So far the orbits in this curriculum have been static — fix the size, fix the shape, watch the body go around. This module is the first useful one. We'll work out exactly how much velocity change it takes to move a spacecraft from one circular orbit to another, using nothing but the vis-viva equation from module 3.
Suppose you're on a circular orbit at radius $r_1$ — say, low Earth orbit — and you'd like to get to a different circular orbit at radius $r_2$. The geostationary belt, perhaps. You can't simply "drive" to the new altitude. The whole apparatus of the orbit — speed, direction, energy — is welded to wherever you happen to be. To change anything, you have to fire an engine.
It turns out that the most fuel-efficient way to do this — when both orbits are coplanar circles around the same body — is a Hohmann transfer, named for the German engineer Walter Hohmann, who worked it out in 1925 in a remarkable little book called Die Erreichbarkeit der Himmelskörper ("The Attainability of Celestial Bodies"). The whole idea is two engine burns, separated by a coast on a transfer ellipse:
The transfer ellipse is the shape that does this with the smallest possible total speed change. Anything fancier — three burns, different arrival points, plane changes — costs more.
The whole story is in one number: the transfer ellipse's semi-major axis. Periapsis is at $r_1$ and apoapsis is at $r_2$, so the long axis of the ellipse — the distance from one end to the other — is $r_1 + r_2$. The semi-major axis is half of that:
$$ a_t = \frac{r_1 + r_2}{2} $$This is just the average of the two radii. Once we have $a_t$ we can hand it to vis-viva and read off the speeds at both ends of the transfer.
Recall the equation from module 3: on any orbit of semi-major axis $a$, the speed at distance $r$ from the central body is $v = \sqrt{GM(2/r - 1/a)}$. We need that twice — once at periapsis ($r = r_1$, on the transfer ellipse) and once at apoapsis ($r = r_2$, on the same ellipse):
Speed at periapsis of the transfer ellipse — where we are right after burn 1:
$$ v_{t,1} = \sqrt{ GM \left( \frac{2}{r_1} - \frac{1}{a_t} \right) } $$Speed at apoapsis — where we are right before burn 2:
$$ v_{t,2} = \sqrt{ GM \left( \frac{2}{r_2} - \frac{1}{a_t} \right) } $$And the circular speeds at the two altitudes are the familiar $v = \sqrt{GM/r}$ from module 2:
$$ v_1 = \sqrt{\frac{GM}{r_1}}, \qquad v_2 = \sqrt{\frac{GM}{r_2}} $$Burn 1 is the speed change from $v_1$ (your old circular speed at $r_1$) to $v_{t,1}$ (the speed you need to start coasting up the ellipse). Burn 2 is the change from $v_{t,2}$ (your speed when you arrive at apoapsis) to $v_2$ (the circular speed at $r_2$):
First burn — the kick-off:
$$ \Delta v_1 = v_{t,1} - v_1 = \sqrt{\frac{GM}{r_1}} \left( \sqrt{ \frac{2 r_2}{r_1 + r_2} } - 1 \right) $$Second burn — the circularization:
$$ \Delta v_2 = v_2 - v_{t,2} = \sqrt{\frac{GM}{r_2}} \left( 1 - \sqrt{ \frac{2 r_1}{r_1 + r_2} } \right) $$Total cost — the number a flight planner cares about:
$$ \boxed{\, \Delta v_{\text{total}} = \Delta v_1 + \Delta v_2 \,} $$The simplified forms come from substituting $a_t = (r_1 + r_2)/2$ into vis-viva and pulling out the circular-velocity factor; both forms are equivalent. Plug in numbers and you get a real budget for a real maneuver.
The coast leg is half of an orbit on the transfer ellipse. The full period of any orbit is $T = 2\pi \sqrt{a^3 / GM}$ from Kepler's third law. Half of that is the transfer time:
$$ \boxed{\, T_t = \pi \sqrt{ \frac{a_t^3}{GM} } \,} $$Notice what's not in the formula: the radii separately. Only $a_t = (r_1 + r_2)/2$ matters. Two transfers with the same average radius take the same time, even if one moves you 100 km and the other moves you 10,000 km.
Let's do the canonical example: get from a 200-km low-Earth orbit ($r_1 = 6{,}571$ km) up to geostationary altitude ($r_2 = 42{,}157$ km), the launch profile that puts every TV-relay and weather satellite where it lives.
The transfer ellipse has $a_t = (6{,}571 + 42{,}157) / 2 \approx 24{,}364$ km. Plugging in $GM_\oplus = 3.986 \times 10^{14}\ \mathrm{m^3/s^2}$:
Circular speed at LEO and GEO:
$$ v_1 \approx 7{,}788\ \mathrm{m/s}, \qquad v_2 \approx 3{,}075\ \mathrm{m/s} $$Speeds at the two ends of the transfer ellipse:
$$ v_{t,1} \approx 10{,}245\ \mathrm{m/s}, \qquad v_{t,2} \approx 1{,}597\ \mathrm{m/s} $$The two burns and the total budget:
$$ \Delta v_1 \approx 2.46\ \mathrm{km/s}, \quad \Delta v_2 \approx 1.48\ \mathrm{km/s}, \quad \Delta v_{\text{total}} \approx 3.93\ \mathrm{km/s} $$Coast time:
$$ T_t \approx 18{,}900\ \mathrm{s} \approx 5.26\ \mathrm{hours} $$Roughly four kilometers per second is the price of admission to GEO from low orbit. By comparison, getting to LEO from the launch pad costs about 9 km/s (more, once you account for atmospheric drag and gravity losses), so once you're in low orbit you've already done two-thirds of the energy work — which is what people mean when they say "low Earth orbit is halfway to anywhere."
The two circles below are an inner orbit at $r_1$ and an outer orbit at $r_2$. The bright ellipse tangent to both is the Hohmann transfer. The spacecraft moves around it at the time-correct rate from Kepler's equation, so it sprints through periapsis and crawls past apoapsis exactly the way a real orbit does. Adjust the altitudes; pick a preset; watch the budget update.
Try the ISS → Hubble preset. Both orbits are within a hundred kilometers of each other, and the total $\Delta v$ comes out to about 74 m/s — a quarter mile per second. A Soyuz capsule's onboard maneuvering propellant could make this transfer twenty times over. Now click LEO → GEO: the budget jumps to nearly four kilometers per second, and the transfer time stretches to over five hours. That's because the ellipse now reaches out to GEO, and apoapsis is far. Most of the time on the transfer is spent crawling near apoapsis, exactly like the slow-aphelion behavior we saw with eccentric orbits in module 4.
Notice too that as $r_2 / r_1$ grows, $\Delta v_1$ keeps climbing but $\Delta v_2$ levels off, then eventually starts coming back down. By the time the destination is far enough away — somewhere past about $r_2 \approx 12 r_1$ — the second burn is barely needed at all, because at the apoapsis of a very elongated transfer ellipse you're already moving very slowly, and the circular speed you need at $r_2$ is also small. There's a famous result in this direction: above a certain ratio of radii, a three-burn "bi-elliptic" transfer becomes cheaper than a single Hohmann, even though it takes drastically longer. (We won't derive that here.)
The Hohmann transfer is the foundational maneuver of orbital mechanics, and you now have everything you need to compute one: pick two radii, average them to get $a_t$, run them through vis-viva twice, subtract from the circular speeds, sum. In module 8 we'll apply the same idea on a much larger scale — between planets — using a trick called patched conics: do one Hohmann around the Sun (Earth's orbit to Mars's orbit) and stitch two short hyperbolic departures and arrivals onto the ends. The math is exactly what's on this page, just with a different central body and bigger numbers.