Orbital elements
98663
219692192
2008-06-16T13:02:56Z
Juliancolton
2793164
Reverted edits by [[Special:Contributions/117.200.163.4|117.200.163.4]] to last version by Portalian (using [[WP:HG|Huggle]])
The '''elements of an orbit''' are the parameters needed to specify that [[orbit]] uniquely, given a model of two point masses obeying the [[Newtonian laws of motion]] and the [[inverse-square law]] of [[gravitational attraction]]. Because there are multiple ways of parameterising a motion, depending on which set of variables you choose to measure, there are several different ways of defining sets of orbital elements, each of which will specify the same orbit.
This problem contains three degrees of freedom (the three [[Cartesian coordinates]] of the orbiting body). Therefore, any given Keplerian (unperturbed) orbit is fully defined by six quantities - the initial values of the Cartesian components of the body's position and velocity - and an '''epoch''', a time at which the elements are valid. For this reason, all sets of orbital elements contain exactly six parameters. For a mathematically accurate explanation of this fact see the Discussion and references therein. (''See also'': [[orbital state vectors]]).
==Keplerian elements==
The traditional orbital element set are the six '''Keplerian elements''', after [[Johannes Kepler]] and his [[Kepler's laws]]:
*[[Inclination]] (<math>i\,\!</math>)
*[[Longitude of the ascending node]] <!--(☊-->(<math>\Omega\,\!</math>)
*[[Argument of periapsis]] (<math>\omega\,\!</math>)
*[[Eccentricity (orbit)|Eccentricity]] (<math>e\,\!</math>)
*[[Semimajor axis]] (<math>a\,\!</math>)
*[[Mean anomaly]] at [[Epoch (astronomy)|epoch]] (<math>M_o\,\!</math>)
Keplerian elements can be obtained from [[orbital state vectors]] using [http://www.mindspring.com/~n2wwd/html/body_state_vectors.html VEC2TLE software] or by some [[Orbital state vectors#Relation to orbital elements|direct computations]]. We see that the first three orbital elements are simply the [[Euler angles|Eulerian angles]] defining the orientation of the orbit relative to some defined inertial coordinate system. The next two establish the size and shape of the orbit, and the last establishes the location of the body within its orbit at the given time (epoch). Unperturbed, [[Two-body problem|two-body]] orbits are always conic sections, so the Keplerian elements define an [[Conic section#Applications|ellipse, a parabola, or a hyperbola]]. Real orbits have perturbations, so a given set of Keplerian elements is valid only at the epoch though the predictions are often adequate at times near the epoch. A real trajectory can be modeled as a sequence of [[Osculating orbit|osculating]] Keplerian element sets defining orbits that osculate ("kiss" or touch) the real trajectory at their respective epoch times.
The last element is "Mean anomaly at Epoch". The mean anomaly steadily increases by 360 degrees per orbit, so we must specify the time (epoch) at which it is measured. As mentioned above, real orbits are generally perturbed by small forces that can cause some or all of the Keplerian elements to change slowly with time, so the other elements are also strictly valid only at the epoch time.
===Alternative expressions===
Instead of the [[mean anomaly]] at [[Epoch (astronomy)|epoch]], <math>M_o\,\!</math>, the [[mean anomaly]] <math>M\,\!</math>, [[mean longitude]], [[true anomaly]] or, rarely, the [[eccentric anomaly]] may also be used. (Sometimes the epoch itself is considered an orbital element.) Other orbital parameters can be computed from the Keplerian elements such as the [[Orbital period|period]], [[apoapsis]] and [[periapsis]]. (When orbiting the earth, the last two terms are known as the [[apogee]] and [[perigee]].) It is common to specify the period instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the [[standard gravitational parameter]], GM, is given for the central body. An orbit can also be described with just five elements if the epoch always represents the moment at which the mean anomaly is zero. (Actually, all six elements are known, we just constrain one to be zero.)
[[Image:Orbit1.svg|thumb|400px|right|Fig. 1: Keplerian '''orbital parameters'''.]]
===Visualizing an orbit===
In Fig. 1, the [[Orbital plane (astronomy)|orbital plane]] (yellow) intersects a reference plane. For earth-orbiting satellites this is usually the earth's equatorial plane, and for satellites in solar orbits it is the [[ecliptic plane]]. The intersection is called the [[Orbital node|line of nodes]], as it connects the center of mass with the ascending and descending nodes. This plane, together with the [[Equinox|Vernal Point]], (<big>♈</big>) establishes a reference frame. The elements can be seen as defining the orbit in this frame by degrees:
* The [[Semimajor axis|semi-major axis]] (violet line in Fig. 1) fixes the size of the orbit. It connects the geometric center of the orbital ellipse with the [[periapsis]], passing through the [[Focus (geometry)|focal point]] where the [[Center of mass|center of mass]] resides. <ref>The semi-major axis is not completely visible in Fig. 1; the segment between the focal point and the geometric center of the orbital ellipse is occluded by the plane of ecliptic</ref>. As noted above, the [[Orbital period|orbital period]] also establishes the size of the orbit.
* The [[Eccentricity (orbit)|eccentricity]] fixes its shape.
* The [[Longitude of the ascending node|longitude of the ascending node]] (green angle <math>\Omega\,\!</math> in Fig. 1) orients the ascending node with respect to the vernal point. Imagine the angle being formed by pivoting the orbital plane through an [[axis of rotation]] perpendicular to the plane of the ecliptic and passing through the [[center of mass]].
