Kepler's Third Law: Proportional Orbit Period to Distance^1.5

[eku_girl83]

Two gravitionally bound stars with equal masses m, separated by a distance d, revolve about their center of mass in circular orbits. Show that the period tau is proportional to d^1.5

Could someone get me started on this? I have no idea where to begin!

Thanks!

 

[Chi Meson]

First of all, state Keppler's Third Law.

Also, realize that d^1.5 is the same as d^(3/2)

And then notice how the radius of the orbits is directly proportional to the distance between them.

 

[quicknote]

Kepler's Third Law states that the square of the orbital period (tau) of a planet is proportional to the cube of its semi-major axis (d) and is expressed mathematically as tau^2 ∝ d^3. In the case of two gravitationally bound stars with equal masses m, the center of mass will be located at the midpoint between them. This means that the distance between the two stars, d, is equal to the semi-major axis of their orbits.

To show that the period tau is proportional to d^1.5, we can start by substituting the value of d into Kepler's Third Law equation:

tau^2 ∝ (d)^3

We can then take the square root of both sides:

tau ∝ √(d^3)

Using the properties of exponents, we can rewrite d^3 as (d^2)^1.5:

tau ∝ √((d^2)^1.5)

Now, we know that the square root and the cube root cancel each other out, leaving us with just d:

tau ∝ d

This means that the period tau is directly proportional to the distance d between the two stars.

In conclusion, Kepler's Third Law states that the orbital period of two gravitationally bound stars with equal masses is directly proportional to the distance between them raised to the power of 1.5. This further supports the idea that the farther apart two objects are, the longer it takes for them to complete one orbit around each other.