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.
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.
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...
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...
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...
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...
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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...
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...
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...
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...
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?
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...
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 ±...
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...
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...