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Mercury’s Orbital Precession. By Gavin Hartnett. 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.

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### Mercury’s Orbital Precession

By Gavin Hartnett

• 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

• 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

• 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

• 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...

• 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

• 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

• 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…

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

• 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