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Register a new userDynamical Friction of Double Perturbers in a Gaseous Medium
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In many astrophysical situations, as in the coalescence of supermassive black hole pairs at gas rich galactic nuclei, the dynamical friction experienced by an object is a combination of its own wake as well as the wakes of its companions. Using a semi-analytic approach, we investigate the composite wake due to, and the resulting drag forces on, double perturbers that are placed at the opposite sides of the orbital center and move on a circular orbit in a uniform gaseous medium. The circular orbit makes the wake of each perturber asymmetric, creating an overdense tail at the trailing side. The tail not only drags the perturber backward but it also exerts a positive torque on the companion. For equal-mass perturbers, the positive torque created by the companion wake is, on average, a fraction ~40-50% of the negative torque created by its own wake, but this fraction may be even larger for perturbers moving subsonically. This suggests that the orbital decay of a perturber in a double system, especially in the subsonic regime, can take considerably longer than in isolation. We provide the fitting formulae for the forces due to the companion wake and discuss our results in light of recent numerical simulations for mergers of binary black holes.
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Following a wave-mechanical treatment we calculate the drag force exerted by an infinite homogeneous background of stars on a perturber as this makes its way through the system. We recover Chandrasekhars classical dynamical friction (DF) law with a modified Coulomb logarithm. We take into account a range of models that encompasses all plausible density distributions for satellite galaxies by considering the DF exerted on a Plummer sphere and a perturber having a Hernquist profile. It is shown that the shape of the perturber affects only the exact form of the Coulomb logarithm. The latter converges on small scales, because encounters of the test and field stars with impact parameters less than the size of the massive perturber become inefficient. We confirm this way earlier results based on the impulse approximation of small angle scatterings.
We investigate the gravitational wake due to, and dynamical friction on, a perturber moving on a circular orbit in a uniform gaseous medium using a semi-analytic method. This work is a straightforward extension of Ostriker (1999) who studied the case of a straight-line trajectory. The circular orbit causes the bending of the wake in the background medium along the orbit, forming a long trailing tail. The wake distribution is thus asymmetric, giving rise to the drag forces in both opposite (azimuthal) and lateral (radial) directions to the motion of the perturber, although the latter does not contribute to orbital decay much. For subsonic motion, the density wake with a weak tail is simply a curved version of that in Ostriker and does not exhibit the front-back symmetry. The resulting drag force in the opposite direction is remarkably similar to the finite-time, linear-trajectory counterpart. On the other hand, a supersonic perturber is able to overtake its own wake, possibly multiple times, and develops a very pronounced tail. The supersonic tail surrounds the perturber in a trailing spiral fashion, enhancing the perturbed density at the back as well as far front of the perturber. We provide the fitting formulae for the drag forces as functions of the Mach number, whose azimuthal part is surprisingly in good agreement with the Ostrikers formula, provided Vp t=2 Rp, where Vp and Rp are the velocity and orbital radius of the perturber, respectively.
We use three-dimensional hydrodynamic simulations to investigate the nonlinear gravitational responses of gas to, and the resulting drag forces on, very massive perturbers moving on circular orbits. This work extends our previous studies that explored the cases of low-mass perturbers on circular orbits and massive perturbers on straight-line trajectories. The background medium is assumed to be non-rotating, adiabatic with index 5/3, and uniform with density rho0 and sound speed a0. We model the gravitating perturber using a Plummer sphere with mass Mp and softening radius rs in a uniform circular motion at speed Vp and orbital radius Rp, and run various models with differing R=rs/Rp, Mach=Vp/a0, and B=G*Mp/(a0^2*Rp). A quasi-steady density wake of a supersonic model consists of a hydrostatic envelope surrounding the perturber, an upstream bow shock, and a trailing low-density region. The continuous change in the direction of the perturber motion makes the detached shock distance reduced compared to the linear-trajectory cases, while the orbit-averaged gravity of the perturber gathers the gas toward the center of the orbit, modifying the background preshock density to rho1=(1 + 0.46B)*rho0 depending weakly on Mach. For sufficiently massive perturbers, the presence of a hydrostatic envelope makes the drag force smaller than the prediction of the linear perturbation theory, resulting in F = 4*pi*rho1*(G*Mp/Vp)^2 * (0.7/etaB) for etaB = B/(Mach^2 -1) > 0.1; the drag force for low-mass perturbers with etaB < 0.1 agrees well with the linear prediction. The nonlinear drag force becomes independent of R as long as R < etaB/2, which places an upper limit on the perturber size for accurate evaluation of the drag force in numerical simulations.
In this paper, we explore the impact of a galactic bar on the inspiral time-scale of a massive perturber (MP) within a Milky Way-like galaxy. We integrate the orbit of MPs in a multi-component galaxy model via a semi-analytical approach including an accurate treatment for dynamical friction generalized to rotationally supported backgrounds. We compare the MP evolution in a galaxy featuring a Milky Way-like rotating bar to the evolution within an analogous axisymmetric galaxy without the bar. We find that the bar presence may significantly affect the inspiral, sometimes making it shorter by a factor of a few, sometimes hindering it for a Hubble time, implying that dynamical friction alone is greatly insufficient to fully characterize the orbital decay. The effect of the bar is more prominent for initially in-plane, prograde MPs, especially those crossing the bar co-rotation radius or outer Lindblad resonance. In the barred galaxy, we find the sinking of the most massive MPs (>~10^7.5 Msun) approaching the galaxy from large separations (>~8 kpc) to be most efficiently hampered. Neglecting the effect of global torques associated to the non-symmetric mass distribution is thus not advisable even within our idealized, smooth Milky Way model, and it should be avoided when dealing with more complex and realistic galaxy systems. This has important implications for the orbital decay of massive black holes in late-type spirals, the natural candidate sources to be detected with the Laser Interferometer Space Antenna (LISA).
We consider the gravitational force exerted on a point-like perturber of mass $M$ travelling within a uniform gaseous, opaque medium at constant velocity $V$. The perturber irradiates the surrounding gas with luminosity $L$. The diffusion of the heat released is modelled with a uniform thermal diffusivity $chi$. Using linear perturbation theory, we show that the force exerted by the perturbed gas on the perturber differs from the force without radiation (or standard dynamical friction). Hot, underdense gas trails the mass, which gives rise to a new force component, the heating force, with direction $+V$, thus opposed to the standard dynamical friction. In the limit of low Mach numbers, the heating force has expression $F_mathrm{heat}=gamma(gamma-1)GML/(2chi c_s^2)$, $c_s$ being the sound speed and $gamma$ the ratio of specific heats. In the limit of large Mach numbers, $F_mathrm{heat}=(gamma-1)GML/(chi V^2)f(r_mathrm{min}V/4chi)$, where $f$ is a function that diverges logarithmically as $r_mathrm{min}$ tends to zero. Remarkably, the force in the low Mach number limit does not depend on the velocity. The equilibrium speed, when it exists, is set by the cancellation of the standard dynamical friction and heating force. In the low Mach number limit, it scales with the luminosity to mass ratio of the perturber. Using the above results suggests that Mars- to Earth-sized planetary embryos heated by accretion in a gaseous protoplanetary disc should have eccentricities and inclinations that amount to a sizeable fraction of the discs aspect ratio, for conditions thought to prevail at a few astronomical units.
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