Mercury's orbit according to Classical VS Modern Physics

[Daniel Sarioglu]

Hello,
I'm a high-school student and I was assigned to do this kind of a paper as a senior (one of the requirements of graduating is a short monograph on a subject of interest.)
My topic includes an analysis of Mercury's orbit using Newton and Kepler's equations and comparing the predicted trajectory vs the correct one. Just to clarify, my intentions are not to delve into relativity to predict the "correct trajectory," but to compare my calculations according to Newtonian Physics and the orbital parameters (i.e major axis, minor axis, distance between foci of ellipse, etc...) found in online sources like NASA or sth -which I suppose corresponds to the trajectory described by relativity.
My problem lies in not knowing which variables are dependent of modern "non Newtonian" conceptions and therefore would have to find myself. For example, deriving from the inverse square law and the polar equation for an ellipse, I got the following:

<br /> \begin{equation*} <br /> L_e = \frac{L_P^2}{Gm_Sm_P^2} <br /> \end{equation*}<br />

Being $L_e$ the semi latus rectum of the elipse, $L_P$, the angular momentum of the planet, $G$ the gravitational constant, and $m_S$ and $m_P$ the masses of the Sun and the planet correspondingly.

I'm not too sure whether I can get a "Newtonian" measurement if I were to pluck in the data found online for the masses and the angular momentum into the equation. Should I instead calculate the masses and the angular velocity myself? If so, how would one proceed?

 

[mfb]

This is difficult.

In Newtonian mechanics, if you have two point-masses orbiting each other you get an ellipse. In general relativity, you do not. But the solar system is not a set of two point-masses. The point of Mercury's perihelion rotates around the Sun by 5.75 arcseconds per year, this corresponds to one full rotation every 220,000 years - a very small effect. And more than 90% of this comes from the gravitational attraction from other planets - which is present in Newtonian mechanics as well.

What NASA and so on are typically publishing is "if the planet would follow a perfect Newtonian ellipse based on its position and speed at time X, how would these parameters look like?" This time is called the epoch. If you want to see an effect of general relativity, you'll need the orbital parameters for different epochs to compare them.

 

[Daniel Sarioglu]

Thanks,
Another couple of questions:
Where could I get the data corresponding to relativity's "epoch" as you say?
And should I assume the planet is orbiting the Sun's mass without the Sun being slightly affected by the gravitational pull form the rest of the solar system, could the orbit predicted by Newton and Relativity be differentiated or does the difference lie in the fact that the Sun is also slightly moving?

 

[mfb]

Where could I get the data corresponding to relativity's "epoch" as you say?

In databases such as HORIZONS.

And should I assume the planet is orbiting the Sun's mass without the Sun being slightly affected by the gravitational pull form the rest of the solar system, could the orbit predicted by Newton and Relativity be differentiated or does the difference lie in the fact that the Sun is also slightly moving?

Neglecting the motion of the Sun provides a reasonable estimate in some cases (e.g. analytic estimates for the perihelion precession), but if you want accurate predictions for the orbits you'll probably have to take it into account.