Question

Prove that for circular orbits around a given gravitational force center (such as the sun) the speed of the orbiting body is inversely proportional to the square root of the orbital radius.

Short answer

The orbital speed is inversely proportional to the square root of the radius.

Step-by-step solution

Understand Gravitational Force

For circular orbits, the gravitational force acting on the orbiting body provides the necessary centripetal force to keep the body in orbit. The gravitational force is given by Newton's law of universal gravitation: \[ F_g = \frac{G \, M \, m}{r^2} \] where \( G \) is the gravitational constant, \( M \) is the mass of the central body (e.g., the sun), \( m \) is the mass of the orbiting body, and \( r \) is the radius of the orbit.

Relate Gravitational Force to Centripetal Force

The centripetal force needed to keep a body in circular motion is given by: \[ F_c = \frac{m \, v^2}{r} \] where \( v \) is the speed of the orbiting body. In a circular orbit, this centripetal force is provided by the gravitational force, hence \[ \frac{G \, M \, m}{r^2} = \frac{m \, v^2}{r} \]

Simplify the Equation

Cancel the mass \( m \) of the orbiting body from both sides of the equation: \[ \frac{G \, M}{r^2} = \frac{v^2}{r} \] Next, multiply both sides by \( r \) to solve for \( v^2 \): \[ G \, M \cdot \frac{1}{r} = v^2 \]

Solve for Velocity

Taking the square root of both sides gives the orbital speed: \[ v = \sqrt{\frac{G \, M}{r}} \] This shows a direct relationship between the speed \( v \) of the orbiting body and the orbital radius \( r \).

Prove Inverse Proportionality

The orbital speed \( v \) is proportional to \( \frac{1}{\sqrt{r}} \), which mathematically expresses that the speed is inversely proportional to the square root of the orbital radius. Hence, \[ v \propto \frac{1}{\sqrt{r}} \]

Key concepts

Gravitational Force

Gravitational force is the invisible force that pulls two masses toward each other. It's fundamental in maintaining the structure of the universe. Every object with mass exerts a gravitational pull on other masses. This concept is captured by Newton's law of universal gravitation, which states that the gravitational force (\( F_g \)) between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them:

• \[ F_g = \frac{G \cdot M \cdot m}{r^2} \]

Here, \( G \) is the gravitational constant, \( M \) is the mass of a large body like the sun, \( m \) is the mass of a smaller object like a planet, and \( r \) is the distance between their centers. In the context of orbital motion, gravitational force is what holds a planet in orbit around the sun. Without it, planets would drift away into space.
Gravitational force not only directs planetary motion but also dictates the orbits and paths that celestial bodies follow.

Centripetal Force

Centripetal force is what keeps an object moving in a circle. It acts toward the center of the circle and is vital in circular motion scenarios. In orbital mechanics, the gravitational force serves as the centripetal force. For an object in circular motion, the required centripetal force (\( F_c \)) can be calculated by:

• \[ F_c = \frac{m \cdot v^2}{r} \]

This equation tells us that the centripetal force depends on the mass of the orbiting body \( m \), its velocity \( v \), and the radius of the orbit \( r \).
In the case of planets or satellites moving in circular orbits, the gravitational pull of the central mass (like a star or planet) provides the necessary force to keep them on their path. The alignment of gravitational force as the centripetal force explains why such motions appear stable and predictable.

Circular Orbits

A circular orbit occurs when an object travels around a massive body in a path that maintains a constant altitude. For this to happen, specific dynamics between gravity and motion must be in place. Circular orbits are a particular case of the more general elliptical orbits described by Kepler's laws.
In a circular orbit, the gravitational force acting on the orbiting body is just right to sustain its circular path. Balancing gravitational and centripetal forces keeps the speed constant along the path, making circular orbits a unique and stable state.

• The mathematical relationship ensures that \( \frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r} \)

This balance shows that the central force (gravity) needed to pull the orbiting body toward the center is perfectly met by the inward force needed for the body to maintain its circular motion.

Orbital Velocity

Orbital velocity is the speed needed for an object to maintain a stable orbit around a central mass without falling into it or escaping its gravitational pull. For circular orbits, this speed is crucial, as it determines how an object like a planet or satellite keeps a steady distance from the body it orbits. The velocity is derived from the equation that equates gravitational force with centripetal force, leading to:

• \[ v = \sqrt{\frac{G \cdot M}{r}} \]

This equation shows how \( v \), the orbital velocity, relates to \( G \), \( M \), and \( r \):

• \( v \) is directly proportional to the square root of \( M \) (the mass of the central body),
• \( v \) is inversely proportional to the square root of \( r \) (the orbital radius).

Thus, a larger central mass results in a higher orbital velocity, and an increase in orbital radius leads to a reduction in speed. This inverse proportionality is pivotal when analyzing how objects navigate through space.