Dynamical friction in a gaseous medium with a large-scale magnetic field
aa r X i v : . [ a s t r o - ph . C O ] N ov Draft version November 8, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
DYNAMICAL FRICTION IN A GASEOUS MEDIUM WITH A LARGE-SCALE MAGNETIC FIELD
F. J. S´anchez-Salcedo Draft version November 8, 2018
ABSTRACTThe dynamical friction force experienced by a massive gravitating body moving through a gaseousmedium is modified by sufficiently strong large-scale magnetic fields. Using linear perturbation theory,we calculate the structure of the wake generated by, and the gravitational drag force on, a bodytraveling in a straight-line trajectory in a uniformly magnetized medium. The functional form of thedrag force as a function of the Mach number ( ≡ V /c s , where V is the velocity of the body and c s thesound speed) depends on the strength of the magnetic field and on the angle between the velocity ofthe perturber and the direction of the magnetic field. In particular, the peak value of the drag force isnot near Mach number ∼ Subject headings: black hole physics — hydrodynamics — ISM: general — waves INTRODUCTION
An object moving in a background medium inducesa gravitational wake. The asymmetry of the mass den-sity distribution upstream and downstream from the per-turber produces a drag on the body, which is often re-ferred to as gravitational drag or dynamical friction (DF)force. A body in orbital motion may undergo a radial de-cay of its orbit due to the loss of angular momentum bythe negative torque caused by DF drag. Chandrasekhar(1943) derived the dynamical friction on a massive par-ticle passing through a homogeneous and isotropic back-ground of light stars. His formula is applied to estimatethe merger timescale of satellite systems or to study theaccretion history of galaxies. Bondi & Hoyle (1944) con-sidered the problem of the mass accretion by a point mass M travelling at velocity V in a collisional homogeneousmedium of sound speed c s in the limit where the per-turber moves at supersonic velocities relative to the am-bient gas (i.e. high Mach numbers). If the perturber is anaccretor, streamlines with small impact parameter maybecome bound because of energy dissipation in shocks,and can be accreted to the perturber. Hence, the force onthe perturber consists of two parts; the gravitational dragand the momentum accretion force. The latter contribu-tion may be decelerating or accelerating (Ruffert 1996).If the size of the perturber is larger than the Bondi-Hoyleaccretion radius defined as R BH ≡ GM/ [ c s (1 + M )]with M = V /c s , the density and velocity structure ofthe wake, at any Mach number, can be inferred analyti-cally in linear theory because the body produces a smallperturbation in the ambient gaseous medium at any lo-cation. The gravitational drag is inferred as the grav-itational attraction between the perturber and its ownwake (e.g., Dokuchaev 1964; Rephaeli & Salpeter 1980; Instituto de Astronom´ıa, Universidad Nacional Aut´onomade M´exico, Ciudad Universitaria, 04510 Mexico City, Mexico;[email protected]
Just & Kegel 1990; Ostriker 1999; Kim & Kim 2007;S´anchez-Salcedo 2009; Namouni 2010).The studies of the gravitational drag in gaseous mediahave enjoyed widespread theoretical application, rangingfrom protoplanets to galaxy clusters. It seems to playa significant role in the growth of planetesimals (Hor-nung, Pellat & Barge 1985; Stewart & Wetherill 1988),the eccentricity excitation of planetary embryo orbits(Ida 1990; Namouni et al. 1996), the orbital decay ofcommon-envelope binary stars (e.g., Taam & Sandquist2000; Nordhaus & Blackman 2006; Ricker & Taam 2008;Maxted et al. 2009; Stahler 2010), the evolution of theorbits of planets around the more massive stars (Villaver& Livio 2009), the evolution of low-mass condensationsin the cores of molecular clouds (Nejad-Asghar 2010), themass segregation of massive stars in young clusters em-bedded in dense molecular cores (Chavarr´ıa et al. 2010),the orbital decay of kpc-sized giant clumps in galaxies athigh redshift (Immeli et al. 2004; Bournaud et al. 2007),or the heating of intracluster gas by supersonic galaxies(El-Zant et al. 2004; Kim et al. 2005; Kim 2007; Conroy& Ostriker 2008). Special work has been devoted to un-derstand the role of gaseous DF in the orbital decay ofstars and supermassive black holes as a result of hydro-dynamic interactions with an accretion flow in galacticnuclei (Narayan 2000). Mergers of supermassive blackhole binaries may be accelerated on sub-parsec scales byangular momentum loss to surrounding gas (Armitage& Natarajan 2005). In particular, gaseous DF expeditesthe growth of SMBH by mergers in colliding galaxies (Es-cala et al. 2004, 2005; Dotti et al. 2006; Mayer et al. 2007;Tanaka & Haiman 2009; Colpi & Dotti 2011).Less developed is the corresponding theory of DF ina magnetized gaseous medium. As far as we know, theanalytic estimate of the gravitational drag for a bodymoving on a rectilinear trajectory parallel to the uniformunperturbed magnetic field lines by Dokuchaev (1964) isthe only work in this area. He concluded that the DF S ´ANCHEZ-SALCEDOforce on a supersonic body is reduced by a factor thatdepends on the ratio between the Alfv´en speed and thesound speed. Since large-scale magnetic fields are ubiq-uitous in many astronomical systems such as molecularclouds (Tamura & Sato 1989; Goodman & Heiles 1994;Matthews & Wilson 2002; Heiles & Crutcher 2005) orgalactic nuclei, it is important to understand how theDF force is affected by the presence of ordered large-scale magnetic fields. In fact, young stellar systems andlow-mass condensations orbiting in the potential of theirbirth clusters can interact with the surrounding denseand magnetized molecular interstellar medium duringthe dispersal of the cluster’s gas. In the Galactic cen-ter, structures associated with ordered magnetic fields,called arches and threads, are detected in radio contin-uum maps (Yusef-Zadeh et al. 1984). The magnetic fieldconfiguration of the Galactic center has been viewed aspoloidal in the diffuse, interstellar (intercloud) mediumand approximately parallel to the Galactic plane only inthe dense molecular clouds (Nishiyama et al. 2010). Onthe scale of 400 pc, fields of 100 µ G have been reported(Chuss et al. 2003; Crocker et al. 2010).The importance of gaseous DF in the evolution andcoalescence of a massive black hole binary is motivatedby both observational and theoretical work that indicatethe presence of large amounts of gas in the central re-gion of merging galaxies. During the merger of galaxies,the inflow of gas material towards the galactic centerdriven by tidal torques associated with bar instabilitiesand shocks will sweep up and amplify the magnetic fieldin the central region (Callegari et al. 2009; Guedes etal. 2011). Observations of gas-rich interacting galaxiessuch as the ultraluminous infrared galaxies (ULIRGs)show that their central regions contain massive and denseclouds of molecular and atomic gas (Sanders & Mirabel1996). ULIRGs are natural locations to expect verystrong magnetic fields (Thompson et al. 2006; Robishawet al. 2008; Thompson et al. 2009).In this paper we will study the DF in a gaseous mediumon a body moving on rectilinear orbit in a homogeneous,uniform magnetized cloud. This is the simplest idealizedextension of the unmagnetized case and is the first step inunderstanding the role of ordered magnetic fields. Previ-ous works have shown that, although the formulae of thegaseous drag force in a unmagnetized gas medium, werederived for rectilinear orbits in homogeneous and infi-nite media (Dokuchaev 1964; Rephaeli & Salpeter 1980;Just & Kegel 1990; Ostriker 1999; S´anchez-Salcedo &Brandenburg 1999; Kim & Kim 2009; Namouni 2010;Lee & Stahler 2011; Cant´o et al. 2011), simple ‘local’extensions have been proven very successful in more re-alistic situations, e.g. smoothly decaying density back-grounds or when the perturber is moving on a circularorbit (S´anchez-Salcedo & Brandenburg 2001; Kim & Kim2007; Kim et al. 2008; Kim 2010). As a useful startingpoint for understanding the role of a large-scale magneticfield, we also consider that the unperturbed medium ishomogeneous and uniformly magnetized. A discussionon the DF in other initial force-free configurations willbe given in a separate paper. Even in the simple caseof a uniformly magnetized medium, the magnetic fieldproduces qualitatively new phenomena.The paper is organized as follows. In §
