Orbit

 ( Celestial Mechanics ) 

In celestial mechanics, an orbit is the curved trajectory of an physical body under the influence of an attracting force. Known as an orbital revolution, examples include the trajectory of a planet around a star, a natural satellite around a planet, or an satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow , with the barycenter being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

revolve around a star, a natural satellite around a planet, or an satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. Furthermore orbits are dynamic, perturbated by all masses, consisting of different trajectory, but most can be approximated as , with the barycenter being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law.

The basis for the modern description 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 the Solar System are elliptical, not circle (or epicycle), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed depends on the planet's distance from the Sun. 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:

$\backslash tfrac\{5.204^3\}\{11.862^2\}\; \backslash approxeq\; 1.002$

$\backslash tfrac\{0.723^3\}\{0.615^2\}\; \backslash approxeq\; 0.999$

Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation,

$T^2\; \backslash propto\; \backslash frac\{a^3\}\{M\}$

Advances in Newtonian mechanics were then used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. Joseph-Louis Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force,

Albert Einstein in his 1916 paper The Foundation of the General Theory of Relativity explained that gravity was due to curvature of space-time and removed Newton's assumption that changes in gravity propagate instantaneously. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory, orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions (except where there are very strong gravity fields and very high speeds) but the differences are measurable. Essentially all the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The original vindication of general relativity is that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion first noted by Le Verrier. However, Newton's solution is still used for most short term purposes since it is significantly easier to use and sufficiently accurate.

Owing to mutual gravitational perturbations, the eccentricities and inclinations of the planetary orbits 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 orbital eccentricities are seen with Venus and Neptune.

As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other. Less properly, "perifocus" or "pericentron" are used. The apoapsis is that point at which they are the farthest, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part.

More specific terms are used for specific bodies. For example, perigee and apogee are the lowest and highest parts of an orbit around Earth, while perihelion and aphelion are the closest and farthest points of an orbit around the Sun. Things orbiting the Moon have a perilune and apolune (or periselene and aposelene respectively).

In the case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at a single point called the barycenter. The individual satellites of that star follow their own elliptical orbits with the barycenter at one focal point of that ellipse. At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and the sum of those two energies is a constant value at every point along its orbit. As a result, as a planet approaches periapsis, the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis, its velocity will decrease as its potential energy increases.

• A body continues in a state of uniform rest or motion unless acted upon by an external force.
• The acceleration produced when a force acts is directly proportional to the force and takes place in the direction in which the force acts.
• To every action there is an equal and opposite reaction.

By the first law of motion, in the absence of gravity, a physical object will continue to move in a straight line due to inertia. According to the second law, a force, such as gravity, pulls the moving object toward the body that is the source of the force and thus causes the object to follow a curved trajectory. If the object has enough tangential velocity, it will not fall into the gravitating body but can instead continue to follow the curved trajectory caused by the force indefinitely. The object is then said to be orbiting the body. According to the third law, each body applies an equal force on the other, which means the two bodies orbit around their center of mass, or barycenter.

Because of the law of universal gravitation, the strength of the gravitational force depends on the masses of the two bodies and their separation. As the gravity varies over the course of the orbit, it reproduces Kepler's laws of planetary motion. Depending on the evolving energy state of the system, the velocity relationship of two moving objects with mass can be considered in four practical classes, with subtypes:

No orbit

Suborbital trajectories

a range of interrupted elliptical paths

Orbital trajectories (or simply, orbits)

escape orbit

To achieve orbit, conventional rockets are launched vertically at first to lift the rocket above the dense lower atmosphere (which causes frictional drag), and gradually pitch over and finish firing the rocket engine parallel to the atmosphere to achieve orbital injection.

If the cannon fires its ball with a low initial speed, the trajectory of the ball curves downward and hits the ground (A). As the firing speed 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 speed, 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 speed (unaffected by the mass of the ball, which is assumed to be very small relative to the Earth's mass) that produces a circular orbit, as shown in (C).

As the firing speed is increased beyond this, non-interrupted 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 non-interrupted elliptical orbits at slower firing speed; these will come closest to the Earth at the point half an orbit beyond, and directly opposite the firing point, below the circular orbit.

At a specific horizontal firing speed called escape velocity, dependent on the mass of the planet and the distance of the object from the barycenter, an open orbit (E) is achieved that has a parabolic path. At even greater speeds 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" potentially never to return. However, the object remains under the influence of the Sun's gravity.

