Orbital mechanics Study Guide

📖 Core Concepts Astrodynamics – Uses Newton’s laws and gravitation to predict spacecraft motion. Two‑body assumption – Orbiting body’s mass ≪ central body; only the central body’s gravity considered. Kepler’s Laws (derived from Newton): Orbits are ellipses with the central body at a focus. Equal areas are swept in equal times (law of areas). \(T^{2} \propto a^{3}\) – orbital period squared proportional to semi‑major axis cubed. Specific orbital energy \(\epsilon = \frac{v^{2}}{2} - \frac{GM}{r}\). \(\epsilon < 0\): bound (elliptical) orbit. \(\epsilon = 0\): parabolic escape trajectory. \(\epsilon > 0\): hyperbolic fly‑by. Vis‑viva equation – Gives speed at any radius: \[v^{2} = \mu\!\left(\frac{2}{r} - \frac{1}{a}\right),\qquad \mu = GM\] Escape velocity – Speed needed for \(\epsilon = 0\): \[v{\text{esc}} = \sqrt{\frac{2GM}{r}}\] (≈ 11 km s⁻¹ from Earth’s surface). --- 📌 Must Remember Circular orbital speed: \(v{\text{c}} = \sqrt{\frac{GM}{r}}\). Escape speed = \(\sqrt{2}\,v{\text{c}}\). Orbital period: \(T = 2\pi\sqrt{\dfrac{a^{3}}{\mu}}\). Hohmann transfer Δv (minimum for coplanar circular‑to‑circular): two impulses at periapsis & apoapsis. Bi‑elliptic is better than Hohmann only when \(r{\text{final}}/r{\text{initial}} \gtrsim 12\). Plane‑change Δv is smallest at apoapsis (lowest velocity). Oberth effect: Δv effectiveness scales with instantaneous speed; burn near periapsis of a deep‑gravity pass yields more energy. Gravity assist changes spacecraft velocity relative to the Sun without propellant; planet’s momentum change is negligible. --- 🔄 Key Processes Solving Kepler’s Equation (for elliptical orbits) Compute eccentric anomaly \(E\) from true anomaly \(\theta\) (or vice‑versa). Iterate \(M = E - e\sin E\) using Newton’s method until convergence. Hohmann Transfer Burn 1: increase velocity to raise apoapsis to target orbit radius. Coast half an orbit to apoapsis. Burn 2: circularize at target radius. Bi‑elliptic Transfer Burn 1: raise apoapsis to a high intermediate radius \(r{b}\). Burn 2 (at \(r{b}\)): adjust periapsis to target radius. Burn 3: circularize at target radius. Plane‑Change Burn Perform at node where planes intersect, preferably near apoapsis: \(\Delta v = 2v\sin\frac{\Delta i}{2}\). Gravity Assist (Patched Conic) Identify dominant gravitating body for each leg (e.g., Earth → Sun → Mars). Compute hyperbolic excess velocity \(v{\infty}\) entering and exiting the planet’s sphere of influence. Apply vector rotation based on turning angle \(\delta = 2\arcsin\!\left(\frac{1}{1 + (r{p}v{\infty}^{2}/\mu)}\right)\). --- 🔍 Key Comparisons Circular vs. Elliptical Orbit Circular: eccentricity \(e = 0\); constant speed \(v = \sqrt{GM/r}\). Elliptical: \(0 < e < 1\); speed varies per vis‑viva; periapsis speed > apoapsis speed. Hohmann vs. Bi‑elliptic Transfer Hohmann: 2 burns, always optimal for modest radius ratios. Bi‑elliptic: 3 burns, lower total Δv only when \(r{f}/r{i} \gtrsim 12\). High‑thrust vs. Low‑thrust Transfer High‑thrust: impulsive burns, short travel time, higher Δv. Low‑thrust: continuous thrust, long travel time, lower propellant use. Parabolic vs. Hyperbolic Trajectory Parabolic: \(e = 1\), \(\epsilon = 0\), speed \(v = \sqrt{2\mu/r}\). Hyperbolic: \(e > 1\), \(\epsilon > 0\), speed approaches \(v{\infty}\) as \(r \to \infty\). --- ⚠️ Common Misunderstandings “Thrust toward a leading spacecraft always catches up.” Wrong: thrust raises the trailing craft’s orbit, slowing its angular rate and causing it to fall behind. “Escape velocity is the same as orbital velocity at infinity.” Escape velocity is the instantaneous speed needed to make \(\epsilon = 0\); once escaped, the spacecraft still has residual