Orbit

What an orbit is

An orbit is the path a body follows around another under the pull of gravity. Newton pictured it with a cannon on a very high mountain: fire the ball faster and it lands farther away, until at about 7.9 km/s it falls towards the ground exactly as fast as the curved ground falls away beneath it. The ball never lands. It is still falling, but it keeps missing the Earth. That is an orbit.

When only two bodies are involved, the path is always a conic section. Bound orbits are ellipses, the circle being the special case with zero eccentricity. A body moving fast enough to escape follows a parabola or a hyperbola and never returns: the interstellar visitor 1I/ʻOumuamua crossed the solar system in 2017 on a hyperbolic trajectory.

Strictly speaking the two bodies orbit their common centre of mass, the barycentre. For the Earth and the Moon it lies about 4,700 km from the Earth's centre, inside the Earth. For the Sun and Jupiter it lies slightly outside the Sun's surface, so the Sun itself traces a small loop as Jupiter goes round.

Kepler's three laws

Johannes Kepler derived three empirical laws from Tycho Brahe's observations of Mars, long before anyone could explain them:

P = 2π × √(a³ / GM)P[years]² = a[AU]³      (bodies orbiting the Sun)

• First law (1609): each planet moves on an ellipse with the Sun at one focus.
• Second law (1609): the line joining the Sun and the planet sweeps out equal areas in equal times, so a planet moves fastest at perihelion and slowest at aphelion.
• Third law (1619): the square of the orbital period is proportional to the cube of the semi-major axis. With the period in years and the distance in astronomical units, P² = a³ for any body orbiting the Sun.

In 1687 Newton showed that all three laws follow from a single inverse-square law of gravitation, and gave the general form of the third law, which works around any central mass M:

The six orbital elements

Six numbers are enough to describe a Keplerian orbit completely and to locate the body on it at any moment. Ephemerides and comet circulars publish orbits in this form:

Placing a planet on its orbit: Kepler's equation

Kepler's second law says that a planet does not move uniformly along its ellipse, so its position cannot be read off directly from the elapsed time. Astronomers use three angles measured from perihelion. The mean anomaly M grows uniformly with time, by 360° per orbit. The eccentric anomaly E is an auxiliary angle. The true anomaly v is the actual angle between the planet and perihelion, seen from the Sun. They are linked by Kepler's equation:

M = E − e × sin E          (E and M in radians)tan(v / 2) = √((1 + e) / (1 − e)) × tan(E / 2)r = a × (1 − e × cos E)

Kepler's equation has no closed-form solution for E, but it is easy to iterate. Take E = M as a first guess and repeat E = M + e × sin E until the value stops changing. For planets the loop converges in a handful of steps. For comets with an eccentricity close to 1, Newton's method or special formulae are needed, and Meeus devotes a full chapter to the different methods.

Worked example: Mars (a = 1.5237 AU, e = 0.0934) with a mean anomaly of 60°. The successive values of E are 64.634°, 64.836°, 64.844° and then 64.8439°, which no longer changes. This gives a true anomaly of v = 69.79° and a distance of r = 1.4632 AU. Mars is almost 10° ahead of where uniform motion would put it, because it is still close to perihelion and moving fast.

Real orbits are not perfect ellipses

A pure ellipse only exists when two bodies are alone. In the real solar system every planet tugs on every other, so the elements slowly change. There are secular drifts (the perihelion of the Earth advances about 0.3° per century) and periodic wobbles (Jupiter and Saturn perturb each other with a period of about 900 years). The elements given for a particular date are called osculating elements: they describe the ellipse the body would follow if all perturbations suddenly vanished.

Perturbations can be revealing. In 1846 Urbain Le Verrier used unexplained deviations in the motion of Uranus to predict the position of an unknown planet, and Johann Galle found Neptune within about one degree of that position on the first night he looked.

For accurate positions a single ellipse is not enough. AstroAlgos uses the VSOP87 planetary theory, which represents each planet's motion as long sums of periodic terms. Meeus gives an abridged version accurate to a few arcseconds for dates within a few thousand years of the present.

Orbital periods and speeds of the planets

Explore the orbits in 3D

The interactive view above draws every planetary orbit in the heliocentric ecliptic frame, computed from the same algorithms. Rotate it to see how the orbits are tilted and offset from the Sun. Open the full heliocentric page to animate the planets and watch Kepler's second law at work: Mercury visibly speeds up as it swings through perihelion.

Explore it on AstroAlgos

• Heliocentric view — animate the planets on their orbits
• Perihelion — the closest point to the Sun
• Aphelion — the farthest point from the Sun
• The ecliptic — the reference plane for orbits