Telescope and Stopwatch for the Mass of a Planet

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SUMMARY

This discussion explores the feasibility of calculating the mass of Saturn using only a stopwatch and a telescope. The primary method suggested involves applying Kepler's 3rd law, utilizing the orbital period of Saturn's moon Titan, which has a known period of 15.9 days and a mean distance of 1,222,000 km from Saturn. The formula M = 4π^2 r^3/GT^2 is recommended for calculating Saturn's mass, where M is the mass of Saturn, r is the estimated orbital distance, and T is the orbital period. The diameter of Saturn (120,000 km) serves as a reference for estimating distances.

PREREQUISITES
  • Understanding of Kepler's 3rd law of planetary motion
  • Familiarity with the formula M = 4π^2 r^3/GT^2
  • Basic knowledge of orbital mechanics
  • Ability to use a telescope for astronomical observations
NEXT STEPS
  • Research the application of Kepler's laws in celestial mechanics
  • Learn how to accurately measure orbital periods using a stopwatch
  • Explore methods for estimating distances in astronomy using known celestial dimensions
  • Investigate the properties and characteristics of Saturn's moons, particularly Titan
USEFUL FOR

Astronomy enthusiasts, amateur astronomers, and students studying celestial mechanics who wish to understand practical methods for calculating planetary masses using observational techniques.

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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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.
 
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.
 
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.)
 
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.
 
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.
 
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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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.
 

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