行星轨道

1. The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any star, not just the Sun, has a periastron and an apastron
2. As the planet moves around its orbit during a fixed amount of time, the line from Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."
3. For each planet, the ratio of the 3rd power of its semi-major axis to the 2nd power of its period is the same constant value for all planets.

Except for special cases like Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies. The 2-body solutions were published by Newton in Principia in 1687. In 1912, K. F. Sundman developed a converging infinite series that solves the 3-body problem, however it converges too slowly to be of much use.

Instead, orbits can be approximated with arbitrarily high accuracy. These approximations take two forms.

One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moon, planets and other bodies are known with great accuracy, and are used to generate tables for 天文导航.

微分方程 form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial ones corresponds to solving an 初值问题. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting the accuracy of this approach.

Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

To analyse the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use 极坐标 with the origin coinciding with the centre of force. In such coordinates the radial and transverse components of the 加速度 are, respectively:

and

.

因为力是完全径向的,加速度与力成正比, 因此横行加速度为0。可以得到,

.

积分之后我们得到

对于任意常数h积分有

我们引入辅助变量

.

If magnitude of the radial force is f(r) per unit mass of the orbiting body, then the elimination of the time variable from the radial component of the equation of motion yields:

.

In the case of gravity, Newton's law of universal gravitation states that the force is proportional to the inverse square of the distance:

其中G使引力常数, m是轨道天体（行星）质量，M是中心天体（太阳）质量。带入前面的等式我们得到：

.

So for the gravitational force – or, more generally, for any inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the 调和方程 (up to a shift of origin of the dependent variable).

The equation of the orbit described by the particle is thus:

,

where φ and e are arbitrary constants of integration,

is the semi-latus rectum, and a is the 半长轴。This can be recognised as the equation of a conic section in 极坐标(r,θ).

[编辑] 轨道参数

参看: 轨道元素

For a general elliptic orbit, the relations between the axis, eccentricity, and least and largest distance are:

半长轴 = (近拱点 + 远拱点)/2 = 极半径

近拱点 = 半长轴 × (1 - 偏心率) = 最小距离

远拱点 = 半长轴 × (1 + 偏心率) = 最大距离

Note that there are alternative definitions for a "mean radius" or "average distance": if you average the radius over time for one orbit (平均近点角), or over the orbital angle as observed by the primary (真近点角), 就会得到不同结果。更多信息参看半长轴#天文。

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. At each periapsis, the object scrapes the air, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. Eventually, the orbit circularises and then the object spirals into the atmosphere.

The bounds of an atmosphere vary wildly. During solar maxima, the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minimums.

Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire.

Another method of artificially influencing an orbit is through the use of 太阳帆 or magnetic sails. These forms of propulsion require no propellant or energy input, and so can be used indefinitely. See 斯达 for one such proposed use.

Orbital decay can also occur due to 潮汐力 for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the solar system are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years.

Finally, orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

Thus the constant has dimension density^-1 time^-2. This corresponds to the following properties.

Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the earth.

When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled.

When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities.

These properties are illustrated in the formula

椭圆轨道的半长轴 a, of a small body around a spherical body with 半径 r and average density σ, where T 是轨道周期.

When atomic structure was first probed experimentally early in the twentieth century, an early picture of the atom portrayed it as a miniature solar system bound by the coulomb force rather than by gravity. This was inconsistent with electrodynamics and the model was progressively refined as quantum theory evolved, but there is a legacy of the picture in the term orbital for the wave function of an energetically bound electron state.