The acceleration of a body is equal to the combination of the forces acting on it, divided by its mass. The gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them. For a two-body problem, defined as an isolated system of two spherical bodies with known masses and sufficient separation, this Newtonian approximation of their gravitational interaction can provide a reasonably accurate calculation of their trajectories.

If the heavier body is much more massive than the smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of a coordinate system that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier. For the case where the masses of two bodies are comparable, an exact Newtonian solution is still sufficient and can be had by placing the coordinate system at the center of the mass of the system.

When only two gravitational bodies interact, their orbits follow a conic section. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total energy (kinetic energy + potential energy) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, the speed is always less than the escape velocity. Since the kinetic energy is never negative if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy.

An open orbit will have a parabolic shape if it has the velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a hyperbola when its velocity is greater than the escape velocity. When two bodies approach each other with escape velocity or greater (relative to each other), they will briefly curve around each other at the time of their closest approach, and then separate and fly apart.

All closed orbits have the shape of an ellipse. A circular orbit is a special case, wherein the foci of the ellipse coincide.

1. The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of that ellipse. This The planet's orbit lies in a plane, called the orbital plane.
2. As the planet moves in its orbit, the line from the Sun to the planet sweeps a constant area of the orbital plane for a given period of time, 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 a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.

The two-body solutions were published by Newton in Principia in 1687. In 1912, Karl Fritiof Sundman developed a converging infinite series that solves the general three-body problem; however, it converges too slowly to be of much use. The restricted three-body problem, in which the third body is assumed to have negligible mass, has been extensively studied. The solutions to this case include the .

$\backslash begin\{align\}$

 \hat{\mathbf{r}}
   &= \cos(\theta)\hat{\mathbf{x}} + \sin(\theta)\hat{\mathbf{y}} \\
 \frac{\delta \hat{\mathbf{r}}}{\delta t} = \dot{\mathbf r}
   &= -\sin(\theta)\dot \theta \hat{\mathbf{x}} + \cos(\theta)\dot \theta \hat{\mathbf{y}} = \dot \theta \hat{\boldsymbol \theta} \\
 \hat{\boldsymbol \theta}
   &= -\sin(\theta)\hat{\mathbf{x}} + \cos(\theta)\hat{\mathbf{y}} \\
 \frac{\delta \hat{\boldsymbol \theta}}{\delta t} = \dot{\boldsymbol \theta}
   &= -\cos(\theta)\dot \theta \hat{\mathbf{x}} - \sin(\theta) \dot \theta \hat{\mathbf{y}} = -\dot \theta \hat{\mathbf r}
     

The velocity and acceleration of the orbiting object can now be determined.

$\backslash begin\{align\}$

  {\mathbf{O}} &= r \hat{\mathbf{r}} \\
  \dot{\mathbf{O}} &= \frac{\delta r}{\delta t} \hat{\mathbf{r}} + r \frac{\delta \hat{\mathbf{r}}}{\delta t}
                    = \dot r \hat {\mathbf r} + r \left[ \dot \theta \hat {\boldsymbol \theta} \right] \\
 \ddot{\mathbf{O}} &= \left[\ddot r \hat {\mathbf r} + \dot r \dot \theta \hat {\boldsymbol \theta}\right] +
                      \left[\dot r \dot \theta \hat {\boldsymbol \theta} +
                            r \ddot \theta \hat {\boldsymbol \theta} -
                            r \dot \theta^2 \hat {\mathbf r}
                      \right] \\
                   &= \left[\ddot r - r\dot\theta^2\right]\hat{\mathbf{r}} +
                      \left[r \ddot\theta + 2 \dot r \dot\theta\right] \hat{\boldsymbol \theta}
     

In the last line, the coefficients of $\backslash hat\{\backslash mathbf\{r\}\}$ and $\backslash hat\{\backslash boldsymbol\; \backslash theta\}$ give the accelerations in the radial and transverse directions. As said, Newton gives the first due to gravity as $-\backslash mu/r^2$ and the second is zero per Newton's first law. Thus:

Equation (2) can be rearranged using integration by parts.

$r\; \backslash ddot\backslash theta\; +\; 2\; \backslash dot\; r\; \backslash dot\backslash theta\; =\; \backslash frac\{1\}\{r\}\backslash frac\{d\}\{dt\}\backslash left(\; r^2\; \backslash dot\; \backslash theta\; \backslash right)\; =\; 0$

Both sides can now be multiplied through by $r$ because it is not zero unless the orbiting object crashes. Having the derivative be zero indicates that the function is a constant.

which is actually the theoretical proof of Kepler's second law (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time). The constant of integration, h, is the angular momentum per unit mass.

