Missions and destinations

Every flight starts on Earth, from a real launch site, into that site's weather. Most of them also end there: in orbit, back on the landing zone, or on the drone ship. A mission can instead name a destination — the Moon or Mars — and an arrival: a flyby, an orbit, or a landing.

Leaving Earth adds geometry, not magic. The vehicle is the same six-degree-of-freedom model, the program is still yours, and every burn is still one your program commanded. What changes is where the planets are, when you are allowed to go, and which body's gravity the flight is measured against. An Earth mission runs exactly the code it ran before destinations existed.

Three destinations

$\mu$ is the body's gravitational parameter in m³/s², and $J_2$ the dimensionless coefficient of its equatorial bulge. Each arrival is an objective the simulator judges:

• Flyby — pass within 10 000 km of the destination's surface.
• Orbit — be captured: a closed orbit about the destination with the engines off and periapsis above 10 km on the Moon or 60 km on Mars.
• Landing — land on it, graded with the same bands as an Earth landing but under the destination's gravity. If the mission names a landing site you are graded against that pad; if it does not, the whole surface counts.

An Earth mission has a 30-hour clock. A destination mission gets the planned time of flight plus fifteen days. It also needs a real launch instant — the ephemerides need a date — so a mission lit by the classic range's notional solar hour is never interplanetary.

No Earth vehicle preset can make the trip. A trans-lunar injection costs about 3.13 km/s after reaching orbit; the best of the Earth presets arrives in orbit with 3.01 km/s left. The destination missions fly Odyssey, a four-stage stack: a thirteen-engine booster, a vacuum second stage, a restartable hydrolox injection stage, and a three-engine lander with legs and a heat shield. Odyssey Mars swaps the lander for a 7 m aeroshell and carries a larger injection stage. The vehicle covers how stages and engine groups are built.

The shape of a mission

Every destination mission follows the same sequence. The simulator names each phase as it happens — the timeline, the report and the audio all hear about it — but it never fires an engine on your behalf.

1. Ascent to a parking orbit, 200 km by default, launched at the window's time along the window's azimuth.
2. Coast round the parking orbit to the injection point, typically 40 to 90 minutes: less than one revolution.
3. Injection — the trans-lunar or trans-Mars burn.
4. Cruise, with mid-course corrections: three to five days to the Moon, nine to ten months to Mars.
5. Sphere-of-influence entry — the frame hands over to the destination.
6. Capture into orbit, or closest approach on a flyby.
7. For a landing: de-orbit, entry or descent, and the landing burn.

The corresponding flight events are injection and injection-cutoff, correction (any burn of 1 m/s or more between injection and capture), soi-entry and soi-exit, flyby (closest approach, with its range and speed), capture and deorbit. A window warning fires if the injection happens outside the planned window: the transfer will still work, but it will cost more to correct.

Injection

Leaving a parking orbit for Mars means leaving Earth altogether, on a hyperbola. What the transfer needs is the speed the vehicle keeps once it has climbed out of Earth's well — the hyperbolic excess speed $v_{\infty }$ — or equivalently its square, the characteristic energy $C_3=v_{\infty }^2$. The burn that delivers it is sized by the energy equation at the burn point:

where $r_p$ is the parking orbit's radius (6 578 km for 200 km), $\mu$ is Earth's gravitational parameter, $v_p$ is the speed needed at the burn and the square root on the right is the circular speed already there, 7.78 km/s.

For the 2026 Mars window the simulator's planner finds $C_3=9.26{\text{km}}^2/{\text{s}}^2$, so $v_{\infty }=3.04$ km/s, $v_p=11.42$ km/s, and the injection costs 3.64 km/s. Escape alone would cost 3.22 km/s from the same orbit. A lunar transfer never escapes — its $C_3$ is negative, about $-1.9{\text{km}}^2/{\text{s}}^2$ — and costs 3.14 km/s. That is why Mars is only about half a kilometre per second dearer than the Moon at this end: most of the price of either is climbing out of Earth's well, and the two differ only in the last few hundred metres per second of it.

The burn is made low, at the parking orbit, for the reason the Oberth effect explains: a given $\Delta v$ adds the most energy where the vehicle is already fastest. For the Moon the planner places the injection point by minimising $\Delta v$ over the transfer angle, which comes out as a tangential burn at the transfer's perigee — what a real trans-lunar injection is.

Launch windows

Why windows exist

Earth and Mars go round the Sun at different rates, so the angle between them repeats with the synodic period

where $T_{\oplus }$ = 365.26 days is Earth's orbital period and $T_{\text{d}}$ is the destination's, 686.98 days for Mars. That gives $S\approx 780$ days, about 25.6 months. A cheap transfer needs one particular arrangement — on the textbook Hohmann transfer, Mars about 44° ahead of Earth at departure — and that arrangement comes round once per synodic period. Miss it and the next chance is two years away.

