# What is Kepler's law: Definition and 47 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.

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1. ### I Plotting the orbits of the planets from Ephemeris data

I would like to plot the position of planets around the Sun for different dates and describe their orbits. I got the Ephemeris data for each planet from the JPL Horizons System, I got: Date__(UT)__HR:MN R.A.___(ICRF)___DEC Ang-diam ObsSub-LON ObsSub-LAT ObsEcLon ObsEcLat however...
2. ### I Difference between d/dt and d(theta)/dt? Why is it dr or ds/dt?

So, first of all, why and how are we taking the derivative of the vector r or s as d/dt if t is not a parameter of the equations? Second question is what is the difference between d/dt(r) and d(theta)/dt(r) and also between d/dt(s) and d(theta)/dt(s)? Like, both of these appear at the bottom of...
3. ### Quantum Books recommendation covering the Hydrogen atom and Kepler's laws (separately)

Hi, I have an interview for masters degree program in 2 weeks and they asked to study two subjects thoroughly, first being Hydrogen atom and second being Kepler's laws. anyone recommends one book about each subject with advanced level questions that would help me understand the subjects to a...
4. ### Calculate the time for a spacecraft to arrive at Mars (No Kepler's law)

So I've seen many similar questions like this on the internet, but they all involve using Kepler's law, which is not in my syllabus in my case. However, they do give me the mass of the Sun. which I assume involves gravitational force/field/circular motion? The question also gave me a hint that I...
5. ### A How to calculate Saturn's mass from Kepler's third law?

This is not a homework. In Chapter 8: Central-Force Motion, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 325, Problem 8-19, we are asked to calculate the mass of the planet Saturn. In the instructor's solution manual, the solution for this...
6. ### How to find Kepler's law from Newton's laws

We were asked to do an experiment where we had to prove the equation: T2=(4π2m)/Mgr Where M is the mass of the mass stack in kg (0.3kg), m is the mass of the rubber bung in kg (0.0226kg), T is the time taken for one rotation in seconds, r is the horizontal radius of rotation in meters, and g is...
7. ### B Re-evaluating Dark Matter and Kepler's Law in Spiral Galaxies

I keep reading articles that assert that spiral galaxies would fly apart if it were not for the presence of a dark matter halo. The observation that rotational speeds vary little with distance from the center is usually cited as the 'proof' of dark matter since this behavior would otherwise...
8. ### B Kepler's Law of Areas: Radially Directed Force

Hi, Just wanted to know how can we attribute the kepler's law of areas to a radially directed force?
9. ### Kepler's Law for finding velocity

Homework Statement [/B] I need to help solving part a) Homework Equations [/B] $$v= \frac{2\pi}{T}$$ $$(\frac{T_1}{T_2})^2 = (\frac{r_1}{r_2})^3$$ The Attempt at a Solution I'm not sure where to begin really. One approach I tried was getting T_1 in terms of v_0 and plugging it into...
10. ### Kepler's first law for any body

I was just wondering, what if an object is placed in our solar system, will it revolve around the sun? If yes, will it have a centripetal acceleration too?
11. ### Derive the equation for the shape of an orbit

Homework Statement I am given this equation: where m and M are masses. I have to obtain the following as a final result: Homework EquationsThe Attempt at a Solution so far, I have done this (This may not be the most efficient or the easiest way, but it is how my professor wants it to be...
12. ### Using Kepler's 3rd Law to find Period of Venus

Homework Statement Deduce, from the equations employed in Q4 and Q5, the exponent n in the equation: T = k rn where k is a constant and T is the period of a satellite which orbits at a radius r from a massive object in space. Hence, how long is the “year” on Venus if its distance from the Sun...
13. ### B Kepler's Laws of planetary motion

Whose observations did Kepler use to prove that planets do in fact orbit in ellipses ?? Was it Tycho's observations? Thanks.
14. ### Law of universal gravitation and Kepler's law

Homework Statement Show that the rate of change of the free-fall acceleration with vertical position near the Earth's surface is \frac{d}{dr} [g] = -\frac{2GM_E}{R_E^3} Assuming h is small in comparison to the radius of the Earth, show that the difference in free-fall acceleration between...
15. ### What is the speed of three stars rotating in an equilateral triangle?

