Kepler's Equation and its choices

[solarblast]

I have a reasonable understanding of Kepler's Equation up to a point. It is written as:
M = E - e sin E, where
M, E and e are the mean anomaly, eccentric anomaly, and eccentricity, resp.

With a few other equations (and a method to solve a transcendental equation), one can use the equation to produce in polar coordinates (r, $\upsilon$), where nu is the true anomaly, and ultimately from these find the ra and dec of the object. This is fine, but problems poised about the use of the equation seem to give variables like E and e values to derive (r,$\upsilon$). To make this a practical, real-world, problem, how would one know, say, E and e? Does the equation itself become useful in some other context? Perhaps in the development of orbital elements?

 

[Chronos]

We have sufficient understanding of Keplerian mechanics to send humans to the moon and back.

 

[mfb]

E and e have to be observed. Without observing an object, you cannot predict where its orbit is.
In general, an object has 6 free parameters. With 3 observations (each as point in the "two-dimensional" sky), you can determine the whole orbit and predict the position of the object for every time in the future with the Kepler formulas, neglecting influences of other objects.

 

[Filip Larsen]

E and e have to be observed.

For the benefit of the OP we perhaps need to clarify, that technically speaking E and e, like many of the other parameters that are used to characterize a Kepler orbit, are in practice never observed directly, but have to be derived from other parameters in the model you are using to describe a particular problem.

Kepler's Equation can be regarded as nothing more than a relationship between M, E and e in a geometric description of an ellipse, noteworthy for being both the only way to derive E from M and e and for being non-trivial to solve for E.

 

[solarblast]

True, but I seriously doubt that Kepler's Equation is used to get us around the solar system. The word I suppose I'm looking for is application. Of what application is it? It seems more of an educational exercise.

 

[solarblast]

Ah, I missed a few responses. My "True" response was a response to Chronos, but I suppose works for the others as well.

 

[Janus]

True, but I seriously doubt that Kepler's Equation is used to get us around the solar system. The word I suppose I'm looking for is application. Of what application is it? It seems more of an educational exercise.

Problems 4.13 and 4.14 show practical examples of how Kepler's Equation is used to solve Orbital mechanics problems.
http://www.braeunig.us/space/problem.htm#4.13

An example of how it is used for getting around the Solar system would be the following:

You are launching a probe to Jupiter, which you plan to use for a gravity slingshot in order to send your probe further out into the Solar system. In order for this to happen, your probe has to intersect Jupiter's orbit at the right time to a high degree of accuracy.

This means that you have to launch the probe when Earth and Jupiter are in the proper relative positions. To do this, you need to know how long it will take from the time you launch to the time that your probe intersects Jupiter's orbit, and how much Jupiter will travel in its orbit in that time. This way, the probe and Jupiter both arrive at the right spots at the right time.

 

[solarblast]

Ah, certainly a good example of modern use, but historically of what application did it have originally, or was it just an exercise by Newton? I see in the examples that there are givens, e.g., semi-major axis. (Note from another web source, Kepler knew the eccentricity of several planets.) I would guess that means some other method was used for those, possibly the methods came later, so his equation didn't become used until later? I suppose I'm wondering why some writers put effort into deriving the equation, and then depart onto other matters without showing a connection.

 

[mfb]

You can predict solar and lunar eclipses. Apart from that and space travel: Which application do you see which requires the prediction of the position of objects in the sky?

 

[solarblast]

Interesting about eclipses. Where's a source that talks about that subject? I'm not sure what you mean by your question.

I may have under estimated Kepler's observing skills and mathematical tools (geometry). He was able to measure the distance to Mars, and possibly find its perihelion.