What is Kepler's third law: Definition and 52 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. M

    B Proof of inverse square law for gravitation?

    Newton arrived at "there is a force that drives a planet around the star by examining kepler's laws but how did he arrive to inverse square law by kepler's third law (##T^2=\frac {4\pi r^3}{GM}##)? Thank you.
  2. M

    I Why does Kepler's Third Law exist?

    Hi; I understand Kepler 3 in terms of the relationship that it demonstrates. I have researched the internet but can't find a reason why this relationship exists. Is it somehow a consequence of some type of gravitational balance, if not is there some other mechanical reason? Thanks Martyn
  3. S

    Kepler's Third Law vs Conservation of angular momentum

    The classic way to go about this problem would be to use Kepler's laws and thus find the new time period of earth. However I encountered this question in a test on rotational motion which deals with conservation of angular momentum. The equation used here would be I1ω1= I2ω2 Replacing I with MR2...
  4. K

    I Separating Masses in Kepler's Third Law

    Hey everyone! I have been looking everywhere to try to find the answer to this question so I thought I'd pose it here. When we discuss finding the mass of orbiting bodies, it's easy to find the combined mass of the system using Kepler's Third Law in the form M1+M2=(4pi^2)(a^3)/((G)(T^2). My...
  5. sams

    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. B

    Apparent (clearly false) contradiction - Kepler's Third Law

    Homework Statement When considering a satellite in geosynchronous orbit, its speed is zero across (relative to) Earth's surface. From Kepler's third Law: T2=(4π2r3)/(GM), we can derive that v2=GM/r This would tell us that as the radius of a satellite to Earth's centre increases, its velocity...
  7. A

    How Can You Calculate Saturn's Mass Using a Telescope and Stopwatch?

    Homework Statement Using only a telescope and a stopwatch, find the mass of Saturn.[/B] (This question may or may not make any sense at all, it was a theoretical lab that my professor said without giving us a chance to copy it down and I am trying to recall the question from memory) If it is...
  8. A

    What Are the Correct Units for Kepler's Third Law?

    I am confused about the units used in Kepler's 3rd law. Is T supposed to be in years or days? Is R supposed to be in kilometers or meters? Is there ever an instance where one combination of units is preferable over another (for example, if you want to use the answers from Kepler's third law to...
  9. TreeLover

    B An exercise related to the mass of the Milky Way, sort of.

    So, in preparation to the Portuguese Astronomy Olympiads, I've stumbled upon this problem (exercise): The sun, which is 8 kpc away from the centre of the Milky Way, has a rotation speed of approximately 220 kms-1 . Whereas a a star that is 15 kpc from the centre of the Galaxy orbits at a speed...
  10. C

    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...
  11. Z

    Kepler's Third Law and Motion of Two Point Masses

    I posted this before but I think it was in the wrong, place, so sorry for the duplicate :O I'm trying to work through some equations in the paper 'Gravitational Radiation and the Motion of Two Point Masses' (Peters, 1964) but I can't get out the right values 1. Homework Statement For a binary...
  12. Z

    Kepler's Third Law and Motion of Two Point Masses

    I'm trying to work through some equations in the paper 'Gravitational Radiation and the Motion of Two Point Masses' (Peters, 1964) but I can't get out the right values 1. Homework Statement For a binary star system with each mass = 1 solar mass, the equations give the results: Separation ~ 10...
  13. D

    Kepler's third law implies force proportional to mass

    Homework Statement Show that Kepler's third law, \tau = a^{3/2}, implies that the force on a planet is proportional to its mass. Homework Equations 3. The Attempt at a Solution [/B] I haven't really attempted anything. I'm not sure what the question is going for. What can we assume and use?
  14. R

    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...
  15. M

    I Calculating semi-major axis and minimum mass of an exoplanet

    Hello guys, I'm doing my physics coursework on kepler's third law and I'm finding the minimum mass and semi-major axis of a unknown planet. I have the following data: Stellar mass Mstar = 1.31 ± 0.05 Msun Orbital period P = 2.243752 ± 0.00005 days Radial velocity semi-amplitude: V = 993.0 ±...
  16. W

    How far would a planet be from the Earth, when its....

