Case Study: Designing a Mars Science Orbiter's Trajectory

"Mars has been flown by, orbited, smacked into, radar examined, and rocketed onto, as well as bounced upon, rolled over, shoveled, drilled into, baked, and even blasted. Still to come: Mars being stepped on." — Buzz Aldrin

Executive Summary

In this case study we design — not merely analyze — the complete interplanetary trajectory for a Track-C science orbiter bound for Mars, and we size the spacecraft to fly it. Starting from mission requirements, we pick a launch window, compute the departure $C_3$ the launch vehicle must deliver, carry the spacecraft across the 259-day cruise, capture it into a science orbit, and then run the whole delta-v budget back through the rocket equation of Chapter 3 to find how much propellant the orbiter must carry and how much mass the launcher must throw. This is the interplanetary leg of your Mission Design Review, worked end to end — the calculation that turns "we want to orbit Mars" into "we need this rocket, this much fuel, and a launch in this two-week window."

Skills applied

• Choosing a launch window from the synodic period (§11.3).
• Computing departure $C_3$ and the trans-Mars injection burn (§11.4).
• Sizing a Mars orbit-insertion burn into a capture ellipse, and choosing aerobraking (§11.5).
• Rolling up an interplanetary delta-v budget and sizing the vehicle via the rocket equation (§11.4, Ch. 3).
• Turning a $C_3$ requirement into a launch-vehicle selection (§11.4).

Background

The mission. A $1{,}200\ \text{kg}$ (dry) science orbiter is to reach a low, near-circular Mars orbit (~$400\ \text{km}$ altitude) suitable for imaging and atmospheric science — a mission in the class of Mars Reconnaissance Orbiter. We will design its trajectory with patched conics and size its propulsion with the rocket equation. Where the real mission would use a numerically integrated, ephemeris-based trajectory (§11.1's caveat), our patched-conic design is exactly what a preliminary study produces: good to a percent or two, and enough to pick a launch vehicle and a propellant load.

The strategy. Three legs, three conics: a hyperbolic departure from Earth (delivered by the launch vehicle's upper stage), a heliocentric transfer ellipse (a free coast), and a hyperbolic arrival at Mars that the orbiter brakes out of with its own engine plus the atmosphere. We size each in turn.

Phase 1: Requirements and the launch window

Mars launch opportunities recur every synodic period,

$$T_{\text{syn}} = \frac{1}{\left|\frac{1}{365.25} - \frac{1}{686.98}\right|} = 780\ \text{days} \approx 26\ \text{months},$$

so the mission must target one of these windows (illustratively, the early-2030s openings roughly two years apart). Missing it costs more than two years — the single hardest schedule constraint on the whole program. We design to a near-minimum-energy window, where the departure geometry gives the low $C_3$ we computed in the chapter. (Real windows vary; a pessimistic year can push $C_3$ toward $15\text{–}17\ \text{km}^2/\text{s}^2$, which we would flag as a design driver — Tier 2/3.)

Phase 2: Departure — the $C_3$ the launcher must deliver

From §11.2 the heliocentric transfer needs a hyperbolic excess of $v_{\infty,\oplus} = 2.95\ \text{km/s}$ at Earth, so the departure characteristic energy is

$$C_3 = v_{\infty,\oplus}^2 = (2.95)^2 \approx 8.7\ \text{km}^2/\text{s}^2.$$

In practice the launch vehicle's upper stage performs the trans-Mars injection, delivering the spacecraft directly to this $C_3$. If it staged from a $300\ \text{km}$ parking orbit, that injection is the burn we sized in §11.4:

$$\Delta v_{\text{TMI}} = \sqrt{v_\infty^2 + \frac{2\mu_\oplus}{r_{\text{park}}}} - \sqrt{\frac{\mu_\oplus}{r_{\text{park}}}} = 11.32 - 7.73 = 3.59\ \text{km/s}.$$

The key design output of this phase is the number we hand the launch provider: "deliver our spacecraft to $C_3 = 8.7\ \text{km}^2/\text{s}^2$." The spacecraft itself carries no propellant for departure — that is the launcher's job. It must, however, carry everything for arrival.

