“Mean motion”的意思、由来-开放百科全书

In orbital mechanics, mean motion (represented by n) is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body.^[1] The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit.

Mean motion is used as an approximation of the actual orbital speed in making an initial calculation of the body's position in its orbit, for instance, from a set of orbital elements. This mean position is refined by Kepler's equation to produce the true position.

Definition

Define the orbital period (the time period for the body to complete one orbit) as P, with dimension of time. The mean motion is simply one revolution divided by this time, or,

with dimensions of radians per unit time, degrees per unit time or revolutions per unit time.^[2]^[3]

The value of mean motion depends on the circumstances of the particular gravitating system. In systems with more mass, bodies will orbit faster, in accordance with Newton's law of universal gravitation. Likewise, bodies closer together will also orbit faster.

Mean motion and Kepler's laws

Kepler's 3rd law of planetary motion states, the square of the periodic time is proportional to the cube of the mean distance,^[4] or

where a is the semi-major axis or mean distance, P is the orbital period as above, and μ is a constant for any particular gravitational system.

where n is in revolutions per unit time. Combining with the above definition of Kepler's 3rd law,

which is another definition of Kepler's 3rd law.^[3]^[5] μ, the constant of proportionality, is a gravitational parameter,^[6]^[7] defined by the masses of the bodies in question and by the Newtonian gravitational constant, G; see below. Therefore, n is also defined^[8]

where M is typically the mass of the primary body of the system and m is the mass of a smaller body.

This is the complete gravitational definition of mean motion in a two-body system. Often in celestial mechanics, the primary body is much larger than any of the secondary bodies of the system, that is, {{nowrap|M ≫ m}}. It is under these circumstances that m becomes unimportant and Kepler's 3rd law is approximately constant for all of the smaller bodies.

Kepler's 2nd law of planetary motion states, a line joining a planet and the Sun sweeps out equal areas in equal times,^[6] or

for a two-body orbit, where {{sfrac|dA|dt}} is the time rate of change of the area swept.

Letting dt = P, the orbital period, the area swept is the entire area of the ellipse, dA = {{pi}}ab, where a is the semi-major axis and b is the semi-minor axis of the ellipse.^[9] Hence,

which is itself constant as a, b, and {{sfrac|dA|dt}} are all constant in two-body motion.

Mean motion and the constants of the motion

Because of the nature of two-body motion in a conservative gravitational field, two aspects of the motion do not change: the angular momentum and the mechanical energy.

The first constant, called specific angular momentum, can be defined as^[9]^[10]

The second constant, called specific mechanical energy, can be defined,^[11]^[12]

From above, the square of mean motion n² = {{sfrac|μ|a³}}. Substituting and rearranging, mean motion can also be expressed,

where the −2 shows that ξ must be defined as a negative number, as is customary in celestial mechanics and astrodynamics.

Mean motion and the gravitational constants

Two gravitational constants are commonly used in Solar system celestial mechanics: G, the Newtonian gravitational constant and k, the Gaussian gravitational constant. From the above definitions, mean motion is

By normalizing parts of this equation and making some assumptions, it can be simplified, revealing the relation between the mean motion and the constants.

Setting the mass of the Sun to unity, M = 1. The masses of the planets are all much smaller, {{nowrap|m ≪ M}}. Therefore, for any particular planet,

The Gaussian gravitational constant k = {{sqrt|G}},^[13]^[14]^[15] therefore, under the same conditions as above, for any particular planet

Mean motion and mean anomaly

Mean motion also represents the rate of change of mean anomaly, and hence can also be calculated,^[16]

where M₁ and M₀ are the mean anomalies at particular points in time, and t is the time elapsed between the two. M₀ is referred to as the mean anomaly at epoch, and t is the time since epoch.

Formulae

For Earth satellite orbital parameters, the mean motion is typically measured in revolutions per day. In that case,

To convert from radians per unit time to revolutions per day, consider the following:

See also

Notes

1. ^{{cite book | editor-last = Seidelmann | editor-first = P. Kenneth | editor2-last = Urban | editor2-first = Sean E. | title = Explanatory Supplement to the Astronomical Almanac | publisher = University Science Books, Mill Valley, CA | year = 2013 | isbn = 978-1-891389-85-6 | edition=3rd|page=648}}
2. ^{{cite book | last = Roy | first = A.E. | title = Orbital Motion | publisher = Institute of Physics Publishing | year = 1988 | ISBN=0-85274-229-0 | edition = third | page =83}}
3. ^¹{{cite book | last1 = Brouwer | first1 = Dirk | last2 = Clemence | first2 = Gerald M. | title = Methods of Celestial Mechanics | publisher = Academic Press | year = 1961 | pages = 20–21}}
4. ^{{cite book | last = Vallado | first = David A. | title = Fundamentals of Astrodynamics and Applications | publisher = Microcosm Press|location= El Segundo, CA | year = 2001 | isbn = 1-881883-12-4 | edition = second | page = 29}}
5. ^{{cite book | last = Battin | first = Richard H. | title = An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition | publisher = American Institute of Aeronautics and Astronautics, Inc. | isbn = 1-56347-342-9 | date = 1999 | page = 119}}
6. ^¹Vallado, David A. (2001). p. 31.
7. ^Do not confuse μ, the gravitational parameter with μ, the reduced mass.
8. ^Vallado, David A. (2001). p. 53.
9. ^¹Vallado, David A. (2001). p. 30.
10. ^{{cite book | last1 = Bate | first1 = Roger R. | last2 = Mueller | first2 = Donald D. | last3 = White | first3 = Jerry E. | title = Fundamentals of Astrodynamics | publisher = Dover Publications, Inc., New York | isbn = 0-486-60061-0 | date = 1971 | page = 32}}
11. ^Vallado, David A. (2001). p. 27.
12. ^Bate, Roger R.; Mueller, Donald D.; White, Jerry E. (1971). p. 28.
13. ^{{cite book | last1 = U.S. Naval Observatory | first1=Nautical Almanac Office | last2 = H.M. Nautical Almanac Office | title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac | publisher = H.M. Stationery Office, London | year = 1961 | page = 493}}
14. ^{{cite book | last = Smart | first = W. M. | title = Celestial Mechanics | publisher = Longmans, Green and Co., London | year = 1953 | page=4}}
15. ^The Gaussian gravitational constant, k, usually has units of radians per day and the Newtonian gravitational constant, G, is usually given in the SI system. Be careful when converting.
16. ^Vallado, David A. (2001). p. 54.