Self-Assessment Quiz: The Three-Body Problem and Lagrange Points

Twenty questions on the restricted three-body problem, the five Lagrange points, stability, halo orbits, and low-energy transfers. Answer each before opening the key. Aim for 16 or more.

Question 1

The "restricted" in restricted three-body problem refers to the assumption that:

A) the two primaries are the same mass B) the third body has negligible mass and does not affect the primaries C) all motion is confined to two dimensions D) the orbits are hyperbolic

Question 2

We analyze the three-body problem in a rotating frame primarily because:

A) it eliminates gravity B) it makes the two primaries stand still, freezing the gravitational landscape C) it removes the need for Newton's laws D) it makes the third body massless

Question 3

How many Lagrange points does a two-primary system have?

A) 3 B) 4 C) 5 D) infinitely many

Question 4

Which Lagrange points lie on the line through the two primaries?

A) L1, L2, L3 B) L4, L5 C) all five D) only L1

Question 5

The three-body problem has no general closed-form solution because:

A) gravity is too weak B) there are too few conserved quantities to pin the motion down algebraically C) the bodies collide D) Newton's laws fail at three bodies

Question 6

The one quantity conserved along trajectories in the circular restricted three-body problem is the:

A) specific orbital energy B) inertial-frame kinetic energy C) Jacobi constant D) angular momentum about the third body

Question 7

Sun–Earth L1 and L2 lie approximately how far from Earth?

A) $384{,}000\ \text{km}$ B) $1.5\times10^6\ \text{km}$ C) $1\ \text{AU}$ D) $150\ \text{km}$

Question 8

The Hill radius (collinear-point distance) scales with the secondary's mass as:

A) $m_2$ B) $m_2^{1/2}$ C) $m_2^{1/3}$ D) $m_2^{2}$

Question 9

L4 and L5 form what shape with the two primaries?

A) a straight line B) an equilateral triangle C) a right triangle D) a square

Question 10

The collinear points L1, L2, L3 are:

A) always stable B) unstable (saddle points) C) stable only for equal masses D) gravity-free

Question 11

The triangular points L4/L5 are stable provided the mass ratio $m_1/m_2$ exceeds about:

A) 2 B) 25 C) 1000 D) there is no threshold — they are always stable

Question 12

What force, absent from a "ball on a hilltop" picture, actually stabilizes L4/L5?

A) drag B) the Coriolis force C) magnetism D) radiation pressure

Question 13

A spacecraft near an unstable collinear point is flown in a halo orbit mainly to:

A) generate artificial gravity B) stay in continuous sunlight/Earth view and avoid sitting on the unstable point C) save mass D) increase its speed

Question 14

JWST is placed at Sun–Earth L2 rather than L1 because at L2:

A) there is no sunlight B) the Sun, Earth, and Moon lie in nearly one direction, so a single sunshield blocks them all C) gravity is stronger D) it is closer to Earth

Question 15 (True/False, justify)

"A Lagrange point is a place where gravity cancels to zero." True or false? Justify in one sentence.

Question 16 (True/False, justify)

"Because L1 and L2 are unstable, keeping a spacecraft there is prohibitively expensive." True or false? Explain briefly.

Question 17 (True/False, justify)

"A low-energy (superhighway) transfer saves propellant but costs time." True or false? Say why.

Question 18 (Short answer)

Define a Trojan asteroid and name the system with the most famous Trojan swarms.

Question 19 (Short answer)

The Hill radius is $r_{\text{H}} = R(m_2/3m_1)^{1/3}$. For Sun–Earth, with $R = 1.496\times10^8\ \text{km}$ and $m_2/m_1 = 3.0\times10^{-6}$, compute $r_{\text{H}}$ (show the cube root step).

Question 20 (Short answer)

In your own words, what is the "interplanetary superhighway," and how does it generalize the gravity assist of Chapter 11?

Answer Key

Q Ans Note
1 B The third body is a massless "pebble" moved by the primaries but not moving them.
2 B Rotating with the primaries freezes them (and the effective potential) in place.
3 C Exactly five: L1–L3 collinear, L4/L5 triangular.
4 A L1, L2, L3 are collinear; L4/L5 are off the line.
5 B Too few integrals of motion (Bruns/Poincaré); the equations don't close.
6 C The Jacobi constant $C_J = 2\Omega - v^2$ (rotating frame).
7 B About $1.5\times10^6\ \text{km}$ — the Hill-radius result.
8 C $r_{\text{H}} \propto (m_2)^{1/3}$ — the cube root.
9 B Each is $60^\circ$ ahead of / behind the secondary, distance $R$ from both.
10 B Saddle points of the effective potential; $\sim$weeks $e$-folding runaway.
11 B Routh criterion: $m_1/m_2 > 24.96$.
12 B The velocity-dependent Coriolis force curves a runaway into a stable loop.
13 B Loop around the empty point for continuous view and to avoid the unstable point.
14 B One sunshield blocks Sun+Earth+Moon; the telescope cools to $\sim 40\ \text{K}$.
15 False Gravity does not cancel (the Sun far outpulls Earth at L1); forces balance in the rotating frame (net gravity + centrifugal), with net gravity supplying the centripetal force for co-rotation.
16 False Instability is slow and predictable; station-keeping runs only a few m/s per year.
17 True Threading gently through the necks/manifolds is fuel-cheap but slow — months to years.
18 — A body librating around a stable L4/L5 point, sharing the larger body's orbit $60^\circ$ ahead/behind; Jupiter has the largest swarms (Greek and Trojan camps).
19 — $r_{\text{H}} = 1.496\times10^8 \times (3.0\times10^{-6}/3)^{1/3} = 1.496\times10^8 \times (1.0\times10^{-6})^{1/3} = 1.496\times10^8 \times 0.0100 = 1.5\times10^6\ \text{km}$.
20 — The network of nearly-free routes woven from the shifting gravity of multiple bodies (invariant-manifold tubes through the Lagrange-point necks); it generalizes the single impulsive gravity assist into continuous, low-thrust transport between realms, trading time for fuel.

Topics to review by question

Questions Topic Section
1, 2, 5 The restricted problem and the rotating frame §15.1–15.2
6 The Jacobi constant §15.2
3, 4, 7, 8, 9, 19 Locating the five points; Hill radius §15.3
10, 11, 12, 15, 16 Stability of collinear vs. triangular points §15.3–15.4
13, 14 Halo orbits and real missions §15.4
17, 20 Low-energy transfers / the superhighway §15.5
18 Trojans and applications §15.6