Getting to Mars: Hohmann transfers and launch windows

Talking points

• NASA's Basics of Space Flight (JPL): to reach Mars using the least propellant possible, the spacecraft's solar orbit is adjusted so that its perihelion is at the distance of Earth's orbit and its aphelion at the distance of Mars's. This is a Hohmann transfer orbit. The burn is tangential, in the direction of Earth's motion round the Sun, and the spacecraft then coasts.
• The model here takes both orbits as circles in one plane at NASA's semi-major axes, 149.598 million km for Earth and 227.956 million km for Mars. Mars's real orbit has an eccentricity of 0.0935 and is inclined 1.848 degrees to the ecliptic (NASA Mars Fact Sheet).
• JPL Horizons: Mars is 44.3 degrees ahead of Earth in heliocentric longitude on 4 December 2026 at about 01:23 UT. The viewer's ephemeris (JPL's approximate elements) puts the same moment at 01:16 UT.
• This view is at true scale, from above the ecliptic. Both planets go round anticlockwise, so "ahead" is anticlockwise of Earth.

Talking points

• a_t = (149.598 + 227.956)/2 = 188.777 million km = 1.2619 au.
• The flight from perihelion to aphelion is half an orbit: T = π√(a_t³/GM☉). With JPL's GM☉ = 1.32712440041 × 10²⁰ m³ s⁻² that is 2.2367 × 10⁷ s, 258.9 days.
• The same from Kepler's third law in years and au: T = ½ × 1.2619^1.5 = 0.7088 years = 258.9 days.

Talking points

• Earth's circular speed: v_E = √(GM☉/r_E) = 29.78 km/s, NASA's mean orbital velocity for Earth.
• Vis-viva at the transfer's perihelion: v_p = √(GM☉(2/r_E − 1/a_t)) = 32.73 km/s. So Δv₁ = 32.73 − 29.78 = 2.95 km/s. This is the spacecraft's speed relative to Earth once it has climbed out of Earth's gravity, v∞, and the launch energy is C₃ = v∞² = 8.67 km² s⁻².
• From a circular parking orbit 200 km up (r = 6378.137 + 200 km, with JPL's GM of Earth, 398,600.4 km³ s⁻²), the circular speed is 7.784 km/s and the escape speed 11.009 km/s, and the burn that leaves Earth at v∞ is √(v∞² + v_esc²) − v_circ = 11.396 − 7.784 = 3.61 km/s.
• This view is at true scale, 40 Earth radii above Earth's north side.

Talking points

• Vis-viva at the transfer's aphelion: v_a = √(GM☉(2/r_M − 1/a_t)) = 21.48 km/s. Mars's circular speed √(GM☉/r_M) = 24.13 km/s, so Δv₂ = 2.65 km/s, and the two burns total 5.59 km/s.
• NASA lists Mars's mean orbital velocity as 24.08 km/s: the average over its eccentric orbit, a little under √(GM/a), as the vis-viva lesson shows for Mercury. Either gives Δv₂ of 2.60 to 2.65 km/s.
• In practice the spacecraft arrives at v∞ = 2.65 km/s relative to Mars and must then slow down relative to Mars to be captured: Basics of Space Flight describes an orbit insertion burn, or aerodynamic braking, a parachute and retro-rockets to land.
• This scene opens on 20 August 2027, 258.9 days after 4 December 2026. With the real, eccentric orbits, Horizons has Mars 162 degrees round the Sun from where Earth was at departure, short of the 180 degrees of the circular model: near aphelion and slow, Mars moves 117.8 degrees in that time against the model's 135.7, at 1.54 to 1.67 au from the Sun against the 1.524 au mean.

Talking points

• Basics of Space Flight compares this with throwing a dart at a moving target: you have to lead the aim point by just the right amount.
• Mars moves 360 × 258.9/686.980 = 135.7 degrees during the transfer, so it must lead Earth by 180 − 135.7 = 44.3 degrees at launch.
• Earth moves 360 × 258.9/365.256 = 255.2 degrees in the same time, so on arrival Earth is 75.2 degrees ahead of Mars. A Hohmann return needs Earth 75.2 degrees behind Mars; Earth gains on Mars at 0.98561 − 0.52404 = 0.46158 degrees a day, so the crew waits (360 − 2 × 75.2)/0.46158 = 454 days at Mars.
• Earth gains 360 degrees on Mars every 360/0.46158 = 779.9 days, NASA's synodic period, so the same launch geometry comes round every 779.94 days. Basics of Space Flight: the opportunity for a minimum-energy transfer to Mars occurs about every 25 months.

Talking points

• JPL Horizons: opposition on 19 February 2027 at 15:50 UT, when the lead is zero.
• Mean rates from NASA's sidereal periods: Earth 360/365.256 = 0.98561 degrees a day, Mars 360/686.980 = 0.52404, so Earth gains 0.46158 degrees a day.

Talking points

• JPL Horizons: Mars is 44.3 degrees ahead on 4 December 2026 at about 01:23 UT, 77.6 days before the opposition, and 70.9 degrees ahead at the start (the viewer's ephemeris).
• The check reads the date through the Sun's longitude of date: it passes within 6 degrees of 251.9 degrees, its value on 4 December, about six days either side (the Sun moves 1.01 degrees a day then), when the lead is between about 41 and 48 degrees. A pause inside it is 72 to 84 days before the opposition.

Talking points

• JPL Horizons: Mars reaches aphelion, 249.2 million km from the Sun, around 5 March 2027 (NASA's aphelion distance: 249.261 million km). USNO: Earth is at perihelion on 3 January 2027 at 02:33 UTC.
• Horizons' daily rates: on 4 December 2026 Earth moves 1.015 degrees a day and Mars 0.458; on 3 January 2027, 1.019 and 0.446; on 19 February, 1.008 and 0.437. Earth gains 0.56 to 0.57 degrees a day, against the mean 0.462: Kepler's second law at work on both planets.
• Mars is also farther out than the model assumes: 1.626 au on 4 December 2026 against the 1.524 au mean. A transfer whose aphelion is at the mean distance would not even reach it. Real trajectories are worked out from where the planets actually are on the launch and arrival dates.
• The clock runs at a day a second from the 44.3 degree moment.

Talking points

• NASA: Curiosity launched on 26 November 2011 and landed on 6 August 2012; Perseverance launched on 30 July 2020 and landed on 18 February 2021. By the calendar those flights took 254 and 203 days.
• JPL Horizons, heliocentric longitudes at 12:00 UTC on each date: for Curiosity Earth was at 64 and Mars at 120 degrees at launch, a lead of 56 degrees, and Mars at 236 degrees on landing day, so the spacecraft went 172 degrees round the Sun. For Perseverance Earth was at 308 and Mars at 334 degrees, a lead of 26 degrees, and Mars at 90 degrees on landing day: 143 degrees round.
• Basics of Space Flight: a trajectory that carries the spacecraft less than 180 degrees round the Sun is Type I, 180 degrees or more Type II. Both rovers flew Type I trajectories, neither of them the 180-degree, 259-day textbook transfer.
• The scene opens at 12:00 UTC on Curiosity's launch date, when the viewer's ephemeris and Horizons both give a lead of 56.4 degrees.