Orbital Mechanics, Kepler’s Laws, and Satellite Energy

For a satellite of mass m in a circular orbit of radius r around a mass M, the kinetic energy is E_k = GMm ÷ 2r, the potential energy is U = −GMm ÷ r, and the total energy is E = −GMm ÷ 2r; for an elliptical orbit of semi-major axis a, the total energy is E = −GMm ÷ 2a, and Kepler’s third law gives T² = 4π²a³ ÷ GM. In brief: a satellite with a negative total energy is bound; at E = 0 it escapes along a parabola, and with E > 0 it follows a hyperbola. Moving to a higher orbit requires an increase in total energy, but the orbital speed decreases. The speed at any point of an orbit is given by the vis-viva equation, v² = GM(2/r − 1/a). Transfer between circular orbits is done efficiently with a Hohmann transfer, an elliptical orbit tangent to both.

Rockets, space probes and satellites follow the laws of orbital mechanics. This guide extends the introductory treatment of orbits with energy methods and Kepler’s laws for ellipses, describes orbit changes and the Hohmann transfer, compares orbit types in a table, works through numerical examples and corrects common mistakes.

What do Kepler’s laws say about ellipses?

An orbit under an inverse-square force is a conic section. For bound orbits it is an ellipse with the central mass at one focus. Parameters: the semi-major axis a (half the longest diameter), the eccentricity e (0 for a circle, close to 1 for a very elongated ellipse), the closest approach (periapsis) r_p = a(1 − e) and the farthest distance (apoapsis) r_a = a(1 + e). Kepler’s laws:

1. The orbit is an ellipse with the central body at one focus.
2. The line from the central body to the orbiting body sweeps out equal areas in equal times, which reflects the conservation of angular momentum: the body moves faster at periapsis and slower at apoapsis (v_pr_p = v_ar_a).
3. T² = (4π² ÷ GM) a³, valid for any ellipse with semi-major axis a.

What is the energy of an orbit?

For a circular orbit, setting the gravitational force equal to the centripetal force, GMm ÷ r² = mv² ÷ r, so the kinetic energy is E_k = ½mv² = GMm ÷ 2r. The potential energy is U = −GMm ÷ r. The total energy is E = E_k + U = −GMm ÷ 2r, which is half the potential energy and the negative of the kinetic energy. For an ellipse, replace r by the semi-major axis a. The energy depends only on a, not on the eccentricity. The vis-viva equation gives the speed at distance r in an orbit with semi-major axis a: v = √[GM(2/r − 1/a)]. For a circle (r = a), this is v = √(GM/r), and for escape (a → ∞), v = √(2GM/r). If a satellite loses energy to atmospheric drag, E becomes more negative and the orbit shrinks, but, paradoxically, the speed increases (the kinetic energy is −E): the satellite speeds up as it spirals inward, because the loss of potential energy is twice the energy dissipated by the drag.

How are orbits changed?

Rocket engines change the velocity (delta-v, Δv). A Hohmann transfer between two coplanar circular orbits of radii r₁ and r₂ uses two burns: (1) at r₁, a burn in the direction of motion raises the apoapsis to r₂ by moving to an elliptical orbit of semi-major axis a = (r₁ + r₂)/2; (2) at apoapsis, a second burn in the direction of motion circularizes the orbit at r₂. The transfer time is half the period of the ellipse: t = π√(a³/GM). To change the plane of the orbit, a much larger Δv is needed, which is why launch sites near the equator are used for geostationary satellites. Other maneuvers: gravity assists (a flyby of a planet gives a spacecraft extra speed relative to the Sun, at the expense of a tiny change in the planet’s motion), aerobraking, and the Oberth effect (burns are more efficient at high speed, near the periapsis).

Orbit types compared

Worked example: energy and a transfer to geostationary orbit

1. Energy of the International Space Station. Mass 4.2 × 10⁵ kg, orbit radius 6.771 × 10⁶ m: E = −GMm ÷ 2r = −3.986 × 10¹⁴ × 4.2 × 10⁵ ÷ (2 × 6.771 × 10⁶) = −1.24 × 10¹³ J. The kinetic energy: +1.24 × 10¹³ J; the potential energy: −2.48 × 10¹³ J.
2. Energy to raise the orbit. Moving the ISS from 400 km to 420 km (r = 6.791 × 10⁶ m): E_new = −3.986 × 10¹⁴ × 4.2 × 10⁵ ÷ (2 × 6.791 × 10⁶) = −1.2326 × 10¹³ J. The energy that must be added: ΔE = 3.6 × 10¹⁰ J (the speed decreases by about 0.01 km/s).
3. Vis-viva for a transfer ellipse. A satellite starts in a circular low Earth orbit (r₁ = 6.671 × 10⁶ m, altitude 300 km) and has to reach the geostationary radius (r₂ = 4.216 × 10⁷ m). The semi-major axis of the transfer ellipse: a = (6.671 × 10⁶ + 4.216 × 10⁷) ÷ 2 = 2.442 × 10⁷ m.
4. First burn. Circular speed at r₁: √(GM/r₁) = 7,730 m/s. Speed at the periapsis of the transfer ellipse: v = √[GM(2/r₁ − 1/a)] = √[3.986 × 10¹⁴ × (2.998 × 10⁻⁷ − 4.095 × 10⁻⁸)] = 10,158 m/s. Δv₁ = 10,158 − 7,730 = 2,428 m/s.
5. Second burn. Speed at the apoapsis of the ellipse: v = √[GM(2/r₂ − 1/a)] = √[3.986 × 10¹⁴ × (4.744 × 10⁻⁸ − 4.095 × 10⁻⁸)] = 1,607 m/s. The circular speed at r₂ is √(GM/r₂) = 3,075 m/s. Δv₂ = 3,075 − 1,607 = 1,468 m/s. The total Δv = 2,428 + 1,468 ≈ 3.9 km/s.
6. Transfer time: half the period of the ellipse, t = π√(a³/GM) = π√[(2.442 × 10⁷)³ ÷ 3.986 × 10¹⁴] = π × 6,043 s = 1.90 × 10⁴ s = 5.3 hours.

How was orbital mechanics developed?

Kepler published his laws in 1609 and 1619, and Newton derived them from the law of gravitation in 1687. Walter Hohmann published the transfer orbit in 1925. The first artificial satellite, Sputnik 1, was launched in 1957, and Luna 3 used a gravity assist in 1959. The Voyager probes (1977) used the alignment of the outer planets for gravity assists. The Apollo missions used a free-return trajectory around the Moon. Modern missions are designed with numerical integration of the equations of motion including the gravity of many bodies.

Which mistakes are common?

Mistake 1: Thinking that a satellite speeds up when it goes to a higher orbit

The satellite speeds up with the engines in the transfer, but in the final higher circular orbit the speed is lower.

Mistake 2: Forgetting the minus sign of the total energy

Bound orbits have negative total energy. E = −GMm/2r, not +.

Mistake 3: Using the altitude instead of the semi-major axis

The distances are measured from the center of the central body. For an ellipse, a is the average of the periapsis and apoapsis radii.

Mistake 4: Thinking that drag always slows a satellite

It lowers the orbit and, as a result, the speed increases. The total energy decreases.