* The [[Inclination|inclination]] (green angle <math>i\,\!</math> in Fig. 1) orients the orbital plane with respect to the plane of the ecliptic. Imagine the angle being formed by pivoting the orbital plane through an [[axis of rotation]] coinciding with the [[Orbital node|line of nodes]].
* The [[Argument of periapsis|argument of periapsis (perihelion)]] (violet angle <math>\omega\,\!</math> in Fig. 1) orients the semimajor axis with respect to the ascending node. Imagine the angle being formed by pivoting the orbital plane through an [[axis of rotation]] perpendicular to itself and passing through the [[center of mass]].
* The [[true anomaly]] (red angle <math>\nu\,\!</math> in Fig. 1) orients the celestial body in space. Imagine this positioning angle being formed by pivoting the body's position vector, starting at periapsis, through an [[axis of rotation]] perpendicular to the orbital plane and passing through the [[center of mass]].
===Variance among Keplerian elements and trajectories of orbiting bodies===
Because the simple Newtonian model of orbital motion of idealised points in free space is not exact, the orbital elements of real objects tend to change over time.
Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, due to the nonsphericity of the primary, due to the
atmospheric drag, relativistic effects, radiation pressure, electromagnetic forces, and so on. This evolution is described by the so-called planetary equations, which come in the form of Lagrange, or in the form of Gauss, or in the form of Delaunay, or in the form of Poincaré, or in the form of Hill. (The last is a very exotic option, emerging in the case when the true anomaly enters the set of six orbital elements. Hill considered this kind of orbit parameterisation back in 1913.) <!-- For more information, see the [[Talk:Orbital elements|Discussion]]. -->
==Two line elements==<!-- This section is linked from [[Epoch (astronomy)]] -->
Keplerian elements parameters can be encoded as text in a number of formats. The most common of them is the [[NASA]]/[[NORAD]] '''"two-line elements"'''(TLE) format[http://celestrak.com/columns/v04n03/] , originally designed for use with 80-column punched cards, but still in use because it is the most common format, and works as well as any other.<br/>
Depending on the application and object orbit, the data derived from TLEs older than 30 days can become unreliable. Orbital positions can be calculated from TLEs through the SGP/[[SGP4]]/[[SDP4]]/SGP8/SDP8 algorithms.<ref>Explanatory Supplement to the Astronomical Almanac. 1992. K. P. Seidelmann, Ed., University Science Books, Mill Valley, California.</ref>
<pre>
Line 1
Column Characters Description
----- ---------- -----------
1 1 Line No. Identification
3 5 Catalog No.
8 1 Security Classification
10 8 International Identification
19 14 YRDOY.FODddddd
34 1 Sign of first time derivative
35 9 1st Time Derivative
45 1 Sign of 2nd Time Derivative
46 5 2nd Time Derivative
51 1 Sign of 2nd Time Derivative Exponent
52 1 Exponent of 2nd Time Derivative
54 1 Sign of Bstar/Drag Term
55 5 Bstar/Drag Term
60 1 Sign of Exponent of Bstar/Drag Term
61 1 Exponent of Bstar/Drag Term
63 1 Ephemeris Type
65 4 Element Number
69 1 Check Sum, Modulo 10
Line 2
Column Characters Description
----- ---------- -----------
1 1 Line No. Identification
3 5 Catalog No.
9 8 Inclination
18 8 Right Ascension of Ascending Node
27 7 Eccentricity with assumed leading decimal
35 8 Argument of the Perigee
44 8 Mean Anomaly
53 11 Revolutions per Day (Mean Motion)
64 5 Revolution Number at Epoch
69 1 Check Sum Modulo 10
</pre>
Example of a two line element:<ref>[http://www.heavens-above.com/orbitdisplay.asp?lat=0&lng=0&alt=0&loc=Unspecified&TZ=CET&satid=27651 SORCE - Orbit Data<!-- Bot generated title -->]</ref>
1 27651U 03004A 07083.49636287 .00000119 00000-0 30706-4 0 2692
2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249
==See also==
* [[Ephemeris]]
* [[Orbital state vectors]]
* [[Proper orbital elements]]
* [[Osculating orbit]]
==References==
{{reflist}}
==External links==
* [http://www.amsat.org/amsat/keps/kepmodel.html Keplerian Elements tutorial]
* [http://marine.rutgers.edu/mrs/education/class/paul/orbits.html another tutorial]
* [http://www.amsat.org/amsat/ftp/docs/spacetrk.pdf Spacetrack Report No. 3], a really serious treatment of orbital elements from [[North American Aerospace Defense Command|NORAD]] (in pdf format)
* [http://celestrak.com/columns/v04n03/ Celestrak Two-Line Elements FAQ]
* [http://ssd.jpl.nasa.gov/?horizons The JPL HORIZONS online ephemeris.] Also furnishes orbital elements for a large number of solar system objects.
* [http://ssd.jpl.nasa.gov/?sat_elem NASA Planetary Satellite Mean Orbital Parameters.]
* [ftp://ssd.jpl.nasa.gov/pub/eph/planets/README.txt Introduction to exporting JPL planetary and lunar ephemerides]
* [http://www.mindspring.com/~n2wwd/html/body_state_vectors.html State vectors: VEC2TLE] Access to VEC2TLE software
{{orbits}}
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