Chapter 34 — Key Takeaways (A Mission to Mars)

A one-page reference. Reread this before an exam, or before you plan a trip to another planet — it is the whole book, applied once.

The mission, end to end

 WINDOW      INJECT        CRUISE         EDL           SURFACE + ISRU      RETURN
 every  -->  TMI       -->  ~259 d    -->  ~7 min   -->  ~500 d, make    -->  MAV ascent
 ~26 mo      ~3.6 km/s      radiation      "terror"      methalox fuel        + TEI (ISRU)
 (Ch. 11)    (§34.2)        (§34.3)        (§34.4)       (§34.5)              (§34.5/34.6)

Two anchors converge: the Hohmann-to-Mars trajectory (Ch. 11) is the skeleton; Starship is the vehicle that tries to fly it with people. All six themes appear at once.

Key equations and numbers (with symbols and units)

Relation / number Gives Note
$1/T_{\text{syn}} = \lvert 1/T_1 - 1/T_2\rvert$ synodic period = launch cadence Earth–Mars $\approx 780$ d $\approx 26$ mo (Ch. 11)
$C_3 = v_\infty^2$ departure characteristic energy ($\text{km}^2/\text{s}^2$) Mars departure $\approx 8.7$ (from $v_\infty = 2.95\ \text{km/s}$)
$\Delta v_{\text{TMI}} = \sqrt{v_\infty^2 + 2\mu_\oplus/r_p} - \sqrt{\mu_\oplus/r_p}$ trans-Mars injection burn $\approx 3.6\ \text{km/s}$ from a $300\ \text{km}$ LEO
$m_p = m_f\left(e^{\Delta v/v_e} - 1\right)$ propellant for a burn rocket equation (Ch. 3); TMI of a $220\ \text{t}$ stack $\approx 358\ \text{t}$
$v_{\text{entry}} = \sqrt{v_\infty^2 + 2\mu_{\text{Mars}}/r_{\text{atm}}}$ Mars entry speed $\approx 5.6\ \text{km/s}$ (relative to Mars)
$v_{\text{term}} = \sqrt{2mg/(\rho C_d A)}$ parachute terminal speed Mars $\sim 84\ \text{m/s}$ vs. Earth $\sim 17\ \text{m/s}$ (same chute)
Sabatier: $\text{CO}_2 + 4\text{H}_2 \rightarrow \text{CH}_4 + 2\text{H}_2\text{O}$ ISRU methane on $96\%$-CO$_2$ air (Ch. 28)
Electrolysis: $2\text{H}_2\text{O} \rightarrow 2\text{H}_2 + \text{O}_2$ ISRU oxygen + recycled H$_2$ product = methalox = Raptor fuel

The round-trip delta-v budget (split by where the propellant comes from)

Leg $\Delta v$ Source of propellant
Earth surface → LEO ~9.4 km/s launch vehicle
Trans-Mars injection 3.6 km/s Earth (refuel in LEO)
Mid-course TCMs 0.1 km/s Earth
Mars landing burn ~0.6 km/s Earth
MAV ascent (surface → low Mars orbit) 4.1 km/s Mars (ISRU)
Trans-Earth injection (TEI) 2.1 km/s Mars (ISRU)
Earth arrival (direct entry) ~0 (atmosphere) —

Margined: ~4.6 km/s launched from Earth, ~6.5 km/s made on Mars. The second column is the architecture.

"Relative to which body?" — still the master check

Speed Frame Value
Spacecraft at Mars's orbit Sun $21.5\ \text{km/s}$
Arrival $v_\infty$ Mars $2.65\ \text{km/s}$
Speed hitting Mars's atmosphere Mars $\approx 5.6\ \text{km/s}$

Decision aids — "which idea when?"

You face… The governing fact The move
"When can we launch?" 26-month synodic window wait for it; patience is a propellant
A big crewed stack in LEO TMI needs $\sim 358\ \text{t}$ refuel in orbit (reuse makes it close)
"Land it with a parachute" Mars terminal speed $\sim 84\ \text{m/s}$ impossible alone — add retropropulsion / sky crane
Heavy (crewed) lander parachutes don't scale (mass wall) propulsive descent, no chutes (Starship)
Save the orbit-insertion burn atmosphere can brake aerocapture (one deep pass) or aerobraking (many)
"Bring the return fuel from Earth" stacked rocket equation make it on Mars (ISRU + MAV)

Why Mars EDL is uniquely hard (the threshold idea)

Mars's atmosphere is $\sim 0.6\%$ of Earth's density: thick enough to need a heat shield, too thin to stop you with parachutes. Earth's air both burns and brakes; the Moon's vacuum does neither (just fire an engine); Mars sits in the cruel middle. Hence the EDL chain: heat shield (removes most of $5.6\ \text{km/s}$ at $\sim 10\ g$) → supersonic parachute (a step, not a solution) → powered descent / sky crane / airbags.

The two architectures at a glance

DRA 5.0 Starship
Reuse expendable fully reusable (goal)
Crew propulsion nuclear thermal (fast transit) methalox + orbital refueling
Arrival aerocapture / aerobrake direct entry, propulsive landing
Return pre-deployed MAV, ISRU-fueled the ship is the MAV, ISRU-fueled
Bet thrift & heritage (don't lose a crew) reuse collapses cost → afford mass margin

Same Hohmann, same window, same ISRU return — opposite bets on the vehicle.

Common pitfalls

Pitfall Reality
Using $21.5\ \text{km/s}$ (heliocentric) as the Mars entry speed. Entry speed is relative to Mars: $\approx 5.6\ \text{km/s}$.
"TMI is small (3.6 km/s), so it needs little fuel." On a $220\ \text{t}$ stack it needs $\sim 358\ \text{t}$ — mass ratio multiplies everything.
"A bigger parachute will land it." Terminal speed falls only as $1/\sqrt{A}$, and chutes can't scale to Mars's flow.
"Bring the return propellant along." It's multiplied by launch + injection + landing; ISRU deletes the chain.
"DRA 5.0 and Starship fly different trajectories." Same Chapter-11 Hohmann on the same window; only the vehicle differs.

Numbers worth memorizing

  • Window every $\approx 26$ months; conjunction-class round trip $\approx 900$ days.
  • TMI $\approx 3.6\ \text{km/s}$; Mars entry $\approx 5.6\ \text{km/s}$; Mars escape $5.03\ \text{km/s}$.
  • One-way cruise radiation $\approx 0.47\ \text{Sv}$ (a near-career dose); comm delay $3$–$22$ min.
  • Mars parachute terminal speed $\sim 84\ \text{m/s}$ ⇒ parachutes alone can't land.
  • Mars ISRU makes methalox (Sabatier + electrolysis) from $96\%$-CO$_2$ air; MOXIE proved it.

Mission / astrotools additions this chapter

  • MDR (Track-C payoff): assembled the full crewed-Mars review — objective → window → trajectory → EDL → ISRU return → systems → launch — with the delta-v budget split by who pays and where the propellant is made (~4.6 km/s Earth-launched, ~6.5 km/s Mars-made).
  • mission.py (reused): roll_up_dv and size_vehicle from Ch. 29 roll up the split budget and size the ISRU-fueled MAV ($\sim 12\ \text{t}$ propellant to low Mars orbit). No new module — Ch. 34 is the application that ties the package together, ahead of the Ch. 40 capstone.