Phase 3: The cruise

The spacecraft coasts the transfer ellipse for

$$t_{\text{trans}} = \pi\sqrt{\frac{a_t^3}{\mu_\odot}} = 259\ \text{days} \approx 8.5\ \text{months},$$

engine off, falling around the Sun. During cruise it performs a handful of small trajectory-correction maneuvers (TCMs) to null out injection errors and refine the aim point at Mars — patched conics gets you close, but you arrive at Mars's SOI needing to hit a capture corridor kilometers wide from a hundred million kilometers away. We budget $0.10\ \text{km/s}$ total for TCMs (a representative allowance — Tier 2). The cruise also sets requirements for the rest of the vehicle: 8.5 months of power, thermal control, and a communications link that grows to hundreds of millions of kilometers (Chapters 24–26).

Phase 4: Arrival — capture and aerobraking

The spacecraft reaches Mars's SOI with $v_{\infty,\text{Mars}} = 2.65\ \text{km/s}$ (§11.2). Braking straight into a low circular orbit would cost the full $2.08\ \text{km/s}$ of §11.5 — far too much propellant. Instead we capture into a highly elliptical orbit and let the atmosphere finish the job.

Target a capture ellipse with periapsis at $400\ \text{km}$ altitude ($r_p = 3{,}790\ \text{km}$) and apoapsis at $30{,}000\ \text{km}$ altitude ($r_a = 33{,}390\ \text{km}$), so $a_{\text{cap}} = (r_p + r_a)/2 = 18{,}590\ \text{km}$. The arrival hyperbola's speed at periapsis is

$$v_{\text{hyp}} = \sqrt{v_{\infty,\text{Mars}}^2 + \frac{2\mu_{\text{Mars}}}{r_p}} = \sqrt{(2.65)^2 + \frac{2(4.283\times10^4)}{3{,}790}} = \sqrt{7.02 + 22.60} = 5.44\ \text{km/s},$$

and the speed we need at periapsis to be on the capture ellipse is

$$v_{\text{ell}} = \sqrt{\mu_{\text{Mars}}\!\left(\frac{2}{r_p} - \frac{1}{a_{\text{cap}}}\right)} = \sqrt{4.283\times10^4\left(\frac{2}{3{,}790} - \frac{1}{18{,}590}\right)} = \sqrt{20.30} = 4.51\ \text{km/s}.$$

So the Mars orbit-insertion (MOI) burn is

$$\Delta v_{\text{MOI}} = v_{\text{hyp}} - v_{\text{ell}} = 5.44 - 4.51 = 0.94\ \text{km/s}.$$

That is less than half the cost of direct circularization. Then, over weeks, we lower the apoapsis by aerobraking — dipping periapsis into the thin upper atmosphere on each of hundreds of passes, shedding energy to drag for almost no propellant, until the orbit shrinks to the $400\ \text{km}$ science orbit. The atmosphere does, for free, the $\sim 1\ \text{km/s}$ of circularization the engine would otherwise have to pay — the same energy transaction as re-entry (Chapter 7), metered out gently.

Phase 5: The budget roll-up and vehicle sizing

Now assemble the spacecraft's own delta-v budget (departure belongs to the launcher) and run it through the rocket equation to size the propellant.

With a storable bipropellant engine ($I_{sp} = 320\ \text{s}$, so $v_e = 3.14\ \text{km/s}$) and a $1{,}200\ \text{kg}$ dry orbiter, the rocket equation gives the propellant:

import math
G0 = 9.80665

v_inf_dep = 2.95                    # km/s, near-minimum Mars window (launch vehicle's job)
C3 = v_inf_dep ** 2                 # km^2/s^2 -> the launcher requirement

moi, tcm, margin = 0.94, 0.10, 0.06
dv_sc = (moi + tcm) * (1 + margin)  # spacecraft's own budget (km/s)

isp = 320.0
ve = isp * G0                       # m/s
m_dry = 1200.0                      # kg, dry orbiter
R = math.exp(dv_sc * 1000 / ve)     # mass ratio (Chapter 3)
m_prop = m_dry * (R - 1)
m_wet = m_dry + m_prop

print(f"departure C3:        {C3:.2f} km^2/s^2")
print(f"spacecraft delta-v:  {dv_sc:.2f} km/s")
print(f"propellant mass:     {m_prop:.0f} kg")
print(f"wet mass to inject:  {m_wet:.0f} kg")
# Expected output:
# departure C3:        8.70 km^2/s^2
# spacecraft delta-v:  1.10 km/s
# propellant mass:     505 kg
# wet mass to inject:  1705 kg

