Subject headings: celestial mechanics—orbit transfer

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1  Motivation

Consider two coplanar, circular, confocal orbits of semimajor axes $a_1$ and $a_2>a_1$. Suppose we wish to transfer an artificial satellite from the inner to the outer orbit and do it in such a way that the transfer orbit is elliptical (as opposed to hyperbolic or parabolic) and confocal to the circular orbits. This type of orbit was first considered by W. Hohmann in 1925. At some point during the inner circular orbit, a thruster firing perpendicular to the radius vector occurs, at which point the satellite's motion switches to the pericenter of the elliptical transfer orbit. The ellipse is such that its apocenter distance coincides with the outer circular orbit distance. After coasting from the pericenter out to the apocenter, another thruster firing (again in the orbital plane and perpendicular to the instantaneous radius vector) places the satellite on the outer circular orbit. This process is symmetric, so one can just as easily transfer from an outer orbit to an inner one, requiring only a few changes in sign in what follows. Hence, what are the semimajor axis $a$ and eccentricity $e$ of the transfer orbit, and what are the changes in energy and velocity magnitude required to perform the transfer?

2  Orbital Elements of the Transfer Orbit

Now, the pericenter of the transfer ellipse (call it P) coincides with the radius of the smaller circle, $a(1-e)=a_1$, and the apocenter (call it A) coincides with the radius of the larger circle, $a(1+e)=a_2$. Thus, we may combine these two relations to find the semimajor axis and eccentricity of the transfer orbit:

Thus, we have a tidy result. The semimajor axis of the transfer ellipse is the average of the radii of the circular orbits, and the eccentricity is the fractional difference of those radii.

The amount of time spent in the transfer orbit is, by definition, half the orbital period of the transfer orbit. Kepler's third law can be written

where $\mu =G(m_1+m_2)$, $n=2\pi /T$ is the mean motion, $T$ is the orbital period. Using (1), we therefore find

3  Energy Considerations

The specific energy $\mathrm{ℰ}$ of a two-body orbit is

where $v$ is the magnitude of the relative velocity vector, and $r$ is the magnitude of the relative position vector. The specific energy is the energy of the two-body system, ${\mathrm{ℰ}}_{\mathrm{tot}}$, divided by the reduced mass,

One can also show that

which is a statement of conservation of energy known as the vis viva integral. Combining (5) and (7) lets us write the specific energy as

Thus, the difference in energy (henceforth we shall assume "energy" means "specific energy") between the two circular orbits is

4  Energy Changes to Accomplish the Transfer

From (8) and (1), we can write the energy of the transfer orbit,

Hence, we discover that the changes in energy that must occur at P and at A are, from (10) and (8),

and

where ${\mathrm{ℰ}}_P=\frac{-\mu }{2a_1}$ and ${\mathrm{ℰ}}_A=\frac{-\mu }{2a_2}$ are the energies of the respective circular orbits. It is quite delightful that the changes in energy are just the respective orbital energies scaled only by the transfer orbit eccentricity. One can easily show that (11) and (12) add up to (9):

5  Changes in Relative Velocity Magnitude to Accomplish the Transfer

Let us assume that the changes in orbital energy are accomplished by thruster firings which are very short compared to the orbital periods involved. Then, since the changes occur at pericenter and apocenter of the elliptical transfer orbit, our assumptions let us put $\dot{r}=0$, and the energy changes involve only the kinetic energies and hence the relative velocity magnitudes.

5.1  Using Physics

Differentiating (5), we have

since during the thruster firings $\dot{r}=0$. Integrating through the thruster firing from (say) times $t_0$ to $t_0+\mathrm{\Delta }t$,

we have

where $v_0=v\left(t_0\right)$ is the magnitude of the velocity the instant before the thruster firing, and $\mathrm{\Delta }v$ is the velocity magnitude impulse arising from a change in energy $\Delta ℰ$. Thus, the changes in relative velocity magnitude are

at the respective orbital locations. At P,

which we obtained from eq. (7) for the interior circular orbit, while at A we similarly find

for the elliptical orbit. Plugging (18) and (19), and (11) and (12), into (17), we find that

and

We must choose the signs in (20) and (21) to match physical circumstances. The first thruster firing is in the same direction as the direction of motion around the inner circular orbit, so the change in velocity magnitude is positive. Likewise, the second thruster firing is also along the direction of motion, so that change, too, is positive. Therefore, since we are considering the case $a_2>a_1$, the changes in velocity magnitude are

and

In each case the change in velocity magnitude is a fraction of the respective circular velocity $v_c=\sqrt{\mu /a}$, the fractions being the expressions in parentheses in eqs. (22) and (23).

5.2  Using Algebra

The previous derivation uses conservation of energy as the starting point. It is therefore satisfying in that is makes use of a fundamental physical principle. If we are willing to forgo physics, then here is a short algebraic derivation. From (7) the circular velocities at P and at A are

The velocities on the transfer orbit at those same points are (again using (7))

Thus, the changes in velocity magnitude are just

and

which yield eqs. (22) and (23).

6  Expansions

Suppose $a_1\ll a_2$. Then we can expand on $ϵ=a_1/a_2$, and (22) and (23) become

which we can also write as

and

which, similarly, we can write as

From (29) and (31) we see that the thruster firings consist of a large kick (of order the circular velocity at the corresponding radius) followed by higher order terms.

The ratio of velocity kicks is

A series expansion of (33) yields

where we have set $u=\sqrt{ϵ}=\sqrt{a_1/a_2}$. We carry out the expansion to an impractical number of terms in order to show the interesting pattern of the expansion coefficients.

7  Effects of Errors in the Velocity Changes

7.1  Impulse Error at Pericenter

7.1.1  Perturbed Transfer Orbit Elements

Suppose, as is always the case in the real world, an error occurs and the change in velocity magnitude imparted by the first thruster firing is in error by some small amount, say ${\mathrm{\Delta }v}_P={\mathrm{\Delta }v}_P^0+{\delta v}_P$, where ${\mathrm{\Delta }v}_P^0$ is the desired change in velocity given by (22) and ${\delta v}_P$ is the error. What are the effects on the transfer orbit semimajor axis and eccentricity, and on the outer orbit?

Using (22), we find the variation in ${\mathrm{\Delta }v}_P$ is

Thus, we can write the resulting error in the radius of the resulting circular orbit (assuming, ideally, that a compensating adjustment of the thrust at A circularizes it),

or