speed (the hyperbolic excess \(v{\infty}\)). “Atmospheric drag can be ignored for all LEO missions.” Drag significantly lowers semi‑major axis and must be compensated, especially below 400 km altitude. “Kepler’s equation can be solved algebraically.” It is transcendental; requires numerical iteration (Newton’s method). --- 🧠 Mental Models / Intuition Energy picture: Negative specific energy = bound (ellipse); zero = parabola; positive = hyperbola. Speed‑radius trade‑off: In an ellipse, the spacecraft “slides” faster when it’s closer to the central body (conservation of angular momentum). Delta‑v budgeting: Think of Δv as “fuel cost.” The cheapest way to change an orbit is where the spacecraft’s speed is lowest (apoapsis for plane changes, periapsis for Oberth burns). Gravity assist as a “slingshot”: The planet’s gravity bends the spacecraft’s trajectory like a cue ball off a billiard cushion; the speed change is the vector projection of the planet’s orbital motion onto the spacecraft’s exit direction. --- 🚩 Exceptions & Edge Cases Binary (comparable‑mass) systems: Two‑body approximation fails; must treat as n‑body problem. Very low Earth orbits (< 200 km): Atmospheric drag dominates; standard Keplerian formulas give poor predictions. High‑eccentricity transfers: For \(e \approx 1\), the vis‑viva equation approaches the escape‑velocity form; careful with rounding errors. Plane‑change at periapsis: Technically possible but incurs huge Δv because velocity is maximal; only justified if combined with an Oberth burn that also changes orbital energy. --- 📍 When to Use Which Choose Hohmann when moving between coplanar circular orbits with radius ratio < 12 and mission time is not critical. Choose Bi‑elliptic for large radius ratios (> 12) where Δv savings outweigh extra burn and time. Use Low‑thrust (electric) transfer for missions where propellant mass is a premium and long transfer times are acceptable (e.g., deep‑space cargo). Apply plane‑change at apoapsis unless the maneuver can be piggy‑backed on an existing periapsis burn (Oberth advantage). Apply patched‑conic gravity assist for interplanetary legs when the spacecraft spends most of the time under a single dominant gravitating body. --- 👀 Patterns to Recognize “Burn‑then‑coast‑then‑burn” pattern → Hohmann or bi‑elliptic transfers. Δv vs. velocity magnitude: Larger Δv needed when the spacecraft’s instantaneous speed is high (Oberth effect). Equal‑area law → If a problem states equal times, the sweep areas are equal → can infer relative speeds at different orbital positions. Energy sign → Positive → escape/hyperbolic; zero → parabolic; negative → bound ellipse. --- 🗂️ Exam Traps Distractor: “Escape velocity is the same as the speed needed to reach a circular orbit at a higher altitude.” Why tempting: Both involve “leaving” a radius, but escape velocity is defined for \(\epsilon = 0\); circular orbit speed at higher altitude is lower. Distractor: “A thrust opposite to motion always shortens the orbit period.” Why tempting: Opposite thrust creates an ellipse, but if applied at the wrong true anomaly the period can increase; only when applied at a point on a circular orbit does the periapsis end up opposite the burn, shortening the period. Distractor: “Plane‑change burns are always best performed at periapsis because of the Oberth effect.” Why tempting: Oberth effect is real, but Δv cost for inclination change scales with velocity; the minimum Δv for pure inclination change is at apoapsis. Distractor: “Bi‑elliptic transfers always use less Δv than Hohmann transfers.” Why tempting: Bi‑elliptic adds a high‑apoapsis leg, which can reduce Δv for very large radius ratios, but not for typical LEO‑to‑GEO moves. ---