Impact of Planetary Shrinkage on Gravitational Force: A Refined Analysis

The gravitational force of attraction between the Earth and the Sun is a fundamental force in our solar system. If the Sun were to shrink to half its original diameter and the Earth to three-fourths its current size, how would this affect the gravitational force between them? In this article, we will explore this scenario using Newton's Law of Universal Gravitation and carefully analyze the implications.

Introduction

Newton's Law of Universal Gravitation states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula for gravitational force is:

F G (m_1 m_2) / r^2

Step 1: Determine the Masses After Shrinking

Let's start by examining how the masses of the Sun and the Earth would change if they shrunk to the specified dimensions.

Sun

When the diameter of the Sun is halved, its radius also becomes half. The volume of a sphere is given by:

V (4/3) π r^3

If the diameter is halved, the new radius is 1/2 of the original, resulting in a new volume:

V (4/3) π (1/2)^3 (4/3) π / 8 1/6 V_{old}

Assuming the density remains constant, the mass of the Sun will decrease proportionally:

m_{Sun} 1/6 m_{Sun, old}

Earth

Similarly, if the diameter of the Earth is reduced to three-fourths, its new radius is 3/4 of the original. The new volume of the Earth is:

V (4/3) π (3/4)^3 (4/3) π 27/64 27/256 V_{old}

Given constant density, the mass of the Earth will also be adjusted accordingly:

m_{Earth} 27/64 m_{Earth, old}

Step 2: Determine the Distance Between the Centers

The average distance between the Earth and the Sun, known as 1 AU, is approximately 1.496 x 1011 meters. This distance would remain largely unchanged even if the planetary diameters changed, as it is the distance between the centers of the two bodies.

Step 3: Calculate the New Gravitational Force

Now, let's substitute the new masses into the gravitational force formula:

F G (m_{Sun} m_{Earth}) / r^2

Substituting the new masses:

F G (1/6 m_{Sun, old} * 27/64 m_{Earth, old}) / r^2 G (27/384) m_{Sun, old} m_{Earth, old} / r^2

Step 4: Compare with Original Gravitational Force

The original gravitational force is:

F_{original} G (m_{Sun, old} m_{Earth, old}) / r^2

To find the ratio of the new force to the original force:

(F / F_{original}) (27/384) 3/32

Conclusion

The gravitational force of attraction between the Earth and the Sun would decrease to 3/32 of its original value if the Sun shrinks to half its original diameter and the Earth shrinks to three-fourths its current size. This calculation highlights the sensitive interplay between planetary dimensions and gravitational forces in our solar system.