Mercury’s Orbital Precession

1. Mercury’s Orbital Precession By Gavin Hartnett

2. Ellipses • Planetary Orbits are ellipses • Earlier lab simplified these orbits to circles • planet moves faster near the sun • Perihelion—closest point to sun • Aphelion—farthest • Two foci—Sun is located at one

3. General Relativity vs. Inverse Square Law • F ≈ GMm/r^2 for low masses/energies • F≈GMm/r^2 + α GMm/r^4 for higher masses • as α 0, we retrieve old, Newtonian expression • larger α will yield a more noticeable departure from Newton

4. Precession of Elliptical Orbit • during precession the orbital axis shift around the center of the ellipse (center of mass) • position of perihelion shifts, but distance to sun remains the same • position of aphelion shifts by same angle • two pair of isosceles triangles—length from extrema to center of mass also remains constant

5. Plan of Attack • How to calculate angle of precession? • We can easily find out the distance from planet to center of mass. • If two successive perihelion’s or aphelion’s are known, then we could calculate the angle using simple trig...

6. How to calculate precession angle… • We can use law of cosines to find φ, or we can break the isosceles triangle into a right triangle to find φ/2 • Works equally well for perihelions or aphelions

7. Putting Ideas into Practice... • can calculate φ if we know the coordinates of two successive perihelion’s • how do we test for perihelion? • Mercury will be: • closest to center of mass • moving fastest • experiencing the maximum force/acceleration • energy is conserved—max force/velocity/min distance values should be same for each revolution • Since we need to know the peri/aphelion values to program an elliptical orbit, let’s create a test to see if the planet is at the perihelion for each iteration

8. Test for Perihelion • perihelion distance equals magnitude of the vector separation between Mercury’s position and the position of the center of mass • Use: • if (merc.pos).mag == (0.313*AU): print merc.pos • Then for each revolution we should have a coordinate corresponding to the perihelion point. We know the magnitude of the position vector will always be the same, and using two such coordinates we can find φ using the trig discussed earlier…

9. Automated φ Calculation • Currently working on… • Seeing if I can get the computer to do the hard work! • Goal: • create a variable that records coordinate of perihelion#1. Update the positions as usual until perihelion#2 occurs. In addition to having a test that prints out the data, have the program take the current coordinate and the old one, and perform the trig. calculations necessary to find φ. Then set perihelion#2=perihelion#1 and repeat. • end result: list of angular shift’s for each revolution printed out for one particular value of alpha

10. Problems… • 1. I’m having a hard time with the task of creating variables for the perihelion coordinates, and then updating them. • 2. Because of numerical rounding errors, the orbit experiences a precession even without G.R. • 3. Due to same errors, perihelion distance doesn’t remain constant, experiences fluctuations. Need to test for 1.00001*perihelion distance---need a big enough window • to catch the perihelion, but not so big that the program records multiple perihelions