2, we discuss thebasic concepts on the ideal problem of a particle travel- ing at constant speed through a uniform gas, both inthe purely hydrodynamic case and when the plasma ismagnetized. In §
3, we outline the linear derivation forcalculating the steady-state density wake generated byan extended body moving along the magnetic fields, givean analytical solution of the problem and compare it withprevious work. The time-dependent linear perturbationtheory is presented in § § §
6, we summarize our results anddiscuss their implications. DYNAMICAL FRICTION IN GASEOUS MEDIA: BASICFORMULAE
Unmagnetized medium
Under assumption of a steady state, Dokuchaev (1964),Ruderman & Spiegel (1971) and Rephaeli & Salpeter(1980) derived the drag force on a point mass M mov-ing at velocity V on a straight-line trajectory through auniform medium with unperturbed density ρ and soundspeed c s . For subsonic perturbers ( M ≡ V /c s < M is the Mach number), these authors found thatthe drag force is zero because of the front-back symme-try of the density distribution about the perturber. Forthe steady-state supersonic case, the drag force takes theform F DF = 4 πρ ( GM ) V ln Λ , (1)where Λ = r max /r min , being r max and r min the maximumand minimum radii of the effective gravitational inter-action of a perturber with the gas. For extended per-turbers, r min is its characteristic size, whereas for point-like perturbers r min is of the order of the Bondi-Hoyleradius R BH (Cant´o et al. 2011), as defined in § r max increases with time inthe supersonic case. More specifically, she found that theCoulomb logarithm is given by:ln Λ = 12 ln (cid:18) M − M (cid:19) − M , (2)for M < t > r min / ( c s − V ), andln Λ = 12 ln (cid:0) − M − (cid:1) + ln (cid:18) V tr min (cid:19) , (3)for M > t > r min / ( V − c s ). The perturber isassumed to be formed at t = 0. The transition be-tween the subsonic to the supersonic regime is smoothwithout any divergence at a Mach number of unity (seeFig. 3 in Ostriker 1999). S´anchez-Salcedo & Branden-burg (1999) tested numerically that Ostriker’s formulais very accurate for non-accreting extended perturbers.In many astrophysical situations, one needs to assigna softening radius to the gravitational potential whichin turn determines r min without any ambiguity. For aYNAMICAL FRICTION 3body described with a Plummer model with core radius R soft ≫ R BH , S´anchez-Salcedo & Brandenburg (1999)found that r min ≃ . R soft .For point-mass accretors, the friction force has beenderived by Lee & Stahler (2011) in the subsonic regimeand by Cant´o et al. (2011) in the hypersonic limit. Magnetized medium
The presence of a small-scale magnetic field tangled atscales below r min will change the speed of sound. Forisotropic compression of a random magnetic field, theeffective sound speed is ( c s + c a ) / (e.g., Zweibel 2002),where c a is the Alfv´en speed of the random small-scalecomponent of the magnetic field. Therefore, in order toinclude the effect of a small-scale magnetic field, one hasto replace the sound speed by the effective sound speedin the definition of M in Eqs. (2) and (3).The extension of the drag force formulae is by no meansstraightforward if the gaseous medium is permeated bya regular magnetic field. Dokuchaev (1964) derived thegravitational drag force in the steady state for perturbersmoving along the lines of the unperturbed magnetic field.He found that the DF drag is F DF = (cid:18) − c A V (cid:19) πρ ( GM ) V ln Λ , (4)at V > ( c s + c A ) / , where c A is the Alfv´en speed of theregular magnetic field, and it is zero for V < ( c s + c A ) / .By comparing Eqs. (1) and (4), we see that the drag in auniform magnetized background is never larger than inthe unmagnetized case. According to Dokuchaev (1964),the gravitational drag on a body with velocity V in auniformly magnetized medium is equal to the drag on abody with velocity V / (1 − c A /V ) / in a unmagnetizedmedium. Therefore, if one naively uses the nonmagneticformulae (1)-(3) by replacing the sound speed c s by themagnetosonic speed ( c s + c A ) / would yield to wrongresults. In the next Section, we will show, however, thatthe paper of Dokuchaev (1964) contains an error and itis not true that the drag force in the magnetized mediumcase is always smaller than in the unmagnetized case.As Ostriker (1999) demonstrated in the field-free case,the steady state result found by Dokuchaev that thenet force is zero at V < ( c s + c A ) / , because of thefront-back symmetry of the density perturbation in themedium, may be misleading. It is also unclear how ln Λvaries in time for the magnetized supersonic case. More-over, it left unexplored how the drag force depends onthe angle between the velocity of the perturber and thedirection of the magnetic field. Before addressing thesequestions, however, it is still worthwhile finding analyt-ical solutions for the perturbed steady density and theresulting drag force in the simplest scenario in which thevelocity of the perturber and the magnetic field are par-allel. Such a exact treatment will allow us to gain insightinto more complicated situations. This will be done inthe next Section. AXISYMMETRIC CASE: VELOCITY OF THEPERTURBER PARALLEL TO THE DIRECTION OF THEMAGNETIC FIELD
We consider a gravitational perturber moving on astraight-line at constant velocity in a medium with un- perturbed density ρ and thermal sound speed c s . Inthe absence of magnetic fields, the linearized equationsof motion can be reduced to a nonhomogeneous waveequation for the relative perturbation ( ρ − ρ ) /ρ (e.g.,Ruderman & Spiegel 1971; Ostriker 1999). Once a uni-form magnetic field, B , parallel to the direction of per-turber’s velocity is included, Dokuchaev (1964) showedthat the relative perturbation obeys an equation of fourthorder in t and solved it using a double Fourier-Hankeltransformation. As it will become clear later, we preferto describe the evolution of the system through wave-equations because it facilitates the physical interpreta-tion of the problem and because the contact with theanalysis of Ostriker (1999) is easier. In addition, theextension of the equations for a case where the angle be-tween B and the velocity of the perturber is arbitrary,becomes straightforward in our approach. Perturbed density distribution
We study first the completely steady flow created by amass on a constant-speed trajectory parallel to the linesof the unperturbed magnetic field B = B ˆ z . To do this,consider a particle at the origin of our coordinate system,surrounded by a gas whose velocity far from the particleis V = − V ˆ z , with V >
0. We will further assumethat the gas evolves under flux-freezing conditions. Ouranalysis begins with the linearized MHD equations todescribe the medium’s response to the perturber’s pres-ence ρ = ρ + ρ ′ , V = V + v ′ and B = B + B ′ , in astationary sate ( ∂/∂t = 0) ρ ∇ · v ′ + V · ∇ ρ ′ = 0 , (5)( V · ∇ ) v ′ = − c s ∇ ρ ′ ρ − ∇ Φ + 14 πρ ( ∇ × B ′ ) × B , (6) ∇ × ( V × B ′ ) + ∇ × ( v ′ × B ) = 0 , (7) ∇ · B ′ = 0 , (8)where Φ is the gravitational potential created by the per-turber. The Poisson equation links the potential with thedensity profile of the perturber ρ p : ∇ Φ = 4 πGρ p . (9)The Lorentz force, which provides the magnetic back-reaction on the flow pattern, is given by( ∇ × B ′ ) × B = B (cid:18)(cid:20) ∂B ′ x ∂z − ∂B ′ z ∂x (cid:21) ˆ x − (cid:20) ∂B ′ z ∂y − ∂B ′ y ∂z (cid:21) ˆ y (cid:19) . (10)Hence, the divergence of the Lorentz force is ∇ · [( ∇ × B ′ ) × B ] == B ∂ B ′ x ∂x∂z − ∂ B ′ z ∂x − ∂ B ′ z ∂y + ∂ B ′ y ∂y∂z ! = − B ∇ B ′ z . (11)In the last equality we have used that ∇ · B ′ = 0. Bysubstituting equations (5) and (11) in the divergence of S ´ANCHEZ-SALCEDOthe equation of motion , we have − V ρ ∂ ρ ′ ∂z = − c s ρ ∇ ρ ′ − ∇ Φ − B πρ ∇ B ′ z . (12)By comparing the second and third terms in the right-hand side of the equation above, we see that, formally,the magnetic back-reaction term ∇ B ′ z is mathematicallyequivalent to having an external potential term. How-ever, whilst Φ is known (Eq. 9), B ′ z is coupled to thefluid motions through the flux-freezing equation (7).Next, we