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. Prove that \(\frac{d}{d t}\left[\frac{\mathbf{r}}{r}\right]=\frac{1}{r^{3}}\left\\{\left[\mathbf{r} \times \mathbf{r}^{\prime}\right] \times \mathbf{r}\right\\}\).

Short answer

After simplifying the left hand side using the quotient rule, replacing the derivative of r using dot product properties, then simplifying the equation using vector algebra, the equation that was expected to be proved was achieved. Therefore, the equation \(\frac{d}{dt}\left[\frac{\mathbf{r}}{r}\right] =\frac{1}{r^{3}}\{\left[\mathbf{r}\times \mathbf{r'}\right] \times \mathbf{r}\) has been verified.

Step-by-step solution

Simplify the derivative

Start by simplifying the left-hand side (LHS) of the equation. As \(\mathbf{r}\) is a vector, the operation \(\frac{\mathbf{r}}{r}\) means to divide each component of \(\mathbf{r}\) by \(r\). The derivative of this operation can then be calculated using the quotient rule:\[\frac{d}{dt}\left[\frac{\mathbf{r}}{r}\right] = \frac{r\frac{d\mathbf{r}}{dt} - \mathbf{r}\frac{dr}{dt}}{r^2}\]

Replace the derivative of r

Recall that \(r=\|\mathbf{r}\|\), the magnitude of vector \(\mathbf{r}\), and any vector dotted with itself results in the square of its magnitude. Therefore we can write:\[\frac{dr}{dt} = \frac{d}{dt}||\mathbf{r}|| = \frac{d}{dt}\sqrt{\mathbf{r}\cdot\mathbf{r}} = \frac{\mathbf{r}\cdot\mathbf{r'}}{r}\]Substitute this into the equation obtained in Step 1 we get:\[\frac{d}{dt}\left[\frac{\mathbf{r}}{r}\right] = \frac{r\mathbf{r'}-\mathbf{r}(\frac{\mathbf{r}\cdot\mathbf{r'}}{r})}{r^2} \]

Simplify the equation

The equation can be simplified by factoring out the orbit function, \(\mathbf{r}\), and then by using the vector triple product identity \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}\). The result would be:\[\frac{d}{dt}\left[\frac{\mathbf{r}}{r}\right] =\frac{1}{r^{3}}\{\left[\mathbf{r}\times \mathbf{r'}\right] \times \mathbf{r}\]This shows the equation that was supposed to be proved. Note that the simplification process used vector algebra extensively.

Key concepts

Vector-Valued Function

Understanding vector-valued functions is crucial when dealing with the motion of planets or any object in space. A vector-valued function assigns a vector to each point in the domain, typically representing some physical quantity like position, velocity, or acceleration.

In the context of Kepler's Laws of Planetary Motion, the position of a planet at any given time is expressed by a vector-valued function, denoted as \(\mathrm{r}\). The function maps a point in time to the planet's position vector in three-dimensional space. When you see an expression like \(\frac{\mathrm{d}\mathbf{r}}{\mathrm{dt}}\), it represents the rate of change of the position vector over time - in other words, the velocity of the planet.

When working with these functions, operations such as differentiation are applied component-wise, allowing us to investigate how each component of the vector changes. This is especially important when verifying the intricate relationships between velocity, position, and acceleration in planetary motion, as described by Kepler's laws.

Universal Gravitational Constant

The concept of the universal gravitational constant, denoted by \(G\), is central to Kepler's Laws of Planetary Motion as well as Newton's Law of Universal Gravitation. This constant provides us with the proportionality factor needed to calculate the gravitational force between two masses.

With a value of approximately \(6.674 \times 10^{-11} \,\text{Nm}^2\text{kg}^{-2}\), \(G\) allows us to understand the gravitational attraction based on the product of the masses and the inverse square of the distance between their centers. It is essential in the derivation and verification of Kepler's laws, as it quantifies the gravitational pull that the Sun exerts on a planet, which determines the shape and behavior of its orbit.

In calculations, \(G\) is often combined with the masses of the two bodies in question, in this case, the Sun \(M\) and a planet \(m\), to form a gravitational parameter that simplifies the equations governing planetary motion.

Vector Algebra

Vector algebra is a branch of mathematics that deals with vectors and operations on them, such as addition, scalar multiplication, and the vector product. It is instrumental in physics, especially when analyzing planetary orbits.

In exploring Kepler's laws, we encounter several vector operations. For example, one critical operation is the cross product \(\mathbf{a} \times \mathbf{b}\), which results in a vector that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\), and has a magnitude equal to the area of the parallelogram they span.

In the solution provided, we see the use of the vector triple product identity, which simplifies to \(\left[\mathbf{r} \times \mathbf{r'}\right] \times \mathbf{r}\). Understanding how to apply these vector products is essential for simplifying the given expressions and ultimately proving aspects of Kepler’s laws. The cross product also provides insight into angular momentum, a conserved quantity in planetary motion that plays a significant role in the characteristics of orbits.