Celestial Mechanics Guide, Meaning , Facts, Information and Description
Celestial mechanics AstrodynamicsCelestial Mechanics is the study of the effects of gravity on celestial bodies. It is distinguished from Astrodynamics, which is the study of the creation of artifical satellite orbits.
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Historical approaches
(this needs to be reworded but I don't exactly understand what is trying to be said) First, you've got Aristotle and Ptolemy.
Next, you have Johannes Kepler and Tycho Brahe, followed by Isaac Newton.
The Keplerian problem was addressed by lots of very able mathematicians, such as (examples here ... Bernoulli ??? Alan Turing ??? Henri Poincaré, Joseph Louis Lagrange, Pierre-Simon Laplace ).
This stuff is useful for calculating the behaviour of planets and comets and such. More recently, it has also become useful to calculate spacecraft trajectories. See spacecraft propulsion, Tsiolkovsky rocket equation.
To compute the position of a satellite at a given time (the Keplerian problem) is a difficult problem. The opposite problem—to compute the time-of-flight given the starting and ending positions—is simpler. We present a derivation for the time-of-flight equation here.
The key construction that will allow us to analyse this situation is the circle (shown in blue) circumscribed on the orbital ellipse. This circle is taller than the ellipse by a factor of in the direction of the minor axis, so all area measures on the circle are magnified by a factor of with respect to the analogous area measures on the ellipse.
Any given point on the ellipse can be mapped to the corresponding point on the circle that is further from the ellipse's major axis. If we do this mapping for the position of the satellite at time , we arrive at a point on the circumscribed circle. Kepler defines the angle to be the eccentric anomaly angle . (Kepler's terminology often refers to angles as "anomalies.") This definition makes the the time-of-flight equation easier to derive than it would be using the true anomaly angle .
To compute the time-of-flight from this construction, we note that Kepler's second law allows us to compute time-of-flight from the area swept out by the satellite, and so we will set about computing the area swept out by the satellite. First, the area is a magnified version of the area :
Kepler's equation
Kepler was the first to successfully model planetary orbits to a high degree of accuracy.Derivation
P to a given point S. Kepler circumscribed the blue circle around the ellipse, and used it to derive his time-of-flight equation in terms of eccentric anomaly.]]
The problem is as follows: We are given that the semimajor axis of the orbit is , and the semiminor axis is . The eccentricity is , and the planet is at , at a distance of from the center of the ellipse. The satellite is at periapsis at time . The goal is to find the time at which the satellite reaches point .
Furthermore, area is the area swept out by the satellite in time . We know that, in one orbital period , the satellite sweeps out the whole area of the orbital ellipse. is the fraction of this area, and substituting, we arrive at this expression for :
Another expression for is found by a simple conglomeration of adjacent areas:
Area is a fraction of the circumscribed circle, whose total area is . The fraction is , thus:
Meanwhile, area is a triangle whose base is the line segment of length , and whose height is :
gives us Kepler's equation. Kepler referred to as the mean motion, and as the mean anomaly. The term "mean" in this case refers to the fact that we have "averaged" the satellite's non-constant angular velocity over an entire period to make the satellite's motion amenable to analysis. All satellites traverse an angle of per orbital period , so the mean angular velocity is always .Substituting into the formula we derived above gives this:
Application
With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of from periapsis is broken into two steps:
Finding the angle at a given time is harder. Kepler's equation is transcendental in , meaning it cannot be solved for analytically, and so numerical approaches must be used. In effect, one must guess a value of and solve for time-of-flight; then adjust as necessary to bring the computed time-of-flight closer to the desired value until the required precision is achieved. Usually, Newton's method is used to achieve relatively fast convergence.
The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits. For near-parabolic orbits, eccentricity is nearly 1, and plugging into the formula for mean anomaly, , we find ourselves subtracting two nearly-equal values, and so accuracy suffers. For near-circular orbits, it is hard to find the periapsis in the first place (and truly circular orbits have no periapsis at all). Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it doesn't hold for parabolic or hyperbolic orbits at all. These difficulties are what led to the development of the universal variable formulation, described below.