We calculate the change in density within a uniform distribution of field
stars (point masses) caused by a single massive body passing through with
a constant velocity. Starting with the simplest case in which the field stars
are initially stationary this leads to an infinite density wake behind the body.
Introducing a small thermalisation within the field stars removes this infinity
whilst leading to similar results off the path of the massive body. Results are
in good agreement with those previously derived. An approximation can be
made for the density in the thermalised case and this can be used to deduce
the force exerted on the massive body due to the drag caused by the accretion
wake

Heggie, Douglas C.

Sweatman, Winston L.

English

Massey Research Online

Res. Lett. Inf. Math. Sci., 2004, Vol.6, pp123-134Available online at http://iims.massey.ac.nz/research/letters/123The density and drag of the accretionwake of a massive body moving througha uniform stellar distributionWinston L. Sweatman1 & Douglas C. Heggie21Institute of Information and Mathematical SciencesMassey University at Albany, Auckland, New Zealandw.sweatman@massey.ac.nz2School of Mathematics, University of Edinburgh,King’s Buildings, Edinburgh, U.K.douglas@maths.ed.ac.ukWe calculate the change in density within a uniform distribution of fieldstars (point masses) caused by a single massive body passing through witha constant velocity. Starting with the simplest case in which the field starsare initially stationary this leads to an infinite density wake behind the body.Introducing a small thermalisation within the field stars removes this infinitywhilst leading to similar results off the path of the massive body. Results arein good agreement with those previously derived. An approximation can bemade for the density in the thermalised case and this can be used to deducethe force exerted on the massive body due to the drag caused by the accretionwake.1 IntroductionWe consider a massive body passing at velocity though a background field of stars.The gravitational attraction of the massive body perturbs the field stars leading toa change in the density of these stars about the body. In particular the field starstend to concentrate towards the axis behind the massive body leading to a wake ofhigher density behind the body. The gravitational interaction effectively leads toa drag on the massive body slowing it down. This drag may also be modelled bycalculating the force backwards due to the enhanced density in the wake.We shall consider simplified models of the process. We assume that prior tothe influence of the massive body, the field stars are distributed uniformly across124 Winston L. Sweatman & Douglas C. Heggiespace. We assume that the single massive body is moving at a constant velocityand that the wake has already been established so that the density of field starsabout and relative to this body is unchanging. For this to be realistic we requirethe velocity of the massive body to be large in comparison with the relative motionof the field stars to one another. By making use of this assumption we can computeapproximate expressions for the density wake and drag.We calculate the increased density directly in the cases when the field stars areinitially stationary and when they have a small initial thermalisation. For the lattercase, we also compute the drag exerted by the wake on the massive body.2 Wake density for a background of stationary field starsWe consider a single massive body mass M moving with speed V in a positive di-rection along the x-axis