need an independent equation for B ′ z to closethe system. This can be accomplished using the thirdcomponent of the induction equation (7), which has theform B (cid:18) ∇ · v ′ − ∂v ′ z ∂z (cid:19) = V ∂B ′ z ∂z . (13)Our strategy is to eliminate v ′ in Equation (13). Thethird component of the equation of motion (6) can bewritten as − V ∂v ′ z ∂z = − c s ρ ∂ρ ′ ∂z − ∂ Φ ∂z . (14)This equation does not depend explicitly on the frozen-inmagnetic field because the z -component of the Lorentzforce vanishes in the linear approximation (see Eq. 10).Substituting Eqs. (5) and (14) in Eq. (13), we obtainthe desired equation B (cid:18) ρ ∂ρ ′ ∂z − ρ M ∂ρ ′ ∂z − V ∂ Φ ∂z (cid:19) = ∂B ′ z ∂z , (15)where we recall that M ≡ V /c s is the (sonic) Mach num-ber. Once again, the magnetic term ∂B ′ z /∂z is formallyidentical to ∂ Φ /∂z , but some caution should be usedwhen interpreting it; the z -component of the Lorentzforce is not ∂B ′ z /∂z but zero.Equations (12) and (15) constitute a system of twocoupled linear differential equations for ρ ′ and B ′ z whichmay be solved once we have chosen suitable bound-ary conditions. Defining the dimensionless perturbations α ≡ ρ ′ /ρ and β ≡ B ′ /B , the equations to solve are: M ∂ α∂z = ∇ α + 1 c s ∇ Φ + Υ ∇ β z , (16)( M − ∂α∂z − c s ∂ Φ ∂z = M ∂β z ∂z , (17)where Υ ≡ c A /c s and c A the Alfv´en speed in the un-perturbed medium. In the limit of vanishing magneticfield, β z = Υ = 0 and Equation (16) reduces to that ofthe wake of a body in a unnmagnetized medium (e.g.,S´anchez-Salcedo 2009).For a point-like perturber of mass M , an analytical so-lution can be derived for the density enhancement, veloc-ity and magnetic fields in the wake. In order to calculate The curl of the equation of motion provides a relationshipbetween the vorticity ω and the current density J ′ : − V ∂ ω ∂z = B cρ ∂ J ′ ∂z . In linear theory, the baroclinic term vanishes and the Lorentz termis the only able to generate vorticity, even if the gravitational forceis irrotational. the dynamical friction force exerted on the body, we onlyneed the gas density enhancement in the wake, which isderived in Appendix A and is given by α ( R, z ) = λ (1 − η ) GMξc s p z + R γ , (18)where R = p x + y is the cylindrical radius and η = ( c A /V ) = (Υ / M ) , (19) ξ = 1 + (1 − M − )Υ = 1 − η + Υ , (20) γ = 1 − M ξ , (21)and λ = M < M crit ;2 if M crit < M < min(1 , Υ) and z/R > | γ | ;1 if min(1 , Υ) < M < max(1 , Υ);2 if M > max(1 , Υ) and z/R < −| γ | ;0 otherwise. (22)Here, the critical Mach number is defined as M crit ≡ (cid:0) − (cid:1) − / . (23)Because of the linear-theory assumption, Equation (18)is properly valid only for ( z + γ R ) / ≫ (1 − η ) GM/ ( ξc s ). The nonmagnetic steady-state solution fordensity in the wake past a gravitating body is recoveredwhen Υ = 0.For clarity, it is convenient to distinguish four intervalsdepending on the value of the Mach number of the body: M < M crit (case or interval I), M crit < M < min(1 , Υ)(case or interval II), min(1 , Υ) < M < max(1 , Υ) (caseIII) and M > max(1 , Υ) (interval IV). γ is a posi-tive number in cases I and III, whereas it is negativein cases II and IV. In the latter cases where γ < α vanishes at some spatial lo-cations. In case IV, for instance, α outside the conedefined by the surface z = −| γ | R is actually zero. Turn-ing to Eq. (15), we see that the magnetic perturbation B ′ z in these regions does not vanish but obeys the fol-lowing relation, ∂B ′ z /∂z = − ( B /V ) ∂ Φ /∂z . Now, fromEq. (14), the axial component of the velocity satisfiesa similar equation, ∂v ′ z /∂z = (1 /V ) ∂ Φ /∂z . Using thefact that ∇ · v ′ = 0 in regions of constant density (seeEq. 5), it is simple to show that B ′ is parallel to v ′ in re-gions where α = 0, and thus the magnetic configurationis force-free in these zones.Once ρ/ρ is known, the gravitational drag can be com-puted; this will be done in Section 3.4. Nevertheless, inorder to gain more insight into the physics of the wake, wewill describe the morphology and structure of the steady-state wake in the next Section. Physical interpretation
Consider first subsonic perturbers. In the limit
M →
0, we have γ → ξ → − Υ / M , (1 − η ) → − Υ / M and λ = 1. Therefore, the density enhancement is GM/ ( c s r ), which is Υ-independent, and correspondsto the linearized solution of the hydrostatic envelope, ρ/ρ = exp (cid:2) GM/ ( c s r ) (cid:3) , around a stationary perturberYNAMICAL FRICTION 5(e.g., Ostriker 1999). In this case, the magnetic field linesremain straight and the whole magnetic configuration isforce-free.The surfaces of constant density for subsonic per-turbers may be either ellipsoids or hyperbolae, depend-ing on the Mach number. At M < M crit it holds that γ > elon-gated along the trajectory of the perturber with eccen-tricity e = M / p | ξ | . This is in sharp contrast to whathappens without any magnetic field where the ellipses areelongated along R for perturbers at any subsonic Machnumber ( γ < z is a clear signatureof curved magnetic fields.In the nonmagnetic subsonic case , v ′ R < z > v ′ z > z , signifying that the incoming fluid is veer-ing towards the perturber, but then turning away againonce it passes the body. The result of v ′ z > v ′ R and v ′ z for a perturber with Mach number inthe interval I.From Eq. (14) and using the result for α in Eq. (18),we find that V ∂v ′ z ∂z = − λ (1 − η ) ξ GM z ( z + R γ ) / + GM z ( z + R ) / , (24)which leads to v ′ z = − GMV (cid:18) z + R ) / − λ (1 − η ) ξ ( z + R γ ) / (cid:19) . (25)It is simple to show that v ′ z > R ∂Rv ′ R ∂R = − ∂v ′ z ∂z + V ∂α∂z = − V M (cid:18) (1 − M ) ∂α∂z + 1 c s ∂ Φ ∂z (cid:19) . (26)Since we already know α and Φ, this equation can besolved to obtain the radial velocity: v ′ R = GM zV R (cid:18) z + R ) / − λ (1 − M )(1 − η ) γ ξ ( z + R γ ) / (cid:19) . (27)From the equation above, it follows that, in case I, v ′ R > R -value at z = 0 and then turning back again. Sincefrozen-in magnetic field lines are dragged by the gas, theupstream magnetic field lines are decompressed radially,resulting in arched magnetic field lines with negative B ′ z -values. In fact, integration of Eq. (15), gives B ′ z = GM B V (cid:18) z + R ) / − λ (1 − M )(1 − η ) ξ ( z + R γ ) / (cid:19) . (28)This clearly states that B ′ z ≤ B > B ′ z arises.At Mach numbers close to M crit , that is, M = M crit − ǫ with ǫ a very small positive number, the density profile, at not extremely large z distances, is α ( R, z ) ≃ Υ / GM/c s √ ǫ (1 + Υ ) / R . (29)Hence, the density enhancement is large and its z -gradient very small.At subsonic Mach numbers in the interval M crit < M < min(1 , Υ), the surfaces of constant density exhibitno front-back symmetry. The isodensity contours corre-spond to hyperbolae in the z − R plane, as occurs forsupersonic perturbers in the absence of magnetic fields,but now the density perturbation is null in the rear Machcone and is non-vanishing in a modified Mach cone lead-ing the perturber. The physical reason is as follows. Thedensity perturbation α is non-positive at any locationbecause the streamlines diverge at the front cone ( η > ξ > α at the edges of the cone is minus infinityfor a point mass. In the front cone, ∂v ′ z /∂z >
0, mean-ing that the flow in that region is being accelerated bythe inward net pressure force. Across the edge of themodified Mach cone, there is a rapid rise in pressure anddensity, and the gas velocity quickly slows. In fact, thecausality criterion used in Appendix A is tantamount toselecting the solution in which a rapid flow is slowed in ashort distance, as occurs in shock waves. Indeed, we willfind numerically further below that the system adoptsthis solution ( § c s < V < c A (if Υ >
