Exercises: Energy in Space — The Vis-Viva Equation

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.

Part A — Warm-ups: speeds and energies (⭐)

6.1 † Find the circular orbital speed at an altitude of $800\ \text{km}$. (First find the orbital radius $r$.)

6.2 What is the specific orbital energy $\varepsilon$ of that same $800\ \text{km}$ circular orbit? Give it in $\text{MJ/kg}$.

6.3 † Use the vis-viva equation to find the speed at $r = 10{,}000\ \text{km}$ on an orbit whose semi-major axis is $a = 15{,}000\ \text{km}$.

6.4 Find the escape velocity at $800\ \text{km}$ altitude, both directly from $\sqrt{2\mu/r}$ and by multiplying your answer to 6.1 by $\sqrt{2}$. Confirm they agree.

6.5 † An orbit has specific energy $\varepsilon = -15\ \text{km}^2/\text{s}^2$. Find its semi-major axis $a$.

6.6 A satellite is in a circular orbit and moving at exactly $6.5\ \text{km/s}$. What is its orbital radius, and its altitude above Earth's surface?

Part B — Vis-viva and the energy ledger (⭐⭐)

6.7 † A Molniya-type orbit has perigee altitude $500\ \text{km}$ and apogee altitude $39{,}900\ \text{km}$. Find (a) its semi-major axis, (b) its speed at perigee, and (c) its speed at apogee. Verify that $r_p v_p \approx r_a v_a$ (conservation of angular momentum).

6.8 Prove algebraically that for any circular orbit, the specific kinetic energy equals $-\varepsilon$ and the specific potential energy equals $2\varepsilon$. (Start from $v_{\text{circ}}^2 = \mu/r$.)

6.9 † A spacecraft leaves Earth on a hyperbolic trajectory with specific energy $\varepsilon = +3.0\ \text{km}^2/\text{s}^2$. Find (a) its hyperbolic excess velocity $v_\infty$, (b) its semi-major axis $a$ (watch the sign), and (c) its speed as it passes $r = 6{,}771\ \text{km}$. Check part (c) against $v^2 = v_{\text{esc}}^2 + v_\infty^2$.

6.10 A satellite in a $400\ \text{km}$ circular orbit ($v_{\text{circ}} = 7.673\ \text{km/s}$) makes a prograde burn of $100\ \text{m/s}$. Find its new speed immediately after the burn, its new specific energy, and its new semi-major axis. Is the point opposite the burn (the new apoapsis) higher or lower than $400\ \text{km}$?

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 $\mu = 3.986\times10^{14}\ \text{m}^3/\text{s}^2$).

6.11 † Write circular_velocity(mu, r) returning $\sqrt{\mu/r}$. Trace circular_velocity(3.986e14, 6.771e6).

6.12 Write specific_energy(mu, a) returning $-\mu/(2a)$. Trace it for the GTO semi-major axis, specific_energy(3.986e14, 2.4468e7), and state whether the orbit is bound.

6.13 † Write vis_viva(mu, r, a) returning $\sqrt{\mu(2/r - 1/a)}$. Trace vis_viva(3.986e14, 6.771e6, 2.4468e7) (the speed at GTO perigee).

Part D — Find the error (⭐⭐)

6.14 A student computes the speed of a satellite circling in GEO by putting $r = 42{,}164\ \text{km}$ into vis-viva but using $a = 24{,}468\ \text{km}$ (the GTO transfer value) "because that's the orbit it arrived on." They get $1.62\ \text{km/s}$ and report it as the GEO orbital speed. What did they compute instead, and what is the correct GEO speed?

