Orbital mechanics exam: Kepler, Hohmann and delta-v

I am an aerospace engineer, I studied at the Politecnico di Milano and I work in space systems. Orbital Mechanics is one of the university exams I help with most often in my tutoring, and it has one feature that sets it apart from Calculus or Physics 1: it is not an exam of complicated calculations, it is an exam of the physical model. Students who treat it as “a handful of formulas to apply” get stuck in the exam as soon as the question changes shape. Students who hold in their heads what really happens to an orbiting body — why the orbit is a conic, what actually changes during a manoeuvre, where the energy sits — solve exercises they have never seen before. Here I take you through the programme with the logic I use in lessons, show you where marks are lost, and we solve three exercises with real numbers together, including a complete Hohmann transfer from low orbit to geostationary orbit.

What the exam really asks

The heart of the course is the two-body problem: a satellite of negligible mass moving under the gravitational pull of a central body (almost always the Earth). The equation of motion is written

where $\mu =GM$ is the gravitational parameter of the central body (for the Earth, ${\mu }_{\oplus }=398600{\text{ km}}^3/{\text{s}}^2$ — a number you must know by heart, because you use it in every exercise). The solution of this equation is a conic: an ellipse, a parabola or a hyperbola, depending on the energy of the system. From here the programme develops in increasingly concrete blocks:

1. Orbital elements: semi-major axis $a$, eccentricity $e$, inclination $i$, true anomaly $\theta$, and the two angles that orient the orbital plane (right ascension of the ascending node $\Omega$ and argument of perigee $\omega$). Six numbers that describe an orbit completely at a given instant.
2. The vis-viva equation, which links speed and position along the orbit:

This is probably the most used formula in the whole course. It works for any conic: all you need is $r$ (the distance from the focus) and $a$ (the semi-major axis, negative for hyperbolas).

3. Orbital period, from Kepler’s third law: $T=2\pi \sqrt{a^3/\mu }$.
4. Orbital manoeuvres, in particular the Hohmann transfer between two coplanar circular orbits: the two-impulse manoeuvre that minimises the total $\Delta v$ between two circular orbits.
5. Plane change and an introduction to patched conics (sphere of influence): how you go from an orbit around the Earth to an interplanetary trajectory, treating the problem as a sequence of local two-body problems rather than a full three-body problem.

The common thread is always the same: the specific energy of the orbit, $\epsilon =-\mu /(2a)$, depends only on $a$. Changing orbit means changing energy, and changing energy costs $\Delta v$. In the end, the whole course is an elegant way of accounting for what it costs to move around in space.

Where marks are lost

I have marked and reviewed enough exercises to recognise the same five mistakes in almost every session.

1. Confusing the radius with the semi-major axis. In the vis-viva equation, $r$ is the instantaneous distance from the focus (it changes along the orbit), while $a$ is the semi-major axis (constant, a property of the orbit). Writing $v^2=\mu (2/r-1/r)$ for an elliptical orbit is the number one mistake I see, and it wipes out the exercise.

2. Forgetting that the Δv of a plane change depends on the speed at which you do it. The formula is $\Delta v=2v\sin (\Delta i/2)$: the lower $v$ is at the point where you perform the manoeuvre, the less it costs to change plane. This is why a plane change is almost always combined with the apogee burn of a transfer (where the speed is lowest), and is not done alone in a circular orbit. I see it written as a separate step in 90% of the exercises I mark, and it costs marks precisely in the “trade-off” part that lecturers want to see.

3. Unit errors. ${\mu }_{\oplus }$ in km³/s² and radii in metres in the same calculation is the most common cause of a result that is “close but wrong by a factor of 1,000”. Fix a single unit system at the start of the exercise (I always use km and seconds) and never change it.

4. Mistaking negative energy for an “impossible orbit”. An elliptical orbit has negative specific energy ($\epsilon <0$): that is correct, it means the body is gravitationally bound. Many students, on seeing a minus sign, think they have made a mistake in the calculation.

5. Treating the transfer time as if it were the period of the final orbit. In a Hohmann transfer the flight time is half the period of the transfer orbit (the ellipse), not of the starting orbit or of the arrival orbit. It is a detail, but it is exactly the kind of detail the exam asks you to show you have understood.

