Orbital Mechanics

The three laws of motion, the inverse-square law and Newton's Moon test

Newton's cannon, circular velocity, free fall and escape speed

Specific angular momentum, planarity of orbits and Kepler's second law

Specific mechanical energy, the vis-viva equation and the sign of ε

Reduction to relative motion, the gravitational parameter μ and the assumptions of the model

The eccentricity vector and the closed-form solution of the two-body problem

Circle, ellipse, parabola and hyperbola: geometry, energy and angular momentum

Turn the equation of motion you derived into a working numerical simulator.

Inertial and rotating frames, ECI, ECEF, perifocal and topocentric coordinates

a, e, i, Ω, ω, ν: conversion between state vectors and elements

Ground-track construction, westward shift, repeat tracks and LEO/MEO/GEO/Molniya

Build the coordinate machinery every flight dynamics system depends on — your way.

From Hipparchus to quasars: the ICRS, its realisations and astronomical time scales

Precession and nutation models, sidereal time, and the MOD, TOD and TEME frames

Earth rotation, polar motion, ITRF, WGS 84, geodetic coordinates and the antenna’s point of view

RTN, VNC, LVLH and body axes; direction cosine matrices, Euler angles and quaternions

Planetary frames, the Moon’s two meridians, rotating frames, and the frame hand-offs of a real mission

Mean, eccentric and true anomaly, and the iterative solution of a transcendental equation

The hyperbolic and parabolic Kepler equations, the universal anomaly, Stumpff functions and the f and g functions

Analytic Kepler propagation, numerical integration, SGP4 and ground-station passes

The public satellite catalogue, the TLE format, mean elements, SGP4 and SDP4, the TEME frame and pass prediction

Predict where a satellite will be, then predict when a ground station can see it.

Tsiolkovsky's equation, specific impulse, mass ratio and staging

The two-impulse transfer between coplanar circular orbits

Three-impulse transfers, the 11.94 crossover ratio and the Oberth effect

Simple and combined plane changes, launch-site geometry and phasing for rendezvous

Compute the two burns that move a satellite between circular orbits — and the fuel they cost.

Nodal regression, apsidal rotation, sun-synchronous and frozen orbits, critical inclination

The drag equation, ballistic coefficient, decay rates and the drag paradox

Radiation pressure, eclipses, lunisolar gravity and which perturbation dominates where

GEO station-keeping, longitude drift, Δv budgets, graveyard orbits and debris mitigation

Add the real-world perturbations that make orbits drift, precess, and eventually re-enter.

Hyperbolic excess speed, C₃, the sphere of influence and an introduction to patched conics

Synodic periods, departure phase angles, interplanetary Hohmann transfers and porkchop plots

Flyby geometry, energy exchange in the Sun's frame, the Voyager Grand Tour and libration points

The boundary-value problem, the transfer triangle, Lambert's theorem and Lagrange's time-of-flight equation

Stumpff functions, one equation for every conic, Newton iteration and verification by propagation

L-similarity, the x-variable, Householder iterations, and the 2N+1 solutions of an N-revolution problem

Δv as a function of departure time and flight time, safety constraints, phasing, and correcting Lambert for real dynamics

Ephemerides, C₃ and arrival v∞ maps, Type I and II transfers, the launch asymptote, and designing a launch period

Find the next three Mars windows from real ephemerides, then design a launch period.

Laplace's sphere, the Hill sphere, the three-phase trajectory and the size of the error

The injection burn, the Oberth effect, hyperbola geometry, DLA and RLA, parking-orbit planes and launch azimuth

Impact parameter, gravitational focusing, the B-plane, capture burns, optimal periapsis and aerobraking

The flyby as a rotation of v∞, energy exchange, leading and trailing passes, powered flybys and the Tisserand parameter

Chaining Lambert legs and flybys, resonant legs, searching sequences and dates, and testing the design against real dynamics

Design a complete 2028 Mars orbiter mission with patched conics — and measure where the model breaks.

Two primaries in a rotating frame, the equations of motion, the Jacobi constant, and the zero-velocity curves that decide where a spacecraft may go

Locating L1 to L5, linear stability, the saddles of L1–L3, Routh’s criterion for L4 and L5, and the Trojans

Linear motion near a collinear point, Richardson’s approximation, differential correction with the state-transition matrix, the monodromy matrix and the halo family

Invariant manifolds, manifold transfers and direct transfers, low-energy routes, station-keeping, and an Earth–Moon L2 relay

Why L1, the Earth-bound phase, trans-L1 injection and cruise, halo-orbit insertion, and the observatory at work

Design the mission from destination to propellant budget — and check every step against Aditya-L1.

Launch energy and flight time, Jupiter as the gateway, inner-planet assists, and why outer-planet windows open and close

Capture Δv and periapsis choice, moon-assisted capture, finite-burn losses, rings and radiation, and aerocapture

v∞ and pump angle, resonant orbits, pump-down sequences, Tisserand graphs in a planetary system, and lowering v∞ to reach a moon

Rotating frames, the Jacobi constant, Lagrange points and zero-velocity curves, Hill spheres of moons, and the Lidov–Kozai instability of polar orbits

Descent budgets on airless worlds, gravity turns and gravity loss, thrust-to-weight trades, and entry and parachute descent at Titan

Ion propulsion, spirals, Edelbaum's approximation, phasing a low-thrust transfer, and Dawn at Vesta and Ceres

Rebuild the mission design that let one spacecraft orbit Vesta and then Ceres.

Choose Europa, Ganymede, Titan, Enceladus or Triton, and design the whole trip — launch pad to touchdown.