Dynamical Mass of the Exoplanet Host Star HR 8799

HR 8799 is a young A5/F0 star hosting four directly imaged giant planets at wide separations (∼16–78 au), which are undergoing orbital motion and have been continuously monitored with adaptive optics imaging since their discovery over a decade ago. We present a dynamical mass of HR 8799 using 130 epochs of relative astrometry of its planets, which include both published measurements and new medium-band 3.1 μm observations that we acquired with NIRC2 at Keck Observatory. For the purpose of measuring the host-star mass, each orbiting planet is treated as a massless particle and is fit with a Keplerian orbit using Markov chain Monte Carlo. We then use a Bayesian framework to combine each independent total mass measurement into a cumulative dynamical mass using all four planets. The dynamical mass of HR 8799 is 1.47 − 0.17 + 0.12 M ⊙ assuming a uniform stellar mass prior, or 1.46 − 0.15 + 0.11 M ⊙ with a weakly informative prior based on spectroscopy. There is a strong covariance between the planets’ eccentricities and the total system mass; when the constraint is limited to low-eccentricity solutions of e < 0.1, which are motivated by dynamical stability, our mass measurement improves to 1.43 − 0.07 + 0.06 M ⊙. Our dynamical mass and other fundamental measured parameters of HR 8799 together with Modules for Experiments in Stellar Astrophysics Isochrones and Stellar Tracks grids yields a bulk metallicity most consistent with [Fe/H] ∼ −0.25–0.00 dex and an age of 10–23 Myr for the system. This implies hot-start masses of 2.7–4.9 M Jup for HR 8799 b and 4.1–7.0 M Jup for HR 8799 c, d, and e, assuming they formed at the same time as the host star.

Non-integrability of a model of elastic dumbbell satellite

We study the integrability of a model of elastic satellite whose centre of mass moves in a circular Keplerian orbit around a gravity centre. The satellite is modelled by two point masses connected by an extensible massless spring that obeys Hooke’s law. It is assumed that the distance between point masses is much smaller than the radius of the orbit, so the orbital motion of the satellite is not perturbed by its rotational motion. The gravity potential of the satellite is expanded into a series with respect to its size up to quadratic terms which describe the gravity gradient torque acting on the satellite. Two cases are considered with Hooke’s centre localised in the centre of mass of the dumbbell and at an arbitrary point along a line connecting both masses. It is shown that the first case appears to be integrable and super-integrable for selected values of the parameter of the system. In the second case, model depends effectively only on one parameter and is non-integrable. In the proof, differential Galois integrability obstructions are used. For the considered sysem, these obstructions are deduced thanks to the recently developed symplectic Kovacic’s algorithm in dimension 4. According to our knowledge, this is the first application of this tool to a physical model.

Constraints on Newtonian Interplanetary Point-Mass Interactions in Multicomponent Systems from the Symmetry of Their Cycles

Interplanetary interactions are the largest forces in our Solar System that disturb the planets from their elliptical orbits around the Sun, yet are weak (<10−3 Solar). Currently, these perturbations are computed in pairs using Hill’s model for steady-state, central forces between one circular and one elliptical ring of mass. However, forces between rings are not central. To represent interplanetary interactions, which are transient, time-dependent, and cyclical, we build upon Newton’s model of interacting point-mass pairs, focusing on circular orbits of the eight largest bodies. To probe general and evolutionary behavior, we present analytical and numerical models of the interplanetary forces and torques generated during the planetary interaction cycles. From symmetry, over a planetary interaction cycle, radial forces dominate while tangential forces average to zero. Our calculations show that orbital perturbations require millennia to quantify, but observations are only over ~165 years. Furthermore, these observations are compromised because they are predominantly made from Earth, whose geocenter occupies a complex, non-Keplerian orbit. Eccentricity and inclination data are reliable and suggest that interplanetary interactions have drawn orbital planes together while elongating the orbits of the two smallest planets. This finding is consistent with conservation principles governing the eight planets, which formed as a system and evolve as a system.

The origin of hotspots around Sgr A*: orbital or pattern motion?

ABSTRACT The Gravity Collaboration detected a near-infrared hotspot moving around Sgr A* during the 2018 July 22 flare. They fitted the partial loop the hotspot made on the sky with a circular Keplerian orbit of radius $\simeq 7.5\, r_{\rm g}$ around the supermassive black hole (BH), where rg is the gravitational radius. However, because the hotspot traversed the loop in a short time, models in which the hotspot tracks the motion of some fluid element tend to produce a best-fitting trajectory smaller than the observed loop. This is true for a circular Keplerian orbit, even when BH spin is accounted for, and for motion along a radiatively inefficient accretion flow (RIAF) streamline. A marginally bound geodesic suffers from the same problem; in addition, it is not clear what the origin of an object following the geodesic would be. The observed hotspot motion is more likely a pattern motion. Circular motion with $r\simeq 12.5\, r_\mathrm{g}$ and a super-Keplerian speed $\simeq 0.8\, c$ is a good fit. Such motion must be pattern motion because it cannot be explained by physical forces. The pattern speed is compatible with magnetohydrodynamic perturbations, provided that the magnetic field is sufficiently strong. Circular pattern motion of radius $\sim 20\, r_{\rm g}$ on a plane above the BH is an equally good alternative; in this case, the hotspot may be caused by a precessing outflow interacting with a surrounding disc. As all our fits have relatively large radii, we cannot constrain the BH spin using these observations.

Determination of the orbits of 62 moons of asteroids based on astrometric observations.

ABSTRACT Based on all published astrometric observations, we have determined the moon orbits for asteroids using a model of the fixed Keplerian orbit. We applied 5–114 observations for each moon. As a result, we have determined the orbits of 62 moons. All results, including the orbital parameters obtained, are presented in the tables that are provided as supplementary material, available online. These data can be used to calculate the ephemerides of the moons of the asteroids. Among the moons considered, 13 belong to asteroids of the main asteroid belt, two are the moons of Jupiter Trojan asteroids, while the rest are trans-Neptunian objects. Our results are in good agreement with the corresponding results published in the literature. We argue that reliable estimates of the accuracy for the ephemerides can be only made using parameter covariance matrices. These matrices that we have obtained are also given in the supporting information.

Lambert's theorem and projective dynamics

We prove that the classical Lambert theorem about the elapsed time on an arc of Keplerian orbit extends without change to the Kepler problem on a space of constant curvature. We prove that the Hooke problem has a property similar to Lambert's theorem, which also extends to the spaces of constant curvature. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.