What is Kepler's second law: Definition and 18 Discussions
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:
The orbit of a planet is an ellipse with the Sun at one of the two foci.
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law helps to establish that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the slower its orbital speed, and vice versa.
Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation.
Many texts state that in an elliptic orbit you can find angular momentum magnitude as
$$ L = r m v = m r^2 \frac {d \theta} {dt} $$
I wonder if
$$ v = r \frac {d \theta} {dt} $$
is valid at every point. I understand this approximation in a circumference or radius r but what about an arc...
I have a lot of questions as usually, but must begin with the difficult to understand moment which have started my explorations of the Kepler's second law. In the book of Steven Wainberg "To Explain The World", in the technical paragraph twenty one I have found the next formula which represents...
Hello.
As I understand it, Kepler's 2nd law of planetary motion can be explained through conservation of energy or conservation of angular momentum.
I am having trouble with the conservation of energy explanation.
We know that the sum of potential and kinetic energy of a planet in orbit around...
Hi,
I'm sorry but I'm not sure if I should post it here or in homework section. It's not homework for sure.
This Wikipedia article, https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion, on Kepler's laws says the following under History section in the last para:
Newton was...
Homework Statement
I am working on the derivation of Kepler's Second Law based on torque and angular momentum. I understand that the vector "L" is equal to the mass (m) times the cross product of the vector "r" and the vector "v." The source I am following then states that
L = mrvtheta. I do...
Hello,
I am completing a research project for differential equations class. I am to derive Kepler's three laws and then compare the results of the derivation with real-world data. For Kepler's second law (a planet sweeps out an equal area in an equal time), I was hoping to find orbital data for...
This is the problem given at my basic astronomy course. If I turn it in within a month correctly, I get 10 extra points in the finals.
Problem:
We have binary star system consisting of Star A and Star B. Astronomers have observed it for 11 years, during which it has moved from point A to point...
I've searched a little bit and found that I can derive kepler's third law from Newton's law of gravitation. That's okay. But I want to deduce kepler's second law too: "An imaginary line joining a planet and the sun sweeps out an equal area of space in equal amounts of time".
I know it's possible...
I have been working on modeling the orbit of a satellite using Kepler's second law of planetary motion, and I have gotten to a point that is really quite bothersome to me. Essentially, my problem boils down to solving this equation for θ (angular position of the planet from the focus of the...
Homework Statement
A particle of mass m moves along a straight line with constant velocity v in the x direction, a distance b from the x axis. (a) Does the particle possesses any angular momentum about the origin? (b) Explain why the amount of its angular momentum should change or should stay...
Kepler's law questions
Homework Statement
What effect will the tangential component of force have on the velocity of the planet?
What effect will the normal component of force have on the velocity of the planet?
Homework Equations
The Attempt at a Solution
I think that the...
Homework Statement
This is just a general question.
When using Kepler's second law, which radius am I supposed to use to sub into r? Is it the radius of the object (ex. Earth's is 6.38e6 m) or the radius of orbit (ex. Earth's is 1.49e11 m)?Homework Equations
C = (GM)/(4pi^2) = (r^3)/(T^2)The...
Deriving Vr and V(sub-theta) from Kepler's second law and...
Homework Statement
Beginning with r=[a(1-e2)]/(1+e*cos \theta) and Kepler's second law, derive general expressions for vr and v\theta for a mass m1 in an elliptical orbit about a second mass m2. The final answers should be...
Homework Statement
I have to prove that A1=A2, or the fact that planets will sweep out equal areas in equal amounts of time. The planet I have to do this for is Pluto. Basically, I need help finding the area it "sweeps out" over any length of time of my choosing.
Homework Equations...
question is http://home.earthlink.net/~urban-xrisis/clip_image002.jpg
I know that an imaginary line adjoining a planet and a sun sweeps out an equal area of space in equal amounts of time.
that means... I know that
\frac{1}{2}b(t_2-t_1)=\frac{1}{2}b(t_4-3_1)
but I don't know that...
While studying the history of classical mechanics I noticed that the primary motor for most of the early equations is the principle of conservation framed firstly as the law of inertia and then as the principle of least time (Fermat) and then as... But while trying to regain for the principle of...