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

(1) (2)

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

(3)

The general solution is u = A cos + B sin + 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 ) as , and that

These will be satisfied by

(4)

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

(5)

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

or

(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

(7)

which is known as the Hoyle-Lyttleton accretion rate.

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

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

(16)

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

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

(17)

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

Material will take a time of about r / v 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

(18)

This makes the gravitational force per unit length

For accreting material, we must have r ~ G M v^-2. If we also assume that the wake is thin (s << r) and roughly conical (ds / s 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:

(19) (20)

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

(21) (22) (23)

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

(24) (25)

We shall now analyse the behaviour of these equations.

We can integrate equation 24 to yield

(26)

for some constant . Since µ is a scaled mass (and hence always positive), we see that the scaled velocity () changes sign when = . That is, is the stagnation point. Material for < will accrete, so knowing will tell us the accretion rate (since the accretion rate will be r₀ where r₀ is the value of r corresponding to ). By writing µ² = µ · , we can use equation 26 to rewrite equation 25 as

(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:

• 1 as
  Which means that v v at large radii

• = 0 at =
  The stagnation point

• d / d > 0 Everywhere

  The velocity must be a monotonic function. This is physically reasonable, if we are to avoid unusual flow patterns

The first two conditions can be satisfied for any value of . 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 = ^-1 . Equation 27 then reads

(28)

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

or, one application of the quadratic roots formula later:

(29)

Since 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

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