Exercises: Perturbations

Work these with a calculator, using $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$, $R_\oplus = 6{,}378\ \text{km}$ (equatorial, for J2), $J_2 = 1.0826\times10^{-3}$, and $g_0 = 9.81\ \text{m/s}^2$. 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 record the result in an # Expected output: comment, exactly as the chapter does. Keep the two J2 secular-rate formulas from §12.2 at hand: $$\dot\Omega = -\tfrac{3}{2}\tfrac{nJ_2R_\oplus^2}{(1-e^2)^2 a^2}\cos i, \qquad \dot\omega = \tfrac{3}{4}\tfrac{nJ_2R_\oplus^2}{(1-e^2)^2 a^2}(5\cos^2 i - 1), \qquad n = \sqrt{\mu/a^3}.$$

Part A — Warm-ups: rank and compute (⭐)

12.1 † A remote-sensing satellite flies a circular orbit at $600\ \text{km}$ altitude ($a = 6{,}971\ \text{km}$) inclined at $i = 40^\circ$. Compute its J2 nodal-regression rate $\dot\Omega$ in degrees per day, and state whether the node drifts east or west.

12.2 A satellite is in a circular polar orbit, $i = 90^\circ$. Without computing, state its J2 nodal regression rate and explain physically why the equatorial bulge cannot swivel this particular plane.

12.3 † For a circular orbit at $700\ \text{km}$ ($a = 7{,}071\ \text{km}$), $i = 45^\circ$, compute the J2 apsidal-precession rate $\dot\omega$ in degrees per day. Is perigee advancing or regressing?

12.4 Solve $5\cos^2 i - 1 = 0$ for the two critical inclinations, and name the Chapter 9 orbit that uses the prograde one and what it achieves.

12.5 † Compute the solar radiation pressure of fully absorbed sunlight at $1\ \text{AU}$ (solar constant $S = 1{,}361\ \text{W/m}^2$, $c = 2.998\times10^{8}\ \text{m/s}$). What is the pressure on a perfectly reflecting surface, and why does it differ?

12.6 A satellite has $m = 200\ \text{kg}$, drag coefficient $C_D = 2.2$, and cross-section $A = 1.5\ \text{m}^2$. Compute its ballistic coefficient $\beta = m/(C_D A)$, and say whether it will decay faster or slower than the ISS ($\beta \approx 105\ \text{kg/m}^2$).

Part B — J2 and the sun-synchronous orbit (⭐⭐)

12.7 † Find the sun-synchronous inclination for a circular orbit at $700\ \text{km}$ altitude ($a = 7{,}071\ \text{km}$). (Require $\dot\Omega = +0.9856^\circ/\text{day} = 1.991\times10^{-7}\ \text{rad/s}$; solve the nodal-regression formula for $\cos i$.)

12.8 The chapter found the sun-synchronous inclination at $800\ \text{km}$ to be $98.6^\circ$. Your answer to 12.7 (at $700\ \text{km}$) should be slightly smaller. Explain physically why a lower sun-synchronous orbit needs an inclination closer to $90^\circ$.

12.9 † Show algebraically that the nodal-regression rate for a circular orbit scales as $a^{-7/2}$. Use this to explain why nodal regression is dramatically stronger in LEO than at GEO.

Part C — Drag and orbital decay (⭐⭐)

12.10 † A 1U CubeSat has $m = 1.3\ \text{kg}$, $C_D = 2.2$, $A = 0.01\ \text{m}^2$, in a $400\ \text{km}$ circular orbit ($v = 7{,}670\ \text{m/s}$, $a = 6.771\times10^{6}\ \text{m}$) where $\rho \approx 3\times10^{-12}\ \text{kg/m}^3$. Compute its ballistic coefficient and its decay rate $\dot a = -\rho v a/\beta$ in m/day. Compare its $\beta$ (and hence its fate) to the ISS.

12.11 In one paragraph, explain — using the fact that drag is strongest at perigee — why an initially eccentric orbit becomes more circular before it spirals in.

12.12 (Back of the envelope.) Using the lifetime table in §12.3 and the idea that thermospheric density falls with a scale height of roughly $50$–$60\ \text{km}$, estimate by what factor the air density (and thus the decay rate) drops between a $400\ \text{km}$ orbit and an $800\ \text{km}$ orbit. Does the huge change in lifetime (months → centuries) make sense?

Part D — Solar radiation pressure and third bodies (⭐⭐)

12.13 † A gossamer solar sail has an area-to-mass ratio $A/m = 20\ \text{m}^2/\text{kg}$ and reflectivity $r = 0.9$. Compute its SRP acceleration $a_{\text{SRP}} = (S/c)(1+r)A/m$. Compare it to the Sun's gravitational acceleration at $1\ \text{AU}$ ($\approx 5.9\times10^{-3}\ \text{m/s}^2$) and comment on whether such a sail could meaningfully change its orbit.