    Homework Statement How far would be a planet from the earth, when its period would be 2 years? T = 2 years/730 days a = 150*106km Homework Equations a3/T2 = C (C is the Kepler-Constant) The Attempt at a Solution I tried inserting T in days and years, but I always get a wrong solution, since C...
  17. D

    Kepler's third law for elliptical orbits

    Kepler's third law states T^2=(4pi^2/GM) x r^3 for CIRCULAR orbits. My question is, in the derivation for this equation ma=GMm/r^2 why can centripetal acceleration be used to replace a at m(v^2/r)=GMm/r^2 yielding v^2/r=GM/r^2 when the orbit is not circular. Planets have elliptical orbits so why...
  18. victorhugo

    I'm not getting what Kepler's Third Law is about

    How R^3/T^2 is a constant, or is it just the simple relationship between the distance between a planet to a star in a solar system and the period for that planet to orbit the star?
  19. A

    Kepler's third law of planetary motion

    Homework Statement Using the equation T = kr^p I drew a loglog graph of orbital time period against orbit radius for the planets Mars to saturn. lnT = plnr + lnk My value for gradient was 5.25/0.48 = 10.94 Meaning p = 10.94 How do I compare this to the actual value? What is it?Homework...
  20. M

    Kepler's third law constant

    Homework Statement Determine Kepler's third-law constant Ku for Uranus using the data for the first four moons. Homework Equations [/B] (r)^3/(T)^2 = K Data: Moon #1 (Ophelia) r=5.38 x 10^4 km T=0.375 Earth Days The Attempt at a Solution [/B] I know the question asks for the K constant...
  21. Calpalned

    Using Kepler's third law, determine the period

    Homework Statement If there were a planet three times farther from the sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun? Assume the orbit is a circle. Homework Equations Kepler's 3rd Law ##= (\frac{r_1}{r_2})^3 = (\frac{T_1}{T_2})^2 ## The Attempt...
  22. A

    Using Kepler's Third Law to find mass

    Homework Statement The fictional planet Snazbort has a fictional moon Pingdol. Pingdol has an orbital period of 7.68 days and a semimajor axis of 92.53×109. Use Kepler's Third Law to estimate the mass of Snazbort. Homework Equations T^2=4pi^2*a^3/(GM) where 2a is the length of major axis T is...
  23. BrainMan

    How Can You Calculate Jupiter's Mass Using Callisto's Orbital Data?

    Homework Statement The planet Jupiter has at least 14 satellites. One of them, Callisto, has a period of 16.75 days and a mean orbital radius of 1.883 x 109 m. From this information, calculate the mass of Jupiter. Homework Equations T2 = (4∏2/ GM)r3 ∏ = pi The Attempt at a...
  24. P

    Can a change in units cancel out G/4π2 in Kepler's third law?

    For two two bodies of mass M1 and M2 in circular orbits of radius a1, a2 about their common centre of mass, the Newtonian modification of Kepler's third law is a3/P2=G(M1+M2)/4π2. Where a=a1+a2. The problem is that I have been told that when using the units of years, solar masses and...
  25. K

    Solving Kepler's Third Law for Period - 6.7E6m

    Homework Statement Find the period of a satellite that is in orbit 6.7×10^6 meters from the center of the earth? Homework Equations P^2=R^3 P=period and R=average distance The Attempt at a Solution so far I have tried P^2=(6.7×10^6)^3 then sqrt of (P^2)=sqrt of (3.00763×10^20)...
  26. S

    Deriving Oort Constants From Kepler's Third Law

    Hello everybody. New here. Hope you are all doing well. I am a sophomore in college, studying physics and astronomy. I am in my first astrophysics class and struggling with how to approach problems. That being said, I need a little homework help if you can afford some time! 1. Starting with...
  27. D

    Find Pluto's year length using Kepler's third law.

