The window that comes back and the cost that does not

The first rung of this anchor built the porkchop plot: a grid of solved Lambert problems, one departure date and one flight time per cell, contoured by the energy the departure costs. It is a picture of one window.

There are many windows, and the fact that everybody knows about them — that they come every twenty-six months — is the least interesting thing about them. That interval is a synodic period and a synodic period is arithmetic: the two planets return to the same relative configuration at a rate that is the difference of their angular rates, and for Earth and Mars that is 780 days. It never varies.

What varies is the price.

It is worth pinning the synodic period down, because it is the one exactly known quantity in the essay. Two bodies with orbital periods $P_1$ and $P_2$ return to the same relative geometry after

$$\frac{1}{P_{\rm syn}} \;=\; \left|\frac{1}{P_1} - \frac{1}{P_2}\right|$$

which for 365.256 and 686.980 days gives 779.9. That number is a difference of reciprocals and nothing else; it does not know about eccentricity, inclination, or where anything is. It is why “twenty-six months” is quoted with no qualification and is correct.

Why the cost has a period of its own #

The synodic period is the interval between successive alignments. It says nothing about where in their orbits the two planets are when they align.

Earth’s orbit is nearly circular, so its position hardly matters. Mars’s is not: an eccentricity of 0.093 means its distance from the Sun varies between 1.38 and 1.67 astronomical units, a range of twenty per cent. A transfer that arrives near Mars’s perihelion has substantially less distance to cover and meets a planet moving faster along its orbit; one that arrives near aphelion has to reach further out and match a slower planet.

Whether a given opportunity does the first or the second depends on where Mars is in its own orbit at the moment of alignment. That advances by a fixed amount each synodic period — 780 days is 1.135 Martian years, so Mars advances by 0.135 of a revolution between one alignment and the next — and returns to the same phase after about 7.4 opportunities, which is fifteen and a half years.

The arithmetic of the fifteen-year beat is worth doing explicitly, because it is the same kind of calculation and gives a second exactly known number. Mars completes 779.9/686.98 = 1.1352 revolutions per synodic period, so it advances 0.1352 of a revolution between one alignment and the next. That returns to the start after 1/0.1352 = 7.40 opportunities, which at 780 days each is 5,770 days, or 15.8 years. Nothing in the figure was told that number; it comes out of the Lambert solutions on their own.

The control experiment #

The attribution to Mars’s eccentricity is not an inference from the shape of the curve. It can be tested by removing it.

That is as clean a demonstration of cause as this subject offers: one number is changed, the effect vanishes, and what is left is accounted for by the other eccentricity in the problem.

It is worth noticing what the control does not change. The synodic period is untouched — it depends only on the two semi-major axes through Kepler’s third law, and the eccentricity does not enter. So the windows still come every 780 days in the circular case; they simply all cost the same. The period and the price are governed by disjoint sets of orbital elements, which is why they have different periodicities and why one is exactly known and the other is not.

There is also a piece of luck in the real solar system that this makes visible. Mars’s eccentricity is unusually large for a planet — larger than any other except Mercury — and the Earth’s is unusually small. Had both been typical, the variation would be smaller and Mars missions would be a routine biennial matter rather than something scheduled against a decade-and-a-half rhythm.

What a factor of two and a half buys #

Departure energy is quoted as C₃, the square of the speed left over at infinity after escaping the Earth. It is a convenient number because it does not depend on the parking orbit, and it is a misleading one because it is a square.

The velocity a launch vehicle must supply at low Earth orbit is $\sqrt{C_3 + v_{\rm esc}^2} - v_{\rm circ}$, and with an escape speed of 11.2 kilometres a second the C₃ term is a correction rather than the whole. Going from C₃ = 5 to C₃ = 13 raises the required departure velocity from about 3.60 to 3.83 kilometres a second — six per cent.

Six per cent in velocity is not six per cent in mass. The rocket equation is exponential, and with an upper stage exhaust velocity of about 4.4 kilometres a second, six per cent more velocity is about five per cent more propellant fraction, which compounds through the staging into something between ten and twenty per cent of the payload delivered. For a mission of fixed launch vehicle that is the difference between carrying an instrument and not.

