Interplanetary Transfers and Launch Windows

Synodic periods, departure phase angles, interplanetary Hohmann transfers and porkchop plots

July 2020

Within eleven days in July 2020, three separate Mars missions left Earth. 19 July — the United Arab Emirates launched Hope from Tanegashima. 23 July — China launched Tianwen-1 from Wenchang. 30 July — NASA launched Perseverance from Cape Canaveral. Three agencies, three countries, three vehicles, no coordination between them. And they left within a fortnight of each other, as did the missions in 2018, and 2016, and 2013, and every Mars mission in history. This is not fashion, or funding cycles, or competition. It is geometry. A minimum-energy transfer to Mars takes 259 days, during which Mars travels a specific distance along its own orbit. For the spacecraft and the planet to arrive at the same point at the same instant, Mars must be at a very specific angle ahead of Earth at the moment of departure. That angle is 44.3 degrees. Earth and Mars hold it for a few weeks, and then the geometry moves on. And it does not come back for 26 months. Miss it and there is no recovery. There is no amount of propellant that buys you a better phase angle — the planets are where they are. A mission that misses its window is not delayed by weeks. It is delayed by two years and two months, at a carrying cost of hundreds of millions of dollars, and every Mars programme in history has been organised around that single number.

Watch first · presented by Shoaib Aziz

The numbers in this film are the labs' own: a Hohmann transfer to Mars has a = 1.26 AU and takes 259 days, in which Mars covers 135.7°, so it must lead the Earth by 44.3° at departure; the synodic period is 2.135 years (26 months) for Mars and 1.09 years (13 months) for Jupiter; departure needs v∞ = 2.95 km/s (C₃ = 8.7 km²/s²), a 3.57 km/s burn from a 400 km orbit; the spacecraft arrives at 21.48 km/s against Mars's 24.13, so v∞ = 2.65 km/s and capture into a 500 km circular orbit costs 2.07 km/s — 5.64 km/s in all, 37 % of it spent on stopping; Venus takes 146 days with a phase angle of −54°, Jupiter 998 days with a 15.06 km/s capture; the porkchop slice runs from C₃ = 19.5 at 150 days to its minimum of 8.67 at 260 days (a 180.6° transfer); and aerobraking saves 607 kg (65 %) of the capture propellant at Mars and 85 % at Venus.

The recurrence interval is the synodic period: how long until two bodies return to the same relative geometry. It is not the period of either one, and it is longer than both.

\frac{1}{T_{syn}} = \left|\frac{1}{T_1} - \frac{1}{T_2}\right|

Synodic period from two orbital periods. The closer the two periods, the longer the wait — which is why Mars, at 1.88 years against Earth's 1, has such an awkward 2.135-year cycle.

Target	Orbital period (yr)	Synodic period	Practical consequence
Venus	0.615	1.60 yr (19 months)	Frequent windows, short transfers
Mars	1.881	2.135 yr (26 months)	The defining constraint of Mars exploration
Jupiter	11.86	1.09 yr (13 months)	Nearly annual — the outer planets are easy to reach often
Saturn	29.46	1.035 yr	Almost every year
Neptune	164.8	1.006 yr	Effectively continuous

💡Synodic periods of the outer planets

This surprises people every time. Jupiter is five times further away than Mars, takes six years to reach rather than nine months, and yet its launch windows come round twice as often. The reason is in the formula. The synodic period depends on the difference of the angular rates. Jupiter moves so slowly — 0.084 revolutions per year — that it is nearly stationary from Earth's point of view. Earth laps it almost exactly once per year, so the geometry repeats every 13 months. Mars is the worst case precisely because it is the closest outer planet: fast enough to move appreciably, slow enough that Earth takes a long time to lap it. The two rates are uncomfortably similar, and the difference is small. The same effect gives the Moon its 29.5-day synodic month against a 27.3-day orbital period, and it is why the inner planets are harder to schedule than the giants.