Homework Statement Three identical stars of mass M = 8.9 x 1030 kg form an equilateral triangle that rotates around the triangle's center as the stars move in a common circle about that center. The triangle has edge length L = 2.8 x 1010 m. What is the speed of the stars? Homework Equations...
16. ### Kepler's Law in Schwarzchild metric

Homework Statement Show Kepler's Third Law holds for circular Schwarzschild orbits. Homework Equations The Attempt at a Solution Setting r' = 0 , \theta' = 0 and \theta = \pi / 2 , where the derivatives are with respect to the variable \lambda and setting c = 1 the Lagrangian is...
17. ### Question on Orbits and Kepler/Uni. Grav. Laws

Homework Statement The picture has the problem question. OK. When the moon is at point B, the distance from the moon to the center of the planet is most nearly: A) (1/25) * rA B) (1/5) * rA C) (1/ sqrt(5) ) * rA D) rA E) sqrt(5) * rA. Homework Equations Newton's law of gravitation...
18. ### Kepler's law of periods problems about orbiting satellite

Homework Statement Question 1: A orbiting satellite stays over a certain spot on the equator of (rotating) Pluto. What is the altitude of the orbit (called a "synchronous orbit")? (The radius of Pluto is 1150 km.) Question 2:A orbiting satellite stays over a certain spot on the equator of...
19. ### Kepler's Law of planetary motion

Homework Statement Two stars of masses M and m, separated by a distance d, revolve in circular orbits around their center of mass.Show that each star has a period given by T^2= (4π^2)(d^3)/ G(M+m) Homework Equations [/B]The Attempt at a Solution I[/B] know the Kepler's Laws can be...
20. ### Kepler's 3rd Law: Solving Questions & Math

Hello I am studying astro independently and have a question on keplers 3rd law and my math. I set myself with a few givens with Kepler's laws My first question is, I remember reading M had to be in solar masses, is this true? For p^2=4π^2/G(M1+M2)/ a^3 Given: a= 1 AU m1= .31...
21. ### Kepler's Law sin[SUP]3[/SUP] i term calculation

It's part c I don't understand, why the expectation value equals to solid angle * sin3 i? I mean, what role does the solid angle play? Thank you
22. ### Kepler's Law to Determine Period of Asteroid

Homework Statement A newly discovered asteroid orbits at twice the Earth-Sun distance. Find its period of orbit (in years). Homework Equations I know that I'm supposed to use Kepler's third law to determine this. T2 = Kr3 K = 4∏2/GM and the Earth-Sun distance is of course 1 Au...
23. ### What is the Mass of the Andromeda Galaxy?

[b]1. The problem statement, all variables and given/known A star at the edge of the Andromeda galaxy appears to be orbiting the center of that galaxy at the speed of about 200km/s. The star is about g*10^9AU from the center of the galaxy. Calculate a rough estimate of the mass of the...
24. ### Third kepler's law error on sun mass calculation

Homework Statement We consider that at the third Kepler's law the Earth mass is 0, that way we are making an error at the Sun mass calculation in comparison to the real Sun mass. In the same calculation we make another error by taking the common error at the gravitational constant, G. Which...
25. ### Using Kepler's Law to find speed of a comet in orbit

Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.5×104 when at a distance of 2.3×1011 from the center of the sun, what is its speed when at a distance of 5.9×1010 . Using Kepler's Law T2 is proportional to R3 T2/R3 is a constant (C)...
26. ### Kepler's Law Program With Planets

Homework Statement Two planets X and Y travel counterclockwise in circular orbits about a star as shown in the figure below. The radii of their orbits are in the ratio 4:3. At one moment, they are aligned as shown in figure (a), making a straight line with the star. During the next five...
27. ### Ellipse and Kepler's Law in Polar Coordinates

Greetings everyone, I am having difficulties grasping the polar form of the ellipse equation, and there seems to be more than one way to express an ellipse in this form, if I am not mistaken. For example on the following webpage http://farside.ph.utexas.edu/teaching/301/lectures/node155.html...
28. ### Kepler's Law vs Newton's Amendment

Hello ..I have a question which I am sure is easy to answer. It has been a long time since I had physics. In a youtube video located here: http://www.youtube.com/watch?v=dAI8BgCudDQ&list=UU1I__Q1k7NgSZE4PByLfQEg&index=54&feature=plcp She writes a relationship at 4 min which is...
29. ### Unravelling Kepler's Law: How F = m/r^2 is Reached

what is it exactly? and in my book they had an equation F = (4)(pi)^2(m)(r)/T^2 then they said by using keplers law... they arrived to a new equation that relates the gravitational force exerted by the sun which is... F = m/r^2 If Keplers law says T = r^3/2 how in the heck did they...
30. ### What is the Relationship between Kepler's law and universal gravitation law?

What is the Relationship between Kepler's law and universal gravitation law? (IT HAS SOMETHING TO DO WITH THE CENTRE OF MASS)I know this; Newton's Law of Universal Gravitation F=mGM/r^2 derived Keplers Third Law r^3/t^2= constant: mGM/r^2 = mv^2/r gives GM/ (2pi)^2 = r^3/t^2. Read more...
31. ### Kepler's Law of Periods and Jupter's Mass

Homework Statement I just finished an experiment asking us to experimentally find Jupiter's mass by using Kepler's Law of Periods T2=((4π2)/GM) * r3. An important property of the law is that period squared is proportional to the radius (circular orbit) or semi-major axis (elliptical orbit)...
32. ### Solve Kepler's Law: Find Satellite's Period

