Question

You are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function \(\mathrm{r}\). Let \(r=\|\mathbf{r}\|,\) let \(G\) represent the universal gravitational constant, let \(M\) represent the mass of the sun, and let \(m\) represent the mass of the planet. Assume that the elliptical orbit \(r=\frac{e d}{1+e \cos \theta}\) is in the \(x y\) -plane, with \(\mathbf{L}\) along the \(z\) -axis. Prove that \(\|\mathbf{L}\|=r^{2} \frac{d \theta}{d t}\)

Short answer

The magnitude of the angular momentum vector \(\mathbf{L}\) is \(\|\mathbf{L}\|=(\frac{e d}{1+e \cos \theta})^2 m \frac{d\theta}{dt}\) when the elliptical orbit formula \(r=\frac{e d}{1+e \cos \theta}\) is used.

Step-by-step solution

Define Given Variables

First, let's define the given variables: \(r\) is the radius vector (distance from the Sun to the planet), \(M\) is the mass of the Sun, \(m\) is the mass of the planet, \(G\) is the universal gravitational constant, \(d\) is the distance from the Sun to the directrix of the ellipse, \(e\) is the eccentricity of the orbit, \(\theta\) is the angle formed by the radius vector and the major axis of the ellipse, and \(t\) is the time.

Derive Formula for \(\mathbf{L}\)

The magnitude of the angular momentum vector \(\mathbf{L}\) for a particle moving under a central force is given by \(\|\mathbf{L}\|=r m v\), where \(v\) is the speed of the particle. Angular momentum is the product of the moment of inertia and the angular velocity \(\omega\). The moment of inertia \(I\) for a particle of mass \(m\) moving in a circle of radius \(r\) is \(I=mr^2\), and the angular velocity \(\omega\) is the rate of change of angle \(\theta\) with respect to time \(t\). Thus, we have \(\omega=\frac{d\theta}{dt}\), so \(\mathbf{L}=I\omega\). Substituting \(I\) and \(\omega\), we get \(\|\mathbf{L}\|=r^2 m \frac{d\theta}{dt}\).

Substitute Given Elliptical Orbit Formula

Now, we will substitute the given elliptical orbit formula \(r=\frac{e d}{1+e \cos \theta}\) into the angular momentum formula from Step 2. After substitution, no changes can be made on the equation and it will remain \(\|\mathbf{L}\|=(\frac{e d}{1+e \cos \theta})^2 m \frac{d\theta}{dt}\). Note that the mass of the planet and the change in angle with respect to time remain factors in the equation for \(\mathbf{L}\).

Key concepts

Vector-Valued Function

A vector-valued function is essentially an equation that provides a vector for each value of a certain variable, typically time. In the context of Kepler's Laws and planetary motion, a vector-valued function can represent the position of a planet as it orbits a star at a given time. This function, denoted as \( \mathbf{r} \), maps out the elliptical path of orbit in a coordinate system.

For instance, consider an orbit shaped like an ellipse in the plane; the position of a planet in its orbit can be a function of time as it moves around the sun. The function \( r=\frac{e d}{1+e \cos \theta} \) is a specific type of vector-valued function that gives us the planet's radial position in polar coordinates, where \( e \) is the eccentricity, \( d \) is a constant related to the shape of the orbit, and \( \theta \) changes over time as the planet moves. These functions are paramount in physics and engineering as they provide a concise mathematical description of motion in space.

Application in Kepler's Laws

In verifying Kepler's Laws of Planetary Motion, the vector-valued function allows us to understand and predict the position of a planet along its orbit at any given moment, which is vital to the analysis of celestial mechanics and the forces exerted by gravity on the motion of the planets.

Universal Gravitational Constant

The universal gravitational constant, denoted by \( G \), is a key factor in the equation that describes the gravitational force between two masses. Its value is determined experimentally and reflects the strength of gravity in the universe. This constant is extremely important in the field of astronomy and astrophysics because it allows us to calculate the gravitational forces and resulting motions that arise in the cosmos.

To elaborate, the universal gravitational constant is used in Newton’s law of universal gravitation, which states that the force of attraction \( F \) between two bodies with masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by \( F = G\frac{m_1 m_2}{r^2} \). This formula illustrates how gravitational force decreases with distance and increases with mass. In the exercise mentioned, \( G \) is used alongside the mass of the sun \( M \) and the mass of the planet \( m \) to analyze the planetary motions based on gravitational forces.

Implications for Planetary Motion

Understanding \( G \) is critical for verifying Kepler's laws because it enables us to quantify the gravitational relationship between planets and stars. It allows us to derive important characteristics of planetary orbits, such as their shapes, sizes, and periods, through comprehensive mathematical modeling.

Angular Momentum

Angular momentum, symbolized by \( \mathbf{L} \), is a conserved quantity in physics, meaning it remains constant within a closed system when no external torques are acting. For a planet orbiting a sun, angular momentum is an essential concept, as it combines the planet's velocity and mass distribution relative to the point of rotation - in this case, the center of its orbit, which we take as the sun. It’s a vector pointing perpendicular to the plane of orbit.

In classical mechanics, the magnitude of angular momentum for a particle (or planet) is the product of its moment of inertia and angular velocity \( \|\bL\| = I \omega \). The moment of inertia \( I \) reflects how mass is distributed relative to the rotation point, while angular velocity \( \omega \) measures how fast the angle \( \theta \) changes with time. In circular motion, \( I = mr^2 \) and \( \omega = \frac{d\theta}{dt} \), so \( \|\bL\|=mr^2\frac{d\theta}{dt} \).

Relevance to Planetary Orbits

In our specific exercise, using the elliptical orbit equation and our understanding of angular momentum, we find that angular momentum remains consistent in a planet's orbit due to the conservation laws, which directly supports Kepler’s second law–a planet sweeps out equal areas in equal times along its orbit. This law implies that the planet’s velocity will vary so that \( r^2\frac{d\theta}{dt} \) stays constant, which is simply another way of saying that angular momentum is conserved. This helps verify Kepler’s Law of Equal Areas for planetary motion.