through a background field of stars which have unperturbeddensity ρ0. There is symmetry about the x-axis. Field stars are perturbed creat-ing an accretion wake. We ignore gravitational interaction between field stars. Wecalculate the density of the wake ρ that will occur if the massive body continuesmoving with the same velocity (we neglect any slow down due to drag).In this section we consider the case where the field stars are initially stationary.We consider the case where they have a non-zero velocity dispersion in the nextsection.The problem is equivalent to that of a stationary body bombarded by a cloud ofstars. We adopt the rest frame of the massive body (which we locate at the origin)for our ensuing calculations.2.1 The trajectories of the field starsIn the rest frame of the massive body the field stars have an initial mean velocity ofmagnitude V towards the negative x-axis. Under gravitational attraction each fieldstar follows a simple hyperbolic orbit about the massive body (at the origin).Consider a field star of impact parameter p at x =∞. Using its polar coordinates(r, θ), with θ measured relative to the positive x-axis,a(e2 − 1) = r(1 + e cos(θ − θ0)) = r + e(x cos θ0 − y sin θ0),where e is eccentricity, a semi-major axis, and θ0 is the angle at which pericentreoccurs.As r →∞ we have x→∞, y → p and θ → 0 so that we deduce that(1 + e cos(θ − θ0)) = 0andep sin θ0 = a(e2 − 1).Properties of a body moving through a uniform stellar distribution 125Figure 1: Creation of a wake behind a fast-moving body (after Heggie & Hut(2003)(6)).Eliminating e and θ0 we obtainr =p2a(1− cos θ) + p sin θ (1)so thatp =12r sin θ ±√14r2 sin2 θ + ra(1− cos θ).Note that two orbits with differing p pass through each off axis point (Figure 1). Wemust add together contributions from both when computing the perturbed density.2.2 Balancing the mass fluxThe mass flux of field stars at x = ∞ in the annulus of impact parameters from pto p + dp is 2piρ0V pdp . Taking ρ1 to be the density of this specific mass stream,the corresponding mass flux at (r, θ) is2piρ1vr sin θdn = 2piρ1vr sin θ cosφdrwhere v is the field star’s speed, dn is the width of the flux along the normal directionto its flow, and φ is the angle which the normal makes with the radial vector r. Themass flux is conserved and hence we deduceρ1ρ0=V | p |vr sin θ cosφ | ∂r∂p| .Now tanφ = 1rdrdθand v =√V 2 + 2GMr= V√1 + 2ar(as the energy of the fieldstar is E = GM2a= 12V 2). So substituting for cosφ and vρ1ρ0=| p |√1 + (1r∂r∂θ)2r sin θ√1 + 2ar| ∂r∂p|.126 Winston L. Sweatman & Douglas C. HeggieFrom our expression (1) for r we derive1r∂r∂θ= − a sin θ + p cos θa(1− cos θ) + p sin θ∂r∂p=2p− r sin θa(1− cos θ) + p sin θand with some further work this leads toρ1ρ0=p2r | sin θ || 2p− r sin θ |=p2r | sin θ |√r2 sin2 θ + 4ra(1− cos θ) . (2)The first of these expressions is given in Bisnovatyi-Kogan et al(1979) (3) (equation(1.1)), however, these authors do not sum together the terms from both of thedifferent streams, with different values of p, that reach the point (r, θ). We do thisnow to obtain the overall density enhancementρρ0=r2 sin2 θ + 2ra(1− cos θ)r sin θ√r2 sin2 θ + 4ra(1− cos θ)=y2 + 2a(r − x)| y |√y2 + 4a(r − x) . (3)An equivalent expression is derived by a different approach in Danby & Camm(1957) (4) (equation 3).Substituting a = GMV 2and using sin2 θ1−cos θ = 1 + cos θρρ0=(1 + V2r(1+cos θ)2GM)√(1 + V2r(1+cos θ)2GM)2− 1=1√1−(1 + V2r(1+cos θ)2GM)−2 . (4)We observe that ρρ0is infinite on the negative x-axis (θ = pi) but is finite elsewhere.For θ 6= pi, we haveρρ0→ 1 as r →∞,so there