Mercury

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Mercurys Orbital Precession

• By Gavin Hartnett

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Ellipses

• Planetary Orbits are ellipses
• Earlier lab simplified these orbits to circles
• planet moves faster near the sun
• Perihelionclosest point to sun
• Aphelionfarthest
• Two fociSun is located at one

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General Relativity vs. Inverse Square Law

• F GMm/r2 for low masses/energies
• FGMm/r2 a GMm/r4 for higher masses
• as a? 0, we retrieve old, Newtonian expression
• larger a will yield a more noticeable departure
  from Newton

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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 triangleslength from
  extrema to center of mass also remains constant

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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 perihelions or aphelions are
  known, then we could calculate the angle using
  simple trig...

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How to calculate precession angle

• We can use law of cosines to find f, or we can
  break the isosceles triangle into a right
  triangle to find f/2
• Works equally well for perihelions or aphelions

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Putting Ideas into Practice...

• can calculate f if we know the coordinates of two
  successive perihelions
• how do we test for perihelion?
• Mercury will be
• closest to center of mass
• moving fastest
• experiencing the maximum force/acceleration
• energy is conservedmax 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, lets create a test
  to see if the planet is at the perihelion for
  each iteration

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Test for Perihelion

• perihelion distance equals magnitude of the
  vector separation between Mercurys position and
  the position of the center of mass
• Use
• if (merc.pos).mag (0.313AU)
• 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 f using the trig
  discussed earlier

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Automated f Calculation

• Currently working on
• Seeing if I can get the computer to do the hard
  work!
• Goal
• create a variable that records coordinate of
  perihelion1. Update the positions as usual
  until perihelion2 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
  f. Then set perihelion2perihelion1 and
  repeat.
• end result list of angular shifts for each
  revolution printed out for one particular value
  of alpha

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Problems

• 1. Im 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
  doesnt remain constant, experiences
  fluctuations. Need to test for
  1.00001perihelion distance---need a big enough
  window
• to catch the perihelion, but not so big that the
  program records multiple perihelions

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