Chapter 9 Exercises — Orbit Types and Their Uses

These problems turn the catalog of Chapter 9 into a working skill: reasoning from a mission's requirement to the orbit it needs, and computing the period, speed, and geometry of that orbit. Everything here uses tools you already have — circular velocity and vis-viva from Chapter 6 and the period relation from Chapter 8.

Difficulty tiers. ⭐ builds fluency with one idea; ⭐⭐ combines ideas or needs a short derivation; ⭐⭐⭐ is a multi-step or open design problem of the kind you would meet in real mission work.

Solutions. Full worked solutions to every odd-numbered problem and every problem marked † are published in appendices/answers-to-selected.md. Work each problem before you look. For the "Simulate it" problems, do not run your code — predict the printed output by hand, as a check on your understanding (this is exactly how flight software is reviewed).

Constants. Earth: $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$, mean radius $6{,}371\ \text{km}$, equatorial radius $6{,}378\ \text{km}$. One sidereal day $= 86{,}164\ \text{s}$; one solar day $= 86{,}400\ \text{s}$. Standard gravity $g_0 = 9.81\ \text{m/s}^2$. Speed of light $c = 3\times10^{5}\ \text{km/s}$.

Part A — Calculate it

Real numbers, one clean computation each. Show units and sanity-check the result.

1. (⭐)† A satellite occupies a circular orbit at $550\ \text{km}$ altitude — one of the Starlink shells. Compute (a) its orbital speed, (b) its period in minutes, and (c) how many orbits it completes in a day. Sanity-check the period against the $\approx 90$–$100\ \text{min}$ you should expect for LEO.

2. (⭐) Galileo navigation satellites orbit at about $23{,}222\ \text{km}$ altitude. Find the orbital radius, the circular speed, and the period (in hours). Is this above or below the GPS altitude, and is it faster or slower than GPS?

3. (⭐⭐)† Derive the geostationary altitude from scratch. Using a period of one sidereal day ($86{,}164\ \text{s}$), invert Kepler's third law to find the semi-major axis, then subtract Earth's equatorial radius to get the altitude. Finally compute the circular speed. You should reproduce $35{,}786\ \text{km}$ and $3.07\ \text{km/s}$.

4. (⭐⭐) A communications satellite is released into a geostationary transfer orbit (GTO) with perigee altitude $300\ \text{km}$ (over the equator) and apogee at GEO. Compute (a) the semi-major axis of the GTO, (b) the satellite's speed at apogee, and (c) the delta-v needed to circularize into GEO. Compare your answer to the "GTO → GEO $\approx 1.5\ \text{km/s}$" figure from the Chapter 3 delta-v map.

5. (⭐⭐⭐)† A Molniya orbit has a period of half a sidereal day ($43{,}082\ \text{s}$) and a perigee altitude of $500\ \text{km}$. Find (a) the semi-major axis, (b) the apogee altitude, (c) the eccentricity, and (d) the speeds at perigee and apogee. By what factor does the satellite slow down between perigee and apogee, and why does that produce the "loiter" that makes the orbit useful?

6. (⭐) A sun-synchronous Earth-observation satellite orbits at $700\ \text{km}$ altitude. Compute its period and the number of orbits it makes per day. (Its inclination, about $98^\circ$, does not enter the period — explain in one line why not.)

Part B — Back of the envelope

Order-of-magnitude reasoning. A factor of two is fine; the goal is the right size and the right story.

7. (⭐)† A LEO satellite at $550\ \text{km}$ sees a ground circle whose radius out to the horizon is about $2{,}560\ \text{km}$. Treating each satellite's footprint as that flat circle, estimate very roughly how many satellites you would need to cover the whole Earth's surface (area $\approx 5.1\times10^{8}\ \text{km}^2$) at once. The real Starlink constellation has thousands of satellites — give two reasons the true number is far larger than your estimate.

8. (⭐⭐) Estimate the round-trip signal delay on a geostationary link where a signal goes ground → GEO → ground → GEO → ground (a two-hop conversation). Use $c = 3\times10^{5}\ \text{km/s}$ and altitude $35{,}786\ \text{km}$. Why is this delay a well-known annoyance on satellite phone calls but a non-issue for one-way television broadcast?

9. (⭐⭐)† Three geostationary satellites are spaced $120^\circ$ apart around the equator. Given that a GEO satellite's coverage reaches out to a ground half-angle of about $81^\circ$, argue qualitatively why three of them cover essentially all longitudes at low and middle latitudes but leave the polar regions poorly served. What orbit from this chapter fills that polar gap?

10. (⭐) A single Molniya satellite is useful to a northern region for only about $8$ hours of its $\approx 12$-hour orbit. How many Molniya satellites, evenly staggered in time, are needed for continuous $24$-hour coverage? Sketch how you would phase them.

Part C — Why can't you just…

Conceptual. Answer in a few sentences of physical reasoning; no arithmetic required.

11. (⭐)† "Why can't you just put every satellite in geostationary orbit, since a single GEO satellite can see nearly half the Earth?" Give three distinct reasons — one about delta-v, one about distance, one about the physical scarcity of the orbit itself.

12. (⭐) "Why can't a geostationary satellite simply be synchronized to the ordinary $24$-hour day?" Explain the difference between the solar day and the sidereal day, and what specifically goes wrong (and how fast) if you use the wrong one.

13. (⭐⭐)† "Why can't a sun-synchronous orbit be exactly polar ($i = 90^\circ$)?" Explain why the required precession of the orbital plane forces the inclination slightly past $90^\circ$ (to about $98^\circ$), and which direction the plane must drift. (You may cite the J2 effect qualitatively; you are not asked to derive it.)