The method I use

For every manoeuvre exercise I follow the same scheme, and I recommend it as a checklist to write at the top of the page before you start calculating:

1. Draw the two orbits (or the orbit and the point), even roughly. If you cannot draw it, you have not understood the geometry of the problem.
2. Identify what is conserved in each phase: away from the impulses, energy and angular momentum are constant along every conic orbit.
3. Calculate the speeds before and after each manoeuvre, always with the vis-viva equation, using the $a$ of the orbit you are on at that instant (not the starting one or the arrival one, but the orbit currently being flown).
4. The Δv of each impulse is the vector difference of the speeds before and after. If the two velocities are parallel (same plane, same direction), it is a simple scalar subtraction; if the plane changes, you need the law of cosines.
5. Add up the Δv of the individual impulses for the total of the manoeuvre, and check that the sign makes physical sense (speed up to climb, slow down to descend, always relative to the instantaneous velocity).

Three worked exercises

Exercise 1 — Low circular orbit: speed and period

A satellite orbits at an altitude of 500 km, so $r=6378+500=6878$ km (mean radius of the Earth 6378 km). Since the orbit is circular, $a=r$.

Circular speed:

Period:

These numbers agree with real data for satellites in low orbit: speeds around 7.6-7.7 km/s, a period a little under two hours.

Exercise 2 — Elliptical orbit given by perigee and apogee

An orbit has $r_p=7000$ km (perigee) and $r_a=25000$ km (apogee). Calculate $a$, $e$, the specific energy and the speeds at the two extremes.

Speed at perigee and at apogee, again with the vis-viva equation (I use the $a$ of the whole orbit, which is the same at every point):

Consistency check (conservation of angular momentum, $r_p{v}_p=r_a{v}_a$): $7000\times 9.433\approx 66028$, and $25000\times 2.641\approx 66028$. It matches: this is the check I always make before writing the final result, and I recommend it as an exam habit — it costs thirty seconds and saves you from handing in a calculation error.

Period of the orbit: $T=2\pi \sqrt{a^3/\mu }\approx 20142$ s, that is about 5 hours 35 minutes.

Exercise 3 — Hohmann transfer from LEO to GEO

This is the exercise I see most often in exams, and it brings together the whole programme. A satellite starts from a low circular orbit at an altitude of 300 km ($r_1=6678$ km) and must reach a circular geostationary orbit ($r_2=42164$ km, altitude 35786 km). Calculate the total $\Delta v$ and the transfer time.

Circular speeds at departure and arrival:

Transfer orbit: an ellipse with perigee at $r_1$ and apogee at $r_2$, with semi-major axis

Speeds on the transfer orbit at perigee and apogee (vis-viva with $a=a_t$):

The two impulses. The first, at perigee, takes the satellite from the low circular orbit onto the transfer orbit: you have to speed up, because $v_p>v_1$.

The second, at apogee (after half a revolution on the ellipse), circularises the orbit at $r_2$: you have to speed up again, because $v_2>v_a$ (at apogee the satellite is “slow” compared with the circular speed needed up there).

Transfer time: half the period of the transfer ellipse.

A real launch towards GEO almost always starts from a non-zero inclination (Kourou is at 5°, Cape Canaveral at 28.5°), so a plane change is needed too. This is where point 2 of the list of mistakes comes in: doing it separately at the apogee of the final orbit would cost, with $\Delta i=28.5°$,

to be added to $\Delta v_2$, for a total of about 2.98 km/s for the second manoeuvre alone. But if you combine the plane change with the circularisation in a single vector impulse (law of cosines, because the plane and the magnitude of the speed change together),

you save almost 1.15 km/s of propellant by doing the same thing in one impulse instead of two, and at exactly the point of the orbit where the speed (and so the cost of the plane change) is lowest. This kind of reasoning — not just “which formula do I use” but “where is it best to do the manoeuvre” — is what separates a 24 from a 30 in this exam (Italian university exams are marked out of 30).

Once the orbit is closed, the next step is the orientation of the satellite: I cover it in attitude dynamics: Euler angles and quaternions. The $\Delta v$ you calculate here instead becomes an input to the propulsion budget, which you will find in space systems engineering. To review the energy and mechanics that these calculations rest on, there is the Physics 1 exam practice test.

How we work in lessons

With Orbital Mechanics students I always start from the drawing: first we understand the geometry of the orbit and what the exercise is really asking, then we write the equations. We work on past exam papers from your own course — every lecturer has a style of question, and practising on that specific style counts for more than redoing generic textbook exercises. When a concept does not click (typically: energy and sign, or when it pays to combine manoeuvres) I stop there until it is clear, before moving on with the programme.

If you are also preparing Calculus 2 or Physics 1 from the same year, have a look at the page on university tutoring with all the practice tests by course, or the one dedicated to physics tutoring, the subject that stands behind all of orbital mechanics. And if you want to find out what you are really stuck on before booking, write to me for a free introductory call: you send me the exercise that does not add up and in twenty minutes we work out whether it is a problem of formula or of physical model — usually it is the second.