Orbital Mechanics
Orbital mechanics (also called astrodynamics) is the branch of aerospace engineering and applied physics that studies the motion of spacecraft, satellites, and celestial bodies under the influence of gravitational forces. It is governed by Newton's law of universal gravitation and Kepler's three laws of planetary motion, and it underpins the planning of satellite launches, orbital transfers, interplanetary trajectories, and re-entry profiles. Mastery of orbital mechanics is essential for mission design, ground-track prediction, and spacecraft manoeuvring.
Key Formula
v_c = sqrt(G * M / r)
LaTeX: v_c = \sqrt{\dfrac{GM}{r}}
| Symbol | Meaning | Unit |
|---|---|---|
| v_c | Circular orbital velocity | m/s |
| G | Gravitational constant (6.674 × 10⁻¹¹) | N·m²/kg² |
| M | Mass of the central body (e.g. Earth) | kg |
| r | Orbital radius from centre of planet | m |
Worked Example
Problem
Calculate the circular orbital velocity of a satellite in low Earth orbit at an altitude of 400 km above Earth's surface. (Earth's radius = 6.371 × 10⁶ m, GM_Earth = 3.986 × 10¹⁴ m³/s²)
Solution
Step 1: Find orbital radius: r = R_Earth + altitude = 6.371 × 10⁶ + 0.4 × 10⁶ = 6.771 × 10⁶ m. Step 2: Apply formula: v_c = sqrt(GM / r) = sqrt(3.986 × 10¹⁴ / 6.771 × 10⁶). Step 3: Compute quotient: 3.986 × 10¹⁴ / 6.771 × 10⁶ = 5.887 × 10⁷ m²/s². Step 4: v_c = sqrt(5.887 × 10⁷) = 7672 m/s.
Answer
v_c ≈ 7,672 m/s (approximately 7.67 km/s)
Key Orbital Parameters for Common Earth Orbits
| Orbit Type | Altitude (km) | Orbital Speed (km/s) | Period | Primary Use |
|---|---|---|---|---|
| Low Earth Orbit (LEO) | 200 – 2000 | 7.8 – 7.2 | 90 – 127 min | ISS, Earth observation |
| Medium Earth Orbit (MEO) | 2000 – 35786 | 6.4 – 3.07 | 2 – 24 h | GPS, navigation |
| Geostationary Orbit (GEO) | 35786 | 3.07 | 23 h 56 min | Communications, weather |
| Sun-synchronous Orbit (SSO) | 600 – 800 | ~7.5 | 97 – 101 min | Remote sensing |
| Transfer orbit (GTO) | 200 – 35786 (elliptical) | Varies | ~10.5 h | GEO deployment |
Interactive Tools
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Related Terms
Tsiolkovsky Rocket Equation
The Tsiolkovsky rocket equation (also called the ideal rocket equation) describes the relationship between the change in velocity (Δv) of a rocket and the logarithmic ratio of its initial to final mass, scaled by the exhaust velocity of the propellant. Derived by Russian scientist Konstantin Tsiolkovsky in 1897, it reveals the fundamental challenge of rocketry: achieving large Δv requires either very high exhaust velocities or carrying propellant many times the mass of the payload. It is the foundational equation for mission planning and launch vehicle design.
Specific Impulse
Specific impulse (I_sp) is a measure of the propellant efficiency of a rocket or jet engine, defined as the thrust produced per unit weight flow rate of propellant consumed; it is expressed in seconds and is independent of gravity field when defined in this way. A higher specific impulse indicates that the engine generates more thrust for each kilogram of propellant burned per second, making it the key figure of merit for comparing propulsion systems. Specific impulse directly relates to exhaust velocity: I_sp = v_e / g_0, where g_0 is standard gravity (9.80665 m/s²).
Engine Thrust
Engine thrust is the reaction force produced by a jet or rocket engine as it expels mass (exhaust gases) at high velocity, propelling the vehicle in the opposite direction in accordance with Newton's third law of motion. For air-breathing jet engines, thrust depends on the mass flow rate of air through the engine and the velocity increase imparted to it; for rocket engines, thrust depends on propellant mass flow and exhaust velocity. Thrust must exceed aerodynamic drag for acceleration and must balance it during steady flight.
From Latin orbita (track, rut, course of a wheel) and Greek mechanikē (of machines, practical). The systematic study of celestial mechanics was founded by Johannes Kepler (1609–1619) and formalised by Isaac Newton in Principia Mathematica (1687).