Move the central body from Earth to the Sun, and the same Hohmann math from module 7 tells you how to fly between planets. The catch is that the planets move while you're en route — so you have to launch at the right moment, into a trajectory that arrives where the destination will be, not where it is now.
From the Sun's point of view, every planet is on a near-circular orbit. Earth at $r_E = 1\ \mathrm{AU}$, Mars at $r_M = 1.524\ \mathrm{AU}$, Jupiter at $5.2\ \mathrm{AU}$, and so on. To go from one to another, the cheapest single maneuver is a Hohmann transfer with periapsis on the Earth-departure orbit and apoapsis on the destination orbit. The transfer ellipse has the same semi-major axis as before:
$$ a_t = \frac{r_E + r_{\text{dest}}}{2} $$The two burns and the transfer time come from the same vis-viva equation, with $GM$ now meaning the Sun's $GM_\odot = 1.327 \times 10^{20}\ \mathrm{m^3/s^2}$ instead of Earth's:
$$ \Delta v_1 = \sqrt{\tfrac{GM_\odot}{r_E}} \left( \sqrt{\tfrac{2 r_{\text{dest}}}{r_E + r_{\text{dest}}}} - 1 \right), \quad \Delta v_2 = \sqrt{\tfrac{GM_\odot}{r_{\text{dest}}}} \left( 1 - \sqrt{\tfrac{2 r_E}{r_E + r_{\text{dest}}}} \right) $$ $$ T_t = \pi \sqrt{ \frac{a_t^3}{GM_\odot} } $$For the canonical Earth-to-Mars Hohmann, plugging in numbers gives $\Delta v_1 \approx 2.95\ \mathrm{km/s}$, $\Delta v_2 \approx 2.65\ \mathrm{km/s}$ (total $\approx 5.6\ \mathrm{km/s}$), with a transfer time of about 259 days — roughly eight and a half months. Jupiter is a 2.7-year trip; Saturn six years; Neptune more than thirty.
Here's what's new at planetary scale. By the time your probe completes its half-ellipse, the destination has to be at the apoapsis. That means at launch, the destination must be ahead of Earth by exactly the right angle — far enough that, while the probe coasts up the ellipse, the planet sweeps around its orbit and meets the probe at the rendezvous point.
Compute how far the destination moves around the Sun during the transfer:
$$ \Delta\theta_{\text{dest}} = n_{\text{dest}} \cdot T_t = \frac{2\pi}{T_{\text{dest}}} \cdot T_t $$For the destination to meet the probe at apoapsis (which is 180° around the Sun from Earth's launch point), it must start ahead by:
$$ \boxed{\, \theta_{\text{lead}} = 180^\circ - \Delta\theta_{\text{dest}} \,} $$For Mars, this works out to a lead angle of about 44°. For Jupiter, about 97°. For closer destinations like Mercury or Venus, the angle is negative, meaning the planet has to be behind Earth at launch — they'll catch up to it on the inward trip.
Earth and Mars line up in this configuration only every synodic period:
$$ T_{\text{syn}} = \frac{1}{\left| \, 1/T_E - 1/T_{\text{dest}} \, \right|} $$For Earth-Mars this comes out to about 780 days — roughly 26 months. That's the famous "Mars launch window" that recurs every two years and change. Miss one and you wait. NASA's Mars-bound spacecraft (Curiosity, Perseverance, the InSight lander, and so on) all launched within a few-week span around the same window in their respective years.
So far we've treated Earth as a moving dot and ignored its gravity entirely. That works for the Sun-centered Hohmann, but a real probe doesn't start in heliocentric orbit — it starts in low Earth orbit, deep inside Earth's gravity well. Likewise it doesn't end in heliocentric orbit; it captures around the destination planet.
The trick that makes all this manageable is called patched conics. The idea is to break the trip into three pieces, each governed by a single dominant body:
The conic in each phase — hyperbola, ellipse, hyperbola — is exact within its own region of dominance. The "patches" between regions are where the math gets approximated; in reality, of course, gravity from all three bodies is acting at all times. For most missions the approximation is excellent, and it's what real flight planners use for first-cut design before refining with full numerical simulation.
One more trick rounds out the basics. Instead of arriving at a planet to capture, you can fly past it on a hyperbolic flyby and use its gravity to steal energy. From the planet's frame, a flyby is symmetric — the probe arrives at speed $v_\infty$ and leaves at speed $v_\infty$, just in a different direction. But from the Sun's frame, that direction change adds (or subtracts) energy because the planet itself is moving. Voyager 2 used flybys at Jupiter, Saturn, and Uranus to reach Neptune in just twelve years — a trip that would have taken thirty by Hohmann. The Cassini mission to Saturn took two Venus flybys, an Earth flyby, and a Jupiter flyby just to get there.
The math of a gravity assist is its own rabbit hole and we won't develop it here, but the concept is: borrow momentum from a moving planet. There's no free lunch — the planet loses an utterly imperceptible amount of orbital energy in trade.
The interactive shows the Sun, Earth's orbit, the destination planet's orbit, and the Hohmann transfer connecting them. Earth and the destination both move at their real angular rates, and the probe launches from Earth's position the moment Mars (or whichever planet you've selected) is at the correct lead angle. The animation loops one transfer at a time so you can watch the simultaneous arrival.
Try Mercury. The total $\Delta v$ is enormous (around 17 km/s) — paradoxically, going inward to Mercury costs more than going all the way out to Saturn, because you have to shed a huge amount of Earth's orbital velocity to drop in. Real Mercury missions like MESSENGER and BepiColombo used multiple Venus and Earth flybys precisely to cheat around this cost.
Try Jupiter. The $\Delta v$ is a little under 15 km/s heliocentric, and the transfer takes just under three years. The lead angle is now huge — about 97° — which means Jupiter has to be more than a quarter-orbit ahead of Earth at launch.
Try Saturn. The transfer takes about six years, and Saturn has to be more than 90° ahead at launch, because Saturn moves so slowly that during the long coast it barely budges. The synodic period is roughly a year — Earth-Saturn alignments come around almost yearly because Saturn is almost stationary on the timescale of Earth's orbit.
You now have the full toolbox to design any single-Hohmann interplanetary mission: pick a destination, run vis-viva twice, sum the burns, compute the transfer time, find the lead angle, wait for the synodic window. What we haven't addressed yet is what those delta-v numbers cost in propellant, or what it actually takes to execute a burn precisely enough to arrive where you planned. The next two modules step into the propulsion engineering — the rocket equation for the fuel cost, then burns, timing, and launch windows for the execution side. After that, the curriculum closes with the strange three-body world where Lagrange points appear.