New Astronomy Reviews 48 (2004) 843–859 www.elsevier.com/locate/newastrev A review of Bondi–Hoyle–Lyttleton accretion Richard Edgar * Stockholms Observatorium, AlbaNova Universitetscentrum, SE-106 91 Stockholm, Sweden Accepted 1 June 2004 Available online 2 July 2004 Abstract If a point mass moves through a uniform gas cloud, at what rate does it accrete material? This is the question studied by Bondi, Hoyle and Lyttleton. This paper draws together the work performed in this area since the problem was first studied. Time has shown that, despite the simplifications made, Bondi, Hoyle and Lyttleton made quite accurate predictions for the accretion rate. Bondi–Hoyle–Lyttleton accretion has found application in many fields of astronomy, and these are also discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 95.30.Lz; 97.10.Gz; 98.35.Mp; 98.62.Mw Keyword: Accretion Contents 1. Introduction..................................................... 844 2. Basics.......................................................... 844 2.1. The analysis of Hoyle and Lyttleton . 844 2.2. Analytic solution. 845 2.3. The analysis of Bondi and Hoyle . 846 2.4. Connection to Bondi accretion . 849 3. Numericalsimulations.............................................. 849 3.1. Summary . 849 3.2. Examples in the literature . 850 4. Issues in Bondi–Hoyle–Lyttleton flow . 852 * Tel.: +46-8-5537-8552; fax: +46-8-5537-8510. E-mail address: [email protected] (R. Edgar). 1387-6473/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2004.06.001 844 R. Edgar / New Astronomy Reviews 48 (2004) 843–859 4.1. The drag force . 852 4.2. Flow stability . 852 4.3. Non-uniform boundary . 854 4.4. Radiation pressure . 854 4.5. Relativity . 855 5. Applications..................................................... 855 5.1. Binary systems . 855 5.2. Protostellar clusters . 856 5.3. Galaxy clusters . 856 5.4. Other applications. 857 6. Summary....................................................... 857 Acknowledgements. 858 References. 858 1. Introduction 2.1. The analysis of Hoyle and Lyttleton In its purest form, Bondi–Hoyle–Lyttleton ac- Hoyle and Lyttleton (1939) considered accre- cretion concerns the supersonic motion of a point tion by a star moving at a steady speed through an mass through a gas cloud. The cloud is assumed infinite gas cloud. The gravity of the star focuses to be free of self-gravity, and to be uniform at the flow into a wake which it then accretes. The infinity. Gravity focuses material behind the point geometry is sketched in Fig. 1. mass, which can then accrete some of the gas. Hoyle and Lyttleton derived the accretion rate This problem has found applications in many in the following manner: Consider a streamline areas of astronomy, and this paper is an attempt with impact parameter f. If this follows a ballistic to address the lack of a general review of the orbit (it will if pressure effects are negligible), subject. then we can apply conventional orbit theory. We I start with a short summary of the original have work of Bondi, Hoyle and Lyttleton, followed by GM a discussion of the numerical simulations per- €r rh_ 2 ; À ¼ 2 ð1Þ formed. Some issues in Bondi–Hoyle–Lyttleton r 2 _ accretion are discussed, before a brief summary r h ¼ fv1 ð2Þ of the fields in which the geometry has proved useful. in the radial and polar directions, respectively. Note that the second equation expresses the con- servation of angular momentum. Setting h ¼ fv1 and making the usual substitution u ¼ rÀ1, we may 2. Basics rewrite the first equation as This section is somewhat pedagogical in nature, d2u GM þ u ¼ : ð3Þ containing a brief summary of the work of Bondi, dh2 h2 Hoyle and Lyttleton. Readers familiar with the basic nature of Bondi–Hoyle–Lyttleton accretion The general solution is u ¼ A cos h þ B sin h þ C may wish to skip this section. for arbitrary constants A, B and C. Substitution of R. Edgar / New Astronomy Reviews 48 (2004) 843–859 845 Fig. 1. Sketch of the Bondi–Hoyle–Lyttleton accretion geometry. this general solution immediately shows that 2.2. Analytic solution C ¼ GM=h2. The values of A and B are fixed by the boundary conditions that u ! 