Exercises: The Three-Body Problem and Lagrange Points

Work these with a calculator. Constants (Appendix B): $m_\odot = 1.989\times10^{30}\ \text{kg}$, $m_\oplus = 5.972\times10^{24}\ \text{kg}$, $m_{\text{Moon}} = 7.342\times10^{22}\ \text{kg}$, $m_{\text{Mars}} = 6.417\times10^{23}\ \text{kg}$, $m_{\text{Jupiter}} = 1.898\times10^{27}\ \text{kg}$, $1\ \text{AU} = 1.496\times10^8\ \text{km}$, Earth–Moon distance $384{,}400\ \text{km}$. 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. Two habits to keep: the Lagrange points are equilibria in the rotating frame, not zero-gravity spots, and sanity-check every number (a Hill radius should be a small fraction of the orbit; a stability ratio should be compared against $\approx 25$).

Part A — Warm-ups: mass parameters and Hill radii (⭐)

15.1 † Compute the mass parameter $\mu^{*} = m_2/(m_1+m_2)$ for the Sun–Earth system and for the Earth–Moon system. Which is closer to the "lopsided" limit $\mu^{*} \to 0$?

15.2 Compute the Hill radius of Mars about the Sun ($R = 1.524\ \text{AU}$). Sanity-check that Mars's moons Phobos and Deimos (orbiting within $\sim 24{,}000\ \text{km}$) sit deep inside it.

15.3 † Compute the Earth–Moon L1/L2 distance from the Moon (one Hill radius). Express it as a fraction of the Earth–Moon distance.

15.4 Sun–Earth L4 forms an equilateral triangle with the Sun and Earth. How far is L4 from Earth, and from the Sun, in AU and in km? (No integration needed — read it off the geometry.)

15.5 † Sun–Earth L1/L2 sit about $1.5\times10^6\ \text{km}$ from Earth. Express that distance in Earth–Moon distances ($384{,}400\ \text{km}$) and as a percentage of $1\ \text{AU}$.

15.6 Compute the mean motion $\omega$ and orbital period of the Earth–Moon pair from $\omega^2 = \mu_{\text{system}}/R^3$ with $\mu_{\text{system}} = \mu_\oplus + \mu_{\text{Moon}} = 3.986\times10^5 + 4.903\times10^3\ \text{km}^3/\text{s}^2$ and $R = 384{,}400\ \text{km}$. Sanity-check against the sidereal month ($\approx 27.3$ days).

Part B — Lagrange-point geometry (⭐⭐)

15.7 † Compute the Hill radius of Jupiter about the Sun ($R = 5.203\ \text{AU}$). Compare it with the orbit of Callisto ($\approx 1.88\times10^6\ \text{km}$) and comment on why all four Galilean moons are safely bound.

15.8 For the Earth–Moon system, confirm the L4 coordinates in the rotating frame (barycenter at origin, primaries on the $x$-axis): $x = R(0.5 - \mu^{*})$, $y = R\sqrt{3}/2$. Evaluate both in km.

15.9 † Show algebraically that the distance of L4 from the barycenter is $d = R\sqrt{1 - \mu^{*} + \mu^{*2}}$, and evaluate it for the Earth–Moon system. Why is $d$ so close to $R$?

15.10 Explain, without a full quintic solution, why Sun–Earth L2 lies slightly farther from Earth than L1 does, even though the leading-order Hill estimate puts them at equal distances.

Part C — Stability of the triangular points (⭐⭐)

15.11 † Apply the Routh criterion ($m_1/m_2 > 24.96$) to decide whether L4/L5 are stable for: (a) Sun–Neptune ($m_1/m_2 \approx 1.9\times10^4$), (b) Earth–Moon, (c) a hypothetical binary with $m_1/m_2 = 20$.

15.12 Pluto and Charon have a mass ratio of about $8.2$. Are Pluto–Charon L4/L5 stable? What does your answer predict about long-lived Trojan bodies at those points?

15.13 † The exact stability threshold is $m_1/m_2 = \tfrac{1}{2}\!\left(25 + \sqrt{621}\right)$. Evaluate it numerically and convert it to a critical mass parameter $\mu^{*}_{\text{crit}}$.

Part D — Jacobi constant and zero-velocity curves (⭐⭐)

15.14 † For the triangular points, the (nondimensional) Jacobi constant is $C_4 = C_5 = 3 - \mu^{} + \mu^{2}$. Evaluate it for the Earth–Moon system.