is no effect at infinite distance as is to be expected. Also for θ 6= pi, we haveρρ0∼ 1V√GMr(1 + cos θ)=1V cos( θ2)√GM2ras r → 0.Properties of a body moving through a uniform stellar distribution 127−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5−2.5−2−1.5−1−0.500.511.522.5xyFigure 2: Density contours in the wake behind a fast-moving body (after Heggie &Hut(2003) (6)).2.3 The contours of constant densityBy rearranging (3) we obtain(ρρ0)2= 1 +4a2(r − x)2y2(y2 + 4a(r − x)) .We may solve this expression as a quadratic in 2a(r−x)y2and then eliminating r yields:κ2ρy2 + 2κρx = 1whereκρ =12a( ρρ0)2− 1 +(ρρ0)√(ρρ0)2− 1 .On the curves of constant density κρ is constant too. Therefore the contours ofconstant density are parabolae (Figure 2).3 Wake density for thermalised field starsWe now consider a more realistic situation with thermalised field stars with aMaxwellian distribution of velocitiesf =(2piσ2)−3/2e−(u2+v2+w2)/(2σ2).128 Winston L. Sweatman & Douglas C. HeggiezxyPQRψφαpi−θFigure 3: The spherical geometry of the wake.Figure 3 illustrates the view from downstream with a sphere of radius r overlaid.As found in the previous section, without thermalisation an infinite density wakeoccurs on the radius through P . For a cohort of field stars with (thermalised)velocity −W = (u − V, v, w) this wake moves to the radius through Q. We shallcompute the density at R where without loss of generality PR lies in the xz-plane.The density enhancement is (cf. non-thermalised result (4))ρρ0=∫f(u, v, w)dudvdw√1−(1 + W2r(1−cosψ)2GM)−2 . (5)Now cosφ = V−uW, tanα = vwand furthermore ψ, φ, pi − θ and α are related bycosψ = − cos θ cosφ+ sin θ sinφ cosαand soW 2 (1− cosψ) =W 2(1 +(V − u)Wcos θ − wWsin θ). (6)Henceforth, we will suppose that the massive body has a large velocity relativeto that of the field stars (V >> σ). As f is small for u, v or w >> σ, we treat theintegrand’s denominator in (5) for u, v, w << V .3.1 Well away from the negative x-axisWell away from the negative x-axis (θ = pi) the dominant term in (6) is constantW 2 (1− cosψ) ≈ V 2 (1 + cos θ) .Properties of a body moving through a uniform stellar distribution 129and (5) reduces to the non-thermalised caseρρ0=∫f(u, v, w)du dv dw√1− (1 + 12rV 2GM(1 + cos θ))−2=1√1−(1 + V2r(1+cos θ)2GM)−2 .3.2 On the negative x-axisOn the negative x-axis expression (6) is dominated byW 2 (1− cosψ) ≈ 12(v2 + w2)and so (5) becomesρρ0=1(2pi σ2)32∫e−12u2+v2+w2σ2 du dv dw√1− (1 + 14rGM(v2 + w2))−2 .Integrating with respect to u and introducing polar coordinatesρρ0=12pi σ2∫ ∞0∫ 2pi0e−n22σ2 ndαdn√1− (1 + 14rGMn2)−2(n =√v2 + w2, α = arctan vw). A further integration with respect to α and by partsleads toρρ0=16G2M2r2 σ4∫ ∞0e−4GMqrσ2√q(1 + q)dq = zezK1(z)where q = 18rn2GM, z = 2GMrσ2and K1 is the modified Bessel function. (We have madeuse of standard integrals and expressions (cf. Abramovitz & Stegun(1965) (1).)This expression agrees with one obtained by Danby & Camm(1957) (4). In contrastto the non-thermalised problem the density enhancement is finite except at r = 0.As we go to an infinite distance away from the massive body r → ∞ (z → 0),K1(z) ∼ 12Γ(1)(z2)−1= z−1 and soρρ0→ 1 as r →∞which is the same as off the axis. As r → 0 (z →∞), K1(z) ∼√pi2ze−z and soρρ0∼ 1σ√piGM2ras r → 0which is only a minor singularity at r = 0 and similar in order to those for θ 6= pi.130 Winston L. Sweatman & Douglas C. Heggie3.3 The general caseClose to but not on the negative x-axis wWsin θ is of a similar magnitude to the sumof the two previously dominant terms of (6) considered in the last two subsections.A hybrid