Chapter 35 — Key Takeaways (Space Debris, Space Law, and Sustainability)

A one-page reference. Reread this before an exam, or before you write your mission's disposal plan.

The one-sentence version

Orbit is a finite, shared resource; debris can breed debris until a shell becomes unusable, so every mission must plan how it leaves — cheaply and reliably — and the law and economics of a commons are only now catching up with the engineering.

Core numbers to memorize

Quantity Value Note
Energy of a $1\ \text{cm}$ Al fleck at $10\ \text{km/s}$ $\sim 70\ \text{kJ}$ ≈ a small car at $50\ \text{km/h}$
Tracked objects ($>10\ \text{cm}$) $\sim 36{,}000$ Tier 2; climbing
"Lethal non-trackable" ($1$–$10\ \text{cm}$) $\sim 1{,}000{,}000$ modeled; the real menace
Objects $\geq 1\ \text{mm}$ $\sim 1.3\times10^{8}$ modeled
Catastrophic-fragmentation threshold EMR $> 40\ \text{J/g}$ any orbital collision clears it $\gg$
Typical closing speed (LEO) $\sim 10\ \text{km/s}$ $v_{\text{rel}} = 2v\sin(\theta/2)$
GEO → graveyard disposal $\sim 11\ \text{m/s}$ go up $\sim 300\ \text{km}$
LEO de-orbit $\sim 100$–$200\ \text{m/s}$ go down; lower perigee
Disposal rule 25 yr (IADC/UN); 5 yr (US FCC, 2022) LEO post-mission
Outer Space Treaty 1967 foundational

Key equations (with meaning and units)

Equation Meaning Symbols
$E = \tfrac{1}{2}mv^2$ impact energy; $v$ is closing speed $m$ [kg], $v$ [m/s], $E$ [J]
$v_{\text{rel}} = 2v\sin(\theta/2)$ closing speed of two orbits crossing at angle $\theta$ $v$ orbital speed, $\theta$ crossing angle
$N = \Phi\,A\,t$; $P = 1-e^{-N}\approx N$ expected impacts and collision probability $\Phi$ [m⁻²yr⁻¹], $A$ [m²], $t$ [yr]
collision/conjunction rate $\propto N^2$ pairs $= N(N-1)/2$ $N$ = number of objects
$\tau \approx \dfrac{H\beta}{\rho_0\,a\,v}$ natural circular-orbit decay lifetime $H$ scale height, $\beta=m/(C_DA)$, $\rho_0$ density, $a$ radius, $v$ speed
$\Delta v \approx \tfrac{v}{2}\tfrac{\Delta h}{a}$ small circular-orbit raise (graveyard) $v$ orbital speed, $\Delta h$ raise, $a$ radius
$m_p = m\left(1-e^{-\Delta v/v_e}\right)$ de-orbit propellant (rocket eq., Ch. 3) $v_e = I_{sp}g_0$

Terms first-defined here

Term One-line definition
Space debris Any defunct human-made object in orbit — dead satellites, spent stages, fragments.
Kessler syndrome Self-sustaining collision cascade; debris breeds debris once production outpaces drag removal.
Collision avoidance Predicting conjunctions and maneuvering when $P_c$ exceeds a threshold ($\sim 10^{-4}$).
Passivation Removing all stored energy at end of life so a derelict cannot later explode.
Active debris removal (ADR) Deliberately capturing/removing existing debris (vs. mitigation, which prevents new debris).
Outer Space Treaty 1967 foundational treaty: no appropriation; states responsible/liable for national (incl. private) activity.
Space traffic management (STM) Coordinating on-orbit traffic to keep operations safe and steward the finite orbital commons.

Decision aid — "which disposal method?"

Situation Method Rough cost
GEO satellite Boost up to graveyard ($\sim 300\ \text{km}$) $\sim 11\ \text{m/s}$
LEO below $\sim 600\ \text{km}$ Let drag de-orbit it (passive) free (decays $\lesssim$ decades)
LEO $\sim 600$–$800\ \text{km}$ Drag sail (meets 25-yr) or propulsive de-orbit (meets 5-yr) sail: $\beta$↓; burn: $\sim 100$–$200\ \text{m/s}$
LEO above $\sim 800\ \text{km}$ Propulsive de-orbit (sails too weak) $\sim 190\ \text{m/s}$+
Any mission Always passivate + design for demise ~free

Space law in one table

Instrument Year What it does
Outer Space Treaty 1967 No appropriation (Art. II); states responsible for private actors → licensing (Art. VI); ownership/jurisdiction retained forever (Art. VIII).
Rescue Agreement 1968 Return astronauts and objects.
Liability Convention 1972 Absolute liability for damage on ground/aircraft; fault-based in space. Invoked once (Cosmos 954).
Registration Convention 1976 States must register objects with the UN → enables ID, liability, STM.
Moon Agreement 1979 "Common heritage"; ratified by no major power — effectively dead.

Common pitfalls

Pitfall Reality
"Space is too big to fill." Useful orbits are thin shells with a finite carrying capacity; cross it and debris self-amplifies.
"It'll decay eventually." At $800\ \text{km}$, "eventually" is centuries — fails both rules; only $<600\ \text{km}$ is passively safe.
"Doubling satellites doubles risk." Risk $\propto N^2$ — doubling quadruples it.
"No one owns space, so anything goes." It's a governed commons: states responsible, liable, and must register objects.
"We'll de-orbit with leftover propellant." Reserve it explicitly; a dead/failed satellite has no leftovers and cannot dispose of itself.
"A drag sail solves disposal anywhere." Only where drag is strong ($\lesssim 600\ \text{km}$); useless high up.

Mission / astrotools additions this chapter

  • MDR: your end-of-life disposal plan — destination, disposal $\Delta v$ (reserved), decay-lifetime check vs. the rule, passivation commitment, and a collision-avoidance conops.
  • debris.py: collision_probability(flux, area, years), orbit_lifetime(alt_km, beta, rho0, H_km), reorbit_dv(v, dh_km, a_km).