Kepler's First Law Mathematical Query

In summary, the author uses the cross product to demonstrate the relationship between acceleration and angular momentum in order to show Kepler's Second Law and the conservation of angular momentum in planetary motion.
  • #1
Michael King
10
0
Hello all! It's been a long summer, and I thought I'd warm up things by going over Kepler's Laws.

I've been following the mathematical derivation in Introduction to Modern Astrophysics, and to be honest I am little stumped on a part of it:

We have the derived definition of angular momentum as

[tex]\vec{L} = \mu r^{2}\hat{r}\times\frac{d}{dt}\hat{r} [/tex]

Then what happens is out of the blue, it seems, the author takes the cross product of the acceleration vector and angular momentum:

[tex]\vec{a}\times\vec{L} = -\frac{GM}{r^{2}}\hat{r}\times \left(\mu r^{2}\hat{r}\times\frac{d}{dt}\hat{r} \right)[/tex]

Ugh, to be honest I am stumped at the physical significance of that cross product. I can understand that the result is a [tex]\vec{v}\times\vec{L}[/tex] expression, but it just seems to have come out of nowhere and I don't know (physically) why we go through that process

For reference it is on page 44 of the red paperback (second) edition. It is under Chapter 2: Celestial Mechanics.
 
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  • #2
Any help would be appreciated! Thanks in advance!The author is basically using the cross product to show the relationship between the acceleration of the planet around its orbit and the angular momentum it contains. By taking the cross product of the acceleration vector and the angular momentum vector, you can see that the magnitude of the two vectors is equal, which is what is needed to show Kepler's Second Law. The physical significance of this is that it shows that the angular momentum of a body orbiting another mass is always conserved, which explains why planets stay in their orbits.
 

1) What is Kepler's First Law?

Kepler's First Law, also known as the Law of Ellipses, states that all planets move in elliptical orbits with the sun at one focus.

2) How does Kepler's First Law differ from the previous concept of circular orbits?

Before Kepler's First Law, it was believed that planets moved in perfect circles around the sun. However, Kepler's Law introduced the idea of elliptical orbits, which allowed for more accurate predictions of planetary motion.

3) What is the mathematical equation for Kepler's First Law?

The mathematical equation for Kepler's First Law is:
r = p / (1 + e * cosθ),
where r is the distance between the planet and the sun, p is the semi-latus rectum, e is the eccentricity of the orbit, and θ is the angle between the planet and the periapsis (closest point to the sun).

4) How did Kepler come up with his First Law?

Kepler came up with his First Law after studying the observational data of his mentor, Tycho Brahe. By analyzing the positions of planets in the sky over time, Kepler discovered that their orbits were not perfect circles, but instead elliptical in shape.

5) How does Kepler's First Law influence our understanding of planetary motion?

Kepler's First Law is a fundamental concept in our understanding of planetary motion. It allows us to accurately predict the positions of planets in their orbits, and has led to the development of other laws and theories, such as Newton's Law of Universal Gravitation. It also helped pave the way for future advancements in astronomy and our understanding of the universe.

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