Exercises: A Mission to Mars

Work these with a calculator. Constants (Appendix B): $\mu_\odot = 1.327\times10^{11}\ \text{km}^3/\text{s}^2$, $\mu_\oplus = 3.986\times10^5\ \text{km}^3/\text{s}^2$, $\mu_{\text{Mars}} = 4.283\times10^4\ \text{km}^3/\text{s}^2$, $1\ \text{AU} = 1.496\times10^8\ \text{km}$, Earth mean radius $6{,}371\ \text{km}$, Mars mean radius $3{,}390\ \text{km}$, Mars surface gravity $3.71\ \text{m/s}^2$, Mars escape velocity $5.03\ \text{km/s}$, $g_0 = 9.80665\ \text{m/s}^2$. Mars surface atmospheric density $\rho \approx 0.020\ \text{kg/m}^3$ (Tier 2). Reuse the Chapter 11 Mars-transfer numbers where a problem says so. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/answers-to-selected.md — try each cold first. For "implement it" problems, do not run the code: hand-trace it and write the result in an # Expected output: comment, exactly as the chapter does. Always ask "relative to which body?" and sanity-check every number.

Part A — Warm-ups: windows, injection, dose (⭐)

34.1 † Earth's and Mars's orbital periods are $365.25$ and $686.98\ \text{days}$. Compute the synodic period and express it in months. Why is it longer than either planet's year?

34.2 A departure needs a hyperbolic excess of $v_\infty = 2.95\ \text{km/s}$. What departure characteristic energy $C_3$ must the launch vehicle deliver?

34.3 † Lay out a conjunction-class mission timeline as three legs — outbound $259\ \text{days}$, surface stay $\sim 500\ \text{days}$, return $259\ \text{days}$. Give the total in days and years, and state what sets the surface-stay length.

34.4 The deep-space cruise radiation rate is $\approx 1.8\ \text{mSv/day}$. Find the dose accumulated on a $259$-day outbound cruise, in mSv and Sv, and compare it with a $\sim 0.6\ \text{Sv}$ career limit.

34.5 † Mars's atmosphere is $\approx 96\%$ carbon dioxide. Write the Sabatier reaction and the water- electrolysis reaction, and state which propellant combination ("_____-lox") the pair manufactures.

34.6 Mars's escape velocity is $5.03\ \text{km/s}$ and the Moon's is $2.38\ \text{km/s}$. In one sentence, explain why the return problem is far harder at Mars than at the Moon, referencing the rocket equation.

Part B — The trajectory: injection, cruise, arrival (⭐⭐)

34.7 † Compute the trans-Mars injection burn from a 400 km parking orbit (Earth radius $6{,}371\ \text{km}$, so $r_{\text{park}} = 6{,}771\ \text{km}$) for $v_\infty = 2.95\ \text{km/s}$, using $\Delta v_{\text{TMI}} = \sqrt{v_\infty^2 + 2\mu_\oplus/r_{\text{park}}} - \sqrt{\mu_\oplus/r_{\text{park}}}$. Compare it with the $300$-km result of the chapter ($3.59\ \text{km/s}$).

34.8 A Starship-class stack has a post-injection mass (dry + payload) of $220\ \text{t}$ and burns methalox at $I_{sp} = 380\ \text{s}$. Find the propellant mass for a $\Delta v_{\text{TMI}} = 3.6\ \text{km/s}$ injection. Then find it again for a faster transfer needing $4.3\ \text{km/s}$. What did the faster trip cost in propellant?

34.9 † The spacecraft reaches the top of Mars's atmosphere ($r_{\text{atm}} \approx 3{,}490\ \text{km}$) with arrival $v_{\infty,\text{Mars}} = 2.65\ \text{km/s}$. Compute the entry speed $v_{\text{entry}} = \sqrt{v_\infty^2 + 2\mu_{\text{Mars}}/r_{\text{atm}}}$, and confirm it matches the chapter's $\approx 5.6\ \text{km/s}$.

34.10 Size a Mars orbit-insertion burn into a low circular $150\ \text{km}$ orbit ($r = 3{,}540\ \text{km}$) for $v_{\infty,\text{Mars}} = 2.65\ \text{km/s}$: find the periapsis speed on the arrival hyperbola, the target circular speed, and the burn. Comment on why real orbiters avoid capturing directly into a low circular orbit.

34.11 † Compute the trans-Earth injection (TEI) burn from a low Mars orbit at $r = 3{,}540\ \text{km}$, taking the required departure excess as $v_\infty = 2.65\ \text{km/s}$ (the Hohmann symmetry). Compare it with the Mars-ascent-to-orbit cost of $\sim 4.1\ \text{km/s}$: which is bigger, and what is the total "surface → Earth-bound" delta-v?

34.12 During the $259$-day cruise the mission budgets $0.1\ \text{km/s}$ of trajectory-correction maneuvers. If the spacecraft's dry-plus-payload mass is $60\ \text{t}$ and its thrusters give $I_{sp} = 320\ \text{s}$, how much propellant is that? (Use $m_p = m_f(e^{\Delta v/v_e} - 1)$.)

Part C — Entry, descent, and landing (⭐⭐)

34.13 † A $3{,}000\ \text{kg}$ capsule hangs under a $20\ \text{m}$-diameter parachute ($A = 314\ \text{m}^2$, $C_d = 0.5$). Compute its terminal velocity at the Martian surface ($\rho = 0.020\ \text{kg/m}^3$, $g = 3.71\ \text{m/s}^2$) and on Earth ($\rho = 1.225\ \text{kg/m}^3$, $g = 9.81\ \text{m/s}^2$). By what factor is Mars worse, and what does the result imply about landing on parachutes alone?