    Homework Statement Pluto is 40 times further from the Sun than we are. How long is a year on Pluto? (Use Kepler's third law.) Homework Equations 4∏^2/Gm=T^2/r^3 The Attempt at a Solution Te^2(Earth period)=x*r^3 Tp^2(Pluto period)=x*(40r)^3 I don't know what to do...
  28. G

    Kepler's Third Law and a comet

    Homework Statement Suppose that a comet that was seen in 545 A.D. by Chinese astronomers was spotted again in year 1937. Assume the time between observations is the period of the comet and take its eccentricity as 0.11. What are (a) the semimajor axis of the comet's orbit and (b) its greatest...
  29. J

    Orbit changes, Kepler's Third Law

    Tidal effects in the Earth-Moon system are causing the Moon's orbital period to increase at a current rate of about 35 ms per century. Assuming the Moon's orbit around the Earth is circular, to what rate of change in the Earth-Moon distance does this correspond? Hint: differentiate Kepler's...
  30. N

    Kepler's Third Law: Homewor Help

    Homework Statement A spy satellite is located one Earth radius above the surface of the Earth. What is its period of revolution? ans. 1.5 x 10^4 s Homework Equations I'm trying to use the kepler's constant: k = T^2/R^3 The Attempt at a Solution
  31. T

    Kepler's Third Law: Mass of Sun and Newton's Law

    I read somewhere that the mass of the sun can be determined using Keplers third law. How is this possible without indulging Newton's law(gravitation) in ?:confused:
  32. Shackleford

    Kepler's Third Law and centripetal acceleration

    I'm not getting very far with these problems. Don't know what my problem is. Given a circular orbit, the centripetal acceleration is given by the gravitational force between the two stars. I chose the center to be halfway between the stars d/2...
  33. N

    Kepler's Third Law orbiting planet

    Dear all, A very simple (almost embarrassingly so) question here, but I just want to double-double check what is going on. Kepler's Third Law states for a planet of mass Mp orbitting star of mass M* at a semi-major axis $a$ with a period P, that: [P/(2pi)]^2 = a^3/G (Mp+M*) Now what I want...
  34. B

    Kepler's Third Law satellite problem

    Homework Statement An orbiting satellite stays over a certain spot on the equator of (rotating) Pluto. What is the altitude (in km) of the orbit (called a "synchronous orbit")? Homework Equations r = [(G*M*T^2)/4pi^2]^(1/3) h= r-radius of pluto The Attempt at a Solution I have...
  35. X

    Derivation of Kepler's Third Law

    Homework Statement I am studying gravitation and I have been trying to derive Kepler's Third Law using Kepler's Second Law.Homework Equations The second law : \frac{dA}{dt} = \frac{L}{2m}The Attempt at a SolutionTo start with, I thought if we take \int \frac{L}{2m}dt from 0 to T; we would...
  36. N

    Help Needed: Solving Kepler's Third Law for a Solar Elliptical Orbit

    Homework Statement I have this problem for homework and I don't know how to even start. Can someone help? Thanks in advance.A neat and exploitive use of the sun would be to put a space probe into a solar elliptical orbit from the Earth on one side of the sun headed towards a rendezvous with...
  37. E

    Troubleshooting Kepler's Third Law with Halley's Comet

    Homework Statement When I plug in all of the parameters for Halley's comet (from Wikipedia) into Kepler's third law a get a semimajor axis of 38.56 AU when it should be about 17? Can someone else try it and see if I am crazy?Homework Equations The Attempt at a Solution
  38. K

    Understanding Kepler's Third Law: Calculating Aphelion Distance of a Comet

    A comet has a perihelion distance of 0.29 A.U. and an orbital period of 77 years. To the nearest thousandth of an A.U., what is its aphelion distance from the Sun? I know that keplers third law plays a part in this p2=a3 but I just can't plug it in right. I know the answer is 35.908 but I...
  39. C

