Help with Kepler's laws and satellite motion.

[Tryan]

Homework Statement

mass of the Earth = 5.97 * 10^24 kg
Polar Radius of Earth = 6.36 * 10^6 m
Satellite = 1.08 * 10^3 Kg
Altitude = 2.02 * 10^7 m

3) for any object orbting around a primary body R^3 ∝ T^2
where R is the radius of the orbit and T is the time period for the orbit.

show that this is true and in doing so:
- state the conditions required for a stable orbit
- show that the conditions do not depend on the mass of the orbiting object.

4) discuss the particular requirements for an orbit that will keep the a satellite vertically above a certain point on Earth's surface.

Homework Equations

1) calculate the net force on a 1.08 * 10^3 Kg Satellite when it is in a polar orbit 2.02 * 10^7 m above the Earth's orbit...

So I think I get this one - as F=GMm/r^2 which gives the net force of (F= 610 N) 3 sig.fig

2)show that the only stable orbit for the satellite orbiting at an altitude of 2.02 * 10^7 m has a period of appoximatly 12 hours.

This next one I think it's correctly done so I've said working out velocity of the satellite I've come up with F(centripetal)=F(gravitational) so mv^2/r=FMm/r^2 so mv^2/r= the non-rounded answer in question 1 which is 609.6320 so 609.6230/1.08 * 10^3= velocity^2 so velocity = 3872.006122. Now with that substituting it into the formula of v = d/t you can find that the T - time period = 12 hours or 11.97 hours.

The Attempt at a Solution

Now these two questions 3) and 4) I really don't understand. I just don't know where to even start. I know however that it has something to do with Kepler's Laws of motion.? What I do think for question 4) is the velocity must be greater than the vertical acceleration... but I am still unsure..

 

[ehild]

3) for any object orbting around a primary body R^3 ∝ T^2
where R is the radius of the orbit and T is the time period for the orbit.

show that this is true and in doing so:
- state the conditions required for a stable orbit
- show that the conditions do not depend on the mass of the orbiting object.

4) discuss the particular requirements for an orbit that will keep the a satellite vertically above a certain point on Earth's surface.

Homework Equations

1) calculate the net force on a 1.08 * 10^3 Kg Satellite when it is in a polar orbit 2.02 * 10^7 m above the Earth's orbit...

So I think I get this one - as F=GMm/r^2 ...

2)show that the only stable orbit for the satellite orbiting at an altitude of 2.02 * 10^7 m has a period of appoximatly 12 hours.

This next one I think it's correctly done so I've said working out velocity of the satellite I've come up with F(centripetal)=F(gravitational) so mv^2/r=FMm/r^2

Now these two questions 3) and 4) I really don't understand. I just don't know where to even start.

Do the same you have done symbolically, without plugging in the numbers. You have got the equation

mv^2/r=GMm/r^2


Can you simplify the equation, by dividing both sides with the common factor?

How is the speed of the satellite related to the radius of the orbit and the time period ? Plug in for v, simplify and arrange the equation with R on one side and T on the other.


As for 4) what is the time period when the satellite is vertically above a fixed point of the equator?

ehild

 

[Tryan]

thanks for that so here's what is got from plugging in v as 2πr/T into mv^2/r. so i got 4π^2r^3=GMT^2 so is that right and can i just say that - pi(π), G and M are all constants?? so therefore r^3 is proportional to T^2. Oh and what are the conditions exactly still don't get it?

 

[ehild]

Yes, pi, M and G are constants, m cancels out, so the r³/T² is the same for all planets and satellites orbiting around the same star.

Question 4) is about geostationary orbits. See http://en.wikipedia.org/wiki/Geostationary_orbit

ehild

 

[Tryan]

thanks heaps ehild just reading the website you gave me about orbital stability...

Although you said it was about geostationary orbits however the time period in the question is 12 hours that's half a sidereal day. So wouldn't that mean the satellite in this context would not actually be geosynchronous.?

 

[ehild]

The satellite in the first part is not geostationary. You answered questions 1-2-3 correctly.

I meant the last question

4) discuss the particular requirements for an orbit that will keep a satellite vertically above a certain point on Earth's surface.

 

[Tryan]

Oh right I understand now. Cool understood.