What is the Correct Formula for Calculating the Velocity of Saturn's Rings?

In summary, the equation for the speed of rotation of the rings around Saturn is related to their distance from the center of Saturn according to the formula ##V=\frac{24.9}{\sqrt{R}}##, where R is the distance in multiples of the radius of Saturn (with R = 1 corresponding to a distance of 60.268 km). The numerical constant 24.9 represents the orbital velocity at Saturn's surface, and there is a typo in the original formula of 29.4 which should actually be 24.9. It is recommended to use the corrected formula for accurate results.
  • #1
nmsurobert
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I'm setting up a lab for my class and I've found this equation, but I can't find where a constant value is coming from.

"...the speed of rotation of the rings is related to their radius (from the center of Saturn) by the following equation:

formulas_4.jpg


where R is the distance from the center of Saturn to the ring in multiples of the radius of Saturn (R = 1 corresponds to a distance of 60.268 km)."

The only relation I can find between 29.4 and Saturn, is that it take 29.4 years for Saturn to complete an orbit. I don't think that has anything to do with this equation though.

Can anyone here explain to me where 29.4 is coming from?
 

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  • #2
Hmmm. I think it has something to do with centripetal acceleration, but I'm not certain.
 
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Since each ring particle follows a Keplerian orbit, this should be just the formula for circular orbital velocity, with constants folded into one. I.e. for a small orbiting particle:
##V_o=\sqrt{\frac{GM}{R}}##
The numerical constant is then just the orbital speed at the planet's surface (let us know if you can't see how to get there).
However, there's something wonky in the state of Denmark. If you calculate it for Saturn's mass and 1 orbital radius, you should get approx. 25 km/s, not 29.4.
The only two sources for the formula I could find is some amateur astronomer's blog that only mentions 'bibliography', and the SpaceMath website: https://spacemath.gsfc.nasa.gov/weekly/10Page28.pdf
I went over the solutions to the problems they gave, and the numbers didn't match for P1 and 2. Why that is became clear with problem 3, where it turns out the 29.4 is just a typo. The correct value (which they were actually using) is 24.9 - and that matches what you get from the orbital velocity formula.

So, yeah. Don't use it. Use:
##V=\frac{24.9}{\sqrt{R}}##
Again, 24.9 km/s is the orbital velocity in circular orbit at Saturn's surface (aka first cosmic velocity).
 
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  • #4
Bandersnatch said:
Since each ring particle follows a Keplerian orbit, this should be just the formula for circular orbital velocity, with constants folded into one. I.e. for a small orbiting particle:
##V_o=\sqrt{\frac{GM}{R}}##
The numerical constant is then just the orbital speed at the planet's surface (let us know if you can't see how to get there).
However, there's something wonky in the state of Denmark. If you calculate it for Saturn's mass and 1 orbital radius, you should get approx. 25 km/s, not 29.4.
The only two sources for the formula I could find is some amateur astronomer's blog that only mentions 'bibliography', and the SpaceMath website: https://spacemath.gsfc.nasa.gov/weekly/10Page28.pdf
I went over the solutions to the problems they gave, and the numbers didn't match for P1 and 2. Why that is became clear with problem 3, where it turns out the 29.4 is just a typo. The correct value (which they were actually using) is 24.9 - and that matches what you get from the orbital velocity formula.

So, yeah. Don't use it. Use:
##V=\frac{24.9}{\sqrt{R}}##
Again, 24.9 km/s is the orbital velocity in circular orbit at Saturn's surface (aka first cosmic velocity).

Awesome! Thank you. After reading your explanation I couldn't figure out where the 25 was coming from. I kept getting 25000... meters. Duh.

Thanks again!
 

FAQ: What is the Correct Formula for Calculating the Velocity of Saturn's Rings?

1. What is the average velocity of Saturn's rings?

The average velocity of Saturn's rings is approximately 14,500 miles per hour, which is equivalent to 7 kilometers per second.

2. Does the velocity of Saturn's rings change over time?

Yes, the velocity of Saturn's rings can vary due to a phenomenon known as orbital resonance, where the gravitational pull of nearby moons affects the orbital speed of the particles in the rings.

3. How does the velocity of Saturn's rings compare to other planets?

The velocity of Saturn's rings is relatively slow compared to other planets, as the rings are composed of small particles that are not held together by strong gravitational forces like the solid surfaces of planets.

4. Can the velocity of Saturn's rings be measured from Earth?

Yes, the velocity of Saturn's rings can be measured from Earth using techniques such as spectroscopy and radar imaging. These methods allow scientists to analyze the motion of the ring particles and calculate their velocity.

5. Is the velocity of Saturn's rings the same at all points on the rings?

No, the velocity of Saturn's rings can vary at different points on the rings due to the varying gravitational forces from nearby moons and the shape of the rings themselves. However, the average velocity of the rings is consistent throughout.

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