The Hale-Bopp Comet Examined
  Kepler Gravitation Astronomy Hale-Bopp 2
Early in 1997 a spectacular object could be seen in the night sky. This was comet Hale-Bopp. As it approached the sun it outshone the stars. On this page some aspects of its orbit are analysed.
Comet orbits are highly elliptical, following Kepler's first law ,approaching close to the Sun which is one focus of the elliptical path. Typically comets then move well beyond the orbit of Pluto. The Independent newspaper (15/3/97) gave the following data for Hale-Bopp:

Time Period of Orbit   4000 years
Closest Approach To Sun   85 million miles.

Elliptical orbits are governed by the mathematics of conic sections. Here is a diagram to show the part of the orbit close to Perihelion.

For an ellipse 0< e < 1 . The data from the newspaper yields a value of  e = 0.99636 . Subsequent accurate observation of the orbit produced a value of e of 0.99512 which gave an orbital time period of 2550 years. Quite a difference.

1.  Newton's Gravitational Law gives the acceleration of the comet as
g = GM / r2  = k / r2
    (i)  Change 85 million miles to metres.
    (ii) Calculate  the constant k. MSun is 2x1030 kg  G is 6.67x10-11 Nkg-2m2
   (iii) Show that the acceleration of Hale-Bopp at Perihelion was 7.19x10-3 ms-2

Conic sections are most readily analysed using the plane polar co-ordinate system where a position is specified using a distance r from the origin and angle q
turned from a fixed line. Unless the conic is a circle, both the angle and distance are continually changing for a particular path and we need to set up radial and transverse equations of motion. (A circular path would have only a transverse equation.) Standard mechanics textbooks yield the following equations for transverse and radial accelerations:
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From these two equations it may be deduced that the angular momentum of the body is constant and therefore a conserved quantity of the motion. (See box below.) Thus a comet must speed up as it approaches the Sun.
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From Conservation of Angular Momentum it follows that comets sweep out equal areas in equal times. This is Kepler's Second Law.
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2. Explain as fully as you can why a comet speeds up as it approaches the sun.
For comet Hale - Bopp we are interested in being able to calculate the time of its orbit and linear speed at different points. Below will be quoted formulae and results which enable us to do this. Full derivations of these may be found in most College/University mechanics texts or they are derived in full in an excellent article by H.R. Corbishley in 'The Mathematical Gazette' July 2000. The book 'Mechanics and Vectors' by T.Heard (Cam.Univ. Press) provides all the necessary mathematical details.
The linear speed v of the comet at the general position (r,q) may be given in terms of speed at Perihelion (Vp)

To be able to use this equation we need to be able to calculate Vp the speed at perihelion. Further analysis of the general conic equation yields this formula:
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p


3. If the eccentricity of Hale-Bopp is 0.99512 , calculate the speed (in m/s) at its closest approach to the Sun and show that this is approximately 100,000 mph.

2.Click here to continue with the analysis of Hale-Bopp's orbit.
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