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Unit 9 · Lesson 9-5
Interplanetary Launch Windows and Porkchop Plots
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Interplanetary Launch Windows and Porkchop Plots

Ephemerides, C₃ and arrival v∞ maps, Type I and II transfers, the launch asymptote, and designing a launch period

Cape Canaveral, July 2020
Three spacecraft leave for Mars in eleven days. The United Arab Emirates' Hope launches from Tanegashima on 19 July; China's Tianwen-1 from Wenchang on 23 July; NASA's Perseverance from Cape Canaveral on 30 July. They come from three different countries and three different launch sites, and they arrive within ten days of one another in February 2021. None of this is coincidence or coordination. Earth and Mars line up favourably for a few weeks roughly every 26 months, and outside those weeks the energy needed rises beyond what any of those rockets can supply. Perseverance's launch period opened on 17 July and closed in mid-August; missing it would have meant waiting 26 months for the next one. This lesson is about how those few weeks are found, and how a launch date is chosen within them.
Watch first · presented by Shoaib Aziz
The numbers in this film are the labs' own: Perseverance, launched 30 July 2020 and landed 18 February 2021, flew 203 days through 143.2° (Type I) with C₃ = 14.46 km²/s² (v∞ = 3.80 km/s), DLA +23.3° and an arrival v∞ of 2.56 km/s; the 2020 porkchop plot (about 1,900 Lambert solves) has its Type I minimum at C₃ = 13.10 (19 July → 26 January, 191 days) and its Type II minimum at 16.44 (20 August, 405 days); and with C₃ ≤ 15, |DLA| ≤ 40° and the landing fixed on 18 February 2021, 5 July fails both tests (21.6, +50.7°), 14 July fails on C₃ (15.21), and the launch period runs from 17 July (14.39) to 1 August.
The interplanetary Lambert problem. Pick a departure date t1t_1 and an arrival date t2t_2. Look up Earth's heliocentric state (r⃗E,v⃗E)(\vec r_E, \vec v_E) at t1t_1 and the target's (r⃗P,v⃗P)(\vec r_P, \vec v_P) at t2t_2 from an ephemeris. Solve Lambert around the Sun from r⃗E\vec r_E to r⃗P\vec r_P in t2−t1t_2 - t_1. The two hyperbolic excess velocity vectors are the mismatches between the transfer and the planets:
v⃗∞,dep=v⃗1−v⃗E(t1),v⃗∞,arr=v⃗2−v⃗P(t2),C3=∣v⃗∞,dep∣2\vec v_{\infty,\text{dep}} = \vec v_1 - \vec v_E(t_1), \qquad \vec v_{\infty,\text{arr}} = \vec v_2 - \vec v_P(t_2), \qquad C_3 = |\vec v_{\infty,\text{dep}}|^2

C₃ is what the launch vehicle must supply (see the lesson on interplanetary transfers and launch windows); arrival v∞ is what the spacecraft must remove, or survive, at the other end. Both are functions of two dates.

💡Where planet positions come from
JPL Development Ephemerides (DE440/DE441, 2021) are the reference: numerically integrated solar-system models fitted to radar ranging, spacecraft tracking and lunar laser ranging, distributed as SPICE kernels and through the Horizons service. Analytical approximations are good enough for design studies: VSOP87 series, the ERFA routines that astropy uses for its offline builtin ephemeris, or JPL's table of Keplerian elements with secular rates (valid 1800–2050), accurate to a few hundred arcseconds — the table the interactives in this lesson use. A design study needs to be self-consistent more than precise; a flight project uses DE-series ephemerides and a full force model from the start.
Loading interactive…
💡Reading a porkchop plot
Two lobes. The lower-left lobe holds Type I transfers (heliocentric transfer angle below 180°); the upper-right holds Type II (above 180°). Each lobe has its own minimum C₃. The ridge. Between them runs a narrow ridge where the transfer angle is 180°. Earth, the Sun and Mars are nearly collinear, the transfer plane is set by tiny out-of-ecliptic offsets, and the required inclination — and C₃ — spikes. This is the singularity met in the lesson on Lambert's problem, written across the plot. Diagonal lines are constant time of flight. Arrival v∞ contours usually have their minimum somewhere other than the C₃ minimum: the best launch is rarely the best arrival, and a mission chooses a compromise.
📐Worked Example: Perseverance, from two dates
Launch 30 July 2020, landing 18 February 2021: 203 days. With astropy's built-in ephemeris and Izzo's solver:
  • heliocentric transfer angle 143.2° → Type I
  • C3=14.46C_3 = 14.46 km²/s² (v∞=3.80v_\infty = 3.80 km/s)
  • declination of the launch asymptote DLA = +23.3°, right ascension RLA = 9.4°
  • arrival v∞=2.56v_\infty = 2.56 km/s
The whole 2020 opportunity, computed on a 4-day × 6-day grid (about 1,900 Lambert solves), has its Type I minimum at C3=13.10C_3 = 13.10 km²/s² (depart 19 July, arrive 26 January, 191 days) and its Type II minimum at 16.44 km²/s² (depart 20 August, arrive 29 September 2021, 405 days).
The launch asymptote. C3C_3 fixes the magnitude of v⃗∞\vec v_{\infty}. Its direction, expressed in Earth's equatorial frame, matters just as much: the declination of the launch asymptote,
DLA=arcsin⁡v∞,z∣v⃗∞∣,RLA=atan2⁡(v∞,y, v∞,x)\text{DLA} = \arcsin\frac{v_{\infty,z}}{|\vec v_\infty|}, \qquad \text{RLA} = \operatorname{atan2}(v_{\infty,y},\, v_{\infty,x})