1) or in therange c A < V < c s (if Υ < γ lies in the range 0 < γ < z . Remind that in the unmagnetizedbackground, a subsonic perturber also produces ellipti-cal density distributions flattened along z (e.g., Ostriker1999); the latter is a particular case of c A < V < c s .For c s < V < c A (which requires that Υ > α ≤ z -direction by the gravitational force plus thepressure gradient (it holds that ∂v ′ z /∂z > z > α is a consequence of the action of the pressuregradient plus the reduction of the radial convergence ofthe flow due to the presence of the ordered magnetic fieldthat preferentially allows motions along z , which lead toan accelerated flow falling towards the perturber. In theradial direction, the magnetic field lines are compressedat z > v R < z < −| γ | R .In this regime, the surfaces of constant density within thewake correspond to similar hyperbolae in the z − R plane,at the rear of the body, with eccentricity e = M / √ ξ .From Eq. (20), it is simple to show that ξ > M , the angularaperture of the cone is larger in the presence of mag-netic fields. In analogy to the unmagnetized medium,one could define the effective speed of propagation ofthe disturbance as v p = √ ξc s in case IV. At large Machnumbers (say M ≫ √ ξ ), the cone is very narrow ( e ≫ v p ≃ p c s + c A . Hence,at these large M values, the stationary flow is similarto that in the nonmagnetic case but replacing the soundspeed by the magnetosonic speed.Now consider case IV but when the perturber movesat the same velocity as the effective speed of the distur-bance, so that V = v p or, equivalently, M = √ ξ . In thenon-magnetic case, this condition corresponds to ξ = 1and, therefore, the velocity of the body is in resonancewith the sound speed in the medium. One could naivelythink that, at M = √ ξ , the response of the mediumis maximum because of the resonance between V and v p . This is not true for Υ > M = √ ξ implies M = Υ (using Eq. 20), η = 1 (from Eq. 19) and, thereby, α = 0. We learn that a mass moving at the Alfv´en speedin a medium with Υ >
1, does not generate any densitydisturbance in the ambient gas because the velocity fieldof the stationary flow is divergence-free ( ∇ · v ′ = 0). A comparison with Dokuchaev (1964)
As already mentioned, Dokuchaev (1964) calculated,for the first time, the properties of the wake created bya star moving along the field lines, by treating it as alinear perturbation. His analysis started from the time-dependent linearized equations of magnetohydrodynam-ics, including a source term Q in the continuity equa-tion, representing the gas replenishment by the star. Al-though he used the time-dependent equations, he tac-itly assumed that the object’s gravitational field is activesince t = −∞ , so that the wake is in a steady state. Forthe case without mass injected by the star ( Q = 0), thephysical stand points used by Dokuchaev (1964) are ex-actly the same as those adopted in § ρ and the radial component v r of the ve-locity. Through Fourier-Hankel transformations, he wasable to solve the differential equation for ρ . He found asimilar expression for ρ as that given in Eq. (18) but hefailed to separate correctly the different intervals for λ (Eq. 22) and the intervals at which the isocontours areellipsoids or hyperboloids. In particular, he claimed thatthe isocontours are ellipsoids at any Mach number below( c s + c A ) / /c s = (1 + Υ ) / , which is misleading. Gravitational drag force in the axisymmetric case
Once we have the gas density enhancement α ( r ) in theambient medium, we can calculate the gravitational forceexerted on the body by its own wake: F DF = 2 πGM ρ Z Z dz dR R α ( r ) z ( z + R ) / ˆ z . (30)The net drag is zero when the isodensity contours areellipsoids, i.e. when γ >
0, because the wake exhibitsfront-back symmetry. At values γ <
0, however, theregion of perturbed density is confined to a cone and thedrag force is nonvanishing. Evaluating the integrals inspherical coordinates ( R = r sin θ and z = r cos θ ), andusing the variable µ defined as µ = cos θ , the drag forcecan be expressed as: F DF = (1 − η ) 4 πG M ρ V I Z r max r min drr ˆ z , (31) where I = 12 Z − dµ µλ M /ξ p − ξ − M + µ ξ − M . (32)As already said, the drag force is nonzero in cases II andIV, where γ < ξ >
0. In case II, λ = 2for all µ between µ lower = ( M − ξ ) / / M and 1, sothat I = 1. In case IV, λ = 2 for all µ between − µ upper = − ( M − ξ ) / / M and thus I = −
1. Since1 − η < − η ) I = | − η | andthe resultant expression for the force is: F DF = ( −| − η |F ln Λˆ z if M > max(1 , Υ)or M crit < M < min(1 , Υ);0 otherwise (33)where F = 4 πG M ρ V , (34)and ln Λ is the Coulomb logarithm. Note that the den-sity diverges in the wake at Mach numbers close to M crit because ξ →
0. However, the drag force is finite becausethe opening angle of the cone becomes very narrow. Still,the drag force peaks at Mach numbers close to M crit be-cause the factor | − η | /V in the formula for α increaseswhen M decreases.Figure 1 shows the DF force felt by the body at t = 100 r min /c s , as a function of the Mach number andfor different values of Υ. In analogy to the unmagnetizedcase, we take Λ = V t/r min = 100 V /c s . Dokuchaev(1964) claimed that the drag force is nonzero only at M > (1 + Υ ) / (see § M > M crit = 0 .
70. If the ratio between the Alfv´en andsound speeds is of Υ = 2, the DF force is nonvanishingin the intervals 0 . < M <
1, and M > .
41. In fact,there exists always a subsonic velocity range at which thedrag force is nonzero.As long as Υ = 0, the DF force has two local max-ima; one located at M crit and the other one at M =max(1 , √ M crit increaseswith Υ, whereas the drag force at the second local maxi-mum decreases with Υ. As Figure 1 clearly shows, at lowΥ-values, the width of the interval with F DF = 0 around M crit becomes very narrow. For instance, the width ofthat interval is only of 4 × − for Υ = 0 .
2. Hence,the drag force at subsonic values is irrelevant for astro-physical purposes when the Alfv´en speed is sufficientlysmall as compared to the sound speed. For Υ > . M crit is alwayslarger than the drag force at the other local maximum M = max(1 , √ . > . < M < M > TIME-DEPENDENT EQUATIONS
The steady state analysis in the axisymmetric case pre-dicts zero drag force at certain Mach numbers becausethe perturber is surrounded by complete ellipsoids thatexert no net force. As Ostriker (1999) demonstrated inthe field-free case, the time-dependent analysis in whichthe body is dropped suddently at t = 0 allows to captureYNAMICAL FRICTION 7 Fig. 1.—
Gravitational drag force as a function of Mach number, at t = 100 r min /c s , as predicted by the steady-state linear-theory in theaxisymmetric case, for different values of Υ. Fig. 2.—
Color map of the density ρ/ρ (left panel) and magneticfield B ′ z /B (right panel), in the ( R, z )-plane for a case with V = 0and Υ = 1 .
41, in a natural logarithmic scale. the asymmetric density shells in the far field which exerta gravitational drag on the body. Other advantage ofthe time-dependent approach is that, contrary to whathappens when assuming steady-state, the ambiguity inthe definition of the maximum cut-off distance r max isfixed.Without loss of generality, it is convenient to use thegas frame of reference in which the ambient gas is initiallyat rest, the initial magnetic field is along the z -axis andthe body moves with velocity V ,y ˆ y + V ,z ˆ z . The firstorder continuity equation is ∂ρ ′ ∂t + ρ ∇ · v ′ = 0 , (35) the MHD Euler equation ∂ v ′ ∂t = − c s ∇ ρ ′ ρ − ∇ Φ + 14 πρ ( ∇ × B ′ ) × B , (36)and the induction equation: ∂ B ′ ∂t = ∇ × ( v ′ × B ) . (37)The medium initially uniform will respond to the grav-itational pull of the body through the emission of fastand slow Alfv´en waves and sound waves. In the follow-ing we will manipulate the above equations to obtain aclosed system of two differential equations for ρ ′ and B ′ z in analogy to the steady-state case.Using Eq. (11) in the divergence of Equation (36) ∂ ( ∇ · v ′ ) ∂t = − c s ρ ∇ ρ ′ − ∇ Φ − B πρ ∇ B ′ z . (38)By substituting Eq. (35) into Eq. (38), we obtain1 ρ ∂ ρ ′ ∂t = c s ρ ∇ ρ ′ + ∇ Φ + B πρ ∇ B ′ z . (39)In terms of α and β z , it yields ∂ α∂t = c s ∇ α + ∇ Φ + c A ∇ β z . (40)Here, the magnetic effect on the density perturbationappears as a inhomogeneous term. We may recover theclassical non-magnetic equation for α by taking c A = 0.On the other hand, the third component of the induc-tion equation (Eq. 37) implies: ∂B ′ z ∂t = − B (cid:18) ∂v ′ x ∂x + ∂v ′ y ∂y (cid:19) . (41)Equations (35) and (41) give ∂B ′ z ∂t = B (cid:18) ρ ∂ρ ′ ∂t + ∂v ′ z ∂z (cid:19) . (42)From the third component of the equation of motion(Eq. 36) ∂v ′ z ∂t = − c s ρ ∂ρ ′ ∂z − ∂ Φ ∂z , (43) S ´ANCHEZ-SALCEDO Fig. 3.—
Color map of the density ρ/ρ , in the ( R, z )-plane, for the cylindrical case with M = 0 .
75 (upper panels) and for M = 0 . .
41, implying M crit = 0 . M = 0 .
