Kilobit

See also: Orbit (homonymy)

In celestial mechanics, a orbit is the Trajectoire which in space a body around an other body under the effect of the Gravitation describes.

The traditional example is that of the Solar system where the Ground, the others Planet S, the Astéroïde S and the Comet S are in orbit around the Sun, just as the the moon S are in orbit around planets.

Nowadays, much of artificial satellite is in orbit around the Ground.

The three Lois of Kepler make it possible to determine by calculation the orbital movement.

Orbital elements

elliptic Orbit

An elliptic orbit can be defined in space according to six parameters making it possible to calculate the complete trajectory very precisely. Two of these parameters (eccentricity and Equatorial radius) define the Trajectoire in a plan, three others (slope, longitude of the ascending node and argument of the péricentre) lay down the orientation of the plan in space and the last (urgent of passage to the péricentre) defines the position of the object. Here the more detailed description of these parameters:

• Equatorial radius has : half of the distance which separates the péricentre from the apocentre (the largeest diameter of the ellipse). This parameter defines the absolute size of the orbit. It actually has direction only in the case of an elliptic or circular trajectory (the half-large-axis is infinite in the case of a Parabole or of a hyperbole)

• Excentricité E : an ellipse is the place of the points whose sum of the distances to two fixed points, the hearths (S on the diagram), is constant. The eccentricity measures the shift of the hearths compared to the center of the ellipse (C on the diagram); it is the report/ratio of the distance center-hearth to the half-large-axis. The type of trajectory depends on the eccentricity:

• 0: circular trajectory

• 0<e< 1: elliptic trajectory

• 1: parabolic trajectory

• e>1: hyperbolic trajectory

• Slope I : the slope (between 0 and 180 degrees) is the angle which the orbital plan with a datum-line forms. This last being in general the plan of the ecliptic in the case of planetary orbits (plan containing the trajectory of the Ground; in black in figure 1). The slope is the orange angle in figure 1.

• Longitude of the ascending node ☊ : it is about the angle between the direction of the vernal Point and the line of the nodes, in the plan of the ecliptic. The direction of the vernal point (in black in figure 1) is the line containing the Sun and the vernal point (astronomical benchmark corresponding to the position of the Sun at the time of the equinox of spring). The line of the nodes (in green in figure 1) is the line to which the nodes belong ascending (the point of the orbit where the object passes on the northern side of the ecliptic) and going down (the point of the orbit where the object passes from the southern part of the ecliptic).
• Argument of the perihelion ω  : it acts of angle formed by line of nodes and direction of perihelion (line to which the Sun and the perihelion of the trajectory of the object belong), in the orbital plan. It is in blue in figure 1. The longitude of the perihelion is the sum of the longitude of the ascending node and the argument of the perihelion.
• Urgent τ of passage to the perihelion : The position of the object on its orbit at a given moment is necessary to be able to predict it for any other moment. There are two ways of giving this parameter. The first consists in specifying the moment of the passage to the Périhélie. The second consists in specifying the mean anomaly M (in red in figure 1) of the object for one conventional moment (the time of the orbit). It should be noted that the mean anomaly is not a physical angle but specifies the fraction of the surface of the orbit swept by the line uniting the hearth with the object since its last passage to the perihelion, expressed in angular form. For example, if the line uniting the hearth with the object traversed the quarter of the surface of the orbit, the mean anomaly is 0,25×360° = 90°. The average longitude of the object is the sum of the longitude of the perihelion and the mean anomaly.

Period

When one speaks about the period of an object, it is in general about its sidereal period, but several possible periods ago:

• sidereal Period - Time which passes between two passages of the object in front of a distant star. It is the “absolute” period with the Newtonian direction of the term.
• anomalistic Period - Time which passes between two passages of the object to its Périastre. According to whether this last precess or recess, this period is shorter or long that the sidereal one.
• draconitic Period - Time which passes between two passages of the object to its ascending Nœud or going down. It will thus depend on the precessions of the two implied plans (the orbit of the object and the datum-line, generally the ecliptic).
• Period tropic - Time which passes between two passages of the object to the Right ascension zero. Because of the Precession of the equinoxes, this period is slightly and systematically shorter than the sidereal one.
• synodical Period - Time which passes between two moments when the object takes the same aspect (conjunction, Quadrature, opposition, etc). For example, the synodical period of Mars is time separating two oppositions from Mars compared to the Earth; as the two planets are moving, their relative angular velocities are withdrawn, and the synodical period of Mars proves to be 779,964 D (1,135 years Martian).

Relations between the anomalies and rays

In what follows, $e$ is the eccentricity, $T$ is the True anomaly, $E$ is the Eccentric anomaly and $M$ is the Mean anomaly.

The ray $r$ of the ellipse (measured since a hearth) is given by:

$r\; =\; has\; (1\; -\; E\; \backslash \; cos\; (E))\; =\; \backslash \; frac\; \{has\; (1\; -\; e^2)\}\{1\; +\; E\; \backslash \; cos\; (T)\}\backslash ,\; \backslash !$

The following relations exist between the anomalies:

$M\; =\; E\; -\; E\; \backslash \; sin\; (E)\; \backslash ,\; \backslash !$

$\backslash \; cos\; (T)\; =\; \backslash \; frac\; \{\backslash \; cos\; (E)\; -\; E\}\; \{1-e\; \backslash \; cos\; (E)\}\backslash ,\; \backslash !$

or

$\backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{T\}\; \{2\}\; \backslash \; right)\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{1+e\}\; \{1-e\}\}\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{E\}\; \{2\}\; \backslash \; right)\; \backslash ,\; \backslash !$

A frequent application consists in finding $E$ starting from $M$. It is then enough to use the expression:

$E\_\; \{i+1\}\; =\; \backslash \; frac\; \{M\; -\; E\; (E\_i\; \backslash \; cos\; (E\_i)\; -\; \backslash \; sin\; (E\_i))\}\{1-e\; \backslash \; cos\; (E\_i)\}\backslash ,\; \backslash !$

If one uses an initial value $E\_0\; =\; \backslash \; pi$, convergence is guaranteed, and is always very fast (ten significant figures in four iterations).

Various types of orbit

Random links:

Ardian Kozniku | Dogliola | Osburga | Aa off Weerijs | Kilobit

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org