# Kepler's laws and proof using angular momentum

• Phy6boii
In summary, Kepler's laws are three scientific principles that describe the motion of planets around the sun. The first law, also known as the law of orbits, states that all planets move in elliptical orbits with the sun at one of the two foci. The second law, known as the law of areas, states that a line connecting a planet to the sun will sweep out equal areas in equal time intervals. The third law, also called the law of periods, states that the square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit. Angular momentum is used to prove Kepler's laws by applying the principle of conservation of angular momentum to the motion of planets around the sun, as the angular
Phy6boii
a) Kepler's first law states that a planet like Earth displays an elliptical orbit with the sun in focus. Using M = dL/dt, prove that a planet cannot leave its plane of orbit. Note: M here is an externally applied torque that the sun exerts on the planet.

diagram of the situation described

b) From the above law of angular momentum, it follows that the angular momentum for the planet is conserved. Use this to prove Kepler's second law, which states that at equal time intervals Δt, the traveling beam of a planet covers equal areas ΔA.
Hint: use the representation: 2ΔA = |r x Δr| (proof?), where the origin lies at the focal point of the ellipse.

c) By what percentage do the orbital velocities of the Earth differ approximately at the closest point to the sun (perihelion, rP = 147.1 million km) and at the most remote point (aphelion, rA = 152.1 Million km)

Can someone guide me through these questions please? I'm lost and don't know where to begin with the thought process here. I really want to figure this out

Phy6boii said:
Can someone guide me through these questions please? I'm lost and don't know where to begin with the thought process here. I really want to figure this out
Yes, but this question doesn't belong here. You should start a new thread in the approriate homework forum, which in this case will be https://www.physicsforums.com/forums/introductory-physics-homework.153/

## 1. What are Kepler's laws?

Kepler's laws are three scientific principles formulated by astronomer Johannes Kepler in the early 17th century. They describe the motion of planets around the sun and are considered fundamental to our understanding of planetary motion.

## 2. What is the first law of Kepler?

The first law, also known as the law of ellipses, states that all planets move in elliptical orbits with the sun at one of the focal points. This means that the distance between a planet and the sun varies throughout its orbit.

## 3. How does angular momentum relate to Kepler's laws?

Angular momentum is a measure of the rotational motion of an object. Kepler's second law, also known as the law of equal areas, states that a planet sweeps out equal areas in equal times as it moves around the sun. This is only possible if the planet's angular momentum remains constant.

## 4. What is the proof for Kepler's laws using angular momentum?

The proof for Kepler's laws using angular momentum involves using mathematical equations to show that the conservation of angular momentum results in elliptical orbits and the equal area law. This mathematical proof provides evidence for the validity of Kepler's laws.

## 5. How have Kepler's laws influenced our understanding of the universe?

Kepler's laws have greatly influenced our understanding of the universe and have paved the way for modern astronomy. They have provided a framework for understanding planetary motion and have led to further discoveries such as the laws of gravity and the laws of motion formulated by Isaac Newton.

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