Exercises: Space Debris, Space Law, and Sustainability

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, Earth's gravitational parameter $\mu = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$, and mean radius $R_\oplus = 6{,}371\ \text{km}$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each cold first. 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. Where a number is uncertain (especially upper-atmosphere density), state your assumption and flag it.

Part A — Warm-ups: the environment and the energy (⭐)

35.1 † A $2\ \text{g}$ fragment closes on a satellite at $9\ \text{km/s}$. What is its kinetic energy in joules? Compare it to a $10\ \text{g}$ rifle bullet at $900\ \text{m/s}$.

35.2 Roughly how many objects larger than $10\ \text{cm}$ are currently tracked, and roughly how many are estimated in the $1$–$10\ \text{cm}$ ("lethal non-trackable") band? Why is the second band considered more dangerous than the first? (Label your numbers Tier 2.)

35.3 † Name the three defensive regimes by debris size (shield / track-and-dodge / neither), give the approximate size boundaries, and say which regime the $1$–$10\ \text{cm}$ band falls in.

35.4 In one sentence each, define space debris, passivation, and active debris removal, and state which two of the three prevent new debris versus remove existing debris.

35.5 † Two satellites orbit at $7.6\ \text{km/s}$ each. Their orbital planes cross at $90^\circ$. Using $v_{\text{rel}} = 2v\sin(\theta/2)$, find their closing speed. Repeat for a $20^\circ$ crossing and comment on why steep crossings are so much more dangerous.

Part B — Collision probability (⭐⭐)

35.6 † A satellite presents $A = 8\ \text{m}^2$ and flies for $t = 7$ years through a shell with debris flux $\Phi = 2\times10^{-5}\ \text{m}^{-2}\,\text{yr}^{-1}$ (objects $> 1\ \text{cm}$; Tier 3 illustrative). Compute the expected number of impacts $N$ and the probability of at least one, $P = 1 - e^{-N}$.

35.7 For the satellite in 35.6, a constellation flies $8{,}000$ identical copies. What is the expected number of catastrophic strikes across the fleet over 7 years? Roughly what is the probability that at least one satellite is lost? (Use $1 - (1-P)^{8000}$, or note that it equals $1 - e^{-8000 N}$.)

35.8 † Show, starting from $N = \Phi A t$ and $P = 1 - e^{-N}$, that for small $N$ the probability is approximately $P \approx \Phi A t$. For what value of $N$ does this linear approximation start to err by more than $5\%$?

Part C — Disposal delta-v and lifetime (⭐⭐ / ⭐⭐⭐)

35.9 † A geostationary satellite ($a = 42{,}164\ \text{km}$, $v = 3{,}070\ \text{m/s}$) is boosted to a graveyard orbit $285\ \text{km}$ above GEO. Using the small-raise approximation $\Delta v \approx \tfrac{v}{2}\tfrac{\Delta h}{a}$, find the disposal delta-v. Then explain in one sentence why de-orbiting the same satellite instead would cost more than a hundred times as much.

35.10 A satellite in a circular orbit at $700\ \text{km}$ ($v = \sqrt{\mu/a}$) de-orbits by lowering its perigee to $100\ \text{km}$ altitude. Set up the transfer ellipse, use vis-viva to find the speed at the $700\ \text{km}$ apogee, and compute the de-orbit burn $\Delta v$.

35.11 † Using the lifetime estimate $\tau \approx H\beta/(\rho_0 a v)$, find the natural decay lifetime of a satellite with $\beta = 80\ \text{kg/m}^2$ at $700\ \text{km}$. Take $H = 75\ \text{km}$ and $\rho_0 = 4\times10^{-14}\ \text{kg/m}^3$ (flag this density as solar-cycle-dependent). Does it satisfy the 25-year rule? The 5-year rule?

35.12 A $200\ \text{kg}$ satellite at $700\ \text{km}$ with compact cross-section $C_D A = 2\ \text{m}^2$ deploys a $12\ \text{m}^2$ drag sail ($C_D = 2.2$). Find its ballistic coefficient before and after, and the factor by which its natural lifetime shortens. If the un-sailed lifetime was 90 years, is the sailed one rule-compliant?

Part D — Implement it in Python (⭐⭐)

Write each function, hand-trace it for the inputs, and record the result in an # Expected output: comment. Do not run it. Use $\mu = 3.986\times10^{14}\ \text{m}^3/\text{s}^2$, $R_\oplus = 6.371\times10^{6}\ \text{m}$, $3.156\times10^{7}\ \text{s/yr}$.

