Research

Theoretical Asteroseismology

In stars like the sun, these normal modes are pressure waves (p-modes). The shapes of these normal modes can be described by spherical harmonics, indexed by quantum numbers \(l\) and \(m\), in addition to their radial order \(n\). Observationally, the p-mode frequencies of stars like the Sun are such that modes of each \(\ell\) are separated by a roughly uniform overtone spacing \(\Delta\nu\): \[\nu_{n\ell m} = \Delta\nu\left(n + {\ell \over 2} + \epsilon_{\ell m}(\nu)\right),\] where \(\Delta\nu\) is called the “large frequency separation”. This is what we would expect for sound waves (“p-modes”, because pressure provides the restoring force) in a spherically-shaped resonant mode cavity, where a more or less constant overtone spacing emerges, given by the sound-crossing time. \(\epsilon_{\ell m}(\nu)\) is a function of frequency that specifies the eigenvalues for a given pair of \(\ell, m\), and containes information about how the star’s internal structure and dynamics differ from being a homogenous and static spherical mode cavity.

On the other hand, other stars possess normal-mode frequencies that satisfy a converse eigenvalue relation, \[{1 \over \nu_{n\ell m}} = \Delta\Pi_\ell\left(n + {\ell \over 2} + \epsilon_{\ell m}(\nu)\right),\] where it is the mode periods, rather than frequencies, which have a characteristic “undertone” period spacing relative to the fundamental mode. Waves of this kind are buoyancy waves (“g-modes”, because gravity provides the restoring force), not unlike those which you might see on the surface of a body of water, or reflected in the undulatory motions of clouds in the atmosphere.

Much of my research pertains to understanding, hopefully analytically, how these two kinds of normal modes relate to the internal structures of stars, react to each other, and respond to perturbations not captured in our computational modelling (e.g. as caused by global magnetic activity, radiative turbulence, or deficiencies in our descriptions of their internal structure). Recent highlights include:

• Constructing integral estimators of various seismic diagnostics for stellar structure and evolution;
• Deriving a weak-interaction limit (and corresponding scattering-matrix elements) to describe the coupling of different mode cavities in evolved stars, via analogies with few-body quantum mechanics.
• Understanding how multicavity oscillations might be used to probe various aspects of stellar rotation and internal structure.

Asteroseismology as a Tool

Astronomers are often more interested in the inverse problem: that is, in inferring properties of a star given a set of observed oscillation frequencies. For example, we might infer masses, radii, and ages by comparing these mode frequencies against computational models of stellar structure. Alternatively, we might infer rotation rates and magnetic field strengths, if we are able to examine how modes of different \(m\) pulsate at different frequencies. Unfortunately, accurate measurement of the normal-mode frequencies requires relatively long observational campaigns. Often, the only quantities that are directly accessible are average values of \(\Delta\nu\), \(\epsilon\), and \(\nu_\text{max}\).

Some of my recent work involves using these diagnostics of internal rotation to make interesting inferences about interactions between stars and their interactions with companions (e.g. planets). Highlights of this include a chemically anomalous, rapidly-rotating, planet-engulfment candidate, as well as a red giant whose core and envelope rotate separately around axes pointing in different directions.

For the past few years, I’ve also been working on results from the NASA Transiting Exoplanet Survey Satellite (TESS) mission through the TESS Asteroseismic Science Consortium (TASC) (in working groups 1 and 2), helping to constrain the global properties and interior structures of these stars in this manner. For this purpose, I am both a methodologist and a practitioner: I develop and investigate methods used in deriving these constraints, as well as actually perform the calculations for actual stars. I’m glad to be able to make contributions to such a large collaboration. Interesting stars I’ve studied include:

• TOI-197, which formed the basis of the first TESS asteroseismology paper since other people found a “hot Saturn” exoplanet around it
• ν Indi, a metal-poor but otherwise chemically enriched star with unusual kinematic properties, which was observed with TESS
• KIC 8144907, perhaps the most metal-poor star whose age we have pinned down using asteroseismology

Extreme Precision Radial Velocity (EPRV) Instrumentation

The tale of exoplanet detection and discovery is long and storied, but the oldest and still most reliable way of detecting planets, and measuring their masses, is through the radial velocity method, which requires spectroscopic characterisation. As planets orbit their stars, they exert an opposite gravitational pull on the star that causes it to also orbit (much more slowly) their common centre of mass, moving towards and then away from us over time. From Earth, we are able to measure this time-varying reflex velocity by the Doppler shift it induces on stellar spectra.

These measurements are made on high-resolution spectrographs. One of them, the EXtreme PREcision Spectrograph, was built at Yale. I wrote parts of its calibration and analysis software, including its template cross-correlation-function (CCF) radial velocity code, and various infrastructural components of its automated optimal-extraction data reduction pipeline.

While the ultimate goal of EXPRES is, of course, to discover long-period, low-mass planets with the radial-velocity method that would otherwise not be accessible to short-baseline space photometry, the high-resolution spectra returned from this process are very useful for other purposes. One example of this is Doppler imaging of active stellar surface, which can be further refined by way of joint constraints with contemporaneous photometry.

More recently, I’ve become a regular observer on the Keck Planet Finder instrument on Maunakea, Hawaiʻi, and have conducted some of the earliest science using it.