Exercises: Kepler's Laws and the Two-Body Problem

Work these with a calculator, using Earth's $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$ and $R_\oplus = 6{,}371\ \text{km}$ unless told otherwise. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each one cold before you peek. For the "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. Keep units consistent within a problem, and sanity-check every number (an orbital radius below $6{,}371\ \text{km}$ is underground; a bound orbit has $0 \le e < 1$).

Part A — Warm-ups: laws, periods, and conics (⭐)

8.1 † State each of Kepler's three laws in one sentence, and for each name the physical quantity or principle it is really an expression of (as established in §8.2).

8.2 A circular orbit has radius $r = 7{,}000\ \text{km}$. Find its period $T$ and its mean motion $n$ (in rad/s).

8.3 † An orbit has perigee radius $r_p = 6{,}678\ \text{km}$ and apogee radius $r_a = 9{,}922\ \text{km}$. Find its semi-major axis $a$ and eccentricity $e$.

8.4 Classify each orbit by its conic type and say whether it is bound: (a) $e = 0$; (b) $e = 0.3$; (c) $e = 1.0$; (d) $e = 2.4$.

8.5 † By Kepler's third law, if orbit B has four times the semi-major axis of orbit A around the same planet, how many times longer is B's period?

8.6 Name the six classical orbital elements and, in a few words each, what each one specifies. Which single element changes as the spacecraft coasts?

Part B — Period, mean motion, and the anomalies (⭐⭐)

8.7 † A GPS satellite orbits at a semi-major axis $a = 26{,}560\ \text{km}$. (a) Find its period in hours. (b) GPS orbits are designed so the satellites repeat their ground track every sidereal day. Show that two GPS periods make (very nearly) one sidereal day ($86{,}164\ \text{s}$).

8.8 A satellite is in an elliptical orbit with $a = 10{,}000\ \text{km}$ and $e = 0.2$. Find (a) its period, (b) its perigee and apogee radii, and (c) its speed at perigee and at apogee (use vis-viva from Chapter 6). Verify $r_p v_p = r_a v_a$.

8.9 † For the orbit of 8.8, find the true anomaly $\nu$ at which the satellite's radius equals its semi-major axis ($r = a = 10{,}000\ \text{km}$). (Hint: use the orbit equation and $p = a(1-e^2)$.)

8.10 A Molniya orbit has $a = 26{,}571\ \text{km}$ and $e = 0.74$. Confirm it is a 12-hour orbit, and find its perigee and apogee altitudes above Earth's surface.

8.11 † An eccentric orbit has $e = 0.6$. At the moment its eccentric anomaly is $E = 60^\circ$, find (a) the mean anomaly $M$ (in degrees), (b) the true anomaly $\nu$, and (c) the radius $r$ if $a = 12{,}000\ \text{km}$.

8.12 A satellite in the orbit of 8.11 ($a = 12{,}000\ \text{km}$, $e = 0.6$) passes perigee at $t = 0$. How long after perigee does it reach the eccentric anomaly $E = 60^\circ$ of the previous problem? (Find the period and mean motion first.)

Part C — Implement it in Python (⭐⭐)

Write each function, then hand-trace it for the given inputs and record the result in an # Expected output: comment. Do not run it. Work in SI (metres, so $\mu = 3.986\times10^{14}\ \text{m}^3/\text{s}^2$) unless told otherwise.

8.13 † Write period(mu, a) returning $2\pi\sqrt{a^3/\mu}$. Trace period(3.986e14, 6.771e6) (the ISS-like orbit) and period(3.986e14, 42.164e6) (GEO). Report both in seconds.

8.14 Write mean_motion(mu, a) returning $\sqrt{\mu/a^3}$. Trace mean_motion(3.986e14, 6.771e6) and confirm it equals $2\pi/T$ for the period you found in 8.13.

8.15 † Write solve_kepler(M, e, tol=1e-8) that solves $M = E - e\sin E$ for $E$ by Newton's method, starting from $E_0 = M + e\sin M$. Hand-trace it (two iterations is enough) for solve_kepler(1.1877, 0.723) — the GTO example of §8.6 — and record the converged $E$.

8.16 Write elements_apsides(a, e) returning the tuple (r_p, r_a) with $r_p = a(1-e)$ and $r_a = a(1+e)$. Trace elements_apsides(10000e3, 0.2) (in metres) and confirm $r_p = 8{,}000\ \text{km}$ and $r_a = 12{,}000\ \text{km}$.

Part D — Find the error (⭐⭐)

8.17 † A student computes the position of a satellite $T/4$ (a quarter period) after perigee and concludes that because a quarter of the period has elapsed, the true anomaly must be $\nu = 90^\circ$. For an eccentric orbit ($e = 0.5$), is $\nu = 90^\circ$ at $t = T/4$? If not, is the true anomaly larger or smaller, and why?

