoplanet discoveries raise questions about how to interpret our data. In particular, what do we do about all those transit finds where we can work out a planet’s radius and need to determine its mass? Andrew LePage returns to Centauri Dreams with a look at a new attempt to derive the relationship between mass and radius. Getting this right will be useful as we analyze statistical data to understand how planets form and evolve. LePage is the author of an excellent blog on exoplanetary science called Drew ex Machina, and a senior project scientist at Visidyne, Inc. specializing in the processing and analysis of remote sensing data.

By Andrew LePage


As anyone with even a passing interest in planetary studies can tell you, we are witnessing an age of planetary discovery unrivaled in the long history of astronomy. Over the last two decades, thousands of extrasolar planets have been discovered using a variety of techniques. The most successful of these to date in terms of sheer number of finds is the transit method – the use of precision photometric measurements to spot the tiny decrease in a star’s brightness as an orbiting planet passes directly between us and the star. The change in the star’s brightness during the transit allows astronomers to estimate the size of the planet relative to the star while the time between successive transits allows the orbital period of the planet to be determined. Combined with information about the properties of the star being observed, other characteristics can be calculated such as the actual size of the planet and its orbit. The most successful campaign to date to search for planets using the transit method has been performed using NASA’s Kepler spacecraft, launched in 2009.

One of the other important bulk properties of a planet that is of interest to scientists is its mass. Unfortunately, the transit method is typically unable to supply us with this information except in special circumstances where planets in a system strongly interact with each other to produce measurable variations in the timing or duration of their transits. The transit timing variation (TTV) or transit duration variation (TDV) methods can be used to estimate the masses of the planets of a system including non-transiting planets that might be present. Based on an analysis of Kepler results to date, however, this method can be used in only about 6% of planetary systems that produce transits.

A more widely applicable method to determine the mass of an extrasolar planet is through the precision measurement of a star’s radial velocity to detect the reflex motion caused by the orbiting planet. Combined with information from transit observations as well as the star’s properties, it is possible to calculate the actual mass of a planet and further refine its orbital properties. Unfortunately, NASA’s Kepler mission has discovered thousands of planets and making precision radial velocity measurements takes a lot of time on a limited number of busy telescopes that are equipped to make the required observations. In addition, many of the stars observed by Kepler are too dim or their planets too small for the current generation of instruments to detect radial velocity variations above the noise. This is especially a problem for sub-Neptune size planets including Earth-size terrestrial planets. Taken as a whole, only a small minority of all of Kepler’s finds currently have had their masses measured.

Puzzling Out a Planetary Mass

While astronomers continue to struggle to measure the masses of thousands of individual extrasolar planets found by Kepler, there have been efforts to derive a mass-radius relationship so that the mass of a planet with a known radius can at least be estimated. In addition to being useful for evaluating the level of accuracy required for detection using radial velocity measurements or other methods, such mass estimates are also valuable for scientists wishing to use Kepler radius and orbit data in statistical studies of planetary properties, dynamics, formation and evolution. Over the past few years, there have been various investigators who have attempted to derive a planetary mass-radius relationship as information on the mass and radius of known planets has expanded. These relationships have taken a mathematical form known as a power law such as M = CRγ where M is the mass of the planet (in terms of Earth mass or ME), R is its radius (in terms of Earth radii or RE) and C and γ are constants determined by analysis.

The latest work to derive a mass-radius relationship for sub-Neptune size planets (i.e. planets whose radii are less than 4RE) is a paper by Angie Wolfgang (University of California – Santa Cruz), Leslie A. Rogers (California Institute of Technology), and Eric B. Ford (Pennsylvania State University), which they recently submitted for publication in The Astrophysical Journal. These sub-Neptune size worlds are of particular interest to the scientific community since they span the size range between the Earth and Neptune where no Solar System analogs exist to provide guidance for deriving a mass-radius relationship.

Earlier work over the last few years on the planetary mass-radius relationship relied on least squares regression analysis of a set of planetary radius and mass measurements – a fairly straightforward mathematical method used to determine the constants of an equation that provides the best fit to a set of data points. Unfortunately, this classic method has some drawbacks. It does not properly take into account the uncertainty in the independent variable (i.e. the planet radius, in this case) or instances where the planet has not been detected using precision radial velocity measurements and only an upper limit of the mass can be derived. Another issue is that the least squares regression method assumes a deterministic relationship where a particular planetary radius value corresponds to a unique mass value. In reality, planets with a given radius can have a range of different mass values, in part reflecting the variation in planetary composition running from massive rocky planets with large iron-nickel cores to less massive, volatile-rich planets with deep atmospheres. These variations are expected to be especially important in sub-Neptune-class worlds.

