Tell me the velocity of a star without telling me its velocity

Title: The 3D Galactocentric velocities of Kepler stars: marginalizing over missing RVs

Authors: Ruth Angus, Adrian M. Price-Whelan, Joel C. Zinn, Megan Bedell, Yuxi (Lucy) Lu, and Daniel Foreman-Mackey

First Author’s Institution: Department of Astrophysics, American Museum of Natural History

Paper Status: Accepted for publication in AAS Journals [open access on arXiv]

One of the best ways to understand stars is to look at their spectra, which are just plots of intensity of light versus the wavelength. More precisely, we can get a lot from the absorption lines of a star. The absorption lines are caused by cool atoms in the atmosphere of a star that preferentially absorb particular wavelengths, causing that dip in the intensity at that particular wavelength. However, there’s another thing that can affect the wavelength too, and it is called the doppler effect. It is the change in wavelength due to the motion of a star along the line of sight. An object moving towards you has its wavelength shifted to shorter wavelengths, in a phenomenon called blue shift, while an object moving away has its wavelength shifted towards longer wavelengths, which is called red shift. So, this means that wavelengths and the radial velocities of a star are related. Therefore, if we know the wavelength, we can get their radial velocity (we can know how the stars move)! Spectroscopic radial velocities (RV) can be measured and, if available, can already provide a ton of information about stars! There’s a problem: they are not measured for all detected stars, just 7 million of them! Don’t be upset just yet! The researchers from today’s paper were able to infer the velocities of stars that don’t have those measurements. What’s more exciting: if we know the radial velocity measurements, then we can get the ages of the stars (stars are really good at hiding their ages!) using the Age-Velocity dispersion Relations.

To have the 3D vector of the velocities, one needs to measure (or infer in this case) the velocities in x-, y-, and z-directions (in cartesian coordinates). The researchers used the data from Gaia, Kepler, LAMOST and APOGEE catalogs. They used Bayes’ theorem, the way you can infer the posterior probability knowing the prior knowledge of your data. The prior knowledge can be anything from some logical reasoning about the target to the data from different catalogs. The prior helps us to constrain our posterior probability. In the paper, the authors constructed their priors over distance and 3D velocities using the Kepler targets which have RV measurements. They calculated the statistical parameters for stars with measured RVs and then used these parameters to construct a prior over the velocity and distance parameters for stars that don’t have RV measurements. 

The goal of the paper is to infer the velocities of stars without RV measurements using a prior calculated from stars with RV measurements. The caveat is that the stars with and without RV measurements can be from different populations: i.e. the stars without RV measurements are more likely to be fainter, less luminous, cooler and potentially older (Figure 1). On Figure 1, you can see the populations of stars with and without RV measurements. You can see that the stars with RVs (left panel) go a little higher which corresponds to upper-main-sequence stars and red giant stars on the Color Magnitude Diagram, while stars without RVs are more likely to be mid- and lower-main-sequence dwarfs. For this reason, a prior based on the velocity distributions of stars with RVs will not necessarily reflect the velocities of those without.

Figure 1: A magnitude-temperature diagram of stars in the Kepler field with (left) and without (right) RVs provided by Gaia, LAMOST and APOGEE. The stars with RVs are generally hotter and more luminous than those without RVs, and include a large number of red clump stars and red giant branch stars. Stars without RVs are mostly concentrated on the main sequence. Figure 3 in the paper.

Nonetheless, the velocities in x- and z- directions ended up to be better constrained due to the fact that the y-direction is similar to the radial direction for observers near the Sun and is therefore poorly constrained for Kepler stars without RV measurements. The authors tested this with different choices of priors and confirmed that x- and z- directions of velocities are not really prior-dependent! Meaning, any choice of a prior can constrain the x- and z- directions of velocities. Looking at Figure 2, the first and the third plots show exactly that: the curves are all on top of each other, which shows that the predicted distributions are not dependent on the prior (in this case the priors are the populations of faint stars and bright stars as shown in a figure legend). However, the distribution of inferred y-direction velocities is prior-sensitive. This is because the y-direction is close to the radial direction for Kepler stars as discussed above.

Figure 2: The distributions of velocity and distance parameters, inferred using three different priors. The orange line represents the distribution of parameters estimated from the bright half of the RV sample. The blue dashed line shows the results from a prior estimated from the faint half of the RV sample. The black line shows the results from a prior calculated from all stars with RV measurements and is the prior the authors adopt in the final analysis. This figure shows that all parameters except the y-direction velocity are relatively insensitive to the three priors that were tested. Figure 6 in the paper.

In the end, to validate their method, they compare their inferred velocities with directly-calculated velocities for 5000 Kepler stars with RV measurements. They found that x- and z-direction velocities were inferred more precisely than the y-direction velocities due to, again, the position of a Kepler field (discussed above). Even despite some inaccuracies in y-direction, the inferred velocities are consistent with the true velocities. These inferred velocities therefore can be used to get the kinematic ages of the stars!

Astrobite edited by Lynnie Saade

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About Sabina Sagynbayeva

I'm a graduate student at Stony Brook University and my main research area is planet formation. I'm currently working on planetary migration using hydrodynamical simulations. I'm also interested in protoplanetary disks but nearly any topic related to planets is fascinating to me! In addition to doing research, I'm also a singer-songwriter. I LOVE writing songs, and you can find them on any streaming platform.

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