The rocket equation in module 9 told you how much propellant a given Δv costs. This module covers the rest of the engineering puzzle: when and how to spend that Δv. Spoiler — the answers are stricter than they look. Burns must happen at specific points on an orbit, with specific timing within seconds, and Earth launches must happen during specific windows that can recur as rarely as every 26 months.
Before getting into burn execution, it's worth pinning down the most important point about a Hohmann transfer (or any orbit-change maneuver): it requires two separate burns at two specific points, not one big push at the start. From module 7:
If you skip burn 2, you don't stay at the destination. You're still on the ellipse: you'll drop right back down to the original orbit, half an orbit later. The transfer ellipse is a closed curve, and without burn 2 you'll trace it forever, brushing the destination altitude only once per orbit before falling back.
If you skip burn 1, you don't go anywhere — you stay on your original circle. Burn 1 is what creates the transfer in the first place; burn 2 is what makes it stick. Both are required, and they have to be sized correctly for each end.
Modules 7 and 8 quietly assumed that each Δv happens in zero time — an "impulsive burn" that instantaneously changes the spacecraft's velocity vector. This is a useful idealization for orbit math; it lets us ignore the burn's internal dynamics and just compare the orbits before and after. Real burns, of course, take time. The heart of a launch vehicle's main engine fires for a few minutes; an upper stage's circularization burn might run 30 seconds; a small reaction-control jet pulses for milliseconds.
During a burn of duration $\tau$, the spacecraft is moving along its orbit. If the burn happens at LEO, where the orbital speed is 7.79 km/s and the orbit period is 88 minutes, here's how far the spacecraft sweeps during burns of various durations:
Angular sweep during a burn of duration $\tau$ at orbital angular speed $\omega = v / r$:
$$ \Delta\theta = \omega \tau, \qquad \text{arc length} = v \cdot \tau $$Plugging in for LEO ($r = 6{,}571$ km, $v = 7{,}788$ m/s):
The implications matter. The Hohmann math from module 7 said "burn at periapsis tangent to the orbit." A 5-minute burn at periapsis is no longer at one point — it sweeps a 20° arc, during which the velocity vector rotates by 20°. To stay tangent to the orbit, the engine has to swivel during the burn ("pitch program"). And no matter how perfectly you steer, some of the burn's $\Delta v$ is now applied non-tangentially, which is wasted. Worse, since the burn is happening over an arc and not at a single instant, the post-burn orbit isn't quite the perfect transfer ellipse we computed.
If you have to spend Δv anyway, where on your orbit should you spend it? The answer, surprisingly, is: at the lowest point — periapsis. This is the Oberth effect, named for the German rocket pioneer Hermann Oberth, who pointed it out in 1929.
The argument is energy. Kinetic energy is $\tfrac{1}{2} m v^2$. If you add a small $\Delta v$ to your current speed $v$ (with the burn aligned along your motion), the change in KE is:
$$ \Delta(\mathrm{KE}) = \tfrac{1}{2} m (v + \Delta v)^2 - \tfrac{1}{2} m v^2 = m v\, \Delta v + \tfrac{1}{2} m (\Delta v)^2 $$For a small burn ($\Delta v \ll v$), the second term is negligible and the energy gain scales as $m v \,\Delta v$. The energy you extract from a kilogram of propellant scales with how fast you're already moving. Burning at high $v$ — meaning at low altitude where orbital speeds are highest — gives you more orbital energy per unit fuel than burning at high altitude where you're moving more slowly.