A Bayesian Approach to the Mass/Radius Problem

Instead of using the least squares regression method, Wolfgang, Rogers and Ford evaluated their data using a hierarchal Bayesian technique which allowed them not only to derive the parameters for a best fit of the available data, but also to quantify the uncertainty in those parameters as well as the distribution of actual planetary mass values. Using their approach, they have derived a probabilistic mass-radius relationship where the most likely mass and the distribution of those values are determined. The team considered a total of 90 extrasolar planets with known radii less than 4RE whose masses have been measured or constrained using radial velocity or TTV methods. Neither unconfirmed planets nor circumbinary planets were considered to keep the sample as homogeneous as possible. The team also truncated the mass distribution to physically plausible values that were no less than zero (since it is physically impossible to have a negative mass) and no greater than the mass of a planet composed of iron (since it is unlikely for a planet to have a composition dominated by any element denser than iron).


Image: This plot shows the available mass and radius data (and associated error bars) used in the latest analysis of the mass-radius relationship for sub-Neptune size planets. Various fits to these data are shown including an earlier analysis by Lauren Weiss and Geoffrey Marcy (black dashed line) as well as fits for radii <8 RE, <4RE and <1.6RE (solid colored lines). (credit: Wolfgang et al.)

The detailed analysis of the dataset by Wolfgang, Rogers and Ford found that the subset of extrasolar planets whose masses were measured using the TTV method has a definite bias towards lower density planets. This bias had been suspected since a low density planet will have a larger radius than a denser planet with the same mass. And all else being equal, a larger planet is more likely to be detected using the transit method than a smaller planet. When only considering the sample of extrasolar planets with masses determined using precision radial velocity measurements, this team found that the best fit for the data set was a power law with C = 2.7 and γ = 1.3 (i.e. M = 2.7R1.3). Based on their statistical analysis, Wolfgang, Rogers and Ford found that the data were consistent with a Gaussian or bell-curve distribution of actual planet masses with a sigma of 1.9ME at any given radius value. Just as has been suspected, planets with radii less than 4RE display a range of compositions that is reflected as a fairly broad distribution of actual mass values.

In earlier work by Rogers, it was found that there seems to be a transition in planet composition at a radius no larger than 1.6 RE, above which planets are unlikely to be dense, rocky worlds like the Earth and much more likely to be less dense, volatile-rich planets like Neptune (see The Transition from Rocky to Non-Rocky Planets in Centauri Dreams for a full discussion of this work). For the sample of planets considered here with radii less than 1.6 RE, the team found that C = 1.4 and γ = 2.3. Unfortunately, the sample considered by Wolfgang, Rogers and Ford has little good data for planets in this size range and the masses with their large uncertainties tend to span the full range of physically plausible values. As a result, this analysis can not rule out the possibility of a deterministic mass-radius relationship where there is only a very narrow range of actual planet masses for any particular radius value. Recent work by others suggests that these smaller planets tend to have a more Earth-like, rocky composition which could be characterized with a more deterministic mass-radius relationship (see The Composition of Super-Earths in Drew Ex Machina for a discussion of this work).

This new work by Wolfgang, Rogers and Ford represents the best attempt to date to determine the mass-radius relationship for planets smaller than Neptune. While more data of better quality for planets in this size range are needed, it does appear that sub-Neptunes can have a range of different compositions and therefore possess a range of mass values at any given radius. This new relation will be most useful to scientists hoping to get the maximum benefit out of the ever-growing list of Kepler planetary finds where only the radius is known. Much more data will be required to determine more accurately the mass-radius distribution of planets with radii less than 1.6 RE and more precisely characterize the transition from large, rocky Earth-like planets to larger, volatile-rich planets like Neptune.

The preprint of the paper by Wolfgang, Rogers and Ford, “Probabilistic Mass-Radius Relationship for Sub-Neptune-Sized Planet”, can be found here.