    A Problem Involving Kepler's Third Law

    Homework Statement A spacecraft on a mission to the outer planets passed through the asteroid belt and imaged an asteroid with a small moon. You are on a proposal team to define a mission to the asteroid’s moon to determine its composition and history and have been requested to estimate the...
  40. B

    Calculating G using Kepler's Third Law: The Relationship Between k, a, and P

    Homework Statement Using astronomical units as the unit of length , years as the time , and the mass of the Sun as the unit of mass , the value for k in kepler's third law is 1. In these units , what is the value of Newton's constant of gravitation G? Homework Equations P^2 =k*a^3...
  41. J

    Newton's Variation of Kepler's third law

    Calculate the period of the Earth's moon if the radius of orbit was twice the actual value of 3.9 x 10^8 m. a. 1.13 x 10^6 s or 13 days b. 2.3 x 10^6 s or 26 days c. 5.14 x 10^6 s or 59 days d. 6.85 x10^6 s or 79 days So far I got: v=velocity; T=time; h= hieght from surface v=? T=...
  42. J

    Question: Kepler's third law of planetary motion

    The mean distance between the Earth and the moon is 3.84x10^8 m, and the moon has an orbital period of 27.3 days. Find the distance from Earth of an artificial satellite that has an orbital period of 8.5 days. a. 1.76x10^8 m b. 1.76x10^4 m c. 1.76x10^10 m d. 5.24x10^8 m So far I got: I...
  43. D

    Kepler's Third Law: Eliptical Orbits & Planet Masses

    I don't know but in school I learned it in a different way. I mean in school the Kepler's third law was thought as if all the orbits were circular. But according to the definition its all elleptical, and in the school equation we won't add the planets mass, but here...
  44. M

    Calculate Period of Planet X with Kepler's Third Law

    Using Kepler's Third Law, give the formula for the period TX of planet X whose orbital radius RX is twice that of earth, RE. A. TX=(1 Year)x(RX/RE)^3 B. TX^2=(1 Year)x(RX/RE)^3 C. TX=(1 Year)x(RE/RX)^3 D. TX^2=(1 Year)x(RE/RX)^3 My book is not giving me a formula to use, so i a totally...
  45. E

    C 12A 2009 - Deriving Kepler's Third Law

    Two gravitaionally bound stars with equal masses m, separated by a distance d, revolve about their cneter of mass in circular orbites. Show that the period is proportional to d^3/2 and find the proportionality constant. I know that in this case, F = mv^2/r and that F=Gm^2/d^2. But where...
  46. E

    Kepler's Third Law: Proportional Orbit Period to Distance^1.5

    Two gravitionally bound stars with equal masses m, separated by a distance d, revolve about their center of mass in circular orbits. Show that the period tau is proportional to d^1.5 Could someone get me started on this? I have no idea where to begin! Thanks!
  47. B

    Comparing Periods of Satellites in Orbit Around a Planet

    not sure what to do here.. Im being asked to compare the periods of 2 different satellites in orbit around a planet. the first one is a circular orbit of radius = r the second one orbits 1r to the left and 3r to the right around the planet. I'll attempt to draw it here :tongue...
  48. Reshma

    Inverse square law and Kepler's third law

    The inverse square law for gravitation was deduced from Kepler's third law. So how was the inverse square law for electrostatics(Coulomb's law) deduced?
  49. P

    PLEASE help Kepler's third law for electrical orbits

    PLEASE help! Kepler's third law for electrical orbits Hi there! I hope someone can help me with this problem. I've been working on this for over 5 hours and I've gotten nowhere! :eek: A positron is a particle with the same mass as an electron but with a positive charge. A positron and an...
  50. M

    Calculating the Mass of the Galactic Center with Kepler's Third Law

    I'm not good at this stuff, so I need some help from some smarter people...:smile: The speed of the Sun in its orbit about the Galactic center is 220 km/sec. Its distance from the center is 8500 pc. Assuming that the Sun's orbit is circular, calculate the mass of the Galaxy (in solar...
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