approximation valid everywhere isW 2 (1− cosψ) ≈ V 2(12(wV−√2(1 + cos θ))2+v22V 2).Substitution into the expression (5) for ρρ0gives1(2pi σ2)32∫e−12u2+v2+w2σ2 du dv dw√1−(1 + 12rV 2GM(12(−√2 + 2 cos θ + wV)2+ 12v2V 2))−2 (7)The integral with respect to u can be done immediately, and we may use scaledpolar coordinates (n =√(v2 + w2)/2V 2, α = arctan vw) to obtainV 2piσ2∫ ∞0∫ pi0(1 + 12rV 2GM(1 + cos θ − 2√1 + cos θ n cosα+ n2)) e−V 2n2σ2 ndαdn√(1 + 12rV 2GM(1 + cos θ − 2√1 + cos θ n cosα+ n2))2 − 1 . (8)For θ 6= pi set n2 = (1 + cos θ)p, κ = V 2(1+cos θ)σ2and k = 12rV 2GM(1 + cos θ) to obtainV 2(1 + cos θ)2pi σ2∫ ∞0∫ pi0(1 + k(1− 2√p cosα+ p)) e−κpdαdp√(1 + k(1− 2√p cosα+ p))2 − 1An alternative approach is to set w = ω + V√2 + 2 cos θ in (7), use another setof polar coordinates (n′, α′) and integrate with respect to α′ to obtainρρ0=e−V 2(1+cos θ)σ22pi σ2∫ ∞0e−12n′2σ2 I0(n′V√1 + cos θ/(√2σ2))n′ dn′√1− (1 + 14rn′2GM)−2 .where I0 is the modified Bessel function of zeroth order. Other general integralexpressions for this density have been given by Danby & Bray(1967) (5).4 Computing the drag due to the accretion wakeAs in the previous section we consider the case where the field star velocity distribu-tion is Maxwellian but the dispersion (σ) is much smaller than the relative velocityof the massive body (V ). Clearly the drag is infinite as well as the wake density forthe non-thermalised case.Properties of a body moving through a uniform stellar distribution 131To derive the drag we take the general expression for ρρ0(8), multiply byGMρ0 cos θ/r2and integrate over space r(r, θ, φ). The spatial integral can be trivially integratedover φ and we obtain2GMρ0V2σ2∫ ∞0∫ pi0ne−V 2n2σ2∫ ∞0∫ pi0[(1 + rV22GM(1 + cos θ − 2√1 + cos θ n cosα+ n2)) cos θ sin θdθdr√(1 + rV22GM(1 + cos θ − 2√1 + cos θ n cosα+ n2))2 − 1 ]dαdn.Consider the inner part of this expression∫ ∞0∫ pi0(1 + 12rV 2GM(1 + cos θ − 2√1 + cos θ n cosα+ n2)) cos θ sin θdθdr√(1 + 12rV 2GM(1 + cos θ − 2√1 + cos θ n cosα+ n2))2 − 1 .We may begin with the integration with respect to r, however, care is required asthere are terms which collectively cancel but individually blow up. (This is to beexpected as in any direction towards an infinite radius the density is uniform andnon-zero.) The inner expression becomes2GMV 2∫ pi0√(1+ rV22GM (1+cos θ−2√1+cos θ n cosα+n2))2−1(1+cos θ−2√1+cos θ n cosα+n2)cos θ sin θdθr=∞r=0.Observe that as r →∞√(1 +rV 22GM(1 + cos θ − 2√1 + cos θ n cosα+ n2))2− 1∼√V 22GM(1 + cos θ − 2√1 + cos θ n cosα+ n2)r + 1 +O(r−1)and as r → 0 √(1 +12rV 2GM(1 + cos θ − 2√1 + cos θ n cosα+ n2))2− 1∼√V 2GM(1 + cos θ − 2√1 + cos θ n cosα+ n2)√r +O(r3/2)→ 0So making use of this in our inner integral we have[2GMV 2∫ pi012√V 2GM (1+cos θ−2√1+cos θ n cosα+n2) r+1(1+cos θ−2√1+cos θ n cosα+n2)cos θ sin θdθ]r=∞.132 Winston L. Sweatman & Douglas C. HeggieWithin the integral with respect to θ the term of order r vanishes leaving2GMV 2∫ pi0cos θ sin θdθ(1 + cos θ − 2√1 + cos θ n cosα+ n2)=4GMV 2∫ √20(s2 − 1)sds(s2 − 2sn cosα+ n2)where we have substituted s =√1 + cos θ (note that s > 0).Reversing the integration order of α and θ in our overall expression for drag andfor brevity letting k = V2σ2, we obtain8G2M2ρ0σ2∫ ∞0ne− kn2∫ √20∫ pi0dα(s2 − 2sn cosα+ n2)(s2 − 1)sdsdn=8G2M2piρ0σ2∫ ∞0ne− kn2∫ √20pi(s2 − 1)s|s2 − n2| dsdn=4G2M2piρ0σ2∫ ∞0ne− kn2[signum(√2− n)(2 + (n2 − 1) ln |2− n2|) + (n2 − 1) lnn2 ]dn.For further simplification we subsitute n2 = p and obtain2G2M2piρ0σ2∫ ∞0e−kp[signum(2− p)(2 + (p− 1) ln |2− p|) + (p− 1) ln p]dp=2G2M2piρ0σ2[∫ 20e−kp(2 + (p− 1) ln |2− p|)dp−∫ ∞2e−kp(2 + (p− 