Orbital Motion

Orbital motion occurs when one object travels continuously around another due to gravity. The Moon orbits Earth, Earth orbits the Sun, and satellites orbit Earth — all following the same underlying physics. Rather than falling straight down, an orbiting object moves fast enough sideways that as it falls toward the central body, the curved surface of that body drops away beneath it at the same rate. The result is a continuous free-fall that never reaches the ground — a closed, repeating path through space. Most natural orbits are ellipses, as Johannes Kepler discovered in the early 1600s. A circle is simply a special case of an ellipse where both foci coincide at the centre. In an elliptical orbit, the central body sits at one focus of the ellipse, not at the geometric centre. This means the orbiting object speeds up when it is closer to the central body and slows down when farther away — a direct consequence of the conservation of angular momentum and gravitational potential energy. Understanding orbital motion requires combining Newton's law of universal gravitation with the concept of centripetal acceleration. Gravity provides the centripetal force that continuously deflects the orbiting object from a straight-line path. By equating gravitational force to the required centripetal force, we can predict orbital speeds, periods, and altitudes — the foundation of modern satellite technology and space exploration.

Key takeaways
  • Gravity as the Centripetal Force
  • Deriving Orbital Speed
  • Orbital Period
  • Elliptical Orbits and Kepler's Laws
  • Geostationary Orbits

Gravity as the Centripetal Force

For any object in a circular orbit, some force must act centripetally — directed toward the centre of the circle — to maintain that curved path. For orbiting bodies, this force is gravity. Newton's law of universal gravitation gives F = GMm/r², where M is the central mass, m is the orbiting mass, and r is the orbital radius. Setting this equal to the centripetal force requirement F = mv²/r allows us to solve for orbital speed.

Deriving Orbital Speed

Equating gravitational force to centripetal force: GMm/r² = mv²/r. The mass m of the orbiting object cancels from both sides — a profound result showing that orbital speed is independent of the orbiting object's own mass. Solving for v gives v = √(GM/r). A heavier satellite does not orbit faster than a lighter one at the same altitude. This mirrors Galileo's finding that all objects fall at the same rate regardless of mass.

Orbital Period

The orbital period T is the time to complete one full orbit. Since the circumference of a circular orbit is 2πr, and the orbital speed is v = √(GM/r), we can write T = 2πr/v = 2πr/√(GM/r) = 2π√(r³/GM). This is Kepler's Third Law in a precise mathematical form: T² ∝ r³. Doubling the orbital radius increases the period by a factor of 2^(3/2) ≈ 2.83. This relationship holds for all bodies orbiting the same central mass.

Elliptical Orbits and Kepler's Laws

Real orbits are ellipses. Kepler's First Law states that planets move in ellipses with the Sun at one focus. Kepler's Second Law (equal areas in equal times) is a consequence of conserved angular momentum: as an object moves closer to the focus, it speeds up to sweep out the same area per unit time. At the closest point (periapsis) kinetic energy is maximum and gravitational potential energy is minimum; the reverse is true at the farthest point (apoapsis). The total mechanical energy E = −GMm/(2a), where a is the semi-major axis, remains constant throughout the orbit.

Geostationary Orbits

A geostationary orbit is a circular orbit above the equator where the satellite's period exactly matches Earth's rotation period (T = 24 hours = 86 400 s). Using T = 2π√(r³/GM), we can solve for the specific radius at which this occurs — approximately 42 200 km from Earth's centre, or about 35 800 km above the surface. Communications and weather satellites use this orbit because they appear stationary relative to the ground. It is a direct application of equating the orbital period formula to a specific required value.

The Key Equation

v=GMrv = \sqrt{\frac{GM}{r}}
  • v — Orbital speed (m s⁻¹): The tangential speed required to maintain a circular orbit at radius r.
  • G — Universal gravitational constant (N m² kg⁻²): G = 6.674 × 10⁻¹¹ N m² kg⁻²; relates gravitational force to mass and distance.
  • M — Mass of central body (kg): The mass of the object being orbited (e.g. Earth, Sun). The orbiting object's own mass cancels out.
  • r — Orbital radius (m): The distance from the centre of the central body to the orbiting object. Larger r means slower orbital speed.

Worked Example

Problem. A communications satellite orbits Earth in a circular orbit at an altitude of 500 km above the surface. Calculate (a) the orbital speed and (b) the orbital period. (Mass of Earth M = 5.97 × 10²⁴ kg; radius of Earth R = 6.37 × 10⁶ m; G = 6.674 × 10⁻¹¹ N m² kg⁻².)

v≈7615 m s−1,T≈5670 sv \approx 7615 \text{ m s}^{-1}, \quad T \approx 5670 \text{ s}

Answer. Orbital speed ≈ 7615 m s⁻¹ (7.6 km s⁻¹); orbital period ≈ 5670 s (about 94.5 minutes).

Figure
Diagram
SunPlanetr (orbital radius)v (tangential)F_gravApoapsisPeriapsisElliptical Orbit — Kepler's First LawCentral body at one focus; orbit is an ellipse.Gravity (F_grav) always points toward the focus.
Origins · 1 minJohannes Kepler and how this was discovered
JK
Johannes Kepler
1571–1630 · German

Kepler analysed Tycho Brahe's meticulous planetary observations to formulate his three laws of planetary motion (published 1609 and 1619). His First Law established that planetary orbits are ellipses, not circles; his Second Law described equal-area sweeping; and his Third Law gave the precise T² ∝ r³ relationship. These laws provided Newton with the empirical foundation needed to derive his law of universal gravitation and to prove mathematically that an inverse-square gravitational force produces exactly the elliptical orbits Kepler had found. Kepler also wrote one of the earliest works of science fiction: Somnium (1608), a story about a journey to the Moon that incorporated his real astronomical knowledge — including the correct description of what Earth would look like from the lunar surface.

  • Centripetal Force and Acceleration — Orbital motion is sustained by centripetal force; understanding a = v²/r and F = mv²/r is essential before analysing gravity as the source of that centripetal force.
  • Newton's Law of Universal Gravitation — Gravity is the centripetal force in orbital motion. The formula F = GMm/r² must be set equal to mv²/r to derive all key orbital equations.
  • Newton's Second Law — Equating net force to ma underpins the derivation of orbital speed and period; Newton's Second Law connects gravitational force to the centripetal acceleration of the orbiting body.
  • Gravitational Potential Energy — In elliptical orbits, kinetic and gravitational potential energy are continuously exchanged. The total orbital energy E = −GMm/2a relies on understanding gravitational potential energy.