Telescope and Stopwatch for the Mass of a Planet

In summary, the conversation discusses the possibility of using a stopwatch and a telescope to find the mass of a planet, specifically Saturn. The initial thought was to use Kepler's 3rd law with Earth's known values to find the range between the sun and Saturn, but it was later realized that this would only give the mass of the sun. Another suggestion was to apply the formula between Saturn and one of its moons, but this would require the knowledge of the speed of light. However, it was mentioned that the mass of Saturn listed in literature was most likely computed by observing its moons and estimating the orbital distance of the moon. It was also suggested to use the diameter of Saturn as a reference for estimating the distance of one of its moons.
  • #1
Abu
I was wondering if it is possible to use only a stopwatch and a telescope to find the mass of a planet, such as Saturn. I've experimented with a couple of things but I keep running into problems. I previously asked this question in the homework section, but it does not involve numbers, is not really homework, and I did not receive any follow-up responses on it in roughly 3 hours, so I felt like maybe my question does not belong there but here instead. I truly apologize if any moderators insist that this question remains in the homework forum, I just thought it did not fit in that forum. Anyways, here is what I tried to come up with:

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? Because even if you knew the speed of light, the formula states v = d*t and you would not be able to solve for d because the time it takes for the speed of light to cover d is not known.

I also tried applying Kepler 3rd law to the moons, but then I would still need the range.

Any ideas? Thanks.
 
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  • #2
The mass of Saturn that is listed in the literature was no doubt computed by observing the moons. It is necessary to have an estimate for the orbital distance of the moon. The speed of light does not enter into this calculation since the moon stays roughly the same distance from Earth at all times.
 
  • #3
Charles Link said:
The mass of Saturn that is listed in the literature was no doubt computed by observing the moons. It is necessary to have an estimate for the orbital distance of the moon. The speed of light does not enter into this calculation since the moon stays roughly the same distance from Earth at all times.
Oh alright, so it seems my second attempt was on the right track, minus the speed of light thought. Do you think that with only a telescope and a stopwatch, the orbital distance of the moon could be calculated? I've been trying to wrap my head around this all day and that is what I have been stumped on.
 
  • #4
Abu said:
Oh alright, so it seems my second attempt was on the right track, minus the speed of light thought. Do you think that with only a telescope and a stopwatch, the orbital distance of the moon could be calculated? I've been trying to wrap my head around this all day and that is what I have been stumped on.
With a telescope, it really isn't cheating to estimate distances by using the known diameter of Saturn as your measuring tool. You can estimate the radius of the orbit of the moon. You would need to track it periodically to see its furthest distance from the planet. For semi-accurate measurements, it would help to get a good photo from your telescope. (I'm assuming you have sufficient optical power that Saturn appears as a sphere/circle with rings.)
 
  • #5
Charles Link said:
With a telescope, it really isn't cheating to estimate distances by using the known diameter of Saturn as your measuring tool. You can estimate the radius of the orbit of the moon. You would need to track it periodically to see its furthest distance from the planet. For semi-accurate measurements, it would help to get a good photo from your telescope. (I'm assuming you have sufficient optical power that Saturn appears as a sphere/circle with rings.)
Oh, so you are saying use the diameter of Saturn as a reference to base an estimate on the distance of one of Saturn's moon to Saturn? Assuming that I could see the moon with the telescope or is there a formula I am missing that I use the diameter of Saturn in? Sorry for the late responses.
 
  • #6
Abu said:
Oh, so you are saying use the diameter of Saturn as a reference to base an estimate on the distance of one of Saturn's moon to Saturn? Assuming that I could see the moon with the telescope or is there a formula I am missing that I use the diameter of Saturn in? Sorry for the late responses.
I think Saturn has at least one moon Titan that should be visible with most telescopes. I looked it up. Titan has a mean distance of 1,222,000 km from Saturn, and the diameter of the sphere of Saturn is 120,000 km. That means you'll find Titan at a maximum of about 10 Saturn diameters away from the center of Saturn. Titan's orbit has a period of 15.9 days. The mass of Saturn is ## M=5.69 \, E +26 \, kg ##. You could do a couple calculations with this info. It would take some patient observation with good weather to verify that Titan has a period of 15.9 days.
 
  • #7
Charles Link said:
I think Saturn has at least one moon Titan that should be visible with most telescopes. I looked it up. Titan has a mean distance of 1,222,000 km from Saturn, and the diameter of the sphere of Saturn is 120,000 km. That means you'll find Titan at a maximum of about 10 Saturn diameters away from the center of Saturn. Titan's orbit has a period of 15.9 days. The mass of Saturn is ## M=5.69 \, E +26 \, kg ##. You could do a couple calculations with this info. It would take some patient observation with good weather to verify that Titan has a period of 15.9 days.
Ok so let me see if I got this. Using the known value of the diameter of saturn, I use it as a measuring tool to estimate the orbital distance from the center of saturn. With this range, and after finding the period of that moon with stop watch like you said, i simply use the formula M = 4π^2 r^3/GT^2 where M is the mass of Saturn, r is the estimated range, and T is the period.
 
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  • #8
I believe I have seen the moon Titan a number of years ago in a 50x Meade telescope. Saturn is large enough to see the rings and Titan was a bright dot nearby. Meanwhile, I computed ## M_{saturn} ## with the numbers I gave you, and it is consistent with the formula.
 

1. What is a telescope?

A telescope is an optical instrument that is used to view distant objects, usually in the night sky. It works by collecting and focusing light from the object and magnifying it so that it can be seen more clearly.

2. How does a telescope help in determining the mass of a planet?

A telescope can help in determining the mass of a planet by allowing scientists to observe the planet's movement and calculate its orbit. By measuring the planet's orbit and the force of gravity acting on it, the mass of the planet can be calculated using Newton's law of gravitation.

3. What is a stopwatch and how is it used in determining the mass of a planet?

A stopwatch is a device used to measure time with precision. In determining the mass of a planet, a stopwatch is used to measure the time it takes for the planet to complete one full orbit around its star. This time, combined with the distance between the planet and its star, can be used to calculate the planet's mass using Kepler's third law of planetary motion.

4. Can a telescope and stopwatch be used to determine the mass of any planet?

Yes, a telescope and stopwatch can be used to determine the mass of any planet, as long as the planet's orbit is observable and the distance between the planet and its star is known.

5. What are the limitations of using a telescope and stopwatch to determine the mass of a planet?

One limitation is that the planet's orbit must be observable, which means that it must be within a certain distance from Earth. Another limitation is that the planet's orbit must be stable and regular, meaning that it is not affected by other large objects in its vicinity. Additionally, the accuracy of the measurement can be affected by factors such as atmospheric conditions and human error in timing the orbit.

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