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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. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the In Physics, an inverse-square law is any Physical law stating that some physical Quantity or strength is inversely proportional Gravitation is a natural Phenomenon by which objects with Mass attract one another 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). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 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). In Astrodynamics or Celestial dynamics orbital state vectors (sometimes state vectors) are vectors of Position (\mathbf{r} and

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Keplerian elements

The traditional orbital element set are the six Keplerian elements, after Johannes Kepler and his Kepler's laws:

Keplerian elements can be obtained from orbital state vectors using VEC2TLE software or by some direct computations. Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. Inclination in general is the Angle between a Reference plane and another plane or axis of direction The longitude of the ascending node (☊ or Ω is one of the Orbital elements used to specify the Orbit of an object in space The argument of periapsis (or argument of perifocus) ( ω) is the Orbital element describing the Angle of an Orbiting body's periapsis In Astrodynamics, under standard assumptions, any Orbit must be of Conic section shape In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae In the study of orbital dynamics the mean anomaly of an orbiting body is the Angle the body would have traveled about the center of the orbit's Auxiliary circle In Astronomy, an epoch is a moment in time used as a reference for the Orbital elements of a Celestial body. In Astrodynamics or Celestial dynamics orbital state vectors (sometimes state vectors) are vectors of Position (\mathbf{r} and In Astrodynamics or Celestial dynamics orbital state vectors (sometimes state vectors) are vectors of Position (\mathbf{r} and We see that the first three orbital elements are simply the Eulerian angles defining the orientation of the orbit relative to some defined inertial coordinate system. The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant 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 orbits are always conic sections, so the Keplerian elements define an ellipse, a parabola, or a hyperbola. In Classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface 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 Keplerian element sets defining orbits that osculate ("kiss" or touch) the real trajectory at their respective epoch times. In Astronomy, and in particular in Astrodynamics, the osculating orbit of an object in space is the gravitational Kepler orbit about a central body that

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, M_o\,\!, the mean anomaly M\,\!, mean longitude, true anomaly or, rarely, the eccentric anomaly may also be used. In the study of orbital dynamics the mean anomaly of an orbiting body is the Angle the body would have traveled about the center of the orbit's Auxiliary circle In Astronomy, an epoch is a moment in time used as a reference for the Orbital elements of a Celestial body. In the study of orbital dynamics the mean anomaly of an orbiting body is the Angle the body would have traveled about the center of the orbit's Auxiliary circle In Astrodynamics or Celestial dynamics mean longitude is the Longitude at which an orbiting body could be found if its orbit were circular and In Astronomy, the true anomaly \nu\\! (Greek nu also written \theta\ or f\) is the angle between the direction z-s of The eccentric anomaly is the angle between the direction of Periapsis and the current position of an object on its Orbit, projected onto the ellipse's circumscribing (Sometimes the epoch itself is considered an orbital element. ) Other orbital parameters can be computed from the Keplerian elements such as the period, apoapsis and periapsis. The orbital period is the time taken for a given object to make one complete Orbit about another object In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from (When orbiting the earth, the last two terms are known as the apogee and perigee. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from ) 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. Small body orbiting a central body Under Standard assumptions in astrodynamics we have m where m \ is the mass 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. )

Fig. 1: Keplerian orbital parameters.
Fig. 1: Keplerian orbital parameters.

Visualizing an orbit

In Fig. 1, the orbital plane (yellow) intersects a reference plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded. For earth-orbiting satellites this is usually the earth's equatorial plane, and for satellites in solar orbits it is the ecliptic plane. The ecliptic is the apparent path that the Sun traces out in the sky during the year The intersection is called the line of nodes, as it connects the center of mass with the ascending and descending nodes. An orbital node is one of the two points where an Orbit crosses a Plane of reference which it is inclined to This plane, together with the Vernal Point, (♈) establishes a reference frame. An equinox is the event of the Sun passing over the Earth's equator in its annual cycle The elements can be seen as defining the orbit in this frame by degrees:

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. )

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. The National Aeronautics and Space Administration ( NASA, ˈnæsə is an agency of the United States government, responsible for the nation's public space program
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. SGP4 (Simplified General Perturbations Satellite Orbit Model 4 is a NASA / NORAD algorithm of calculating near earth Satellites (i SDP4 is a NASA / NORAD orbital model used with deep space Satellites. [2]

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

Example of a two line element:[3]

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

References

  1. ^ The semi-major axis is not completely visible in Fig. An ephemeris (plural ephemerides; from the Greek word ἐφήμερος ephemeros "daily" is a table of values that gives the positions of In Astrodynamics or Celestial dynamics orbital state vectors (sometimes state vectors) are vectors of Position (\mathbf{r} and The proper orbital elements of an orbit are constants of motion of an object in space that remain practically unchanged over an astronomically long timescale In Astronomy, and in particular in Astrodynamics, the osculating orbit of an object in space is the gravitational Kepler orbit about a central body that 1; the segment between the focal point and the geometric center of the orbital ellipse is occluded by the plane of ecliptic
  2. ^ Explanatory Supplement to the Astronomical Almanac. 1992. K. P. Seidelmann, Ed. , University Science Books, Mill Valley, California.
  3. ^ SORCE - Orbit Data

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