orbit equation について

, 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 (i.e. circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory, 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).
== 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 polar coordinates, 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 (also called the ''true anomaly''). The parameter

\ell

is the angular momentum of the orbiting body about the central body, and is equal to

mr^2\dot

.〔There is a related parameter, known as the specific relative angular momentum,

h

. It is related to

\ell

h = \ell/m

.〕 The parameter

\gamma

is the constant for which

\gamma/r^2

equals the acceleration of the smaller body (for gravitation,

\gamma

is the standard gravitational parameter,

GM

). For a given orbit, the larger

\gamma

, 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 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.〔 The value of

e

controls what kind of conic section the orbit is. When

e<1

, the orbit is elliptic; when

e=1

, the orbit is parabolic; and when

e>1

, the orbit is hyperbolic.
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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
■ウィキペディアで「orbit equation」の詳細全文を読む

スポンサードリンク

翻訳と辞書 : 翻訳のためのインターネットリソース