Chapter 11 — Key Takeaways (Interplanetary Trajectories)

A one-page reference. Reread this before an exam, or before you design a trip to another planet.

The big picture

Interplanetary flight is orbital mechanics with the Sun at the center. A whole mission is stitched from three two-body conics — patched conics:

 Earth departure     heliocentric        Mars arrival
   HYPERBOLA    -->    ELLIPSE     -->    HYPERBOLA
 (inside Earth SOI)  (Sun's domain)   (inside Mars SOI)

Key equations (with symbols and units)

Equation Gives Symbols
$r_{\text{SOI}} \approx a\left(\dfrac{m_{\text{pl}}}{m_\odot}\right)^{2/5}$ sphere-of-influence radius (km) $a$ = planet's Sun-distance; mass ratio to Sun
$a_t = \dfrac{r_1 + r_2}{2}$ transfer-ellipse semi-major axis (km) $r_1,r_2$ = the two orbital radii
$v = \sqrt{\mu\left(\dfrac{2}{r} - \dfrac{1}{a}\right)}$ speed on the transfer (km/s) vis-viva (Ch. 6); $\mu = \mu_\odot$
$v_\infty = \lvert v_{\text{transfer}} - v_{\text{planet}}\rvert$ hyperbolic excess speed (km/s) at each planet's orbit
$t_{\text{trans}} = \pi\sqrt{\dfrac{a_t^3}{\mu_\odot}}$ Hohmann coast time (s) half the ellipse period (Kepler III)
$\dfrac{1}{T_{\text{syn}}} = \left\lvert\dfrac{1}{T_1} - \dfrac{1}{T_2}\right\rvert$ synodic period / launch cadence $T_1,T_2$ = orbital periods
$C_3 = v_\infty^2 = 2\varepsilon = -\dfrac{\mu}{a}$ characteristic energy ($\text{km}^2/\text{s}^2$) departure hyperbola's energy
$\Delta v_{\text{inj}} = \sqrt{v_\infty^2 + \dfrac{2\mu}{r_p}} - \sqrt{\dfrac{\mu}{r_p}}$ departure (or capture) burn (km/s) $r_p$ = parking-orbit radius, $\mu = \mu_\oplus$
$\sin\!\left(\dfrac{\delta}{2}\right) = \dfrac{1}{e},\ e = 1 + \dfrac{r_p v_\infty^2}{\mu}$ gravity-assist turn angle $r_p$ = flyby closest approach

The Earth→Mars Hohmann, worked (memorize the shape, not the digits)

Quantity Value Where
Earth / Mars heliocentric speed $29.78$ / $24.13\ \text{km/s}$ §11.2
Transfer $a_t$ $1.888\times10^8\ \text{km}$ ($1.262\ \text{AU}$) §11.2
Speed at perihelion / aphelion $32.73$ / $21.48\ \text{km/s}$ §11.2
$v_\infty$ at Earth / Mars $2.95$ / $2.65\ \text{km/s}$ §11.2
Departure $C_3$ $\approx 8.7\ \text{km}^2/\text{s}^2$ §11.4
Heliocentric $\Delta v$ $\approx 5.6\ \text{km/s}$ §11.2
Cruise time $\approx 259$ days ($\sim 8.5$ months) §11.2
Synodic period (window) $\approx 780$ days ($\sim 26$ months) §11.3
TMI burn from $300\ \text{km}$ LEO $\approx 3.59\ \text{km/s}$ §11.4
MOI (low circular / loose ellipse) $\approx 2.08$ / $0.69\ \text{km/s}$ §11.5

"Relative to which body?" — the master check

Every velocity in this chapter is measured relative to some body. Confusing frames is the #1 error.

Speed Frame Example value
Transfer speed at Mars's orbit Sun $21.48\ \text{km/s}$
Arrival $v_\infty$ Mars $2.65\ \text{km/s}$
Speed hitting Mars's upper atmosphere Mars $\approx 5.6\ \text{km/s}$

Decision aid — "which tool when?"

You know… You want… Use
two orbital radii + $\mu_\odot$ transfer speeds & $v_\infty$ vis-viva at each end, then subtract planet speed
the transfer ellipse trip time $t = \pi\sqrt{a_t^3/\mu_\odot}$
two orbital periods launch cadence $1/T_{\text{syn}} = \lvert 1/T_1 - 1/T_2\rvert$
$v_\infty$ launcher requirement $C_3 = v_\infty^2$
$v_\infty$ + parking orbit departure/capture burn $\Delta v = \sqrt{v_\infty^2 + 2\mu/r_p} - \sqrt{\mu/r_p}$
flyby $v_\infty$ + closest approach turn angle $e = 1 + r_p v_\infty^2/\mu$, $\sin(\delta/2)=1/e$

Common pitfalls

Pitfall Reality
Using the heliocentric arrival speed ($21.5$ km/s) as $v_\infty$ at Mars. $v_\infty$ is relative to Mars — $2.65$ km/s. The two heliocentric velocities nearly cancel.
"Just launch anytime and steer." Off-window transfers cost ruinous extra $\Delta v$; almost always cheaper to wait.
"A gravity assist creates free energy." Energy is transferred from the planet; the planet slows imperceptibly.
"Lower parking orbit → always smaller injection burn." The injection burn is nearly flat and slightly rises as you go lower; Oberth's win is burning deep-and-fast vs at the slow SOI edge.
Treating patched conics as exact. It is a design approximation (good to ~1–2%); real flight uses full numerical integration.

Numbers worth memorizing

  • Earth heliocentric speed $\approx 29.8\ \text{km/s}$; Mars $\approx 24.1\ \text{km/s}$.
  • Earth→Mars Hohmann: cruise $\approx 8.5$ months; window every $\approx 26$ months; $C_3 \approx 8.7$.
  • TMI from LEO $\approx 3.6\ \text{km/s}$ (matches the Ch. 3 delta-v map).
  • Gravity-assist maximum heliocentric boost from one flyby: up to $2v_\infty$.

Mission / astrotools additions this chapter

  • MDR: added the interplanetary transfer block — $v_\infty$, $C_3$, cruise time, launch window, insertion $\Delta v$ (Tracks C/D). Fed the TMI and MOI into the Chapter-3 delta-v budget.
  • interplanetary.py: hohmann_transfer(mu, r1, r2), synodic_period(T1, T2), c3_required(v_inf).