Orbital Evolution of Eccentric Low-mass Companions Embedded in Gaseous Disks: Testing the Local Approximation

Abstract

We study the tidal interaction between a low-mass companion (e.g., a protoplanet or a black hole) in orbit about a central mass, and the accretion disk within which it is submerged. We present results for a companion on a coplanar orbit with eccentricity e between 0.1 and 0.6. For these eccentricities, dynamical friction arguments in its local approximation, that is, ignoring differential rotation and the curvature of the orbit, provide simple analytical expressions for the rates of energy and angular momentum exchange between the disk and the companion. We examine the range of validity of the dynamical friction approach by conducting a series of hydrodynamical simulations of a perturber with softening radius R soft embedded in a two-dimensional disk. We find close agreement between predictions and the values in simulations provided that R soft is chosen sufficiently small, below a threshold value ∼ R soft , which depends on the disk parameters and on e. We give ∼ R soft for both razorthin disks and disks with a finite scaleheight. For point-like perturbers, the local approximation is valid if the accretion radius is smaller than ∼ R soft . This condition imposes an upper value on the mass of the perturber.

Orbital evolution of gas-driven inspirals with extreme mass-ratios: retrograde eccentric orbits

“…In Sánchez-Salcedo et al (2018), we have studied the range of validity of the LA for perturbers in retrograde and circular orbit. On the other hand, the case of prograde and eccentric orbits was presented in Sánchez-Salcedo (2019). These studies demonstrate that the LA can predict the power and the torque provided that q is small enough.…”

“…Suppose that the LA predicts the power or the torque with some permissible error if R soft ≤ Rsoft ≡ E max H. Once we know Rsoft in a 2D disk, denoted by R(2D) soft , we can obtain R(3D) soft , the maximum softening radius in a 3D disk. In fact, Figure 13 in Sánchez-Salcedo (2019) shows the relationship between E (2D) max and E (3D) max . In particular, for h ≥ 0.025 and e ≤ 0.6, we have found in Section 4.2.2 that E (2D) p;1.5 = 0.1.…”

“…For that reason, a DF approach has been invoked to describe the interaction between the disk and a body in inclined or eccentric orbits (e.g., Papaloizou 2002;Muto et al 2011;Rein 2012;Amaro-Seoane et al 2016). In Sánchez-Salcedo (2019), we find that a simple model based on DF describes the orbital evolution of bodies in coplanar (i = 0) eccentric orbits (h < e 0.6), provided that the ratio between the mass of the perturber and the mass of the central object (denoted by q) is sufficiently small. For typical protoplanetary disks, this occurs for planets with q 10 −4 .…”

Estimating the depth of gaps opened by planets in eccentric orbit

“…In particular, for ℎ = 0.025 and 𝑒 = 0.16, rings at 𝑥 0.2 lose angular momentum. This happens at distances slightly larger than apocentre, where the planet feels a tail wind (e.g., Cresswell et al 2007;Bitsch & Kley 2010;Muto et al 2011;Sánchez-Salcedo 2019).…”

“…Less well developed is the description of the gap opened by a planet in eccentric orbit. Most of the studies of the interaction between an eccentric planet and the disc have focused on the planetary orbital evolution, either using an analytical approach (e.g., Papaloizou & Larwood 2000;Goldreich & Sari 2003;Tanaka & Ward 2004;Moorhead & Adams 2008;Muto et al 2011;Teyssandier & Ogilvie 2016) or hydrodynamical simulations (e.g., Cresswell & Nelson 2006;Cresswell et al 2007;Bitsch & Kley 2010;Bitsch et al 2013;Fendyke & Nelson 2014;Duffell & Chiang 2015;Ragusa et al 2018;Sánchez-Salcedo 2019;Lega et al 2021).…”