Two windows are a difference and three begin to be a pattern. The Mars opportunities do not alternate cheap and dear: they drift, because the synodic period is 25.6 months and the geometry only comes back into register after seven of them, so the sequence of costs has a period nearer fifteen years than two.

There is a second and quite separate reason some windows are better, and it is one this planar calculation does capture. The flight time at the optimum is not the same in every window: the cheap windows are cheap partly because a shorter transfer is available, and a shorter transfer means less exposure for the spacecraft and less consumables for a crew. The 2003 opportunity offered arrivals about six months after departure; the dearest offer nearer nine. For an uncrewed mission that is a minor consideration and for a crewed one it is close to decisive.

What the plot leaves out #

The calculation above is planar, and Mars’s orbit is inclined by 1.85 degrees to the ecliptic. That is a small angle and it is not a small effect.

A transfer between two orbits that are not coplanar must change its plane, and the cheapest place to do that is where the two planes cross. If the transfer’s geometry puts the arrival far from a node, the plane change has to be paid somewhere expensive — and the cost peaks when the transfer angle passes through 180 degrees, where the departure and arrival points and the Sun are nearly collinear and the plane of the transfer is undefined.

That produces a ridge running through the middle of a real porkchop plot, dividing it into a Type I region with transfer angles below 180 degrees and a Type II region above. The ridge is not a small feature: near it the plane-change cost diverges, so the plot has an impassable band across it rather than a bump, and the two regions are genuinely separate families of trajectory rather than two ends of one.

The practical escape is the broken-plane manoeuvre: instead of doing the whole plane change at departure or arrival, a small burn is made partway along the transfer, at a point chosen so that the two halves are each cheaper than the single change would have been. It is a strictly better solution and it costs a mid-course engine firing, which is why it is used on missions that have an engine and not on those that do not. It also cannot be represented on a porkchop plot at all, because the plot’s axes are two dates and the manoeuvre is a third free parameter — so the figure a mission designer uses is a projection of something with more dimensions than it can show. Neither figure here has that ridge, because neither has the inclination. The consequence is that a real mission chooses between a shorter Type I trajectory and a longer Type II one, and the choice is not always the same as the choice this planar calculation would make.

Where the departure energy actually goes #

It is worth following the number through the vehicle, because C₃ is quoted so often and understood so rarely.

A spacecraft in a low parking orbit is moving at 7.8 kilometres a second and needs 11.2 to escape. The extra it needs beyond escape is not added arithmetically but through the vis-viva relation: the speed required in the parking orbit is $\sqrt{C_3 + v_{\rm esc}^2}$, so a C₃ of 9 km²/s² — 3 kilometres a second at infinity — requires 11.6 in the parking orbit rather than 14.2. Three kilometres a second at infinity costs four hundred metres a second at departure, and that compression is the Oberth effect: a burn deep in a gravity well buys far more energy than the same burn far from one.

The consequence for this essay is that the fractional variation in the burn is much smaller than the fractional variation in C₃. A factor of 2.6 in C₃ is a six per cent difference in the departure burn. It is still decisive, because the exponential that decides what can be flown turns six per cent in velocity into something like fifteen per cent in delivered mass — but the mechanism is the exponential rather than the energy.

One more consequence of the exponential deserves its own line, because it explains a pattern in the record. A mission designed for a cheap window and slipped to the next one does not merely cost more propellant; it usually cannot fly at all on the same vehicle, because the payload has been sized to the vehicle’s capability in the window it was designed for. So a slip of one opportunity is a redesign, not a delay — which is why missions that miss a window are cancelled about as often as they are rescheduled, and why the twenty-six-month cadence appears in programme schedules as a hard wall rather than as a preference.

What is actually flown #

The trajectories flown are not the minima on these plots, and the reasons are worth naming because they are all of the same kind. Each is a constraint from outside the two-body geometry, and each one stitches another patch onto a trajectory the conics do not describe.

The arrival matters too. C₃ is a departure cost. A mission that must enter orbit around Mars has to pay a second, larger number to slow down, and the transfer that minimises the first does not minimise the sum. Missions that land directly rather than orbiting have a different optimum again, because they can arrive fast and let the atmosphere do the braking.