Kepler's Law help! Use this equation: T^2 of A / R^3 of A = T^2 of B / R^3 of B The moon has a period of 27.3 days and has a mean distance of 3.9*10^5 km from the center of the earth. Find the period of an artifical satellite that is 7.5*10^3 km from the center of the earth. How do i...
33. ### Gravitational Energy and Kepler's Law

[b]1. In a double-star system, two stars of mass 6.0 *10^30 kg each rotate about the system's center of mass at a radius of 2.0 * 10^11 m. (a) What is their common angular speed? (b) If a meteoroid passes through the system's center of mass perpendicular to their orbital plane, what minimum...
34. ### From Kepler's law to gravitation force

Hi all! It is often said that Newton deduced from Kepler's laws the theory of gravitation. Particularly from (T^2)/(a^3)=const for differnet planets he deduced that gravitational force must look like '1/(r^2)'. I can also do that by writing F=ma and integrating it in spherical coordinates, but...
35. ### How Does Doubling a Satellite's Mass Affect Its Orbital Speed and Stability?

Homework Statement 1. If the mass of a satellite is doubled, the radius of its orbit can remain constant if the speed of the satellite: a. increases by a factor of 8 b. increases by a factor of 2 c. does not change d. is reduced by a factor of 8 e. is reduced by a factor of 2 2...
36. ### Kepler's Law and Non-terrestrial Orbits (Not Earth)

Homework Statement A satellite orbits a planet at a distance of 6.80 multiplied by 10^8 m. Assume that this distance is between the centers of the planet and the satellite and that the mass of the planet is 3.08 multiplied by 10^24 kg. Find the period for the moon's motion around the earth...
37. ### I'm confused about 1st Kepler's law and the 2 body problem

Hi PF, According to Kepler's first Law any planet in orbit of the Sun describes an elliptical orbit. I don't know why I can't go on wikipedia since days. But if I remember well I've seen that when there are 2 bodies like 2 stars for example then the center of mass will remain quiet (since...
38. ### What is the semi-major axis and distance of Icarus' highly eccentric orbit?

The asteroid Icarus, discovered in 1949, was so named because its highly eccentric elliptical orbit brings it close to the sun at perihelion. The eccentricity e of an ellipse is defined by the relation rp=a(1-e) where rp is the perihelion distance and a is the semi-major axis. Icarus has an...
39. ### Problem involving Kepler's Law

1. Halley's comet orbits the sun roughly once every 76 years. It comes very close to the surface of the sun on its closest approach. Roughly how far our from the sun is it at its closest? Is it still 'in' the solar system? What planet's orbit is nearest when it is out there? [Hint: the mean...
40. ### What is Saturn's Mass Using Kepler's Laws and Moon Data?

Homework Statement I am told to find the mass of Saturn using Kepler's Laws and the following information on Saturn's moons: Mimas r = 185.52 x 106 t = 8.14 x 104 Enceladus r = 238.02 x 106 t = 11.839 x 104 Tethys r = 294.66 x 106 t = 16.311 x 104 Dione r = 377.4 x 106...
41. ### Kepler's Law: Error Analysis & Mass of Earth

hello everyone, I have a couple of questions from my astonomy course that are confusing me. 1. Which do you think would cause the larger error: a 10% error in "p" (period of orbit) or "a" (length of semi-major axis in units)? I would think that the "p" would produce the larger error, because...
42. ### Disproving Kepler's Law: Evidence from Earth's Orbit

Hello all! I read this post on another physics forum. I thought Keplers laws were accurate for our earth/sun system? "Keplers law is so easily disproven Keplers Law states that the planets move in elliptical orbits with the sun at one focus of the ellipse so that a line connecting...
43. ### A question on the Kepler's law

According to Kepler's law of areas, the area swept by the line joining a planet(say earth) and the sun is constant with time. i.e., dA/dt=constant Also, while deriving we get that, dA/dt=L/2m where L is the angular momentum of the planet, and m is its mass. Now, in order for dA/dt...
44. ### Bohr's Atom Model: Kepler's Law & Elliptical Orbits

When bohr gave his model of atom kepler's law already existed ... What made him think that the electrons revolve around the neucleus in circular orbits ? The obvious assumption that should have come to his mind looking at the analogy between the gravitational and electrostatic force should have...
45. ### Proving Kepler's Law: A Math Challenge

kepler's law! hi, does anybody know how to prove that the eccentricity multiplied by the directrix is equal to [itex] \frac {b^2}{a} [\itex]? i found that the eccentricity of an ellipse is equal to c/a. i also found that the directrix is equal to a/e. the way i see it, if i multiply e...
46. ### Help with kepler's law and gravitation

guys i need help with these problems.I have been trying to solve then for a long time and still cannot as i am very confused and have no where to go for help 1) On the way to the Moon , Apollo astronauts passed a point after which the moon's gravitational pull became stronger than the...
47. ### Newton's version of kepler's law

Question: Use Newton's version of Kepler's third law to find the mass of Earth. The moon is 3.9*10^(8) m away from Earth, and the moon has a period of 27.33 days. I just have trouble with using the correct units...thanks for any help!