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. Using Newton's Second Law of Motion, \(\mathbf{F}=m \mathbf{a},\) and Newton's Second Law of Gravitation, \(\mathbf{F}=-\left(G m M / r^{3}\right) \mathbf{r},\) show that \(\mathbf{a}\) and \(\mathbf{r}\) are parallel, and that \(\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)=\mathbf{L}\) is a constant vector. So, \(\mathbf{r}(t)\) moves in a fixed plane, orthogonal to \(\mathbf{L}\).

Short answer

After applying Newton's Second Law of Motion and Law of Gravitation, it is established that acceleration \( \mathbf{a} \) and radius vector \( \mathbf{r} \) are parallel, and that the cross product of \( \mathbf{r}(t) \) and its derivative \( \mathbf{r}'(t) \) gives constant vector \( \mathbf{L} \). This means the planet moves in a fixed plane around the Sun, orthogonal to \( \mathbf{L} \), thus verifying Kepler's Laws.

Step-by-step solution

Application of Newton's Laws

Start by setting Newton's Second Law equal to the attractive force between the Sun and Planet, since the force is the planet's mass times acceleration: \( m \mathbf{a} = -\left( GmM/r^{3} \right) \mathbf{r} \). Since the mass of the planet appears on both sides, it can be cancelled.

Determination of Acceleration

Now we have the acceleration \( \mathbf{a} = -\left( GM/r^{3} \right) \mathbf{r} \). Here, acceleration is directly proportional to the radius vector, but in the opposite direction; therefore, \( \mathbf{a} \) and \( \mathbf{r} \) are parallel.

Derivation Part 1

To show that \( \mathbf{r}(t) \times \mathbf{r}'(t) = \mathbf{L} \) is constant, we take the derivative of \( \mathbf{r}(t) \) with respect to time which gives us \( \mathbf{r}'(t) = \mathbf{v}(t) \) as the velocity function.

Derivation Part 2

We can cross-multiply \( \mathbf{r}(t) \times \mathbf{r}'(t) \) to obtain \( \mathbf{r}(t) \times \mathbf{v}(t) \). The right hand side is the definition of angular momentum \( \mathbf{L} = m \times (\mathbf{r} \times \mathbf{v}) \), which is constant because the total external torque acting on the system is zero.

Final Proof

Lastly, given \( \mathbf{r}(t) \times \mathbf{r}'(t) = \mathbf{L} \) is a constant vector, \( \mathbf{r}(t) \) moves in a fixed plane, orthogonal to \( \mathbf{L} \). This determines the plane of orbit around the Sun.

Key concepts

Newton's Second Law of Motion

When we talk about Newton's Second Law of Motion, we're referring to the relationship between the forces acting upon an object, its mass, and the acceleration that it experiences as a result of those forces. This law is often expressed with the equation
\[ \mathbf{F}=m\mathbf{a} \]
What this tells us is that the force (\(\mathbf{F}\)) applied to an object is equal to the mass (\(m\)) of the object multiplied by its acceleration (\(\mathbf{a}\)). The key to understanding this equation is recognizing that acceleration is the rate of change of velocity with respect to time, and it is a vector quantity, which means it has both magnitude and direction.

In our planetary motion scenario, when applying this law, we equal the force of the planet's acceleration to the gravitational force acting between the planet and the Sun. This helps us understand how the planet's velocity changes over time as it orbits the Sun.

Newton's Law of Gravitation

Newton's Law of Gravitation explains the gravitational attraction between two objects with mass. It is formulated as:
\[ \mathbf{F}=-\left(\frac{GmM}{r^{3}}\right)\mathbf{r} \]
In this formula, \(\mathbf{F}\) is the gravitational force, \(G\) is the gravitational constant, \(m\) and \(M\) are the masses of two objects (typically a planet and the Sun in our case), and \(r\) is the distance between the centers of the two objects. The negative sign indicates that the force is attractive and points in the direction opposite to the radius vector \(\mathbf{r}\). This law forms the foundation of orbital dynamics by explaining why planets remain in orbit around the Sun, and in combination with Newton's Second Law of Motion, it allows us to deduce the nature of a planet's orbit.

Planetary Orbit and Acceleration

The orbit of a planet is the path it takes around a star like the Sun. From the perspective of our planetary motion exercise, we see that a planet's acceleration is directly related to its orbit. The acceleration of a planet is not constant but varies depending on its position in the orbit due to the Law of Gravitation. As shown in the exercise,
\[ \mathbf{a}=-\left(\frac{GM}{r^{3}}\right)\mathbf{r} \]
This equation indicates that the acceleration \(\mathbf{a}\) is parallel to the radius vector \(\mathbf{r}\), pointing towards the center of the orbit -- the Sun in this case. It's important to note that while the magnitude of acceleration changes, it always points inward, and this inward force is what keeps the planet moving along a curved orbital path instead of flying off in a straight line.

Angular Momentum in Planetary Motion

Angular momentum in planetary motion is an essential principle that helps in understanding how planets orbit. It is a measure of the quantity of rotation of an object and is given by the cross product of the position vector \(\mathbf{r}(t)\) and the linear momentum (product of mass and velocity) of the object. As per our exercise, angular momentum is expressed as:
\[ \mathbf{L}=\mathbf{r}(t) \times m\mathbf{v}(t) \]
The remarkable trait about angular momentum in a closed system -- such as a planet orbiting around the Sun with no external torques -- is that it remains constant over time. This is known as the conservation of angular momentum and it explains why planets continue to orbit in predictable paths. The fixed orientation of the angular momentum vector also ensures that the planet's orbit maintains a constant inclination, resulting in a fixed orbital plane.