The figure is the idealised picture. Move the destination outward and the synodic period shortens towards a year, because the outer planet barely moves while Earth laps it; bring it close to 1 AU and the wait grows without bound. Real Mars windows are not evenly spaced. The simulator's three next departures — 31 October 2026, 22 November 2028 and 24 December 2030 — are 753 and 762 days apart, because Mars's orbit is eccentric and inclined, and the cheapest real transfer is not a Hohmann.

The Moon has no such wait. It goes round Earth every 27.3 days and is reachable on any day; what varies is the time of day.

How the simulator finds them

The planner works from real planetary positions (see the physics model for the ephemerides) and a real solver. It does not use the phase-angle rule above.

1. A grid of transfers. For Mars it scans departure dates and times of flight from 120 to 400 days, and for each pair solves Lambert's problem — the orbit that joins Earth's position at departure to Mars's at arrival in exactly that time — around the Sun, using Izzo's solver. For the Moon the same is done around Earth, from the parking orbit's radius to the Moon's position, over times of flight from 3 to 5 days. Lambert's problem has a course of its own.
2. A price for each cell. Injection $\Delta v$ as above, plus the arrival burn: nothing for a flyby, and for an orbit or a landing a capture into a 2-hour ellipse at the Moon (periapsis 100 km) or a one-sol ellipse at Mars (periapsis 250 km, apoapsis about 33 800 km). A landing is priced as the capture only; the descent belongs to the vehicle, not the window.
3. Minima become windows. One per Mars opposition — minima within 120 days of each other are merged. The Moon's cost varies by under a percent across a month, so its candidates are thinned to roughly one a day. The soonest opportunity is always offered alongside the cheapest.
4. The daily window at the site. A launch can only reach a parking orbit whose plane contains the pad, and the transfer fixes one direction that plane must also contain: the outgoing asymptote for Mars, the Moon's arrival position for the Moon. The planner takes the lowest inclination that works, $i=\max (|\delta |,|\varphi |)$, where $\delta$ is that direction's declination and $\varphi$ the site's latitude. The pad sweeps through the plane twice per sidereal day; the window opens and closes where it is within 2° of it.
5. The azimuth. The relation $\sin \beta =\cos i/\cos \varphi$ gives the heading $\beta$ in inertial space. A vehicle on the pad already moves east with the ground — about 410 m/s at Cape Canaveral — so the planner removes that motion and reports the heading the vehicle actually flies. At Kourou, for the Moon, that is 61.4° rather than 63.1°: a 1.6° difference that would otherwise put the vehicle some ten thousand kilometres out of plane by the time it reaches the Moon.
6. Coast, then re-solve. Liftoff is at the crossing. The vehicle coasts to the injection point, and the transfer is solved again at liftoff plus coast, so that the plan you fly is consistent from pad to arrival.

When the pad's latitude is higher than the direction's declination — the usual case at Cape Canaveral — the required inclination equals the latitude and the pad only grazes the plane instead of crossing it. The two daily crossings merge into one long window, about ±94 minutes at the Cape. A window the site cannot reach at all is not dropped: it is returned marked infeasible, with the reason, such as a declination out of reach from that latitude.

Three things about the planner are worth knowing before you trust it:

• The lunar search stops at five days, and the cheapest lunar transfers are slower than that, so lunar solutions sit on the five-day edge.
• A search span with no Mars opposition in it returns poor local minima alongside the good ones. They sort last, but read the total $\Delta v$ rather than trusting a window's place in the list.
• It does not model the drift of the parking orbit's plane during the coast. At 200 km and 28.6° Earth's bulge turns the plane about half a degree in 90 minutes — well inside the ±2° window, but the program has to absorb it.

The chosen window is frozen into the mission at launch, as the live weather is, and the program reads it as fc.plan: when to inject, the $\Delta v$, the parking inclination and azimuth, and the departure asymptote.

Cruise

Between injection and arrival the vehicle coasts through the full gravity field: the central body's point mass and $J_2$, plus the Sun, Earth and Moon — and Mars on a Mars mission — as third bodies. Nothing is approximated by a chain of conics.

Integrating that at the flight model's 20 ms step would take the whole cruise to Mars about 1.3 billion steps. A program that calls fc.cruise(seconds) instead allows cruise rails: one simulation step may then cover up to an hour, integrated by fourth-order Runge–Kutta with the step sized by how fast the trajectory turns. The step is capped so the trajectory bends by no more than half a degree in it, and it shrinks as the vehicle approaches anything. Checked against an adaptive Dormand–Prince integrator at a tolerance of $10^{-12}$, it lands within centimetres after a 280-day Mars cruise. The rails drop back to 20 ms before the vehicle meets an atmosphere or comes within 2 000 km of any surface, they never step across a sphere-of-influence boundary, and they only engage when every live vehicle is coasting. fc.sleep keeps its old meaning — 20 ms steps on a cheaper derivative — so a parking-orbit coast is never stretched. The acceptance test flies the heliocentric leg to Mars in under two minutes of wall time and requires it to land within 100 km of a finer reference integration.