The phase angle follows exactly as in Lesson 6-4, with a planet playing the role of the target. The transfer takes

t_t

; the destination sweeps

n_2 t_t

during that time; and you must arrive together at the far apsis:

\phi_{\text{departure}} = 180° - n_{2}\,t_{t} = 180°\left(1 - \left(\frac{a_t}{a_2}\right)^{3/2}\right)

The required lead angle at launch. For Mars it comes to 44.3°; the target must be that far ahead of Earth in its orbit at the moment you leave.

Figure 8.4

Earth and Mars at departure and at arrival. Mars leads by 44.3° when the spacecraft leaves, sweeps 135.7° during the 259-day cruise, and the two meet at the transfer ellipse's aphelion. Get the angle wrong by 10° and the spacecraft arrives at Mars's orbit with Mars 20 million kilometres away.

📐Worked Example: Earth to Mars, end to end

Heliocentric:

\mu_\odot = 1.32712\times10^{11}

km³/s²,

r_1 = 1

= 1.496\times10^8

km,

r_2 = 1.524

= 2.2799\times10^8

km. Transfer ellipse.

a_t = (1.496 + 2.2799)/2 \times 10^8 = 1.8879\times10^8

km = 1.262 AU. Flight time.

t_t = \pi\sqrt{a_t^3/\mu_\odot} = 2.24\times10^7

s = 259 days. Phase angle. Mars's period is 1.881 yr, so it sweeps

360 \times (259/687) = 135.7°

. Required lead:

180 - 135.7 = \mathbf{44.3°}

. Departure energy. Earth's orbital speed is 29.78 km/s. Required heliocentric speed:

v_{t1} = \sqrt{\mu_\odot(2/r_1 - 1/a_t)} = 32.73

km/s.

v_\infty = 32.73 - 29.78 = \mathbf{2.95}

km/s, so

C_3 = 8.70

km²/s². From a 400 km parking orbit:

\Delta v = \sqrt{8.70 + 117.74} - 7.673 = \mathbf{3.57}

km/s. Arrival.

v_{t2} = \sqrt{\mu_\odot(2/r_2 - 1/a_t)} = 21.48

km/s, while Mars moves at 24.13 km/s. The spacecraft arrives slower than Mars, so Mars catches up from behind:

v_\infty = 24.13 - 21.48 = \mathbf{2.65}

km/s. Capture. To enter a 500 km circular Mars orbit (

\mu_{\text{Mars}} = 42{,}828

km³/s²,

r_p = 3896

km): Arrival periapsis speed

= \sqrt{2.65^2 + 2(42828)/3896} = \sqrt{7.02 + 21.98} = 5.386

km/s. Circular speed there

= \sqrt{42828/3896} = 3.316

km/s.

\Delta v_{\text{capture}} = \mathbf{2.07}

km/s. Total: 5.64 km/s from LEO. And note where it goes — the capture burn is 37% of the whole budget, spent entirely on stopping.

🛰Aerobraking: getting the capture burn for free

That 2.07 km/s capture burn is the single largest item after departure, and Mars has an atmosphere. Thin — about 1% of Earth's — but not nothing. Aerocapture would use a single deep atmospheric pass to remove all the excess energy at once. It is enormously attractive and has never been flown, because a pass deep enough to work is a pass deep enough to destroy the spacecraft if the atmosphere is denser than predicted, and Mars's atmosphere is notoriously variable. Aerobraking is the flown compromise. Capture into a very elliptical orbit with a small burn, then dip through the upper atmosphere at each periapsis — hundreds of times, over months — shaving a little energy from apoapsis on every pass. Mars Global Surveyor spent four months aerobraking. Mars Reconnaissance Orbiter spent five months and 445 passes, saving roughly 600 kg of propellant — a saving that made the mission fit its launch vehicle at all. It is also nerve-racking. The spacecraft is deliberately flown into an atmosphere whose density is forecast rather than known, several hundred times, with a solar array acting as the brake. MGS aerobraked with a partially-deployed panel that had been damaged at launch, at a reduced rate, for exactly that reason.

