suppose the distance between the sun and the earth were quadrupled. what exactly (numerically) would happen to the gravitational force between earth and the sun

The gravitational force between the sun and the earth drops to (1/16) of its original value.

Explanation:

According to the gravitational law, gravitational force between two bodies is directly proportional the product of the masses of the two bodies and inversely proportional to the square of their distances apart.

If the two masses and the other constants of proportionality are held constant, the gravitational force between two bodies is inversely proportional to the square of their distances apart.

F ∝ (1/R²)

If the initial gravitational force between the sun and the earth is given by F₀ and the distance between the sun and the earth is R₀.

F₀ = (k/R₀²)

where k is constant regardless of the values of corresponding gravitational forces and distances apart.

Then, let the new distance apart be R₁,

R₁ = 4 R₀

The new gravitational force between the sun and the earth, F₁ will be given by

F₁ = k / (R₁²)

Recall, R₁ = 4 R₀

F₁ = k / (4R₀) ² = k/16 (R₀²) = F₀/16

Hence, the gravitational force between the sun and the earth drops to (1/16) of its original value.