Same 50 m/s prograde burn, applied to a circular orbit at different altitudes:
| Altitude | Orbital speed | Energy gain per kg |
|---|---|---|
| 200 km (LEO) | 7.79 km/s | 391 kJ/kg |
| 2,000 km | 6.90 km/s | 346 kJ/kg |
| 10,000 km | 4.93 km/s | 248 kJ/kg |
| 35,786 km (GEO) | 3.07 km/s | 155 kJ/kg |
The same 50 m/s of $\Delta v$ produces 2.5× more energy gain at LEO than at GEO. This is why every interplanetary mission performs its trans-X injection burn from low Earth orbit, not from a high parking orbit. From module 8: a Mars mission needs an outbound $v_\infty$ of about 2.95 km/s. Done from LEO, the burn costs about 3.6 km/s. Done from GEO instead, after a Hohmann to GEO has already been completed, the same $v_\infty$ would cost an additional 1+ km/s on top of the 3.93 km/s already spent reaching GEO. Total: more than 5 km/s, vs 3.6 km/s from LEO.
The same logic explains why both Hohmann burns "should" be at periapsis of the transfer ellipse — but only one of them can be. Burn 1 happens at the transfer ellipse's periapsis (where you're moving fastest, so it's Oberth-favorable). Burn 2 has to happen at apoapsis (slowest point), which is Oberth-unfavorable — but you have no choice, because that's where you need to be at the moment of arrival. The geometry constrains the timing. Real spacecraft sometimes use slightly more elaborate transfers (bi-elliptic, low-thrust spirals) when the simple Hohmann's apoapsis burn turns out to be expensive enough that a longer-but-cheaper alternative wins.
It's not enough to do the burn — you have to do it at the right moment in your orbit. A perfect Hohmann burn happens precisely at periapsis (for burn 1) or apoapsis (for burn 2), with $\Delta v$ aligned exactly tangent to the velocity vector. Burning a few seconds early or late, or off-axis by a few degrees, doesn't just reduce efficiency — it puts you on the wrong orbit entirely.
The interactive below shows what happens. The faint blue ellipse is the planned Hohmann transfer; the bright blue ellipse is what you actually fly when burn 1 happens at the wrong angular position. Both ellipses have the same size and shape, but the actual one is rotated, so its apoapsis is at the wrong angular location — the spacecraft arrives at the destination altitude, but at the wrong place. For a mission targeting a specific GEO slot or planetary rendezvous, that's mission failure.
Slide the timing offset and notice: the apoapsis altitude doesn't change (the burn's $\Delta v$ is the same), but the angular position of arrival drifts directly with the timing error. A 15° timing error becomes a 15° angular miss at the destination. For a GEO satellite, 15° around the geostationary belt is about 11,000 km from where the operator wanted it. The fix: a small additional burn after arrival to slowly drift to the right slot — which costs more delta-v, which costs more fuel, which the rocket equation will not forgive lightly.
One more piece of the burn-execution puzzle deserves a mention: gravity loss, which dominates the ascent from Earth's surface to orbit. As a launching rocket climbs vertically out of the atmosphere, every second it spends fighting gravity is a second when its thrust is doing useful work against gravity rather than purely accelerating sideways into orbit. A rocket with high thrust-to-weight (think solid boosters) gets out of the dense atmosphere fast and minimizes gravity loss; a slower-accelerating rocket pays more.
For a typical surface-to-LEO launch, the breakdown of total $\Delta v$ looks like:
The 1.5 km/s of gravity loss is real, irrecoverable energy lost to working against Earth's gravity. It's why getting to orbit is hard — not because LEO speed is enormous (it's only Mach 23 in some sense), but because you have to first climb out of a 200 km hole while being pulled down at $g$. This is also why all real launch vehicles do a pitch-over maneuver early in flight: the sooner you can convert vertical climb into horizontal acceleration, the less gravity loss you eat.
One geometric subtlety from where rockets launch: the Earth is rotating, and a launch site is moving eastward at the surface speed of Earth's rotation. At the equator, that's $R_\oplus \omega_\oplus = 6378\ \mathrm{km} \times 7.29 \times 10^{-5}\ \mathrm{rad/s} = 465\ \mathrm{m/s}$. A rocket launched eastward gets that 465 m/s for free (it's already moving that fast at liftoff); a rocket launched westward has to overcome it, costing 465 m/s of extra $\Delta v$. Hence: virtually every orbital launch goes east.