The design closes like this. The orbiter must carry about $505\ \text{kg}$ of propellant, for a wet mass of about $1{,}705\ \text{kg}$ — and that is the mass the launch vehicle must deliver to $C_3 = 8.7\ \text{km}^2/\text{s}^2$. Consulting an (illustrative) launch-vehicle performance chart, a medium-lift vehicle able to send $\sim 2{,}000\text{–}2{,}500\ \text{kg}$ to $C_3 \approx 10\ \text{km}^2/\text{s}^2$ closes the design with comfortable margin. Sanity check: the propellant fraction is $505/1705 \approx 30\%$ — entirely reasonable for a spacecraft whose biggest burn is a sub-km/s capture, and far below the $\sim 90\%$ of a launch vehicle. The arithmetic hangs together.

🔧 Engineering Reality: Notice how the choice to aerobrake rippled through the whole design. Had we circularized propulsively ($2.08\ \text{km/s}$ instead of $0.94$), the spacecraft delta-v would jump to $\sim 2.3\ \text{km/s}$, the mass ratio to $\sim 2.1$, and the propellant from $505$ to over $1{,}300\ \text{kg}$ — nearly doubling the wet mass and likely forcing a bigger, costlier launch vehicle. A single trajectory decision, made with this chapter's arithmetic, moved the mission from one rocket class to another. This is why trajectory design happens first in mission design (Chapter 29): it sizes everything downstream.

Discussion Questions

1. The launch vehicle, not the spacecraft, performs the trans-Mars injection. Why is it advantageous to put the big departure burn on the launcher's upper stage rather than the spacecraft's own engine?
2. Aerobraking saved roughly $1\ \text{km/s}$ of propulsive delta-v. Trace, through the rocket equation, how that saving shrank the wet mass — and explain why the effect is more than linear.
3. If the mission had to fly in a poor window with $C_3 = 16\ \text{km}^2/\text{s}^2$ instead of $8.7$, what changes — for the launch vehicle, and for the spacecraft's own budget? (Careful: which burn does the higher $C_3$ affect?)
4. Why does the 259-day cruise, though "free" in delta-v, still drive spacecraft mass and cost?

Your Turn: Extensions

• Option A (analysis). Recompute the MOI burn for a capture ellipse with a higher apoapsis of $80{,}000\ \text{km}$ altitude. Does the insertion get cheaper or more expensive, and why? (Hint: a looser capture ellipse has a periapsis speed closer to the arrival hyperbola's.)
• Option B (computation). Extend the sizing code to sweep $I_{sp}$ from $220\ \text{s}$ (monopropellant) to $320\ \text{s}$ (bipropellant) and report the propellant mass at each. By hand-tracing two or three points, show how a better engine buys back payload. (Do not run it — add # Expected output:.)
• Option C (design). Redesign the mission for Track D, an asteroid rendezvous at $2.7\ \text{AU}$. Recompute the transfer $v_\infty$, $C_3$, and cruise time (use Exercise 11.9's method), and note why a rendezvous — matching the asteroid's velocity — is harder than a flyby. Which is the dominant delta-v?

Key Takeaways

1. Trajectory design sizes the whole mission. The departure $C_3$ picks the launch vehicle; the arrival delta-v, through the rocket equation, sets the propellant and wet mass.
2. Put departure on the launcher, arrival on the spacecraft. The upper stage delivers $C_3$; the orbiter carries only what it needs to capture and correct.
3. Aerobraking is a force multiplier. Trading $\sim 1\ \text{km/s}$ of circularization for atmospheric drag can nearly halve the wet mass — one trajectory decision that moves the mission a whole rocket class.
4. The window rules the schedule. A $780$-day synodic period means a design must target a specific two-week opportunity, or wait more than two years.