What equations are needed for calculating Keplerian orbital mechanics?

[Philosophaie]

I have some data about the elliptical orbits of the planets. I have a,e,i,ML,LP,N @ for a specific Julian Century, J2000. I need an equation check:

w = LP - N
EA = MA + e * sin(MA) * (1 + e * cos(MA))
x = a * (cos(EA) - e)
y = a * sin(EA) * (1 - e^2)^0.5
r = (x^2 + y^2)^0.5
v = atan2(y , x)

where

a - Semi-Major Axis
e - Eccentricity
i - Inclination
ML - Mean Anomaly
LP - Longitude of the Perihelion
N - Longitude of the Ascending Node
w - Argument of the Perihelion
EA -= Eccentric Anomaly
r - Radius of the Sun to the Planet
v - True Anomaly

 

[D H]

All of these equations are correct except for the expression for eccentric anomaly. Kepler's equation is

$$M=E-e\sin E$$

The inverse function, $E=f(M)$, does not have a solution in the elementary functions. Newton's method works quite well for small eccentricities. All of the planets have small eccentricities.

Note well: The planets do not follow Keplerian orbits. Kepler's laws are approximately correct. They are not exact.

 

[Philosophaie]

Is there a solar system elliptical orbital program that incorporates all the above terminology into one computer program that gives the correct orientation of the angles not just the correct placement of the planets and moons? Maybe on a main frame?

 

[D H]

One more time: Kepler's laws are only approximately correct. Newton's laws provide a better description of what is going on. General relativity is even better.

If you want an accurate picture of how the planets move over time you will not use Kepler's laws.

 

[spacester]

I do not disagree, but in an effort to clarify . . .

It is my understanding that the only reason Kepler's laws do not produce exact results is due to the presence of other bodies and the perturbations they cause. IOW a solar system with one planet and nothing else would be perfectly predictable. (Also ignoring solar wind and the "pressure" caused by solar radiation.)

Certainly the perturbation effects are quite significant . . .

 

[D H]

I do not disagree, but in an effort to clarify . . .

It is my understanding that the only reason Kepler's laws do not produce exact results is due to the presence of other bodies and the perturbations they cause. IOW a solar system with one planet and nothing else would be perfectly predictable. (Also ignoring solar wind and the "pressure" caused by solar radiation.)

Certainly the perturbation effects are quite significant . . .

Kepler's laws implicitly assume the planets have negligible mass compared to that of the Sun. While this is a reasonably good assumption for the inner planets, it is not all that reasonable for the gas giants.

Those perturbations are very significant, especially over the long haul. The solar system is a chaotic system.

Finally, don't forget about general relativity. One of the reasons it was accepted fairly quickly was because it solved a known problem with Newtonian mechanics, Mercury's anomalistic precession.

 

[Chronos]

Don't forget, Einstein corrected kepler's law when he solved the orbit of mercury, which had been a mystery for many years - as D H noted. I don't see an issue here.

 

[D H]

Newton corrected Kepler's laws, too. Kepler's first two laws are approximations that ignore the perturbations of other planets. Kepler's third law, [tex]P^2 \propto a^3[/itex] is an approximation that ignores the mass of the orbiting body.