Orbital elements
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The elements of an orbit are the parameters needed to specify that orbit uniquely, given a model of two ideal masses obeying the Newtonian laws of motion and the inverse-square law of gravitational attraction. Because there are multiple ways of defining a motion, depending on which set of variable you choose to measure, there are several different ways of defining sets of orbital elements, each of which will define the same orbit.
There are seven degrees of freedom in this model (time, position in 3-space at that time, velocity in 3-space at that time) so all sets of orbital elements have seven parameters. (See also: orbital state vectors).
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Keplerian elements
The traditionally used set of orbital elements is called the set of Keplerian elements, after Johannes Kepler and his Kepler's laws. The Keplerian elements are:
- Epoch (time)
- Inclination (
)
- Longitude of the ascending node (☊)
- Argument of periapsis (
)
- Eccentricity (
)
- Orbital period (
)
- Mean longitude (Mean anomaly (
) at epoch)
The true anomaly or, rarely, the eccentric anomaly are sometimes used instead of the mean anomaly.
Keplerian elements can be obtained from orbital state vectors using VEC2TLE software or by some direct computations.
Other orbital parameters, such as the semi-major axis, can then be calculated from the Keplerian elements. In many cases, the semi-major axis is used as an orbital element instead of period. The elements can be seen as defining the orbit by degrees:
- The semi-major axis (or the period, interchangeably) fixes the size of the orbit.
- The eccentricity fixes its shape.
- The inclination (orange in Fig. 1) and longitude of the ascending node (green) fix its plane.
- The argument of perihelion (blue) orients the orbit within its plane.
- The epoch (or mean anomaly, interchangeably) (red) fixes the object in time on its orbit.
Because the simple Newtonian model of orbital motion of idealized points in free space is not exact, the orbital elements of real objects tend to change over time. For artificial satellites grazing the fringes of the atmosphere, an eighth parameter (" drag") may be added. For the planets and moons it is more usual simply to add a time term to the elements.
Two line elements
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[1] , 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.
TLEs older than 30 days become considerably inaccurate. Orbital positions and heights can be calculated from TLEs through the SGP/SGP4/SDP4/SGP8/SDP8 algorithms.
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 Derative 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
Reference:
- Explanatory Supplement to the Astronomical Almanac. 1992. K. P. Seidelmann, Ed., University Science Books, Mill Valley, California.
See also
External links
- Keplerian Elements tutorial
- another tutorial
- Spacetrack Report No. 3, a really serious treatment of orbital elements from NORAD (in pdf format)
- Celestrak Two-Line Elements FAQde:Bahnelement
