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This section is somewhat pedagogical in nature, containing a brief summary of the work of Bondi, Hoyle and Lyttleton. Readers familiar with the basic nature of Bondi-Hoyle-Lyttleton accretion may wish to skip this section.

2.1. The Analysis of Hoyle & Lyttleton

Hoyle and Lyttleton (1939) considered accretion by a star moving at a steady speed through an infinite gas cloud. The gravity of the star focuses the flow into a wake which it then accretes. The geometry is sketched in figure 1.

Figure 1

Figure 1. Sketch of the Bondi-Hoyle-Lyttleton accretion geometry

Hoyle and Lyttleton derived the accretion rate in the following manner: Consider a streamline with impact parameter zeta. If this follows a ballistic orbit (it will if pressure effects are negligible), then we can apply conventional orbit theory. We have

Equation 1 (1)


in the radial and polar directions respectively. Note that the second equation expresses the conservation of angular momentum. Setting h = zeta vinfty and making the usual substitution u = r-1, we may rewrite the first equation as

Equation 3 (3)

The general solution is u = A costheta + B sintheta + C for arbitrary constants A, B and C. Substitution of this general solution immediately shows that C = GM / h2. The values of A and B are fixed by the boundary conditions that u --> 0 (that is, r --> infty) as theta --> pi, and that

Equation 4

These will be satisfied by

Equation 4 (4)

Now consider when the flow encounters the theta = 0 axis. As a first approximation, the theta velocity will go to zero at this point. The radial velocity will be vinfty and the radius of the streamline will be given by

Equation 5 (5)

Assuming that material will be accreted if it is bound to the star we have

Equation 6


Equation 6 (6)

which defines the critical impact parameter, known as the Hoyle-Lyttleton radius. Material with an impact parameter smaller than this value will be accreted. The mass flux is therefore

Equation 7 (7)

which is known as the Hoyle-Lyttleton accretion rate.

2.2. Analytic Solution

The Hoyle-Lyttleton analysis contains no fluid effects, which makes it ripe for analytic solution. This was performed by Bisnovatyi-Kogan et al. (1979), who derived the following solution for the flow field:

Equation 8 (8)




The first three equations are fairly straightforward, and follow (albeit tediously) from the orbit solution given above. The equation for the density is rather less pleasant, and involves solving the steady state gas continuity equation under conditions of axial symmetry.

Equation 4 may be rewritten into the form

Equation 12 (12)

where e is the eccentricity of the orbit, r0 is the semi-latus rectum, and theta0 is the periastron angle. These quantities may be expressed as

Equation 13 (13)



which may be useful as an alternative form to equation 10.

Note that these equations do not follow material down to the accretor. Accretion is assumed to occur through an infinitely thin, infinite density column on the theta = 0 axis. This is not physically consistent with the ballistic assumption, since it would not be possible to radiate away the thermal energy released as the material loses its theta velocity. Even with a finite size for the accretion column, a significant trapping of thermal energy would still be expected. For now we shall neglect this effect.

2.3. The Analysis of Bondi and Hoyle

Bondi and Hoyle (1944) extended the analysis to include the accretion column (the wake following the point mass on the theta = 0 axis). We will now follow their reasoning, and show that this suggests that the accretion rate could be as little as half the value suggested in equation 7. Figure 2 sketches the quantities we shall use.

Figure 2

Figure 2. Sketch of the geometry for the Bondi-Hoyle analysis

From the orbit equations, we know that material encounters the theta = 0 axis at

Equation 16

This means that the mass flux arriving in the distance r to r + dr is given by

Equation 16 (16)

which defines Lambda. Note that it is independent of r. The transverse momentum flux in the same interval is given by

Equation 17

which is the mass flux, multiplied by the transverse velocity, divided over the approximate area of the wake. Applying the orbit equations once more, and noting that a momentum flux is the same as a pressure, we find

Equation 17 (17)

as an estimate of the pressure in the wake. The longitudinal pressure force is therefore

Equation 18

Material will take a time of about r / vinfty to fall onto the accretor from the point it encounters the axis. This means that we can use the accretion rate to estimate the mass per unit length of the wake, m, as

Equation 18 (18)

This makes the gravitational force per unit length

Equation 20

For accreting material, we must have r ~ G M vinfty-2. If we also assume that the wake is thin (s << r) and roughly conical (ds / s approx dr / r), then taking the ratio of the pressure and gravitational forces, we find that pressure force is much less than the gravitational force. We can therefore neglect the gas pressure in the wake.

