Observations and the Solar System : Lecture 3

Kepler's Laws of Planetary Motion

• Kepler 1: The orbit of the planets around the sun follow the path of an ellipse with the sun at one focus.

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  We described how to draw ellipses using string and defined the focus of an ellipse. We defined the
  • focus and
  • eccentricity, denoted e
  of an ellipse. The eccentricity is a number between 0 and 1. If e=0 then the ellipse reduces to a circle.

  Planet e earth, 0.017 venus 0.007 mars 0.093 jupiter 0.048 saturn 0.056 mercury 0.206

  Evidence for eccentricity?

  The fact that planets move in eclipses explains two phenomena.

  Different separation of Mercury from Sun at Greatest Elongations

  The height which Mercury rises to at elongation is not the same but varies between 18 and 28 degrees.

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• Kepler 3: This describes how quickly the planets orbit the sun. If we take the period of rotation around the sun (1yr for the earth) and square it then this is the same as the distance from the sun cubed. To see this

  Planet	a (A.U.)	P (years)	a3	P2
  Mercury	0.39	0.24	0.06	0.06
  Venus	0.72	0.62	0.37	0.37
  Earth	1	1	1	1
  Mars	1.52	1.88	3.51	3.53
  Jupiter	5.20	11.86	140.6	140.7
  Saturn	9.54	29.46	868.3	867.9

  In this table distances are in Astronomical Units a useful distance measure thich is defined to be the mean earth-sun distance 1 A.U. = 149.6 x 10⁶ km. From this we can deduce

  P² = a³

• Kepler 2: This law tells how a planet goes round its orbit. This law states that The planet sweeps out equal areas in equal times
  for a planet travelling around an elipse this means that it is travelling faster when closest to sun and slower when furthest away

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  We discussed how we can use the observational data to measure periods and distances within the copernican model. Let w_earth be the earths angular velicity measured in degrees per year. This has the value 360. Let w_jupit be the angular velicity of Jupiter - this we wish to calculate. The Period, P, of Jupiter is 360/w_jupit. Let T be the time between oppositions for Jupiter. This is 398.9 days or 1.092 years. Between oppositions the earth goes round 360⁰ plus an extra angle d. Jupiter only orbits d degrees. From this we can deduce
  

  This gives the Jupiter period of 11.89yr (approx.)

  © Dave Dunbar 2020