The force which drives the motion of the planets around the sun
is gravity and Newton showed that this force is inversely
proportional to the square of the distance of the planet from the
sun. The constant of proportionality is called Newton's constant
G.
If the mass of the planet is m and the mass of the sun is M then
the force between the two is
|
F = - |
GmM
r2
|
G = 6.674 ×10-11m3kg-1s-2 |
|
In polar coordinates in the plane of motion, the equation of
motion is
where the angular momentum h of the planet is given by
This equation is very tricky to solve directly, but making the
substitution
leads to an inhomogenous linear second order differential
equation.
Find this equation and show that its solution is
Show that numbers e
and f can be found so
that
Hence sketch the solutions for
|
|e|=1, |e| = 1 and |e| = 1 |
|
For a fun and very much simpler
practical activity, why not try the problem Making
Maths: Planet paths
Extension: Investigate the
elliptic paths observed in the solar system using the real data
below. Why not try to draw a scale model of the solar system and
send it in to us using the 'submit a solution' link?
NOTES AND BACKGROUND
Solving the problem of the motion of the planets around the sun
required the invention of differentiation and integration,
credited to Newton about 350 years ago. The fact that orbits
follow such beautiful, pure paths shows how elegantly the
universe is put together. The results are incredibly accurate
and were only challenged by Einstein whose theory of General
Relativity provides very small corrections to the orbits.
Essentially, this mathematics is sufficient to send space
probes all the way from earth to the far reaches of the solar
system with sufficient accuracy to meet up with various planets
and moons along the way.
The following table comprises real astronomical data (compiled
from Wikipedia) which describe the elliptical paths taken by
some key objects in our solar system.
| Name |
Diameter relative to Earth |
Mass relative to Earth |
Orbital radius |
Orbital period |
Inclination to sun's equator |
Orbital Eccentricity
e
|
Rotation period (days) |
| The Sun |
109 |
332946 |
-- |
-- |
-- |
-- |
26.38 |
| The Moon |
0.273 |
0.0123 |
-- |
29.5 days |
-- |
0.0549 |
-- |
| Halleys Comet |
-- |
-- |
-- |
73.3 |
162.3 |
0.967 |
-- |
| Mercury |
0.382 |
0.06 |
0.39 |
0.24 |
3.38 |
0.206 |
58.64 |
| Venus |
0.949 |
0.82 |
0.72 |
0.62 |
3.86 |
0.007 |
-243.02 |
| Earth |
1.00 |
1.00 |
1.00 |
1.00 |
7.25 |
0.017 |
1.00 |
| Mars |
0.532 |
0.11 |
1.52 |
1.88 |
5.65 |
0.093 |
1.03 |
| Jupiter |
11.209 |
317.8 |
5.20 |
11.86 |
6.09 |
0.048 |
0.41 |
| Saturn |
9.449 |
95.2 |
9.54 |
29.46 |
5.51 |
0.054 |
0.43 |
| Uranus |
4.007 |
14.6 |
19.22 |
84.01 |
6.48 |
0.047 |
-0.72 |
| Neptune |
3.883 |
17.2 |
30.06 |
164.8 |
6.43 |
0.009 |
0.67 |
The actual numbers for the earth are
| Diameter |
Mass kg |
Distance from sun |
Orbital period |
Rotation time |
| 12756 km |
5.9736 x 10^24 |
147.1-152.1 million km |
365.256366 days |
23 hours 56 minutes |