Question

Assume that a planet of mass \(m\) is revolving around the sun (located at the pole) with constant angular momentum \(m r^{2} d \theta / d t\). Deduce Kepler's Second Law: The line from the sun to the planet sweeps out equal areas in equal times.

Short answer

The constant angular momentum implies the area swept out per unit time is constant, confirming Kepler's second law.

Step-by-step solution

Define Angular Momentum

The angular momentum of a planet revolving around the sun can be expressed as \( L = m r^2 \frac{d\theta}{dt} \). In this situation, it is given that the angular momentum \( L \) is constant.

Express Area Swept in Terms of Angular Variables

The area \( A \) swept out by the line connecting the sun and the planet over a time interval \( dt \) is given by \( dA = \frac{1}{2} r^2 d\theta \). This formula is derived from the sector area formula \( A = \frac{1}{2} \times r^2 \times \theta \).

Relate Area Swept to Angular Momentum

We can substitute \( d\theta = \frac{d\theta}{dt} dt \) into the area formula to obtain \( dA = \frac{1}{2} r^2 \frac{d\theta}{dt} dt \).

Incorporate Angular Momentum into Area Swept Formula

Substituting the expression for angular momentum, we get \( L = m r^2 \frac{d\theta}{dt} \), or \( \frac{d\theta}{dt} = \frac{L}{mr^2} \). Substituting this in gives us \[ dA = \frac{1}{2} r^2 \cdot \frac{L}{mr^2} \cdot dt = \frac{L}{2m} dt \].

Simplify the Equation

Since \( \frac{L}{2m} \) is a constant as both \( L \) and \( m \) are constant, \( dA/dt = \frac{L}{2m} \). This result implies that the rate of change of area with respect to time is constant.

Conclude Kepler's Second Law

As the rate of sweeping out area \( dA/dt \) is constant, this supports Kepler's second law that a planet sweeps out equal areas in equal times as it revolves around the sun.

Key concepts

Angular Momentum

Angular momentum is a fundamental concept in physics, especially when it comes to planetary motion. It describes the quantity of rotation of an object, and in our scenario, it applies to how a planet revolves around the sun. The equation for angular momentum of a planet orbiting the sun is given by \[ L = m r^2 \frac{d\theta}{dt} \]where:

• \( L \) is the angular momentum,
• \( m \) is the mass of the planet,
• \( r \) is the distance between the planet and the sun,
• \( \frac{d\theta}{dt} \) is the rate of change of the angular position.

In our case, the angular momentum \( L \) is constant. This constancy stems from the absence of external torques acting on the planet. When no torque acts on a system, as in the case of planetary motion, the angular momentum remains unchanged. This consistency is crucial for proving Kepler's Second Law.

Planetary Motion

Planetary motion refers to the intricate path or orbit a planet follows as it travels around the sun. These orbits are typically elliptical in shape. Johannes Kepler, a famous astronomer, developed laws that explain how planets move in these paths. His second law, in particular, is directly related to angular momentum. One of the most critical aspects of planetary motion is the conservation of angular momentum; as the planet moves along its orbit, its speed and distance from the sun vary, but the angular momentum remains constant. This constant angular momentum results in the planet speeding up when it is closer to the sun and slowing down when it's farther away. This principle is a direct result of the conservation laws in physics, showing that the physical properties and conditions remain constant within the system unless altered by external forces.

Areas Swept by Planets

The area swept refers to the sector created by connecting a line from the sun to a planet (radial line) and tracing the planet's path over time. Keeping track of these areas as a planet moves along its orbit helps us understand how it navigates space.The formula that gives the area \( A \) swept by the planet in a short time period \( dt \) is:\[ dA = \frac{1}{2} r^2 d\theta \]This expression arises from the geometry of a circular sector, based on the familiar formula for the area of a sector of a circle: \( A = \frac{1}{2} r^2 \theta \).As the planet travels, the area it's sweeping out per unit time \( \frac{dA}{dt} \) remains constant, which is essential in proving Kepler's Second Law.

Calculating Area in Polar Coordinates

The polar coordinate system is a powerful mathematical framework ideal for analyzing motion in circular or elliptical paths. Here, the position of a point (such as a planet) is determined by a radius \( r \) and an angle \( \theta \) relative to a fixed point, the pole (like the sun in our scenario).To calculate the area swept by a planet moving along its orbit, we break it down as follows:1. **Identify Variables**: - \( r \): the distance from the sun to the planet. - \( \theta \): the angle from a reference direction.2. **Sector Area Formula**: - The formula for a sector in polar coordinates is \( dA = \frac{1}{2} r^2 d\theta \).3. **Connecting to Time**: - By incorporating the rate of angle change \( \frac{d\theta}{dt} \), which links directly to angular momentum, we see how \( \frac{dA}{dt} = \frac{L}{2m} \) stabilizes our understanding of how constant area sweep rates arise, thus aligning with Kepler's Second Law.This approach highlights the intersection of geometry, motion, and physics, revealing the beautifully predictable patterns in our solar system's dance.