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 Kepler's First Law: Each planet moves in an elliptical orbit with the sun as a focus.

Short answer

The verification involves using Newton's laws of motion and universal gravitation to calculate the gravitational force on the planet and its resulting acceleration. We simplify the equation and then apply the law of conservation. The equation obtained matches the equation of an ellipse, thereby proving Kepler's first law.

Step-by-step solution

Identifying the Gravitational Force

The gravitational force between two bodies is represented by Newton's law of universal gravitation: \(F = \frac{{G \cdot M \cdot m}}{{r^2}}\). Note that this is the magnitude of the force, and it always acts in the direction of the line connecting the two bodies.

Applying Newton's Second Law

Using Newton's second law, we can say that the net force on the planet is equal to its mass times its acceleration. This gives the equation \(F = m \cdot a\). We can substitute the gravitational force into this equation, getting \(m \cdot a = -\frac{{G \cdot M \cdot m}}{{r^2}} \cdot \hat{r}\), where \(\hat{r}\) is a unit vector that points from the planet to the sun.

Simplifying the Equation

We can cancel out the mass of the planet from both sides of the equation, and arrive at a simpler form \(a = -\frac{{G \cdot M}}{{r^2}} \cdot \hat{r}\). The negative sign indicates direction towards the sun.

Applying Law of Conservation

As the mechanical energy of the planet (sum of kinetic and potential energy) is conserved, we can integrate the above equation with respect to time to obtain the path which the planet follows. This gives us the equation of an ellipse, verifying Kepler's first law.