The mass per unit length of the wake, m, was introduced above. If we assume the mean velocity in the wake is v, we can write two conservation laws, for mass and momentum:

Equation 19 (19)


Recall that Lambda vinfty is the momentum supply into the wake, since r = vinfty on axis for all streamlines. We can declutter these equations by introducing dimensionless variables for m, r and v:

Equation 21 (21)



Note that chi = 2 corresponds to material arriving from the streamline characterised by zetaHL. Substituting these definitions into equations 19 and 20, we obtain

Equation 24 (24)


We shall now analyse the behaviour of these equations.

We can integrate equation 24 to yield

Equation 26 (26)

for some constant alpha. Since µ is a scaled mass (and hence always positive), we see that the scaled velocity (nu) changes sign when chi = alpha. That is, alpha is the stagnation point. Material for chi < alpha will accrete, so knowing alpha will tell us the accretion rate (since the accretion rate will be Lambda r0 where r0 is the value of r corresponding to alpha). By writing µnu2 = µnu · nu, we can use equation 26 to rewrite equation 25 as

Equation 27 (27)

This has not obviously improved matters, but we can now study the general behaviour of the function, without trying to solve it. First we need some boundary conditions. These are as follows:

The first two conditions can be satisfied for any value of alpha. Fortunately, the third implies as restriction. The next set of manipulations may seem a little obscure at first, but they do lead in the desired direction.

Substitute xi = alpha-1 chi. Equation 27 then reads

Equation 28 (28)

Now, suppose the derivative is zero. This leads to the condition

Equation 29

or, one application of the quadratic roots formula later:

Equation 29 (29)

Since nu ultimately represents a physical quantity (the velocity), it's obviously desirable that it remain real. We therefore need to look at when the discriminant can become zero. This is another quadratic equation, leading to

Equation 30

which means that something must happen when alpha = 1. To determine this `something,' it is best to plot equation 29.

Figures 3 and 4 demarcate the regions where dnu / dxi changes sign, as dictated by equation 29. These are not possible solutions for nu. However, any suitable solution for nu must remain within the region marked `a' if it is to remain monotonic and increasing. This is only possible when alpha > 1. Unwinding our rescaled variables, we see that an alpha value of unity puts the stagnation point halfway between the accretor and the original value of Hoyle and Lyttleton. This in turn implies a minimum accretion rate of 0.5 MHL.

Figure 3

Figure 3. Curves where dnu / dxi = 0 for alpha < 1. In the regions marked `a,' the derivative is greater than zero. It is less than zero in the `b' region.

Figure 4

Figure 4. Curves where dnu / dxi = 0 for alpha > 1. In the region marked `a,' the derivative is greater than zero. It is less than zero in the `b' regions.

Again, I would like to remind the reader that the flow has been assumed to remain isothermal with negligible gas pressure throughout this discussion. This assumption is likely to be violated in the wake, where densities will be high and radiative heat loss inefficient. At the very least, thermal effects should be important close to the stagnation point in the wake. Horedt (2000) details an analysis similar to that given above, but with a pressure term included. The value of alpha (which Horedt calls x0) was found to lie between 0.6 and 3.5 for flows which were supersonic at infinity and subject to Newtonian physics (the polytropic and adiabatic indices were also free parameters in this analysis).

Will the flow be stable? Bondi and Hoyle asserted that if alpha > 2 (note that alpha = 2 gives the solution of Hoyle and Lyttleton), then the wake would become unstable to perturbations which preserve axial symmetry. However, later analysis by Cowie (1977) suggested that the wake should be unstable, regardless of the value of alpha. Subsequent numerical simulations and analytic work have shown that Bondi-Hoyle-Lyttleton flow is far from stable, and we will discuss the subject in section 4.2.

2.4. Connection to Bondi Accretion

Bondi (1952) studied spherically symmetric accretion onto a point mass. The analysis shows (see e.g. Frank et al. (2002)) that a Bondi radius may be defined as

Equation 30 (30)

Flow outside this radius is subsonic, and the density is almost uniform. Within it, the gas becomes supersonic and moves towards a freefall solution. The similarities between equations 6 and 30 led Bondi to propose an interpolation formula:

Equation 31 (31)

This is often known as the Bondi-Hoyle accretion rate. On the basis of their numerical calculations, Shima et al. (1985) suggest that equation 31 should acquire an extra factor of two, to become

Equation 32 (32)

which then matches the original Hoyle-Lyttleton rate as the sound speed becomes insignificant. The corresponding zetaBH is formed by analogy with equation 7.

Nomenclature in this field can be a little confused. When papers refer to `Bondi-Hoyle accretion rates,' they may mean equation 7, 31 or 32. In this review, I shall refer to pressure-free flow as `Hoyle-Lyttleton' accretion and use MHL and zetaHL. When there is gas pressure, I will talk about `Bondi-Hoyle accretion' and use MBH and zetaBH, in the sense defined by equation 32. I shall use `Bondi-Hoyle-Lyttleton' accretion to refer to the problem in general terms.

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