Real mission design does not use a single Hohmann transfer. Departure and arrival dates are both free variables, and every combination has its own cost. The tool for exploring that two-dimensional space is the porkchop plot.

Figure 8.5

A porkchop plot: departure date on one axis, arrival date on the other, and contours of departure C₃. Each contour encloses a region of feasible missions, and the nested closed curves give the chart its name. The minimum sits inside the innermost contour. Diagonal lines are constant flight time; the ridge running through the middle is where Earth and Mars sit on opposite sides of the Sun: a transfer of exactly 180° has no unique plane, and Unit 9 shows why.

💡How to read a porkchop plot

Every point on the chart is the solution of a boundary-value problem — Lambert's problem, the subject of Unit 9 — with

\vec{r}_1

= Earth on the departure date and

\vec{r}_2

= Mars on the arrival date. The contours are constant departure

C_3

. Pick your launch vehicle, read off its maximum

C_3

for your payload mass, and everything inside that contour is a mission you can fly. Two lobes appear in every window: a Type I transfer with

\Delta\theta < 180°

(shorter, usually cheaper) and a Type II with

\Delta\theta > 180°

(longer, sometimes lower arrival energy). The ridge between them is the 180° singularity. The launch period is the horizontal width of the feasible region — typically two to four weeks. That is the actual number that drives the launch campaign, and it is why a scrubbed launch is a genuine crisis rather than an inconvenience. And the arrival matters too. A cheap departure may produce a brutal arrival

v_\infty

, so designers usually plot both and choose from the intersection. Perseverance's launch period ran from 17 July to 15 August 2020; processing delays meant it launched on 30 July, with about two weeks of margin remaining.

Figure 8.6

The full patched-conic picture in one frame: departure hyperbola inside Earth's SOI, heliocentric transfer ellipse, arrival hyperbola inside Mars's SOI. The velocity triangles at each seam show the vector addition — heliocentric velocity equals planetary velocity plus hyperbolic excess.

⚠What the simple picture leaves out

The 44.3° phase angle assumes both planets are on coplanar circular orbits. Neither is true. Mars is eccentric.

e = 0.0934

, so its distance from the Sun varies between 1.38 and 1.67 AU. Windows are therefore not equal: the 2018 opposition was far more favourable than 2027 will be, and

C_3

requirements vary by nearly a factor of two across the cycle. Missions are planned years ahead around which windows are cheap. Mars's orbit is inclined 1.85°. Small, but at 200 million kilometres a 1.85° plane change is expensive, and the transfer must be designed to cross Mars's orbital plane at the right place — usually by departing slightly out of the ecliptic. The ephemeris is not circular. Real design uses JPL's numerically integrated planetary ephemeris, not Kepler orbits, because a hundred metres per second of error is a hundred metres per second of propellant. Which is why the honest description of the method is: the patched-conic Hohmann gives you the window, the porkchop plot gives you the dates, and a numerical optimiser gives you the trajectory you actually fly.

↓But there is a problem

So we can reach Mars. 3.57 km/s to leave, 259 days to cross, 2.07 km/s to stop — and a two-week window every 26 months in which any of it is possible. Now try Jupiter directly.

C_3 \approx 80

km²/s², a departure Δv of 6.3 km/s, and a launch vehicle that delivers a few hundred kilograms. Now try Neptune, at 30 AU. The direct transfer takes thirty years and the departure energy is beyond any vehicle ever built. And yet in August 1977 a Titan IIIE — a rocket with barely enough energy to reach Jupiter — launched a 722 kg spacecraft that visited Jupiter, Saturn, Uranus and Neptune, arriving at Neptune in 1989, twelve years later. Voyager 2 did not carry the energy for that trajectory. Nothing carried the energy for that trajectory. It stole the difference — from the planets themselves, four times, taking a little orbital momentum from each and leaving them very slightly slower than it found them. And the explanation for how that is possible is the single best idea in astrodynamics.