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astrodynamics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the Newton's law of univ ...
, an
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
equation defines the path of orbiting body m_2\,\! around central body m_1\,\! relative to m_1\,\!, without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance (such as gravity), has an orbit that is a
conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
(i.e.
circular orbit A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. In this case, not only the distance, but also the speed, angular speed, Potential energy, potential and kinetic energy are constant. T ...
,
elliptic orbit In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. Some orbits have been referre ...
,
parabolic trajectory In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the Orbital eccentricity, eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away f ...
,
hyperbolic trajectory In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the ...
, or radial trajectory) with the central body located at one of the two foci, or ''the'' focus ( Kepler's first law). If the conic section intersects the central body, then the actual trajectory can only be the part above the surface, but for that part the orbit equation and many related formulas still apply, as long as it is a freefall (situation of
weightlessness Weightlessness is the complete or near-complete absence of the sensation of weight, i.e., zero apparent weight. It is also termed zero g-force, or zero-g (named after the g-force) or, incorrectly, zero gravity. Weight is a measurement of the fo ...
).


Central, inverse-square law force

Consider a two-body system consisting of a central body of mass ''M'' and a much smaller, orbiting body of mass m, and suppose the two bodies interact via a central, inverse-square law force (such as
gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
). In
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
, the orbit equation can be written as r = \frac\frac where * r is the separation distance between the two bodies and * \theta is the angle that \mathbf makes with the axis of
periapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...
(also called the ''
true anomaly In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus ...
''). * The parameter \ell is the
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
of the orbiting body about the central body, and is equal to mr^2\dot, or the mass multiplied by the magnitude of the cross product of the relative position and velocity vectors of the two bodies.There is a related parameter, known as the
specific relative angular momentum In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative positi ...
, h. It is related to \ell by h = \ell/m.
* The parameter \mu is the constant for which \mu/r^2 equals the acceleration of the smaller body (for gravitation, \mu is the
standard gravitational parameter The standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of that body. For two bodies, the parameter may be expressed as , or as when one body is much larger than the ...
, -GM). For a given orbit, the larger \mu, the faster the orbiting body moves in it: twice as fast if the attraction is four times as strong. * The parameter e is the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
of the orbit, and is given by *:e = \sqrt *:where E is the energy of the orbit. The above relation between r and \theta describes a
conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
. The value of e controls what kind of conic section the orbit is: * when e<1, the orbit is elliptic (circles are ellipses with e=0); * when e=1, the orbit is parabolic; * when e>1, the orbit is
hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
. The minimum value of r in the equation is: r= while, if e<1, the maximum value is: r= If the maximum is less than the radius of the central body, then the conic section is an ellipse which is fully inside the central body and no part of it is a possible trajectory. If the maximum is more, but the minimum is less than the radius, part of the trajectory is possible: *if the energy is non-negative (parabolic or hyperbolic orbit): the motion is either away from the central body, or towards it. *if the energy is negative: the motion can be first away from the central body, up to r= after which the object falls back. If r becomes such that the orbiting body enters an atmosphere, then the standard assumptions no longer apply, as in
atmospheric reentry Atmospheric entry (sometimes listed as Vimpact or Ventry) is the movement of an object from outer space into and through the gases of an atmosphere of a planet, dwarf planet, or natural satellite. Atmospheric entry may be ''uncontrolled entry ...
.


Low-energy trajectories

If the central body is the Earth, and the energy is only slightly larger than the potential energy at the surface of the Earth, then the orbit is elliptic with eccentricity close to 1 and one end of the ellipse just beyond the center of the Earth, and the other end just above the surface. Only a small part of the ellipse is applicable. If the horizontal speed is v\,\!, then the periapsis distance is \frac. The energy at the surface of the Earth corresponds to that of an elliptic orbit with a=R/2\,\! (with R\,\! the radius of the Earth), which can not actually exist because it is an ellipse fully below the surface. The energy increase with increase of a is at a rate 2g\,\!. The maximum height above the surface of the orbit is the length of the ellipse, minus R\,\!, minus the part "below" the center of the Earth, hence twice the increase of a\,\! minus the periapsis distance. At the top the potential energy is g times this height, and the kinetic energy is \frac. This adds up to the energy increase just mentioned. The width of the ellipse is 19 minutes times v\,\!. The part of the ellipse above the surface can be approximated by a part of a parabola, which is obtained in a model where gravity is assumed constant. This should be distinguished from the parabolic orbit in the sense of astrodynamics, where the velocity is the
escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming: * Ballistic trajectory – no other forces are acting on the object, such as ...
. See also
trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tra ...
.


Categorization of orbits

Consider orbits which are at one point horizontal, near the surface of the Earth. For increasing speeds at this point the orbits are subsequently: *part of an ellipse with vertical major axis, with the center of the Earth as the far focus (throwing a stone,
sub-orbital spaceflight A sub-orbital spaceflight is a spaceflight in which the spacecraft reaches outer space, but its trajectory intersects the surface of the primary (astronomy), gravitating body from which it was launched. Hence, it will not complete one orbital ...
,
ballistic missile A ballistic missile is a type of missile that uses projectile motion to deliver warheads on a target. These weapons are powered only during relatively brief periods—most of the flight is unpowered. Short-range ballistic missiles (SRBM) typic ...
) *a circle just above the surface of the Earth (
Low Earth orbit A low Earth orbit (LEO) is an geocentric orbit, orbit around Earth with a orbital period, period of 128 minutes or less (making at least 11.25 orbits per day) and an orbital eccentricity, eccentricity less than 0.25. Most of the artificial object ...
) *an ellipse with vertical major axis, with the center of the Earth as the near focus *a parabola *a hyperbola Note that in the sequence above, h, \epsilon and a increase monotonically, but e first decreases from 1 to 0, then increases from 0 to infinity. The reversal is when the center of the Earth changes from being the far focus to being the near focus (the other focus starts near the surface and passes the center of the Earth). We have :e=\left , \frac-1\right , Extending this to orbits which are horizontal at another height, and orbits of which the extrapolation is horizontal below the surface of the Earth, we get a categorization of all orbits, except the radial trajectories, for which, by the way, the orbit equation can not be used. In this categorization ellipses are considered twice, so for ellipses with both sides above the surface one can restrict oneself to taking the side which is lower as the reference side, while for ellipses of which only one side is above the surface, taking that side.


See also

* Kepler's first law *
Circular orbit A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. In this case, not only the distance, but also the speed, angular speed, Potential energy, potential and kinetic energy are constant. T ...
*
Elliptic orbit In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. Some orbits have been referre ...
*
Parabolic trajectory In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the Orbital eccentricity, eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away f ...
*
Hyperbolic trajectory In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the ...
*
Tsiolkovsky rocket equation The classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part o ...
*
Orbital speed In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter (the combined center of mass) or ...
*
Escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming: * Ballistic trajectory – no other forces are acting on the object, such as ...
*
Celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...


Notes


References

{{orbits Orbits