Self-Assessment Quiz: Interplanetary Trajectories

Twenty questions on patched conics, the interplanetary Hohmann, launch windows, $C_3$, arrival, and gravity assists. Answer each before opening the key. Aim for 16 or more.

Question 1

The characteristic energy $C_3$ has units of:

A) km/s B) $\text{km}^2/\text{s}^2$ C) km D) seconds

Question 2

The hyperbolic excess velocity $v_\infty$ is the spacecraft's speed:

A) at periapsis of the parking orbit B) relative to the Sun during cruise C) retained relative to a body after climbing out of its gravity well (at the SOI edge) D) at the moment of launch

Question 3

The synodic period of two planets is given by:

A) $T_{\text{syn}} = T_1 + T_2$ B) $1/T_{\text{syn}} = |1/T_1 - 1/T_2|$ C) $T_{\text{syn}} = \sqrt{T_1 T_2}$ D) $T_{\text{syn}} = |T_1 - T_2|$

Question 4

In the patched-conic model, an Earth-to-Mars trajectory is built from how many two-body conic arcs?

A) one B) two C) three D) six

Question 5

Inside a planet's sphere of influence, the patched-conic method treats which body as the sole source of gravity?

A) the Sun B) the planet C) both equally D) neither — it uses the full three-body equations

Question 6

The minimum-energy (Hohmann) Earth-to-Mars transfer takes roughly:

A) 3 weeks B) 3 months C) 8–9 months D) 3 years

Question 7

Earth-to-Mars launch opportunities recur about every:

A) 6 months B) 12 months C) 26 months D) 10 years

Question 8

When a Hohmann spacecraft reaches Mars's orbit, its speed relative to the Sun is:

A) greater than Mars's orbital speed B) less than Mars's orbital speed C) exactly equal to Mars's orbital speed D) zero

Question 9

In the planet's reference frame, a gravity assist changes the spacecraft's:

A) speed only B) direction only C) both speed and direction D) neither

Question 10

The energy a spacecraft gains in a gravity assist comes from:

A) nowhere — energy is created B) the spacecraft's own fuel C) the planet's orbital motion around the Sun D) sunlight

Question 11

A departure trajectory with $C_3 = 0$ is:

A) still bound to Earth B) exactly parabolic — just barely escaping C) hyperbolic with large excess speed D) impossible

Question 12

Interplanetary missions inject from a low parking orbit, burning at perigee, mainly to exploit:

A) lower air resistance B) the Oberth effect (a burn while moving fast, deep in the well, buys the most energy) C) a shorter countdown D) the Coriolis effect

Question 13

The main reason you cannot launch to Mars on an arbitrary date is:

A) the rocket needs a full Moon B) Earth and Mars must be correctly phased so the spacecraft and Mars arrive together C) Mars's atmosphere changes daily D) the Sun blocks the trajectory

Question 14

Earth's sphere of influence ($\approx 924{,}000\ \text{km}$) is what fraction of the Earth–Sun distance?

A) about half B) about 10% C) about 0.6% D) about 50 times

Question 15 (True/False, justify)

"Aerocapture requires a large propulsive burn to enter orbit." True or false? Justify in one sentence.

Question 16 (True/False, justify)

"The higher the arrival $v_\infty$, the more expensive a propulsive orbit insertion becomes." True or false? Explain briefly.

Question 17 (True/False, justify)

"A closer flyby (smaller periapsis) bends the trajectory more and produces a larger velocity change." True or false? Say why.

Question 18 (Short answer)

Explain in one or two sentences why the interplanetary Hohmann transfer uses exactly the same mathematics as the LEO-to-GEO Hohmann of Chapter 10.

Question 19 (Short answer)

A mission's departure $C_3$ is $9\ \text{km}^2/\text{s}^2$. What is its hyperbolic excess velocity $v_\infty$?

Question 20 (Short answer)

In your own words: what is the patched-conic approximation, and what hard problem is it a practical response to?

Answer Key

Q Ans Note
1 B $C_3 = v_\infty^2$; energy per unit mass, $\text{km}^2/\text{s}^2$.
2 C Speed "at infinity" — retained at the edge of the sphere of influence.
3 B Set by the difference of angular rates $1/T_1 - 1/T_2$.
4 C Earth departure hyperbola, Sun transfer ellipse, Mars arrival hyperbola.
5 B Inside the SOI, the planet dominates; outside, the Sun.
6 C The Hohmann Earth–Mars cruise is $\approx 259$ days.
7 C Synodic period $\approx 780$ days $\approx 26$ months.
8 B It slowed climbing the Sun's well; Mars ($24.1$) overtakes it ($21.5\ \text{km/s}$).
9 B Speed ($v_\infty$) is preserved; only the direction rotates.
10 C Energy is transferred from the planet's orbital motion (imperceptibly).
11 B $C_3 = 0 \Rightarrow \varepsilon = 0 \Rightarrow$ parabola (marginal escape).
12 B Oberth: $\Delta\varepsilon = v\,\Delta v + \tfrac12\Delta v^2$ rewards large $v$.
13 B The launch window is set by planetary phasing.
14 C $9.24\times10^5 / 1.496\times10^8 \approx 0.006 = 0.6\%$.
15 False Aerocapture uses atmospheric drag to shed energy — little or no propulsive burn (a heat shield instead).
16 True Higher $v_\infty$ means a faster arrival hyperbola, so more energy must be removed — the insertion burn grows.
17 True Smaller $r_p$ lowers the flyby eccentricity toward 1, raising $\sin(\delta/2)=1/e$ and the turn angle $\delta$.
18 — Both are Hohmann transfers between two coplanar circular orbits — same tangential-burn ellipse, vis-viva for speeds, Kepler's third law for time. Only the central body ($\mu$) and the two "orbits" change (Sun + planets instead of Earth + LEO/GEO).
19 — $v_\infty = \sqrt{C_3} = \sqrt{9} = 3\ \text{km/s}$.
20 — Modeling the trajectory as a sequence of one-body conic arcs (one per gravitating body), patched at the SOI boundaries — a practical response to the three-body problem, which has no closed-form solution.

Topics to review by question

Questions Topic Section
4, 5, 14, 20 Patched conics & spheres of influence §11.1
6, 8, 18 The interplanetary Hohmann §11.2
3, 7, 13 Launch windows & synodic period §11.3
1, 2, 11, 12, 19 $C_3$ & hyperbolic departure §11.4
15, 16 Arrival: insertion & aerocapture §11.5
9, 10, 17 Gravity assists §11.6