0 (that is, r !1) The Hoyle–Lyttleton analysis contains no fluid as h ! p, and that effects, which makes it ripe for analytic solution. du This was performed by Bisnovatyi-Kogan et al. r_ ¼h !v as h ! p: (1979), who derived the following solution for the dh 1 flow field: These will be satisfied by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2GM f v1 GM v1 v ¼ v2 þ À ; ð8Þ u ¼ ð1 þ cos hÞ sin h: ð4Þ r 1 2 h2 h r r Now consider when the flow encounters the fv1 vh ¼ ; ð9Þ h ¼ 0 axis. As a first approximation, the h velocity r will go to zero at this point. The radial velocity will f2v2 r ¼ 1 ; ð10Þ be v1 and the radius of the streamline will be given 2 GMð1 þ cos hÞþfv1 sin h by 2 q1f 1 2GM q ¼ : ð11Þ ¼ : ð5Þ r sin hð2f À r sin hÞ r h2 The first three equations are fairly straightforward, Assuming that material will be accreted if it is and follow (albeit tediously) from the orbit solu- bound to the star we have tion given above. The equation for the density is 1 GM rather less pleasant, and involves solving the v2 À < 0 2 1 r steady state gas continuity equation under condi- tions of axial symmetry. or Eq. (4) may be rewritten into the form 2GM f < f ; r0 HL ¼ 2 ð6Þ r ¼ ; ð12Þ v1 1 þ e cosðh À h0Þ which defines the critical impact parameter, known where e is the eccentricity of the orbit, r0 is the as the Hoyle–Lyttleton radius. Material with an semi-latus rectum, and h0 is the periastron angle. impact parameter smaller than this value will be These quantities may be expressed as accreted. The mass flux is therefore 2 À1 fv1 4pG2M 2q h0 ¼ tan ; ð13Þ M_ ¼ pf2 v q ¼ 1 ; ð7Þ GM HL HL 1 1 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v1 f2v4 which is known as the Hoyle–Lyttleton accretion e ¼ 1 þ 1 ; ð14Þ rate. G2M 2 846 R. Edgar / New Astronomy Reviews 48 (2004) 843–859 2 2 2pGMq f v1 1 r0 ¼ ; ð15Þ 2pfdfq1v1 ¼ dr ¼ Kdr; ð16Þ GM v1 which may be useful as an alternative form to Eq. which defines K. Note that it is independent of r. (10). The transverse momentum flux in the same inter- Note that these equations do not follow mate- val is given by rial down to the accretor. Accretion is assumed to 1 occur through an infinitely thin, infinite density Kvhðh ¼ 0Þ ; 2ps column on the h ¼ 0 axis. This is not physically consistent with the ballistic assumption, since it which is the mass flux, multiplied by the transverse would not be possible to radiate away the thermal velocity, divided over the approximate area of the energy released as the material loses its h velocity. wake. Applying the orbit equations once more, Even with a finite size for the accretion column, a and noting that a momentum flux is the same as a significant trapping of thermal energy would still pressure, we find rffiffiffiffiffiffiffiffiffiffiffi be expected. For now we shall neglect this effect. K 2GM Ps ð17Þ 2.3. The analysis of Bondi and Hoyle 2ps r as an estimate of the pressure in the wake. The Bondi and Hoyle (1944) extended the analysis longitudinal pressure force is therefore to include the accretion column (the wake fol- rffiffiffiffiffiffiffiffi lowing the point mass on the h ¼ 0 axis). We will GM s dðps2P Þ¼K d pffiffi : now follow their reasoning, and show that this s 2 r suggests that the accretion rate could be as little as half the value suggested in Eq. (7). Fig. 2 sketches Material will take a time of about r=v1 to fall the quantities we shall use. onto the accretor from the point it encounters the From the orbit equations, we know that mate- axis. This means that we can use the accretion rate to estimate the mass per unit length of the wake, m, rial encounters the h ¼ 0 axis at as 2 2 f v1 r ¼ : GM 2GM m K : ð18Þ v3 This means that the mass flux arriving in the dis- 1 tance r to r þ dr is given by This makes the gravitational force per unit length s r Fig. 2. Sketch of the geometry for the Bondi–Hoyle analysis. R. Edgar / New Astronomy Reviews 48 (2004) 843–859 847 GMmdr G2M 2 dr so knowing a will tell us the accretion rate (since Fgrav ¼ K : 2 3 2 the accretion rate will be Kr0 where r0 is the value r v1 r of r corresponding to a). By writing lm2 ¼ lm Á m, For accreting material, we must have we can use Eq. (26) to rewrite Eq. (25) as À2 r GMv1 . If we also assume that the wake is thin (s r) and roughly conical (ds=s dr=r), then dm mð1 À mÞ 1 m ¼ À : ð27Þ taking the ratio of the pressure and gravitational dv v À a v2 forces, we find that pressure force is much less than This has not obviously improved matters, but we the gravitational force. We can therefore neglect can now study the general behaviour of the func- the gas pressure in the wake. tion, without trying to solve it. First we need some The mass per unit length of the wake, m, was boundary conditions.