Launch windows: the road to Mars

In July 2020 three missions left Earth for Mars within twelve days: the United Arab Emirates’ Hope orbiter, China’s Tianwen-1 and NASA’s Perseverance rover. That was no coincidence. Mars can only be reached cheaply when the planets are lined up just right, and that happens about every 26 months.

You do not aim at a planet. You aim at where it will be.

Look

Earth and Mars both circle the Sun, Earth on the inside track and faster. A spacecraft leaving Earth does not fly in a straight line: it follows its own orbit around the Sun, a half-ellipse that starts at Earth’s orbit and ends at Mars’s.

• The trip takes about 259 days on the cheapest path.
• In that time Mars moves on too, so it must be about 44 degrees ahead of Earth at departure.
• Earth overtakes Mars and the same lineup repeats only every 780 days, about 26 months.

Miss the window by a few weeks and the trip costs far more fuel, or becomes impossible for the rocket you have. Wait, and the next window is two years away.

Understand

Treating both orbits as circles (Earth at 1 AU, Mars at 1.524 AU) and using the Sun’s μ = 1.327 × 10¹¹ km³/s²:

• The transfer ellipse has a = (1 + 1.524)/2 = 1.262 AU; half its period is 259 days.
• Leaving Earth, the spacecraft needs 2.94 km/s more than Earth’s own 29.8 km/s around the Sun.
• It arrives at Mars’s orbit 2.65 km/s slower than Mars and must be captured (next lesson).
• Phase angle: in 259 days Mars covers 360° × 259/687 ≈ 136°, so it must start 180° − 136° ≈ 44° ahead.

Real orbits are slightly elliptical and tilted, so each window is different; mission designers search thousands of launch and arrival dates at once with “porkchop plots”. Paying a little more Δv buys a shorter trip: Perseverance took 203 days.

Master

Synodic period:

1/S = 1/P_Earth − 1/P_Mars  →  S = 1 / (1/365.25 − 1/686.98) ≈ 780 days

Departure from low Earth orbit is not simply 2.94 km/s: that is the hyperbolic excess speed v∞ Earth must be left with. Burning at 200 km altitude, the required speed is √(v∞² + 2μ_E/r), so the burn from a 7.79 km/s parking orbit is about 3.6 km/s, and the Oberth effect makes it cheaper than adding 2.94 km/s far from Earth. Launch vehicles quote this as C3 = v∞² ≈ 8.7 km²/s², the minimum for a Mars departure.