Calculating Altitude for a Satellite in Synchronous Orbit with Saturn

In summary, the satellite in synchronous orbit with the equator of Saturn has an altitude of 5.28E4 km due to its lower density than Earth. This answer is correct, but the student may be thinking incorrectly about the rotational period.
  • #1
synchronous
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Homework Statement


Satellite in synchronous orbit with equator of Saturn. Find altitude in km.

rotational period = 38826 s
radius of Saturn = 6.03E7 m
mass of Saturn = 5.68E26 kg
Gravity constant = 6.673E-11

Homework Equations



T^2 = (4pi^2)(r^3)/(Gm)

The Attempt at a Solution



I plugged and chugged and arrived at an answer of 113114516 m. Then I subtracted the planet's radius from this answer to find the altitude of 52797562 m. I then concluded that the synchronous orbit had to have an altitude of 5.28E4 km. I also solved the problem using F(g) = F(c) and arrived at the same solution. What am I doing wrong? Are my constants incorrect?
 
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  • #2
There is an interesting version of Keplers law (the equation you have) based on the density of the body.
this is possible since you have r^3 and the mass in the equation anyway.
One interesting result of this is that the orbital period at the surface of a body only depends on the density - for water it's a period of 3.3hours.

What's also interesting about the density of Saturn?
 
  • #3
synchronous said:
I plugged and chugged and arrived at an answer of 113114516 m. Then I subtracted the planet's radius from this answer to find the altitude of 52797562 m. I also solved the problem using F(g) = F(c) and arrived at the same solution. What am I doing wrong? Are my constants incorrect?
The only things you are doing wrong are (a) expressing your answer with too much precision, and (b) writing G without units (G is not just a number. Change your units and you get a different numerical value for the speed of light -- and for G.)

That said, your answers are correct. What makes you think you they are wrong?
 
  • #4
Vol of sphere = 4/3[tex]\pi[/tex]r[tex]^{3}[/tex]

so density of sphere = vol(mass)

Orbital period at surface? Isn't that just the rotational period?
 
  • #5
My "incorrect" answer was actually correct. I switched the rotational period to 10.23 hrs and came up with an answer that the online homework accepted. I certainly wasn't using significant digits...thanks for the reminder! BTW, I just left out the units for G to ease the typing requirements...lazy on my part, my bad!

The density of Saturn should be less than water, right? It's a gaseous planet...
 
  • #6
Sorry - I thought that for Saturn the geostationary orbit would be below the surface.
I had used GM for Earth (400,000km - a useful figure to know) and got an orbital radius of 24,000km - which is below the surface.

That was the reason for the point about density and water, I assumed that if the rotation speed at the surface of water is 3.3hr and Saturn is less dense than water it would be less, and so less than the 10hrs needed. I had the reasoning the wrong way around - sorry.
 

What is a synchronous orbit with Saturn?

A synchronous orbit with Saturn is a type of orbit where a satellite or spacecraft orbits around Saturn at the same speed as the planet rotates. This allows the satellite to maintain a fixed position above a specific location on Saturn's surface.

What is the distance between Saturn and its synchronous orbit?

The distance between Saturn and its synchronous orbit varies depending on the specific orbit and the size of the satellite. However, on average, the distance is approximately 238,000 kilometers.

How is a synchronous orbit with Saturn achieved?

A synchronous orbit with Saturn is achieved by carefully calculating the satellite's trajectory and velocity so that it matches the rotational speed of Saturn. This requires precise calculations and adjustments to ensure the satellite remains in the desired orbit.

What are the benefits of a synchronous orbit with Saturn?

One of the main benefits of a synchronous orbit with Saturn is that it allows for continuous observation of a specific location on the planet's surface. This is particularly useful for studying Saturn's weather patterns, moons, and other features.

What are the challenges of maintaining a synchronous orbit with Saturn?

Maintaining a synchronous orbit with Saturn can be challenging due to the planet's strong gravitational pull and the potential for disruptions from other objects, such as Saturn's rings or other moons. This requires careful monitoring and adjustments to keep the satellite in the correct orbit.

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