Hohmann Transfers

Introduction

This will be a paraphrase of the main reference 1 that I found a little hard to follow. Figure 1 shows the pertinent variables that will be used.

Figure 1: Circular orbits r₁ and r₂ as well as the elliptical Hohmann transfer orbit.

Obviously the semi-major axis of the ellipse is $a = \frac{r_{1} + r_{2}}{2}$ . This means that the energy needed for the elliptical transfer orbit is more than that of the inner orbit and smaller than that of the outer orbit. The velocity of any circular orbit can be obtained from the requirement that the centripetal force equal the gravitational attraction

(1.1)

$m ω^{2} r = \frac{G M m}{r^{2}}$

(1.2)

Where $ω$ is the angular rate of the mass m in its orbit, m is its mass, r is the radius of the circular orbit, and GM is the gravitational constant, G, times the mass, M, of the planet at the center of the orbit. For a circular orbit the speed, v, of mass m is $v = ω r$ . We can now evaluate the kinetic energy, E, of the circular orbit using equation (1.2)

$E = \frac{m v^{2}}{2} = \frac{G M m}{2 r}$

(1.3)

For convenience we will now use specific kinetic energy which is energy per unit mass.

Using equation (1.3) we can state the specific energies, e, of the inner and outer circular orbits as

$\begin{array}{l} e_{1} = \frac{G M}{2 r_{1}} \\ e_{2} = \frac{G M}{2 r_{2}} \end{array}$

(1.4)

The corresponding speeds of these 2 orbits are

$\begin{array}{l} v_{1} = \sqrt{\frac{G M}{r_{1}}} \\ v_{2} = \sqrt{\frac{G M}{r_{2}}} \end{array}$

(1.5)

For the transfer orbit, conservation of angular momentum requires that

$l = r_{p e r i g e e} v_{p e r i g e e} = r_{a p o g e e} v_{a p o g e e}$

(1.6)

Where l is the mass specific angular momentum.

At perigee the total specific energy is

$e = e_{p e r i g e e} = \frac{v_{p e r i g e e}^{2}}{2} - \frac{G M}{r_{p e r i g e e}}$

(1.7)

And this energy must be the same at the apogee

$e = e_{a p o g e e} = \frac{v_{a p o g e e}^{2}}{2} - \frac{G M}{r_{a p o g e e}}$

(1.8)

We can multiply equation (1.7) by $r_{p e r i g e e}^{2}$ and equation (1.8) by $r_{a p o g e e}^{2}$ and obtain

$r_{p e r i g e e}^{2} e = \frac{v_{p e r i g e e}^{2} r_{p e r i g e e}^{2}}{2} - G M r_{p e r i g e e} = \frac{l^{2}}{2} - G M r_{p e r i g e e}$

(1.9)

$r_{a p o g e e}^{2} e = \frac{v_{a p o g e e}^{2} r_{a p o g e e}^{2}}{2} - G M r_{a p o g e e} = \frac{l^{2}}{2} - G M r_{a p o g e e}$

(1.10)

Dividing both equations by their radii squared we obtain the following:

$e = \frac{l^{2}}{2 r_{p e r i g e e}^{2}} - \frac{G M}{r_{p e r i g e e}}$

(1.11)

$e = \frac{l^{2}}{2 r_{a p o g e e}^{2}} - \frac{G M}{r_{a p o g e e}}$

(1.12)

We may now set equations (1.11) and (1.12) equal obtaining

$\frac{l^{2}}{2 r_{p e r i g e e}^{2}} - \frac{G M}{r_{p e r i g e e}} = \frac{l^{2}}{2 r_{a p o g e e}^{2}} - \frac{G M}{r_{a p o g e e}}$

(1.13)

Now solve for l²

$l^{2} = 2 G M \frac{(\frac{1}{r_{p e r i g e e}} - \frac{1}{r_{a p o g e e}})}{(\frac{1}{r_{p e r i g e e}^{2}} - \frac{1}{r_{a p o g e e}^{2}})} = 2 G M \frac{r_{p e r i g e e} r_{a p o g e e} (r_{a p o g e e} - r_{p e r i g e e})}{(r_{a p o g e e}^{2} - r_{p e r i g e e}^{2})} = \frac{2 G M r_{p e r i g e e} r_{a p o g e e}}{r_{a p o g e e} + r_{p e r i g e e}}$

(1.14)

Now we use equation (1.6) and equation (1.14) to solve for v_perigee and v_apogee .