8.18 A blog states: "A geostationary satellite orbits once every 24 hours, matching Earth's rotation, so it stays over the same point on the ground." One number is subtly wrong in a way that would make a real satellite drift. Identify and correct it, and explain the consequence of using 24 h instead.

8.19 † A homework solution finds the period of an elliptical orbit ($a = 20{,}000\ \text{km}$, $e = 0.5$) by "averaging" the perigee and apogee circular periods. Explain why this is conceptually wrong, and compute the correct period. What does Kepler's third law say the eccentricity contributes to the period?

8.20 In solving Kepler's equation, a student writes the iteration as $E_{k+1} = M + e\sin E_k$ (a fixed-point iteration) but then reports that for a near-parabolic orbit ($e = 0.97$) it "barely moves between steps and never converges." Without doing the full analysis, explain why high eccentricity makes this simple iteration struggle, and name the method the chapter used instead.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

8.21 † Design a Molniya orbit. You want a communications orbit with a 12-hour period ($T = 43{,}082\ \text{s}$, half a sidereal day) and a perigee altitude of $500\ \text{km}$. (a) Find the required semi-major axis $a$ from Kepler's third law. (b) Find the eccentricity $e$ and the apogee altitude. (c) Which orbital element would you set to $270^\circ$, and why, to make apogee hang over the northern hemisphere?

8.22 Sun-synchronous sizing (preview). A remote-sensing satellite needs a low, near-circular orbit at $700\ \text{km}$ altitude. Find its semi-major axis, period, and the number of orbits it completes per day. (In Chapter 9 you will add the inclination that makes it sun-synchronous; here, just size it.)

8.23 † (Mission project) For your mission's target orbit (Track A/B/C/D), write down the full six-element set $(a, e, i, \Omega, \omega, \nu)$ and compute the period and mean motion, using the correct primary body's $\mu$ (Earth $3.986\times10^{5}$; Moon $4.903\times10^{3}$; Mars $4.283\times10^{4}\ \text{km}^3/\text{s}^2$). State which elements are constrained by the mission and which you are free to choose. Record it all in your MDR.

8.24 Position report. For your mission's orbit from 8.23 (or, if it is circular, use the GTO of §8.6), compute where the spacecraft is $30\ \text{minutes}$ after perigee: find $M$, solve Kepler's equation for $E$, and report $\nu$ and $r$. Show your Newton iteration.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

8.25 † Why can't you just read the true anomaly straight off a clock, the way you read the minute hand of a watch — multiply the elapsed fraction of the period by $360^\circ$? Explain what breaks, for which orbits it matters most, and for which orbit it would actually work.

8.26 Estimate how far a geostationary satellite is from Earth's center as a multiple of Earth's radius, and use that plus Kepler's third law to explain, in order-of-magnitude terms, why GEO takes about $24\times$ longer than a low orbit even though it is only about $6\times$ higher.

8.27 † Back of the envelope. Mars orbits the Sun at about $1.52\ \text{AU}$. Using Kepler's third law and Earth's 1-year period at $1\ \text{AU}$ (no need for $\mu_\odot$ — use the ratio form $T^2 \propto a^3$), estimate Mars's orbital period in Earth years. Compare with the real value of $687\ \text{days}$.

Part G — Interleaved & synthesis (from Chapters 2 and 6) (⭐⭐ / ⭐⭐⭐)

8.28 (Ch. 2) Chapter 2 derived the gravitational parameter $\mu = GM$. Kepler's third law lets you measure it. Given that the Moon orbits Earth with a period of $27.3\ \text{days}$ at a semi-major axis of $384{,}400\ \text{km}$, solve $T^2 = (4\pi^2/\mu)a^3$ for $\mu_\oplus$ and compare with the standard $3.986\times10^{5}\ \text{km}^3/\text{s}^2$.

8.29 † (Ch. 6) A satellite is in the GTO of §8.3 ($a = 24{,}468\ \text{km}$, $e = 0.723$). Using Chapter 6's vis-viva and this chapter's geometry, find its speed at the true anomaly $\nu = 147.3^\circ$ computed in §8.6 (where $r = 29{,}802\ \text{km}$). Then find its specific orbital energy $\varepsilon$ two ways — from $-\mu/(2a)$ and from $v^2/2 - \mu/r$ — and confirm they agree.

8.30 (Ch. 6, synthesis) Chapter 6 said "a higher orbit is a slower orbit" in terms of speed. This chapter added Kepler's third law. For two circular orbits at $a_1 = 7{,}000\ \text{km}$ and $a_2 = 14{,}000\ \text{km}$, compute the ratio of their speeds and the ratio of their periods, and explain in one sentence how both ratios express the same physical truth.

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: correct use of Kepler's third law for $a \leftrightarrow T$, a clean distinction between $r$ and $a$ and between the three anomalies, explicit units, a sanity check on every radius and period, and — for the Kepler's-equation problems — a visible iteration with a stated starting guess.