6.15 † A blog post claims: "The ISS is losing energy to atmospheric drag. Losing energy makes it go slower, so it gradually slows down and drifts lower until it re-enters." One clause is exactly backwards. Identify it and correct it, using this chapter's relationship between orbit size and speed.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

6.16 † Design a Hohmann-style speed budget to raise a spacecraft from a $400\ \text{km}$ circular LEO to a circular navigation orbit at $20{,}200\ \text{km}$ altitude (a GPS/MEO orbit, $r_2 = 26{,}571\ \text{km}$). Compute, in order: the LEO circular speed; the transfer ellipse's semi-major axis; the transfer speeds at perigee and apogee; the MEO circular speed; and the two burns (perigee raise and apogee circularization). Report the total delta-v. (You are reconstructing, from energy alone, a maneuver Chapter 10 will formalize.)

6.17 (Mission project) For your mission's target orbit (Track A/B/C/D), compute and record in your MDR: the specific orbital energy $\varepsilon = -\mu/(2a)$; the circular speed $v_{\text{circ}}$ at your operating radius; and, if your mission must escape its primary body, the local escape velocity and the delta-v from your parking orbit to reach it. Use the $\mu$ you recorded in Chapter 2.

6.18 † You are in a $300\ \text{km}$ circular parking orbit ($r = 6{,}671\ \text{km}$) and must inject a probe onto an Earth-escape trajectory with hyperbolic excess velocity $v_\infty = 3.5\ \text{km/s}$. Find the speed you must reach at $300\ \text{km}$ and the size of the injection burn from the parking orbit. (Hint: $v^2 = v_{\text{esc}}^2 + v_\infty^2$.)

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

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

6.20 Estimate the kinetic energy of the ISS (mass $\approx 420\ \text{t}$, speed $\approx 7.66\ \text{km/s}$), and express it as an equivalent mass of TNT (take TNT at $4.6\ \text{MJ/kg}$). What does your answer say about why re-entry (Chapter 7) is so violent?

6.21 † Estimate the departure burn from a $400\ \text{km}$ LEO needed to send a spacecraft coasting out to the Moon's distance ($r \approx 384{,}400\ \text{km}$) on a transfer ellipse — i.e., find the perigee speed of an ellipse with $r_p = 6{,}771\ \text{km}$ and $r_a = 384{,}400\ \text{km}$, and subtract the LEO circular speed. Compare with the delta-v map's "LEO → trans-lunar injection $\approx 3.1\ \text{km/s}$" from Chapter 3.

Part G — Interleaved & synthesis (from Chapters 2–3) (⭐⭐ / ⭐⭐⭐)

6.22 (Ch. 2) Chapter 2 derived escape velocity from the surface as $\sqrt{2\mu/R} \approx 11.2\ \text{km/s}$. Using this chapter's energy definition, show that escape velocity from any radius $r$ is $\sqrt{2\mu/r}$, and explain in one sentence why it is always $\sqrt{2}$ times the circular velocity there.

6.23 † (Ch. 3) A kick stage must supply the $1.45\ \text{km/s}$ GTO → GEO circularization burn from §6.4. Using the rocket equation from Chapter 3 with exhaust velocity $v_e = 3.05\ \text{km/s}$ ($I_{sp} \approx 311\ \text{s}$), what mass ratio must the kick stage have? What propellant fraction is that?

6.24 (Ch. 3, synthesis) Chapter 3 claimed that "once you are in orbit, you are halfway to anywhere." Make it quantitative with energy: compare the specific energy added to lift a payload from rest at Earth's surface to a $400\ \text{km}$ LEO (ignore losses) with the additional specific energy needed to go from that LEO to escape ($\varepsilon = 0$). Is "halfway" a fair description?

6.25 † (Synthesis) Two satellites share the same $400\ \text{km}$ circular orbit. Satellite A fires $50\ \text{m/s}$ prograde; satellite B fires $50\ \text{m/s}$ retrograde. For each, state qualitatively what happens to its orbital energy, its semi-major axis, its period, and — one full orbit later — whether it is ahead of or behind the point where it started relative to the other. Tie your answer to the threshold concept of §6.6.

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 speed, and a clear statement of which burns are prograde or retrograde.