The launch period is negotiated. A vehicle cannot be guaranteed to launch on one day. The mission is designed for a period of two to four weeks, and the trajectory is chosen so that every day in that period has an acceptable cost — which means starting from a point up the slope rather than at the minimum.

Arrival conditions are constrained. Entry angle, lighting at the landing site, the geometry for communications through an orbiting relay, and the Martian season all constrain the arrival date, and each constraint cuts a slice out of the plot. Dust-storm season is a real one: a landing during the global storm season carries an atmosphere whose density profile is different by tens of per cent, and the entry corridor is a degree and a half wide to begin with.

Mass is traded against time. A slower transfer is usually cheaper, and there is always a slower transfer available; what stops a mission taking it is the cost of keeping a spacecraft alive and staffed for the extra months, and the fact that a longer flight arrives at a worse Martian season. The optimum on the plot is a minimum in one variable of a problem with four.

The result is that a flown trajectory typically costs one to three km²/s² above the minimum on a plot like these, and that penalty is smaller than the window-to-window variation the first figure shows. The choice of which opportunity to use matters more than the choice of trajectory within it.

The other planets, briefly #

The same arithmetic runs for every target and produces two numbers that between them describe an entire class of missions.

Venus. Synodic period 584 days, eccentricity 0.0068 — the most nearly circular orbit in the solar system. So Venus windows come more often than Mars’s and cost very nearly the same every time. The window-to-window variation is a few per cent rather than a factor of two and a half, and Venus mission planning has no fifteen-year cycle to wait for.

Jupiter. Synodic period 399 days, barely longer than a year, because Jupiter moves so slowly that the Earth almost laps it annually. Windows are frequent; they are also expensive, at a C₃ near 80 for a direct transfer, which is why stealing speed from a planet is the usual route and a direct one is a luxury.

Mercury. Synodic period 116 days — windows three times a year — and a departure that is cheap while the arrival is ruinous, because arriving at Mercury means shedding the enormous speed picked up falling towards the Sun. Every mission to Mercury has spent years on gravity assists rather than taking one of the frequent direct windows.

The pattern across the three is that the synodic period says how often to look and nothing at all about whether to go.

The generalisation #

The structure worth extracting is that a periodic opportunity and a periodic cost are different things, and the second inherits the periods of everything the geometry touches.

The synodic period is a beat between two angular rates and nothing else enters it. The cost involves the actual positions, so it inherits the eccentricities, the orientations of the two lines of apsides, and the inclination — and a quantity built from several periodicities has a period that is their common multiple, which is generally long.

The same shape appears wherever this collection finds a repeating configuration whose consequences do not repeat. An eclipse repeats a third of a world away, because the saros is not a whole number of days and the alignment recurs over a rotated Earth. A month has to be tabulated because the lunar orbit’s own periods are incommensurate. In each case an alignment is exactly periodic and its appearance is not.

There is a second reading which is about how a constraint becomes invisible. Nobody plans a Mars mission around the synodic period, because it is not a decision — it is the shape of the calendar. What gets decided is which opening to use, and that decision is made against the curve in the first figure. So the quantity everybody knows about is the one nobody optimises, and the quantity that determines the schedule is one that does not appear in any popular account of how missions are planned.

The corollary is a practical one. When a window repeats and its cost does not, the schedule is set by the cost. Mars mission planning is not organised around the twenty-six-month cadence, which is a given, but around which of the next several openings is on the favourable side of a fifteen-year cycle — and that is why the launches cluster.

Where the ladder goes next #

The next rung puts the inclination back and draws the ridge. The Type I and Type II branches, the broken-plane manoeuvre that splits the plane change into two smaller ones performed away from the nodes, and the reason a mission occasionally accepts a transfer angle very close to 180 degrees despite the singularity, are one connected argument and they are the difference between the plot above and the plot a mission designer uses.

Further rungs on this anchor: multi-revolution transfers, where one time of flight admits five ways round, so the Lambert problem acquires several solutions and the plot acquires several minima; where the spacecraft goes round more than once and the Lambert problem acquires several solutions; windows to targets whose orbits are steeply inclined, where the plane change dominates everything; the use of a gravity assist to convert a bad window into an acceptable one, which is what makes the outer planets reachable at all; and the low-thrust case, where there is no window in the same sense because the spacecraft is under continuous thrust and the whole framing changes.