Slope

In Celestial mechanics, the slope is a orbital element of a body in Orbite around an other. It describes the Angle between the plan of the orbit and the datum-line (generally the Plan of the ecliptic, i.e. the average plan of the orbit of the Ground, or the equatorial plan). The slope is usually noted by the letter I .

Examples

In the Solar system, the slope of the orbit of a celestial body orbiting around the Sun (Planet S, Asteroid S, etc) is defined like the angle between its orbital plan and that of the ecliptic. It could be measured compared to another plan, such as for example the equatorial plan of the Sun or the orbital plan of Jupiter, but the plan of the ecliptic is most practical for terrestrial observers. The majority of the orbits of the bodies of the solar system have a weak slope, whether it is compared to the ecliptic or in the solar equatorial plan. Among the notable exceptions, one finds the dwarf planets Pluton (17°) and Éris (44°) and the asteroid (2) Pallas (34°).

For a natural satellite or artificial, the slope is measured relative in the equatorial plan around whose the object orbits, if it of it is rather close (the equatorial plan being the plan perpendicular to the axis of rotation of the central body). In this case:

  • a slope of 0° means that the object orbits in the equatorial plan and the same direction as the rotation of the central body;
  • a slope of 90° means that the object is on a polar Orbite and passes to the zenith of the north poles and south of the central body;
  • a slope higher than 90° means than the object orbits in way Rétrograde. If it is equal to 180°, it is located on a equatorial Orbite retrograde.

For the objects located far from the central body, it is possible to use the Plan of Laplace. This one is confused with the equatorial plan close to the central body and is inclined gradually with the distance to end up merging with the orbital plan.

If the rotation of the central body is not known with precision, the slope of a satellite will be given compared to the ecliptic or sometimes compared to the plan of the celestial sphere (or, in an equivalent way, like the angle between the axis of the orbit and the direction of the observer). This last measurement is used for the external objects with the solar system, like the double stars. It thus depends on the direction in which the observer looks at, i.e. of the area of the celestial Sphère in which the object is located. The double stars having a slope close to 90° are often Binaire with eclipses.

In the case of the the Moon, to measure the slope compared to the equatorial plan of the Earth leads to a value varying quickly (between 18,29° and 28,58°). It is more useful to measure it compared to the ecliptic, which gives a relatively constant value (5,145°).

Calculation

In Astrodynamique, the slope i can be calculated in the following way:

i = \ arccos {\ frac {h_z} {\|\ vec H \|} }

where:

  • \ vec H is the vector of the Orbital moment, perpendicular to the plan of the orbit,
  • h_z is the component on Z of this vector.

See too

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