15.15 A spacecraft is momentarily at rest ($v = 0$) at nondimensional position $(x, y) = (0.5, 0)$ in the Earth–Moon rotating frame (units: $R = 1$, barycenter at origin, Earth at $(-\mu^{*}, 0)$, Moon at $(1-\mu^{*}, 0)$, $\mu^{*} = 0.0121$). Compute its Jacobi constant $C_J = 2\Omega - v^2$, where $\Omega = \tfrac12(x^2+y^2) + (1-\mu^{*})/r_1 + \mu^{*}/r_2$.

15.16 The critical Jacobi values satisfy $C_1 > C_2 > C_3 > C_4 = C_5$. As a spacecraft's energy rises (its $C_J$ falls), which "neck" opens first, and what does opening it physically allow?

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.

15.17 † Write hill_radius(m1, m2, R) returning $R(m_2/3m_1)^{1/3}$. Trace it for Sun–Earth: hill_radius(1.989e30, 5.972e24, 1.496e8).

15.18 Write routh_stable(m1, m2) returning True if m1/m2 > 24.96. Trace it for Earth–Moon (5.972e24, 7.342e22) and Pluto–Charon (1.303e22, 1.586e21).

15.19 † Write jacobi_C4(mu_star) returning $3 - \mu^{*} + \mu^{*2}$. Trace it for the Earth–Moon system (mu_star = 0.0121).

Part F — The superhighway and applications (⭐⭐ / ⭐⭐⭐)

15.20 † (Why can't you just…) A cargo tug must move propellant to lunar orbit and is not in a hurry. Argue in a short paragraph why a low-energy (weak-stability-boundary) transfer beats a direct Hohmann here, and name the one resource it spends to save fuel.

15.21 (Design it) In two or three sentences, justify placing a cryogenically cooled infrared telescope at Sun–Earth L2 rather than L1 — argue from where the Sun, Earth, and Moon are as seen from each point, and what a single sunshield can and cannot block.

15.22 † (Back of the envelope) A libration-point observatory station-keeps at about $3\ \text{m/s}$ per year for a $20$-year life. Estimate its total station-keeping delta-v and compare it with a single LEO→GTO leg ($\approx 2{,}500\ \text{m/s}$). What does the comparison say about the cost of living at an unstable point?

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

15.23 † Find the error. A student writes: "Sun–Earth L1 is the point between the Sun and Earth where their gravitational pulls exactly cancel, so an object there feels zero net force and just floats." Identify the two things wrong with this and give the correct picture.

15.24 Why can't you just fly to Sun–Earth L2, stop dead on the point, and stay there indefinitely with no fuel? Give both reasons the chapter raised.

15.25 † (Interleaved, Ch. 11) Earth's sphere of influence is $\approx 924{,}000\ \text{km}$ and its Hill radius is $\approx 1.5\times10^6\ \text{km}$. State the formula behind each, explain why they differ, and confirm that the Moon ($384{,}400\ \text{km}$) is bound to Earth by both measures.

15.26 (Interleaved, Ch. 6) A zero-velocity curve is where a spacecraft's speed drops to zero in the rotating frame. Relate this to the turning point at apoapsis of a bound orbit (where speed is minimum) and to the $\varepsilon = 0$ escape threshold you used in Chapter 6. In what sense is a zero-velocity curve a "wall for a given energy"?

15.27 † (Interleaved, Ch. 3) Explain, invoking the tyranny of the rocket equation, why the interplanetary superhighway is best understood as a response to the rocket equation rather than a way around physics. (Hint: what does the exponential punish, and what does a low-energy transfer refuse to spend?)

15.28 (Design it — Mission project) For your chosen track, write the "Multi-body dynamics" note for your MDR: does your mission use a Lagrange point, and if so which and how (relay, staging, capture, Trojan target)? If not, state in one line why not. Tracks: A (GEO comsat — almost certainly not), B (lunar lander — far-side relay at Earth–Moon L2 and/or a low-energy delivery), C (Mars orbiter — a possible ballistic-capture option at arrival), D (asteroid — is the target a Trojan?).

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 primaries and mass parameter, explicit units, a "rotating frame, not zero gravity" check on any force-balance claim, and a sanity check on the final number (is the Hill radius a small fraction of the orbit? is the stability ratio on the right side of $\approx 25$?).