DYNAMICS OF PLANETS AND DEBRIS DISKS
Exoplanetary systems in general contain not only planets, but also belts of debris. `Debris' refers to any component of a planetary system which is not an actual planet (or a moon). For instance, in our own Solar System, the asteroids and comets in the Asteroid and Kuiper belts, as well as the dust generated from them, as in the Zodiacal cloud near the Earth's orbit, can all be classified as debris. Since their first discovery in the early 1980s, the number of known such debris discs has dramatically increased, with current detection rates of ~ 20 - 30% around main-sequence stars in the Solar neighbourhood. With the advent of astronomical instruments, astronomers have also been able to image some of these dusty debris disks with high resolution. Such observations reveal a rich variety of disk structures such as gaps, spirals, and warps.
Debris disks are affected by several processes. One important process involves gravitational interactions. Indeed, planets exert gravitational forces on debris particles, which also exert gravitational forces on the planets. Additionally, the debris particles exert gravitational forces amongst themselves, perturbing each other's orbits. Taken together, such planet-debris interactions can affect the orbits of the planets, as well as the spatial distribution and appearance of the debris, potentially producing structures that are akin to those observed. Thus, detailed modelling of planet-debris interactions and comparing the outcomes with observations can improve our overall understanding of the formation, evolution, and architecture of planetary systems. Moreover, and perhaps more importantly, examining the structures of debris discs can even reveal the presence of planets that might otherwise not be detected with current technologies. This is where much of my work comes in.
Interactions between Planets and Debris Disks: The Role of Disk Gravity
Investigating the structure of debris disks can provide unique insights into the architecture and evolution of exoplanetary systems, and may even help to infer the presence and parameters of otherwise-undetectable planets. The Solar System provides a case in point to this end, whereby studies of the asteroid and Kuiper belts have enabled the re-construction of the Solar System's dynamical history, including Neptune's migration, the origin of the cold and hot populations in the Kuiper belt, the sculpting of the Kirkwood gaps in the asteroid belt, to name a few. More recently, studies of the Kuiper belt have also led to a suggestion of a ninth - yet undetected - planet beyond Neptune (for more on this, click here).
In analogy with the Solar System studies, much effort has been put into understanding how planets and/or stellar companions sculpt debris disk morphologies though their gravitational perturbations. One of the triumphs of such planet-disk interaction studies is the discovery of a giant planet around the star β Pictoris in 2010, which was first predicted mathematically based on the observed vertical structure in the debris disk. However, studies of planet-debris disk interactions usually ignore the gravitational effects of the disk itself. That is, debris disks are usually treated as a collection of massless particles subject only to the gravity of the star and (invoked) planets. This treatment, while useful, may not always be justified, especially since observations suggest that debris disks could contain tens if not hundreds of Earth masses in large planetesimals.
In Sefilian et al. (2021) and Sefilian et al. (2023), we investigated the long-term, secular interaction between a single eccentric planet and an external debris disk, accounting for the oft-neglected gravitational effects of the disk. We demonstrated that even when the disk is less massive than the planet, the disk's gravity can have a considerable impact on the evolution of both constituent planetesimals and the planet. Namely, we found that the system may feature secular resonances at two locations within the disk (contrary to what may be naively expected), where planetesimals eccentricities get significantly excited. In our single-planet systems, the secular resonances occur where the apsidal precession rate of debris (due to the disk and planet) is equal to that of the planet (due to the disk). Based on this we proposed that gaps (i.e. depleted regions) in debris disks, akin to those observed in HD 107146, HD 92945, and HD 206893, could be the result of secular resonances with a yet-undetected planet interior to the disk (rather than within the gap itself as is otherwise commonly expected). An example simulation is shown in the animation.
Our results may be used to infer the presence of a yet-undetected planet based on the observed gap features (as we do for HD 107146 and HD 92945). Additionally - and more importantly - if a companion is already known (or discovered later), our results may be used to indirectly measure the total mass of the debris disk based on the gap features. We demonstrated this for HD 206893, for which we infer a total disk mass of approximately 170 Earth masses based on the properties of the known brown dwarf companion in that system.
In Sefilian et al. (2023), we also showed that the gravitational coupling between the planet and the disk leads to the circularization of the planetary orbit. This happens due to the strong torque that planetesimals orbiting at (and around) the secular resonance location exert on the planet. This torque transports angular momentum from the resonant planetesimals to the planet, without affecting the system's total angular momentum budget. As a result, the planet's orbital eccentricity is damped, while its semimajor axis remains unchanged. This process is known as resonant friction.
All simulations in Sefilian et al. (2023) (as in the animation above) are carried out using what we refer to as the "N-ring" tool that we developed. This is a semi-analytic code that allows the study of the secular evolution of self-gravitating particulate disks and their response to external perturbations in general (coplanar) astrophysical setups. The underlying framework is built upon the continuum version of the classical Laplace-Lagrange theory, whereby the disk is modelled as a series of N massive rings interacting with each other and with (any) external perturbers. Unlike the classical Laplace-Lagrange theory, however, here the ring-ring interaction is softened by spatially smoothing the Newtonian point-mass potential (based on Sefilian & Rafikov 2019). For more details on this, check out Sefilian & Rafikov (2019) and Sefilian et al. (2023). A copy of the code is made publicly available.
Laplace Surface Dynamics: Application to HD 106906
In the early 1800s, Laplace had studied how the orbital architecture of Jupiter’s moons is shaped by the combined gravitational perturbations due to Jupiter’s bulge and the Sun. In his seminal work, Laplace had posited that the competing effects of Jupiter and the Sun can balance each other out, giving rise to the so-called Laplace equilibria. These equilibria, representing fixed circular orbits with proper orbital inclinations, coincide with the planet's equator at small distances from the planet and tend to the planet's orbital plane at larger distances, giving rise to a warped surface known as the Laplace surface. More recently, and with a view to greater realism, two studies generalized Laplace's framework by allowing for perturbations due to planets on eccentric and inclined orbits (Tremaine et al. 2009, Farhat & Touma 2021), ushering a stream of studies related to the early evolution of the Moon around a fast-spinning oblique Earth, evolution of the Uraniun satellites, and warps in circumplanetary discs, to name a few.
In Farhat, Sefilian & Touma (2023), using these newly developed theoretical frameworks of eccentric Laplace surface, we investigated the dynamical evolution of the extra-solar system HD 106906, a system whose architecture provides a unique testbed for studying the implications of Laplace surface dynamics. Indeed, the HD 106906 system features a young (~13 Myr old) spectroscopic stellar binary which is surrounded by a low-mass debris disk interior to the directly imaged planet on a wide, eccentric, and inclined orbit.
Previous studies of this system had ignored the gravitational perturbations of the stellar binary. In our study, we showed that the binary is strong enough, that in tandem with the external planet, it may give rise to the so-called Laplace dynamics. Indeed, we showed that within the observational uncertainties on the planetary orbit, the disk can be fully dominated by the inner binary and thus remain vertically thin, or be dominated by the binary in the inner regions and by the inclined planet in the outer parts, thus exhibiting a vertically warped structure. This interplay results in distinctive observational signatures such as asymmetries and warps which we render via simulated surface density/brightness maps and explore as a function of the outer planet's orbit, with and without radiation pressure due to the central binary. We then used these results not only to put constraints on the planetary orbit, but also to explain the differences in the disk structure as seen by ALMA at millimeter wavelengths and by the Hubble Space Telescope in scattered light images.