The
Hale-Bopp Comet Examined Further
Kepler
Gravitation Astronomy
Hale-Bopp 1
Having looked at what
happens at perihelion, we now consider the full orbit of Hale-Bopp. We
will then be able to calculate the time for the orbit and the speed at
hte furthest point from the sun (aphelion) as well as at other points on
the orbit. By conservation of angular momentum, the speed at aphelion will
be considerably less than at perihelion. A diagram shows the relevant geometry:
.
.
By analysis of the full
ellipse, it may be shown that the semi-major axis of the ellipse is given
in terms of eccentricity and perihelion distance by:
.
.
The time for a complete
orbit in terms of the semi- major axis is given by:
.
.
Note that this is in accordance
with Kepler's
Third Law, which is used in 'Weighing
The Planets'.
1. (i) Show
that a = 2.79x1013m and this 186 Astronomical Units where 1A.U.
is 1.5x1011 m.
(ii)
Show that the time for Hale-Bopp to orbit the Sun is approximately 8.03x1010
s.
How many years is this ?
The furthest point
from the Sun is termed the Aphelion. By Conservation of Angular Momentum
its speed at this point will be a minimum, vmin and it
may be shown that:
.
2. (i) Show that Hale-Bopp's
minimum speed is 243 mph.
(ii) At what dates will it next be at Aphelion and Perihelion ?
An improved equation
for the speed of Hale-Bopp at any position in its orbit is given by:
.
3. Show that the speed
of Hale-Bopp at B and B' is 4900mph.
Full analysis of the
orbit using Kepler's Second Law and an accurate Time Period of 2540 years
for the orbit yields the following information:
.
.
For fuller details of
the mathematics governing the orbit of Hale-Bopp see 'The Mathematical
Gazette' - July 2000.
4. Pluto has an elliptical
orbit with a = 39.5 A.U. , e = 0.25 Calculate the following:
(i) Time Period of Orbit.
(ii) Aphelion and Perihelion distances.
(iii)Perihelion Speed.
(iv)Aphelion Speed.
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