75 falls into theinterval I, while M = 0 . § R = z = 0 is shown in the left panels. The central and right panels display the density in the wake at two snapshots, when the perturberis dropped suddenly at t = 0 at the origin of the coordinate system. The time of the snapshots is given in the lower right hand corner ofeach panel in units of t cross . we know that ∂∂t (cid:18) ∂v ′ z ∂z (cid:19) = − c s ρ ∂ ρ ′ ∂z − ∂ Φ ∂z . (44)Inserting Eq. (44) into the temporal derivative ofEq. (42), one finds1 B ∂ B ′ z ∂t = 1 ρ ∂ ρ ′ ∂t − c s ρ ∂ ρ ′ ∂z − ∂ Φ ∂z . (45)In dimensionless form: ∂ β z ∂t = ∂ α∂t − c s ∂ α∂z − ∂ Φ ∂z . (46) Putting together, the equations (40) and (46) to solvecan be written as ✷ s α = ∇ ˜Φ + Υ ∇ β z , (47) ✷ A β z = Υ − (cid:18) ∂ ∂x + ∂ ∂y (cid:19) ( α + ˜Φ) , (48)where ˜Φ is the gravitational potential in units of c s (i.e. ˜Φ = Φ /c s ) and we have used the Lorentz invariantD’Alembertian ✷ , defined as: ✷ l φ = (cid:18) c l ∂ ∂t − ∇ (cid:19) φ. (49)YNAMICAL FRICTION 9 Fig. 4.—
Same as Fig. 3 but for M = 1 . M = 1 . . The first equation (Eq. 47) governs the evolution of thedensity in the presence of a gravitational potential andmagnetic fields. The second equation (Eq 48) describesthe evolution of a frozen-in magnetic field when the gasis subject to pressure gradients and to an external gravi-tational potential. The inhomogeneous term in Eq. (48)does not have z -derivatives because gas motions in thatdirection does not compress, stir or stretch the back-ground magnetic field. The resulting equations (47) and(48) conform to a set of two coupled non-homogeneouswave equations. For a point-mass perturber, it is sim-ple to find α in the Fourier-Laplace space, ˆ α ( k , ω ), butthe inverse Fourier-Laplace integral cannot be given in aclosed analytic form.In order to gain physical insight, consider first a two-dimensional example. If Φ = Φ( x, y ), that is, if the per- turber is an infinite cylinder with a certain radial den-sity profile ρ p = ρ p ( R ), then β z = α because of the flux-freezing condition, and α satisfies a simple wave equationwith magnetoacoustic speed: (cid:18) c s + c A ∂ ∂t − ∇ (cid:19) α = 11 + Υ ∇ ˜Φ . (50)The physical reason is that motions are always perpen-dicular to the frozen-in magnetic field lines. Magneto-hydrodynamical equilibrium is reached within the mag-netosonic cylinder. At a later stage, Parker instabilitiescan develop (S´anchez-Salcedo & Santill´an 2011).In the purely hydrodynamical problem, the equationgoverning the evolution of α is ✷ s α = ∇ ˜Φ. If the per-turber is a point source, we have ✷ s α = (4 πGM/c s ) δ ( x − Fig. 5.—
Same as Fig. 3 but for M = 1 .
7. Again Υ = 1 .
41. This Mach number lies in the interval IV.
Fig. 6.—
Distributions of the perturbed density α (solid lines) and the z -component of the perturbed magnetic field (dashed lines) alonga cut at R = 0 at two different times, for the same model as that shown in Fig. 5 ( M = 1 . . YNAMICAL FRICTION 11
Fig. 7.—
Temporal evolution of the gravitational drag force in the axisymmetric model, at six different Mach numbers between 0 .
55 and2, for Υ = 1 (left panel), and for Υ = 1 .
41 (right panel). The number on each curve is the Mach number M . V ,x t ) δ ( y − V ,y t ) δ ( z − V ,z t ) H ( t ). Hence, the density re-mains unperturbed outside the causal region for soundwaves (see Ostriker 1999). In a magnetized medium,however, the situation is different because Equation (48)for the perturbed magnetic field β z has a source term( ∂ Φ /∂x + ∂ Φ /∂y ) which does not vanish even out-side the causal region for magnetosonic waves.In the next Section we will solve the coupled wave-equations numerically. To do so, the perturber gravita-tional potential will be modeled by a smooth core Plum-mer potential:Φ( r , t ) = − GM H ( t ) p x + ( y − V ,y t ) + ( z − V ,z t ) + R , (51)where R soft is the softening radius and H is a Heavisidestep function. Stellar and globular clusters can be ac-curately described by Plummer potentials. These typeof models were also used in S´anchez-Salcedo & Branden-burg (1999, 2001), Kim & Kim (2009), and Kim (2010)to study DF. RESULTS
The coupled inhomogeneous wave equations (47) and(48) were solved using a finite difference scheme in a uni-form grid. The scheme is second order in space and thirdorder in time. The temporal algorithm was describedin S´anchez-Salcedo & Brandenburg (2001). Calculationsstart with a uniform background density and magneticfield, and the body is initially placed at the origin of thecoordinate system with velocity V ,y ˆ y + V ,z ˆ z . For theaxisymmetric case, which occurs when V ,y = 0, the cal-culations were carried out on a two-dimensional ( R, z )-plane in cylindrical symmetry. In the general three-dimensional case ( V ,y = 0), the variables α and β z aresymmetric about the plane x = 0. Hence, we considereda finite domain with x ∈ [0 , L x, max ] and used symmet-ric boundary conditions at x = 0, and outflow boundaryconditions in the other five caps of the computational do-main. However, the size of the domain was taken largeenough to ensure that the perturbed density and mag-netic field do not reach the boundaries.As a test of the algorithm, we studied the convergenceof homogeneous wave modes by perturbing a uniformbackground medium. We further tested convergence ofour models for several resolutions and found that fourzones per R soft suffice to have converged results.We take R soft , c s , and t cross = R soft /c s as the units oflength, velocity, and time, respectively. A model canthus be specified with four dimensionless parameters: GM/ ( c s R soft ), M , Υ and Θ ≡ atan( V ,y /V ,z ). Θ is theangle between V and B . Fixed M , Θ and Υ, the vari-ables α and β z depend linearly on GM/ ( c s R soft ). Hence,in our calculations we always take GM/ ( c s R soft ) =0 . / Axisymmetric case
We first run models with the magnetic field terms swichoff and compare the density enhancement and the gravi-tational drag with previous linear calculations in Ostriker(1999) and S´anchez-Salcedo & Brandenburg (1999). Wefound excellent agreement, backing up our numerical model. In the following, we will present results for a bodymoving along the field lines of the unperturbed magneticfield, which corresponds to Θ = 0.The simplest scenario occurs when the gravitationalperturber is dropped at t = 0 at rest ( V = 0). As dis-cussed at the begining of § B ′ z for a case with Υ = 1 .
41. Wesee that the density distribution in the vicinity of thebody is indeed spherically symmetric and the magneticfield takes essentially its initial value, implying that thispart has reached hydrostatic equilibrium. However, thereare two symmetric underdense regions along the z -axis inthe outer parts. Physically, the origin of them is that themagnetic field reduces the flow convergence toward thesymmetry axis because magnetic forces mainly affect theradial component of the velocity v R . This loss of radialconvergence produces a wave with negative density en-hancement ( α <
0) but a positive magnetic enhancement( β z >
0) because of the compression of the magnetic fieldlines in the radial direction.Figure 3 shows snapshots of the density at the (
R, z )plane for Υ = 1 .
41 and two subsonic velocities ( M = 0 . M = 0 . α <
0) at the head of the perturber. As M increasesfrom 0 to 0 .
9, the underdense region at the head of thebody remains and gets deeper, while the underdensity inthe downstream region becomes less pronounced. Thisis simply consequence of the Doppler effect; gradientsbecome steeper upstream (remind that this also occursin the purely hydrodynamic case). By comparing thedensity at two times (the central and the right panels ofFigure 3), we see that the evolution of the density looksself-similar.At M = 0 .
75, the steady-state analysis predicts a nulldrag force because of the front-back symmetry in thewake (see upper left panel of Fig. 3). In the finite-timecase, complete ellipsoids are visible only in the vicinity ofthe body. For instance, at t = 173 . t cross (with Υ = 1 . M = 0 . ≃ R soft from the body’scenter. For comparison, in the absence of magnetic fields(Υ = 0) and at M = 0 .
75, ellipsoids within a radius of43 R soft are complete at t = 173 . t cross . This differenceis a consequence of the coupling between α and β z . At z = 136 R soft and R = 0 (i.e. in the symmetry axis),a steep front separates the fluid into two regions; onewith α > β z <
0, from another with α < β z > M = 0 .
9, a cone of negative density enhancementis located at the head of the perturber (see Fig. 3), asthat predicted in § Fig. 8.—
Gravitational drag force for the time-dependent axisymmetric models against the Mach number at t = 40 t cross (left panel) andat t = 200 t cross (right panel). Symbols correspond to the numerical models. The solid curves plot Ostriker’s formula, while the remaindercurves draw the drag force as predicted by Equation (33), adopting r min = 2 . R soft and r max = V t . detached from the perturber. It is important to remarkthat, according to the analysis in § .