14. (⭐⭐) "Why can't you photograph the whole Earth from a low-inclination orbit, say $28^\circ$ (a straight-east launch from Florida)?" Describe exactly which latitudes such an orbit can and cannot reach, and state the inclination you would need for global coverage.

15. (⭐⭐⭐)† Make the full two-sided argument for why MEO — not LEO, not GEO — is the natural home for a satellite-navigation system. Explain why you would not go lower (LEO) and why you would not go higher (GEO), grounding each in coverage geometry and constellation size.

Part D — Simulate it

Write the Python, then predict its printed output by hand. Do not execute it. Each mirrors a helper in code/exercise-solutions.py.

16. (⭐)† Write a short program that computes and prints the circular speed (km/s) and period (min) of a $550\ \text{km}$ circular orbit. Predict the exact printed output to the precision your round() calls imply.

17. (⭐⭐) Write a program that computes the geostationary radius twice — once from the sidereal day ($86{,}164\ \text{s}$) and once from the solar day ($86{,}400\ \text{s}$) — and prints both plus their difference in kilometers. Predict the output, and state what that difference means physically for a satellite synchronized to the wrong day.

18. (⭐⭐)† Write a loop that prints, for altitudes $400$, $800$, $20{,}200$, and $35{,}786\ \text{km}$, the horizon coverage half-angle $\lambda = \arccos\!\big(R_{\text{eq}}/(R_{\text{eq}}+h)\big)$ in degrees. Predict the four printed values, and explain what the trend shows about the coverage-versus-altitude relationship.

19. (⭐⭐⭐) Write a program that, for a Molniya orbit (half-sidereal-day period), computes the eccentricity and apogee altitude for perigee altitudes of $600$ and $1{,}000\ \text{km}$. Predict the output, and explain physically why raising the perigee lowers the eccentricity.

Part E — Design it

Mission-design tasks. State the requirement, choose the orbit, and justify it in the language of this chapter. There is rarely one "right" answer — there is a defensible one.

20. (⭐⭐) A national agency wants continuous television and data relay to all of Norway (latitudes $58^\circ$ to $81^\circ$ N), including the Svalbard archipelago. Choose between a geostationary system and a Molniya/Tundra system, justify your choice with the geometry of this chapter, and specify the orbit parameters and the number of satellites you would fly.

21. (⭐⭐)† A startup wants to sell low-latency (under $50\ \text{ms}$ round trip) global broadband internet. Which orbital regime must they use, roughly how many satellites will it take, and what is the single hardest operational consequence of that choice? Contrast explicitly with putting one big satellite in GEO.

22. (⭐⭐) A radar-imaging satellite needs continuous electrical power (its radar is power-hungry) and a stable thermal environment. Argue for a dawn–dusk sun-synchronous orbit, and specify its altitude, inclination, and the local time of its ascending node. What does "dawn–dusk" buy the power and thermal subsystems?

23. (⭐⭐⭐)† Your mission. Complete the Orbit Selection block for your chosen Track (A/B/C/D) from the Mission Design Checkpoint. Record the altitude (or apogee/perigee if elliptical), the inclination, and the period, and write one sentence that traces from the mission's requirement to this orbit. Compute the period from your chosen altitude using $T = 2\pi\sqrt{a^3/\mu}$ (use the correct body's $\mu$).

Part F — Find the error

Each contains a specific mistake. Identify it, explain why it is wrong, and give the correct result.

24. (⭐⭐) A student computes the geostationary speed as $v = \sqrt{\mu/r}$ but plugs in $r = 35{,}786\ \text{km}$ (the altitude) and reports $v = 3.34\ \text{km/s}$. Diagnose the error and give the correct speed.

25. (⭐⭐)† A satellite brochure claims: "Our spacecraft is in a polar orbit, so it always passes over each city at the same local time of day." Is that claim necessarily true? Identify the missing condition, name the orbit actually being described, and explain the difference between a polar orbit and a sun-synchronous one.

Part G — Interleaved

These deliberately reach back to earlier chapters. Naming which tool to use is half the problem.

26. (⭐⭐) [Chapter 6 — vis-viva] For the Molniya orbit of Problem 5, compute the satellite's speed as it passes through $800\ \text{km}$ altitude, and compare it to the circular speed at $800\ \text{km}$. Which is faster, and explain the result using the semi-major axes in the vis-viva equation.

27. (⭐⭐)† [Chapter 3 — rocket equation] A satellite must perform the $1.47\ \text{km/s}$ GTO → GEO circularization burn. Its dry mass with empty tank is $1{,}800\ \text{kg}$, and its apogee engine has $I_{sp} = 320\ \text{s}$. Use the rocket equation to find the propellant mass required. Comment on how large a fraction of the satellite that propellant is.

28. (⭐⭐) [Chapter 2 / 6 — escape energy] Compute the escape velocity from geostationary altitude and compare it to the GEO circular speed. By what factor do they differ (you should recognize the number), and what delta-v would it take to escape Earth entirely starting from GEO?

29. (⭐⭐⭐)† [Chapter 8 / 6 — elements and period] A tracking station reports an orbit with semi-major axis $a = 26{,}562\ \text{km}$, eccentricity $e = 0.74$, and inclination $i = 63.4^\circ$. (a) Name the orbit type. (b) Compute its period. (c) Compute the perigee and apogee radii ($r_p = a(1-e)$, $r_a = a(1+e)$). (d) Explain the special significance of the $63.4^\circ$ inclination.

When you have finished, check the odd-numbered and †-marked solutions in appendices/answers-to-selected.md. Then take the quiz and work the two case studies — the GPS constellation audit and the high-latitude communications design — which put several of these skills together on real systems.