Mid-course corrections are the cruise's real work. A transfer solved on the ground is only as good as the injection that flew it, and a small error at the start is a large miss at the end. The reference programs show one way to keep the corrections small:

• Predict by integrating, not by conics. An Earth-centred conic carried out to the Moon answers the wrong question, and even patching to a lunar conic at the sphere of influence is about 1 000 km out, because Earth's tidal pull there is half the Moon's own.
• Correct with the smallest burn that works. A minimum-norm Newton step on the predicted miss, rather than a fresh Lambert solve. A flyby does not care when it arrives, and pinning the arrival time was most of the cost: 56 m/s to move perilune 150 km against 1.5 m/s along the gradient.
• Re-plan at ignition. A solution is the $\Delta v$ for one place and one instant; fired after a two-minute slew it flies something else.

The flight computer gives you what you need for this: fc.state for the state vector, fc.ephemeris(id, t) for where another body is or will be, fc.soi and fc.soiTime() for the handover, and fc.target.closestApproach. The reference lists them all.

Handing over to another body

Near a body, its own gravity dominates; far away, its parent's does. The usual boundary is the sphere of influence,

where $a$ is the semi-major axis of the body's orbit about its parent, $m$ the body's mass and $M$ the parent's. It puts the Moon's boundary 66 183 km from its centre, Earth's 924 630 km out towards the Sun, and Mars's at 577 460 km. A flight to Mars crosses two: out of Earth's sphere into the Sun's, then into Mars's. Spheres of influence and patched conics derives the radius and what it leaves out.

In a patched-conic model the boundary is where the physics changes: inside, only the body pulls. Here it is only where the bookkeeping changes. Every body pulls throughout the flight. What the crossing switches is the frame — the flight is re-centred on the new body, with its pole as the frame's axis so that body's $J_2$ is applied correctly — along with which surface, atmosphere and weather the world answers for. The change is a fixed rotation and a velocity shift applied to every vehicle in the same step; the inertial state is conserved exactly, and nothing happens to the trajectory at the line.

To stop a trajectory that grazes the boundary from flipping frames back and forth, the switch has a 1 % band: a vehicle leaves a sphere at 1.01 times its radius and enters at 0.99 times. It is tested at most once a second. An Earth-fixed guidance plane means nothing about another body, so a vehicle on the 'launch' plane falls back to 'orbit' at the crossing.

Arriving

The Moon

There is no air, so everything at the far end is a burn: capture, the de-orbit and the whole descent. The Moon is a sphere of 1 737.4 km with no terrain, turning once every 27.3 days — slowly, but not negligibly: its ground moves at 2.7 m/s at 54° north, and a lander that nulls only its in-plane speed touches down sliding. The reference landing program captures into a low orbit, then aims its hoverslam to arrive at 1.5 m/s rather than at rest, because its three engines at their lowest throttle still push 32 kN against an 18 kN lander: it cannot hover, only arrive.

Mars

Mars has an atmosphere too thin to land on and too thick to ignore. The simulator's is a mean model: no dust storms, no daily cycle, no variation with latitude, and uncertain by a factor of about two between 75 and 105 km. Fidelity and its limits has the details. The arrival is an entry behind an aeroshell followed by a powered landing; there is no parachute model. How much an aeroshell helps is set by the ballistic coefficient, $\beta =m/(C_{D}A)$ — mass over drag area. Odyssey Mars's 7 m shell gives 507 kg/m², against Curiosity's 146 and Viking's 64. Entry, descent and landing on Mars is the course for this.

What works today, and what does not

The four destination programs were flown from four launch sites over five seeds each, after the third physics audit. The results are recorded as they came out.

Until the third physics audit the lunar landing failed from Kourou (0/5) and the Mars flyby arrived on 2 flights in 20. Both were bugs, not physics: the window planner always gave the north-easterly launch heading, so from Kourou, whose crossing of the lunar plane is the south-easterly one, the parking orbit came out 41° from the Moon; and the programs' circularisation never stopped, leaving a 185 × 10,300 km parking orbit and 1.6 km/s of the injection stage spent. With those fixed — and a gravity term with the wrong sign, the programs' own trajectory predictor and the ephemeris's velocities — what remains is:

• The Mars lander does not land. It now reaches the entry corridor with its full 1.96 km/s, and the aeroshell is what falls short: at a ballistic coefficient of 507 kg/m² the air leaves it at 2.6 km/s at 30 km, and once the engines light their plume collapses the drag, so no ignition point stops it. A 9 m aeroshell brought a test lander to 40 m/s at 118 m with 2.7 t to spare; the reference vehicle has not been changed.
• The lunar landing does not aim for a site. It comes down under its capture orbit, about 5,000 km from any chosen point, in whatever light is there.
• Cold gas runs short from Kourou. A large slew in vacuum uses far more reaction-control gas than it should; on three Kourou flights the gas ran out mid-cruise and a later correction cost 391–492 m/s.
• The aeroshell's sizing is analytic. The claim that 507 kg/m² is enough rests on an estimate; it has not been verified in the full six-degree-of-freedom aerodynamics.

Some limits belong to geography rather than software. A site cannot reach an orbit inclined less than its own latitude, so a destination whose required plane is shallower than that is out of reach from there; the planner says so rather than hiding the window.