41 occurs at a Machnumber close to 0 . .
41 and twosupersonic Mach numbers: M = 1 . M = 1 . M = 1 .
4, a teneous ellipsoidal under-dense envelop is still visible.The overdensity behind the body has the shape of atulip. This tulip-shaped overdensity also appears at therear of the body at M = 1 . M = 1 . Fig. 9.—
Snapshots of the density map along a cut-off through the ( y, z )-plane at x = 0, in natural logarithmic scale, for Θ = 45 ◦ (upperpanels) and for Θ = 90 ◦ (lower panels), at three different Mach numbers. The Mach number M is indicated at the right corner in eachpanel. The perturber was dropped at t = 0 and moves on a rectilinear orbit in the ( y, z )-plane. In all panels, Υ = 1 . Fig. 10.—
Gravitational DF force as a function of Mach numberfor different Υ-values at t = 40 t cross . The perturber moves per-pendicular to the magnetic field lines (i.e. Θ = 90 ◦ ). The solidcurve plots Ostriker’s formula, which was derived for unmagne-tized media, with r min = 2 . R soft and r max = V t . To make theplot readable, the symbols at M = 0 . , . , .
75 and 2 have beenslightly shifted in the horizontal direction.
Fig. 11.—
Temporal evolution of the gravitational drag force forΥ = 1. The perturber moves in a direction perpendicular to themagnetic field lines. The numbers given at each curve representsthe Mach number. is that it has a negative magnetic enhancement. Thetulip-shaped overdensity is a consequence of our initialconditions and, as expected, it is detached from the body.In order to illustrate the birth of the tulip-shaped wave,Figure 6 shows the density and magnetic perturbationsalong the symmetry axis for M = 1 .
7, at two early times.Initially, β z increases due to the compression of mag-netic field lines (see the panel at t = 2 . t cross ). At thefar edge of the tail, z ≃ − R soft , an underdense regionYNAMICAL FRICTION 15with positive β z appears. Later on (see the profiles at t = 6 . t cross ), the overdensity loses gravitational sup-port and expands behind the body, decreasing the mag-netic field strength, until the magnetic pressure plus themagnetic tension provides sufficient radial confinementto the tulip-shaped structure, allowing it to remain overlong times.In Figure 5 it is simple to identify the modified Machcone dragged by its point by the perturber. At t =64 . t cross , the Mach cone at the rear of the body is well-defined. Clearly, the timescale for the development of theMach cone at the rear for M = 1 . M = 0 . M = 0 . t = 57 . t cross .Figure 7 shows the gravitational DF drag as a func-tion of time for M = 0 . , . , . , . , . M = 0 .
55, the drag force clearly saturatesin ∼ t cross . In the remainder cases the drag force in-creases with time. However, the drag force on a per-turber moving in a medium with Υ = 1 .
41, saturates for M = 0 . , . , .
1, and 1 .
4. This means that the DFforce saturates to a constant value at those Mach num-bers that the steady-state analysis predicts a null dragforce.In Fig. 8 we plot the drag force at t = 40 t cross and t = 200 t cross for different values of Υ, together with thepredicted force with r min = 2 . R soft . We see that forΥ = 0, there is a perfect agreement between the Os-triker formula and the inferred values, confirming theresult that r min = 2 . R soft reported in S´anchez-Salcedo& Brandenburg (1999). The drag force formula given inEq. (34), with r min = 2 . R soft , overestimates the dragforce in a neighbourhood of M crit , where the drag forceas a function of M , becomes very cuspy. As expected, thesteady-state formula is more accurate at long timescales,except when it predicts a zero net force. Roughly speak-ing, we may say that, for the axisymmetric case, the grav-itational drag in a magnetized medium is always smalleror equal as the drag force in the unmagnetized case forsupersonic perturbers ( M > M < Magnetic field perpendicular to the direction ofmotion of the perturber In § π/ π/ π/ π/ π/ M = 0 . π/
2, the perturber is surrounded by a ellipsoid-like envelope but also presents a tail with positive and neg-ative α -values separated by a sharp front (see Fig. 9 forΥ = 1 . y -axis, the plowing up of field lines in-creases the total pressure. At low Mach numbers, say M = 0 .
3, underdense regions are now formed in the di-rection of the ambient field lines, which are lagged behindthe body (remind that regions with negative α appearalong the field lines; see Fig. 2). At M = 0 .
9, even ifthe motion is subsonic and sub-Alfv´enic, a magnetic bowwave with sharp edges and opening angle arctan[ c A /V ]is apparent in Fig. 9. Note that when we say that it issub-Alfv´enic, we only mean c A /V <
1. However, somecaution should be used when interpreting this ratio be-cause the velocities are oriented in different directions.Since the velocity of the perturber is always orthogonalto the ambient direction of propagation of Alfv´en waves,the Alv´en speed in the direction of motion of the per-turber is zero and, thus, the body is always infinitelysuper-Alfv´enic in the direction of motion. The morphol-ogy of the wake is the result of a competition betweenthe gravitational focusing of gas by the perturber andthe drainage of gas along magnetic field lines. We shouldwarn here that the wake is not axisymmetric and thusthe density map in the ( x, y )-plane is different than themap in the ( y, z )-plane.When the perturber travels faster than the magne-tosonic velocity c s (1+Υ ) / , a magnetosonic Mach coneis formed at the rear of the pertuber; the entire perturbeddensity distribution lags the perturber. In the ( y, z )-plane, the perturber creates two magnetic bow waves;the Alfv´enic wave with opening angle arctan[ c A /V ] andthe magnetosonic wave with opening angle arctan[( c A + c s ) / /V ].The gravitational drag force is the result of the con-tribution of all the parcels in the domain and it is notpossible to estimate its value just by comparing the den-sity structure by eye. Figure 10 shows the gravitationaldrag as a function of perturber’s Mach number for dif-ferent values of Υ, together with the gravitational dragin the unmagnetized case using Ostriker’s formula. Allthe points at Mach number larger than 0 . M > . ≤ .
5, the effect of including the magneticfield on the drag force is rather small. Interestingly, atΥ ≥ .
5, the strength of the drag for Mach numbers > (1 . ) / is identical as it is in the unmagnetizedcase.For Υ = 1 .
41, the drag force shows a plateau between M = 0 . . ∼ > . . ≤ M ≤ . . ≤ M ≤ . . ≤ M <
1, the drag forcesreach a steady-state value either the medium is magne-tized or not. However, we know that the unmagnetizeddrag force increases logarithmically in time for supersonicperturbers. This implies that the drag force may be sup-6 S ´ANCHEZ-SALCEDO
Fig. 12.—
Components of the gravitational DF force parallel (left panel) and perpendicular (right panel) to the direction V (at t = 40 t cross ) versus M , for three different values of Υ (Υ = 0 .
71, asterisks; Υ = 1, crosses; Υ = 1 .
41, diamonds). The angle between theperturber velocity and the magnetic field is 45 ◦ . The key to symbols is the same as in Figure 10. Fig. 13.—
Component of the DF force perpendicular to the di-rection of motion as a function of time for Υ = 1 .
41 and Θ = 45 ◦ .The number on each curve is the Mach number. pressed by one order of magnitude at 1 < M < . M < .
75 (againΥ = 1) there is no indication that the drag force satu-rates, at least up to t = 40 t cross , implying that the DFforce in a magnetized medium may be larger by a factorof a few than the drag in the unmagnetized case.In summary, when the magnetic fields are relevant,that is for Υ > .
5, we distinguish three ranges. Athigh Mach numbers [i.e. M > (1 . ) / ], the drag isthe same as in the unmagnetized case. At intermediateMach numbers (1 < M ≤ M − ), the drag is highly sup-pressed. Finally, at low Mach numbers, the drag force isstronger in the MHD case than in a purely hydrodynam-ical medium. Intermediate angle between perturber’s velocity andmagnetic field.
Θ = 45 ◦ The upper panels of Figure 9 exhibit the complex mor-phology of the wake when Θ = π/
4. As it is obvious from these panels, the gravitational drag will have two compo-nents: one parallel to V , which produces the drag andloss of kinetic energy by the perturber, and one compo-nent perpendicular to V , which would change the di-rection of V . Given the symmetry of the problem, bothcomponents lie in the ( y, z )-plane. We will start our dis-cussion by considering the gravitational drag force.For those simulations presented in Figure 12 with Θ = π/
4, the drag force saturates within t = 40 t cross only forthe model with M = 0 . .
