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

[Abu]

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 impossible then let me know, I am taking a chance because this may be a very dumb question haha.

Homework Equations

T^2/R^3 = T^2/R^3
v = 2πr/τ
d = vt
v = √GM/R
M = 4π^2 r^3/GT^2

3. The Attempt at a Solution
At first, I thought that:
If it is possible to find the orbital period of Saturn around the sun using a telescope and stopwatch along with an extreme amount of patience, then you could simply use Kepler's 3rd law with Earth's known values to find the range between the sun and Saturn. Then you could use this range in the formula M = 4π^2 r^3/GT^2, but then I realized that M would be the mass of the Sun and not of Saturn.

So then I thought if it is possible to apply the M = 4π^2 r^3/GT^2 formula between Saturn and one of its many moons. Then, M would be the mass of Saturn, r would be the distance between the moon and Saturn, and T would be how long it takes for a moon to revolve around Saturn, using the stop watch and patience.

The only problem is R, which you need the speed of light for. Is it possible to somehow calculate the speed of light with only what was given, and then apply it to find the range, and if so, how? If it isn't possible, then I assume my professor would just simply give the speed of light?

 

[haruspex]

The only problem is R, which you need the speed of light for.

Please explain your thinking there.

 

[Abu]

Please explain your thinking there.

I thought that you need the speed of light first so that you can use the formula:
v = d*t where v is the speed of light, d is the distance between the moon and Saturn, and t is the time it takes for the speed of light to cover that distance. Then you would use d in the formula mentioned.

As I am typing this out though, that hardly makes any sense because even if I did know the speed of light, I wouldn't know the time it takes for the speed of light to cover that distance. Does that mean it is impossible with what is given? It seems to me like either way there will be difficulties.

 

[Abu]

Please explain your thinking there.

I just realized now that keplers laws could also apply to the moons revolving around saturn. Do you think i could use that, in theory? But then range would be a problem again, right. Or is there something else I am missing.

 

[haruspex]

is there something else I am missing.

You can see the moons revolving around Saturn. What parameter do you need to figure out from that how far apart they are?

 

[Abu]

You can see the moons revolving around Saturn. What parameter do you need to figure out from that how far apart they are?

The only other parameter i could think of that would help in solving the range of the moon from Saturn is the radius of Saturn, because the total range is how far the moon is from the surface plus the radius. Sorry if it is super obvious, I am just confused.

 

[Gigaz]

It's difficult obtain the mass with just a stopwatch and a telescope.
In Newtons time, people could predict planetary motions but they didnt really know the distance between the Earth and the Sun. And they also didnt know the masses of these objects. To obtain it, you need a local measurement of the gravitational constant G and the surface gravity g of the Earth. Then the other stuff follows.
Without knowledge of Saturns "surface" gravity or the distance to Saturn you can't get it's mass. A speed of light approach may work, assuming you already know the speed of light and the distance from the Earth to the Sun. But then, if you know the latter you can already pretty much calculate the distance to Saturn and the distance between Saturn and its moons from astronomical observations alone.

 

[Abu]

It's difficult obtain the mass with just a stopwatch and a telescope.
In Newtons time, people could predict planetary motions but they didnt really know the distance between the Earth and the Sun. And they also didnt know the masses of these objects. To obtain it, you need a local measurement of the gravitational constant G and the surface gravity g of the Earth. Then the other stuff follows.
Without knowledge of Saturns "surface" gravity or the distance to Saturn you can't get it's mass. A speed of light approach may work, assuming you already know the speed of light and the distance from the Earth to the Sun. But then, if you know the latter you can already pretty much calculate the distance to Saturn and the distance between Saturn and its moons from astronomical observations alone.

Alright, so if I were to use the speed of light approach, how would I go about it? I know that you need to use the distance equals speed multiplied by time formula for that but time would be an issue. Would you somehow need to reflect something to see how fast the light travels from Saturn, as ridiculous as that sounds, to see the distance between Earth and Saturn? I feel like I am on the wrong track... by astronomical observations you mean using the distance between the Sun and Saturn as a reference to base an estimate for the range of the moon to Saturn?

 

[Gigaz]

What I think you should do is this: Assume you know the mass of the Sun. Then you also know the distance between the Sun and, let's say, Mars, from it's orbital period. From that point you should be able to derive the distance between Saturn and one of its moons by comparing it to the distance between Saturn and Mars. (The geometry may be a bit complicated but it's clearly doable)

 

[haruspex]

The only other parameter i could think of that would help in solving the range of the moon from Saturn is the radius of Saturn, because the total range is how far the moon is from the surface plus the radius. Sorry if it is super obvious, I am just confused.

If you see a vertical pole in the distance, what do you need to know to figure out how tall it is?