     \frac{\delta u}{\delta \theta} &= \frac{\delta}{\delta t}\left(\frac{1}{r}\right)\frac{\delta t}{\delta \theta } = -\frac{\dot{r}}{r^2\dot{\theta }} = -\frac{\dot{r}}{h} \\
 \frac{\delta^2 u}{\delta \theta^2} &= -\frac{1}{h}\frac{\delta \dot{r}}{\delta t}\frac{\delta t}{\delta \theta } = -\frac{\ddot{r}}{h\dot{\theta}} = -\frac{\ddot{r}}{h^2 u^2}
 \ \ \ \text{ or } \ \ \  \ddot r = - h^2 u^2 \frac{\delta^2 u}{\delta \theta^2}
     

                                                         \ddot r - r\dot\theta^2 &= -\frac{\mu}{r^2} \\
  -h^2 u^2 \frac{\delta^2 u}{\delta \theta^2} - \frac{1}{u} \left(h u^2\right)^2 &= -\mu u^2
     

                  \frac{\delta r}{\delta \theta} &= -\frac{1}{u^2} \frac{\delta u}{\delta \theta} = -\frac{h}{m} \frac{\delta u}{\delta \theta} \\
              \frac{\delta^2 r}{\delta \theta^2} &= -\frac{h^2u^2}{m^2} \frac{\delta^2 u}{\delta \theta^2} - \frac{hu^2}{m^2} \frac{\delta h}{\delta \theta} \frac{\delta u}{\delta \theta} \\
 \left(\frac{\delta \theta}{\delta t}\right)^2 r &= \frac{h^2 u^3}{m^2}
     

$\backslash bar\{r\}\; =\; a\; \backslash left(1\; +\; \backslash frac\{e^2\}\{2\}\; \backslash right)$

which only equals a when e is zero, for a circular orbit.

The region for experiencing atmospheric drag varies by planet; a re-entry vehicle needs to draw much closer to Mars than to Earth,

Orbits can be artificially influenced through the use of rocket engines, which change the kinetic energy of the body at some point in its path. In this way, changes in the orbit shape or orientation can be facilitated. or are forms of propulsion that require no propellant or energy input other than that of the Sun, and so can be used indefinitely for station keeping. (See statite for one such proposed use.) Satellites with long conductive tethers can experience orbital decay because of electromagnetic drag from the Earth's magnetic field. As the wire cuts the magnetic field it acts as a generator, moving electrons from one end to the other. The orbital energy is converted to heat in the wire.

For objects below the synchronous orbit for the body they're orbiting, orbital decay can occur due to . The gravity of the orbiting object raises in the primary, and since it is below the synchronous orbit, the orbiting object is moving faster than the body's surface so 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 direction of 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.

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 that are orbiting each other closely.

From the point of view of satellite dynamics, of particular relevance are the so-called even zonal harmonic coefficients, or even zonals, since they induce secular orbital perturbations which are cumulative over time spans longer than the orbital period. They do depend on the orientation of the body's symmetry axis in the space, affecting, in general, the whole orbit, with the exception of the semimajor axis.

In the case where a tidally locked body possesses synchronous rotation, the object takes just as long to rotate around its own axis as it does to revolve around its partner. In this case, one side of the celestial body is permanently facing its host object. This is the case for the Earth's Moon and for both members of the Pluto-Charon system.

A long-term impact of multi-body interactions can be apsidal precession, which is a gradual rotation of the Apse line. For an elliptical system, the result is a rosetta orbit. The ancient Greek astronomer Hipparchus noted just such an apsidal precession of the Moon's orbit, as the revolution of the Moon's apogee with a period of approximately 8.85 years. Apsidal precession can result from tidal perturbation, rotational perturbation, general relativity, or a combination of these effects. The detection of apsidal precession in a distant binary star system can be an indicator of the purturbative effect of an unseen third stellar companion.

When there are more than two gravitating bodies it is referred to as an n-body problem. Most n-body problems have no closed form solution, although some special cases have been formulated.

An incremental approach uses differential equations for scientific or mission-planning purposes.For example: According to Newton's laws, each of the gravitational forces acting on a body will depend on the separation from the sources. 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 values of position and velocity corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad nauseam. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the long-term 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 assemblages of objects have been simulated.