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Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly 0.003 4 to almost 0.058 as a result of gravitational attractions among the planets. [4] Luna's value is 0.054 9, the most eccentric of the large moons in the Solar System.
The eccentric anomaly E is related to the mean anomaly M by Kepler's equation: [3] = This equation does not have a closed-form solution for E given M. It is usually solved by numerical methods, e.g. the Newton–Raphson method. It may be expressed in a Fourier series as
Diagram of 2006 QH 181 's orbit. 2006 QH 181 orbits the Sun beyond Neptune with an orbital period of 541 years. It has an highly elliptical orbit with a semi-major axis of 66.4 astronomical units (AU) and an orbital eccentricity of 0.42. In its eccentric orbit, 2006 QH 181 comes within 38.8 AU from the Sun at perihelion and 94.0 AU at aphelion.
However, the actual solution, assuming Newtonian physics, is an elliptical orbit (a Keplerian orbit). For these, it is easy to find the mean anomaly (and hence the time) for a given true anomaly (the angular position of the planet around the sun), by converting true anomaly f {\displaystyle f} to " eccentric anomaly ":
A common problem in orbital mechanics is the following: Given a body in an orbit and a fixed original time , find the position of the body at some later time . For elliptical orbits with a reasonably small eccentricity, solving Kepler's Equation by methods like Newton's method gives excellent results.
From a circular orbit, thrust applied in a direction opposite to the satellite's motion changes the orbit to an elliptical one; the satellite will descend and reach the lowest orbital point (the periapse) at 180 degrees away from the firing point; then it will ascend back. The period of the resultant orbit will be less than that of the original ...
The body does not actually have to be in orbit for its state vectors to determine its trajectory; it only has to move ballistically, i.e., solely under the effects of its own inertia and gravity. For example, it could be a spacecraft or missile in a suborbital trajectory. If other forces such as drag or thrust are significant, they must be ...
According to Copernicus: [3] [4] The planetary orbit is a circle with epicycles. The Sun is approximately at the center of the orbit. The speed of the planet in the main orbit is constant. Despite being correct in saying that the planets revolved around the Sun, Copernicus was incorrect in defining their orbits.