34.14 To halve the Mars terminal velocity of 34.13 (from $\sim 84$ to $\sim 42\ \text{m/s}$) using a bigger parachute alone, by what factor must the area — and hence roughly the diameter — increase? Comment on why "just use a bigger chute" fails for heavy landers.

34.15 † A heavy lander must decelerate from an entry speed of $5.6\ \text{km/s}$ to a soft landing. Roughly what fraction of that speed can a heat shield remove (hypersonic entry), and why must the last few hundred m/s be removed by something other than a parachute? (Conceptual — argue from the chapter's terminal-velocity result.)

34.16 Aerocapture at Mars can save $\sim 2\ \text{km/s}$ of orbit-insertion delta-v. For a $20\ \text{t}$ arriving spacecraft with a capture engine at $I_{sp} = 340\ \text{s}$, estimate the propellant that $2\ \text{km/s}$ would have cost (so the mass aerocapture saves), using $m_p = m_f(e^{\Delta v/v_e} - 1)$.

Part D — ISRU and the return problem (⭐⭐)

34.17 † Size a Mars ascent vehicle: lift a $5\ \text{t}$ crew capsule from the surface to a low Mars orbit at $\Delta v_{\text{asc}} = 4.1\ \text{km/s}$ with an ISRU methalox engine at $I_{sp} = 360\ \text{s}$. Find the propellant mass and the propellant fraction. Confirm it matches the chapter's $\sim 12\ \text{t}$.

34.18 MOXIE produced oxygen at a few grams per hour. A MAV needs $\sim 8.5\ \text{t}$ of oxygen made over a $480$-day surface stay. What average production rate (g/hr) does that require, and roughly how many times MOXIE's $\sim 8\ \text{g/hr}$ is it?

34.19 † The methalox mixture ratio is about $3.5\ \text{kg}$ of oxygen per $1\ \text{kg}$ of methane. For the $\sim 12\ \text{t}$ of MAV propellant in 34.17, how much is oxygen and how much is methane? Which does Mars's $96\%$-CO$_2$ atmosphere supply most directly?

34.20 (Back of the envelope) The chapter argues that $11\ \text{t}$ of return propellant brought from Earth would cost "many tens of tonnes in LEO." Sketch the stacked rocket-equation reasoning in three steps (land → inject → launch), and estimate an order-of-magnitude multiplier, stating your assumptions.

Part E — Implement it in Python (⭐⭐)

Write each function, hand-trace it for the given inputs, and record the result in an # Expected output: comment. Do not run it.

34.21 † Write injection_prop(dv_kms, isp_s, m_post_t) returning the propellant (t) for a burn of dv_kms with engine isp_s, pushing a post-burn mass m_post_t. Trace it for injection_prop(3.6, 380, 220).

34.22 Write terminal_velocity(m, g, rho, Cd, A) returning $\sqrt{2mg/(\rho C_d A)}$. Trace it for a Mars case: terminal_velocity(3000, 3.71, 0.020, 0.5, 314).

34.23 † Write synodic_period(T1, T2) (from Chapter 11's interplanetary.py) and trace it for Earth–Mars, synodic_period(365.25, 686.98) days. Then trace a hypothetical planet with $T_2 = 400\ \text{days}$ and comment on why the synodic period is now long.

Part F — Architectures and design (⭐⭐ / ⭐⭐⭐)

34.24 (Design it) Build the split delta-v budget for a crewed Mars mission as two tables — "launched from Earth" and "made on Mars (ISRU)" — using the chapter's legs (TMI $3.6$, TCM $0.1$, landing $0.6$; MAV ascent $4.1$, TEI $2.1$, all km/s). Add a 5–20% margin per leg (justify each), roll up both totals, and state in one sentence why the split matters.

34.25 † (Design it) Your Track-C mission may launch in a poor window with departure $C_3 = 16\ \text{km}^2/\text{s}^2$ instead of $8.7$. Find the new $v_\infty$ and the new TMI burn from a $300\ \text{km}$ parking orbit, and state which part of the architecture the higher $C_3$ hits (the launcher, the spacecraft's own budget, or both).

34.26 (Design it — trade study) Score DRA 5.0 against Starship on five weighted criteria (mass, cost, schedule, crew safety, technical maturity) as a Chapter-29-style trade table. Pick weights for a risk-averse government and again for a cost-driven startup, and show the "winner" can flip. What does that prove about trade studies?

34.27 † (⭐⭐⭐, back of the envelope) A short-stay opposition-class mission cuts the surface stay to $60\ \text{days}$ but flies a faster transit and a Venus swingby, so the crew spends far more total time in deep-space radiation. Argue qualitatively why opposition-class missions are usually judged worse for the crew despite being shorter overall — name the two competing effects.

Part G — Find the error, "why can't you just…", interleaved (⭐⭐⭐)

34.28 Find the error. A student writes: "The spacecraft arrives at Mars moving $21.5\ \text{km/s}$ (its heliocentric aphelion speed from Chapter 11), so the heat shield must survive entry at $21.5\ \text{km/s}$." What is wrong, and what is the correct entry speed?

34.29 † Why can't you just bring the return propellant from Earth and skip the complexity of ISRU? Frame your answer in terms of the stacked rocket equation (land, inject, launch) and what ISRU deletes.

34.30 (Interleaved, Ch. 28 & Ch. 29) For a crew of six on a $900$-day mission at $\sim 5\ \text{kg}$ of consumables per person-day, compute the open-loop consumables mass. Then explain, using the Chapter 28 break-even idea and the Chapter 29 "driving requirement," why closed-loop life support is not optional for this mission.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: the correct central body and $\mu$, explicit units, a "relative to which body?" check on every velocity, a clear split of Earth-launched vs. Mars-made propellant, and a sanity check on every final number.