1) ln |2− p|)dp+∫ ∞0e−kp(p− 1) ln pdp].The three integrals on the right-hand side may be calculated in terms of the standardexponential integralsEi(x) = PV∫ x−∞ettdt and E1 (x) = PV∫ ∞xe−ttdtand Euler’s constant γ ' 0.577215665. In the first integral by setting q = 2− p wefind ∫ 20e−kp(2 + (p− 1) ln |2− p|)dp= e−2k∫ 20ekq(2 + (1− q) ln q)dq=e−2kk2[(1 + (1− q)k)ekq ln q + (1 + 2k)ekq − (1 + k)Ei(kq)]20=1k2[(1− k) ln 2 + (1 + 2k) + e−2k((1 + k)(γ + ln k − Ei(2k))− (1 + 2k))].Properties of a body moving through a uniform stellar distribution 133In the second integral set q = p− 2:−∫ ∞2e−kp(2 + (p− 1) ln |2− p|)dp= − e−2k∫ ∞0e−kq(2 + (1 + q) ln q)dq=e−2kk2[(1 + (1 + q)k)e−kq ln q + (1 + 2k)e−kq + (1 + k)E1 (kq)]∞0= − e−2kk2[(1 + 2k) + (1 + k)(−γ − ln(k))] .The third integral can be calculated immediately,∫ ∞0e−kp(p− 1) ln pdp=1k2[(−1 + (1− q)k)e−kq ln q − e−kq − (1− k)E1 (kq)]∞0=1k2[1 + (1− k)(−γ − ln(k))] .Adding these terms the expression for drag becomes2piG2M2ρ0σ21k2[(1− k)(ln 2− γ − ln(k)) + 2(1 + k)+ e−2k((2(γ + ln(k))− Ei(2k))(1 + k)− 2(1 + 2k))]=2piG2M2ρ0V 2[(− ln(2) + ln(k) + γ + 2)+32+ ln(2)− ln(k)− γk− 34k−2 − 12k−3 +O(k−4)].We recall that k = V2σ2and by assumption k is large (higher order terms wereneglected in our earlier approximations for the density). Hence it is as well that withcomparatively small values of k, the first term in the expression for drag is dominant.For example with the massive body’s velocity ten times the velocity dispersion ofthe field stars (V = 10σ), the higher order terms make a difference of under half apercent.Removing the higher order terms our final expression for the drag force exertedby the density enhancement is2piG2M2ρ0V 2[− ln(2) + ln(k) + γ + 2] = 4piG2M2ρ0V 2ln(Vσe1+γ/2√2)The dependency of the drag upon the field star velocity dispersion σ is slightas would be expected. The direct proportionality to initial field density ρ0 is also134 Winston L. Sweatman & Douglas C. Heggienot surprising. The drag scales approximately with the square of the ratio of themassive body’s mass divided by its velocity, the exponential term providing a slightcorrection. The factor e1+γ/2√2' 2.565 is negligable for large ratios Vσ.This result is of the same order as previous expressions derived for the drag byalternative methods such as the Chandrasekhar dynamical friction formula (equa-tion 7-17 in Binney & Tremaine(1987) (2)) and the similar expression in Binney& Tremaine’s problem 7-11 although their expression’s dependency upon velocitydispersion differs slightly from ours.References[1] Abramowitz, M., and Stegun, I.A.(1965), Handbook of Mathematical Functions,Dover Publications, New York[2] Binney, J., and Tremaine, S. (1987), Galactic Dynamics, Princeton UniversityPress, Princeton[3] Bisnovatyi-Kogan, G.S., Kazhdan, Ya.M., Klypin, A.A., Lutskii, A.E., andShakura, N.I.(1979), ”Accretion onto a rapidly moving gravitational center”,Sov.Astron.23, 201[4] Danby, J.M.A., and Camm, G.L.(1957), ”Statistical Dynamics and Accretion”,Monthly Notices Roy. Astron. Soc. 117, 50[5] Danby, J.M.A., and Bray, T.A.(1967), ”Density of Interstellar Matter near aStar”, Astron.J. 72, 219[6] Heggie D.C., and Hut P.(2003), The Gravitational Million-Body Problem, Cam-bridge University Press, Cambridge

The density and drag of the accretion wake of a massive body moving through a uniform stellar distribution

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The density and drag of the accretion wake of a massive body moving through a uniform stellar distribution

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