Question

Derivation of keplers 3rd law without gravitational formula

Answer 1

Period of revolution is T and the radius of the orbit is r.
And r = R + h = Radius of the earth + height of the satellite from the surface of the earth.

So period of revolution = T = (2 π r) /V, where V is the Orbital Velocity of the satellite. ……………………….. (1)

Now from the previous post on orbital velocity of Satellite, we know that
V = R √(g/r) = R(g/r)^(1/2) .

Here g is the acceleration due to gravity and R is the radius of the earth …………….(2)

From equation 1 and 2, we get,

T = (2 π r)/V
= [(2 π r)] /[R (g/r)^(1/2)]
=[ 2π / R]. √[r ^3/g]

Here we get an important equation for Period of Revolution of earth satellite:

T =[ 2π / R]. √[r^3/g]..(3)

here R is the radius of the earth and r = R + h

In the next section we will use the above equation to derive Kepler’s Third Law

Derivation of Kepler’s Third Law

Now if we square both side of equation 3 we get the following:

T^2 =
[ (4 π^2)/(R^2)]. r^3/g...(4)

Here, (4.π^2)/(R^2)
and g are constant as the values of π (Pi), g and R are not changing with time.

So we can say, T^2 ∝ r ^3. …………….. (5) .

[Kepler’s Third Law equation]

Here this Kepler’s Third Law equation says that square of the Orbital Period of Revolution is directly proportional to the cube of the radius of the orbit.

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