71. The maximumof the drag force occurs at Mach numbers near ≃ M − .At intermediate Mach numbers, say at 1 . . .
41, the drag force may decrease by a factor of 2–3 ascompared to the force without magnetic fields. At highMach numbers, M > M − , the drag force is slightlysuppressed as compared to the unmagnetized case, butthis reduction is more modest than for Θ = 0 ◦ . Onceagain, at low Mach numbers ( M < . F perp , will tend toinduce a change in the direction of the velocity (note thatwe force the body to move along a straight line). We willuse the following sign convention for F perp . For an angleΘ in the interval 0 ≤ Θ ≤ π/ F perp > >
0, in our convention. In Figure 12, F perp is shownas a function of Mach number. The magnitude of F perp may be comparable to the drag force. For instance, at M = 1 .
2, the perpendicular force is only a factor of 2smaller for Υ = 1 .
41 and a factor of 3 for Υ = 1. Given acertain supersonic velocity, F perp increases with Υ, while F DF shows the opposite behaviour. For supersonic mo-tions with angle Θ = π/ F perp is always positive andincreases monotonically in time (see Fig. 13). This im-plies that F perp will tend to redirect perturber’s velocityto a higher Θ. For the cases shown in Figure 12, theperpendicular force saturates in the run of the calcula-YNAMICAL FRICTION 17tion ( t = 40 t cross ) only in two cases; for M = 0 . .
71, and for M = 0 .
75 and Υ = 1 .
41 (this lattercase is shown in Fig. 13). In some cases with subsonicMach numbers, the perpendicular component of the forceis initially positive, achieves a maximum and then starsa linear decline up to negative values (see Fig. 13).
Dependence of the drag force on
ΘIn Figure 14 we plot the drag force as a function of Θ,for Υ = 1 and 1 .
41. The dependence of F DF on Θ isnot always monotonic. The strongest variation of F DF with Θ occurs for M = 0 .
9. For this Mach number, thedrag force may decrease by a factor of 2–3 from Θ = 0to Θ = 30 ◦ . For M = 1 .
2, the drag force may change upto a factor of 2 depending on the Θ-value. For M = 0 . .
4, the drag force depends gently on the angle.In many astrophysical scenarios, the perturber will besubject to an external gravitational potential and willdescribe a nonrectilinear orbit. S´anchez-Salcedo & Bran-denburg (2001) numerically treated the orbital decayof a perturber in orbit around a unmagnetized gaseoussphere. They found that the “local approximation”, thatis estimating the drag force at the present location of theperturber as if the medium were homogeneous but takingappropriately the Coulomb logarithm, is very successful.Consider now a perturber on a circular orbit in a magne-tized medium. If the orbit lies in a plane perpendicular tothe magnetic field, the attack angle Θ is always π/
2. Inthe local approximation, the maximum drag for Θ = π/ M ≈
M ≈ . + Ω t . Therefore, if thelocal approximation is valid, one can estimate the meandrag force over a rotation period, which is approximatelyequivalent to take the mean value of F DF over Θ. In par-ticular, for Υ = 1, the Θ-average drag force is maximumat M ≈ .
4. This example illustrates how F DF may de-pend on Υ and on the inclination of the orbit respect tothe magnetic field lines. A more detailed analysis of thedrag force on a body on a circular orbit will be givensomewhere else. SUMMARY AND DISCUSSION
Understanding the nature of the DF force experiencedby a gravitational object that moves against a mass den-sity background is of great importance to describe theevolution of gravitational systems. In this work, we in-vestigated the DF on a body moving in rectilinear tra-jectory through a gaseous medium with a magnetic fielduniform on the scales considered. In linear theory, theproblem is largely characterized by three dimensionlessparameters, M , which is defined as the ratio of the par-ticle velocity to the sound speed of the uniform gas, Υ,defined as the ratio between the Alfv´en and sound speeds,and Θ, the angle between the magnetic field direction andthe particle velocity. We find that magnetic effects mayalter the drag force, especially for Υ > .
5, because themagnetic field affects the flow velocity field in the perpen-dicular direction of the ambient field lines, and therebythe morphology of the wake. Note that the plasma beta,defined as the ratio of gas to magnetic pressure, is 2 / Υ for an isothermal system. Fig. 14.—
Dependence of the drag force on the angle Θ for Υ = 1(upper panel) and Υ = 1 .
41 (lower panel). The numbers given ateach curve represent the Mach number M . The drag force wascomputed at t = 40 t cross . There are two major differences between the magne-tized and unmagnetized case. One conceptual differenceis that, while gravitational focusing in a unmagnetizedmedium always generates a positive density enhance-ment, this is not the case in a magnetized medium (see,e.g., Figs. 2, 3 and 9). A second result is that the peakvalue of the drag force is not near M = 1 for a mass mov-ing in a magnetized medium. In fact, the sharp peak of F DF at M = 1 found in the Υ = 0 case is no longerpresent in a magnetized medium with Υ > .
5. For in-stance, for a perturber in perpendicular motion to thefield lines (Θ = π/
2) in a medium with Υ = 1 .
41, thedrag force is essentially constant from M = 0 . . M = 2 (see Fig. 10).The flat plateau in the drag force between M = 0 . . x and y directions.For a body traveling along the field lines, i.e. Θ = 0,the steady-state problem can be treated analytically. Wefocus first on this case. For Υ = 0, the drag force presentstwo local maxima (see Fig. 1); one is located in the sub-sonic branch (at M crit ) and the other peak value is atthe supersonic branch [at M = max(1 , √ − η ), with η = ( c A /V ) . The physi-cal reason is that the medium becomes more rigid in theradial direction and, hence, the opening aperture of themodified Mach cone is the same as that in a unmagne-tized medium with effective sound speed ( c s + c A ) / ,but the density enhancement is smaller by a factor of(1 − η ). By contrast, the drag force for subsonic veloc-ities is stronger if the medium is uniformly magnetized.For Θ = 0, an underdense region is formed upstreambecause of the gas channeling along the direction of themagnetic field, following the path of less resistance. Thesteady-state theory predicts that the gravitational dragon a body with Θ = 0 vanishes at Mach numbers in8 S ´ANCHEZ-SALCEDOthe following two ranges: (1) at M < M crit and (2)at min(1 , Υ) < M < max(1 , Υ). However, using time-dependent analysis we find that the DF force asymptoti-cally approaches to a nonzero steady-state value at theseMach numbers. For Υ > . M crit (Fig. 8). At Mach numbers around M crit ,the density enhancement is large but negative in a cone infront of the body. At those Mach numbers, the DF maybe even more efficient than in the stellar case. For exam-ple, for a medium with Υ = 1 .
41, the drag force peaksbetween M = 0 . M = 1 .
1. As a consequence of thestronger DF force, subsonic massive objects in a orbitelongated along the magnetic field lines in a constant-density core of a nonsingular gaseous sphere will suffera orbital decay faster if the medium is pervased by alarge-scale magnetic field.We have also explored the Θ-dependence of the DFdrag. For Mach numbers around M crit , the drag forceexhibits the strongest variations with Θ (see Fig. 14). Formagnetized media with Υ ≥ . M > . ≤ M ≤ .
4, the drag force is a factor of 2–3weaker than it is in the absence of magnetic fields . Athigh Mach numbers, M > (1 . ) / , the suppresionof the drag force is more important at small values of Θ(Fig. 14). At these high Mach numbers and for an angleof Θ = π/
2, the drag forces are similar with and withoutmagnetic fields (Fig. 10).As a consequence, supersonic massive objects may make their way more slowly to the center of the sys-tem if the medium is pervased by a large-scale mag-netic field. As a model problem, consider a singularisothermal spherical cloud threaded by a uniform mag-netic field and a small-scale random magnetic field withAlfv´en speed c a everywhere constant. The density profileof the cloud is given by ρ ( r ) = ( c s + c a / / πGr , where c s is the isothermal sound speed. The circular speedis V = p c s + c a . Since the effective sound speed is p γc s + 2 c a /
3, the Mach number of a body on a quasi-circular orbit is M = (cid:18) c s + c a γc s + c a (cid:19) / , (52)which varies from 1 . . c a and whether the perturbations are isothermal or adi-abatic . If the uniform magnetic field component has aΥ-value between 1 and 1 .
41, the time for the perturber’sorbit to decay will be a factor of 2–5 larger than thecorresponding decay time for Υ = 0. Our results demon-strate that, in the presence of ordered magnetic fieldswith Υ > .
7, the role of the magnetic field on the dragforce should be taken into account to have accurate esti-mates of the timescales of orbital decay via gravitationalDF.The author would like to thank Miguel Alcubierre andJuan Carlos Degollado for interesting discussions. Con-structive comments by an anomymous referee are greatlyappreciated. This work has been partly supported byCONACyT project 60526.