12.14 Compute the Moon's and the Sun's differential (tidal) accelerations, $a_{3} \sim 2\mu_3 r/d^3$, on a satellite at MEO ($r = 20{,}000\ \text{km}$; $\mu_{\text{Moon}} = 4{,}903$, $d_{\text{Moon}} = 384{,}400\ \text{km}$; $\mu_{\text{Sun}} = 1.327\times10^{11}$, $d_{\text{Sun}} = 1.496\times10^{8}\ \text{km}$). Which is larger, and by roughly what factor?

Part E — 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.

12.15 † Write j2_nodal_rate(a, e, i_deg) returning $\dot\Omega$ in deg/day (use $n = \sqrt{\mu/a^3}$). Trace it for j2_nodal_rate(7078.0, 0.0, 98.2) — a $700\ \text{km}$ orbit at $i = 98.2^\circ$. (Does it come out near $+0.9856$?)

12.16 Write sun_sync_inclination(a) returning the inclination (deg) that makes a circular orbit sun-synchronous. Trace it for sun_sync_inclination(7078.0).

12.17 † Write decay_rate(rho, v, a, beta) returning $\dot a = -\rho v a/\beta$ in m/s, and a wrapper that reports m/day. Trace it for the ISS: decay_rate(3e-12, 7670.0, 6.771e6, 105.0).

Part F — Find the error & "why can't you just…" (⭐⭐⭐)

12.18 † A design memo proposes making an orbit sun-synchronous by firing thrusters to rotate the orbital plane $0.9856^\circ$ per day. Estimate the delta-v this would cost per year (a plane rotation of angle $\theta$ at orbital speed $v$ costs $\Delta v \approx v\,\theta_{\text{rad}}$; use $v = 7.5\ \text{km/s}$), and compare it to the zero delta-v that J2 provides for free at the right inclination. Why is this the whole point of §12.2?

12.19 Why can't you just put a satellite at exactly $i = 90^\circ$ to make it sun-synchronous? Answer in terms of the sign and size of $\dot\Omega$ at $90^\circ$.

12.20 † Why can't you just ignore drag for a $400\ \text{km}$ science satellite intended to operate for three years? Estimate how far it would decay in that time (using $\sim 128\ \text{m/day}$ at moderate solar activity) and explain what "moderate solar activity" hides.

Part G — Design it & interleaved (⭐⭐ / ⭐⭐⭐)

12.21 (Mission project.) Add a station-keeping budget line to your MDR for the mission you chose in Chapter 1. Identify the dominant perturbation for your orbit (drag / luni-solar / J2 / SRP), estimate its annual delta-v from §12.6, multiply by your design life, and record the total.

12.22 † (Design.) A GEO comsat has a wet mass of $m_0 = 2{,}500\ \text{kg}$ and a $12$-year design life. Using north–south $= 48\ \text{m/s/yr}$ and east–west $= 3\ \text{m/s/yr}$, compute the total station-keeping delta-v, then the propellant mass with (a) a chemical thruster $I_{sp} = 300\ \text{s}$ and (b) an electric thruster $I_{sp} = 1{,}800\ \text{s}$. How much mass does electric free up for payload?

12.23 (Ch. 8 interleaved.) An orbit has $a = 7{,}000\ \text{km}$, $e = 0$, $i = 98^\circ$. Compute its J2 nodal-regression rate in deg/day. Is this orbit sun-synchronous (is the rate $+0.9856$)? If not, should the inclination be increased or decreased to make it so?

12.24 (Ch. 9 interleaved.) A Molniya orbit is specified at $i = 63.4^\circ$. Which perturbation effect does this inclination defeat, and qualitatively what would happen to the orbit's apogee over a year if the inclination were mistakenly set to $70^\circ$ instead?

12.25 † (Ch. 6 / 8 interleaved.) A satellite orbits with perigee altitude $300\ \text{km}$ and apogee altitude $3{,}000\ \text{km}$. Using vis-viva reasoning, state where in the orbit the satellite is fastest and where the air is densest, and hence explain which apsis (perigee or apogee) drag lowers first and why the orbit circularizes.

12.26 (Ch. 3 interleaved.) The 15-year GEO station-keeping budget in §12.6 came to $\sim 0.8\ \text{km/s}$. Compare that to the GTO → GEO insertion burn ($\sim 1.5\ \text{km/s}$) from the Chapter 3 delta-v map. What fraction of a comsat's own lifetime delta-v is spent merely holding station, and why does this make electric propulsion so attractive?

12.27 † (Synthesis.) A dead satellite is left in a circular orbit at $1{,}000\ \text{km}$. Using the §12.3 lifetime table, roughly how long until drag removes it, and why does this altitude sit at the heart of the space-debris debate you will meet in Chapter 35?

12.28 (Design / synthesis.) Your Track-A comsat can either (a) carry chemical propellant for 15 years of station-keeping, or (b) carry an electric system that saves $\sim 700\ \text{kg}$ of propellant but draws $5\ \text{kW}$ and adds $80\ \text{kg}$ of thrusters and power. Argue which you would choose and what additional subsystem chapters (power, Chapter 25; propulsion, Chapter 20) the decision pulls in. There is no single right answer — the rubric rewards a clear trade.

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: correctly identifying the dominant perturbation, explicit units, a sanity check on every number, and a clear statement of the trade being made.