Exercises: Orbital Maneuvers

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 appendices/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 delta-v against the chapter's numbers (LEO→GEO Hohmann $\approx 3.9\ \text{km/s}$; plane change $\approx 134\ \text{m/s}$ per degree in LEO).

Part A — Warm-ups: single maneuvers (⭐)

10.1 † Compute the two-burn Hohmann transfer from a $300\ \text{km}$ circular orbit ($r_1 = 6{,}671\ \text{km}$) to a $2{,}000\ \text{km}$ circular orbit ($r_2 = 8{,}371\ \text{km}$). Find the transfer semi-major axis, both burns, and the total delta-v.

10.2 A satellite in a $2{,}000\ \text{km}$ circular orbit ($v_{\text{circ}} = 6.90\ \text{km/s}$) performs a $15^\circ$ plane change. What does it cost? Compare with the total delta-v you found in 10.1.

10.3 † A spacecraft in a $400\ \text{km}$ circular orbit ($v = 7.673\ \text{km/s}$) fires a retrograde burn of $50\ \text{m/s}$. Find its new semi-major axis, decide whether the burn point is now apoapsis or periapsis, and compute the altitude of the orbit's opposite end.

10.4 Compute the time for the Hohmann transfer of problem 10.1 (from $300$ to $2{,}000\ \text{km}$). Express it in minutes.

10.5 † A mission must raise a circular orbit by a factor of $r_2/r_1 = 25$ in radius. Should you even consider a bi-elliptic transfer instead of a Hohmann? State the rule you are using.

10.6 Compute the orbital period of a $400\ \text{km}$ circular orbit and of a $500\ \text{km}$ circular orbit. By how many minutes does one lap differ? (This gap is the raw material of phasing.)

Part B — Hohmann transfers in depth (⭐⭐)

10.7 † Compute the full Hohmann transfer from a $400\ \text{km}$ LEO ($r_1 = 6{,}771\ \text{km}$) to a GPS/MEO orbit at $20{,}200\ \text{km}$ altitude ($r_2 = 26{,}571\ \text{km}$): both burns and the total. (You reconstructed part of this from energy alone in Chapter 6, Exercise 6.16 — check your two answers agree.)

10.8 You need to lower a spacecraft from GEO ($r = 42{,}164\ \text{km}$) back to a $400\ \text{km}$ LEO. Using the reversibility of the Hohmann transfer, state the two burns (magnitude and direction) and the total delta-v. Why is it the same total as going up?

10.9 † Compute the transfer time for the LEO→MEO Hohmann of problem 10.7. Express it in hours.

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 km and km/s ($\mu = 3.986\times10^{5}$).

10.10 † Write hohmann(mu, r1, r2) returning (dv1, dv2, total) per Section 10.2. Trace hohmann(3.986e5, 6771, 42164).

10.11 Write plane_change(v, di_deg) returning $2v\sin(\Delta i/2)$. Trace plane_change(7.673, 28.5) and state, in one line, what real maneuver this represents.

10.12 † Write dv_budget(legs) that sums a list of (label, dv) tuples. Trace it on [("perigee", 2.40), ("apogee", 1.46), ("plane", 1.51)] and say which of these three legs Section 10.6 would eliminate by combining.

Part D — Find the error (⭐⭐)

10.13 A student prices a GEO insertion by adding the circularization burn ($1.46\ \text{km/s}$) and a separate $28.5^\circ$ plane change ($1.51\ \text{km/s}$) to get $2.97\ \text{km/s}$, and reports that as the minimum. A colleague insists the real number is about $1.83\ \text{km/s}$. Who is right, and what did the student miss?