35.13 † Write collision_probability(flux, area, years) returning $1 - e^{-\Phi A t}$. Trace it for collision_probability(1e-5, 10, 5).

35.14 Write orbit_lifetime(alt_km, beta, rho0, H_km) returning the lifetime in years from $\tau = H\beta/(\rho_0 a v)$, with $a = R_\oplus + h$ and $v = \sqrt{\mu/a}$. Trace it for orbit_lifetime(600, 100, 2e-13, 70).

35.15 † Write graveyard_dv(v, dh_km, a_km) returning $\tfrac{v}{2}\tfrac{\Delta h}{a}$. Trace it for graveyard_dv(3070, 300, 42164).

Part E — Design it (⭐⭐ / ⭐⭐⭐)

35.16 † (Mission project) Write the end-of-life disposal plan for your chosen mission (Track A/B/C/D): destination, disposal delta-v (added to your budget), decay lifetime if applicable, and a passivation commitment. State which rule (25-year / 5-year / planetary protection) governs it and show your mission complies.

35.17 Your smallsat must operate at $800\ \text{km}$ but comply with the 5-year disposal rule. Natural decay there is centuries, so you need active de-orbit. Estimate the de-orbit $\Delta v$ (lower perigee to $\sim 150\ \text{km}$), then, for a $120\ \text{kg}$ satellite with an $I_{sp} = 220\ \text{s}$ thruster, the propellant mass required (use the rocket equation). What fraction of the satellite is disposal propellant?

35.18 † Design a collision-avoidance concept of operations for a $500$-satellite constellation: what conjunction-probability threshold triggers a maneuver, why the process must be autonomous at this scale, and one failure mode of autonomous avoidance you would guard against.

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

35.19 † Why can't you just sweep up small debris with a big net or a "space vacuum"? Frame your answer in terms of the flux, the closing speeds, and the number of objects in the $1$–$10\ \text{cm}$ band.

35.20 Estimate, to an order of magnitude, the total kinetic energy stored in the tracked debris population: take $\sim 5{,}000$ tonnes of trackable debris (Tier 2) all moving at $7.5\ \text{km/s}$. Express the result in tonnes of TNT ($1\ \text{t TNT} = 4.18\times10^{9}\ \text{J}$). What does this say about why even one collision is so productive of fragments?

35.21 † Why can't you just leave a defunct satellite where it is if space is so empty? Answer using both the $N^2$ collision scaling and the idea of an orbit's carrying capacity.

Part G — Space law and reasoning (⭐⭐)

35.22 Under the Liability Convention, a piece of a re-entering satellite damages a house, and — separately — a maneuvering satellite collides with another operator's satellite in orbit. Which liability standard (absolute vs. fault-based) applies to each, and why is the second case far harder to litigate?

35.23 † Explain, citing the relevant Outer Space Treaty articles, (a) why a commercial operator must be licensed by its government, and (b) why you cannot legally remove another country's derelict satellite without permission.

Part H — Interleaved & synthesis (Chapters 3, 7, 12, 33) (⭐⭐ / ⭐⭐⭐)

35.24 (Ch. 12) Chapter 12 gave the per-orbit decay $\Delta a_{\text{rev}} = -2\pi\rho a^2/\beta$. For a satellite at $500\ \text{km}$ ($a = 6{,}871\ \text{km}$, $\rho = 5\times10^{-13}\ \text{kg/m}^3$, $\beta = 100\ \text{kg/m}^2$), find the altitude lost per orbit and per day ($\sim 15$ orbits/day). Is this consistent with the "self-cleaning low orbit" story?

35.25 † (Ch. 7, synthesis) "Design for demise" wants a satellite to burn up completely on re-entry. Using what Chapter 7 taught about re-entry heating, explain which components tend to survive and why, and name one design change that helps a part demise.

35.26 (Ch. 33, synthesis) A rideshare launch deploys $50$ CubeSats into a single $600\ \text{km}$ orbit. Argue why the deployment itself is a debris-and-traffic concern, and how it connects to the collision-probability arithmetic of §35.2.

35.27 † (Ch. 3, synthesis) Your GEO comsat reserves $11\ \text{m/s}$ for graveyard disposal on a $3{,}000\ \text{kg}$ wet mass with an $I_{sp} = 300\ \text{s}$ thruster. How much propellant must you set aside? Why is it a design mistake to plan to use "leftover" propellant for disposal rather than a reserved amount?

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 set-up, explicit units, an honest uncertainty flag on any density-dependent number, and a sanity check on the final result.