Computed in the equatorial (ICRF) frame — not the ecliptic frame the Lambert problem is usually set up in. Mixing the two is a classic error worth 23.4° of declination.

A departure hyperbola lies in a plane through Earth's centre; the parking orbit must lie in the same plane. A plane with inclination ii only reaches declinations up to ii, so the parking orbit needs i≥∣DLA∣i \ge |\text{DLA}|, and a launch site at latitude ϕ\phi cannot reach inclinations below ϕ\phi. Range-safety limits on the launch azimuth cap the inclination from above. A large |DLA| can close a launch period even when C₃ is fine — the lesson on departure, in the unit on patched conics, builds the parking orbit and the hyperbola explicitly.
💡Launch period and launch window
The launch period is the run of consecutive days on which the mission can launch; on each of those days the launch window is the span of minutes or hours during which the rocket can lift off. They are different constraints:
  • The launch period is set by the porkchop: C₃ within the launcher's capability for this payload, |DLA| within reach of the site, arrival v∞ and arrival date acceptable.
  • The daily window is set by Earth's rotation: the launch site must pass under the plane that contains the asymptote. A coast in the parking orbit stretches a few instants into a usable window.
Mission designers often fix the arrival date across the whole period (Perseverance landed on 18 February 2021 whichever day it launched), which costs a little C₃ but simplifies everything downstream: entry, descent and landing design, relay-satellite coverage, operations staffing.
📐Worked Example: designing the 2020 launch period
Assume a launcher that can deliver this spacecraft to C3≤15.0C_3 \le 15.0 km²/s² and an ascent that can reach ∣DLA∣≤40°|\text{DLA}| \le 40°, with arrival fixed on 18 February 2021. Daily values (every 3 days): 5 July C3=21.6C_3 = 21.6, DLA +50.7°+50.7° (fails both); 14 July C3=15.21C_3 = 15.21 (fails C₃); 17 July 14.39 — the period opens; 23 July 13.77 (lowest); 1 August 14.87; 4 August 15.64 — closed. The period is 17 July to 1 August, and C₃ limits both ends. Perseverance's actual launch period opened on 17 July 2020. This model has the right physics but a simplified launcher and constraint set, so it reproduces the opening date and not the exact closing date.
MissionLaunchArrivalNote
Mars Science Laboratory (Curiosity)26 Nov 20116 Aug 2012Type I
MAVEN18 Nov 201322 Sep 2014Orbiter
InSight5 May 201826 Nov 2018First interplanetary launch from the US West Coast
Hope / Tianwen-1 / Perseverance19–30 Jul 20209–18 Feb 2021Three nations, one window
Next opportunitieslate 2026, late 2028, 2031—Found in this lesson's assignment
💡Why every Mars window is different
The synodic period of Earth and Mars is 780 days, but Mars's orbit has e=0.093e = 0.093 and is inclined 1.85° to the ecliptic, so each opportunity meets Mars at a different point of its orbit. The minimum C₃ varies from roughly 8 to 16 km²/s² over the 15–17-year cycle in which the geometry repeats. Some windows are cheap and some are not, and mission planners know which is which a decade in advance.
🕰Why "porkchop"?
The contours of C₃ in departure–arrival space close around two lobes that, to engineers in the 1960s with a sense of humour, looked like a pork chop. The plots were drawn at JPL and elsewhere from the 1960s, first by hand and then by early computers; mission-design teams still produce them, in essentially the same form, for every planetary mission.
↓But there is a problem
You can now find a window, compute its porkchop, and choose a launch period with real constraints. But look again at what was assumed: the spacecraft left from the centre of the Earth with velocity v⃗1\vec v_1 and arrived at the centre of Mars with v⃗2\vec v_2. In reality it starts 200 km up in a parking orbit, climbs out of Earth's gravity on a hyperbola, crosses a sphere of influence, coasts around the Sun, falls into Mars's gravity on another hyperbola and must be captured or land. The heliocentric arc is only the middle third of the trajectory. The next unit adds the other two.