10.14 † A blog claims: "To save fuel, correct your orbital plane early — do the plane change in LEO right after launch, while you still have plenty of propellant, instead of waiting until you're way out at GEO." Identify the physics error and correct it with a number.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

10.15 † Design the apogee burn for a geostationary satellite. It arrives at GTO apogee (GEO altitude) at $v_a = 1.617\ \text{km/s}$ in a plane inclined $\Delta i = 28.5^\circ$, and must reach the equatorial GEO circular speed $v_{c2} = 3.075\ \text{km/s}$. Compute (a) the cost of circularizing and changing plane separately, and (b) the combined burn. Report the saving.

10.16 (Mission project) For your mission's target orbit (Track A/B/C/D), compute the transfer delta-v from your parking orbit and record it as a line in your MDR delta-v budget. Track A: the LEO→GEO transfer with the combined apogee burn. Tracks B/C/D: the trans-lunar or Earth-escape departure burn (you will refine these in Chapter 11 — for now, use the transfer-ellipse method of Section 10.2 with the Moon's distance or a first-guess apogee).

10.17 † Your mission raises a circular orbit by a radius factor of $20$. (a) Is a bi-elliptic transfer worth considering, and why? (b) Name the two concrete costs you would pay for the small delta-v it might save.

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

10.18 † Why can't you just point your thruster at a satellite floating a few kilometers ahead of you in the same orbit and fire, to close the gap? Explain what happens to your orbit, your speed, and your position, and state the maneuver that does let you catch up.

10.19 Estimate the delta-v for a $90^\circ$ plane change in LEO ($v = 7.673\ \text{km/s}$). Compare it with the $\approx 9.4\ \text{km/s}$ needed to reach orbit in the first place, and with escape from LEO ($\approx 3.2\ \text{km/s}$ more than circular). What does this say about ever launching into the "wrong" plane?

10.20 A chaser trails its target by $8^\circ$ in a $400\ \text{km}$ circular orbit. It can close the gap in $1$ revolution or spread it over $4$. Without full calculation, argue which option costs less delta-v and why, and name the cost of the cheaper option.

Part G — Interleaved & synthesis (Chapters 3, 6, 8, 9) (⭐⭐ / ⭐⭐⭐)

10.21 † (Ch. 6) Use the vis-viva equation to find the speed at $r = 10{,}000\ \text{km}$ on a transfer orbit whose semi-major axis is $a = 15{,}000\ \text{km}$. Then state which chapter-10 maneuver you would use this speed to price.

10.22 (Ch. 8) The Hohmann transfer time is $\pi\sqrt{a_t^3/\mu}$, which comes from Kepler's third law. For the LEO→GEO transfer ($a_t = 24{,}467.5\ \text{km}$), compute the full period of the transfer ellipse and confirm the one-way transfer time is half of it ($\approx 5.3\ \text{h}$).

10.23 (Ch. 9) A geostationary satellite can be launched from Cape Canaveral ($28.5^\circ$ latitude) or from Kourou ($5.2^\circ$). Explain, in the language of Section 10.4, why Kourou is the better site for a GEO mission, and estimate the difference in the required plane change at GEO speed ($3.075\ \text{km/s}$).

10.24 † (Ch. 3 synthesis) A comsat must fly the smart LEO→GEO transfer, $\Delta v = 4.23\ \text{km/s}$, using an apogee engine with $I_{sp} = 320\ \text{s}$ ($v_e = 3.14\ \text{km/s}$). Using the rocket equation from Chapter 3, find the mass ratio and the propellant mass fraction. Comment on how much of the satellite is propellant just to fly this one transfer.

10.25 (Ch. 6 synthesis) Track the specific orbital energy $\varepsilon = -\mu/(2a)$ through the LEO→GEO Hohmann: compute $\varepsilon$ for the parking orbit, the transfer ellipse, and the final GEO orbit. Verify that energy rises only at the two burns and is unchanged during the coast, and state the total energy added per kilogram.

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 identification of $r$ versus $a$, explicit units, a sanity check on each delta-v against the chapter's benchmarks, a clear statement of each burn's direction (prograde/retrograde/normal), and — for combined maneuvers — the vector (law-of-cosines) reasoning rather than a scalar sum.