$\begin{array}{l} v_{p e r i g e e} = \sqrt{\frac{2 G M r_{a p o g e e}}{r_{p e r i g e e} (r_{p e r i g e e} + r_{a p o g e e})}} \\ v_{a p o g e e} = \sqrt{\frac{2 G M r_{p e r i g e e}}{r_{a p o g e e} (r_{p e r i g e e} + r_{a p o g e e})}} \end{array}$ $δ v_{1}$ (1.15)

Equation (1.15) is the solution for any elliptical orbit. For the Hohmann transfer orbit we have referring to Figure 1:

$\begin{array}{l} v_{p e r i g e e} = \sqrt{\frac{2 G M r_{2}}{r_{1} (r_{1} + r_{2})}} \\ v_{a p o g e e} = \sqrt{\frac{2 G M r_{1}}{r_{2} (r_{1} + r_{2})}} \end{array}$

(1.16)

The speed change needed for perigee and apogee can be calculated using equations (1.5) and equations (1.16)

$\begin{array}{l} δ v_{1} = \sqrt{\frac{2 G M r_{2}}{r_{1} (r_{1} + r_{2})}} - \sqrt{\frac{G M}{r_{1}}} = \sqrt{\frac{G M}{r_{1}}} (\frac{2 r_{2}}{r_{1} + r_{2}} - 1) \\ δ v_{2} = \sqrt{\frac{G M}{r_{2}}} - \sqrt{\frac{2 G M r_{1}}{r_{2} (r_{1} + r_{2})}} = \sqrt{\frac{G M}{r_{2}}} (1 - \frac{2 r_{1}}{r_{1} + r_{2}}) \end{array}$

(1.17)

Here $δ v_{1}$ is the speed change needed to go from the circular orbit of radius r₁ to the Hohmann transfer orbit and $δ v_{2}$ is the speed change needed to go from the Hohmann transfer orbit to the circular orbit of radius r₂. These speed changes need to be provided by very short burns when the spacecraft is very near the perigee and apogee, respectively, of the transfer orbit. Ideally the apogee burn should start a short time before reaching apogee and end at the same short time after apogee.

The time between the start of the transfer orbit and reaching apogee is the Kepler expression for $½$ of an orbital period.

$T = \frac{π {(r_{1} + r_{2})}^{\frac{3}{2}}}{\sqrt{G M}}$

(1.18)

Now we can write eccentricity as

$ε = \frac{\frac{r_{2}}{r_{1}} - 1}{\frac{r_{2}}{r_{1}} + 1}$

(1.19)

And semi-minor axis, b, as

$b = a \sqrt{1 - ε^{2}}$

(1.20)

Where

$a = \frac{r_{1} + r_{2}}{2}$

(1.21)

The rate of angular rotation with respect to M is given by

$ω = \frac{2 π a b}{T r^{2}}$

(1.22)

Where

$r = a \frac{1 - e^{2}}{1 + e \cos θ}$

(1.23)

Finally we can write equation (1.22) as

$ω = \frac{d θ}{d t} = \frac{\sqrt{G M}}{{[a (1 - ε^{2})]}^{\frac{3}{2}}} {(1 + ε \cos θ)}^{2}$

(1.24)

Therefore we can compute the time variation of θ and r by numerically integrating equation (1.24).

Having obtained θ and r for any time t, we can use the following equations to compute the Cartesian coordinates (x,y).

$\begin{array}{l} x = r \cos θ \\ y = r \sin θ \end{array}$

(1.25)