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The factor a in the preceding formula is the main amplitude, the factor q the main angular velocity, which is directly related to a harmonic of the driving force, that is a planetary position. For example: q = 3×(length of Mars) + 2×(length of Jupiter).
The Lunar Laser Ranging data is collected in order to extract numerical values for a number of parameters. Analyzing the range data involves dynamics, terrestrial geophysics, and lunar geophysics. The modeling problem involves two aspects: an accurate computation of the lunar orbit and lunar orientation, and an accurate model for the time of ...
The central bodies are the sources of the gravitational forces, like the Sun, Earth, Moon and other planets. The orbiting bodies, on the other hand, include planets around the Sun, artificial satellites around the Earth, and spacecraft around planets. Newton's laws of motion will explain the trajectory of an orbiting body, known as Keplerian orbit.
For example, to view the eccentricity of the planet Mercury (e = 0.2056), one must simply calculate the inverse sine to find the projection angle of 11.86 degrees. Then, tilting any circular object by that angle, the apparent ellipse of that object projected to the viewer's eye will be of the same eccentricity.
To calculate the accelerations the gravitational attraction of each body on each other body is to be taken into account. As a consequence the amount of calculation in the simulation goes up with the square of the number of bodies: Doubling the number of bodies increases the work with a factor four.
Figure 1: Rangekeeper Coordinate System. The coordinate system has the target as its origin. The y axis value range to the target. US Navy rangekeepers during World War II used a moving coordinate system based on the line of sight (LOS) between the ship firing its gun (known as the "own ship") and the target (known as the "target").
Data may be based on each planet's geometric center or a planetary-system barycenter. The use of Chebyshev polynomials enables highly precise, efficient calculations for any given point in time. DE405 calculation for the inner planets "recovers" accuracy of about 0.001 seconds of arc (arcseconds) (equivalent to about 1 km at the distance of ...
Johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy, publishing his laws in 1605. Isaac Newton published more general laws of celestial motion in the first edition of Philosophiæ Naturalis Principia Mathematica (1687), which gave a method for finding the orbit of a body following a parabolic path ...