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Kepler’s Third Law

The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun. In equation form, this is

T 1  2 T 2  2 = r 1  3 r 2  3 , size 12{ { {T rSub { size 8{1} } rSup { size 8{2} } } over {T rSub { size 8{2} } rSup { size 8{2} } } } = { {r rSub { size 8{1} } rSup { size 8{3} } } over {r rSub { size 8{2} } rSup { size 8{3} } } } } {}

where T size 12{T} {} is the period (time for one orbit) and r size 12{r} {} is the average radius. This equation is valid only for comparing two small masses orbiting the same large one. Most importantly, this is a descriptive equation only, giving no information as to the cause of the equality.

In the figure, the elliptical path of a planet is shown. The Sun is at the left focus. Three shaded regions M A B, M C D and M E F are marked on the figure by joining the Sun to the three pairs of points A B, C D, and E F on the elliptical path. The velocity of the planet is shown on the planet in a direction tangential to the path.
The shaded regions have equal areas. It takes equal times for m size 12{m} {} to go from A to B, from C to D, and from E to F. The mass m size 12{m} {} moves fastest when it is closest to M size 12{M} {} . Kepler’s second law was originally devised for planets orbiting the Sun, but it has broader validity.

Note again that while, for historical reasons, Kepler’s laws are stated for planets orbiting the Sun, they are actually valid for all bodies satisfying the two previously stated conditions.

Find the time for one orbit of an earth satellite

Given that the Moon orbits Earth each 27.3 d and that it is an average distance of 3.84 × 10 8 m size 12{3 "." "84" times "10" rSup { size 8{8} } m} {} from the center of Earth, calculate the period of an artificial satellite orbiting at an average altitude of 1500 km above Earth’s surface.


The period, or time for one orbit, is related to the radius of the orbit by Kepler’s third law, given in mathematical form in T 1  2 T 2  2 = r 1  3 r 2  3 size 12{ { {T rSub { size 8{1} } rSup { size 8{2} } } over {T rSub { size 8{2} } rSup { size 8{2} } } } = { {r rSub { size 8{1} } rSup { size 8{3} } } over {r rSub { size 8{2} } rSup { size 8{3} } } } } {} . Let us use the subscript 1 for the Moon and the subscript 2 for the satellite. We are asked to find T 2 size 12{T rSub { size 8{2} } } {} . The given information tells us that the orbital radius of the Moon is r 1 = 3 . 84 × 10 8 m size 12{r rSub { size 8{1} } =3 "." "84" times "10" rSup { size 8{8} } m} {} , and that the period of the Moon is T 1 = 27.3 d size 12{T rSub { size 8{1} } ="27" "." 3} {} . The height of the artificial satellite above Earth’s surface is given, and so we must add the radius of Earth (6380 km) to get r 2 = ( 1500 + 6380 ) km = 7880 km size 12{r rSub { size 8{2} } = \( "1500"+"6380" \) "km"="7880"`"km"} {} . Now all quantities are known, and so T 2 size 12{T rSub { size 8{2} } } {} can be found.


Kepler’s third law is

T 1  2 T 2  2 = r 1  3 r 2  3 . size 12{ { {T rSub { size 8{1} } rSup { size 8{2} } } over {T rSub { size 8{2} } rSup { size 8{2} } } } = { {r rSub { size 8{1} } rSup { size 8{3} } } over {r rSub { size 8{2} } rSup { size 8{3} } } } } {}

To solve for T 2 size 12{T rSub { size 8{2} } } {} , we cross-multiply and take the square root, yielding

T 2  2 = T 1  2 r 2 r 1 3 size 12{T rSub { size 8{2} } =T rSub { size 8{1} } times left ( { {r rSub { size 8{2} } } over {r rSub { size 8{1} } } } right ) rSup { size 8{3/2} } } {}
T 2 = T 1 r 2 r 1 3 / 2 . size 12{T rSub { size 8{2} } =T rSub { size 8{1} } times left ( { {r rSub { size 8{2} } } over {r rSub { size 8{1} } } } right ) rSup { size 8{3/2} } } {}

Substituting known values yields

T 2 = 27.3 d × 24.0 h d × 7880 km 3.84 × 10 5 km 3 / 2 = 1.93 h.

Discussion This is a reasonable period for a satellite in a fairly low orbit. It is interesting that any satellite at this altitude will orbit in the same amount of time. This fact is related to the condition that the satellite’s mass is small compared with that of Earth.

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People immediately search for deeper meaning when broadly applicable laws, like Kepler’s, are discovered. It was Newton who took the next giant step when he proposed the law of universal gravitation. While Kepler was able to discover what was happening, Newton discovered that gravitational force was the cause.

Derivation of kepler’s third law for circular orbits

We shall derive Kepler’s third law, starting with Newton’s laws of motion and his universal law of gravitation. The point is to demonstrate that the force of gravity is the cause for Kepler’s laws (although we will only derive the third one).

Let us consider a circular orbit of a small mass m size 12{m} {} around a large mass M size 12{m} {} , satisfying the two conditions stated at the beginning of this section. Gravity supplies the centripetal force to mass m size 12{m} {} . Starting with Newton’s second law applied to circular motion,

Questions & Answers

How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?
Larissa Reply
How do you convert 0.0045kgcm³ to the si unit?
how many state of matter do we really have like I mean... is there any newly discovered state of matter?
Falana Reply
I only know 5: •Solids •Liquids •Gases •Plasma •Bose-Einstein condensate
Alright Thank you
Which one is the Bose-Einstein
can you explain what plasma and the I her one you mentioned
u can say sun or stars are just the state of plasma
but the are more than seven
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Akhigbe Reply
What state of matter is fire
Thapelo Reply
fire is not in any state of matter...fire is rather a form of energy produced from an oxidising reaction.
Isn`t fire the plasma state of matter?
all this while I taught it was plasma
How can you define time?
Thapelo Reply
Time can be defined as a continuous , dynamic , irreversible , unpredictable quantity .
unpredictable? but I can say after one o'clock its going to be two o'clock predictably!
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Paul Reply
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Anthony Reply
V=u+at V²=u²-2as
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v=u+at s=ut+at^\2 v^=u^+2as where ^=2
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Jennifer Reply
Not yet learnt
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Alabi Reply
Scalar quantity Because acceleration has only magnitude
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its a scalar quantity
velocity is speed and direction. since velocity is a part of acceleration that makes acceleration a vector quantity. an example of this is centripetal acceleration. when you're moving in a circular patter at a constant speed, you are still accelerating because your direction is constantly changing.
acceleration is a vector quantity. As explained by Josh Thompson, even in circular motion, bodies undergoing circular motion only accelerate because on the constantly changing direction of their constant speed. also retardation and acceleration are differentiated by virtue of their direction in
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Nworu Reply
sound wave is a mechanical longitudinal wave that transfers energy from one point to another
its a longitudnal wave which is associted wth compresion nad rearfractions
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Cheyanne Reply
strengthen and whiten teeth.
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Nansing Reply
length of pendulm?

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