History

In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after measurements of the exact motion of the planets theoretical mechanisms such as the deferent and epicycles were later added. Although it was capable of accurately predicting the planets position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy.

The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed, and that the sun is not located at the center of the orbits, but rather at one focus.^[5] Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed of the planet depends on the planet's distance from the sun. And third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the sun. For the planets, the cubes of their distances from the sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.2³/11.86², is practically equal to that for Venus, 0.723³/0.615², in accord with the relationship.

While the planetary bodies do have elliptical orbits about the Sun, the eccentricity of the orbits is often not large. A circle has an eccentricity of zero, Earth's orbit's eccentricity is 0.0167 meaning that the ratio of its semi-minor (b) to semi-major axis (a) is 99.99%. Mercury has the largest eccentricity of the planets with an eccentricity of 0.2056, b/a=97.86%. (Eris has an eccentricity of 0.441 and Pluto 0.249. For the values for all planets in one table, see Table of planets in the solar system.)

The lines traced out by orbits dominated by the gravity of a central source are conic sections: the shapes of the curves of intersection between a plane and a cone. Parabolic (1) and hyperbolic (3) orbits are escape orbits, whereas elliptical and circular orbits (2) are captive.

Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections, if the force of gravity propagated instantaneously. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, and that the bodies revolve about their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Albert Einstein was able to show that gravity was due to curvature of space-time and was able to remove the assumption of Newton that changes propagate instantaneously. In relativity theory orbits follow geodesic trajectories which approximate very well to the Newtonian predictions. However there are differences that can be used to determine which theory describes reality more accurately. Essentially all experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measuremental accuracy, but the differences from Newtonian mechanics are usually very small (except where there are very strong gravity fields and very high speeds).

However, Newtonian mechanics is still used for most purposes since Newtonian mechanics is significantly easier to use.

Planetary orbits

Within a planetary system; planets, dwarf planets, asteroids (a.k.a. minor planets), comets, and space debris orbit the central star in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. To date, no comet has been observed in our solar system with a distinctly hyperbolic orbit. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet.

Owing to mutual gravitational perturbations, the eccentricities of the orbits of the planets in our solar system vary over time. Mercury, the smallest planet in the Solar System, has the most eccentric orbit. At the present epoch, Mars has the next largest eccentricity while the smallest eccentricities are those of the orbits of Venus and Neptune.

As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest from each other. (More specific terms are used for specific bodies. For example, perigee and apogee are the lowest and highest parts of an Earth orbit, respectively.)

In the elliptical orbit, the center of mass of the orbiting-orbited system will sit at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in speed, or velocity. As a planet approaches apoapsis, the planet will decrease in velocity.

See also:

• Kepler's laws of planetary motion
• Secular variations of the planetary orbits

Understanding orbits

There are a few common ways of understanding orbits.

• As the object moves sideways, it falls toward the central body. However, it moves so quickly that the central body will curve away beneath it.
• A force, such as gravity, pulls the object into a curved path as it attempts to fly off in a straight line.
• As the object moves sideways (tangentially), it falls toward the central body. However, it has enough tangential velocity to miss the orbited object, and will continue falling indefinitely. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of the three one-dimensional coordinates oscillating around a gravitational center.

As an illustration of an orbit around a planet, the Newton's cannonball model may prove useful (see image below). This is a 'thought experiment', in which a cannon on top of a tall mountain is supposed to be able to fire a cannonball horizontally at any chosen muzzle velocity. The effects of air friction on the cannonball are ignored (or perhaps the mountain is high enough that the cannon will be above the Earth's atmosphere, which comes to the same thing.)^[6]

Newton's cannonball, an illustration of how objects can "fall" in a curve.

If the cannon fires its ball with a low initial velocity, the trajectory of the ball curves downward and hits the ground (A). As the firing velocity is increased, the cannonball hits the ground farther (B) away from the cannon, because while the ball is still falling towards the ground, the ground is increasingly curving away from it (see first point, above). All these motions are actually "orbits" in a technical sense — they are describing a portion of an elliptical path around the center of gravity — but the orbits are interrupted by striking the Earth.

If the cannonball is fired with sufficient velocity, the ground curves away from the ball at least as much as the ball falls — so the ball never strikes the ground. It is now in what could be called a non-interrupted, or circumnavigating, orbit. For any specific combination of height above the center of gravity and mass of the planet, there is one specific firing velocity (practically unaffected by the mass of the ball where that is as usual very small relative to the Earth's mass) that produces a circular orbit, as shown in (C).

As the firing velocity is increased beyond this, a range of elliptic orbits are produced; one is shown in (D). If the initial firing is above the surface of the Earth as shown, there will also be elliptical orbits at slower velocities; these will come closest to the Earth at the point half an orbit beyond, and directly opposite, the firing point.

At a specific velocity called escape velocity, again dependent on the firing height and mass of the planet, an open orbit such as (E) results — a parabolic trajectory. At even faster velocities the object will follow a range of hyperbolic trajectories. In a practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space".

The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes:

1. No orbit
2. Suborbital trajectories
   • Range of interrupted elliptical paths
3. Orbital trajectories (or simply "orbits")
   • Range of elliptical paths with closest point opposite firing point
   • Circular path
   • Range of elliptical paths with closest point at firing point
4. Open (or escape) trajectories
   • Parabolic paths
   • Hyperbolic paths