Question

Show how, guided by Kepler's third law, Newton could deduce that the force holding the Moon in its orbit, assumed circular, must vary as the inverse square of the distance from the center of the Earth.

Short answer

Newton deduced from Kepler's third law and the concept of centripetal force that the force holding the Moon in its orbit around the Earth, assumed circular, varies as the inverse square of the distance from the center of the Earth. That means gravitational force decreases with the square of the distance between the two objects.

Step-by-step solution

Understand Kepler's Third Law

Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Mathematically, this can be written as: \(T^2 ∝ r^3\), where \(T\) is the orbital period and \(r\) is the distance from the center of the orbit.

Apply Kepler's Third Law to the Moon and Earth

Considering the Earth and the Moon, the relation between their orbital period \(T\) and the distance \(r\), according to Kepler's Third Law, is: \(T_{moon}^2 ∝ r_{moon}^3\). The Earth orbits the Sun in one year and the distance between them \(r_{earth}\), is one Astronomical Unit (AU), therefore: \(T_{earth}^2 ∝ r_{earth}^3\). We can now make the ratio between the Moon's and Earth's equation, which results in: \((\frac{T_{moon}}{T_{earth}})^2 ∝ (\frac{r_{moon}}{r_{earth}})^3\)

Use Centripetal Force Concept

Due to the circular movement of the Moon around the Earth, there is a centripetal force that keeps the Moon in its orbit and prevents it from going off straight. The centripetal force is given by: \(F = \frac{m*v^2}{r}\), where \(m\) is the mass of the object, \(v\) is the velocity and \(r\) is the radius of the motion. By expressing the velocity \(v\) in terms of the radius \(r\) and the period \(T\), we find: \(v = \frac{2*π*r}{T}\). Substituting this into the centripetal force equation gives: \(F ∝ \frac{1}{r^2}\)

Deduce the Inverse Square Law

By comparing the laws of the forces obtained from Kepler's Third Law and the concept of centripetal force, it can be noticed that the gravitational force must fall off as the square of the distance, known as the Inverse Square Law. This law states that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object's centers.

Key concepts

Kepler's Third Law

Kepler's Third Law is a fundamental concept in the study of celestial mechanics. It states that the square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit. In simpler terms, as distance increases, the time it takes for a celestial body to complete an orbit also increases. This relationship is mathematically represented as \(T^2 \propto r^3\). For the Moon orbiting the Earth, this means that you can predict how long it takes the Moon to orbit based on its average distance from Earth.
This law helps explain why bodies further from the central mass take longer to orbit, which was pivotal in guiding Newton's work. By understanding this, you can start to see why the relationship between distance and time is crucial to defining gravitational forces.

Centripetal Force

Centripetal force is the necessary force that keeps a body moving in a circular path. Without it, the Moon or any other object would move in a straight line off its orbit. The formula for centripetal force is given by \(F = \frac{m \cdot v^2}{r}\), where \(m\) is the mass, \(v\) is the velocity, and \(r\) is the radius of the orbit.
The velocity, \(v\), can be expressed in terms of the orbital period, \(T\), and the radius, \(r\), as \(v = \frac{2 \cdot \pi \cdot r}{T}\). Substituting back into the expression for centripetal force, it becomes evident that \(F \propto \frac{1}{r^2}\). This relation shows that centripetal force also tends to decrease rapidly with increased distance in a circular orbit, aligning with the inverse square nature of gravitational pull.

Gravitational Force

The concept of gravitational force is a cornerstone in physics and astronomy. It is the attractive force that acts between any two masses. According to Newton’s formulation, gravitational force follows the Inverse Square Law. This means that the force of gravity between two objects is inversely proportional to the square of the distance between their centers.
Mathematically, it is expressed as \(F \propto \frac{1}{r^2}\). This inverse relationship implies that if you double the distance between two objects, the gravitational force becomes four times weaker. This principle was crucial for Newton to connect the force keeping the Moon in orbit to the broader universe's mechanics, supporting his formulation of universal gravitation.

Newton's Laws of Motion

Newton's Laws of Motion lay the foundation for understanding movement and force. They describe how objects respond to forces and are essential in understanding celestial dynamics. The three laws are:

• First law: An object in motion will remain in motion unless acted upon by an external force (inertia).
• Second law: The acceleration of an object is dependent on the net force acting on it and is inversely proportional to its mass, expressed as \(F = m \cdot a\).
• Third law: For every action, there is an equal and opposite reaction.

In the context of the Moon's orbit, these laws underpin the movement around Earth. The second law, particularly, links the gravitational pull (acting as the centripetal force) to the amount of mass and distance involved. Together with Kepler's Law and the concept of gravitational force, Newton's Laws help illustrate the natural order underlying celestial movements.