Let us find the distance between the earth and the sun. It won't be simple! Earth and sun both are moving, there are many types of motion they are performing like earth is spinning about its own axis,revolving around the sun, both are moving in galaxy and the universe is expanding ufffffffffff. Just hold on, we have to simplify this situation, otherwise the solution won't be that simple. For simplification we consider following assumptions;
1) The mass of the earth and the sun are concentrated at their centers, we can consider them point masses.
2) The sun is static and at any instant, earth is at a distance R from the sun.
3) The orbit of the earth around the sun is circular not elliptical. The above distance R is also the radius of that orbit.
4) Neglect the spin of the earth.
credit
Lets start, the gravitational force of attraction between the earth and the sun can be expressed by using Universal Law of gravitation discovered by Newton,
F = G mems/R2 ..............→(Equation-1)
where
me = mass of the earth
ms = mass of the sun
G = universal gravitational constant
The earth is revolving around the sun with uniform speed "v", experiencing centripetal force which is given as,
F = meac = mev2/R
where
ac is the centripetal acceleration of the earth while revolving around the sun.
Let T is the time required by the earth to complete one revolution around the sun. The earth will cover the cicumference(2πR) of the orbit in which it is moving,
As we know that:
S = vT → 2πR = vT → v = 2πR/T
The formula for centripetal acceleration becomes:
F = me (2πR/T)2/R
F= me4π2R/T2 ..............→(Equation-2)
In the case of the earth revolving around the sun, gravitational force behaves as a centripetal force, compairing equation-1 and equation-2