APPENDIX
A. FOURIER TRANSFORMATION: AXISYMMETRIC CASE
The three-dimensional Fourier transform of a perturbed variable f ( r ) is given byˆ f ( k ) = 1(2 π ) / Z R f ( r ) e − i k · r d r . (A1)In the Fourier space, Equations (16) and (17) are transformed into: M k z ˆ α = k ˆ α − πGc s ˆ ρ p + Υ k ˆ β z , (A2) i ( M − k z ˆ α = ic s k z ˆΦ + i M k z ˆ β z , (A3)where ρ p is the mass density of the perturber, thus ∇ Φ = 4 πGρ p . In order to have an equation for ˆ α , we will eliminateˆ β z . From Eq. (A3), we have ˆ β z = 1 M " ( M − α − ˆΦ c s , (A4)and substituting into Eq. (A2) we find (cid:2) M k z − k (cid:0) − M − )Υ (cid:1)(cid:3) ˆ α = 4 πGc s (cid:18) Υ M − (cid:19) ˆ ρ p , . (A5) This factor may be larger at later times because the magnetizeddrag force saturates, whereas it increases logarithmically in timein the unmagnetized case. See Figure 8 for an evolved stage. In the nonmagnetic simulations of the orbital decay of a single black hole due to gaseous DF in Escala et al. (2004), the velocity ofthe black hole is initially supersonic ( M = 1 .
4) and remains barelysupersonic through most of the simulation.
YNAMICAL FRICTION 19In the absence of magnetic fields (Υ = 0), the above equation reduces to( M k z − k )ˆ α = − πGc s ˆ ρ p , (A6)and the standard steady-state equation for the wake past a gravitating body is recovered. At velocities much largerthan the Alfv´en speed, M ≫ c A /c s = Υ, Equation (A5) is simplified to h(cid:0) (cid:1) − M k z − k i ˆ α = − πG (1 + Υ ) c s ˆ ρ p . (A7)By comparing the above equation with Equation (A6), we see that the response of the medium in this case is indis-tinguishable to that of an unmagnetized medium with sound speed ( c s + c A ) / = c s (1 + Υ ) / .It is interesting to note that when V = c A = 0, the right-hand-side of Equation (A5) vanishes and thereby thesolution is α = 0, implying that the steady-state configuration satisfies ∇ · v ′ = 0. Obviously, the drag force is exactlyzero in this configuration.There exist two situations where the differential equation (A5) is not well-posed: (1) at M = 1 and (2) at the criticalMach number, M crit , satisfying that 1 + (1 − M − )Υ = 0 . (A8)So that M crit ≡ (cid:0) − (cid:1) − / . (A9)It is clear that M crit <
1. If the dynamics is dominated by the magnetic field, i.e. when Υ ≫
1, then M crit →
1. Ifnot specified, we will consider
M 6 = 1 and
M 6 = M crit throughout this section.We will now calculate the solution of Eq. (A5) when the perturber is a point mass M , so that ˆ ρ p = M/ (2 π ) / , whichcorresponds to the Fourier transformation of ρ p ( r ) = M δ ( r ), to obtain the Green’s function. Using the convolutiontheorem, it is possible to evaluate α for any general distribution ρ p . Hence, we solve for α ( r ) = (1 − η ) GM π c s Z ξk − M k z e i k · r d k , (A10)where η = (cid:18) Υ M (cid:19) = (cid:18) c A V (cid:19) , (A11)and ξ = 1 + (1 − M − )Υ . (A12)The integral (A10) along k z is evaluated by transforming to the complex plane. It is convenient to define γ ≡ −M /ξ .Either if M stands in the range 0 < M < M crit or in the range min (1 , Υ) < M < max (1 , Υ), then γ > α ( r ) = (1 − η ) GMξc s p z + R γ . (A13)For Mach numbers in any of the two ranges: M crit < M < min (1 , Υ) and M > max (1 , Υ), the integrand haspoles on the real axis. Hence we make ‘indentations’ in the contour at the position of the poles. We consider firstMach numbers larger than max(1 , Υ). Then, for z >
0, we close the contour at + i ∞ , leaving the poles outside thecontour to preserve causality, whereas for z <
0, we consider a domain containing the lower half-plane, that is whereIm( k z ) <
0, and the contour slightly above the real axis, so that the two poles lie inside the contour. More specifically,for z <
0, the integration over k z can be evaluated as Z ∞−∞ e ik z z k − M ξ − k z dk z = Z ∞−∞ e ik z z k x + k y − γ k z dk z = − πγ k R sin (cid:18) k R zγ (cid:19) , (A14)where γ ≡ − γ = ξ − M − k R = k x + k y . The integration over k x and k y can be carried out in polar coordinates( k x = k R cos φ and k y = k R sin φ ): − πγ Z ∞ Z π sin (cid:18) k R zγ (cid:19) e ik R cos φ dφ dk R = − π γ Z ∞ sin (cid:18) k R zγ (cid:19) J ( k R R ) dk R = ( π √ z − γ R if z < − γ R ;0 otherwise. (A15)0 S ´ANCHEZ-SALCEDOAt Mach numbers in the interval M crit < M < min(1 , Υ), causality is perserved at z > α ( r ) = λ (1 − η ) GMξc s p z + R γ , (A16)where λ = M > max(1 , Υ) and z/R < −| γ | ;2 if M crit < M < min(1 , Υ) and z/R > | γ | ;1 if M < M crit or if min(1 , Υ) < M < max(1 , Υ);0 otherwise.This result can be compared with that in Dokuchaev (1964) by noting that he used the variable A to denote thecombination (1 − η ) /ξ . Dokuchaev (1964) found the same functional form for α , but failed to divide correctly the casesaccording to the Mach number. In the absence of a background magnetic field, which corresponds to ξ = 1 and η = 0,we recover the classical form derived by previous authors. B. FOURIER TRANSFORMATION: PERTURBER’S VELOCITY PERPENDICULAR TO THE MAGNETIC FIELD
We consider a gravitational perturber moving at constant velocity v ˆ y in an unperturbed medium with density ρ ,sound speed c s and a magnetic field B = B ˆ z . Therefore, the velocity of the perturber and the magnetic field areperpendicular. We are interested in the stationary wake formed behind the body. To do this, we assume that perturberis at rest and feels a wind with velocity − V ˆ y at infinity. Following the same approach as in Appendix A, the twocoupled governing equations in this geometry are given by M ∂ α∂y = ∇ α + 1 c s ∇ Φ + Υ ∇ β z , (B1) M ∂ α∂y = ∂ α∂z + 1 c s ∂ Φ ∂z + M ∂ β z ∂y . (B2)In the Fourier space, these equations are transformed into − M k y ˆ α = − k ˆ α + 4 πGc s ˆ ρ p − Υ k ˆ β z , (B3) M k y ˆ α = k z ˆ α + 1 c s k z ˆΦ + M k y ˆ β z . (B4)In order to have an equation for ˆ α , we eliminate ˆ β z usingˆ β z = 1Υ k (cid:20) πGc s ˆ ρ p + ( M k y − k )ˆ α (cid:21) , (B5)and we obtain the solution of the density disturbance in Fourier space: (cid:18) (1 + Υ ) k y k − M k y − Υ M k z k (cid:19) ˆ α =4 πGc s (cid:18) k y − Υ M k z (cid:19) ˆ ρ p . (B6)Of course, the structure of α in this case is different than in the axisymmetric case (Eq. A5). The inverse Fouriertransform cannot be derived analytically but since the above equation is well-posed for any M -value if Υ = 0, weexpect the drag force to be a continuous function of M . REFERENCESArmitage, P. J., & Natarajan, P. 2005, ApJ, 634, 921Bondi, H., & Hoyle, F. 1944, MNRAS, 104, 273Bournaud, F., Elmegreen, B. G., & Elmegreen, D. M. 2007, ApJ,670, 237Callegari, S., Mayer, L., Kazantzidis, S., Colpi, M., Governato,F., Quinn, T., & Wadsley, J. 2009, ApJ, 696, L89Cant´o, J., Raga, A. C., Esquivel, A., & S´anchez-Salcedo, F. J.2011, MNRAS, arXiv:1108.3032Chandrasekhar, S. 1943, ApJ, 97, 255Chavarr´ıa, L., Mardones, D., Garay, G., Escala, A., Bronfman, L.,& Lizano, S. 2010, ApJ, 710, 583 Chuss, D. T., Davidson, J. A., Dotson, J. L., Dowell, C. D.,Hildebrand, R. H., Novak, G., & Vaillancourt, J. E. 2003, ApJ,599, 1116Colpi, M., & Dotti, M. 2011, Advanced Science Letters, 4, 181Conroy, C., & Ostriker, J. P. 2008, ApJ, 681, 151Crocker, R. M., Jones, D. I., Melia, F., Ott, J., & Protheroe, R.J. 2010, Nature, 463, 65Dokuchaev, V. P. 1964, Soviet Astron., 8, 23Dotti, M., Colpi, M., & Haardt, F. 2006, MNRAS, 367, 103El-Zant, A. A., Kim, W.-T., & Kamionkowski, M. 2004, MNRAS,354, 169