Latitude reduces the boost as $\cos(\text{lat})$, since the eastward velocity at latitude $\phi$ is $R_\oplus \omega_\oplus \cos\phi$:
| Launch site | Latitude | Eastward boost |
|---|---|---|
| Equator (theoretical) | 0° | 465 m/s |
| Kourou (French Guiana) | 5.2° | 463 m/s |
| Cape Canaveral (USA) | 28.5° | 409 m/s |
| Vandenberg (USA) | 34.7° | 382 m/s |
| Baikonur (Kazakhstan) | 45.6° | 325 m/s |
| Plesetsk (Russia) | 62.9° | 212 m/s |
This is why ESA's Kourou launch site in French Guiana is so prized: at 5.2° N latitude, it's almost as good as launching from the equator. SpaceX, NASA, and ULA launch from Cape Canaveral (28.5°) for historical and political reasons; Vandenberg is reserved for polar and retrograde launches that need to fly south or west and can't safely be done from the Cape.
Latitude also constrains which inclinations you can reach directly. A rocket launched due east from latitude $\phi$ ends up in an orbit with inclination equal to $\phi$. (The launch site sits in the orbital plane, so the orbit's intersection with Earth's equator is at the launch latitude.) To reach a lower inclination — like equatorial or geostationary — you have to do an expensive plane-change burn after launch. Rockets launching from KSC for GEO missions burn off about 1 km/s of $\Delta v$ removing that 28.5° of inclination over their first few orbits. Kourou launches save most of that.
Some satellite missions need to look at Earth at the same local solar time on every pass — useful for imaging, since shadows and lighting are consistent. A sun-synchronous orbit achieves this by exploiting Earth's equatorial bulge, which causes the orbital plane to precess slowly. At an inclination near 98° (slightly retrograde) and an altitude around 800 km, the precession rate happens to equal Earth's annual motion around the Sun (about 0.986°/day), so the orbit's geometry relative to the Sun stays fixed indefinitely.
Sun-synchronous launches from Vandenberg therefore go southward (because the orbit is retrograde) and at very specific times of day to hit the right ascending-node longitude. A mistimed sun-synchronous launch by 30 minutes lands the satellite in an orbit whose ground track shifts an hour later through every pass — usually unacceptable.
For interplanetary missions, the launch-window problem combines Earth-based timing with the planetary alignment from module 8. Earth's rotation determines the time of day; the destination planet's position determines the time of year.
Recall the Mars synodic period: Earth and Mars line up for a Hohmann transfer once every ~26 months. Each "alignment" is actually a window about three to five weeks wide, during which the $\Delta v$ cost is close to the theoretical minimum (around 3.6 km/s from LEO to Mars). Outside that window, the $\Delta v$ cost grows rapidly — by 30 days off-window, you've added a kilometer per second or more, which the rocket equation translates into a doubling of propellant. Beyond that, you're better off waiting 26 months for the next alignment. That's exactly what NASA does. Curiosity in 2011, InSight in 2018, Perseverance in 2020, and the planned ESA Rosalind Franklin (delayed to 2028) are all on the synodic calendar.
And the same logic applies inside each window: the launch on a given day has to happen during a "daily launch window" that's typically 1–2 hours wide, when Earth has rotated to put the launch site in the correct orbital plane for the interplanetary trajectory. Miss the daily window, you wait until tomorrow. Miss the seasonal window, you wait two years.
The kinematic delta-v's of module 7 and module 8 told you how much velocity change a maneuver requires. The rocket equation in module 9 told you what each kilometer per second costs in propellant. This module added the real-world constraints: those burns must happen at specific points (Hohmann needs two burns at two specific positions on the orbit), with timing accurate to seconds, at altitudes that exploit the Oberth effect, launched from sites that exploit Earth's rotation, on dates that exploit planetary alignment. The single number "$\Delta v = 3.6$ km/s" hides an enormous amount of engineering and astrodynamics.
The final module steps off the two-body problem entirely, into the strange world where three masses interact and the Lagrange points appear.