MathDB

Newton's Law of Gravity from Kepler's Laws? [*] Planets in the solar system move in elliptic orbits with the sun at one of the foci. [/*] [*] The line joining the sun and the planet sweeps out equal areas in equal times. [/*] [*] The period of revolution (

T

) and the length of the semi-major axis

(a

) of the ellipse sit in the relation

T^2/a^3 = constant

. [/*]
Now answer the following questions:
[*] Starting from Newton's Law of Gravitation and Kepler's first law, derive the second and third law. It is possible to derive the first law but that is beyond the scope of this exam. [/*] [*] For convenience work in the complex (Argand) plane and take the sun to be at the origin

(z=0)

. Show that points on the ellipse may be represented by,

z(\theta) = \frac{a(1-\epsilon^2)}{1+\epsilon\cos\theta}\exp(i\theta) = r(\theta) e^{i\theta}

where

a

is the length of the semi-major axis,

\epsilon

is the eccentricity of the ellipse and

\theta

is called the \emph{true anomaly} in celestial mechanics. [/*] [*] Show that Kepler's second law leads to,

\frac{1}{2}r^2 \dot{\theta} = constant

where

r

and

\theta

are defined as in part (b) and a dot

(.)

over a variable denotes its time derivative. What is this constant in terms of the other variables of the problem? [/*] [*] Using the results of parts (b) and (c) along with Kepler's third law obtain Newton's Law of Gravitation. [/*] [*] Can the above exercise truly be called a "derivation" of Newton's Law of Gravitation? State your reasons. [/*]

Problem Statement