Explain Kepler's 3rd Law of Planetary Motion

[AstrophysicsX]

Can anyone help explain Kepler's 3rd Law of Planetary Motion to me?

 

[Fewmet]

Can anyone help explain Kepler's 3rd Law of Planetary Motion to me?

I understand the basics of it.

First, the orbital period T is the time for a body to complete one orbit. The orbit has a radius of R.

Kepler discovered that the planets (at least the ones he knew about) follow a pattern: if you divide the square of the period by the cube of the radius, you always get the same number:

R³/T²=K​

The value of K depends on units, course. The easiest calculation is for the Earth using T = 1 year and R = 1 Astronomical Unit:

(1 au)³/(1 yr)²=1 au³/yr²​

That means the the cube of the radius (in au) of anything orbiting the sun divided by the square of its period (in years) will equal
1 au³/yr².

The law still holds for bodies orbiting a different body (like the moons of Jupiter), but the value of K is different.

You can derive the law from the Law of Gravity. In fact I've read that the derivation was one of the persuasive arguments in favor of the Law of Gravity when Newton first proposed it. The derivation makes clear that the mass of the body being orbited is hidden in the constant.

 

[Creator]

Can anyone help explain Kepler's 3rd Law of Planetary Motion to me?

As 'Fewmet" explained ... FOR ANY planet in the solar system, its ratio of orbital period squared to radius cubed is a constant.

T^2 / R^3 = K

...but he didn't tell you what that constant is.

It turns out that the constant is always 4(pi)^2 / GM...where G is the gravitational constant and M is the mass of the sun. (using standard units..kgs system)
And BTW, it matters not if the orbit is highly elliptical; simply substitute the 'semi-major" axis for radius and the formula is still valid.

Similarly ALL satellites in orbit about the Earth will have the same ratio T^2 / R^3 using M as the Earth's mass.

Knowing the constant in terms of its central mass allows you to use the formula for finding orbits about any central mass...or for that matter you can find the mass of anybody by measuring the orbital period and semi-major axis of any of its satellites...which is exactly how Kepler "measured" the (approx.) mass of the sun in the early 1600's !

...

 

[Fewmet]

which is exactly how Kepler "measured" the (approx.) mass of the sun in the early 1600's !

It looks like you are saying the Kepler used a result from Newton's Law of Gravity to find the mass of the Sun, but Kepler died before Newton's birth. If I am misreading you and Kepler did do a rough calculation I'd really be interested in seeing details. Can you provide any?

Thanks.

 

[Creator]

It looks like you are saying the Kepler used a result from Newton's Law of Gravity to find the mass of the Sun, but Kepler died before Newton's birth. If I am misreading you and Kepler did do a rough calculation I'd really be interested in seeing details. Can you provide any?

Thanks.

Opps , my bad; it should say: "that is how "Newton" determined the mass of sun in 1600's".

Sorry.

 

[Fewmet]

Opps , my bad; it should say: "that is how "Newton" determined the mass of sun in 1600's".

Sorry.

I've had far fewer posts that you and made worse misstatements.

Too bad, really: I was hoping you had something clever Kepler did that I hadn't encountered.

 

[nasu]

Opps , my bad; it should say: "that is how "Newton" determined the mass of sun in 1600's".

Sorry.

I won't be so sure about this either. The constant G was measured in 1797, more than 50 years after Newton died.

 

[starfish99]

Kepler could have known the approximate distance from the Earth to the Sun( the ancient Greeks may have known this). However, I think he had no way of knowing the distance of the Sun to the other five known planets. So how could he have established his Law with only one point(Earth-Sun distance)?

 

[Fewmet]

Kepler could have known the approximate distance from the Earth to the Sun( the ancient Greeks may have known this). However, I think he had no way of knowing the distance of the Sun to the other five known planets. So how could he have established his Law with only one point(Earth-Sun distance)?

The Copernican system let's you calculate the orbital radii. I recall that from my eighth grade report on Copernicucs, but looked it up to be sure. Here is a good http://astro.unl.edu/naap/ssm/ssm_advanced.html" .

 

[nasu]

These are relative values of the distances. And is all that Kepler needed to deduce his 3rd law.
Absolute values of the distances were determined in late 18th century by observing the transits of Venus.
Some crude estimate was done during some 17th century transits (by Jeremiah Horrocks) but they were published much later, after Kepler's death.

 

[Fewmet]

These are relative values of the distances. And is all that Kepler needed to deduce his 3rd law.
Absolute values of the distances were determined in late 18th century by observing the transits of Venus.
Some crude estimate was done during some 17th century transits (by Jeremiah Horrocks) but they were published much later, after Kepler's death.

I wonder: would Kepler have had access to the ancient Greek estimate of the Earth-Sun distance? I recall that they were reasonable accurate on the Earth-Moon distance, and the diameters of the Earth and Moon, but less on-target with the Earth-Sun distance.

 

[AstrophysicsX]

And what is that equation describing?

 

[Fewmet]

And what is that equation describing?

Do you mean R³/T²=K? It is describing a relationship between the period and radius of orbits: the cube of the radius divided by the square of the period always equals the same number for a given mass (like the sun) around which other bodies (like the planets) orbit.

That's not very different from what was said in previous posts... I feel like I might be misunderstanding you. Does that answer your question? If not, can you say more about what you want to know?

 

[nasu]

I wonder: would Kepler have had access to the ancient Greek estimate of the Earth-Sun distance? I recall that they were reasonable accurate on the Earth-Moon distance, and the diameters of the Earth and Moon, but less on-target with the Earth-Sun distance.

The estimate of the Earth-Sun distance according to Aristarchus was about 20 times smaller than the actual value.

 

[Disinterred]

As 'Fewmet" explained ... FOR ANY planet in the solar system, its ratio of orbital period squared to radius cubed is a constant.

T^2 / R^3 = K

...but he didn't tell you what that constant is.

It turns out that the constant is always 4(pi)^2 / GM...where G is the gravitational constant and M is the mass of the sun. (using standard units..kgs system)
And BTW, it matters not if the orbit is highly elliptical; simply substitute the 'semi-major" axis for radius and the formula is still valid.

Similarly ALL satellites in orbit about the Earth will have the same ratio T^2 / R^3 using M as the Earth's mass.

Knowing the constant in terms of its central mass allows you to use the formula for finding orbits about any central mass...or for that matter you can find the mass of anybody by measuring the orbital period and semi-major axis of any of its satellites...which is exactly how Kepler "measured" the (approx.) mass of the sun in the early 1600's !

...

Well if we are getting technical about this, the Earth actually revolves around the center of mass of the Sun-Earth system in an elliptical orbit. If I remember correctly, the usual derivations don't include this fact and thus somewhere along the way you might need to make some correction.