# Help with Kepler's laws and satellite motion.

• Tryan
In summary: So for an orbit to be stable the velocity of the satellite has to be greater than the vertical acceleration.
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..

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Tryan said:
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

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?

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

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.?

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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.

Oh right I understand now. Cool understood.

## 1. What are Kepler's laws of planetary motion?

Kepler's laws of planetary motion are three scientific principles that describe the motion of planets around the sun. These laws were developed by the German astronomer Johannes Kepler in the early 17th century and are considered some of the most important and influential laws in the history of astronomy.

## 2. What is the first law of Kepler?

The first law of Kepler, also known as the law of elliptical orbits, states that all planets move in elliptical orbits around the sun with the sun at one focus of the ellipse. This means that the distance between the planet and the sun changes as the planet moves along its orbit, with the closest distance being called perihelion and the farthest distance being called aphelion.

## 3. How does Kepler's second law explain the speed of a planet in its orbit?

Kepler's second law, also known as the law of equal areas, states that a line that connects a planet to the sun will sweep out equal areas in equal amounts of time. This means that a planet will move faster when it is closer to the sun (at perihelion) and slower when it is farther away (at aphelion). This allows the planet to cover the same amount of area in the same amount of time, resulting in a consistent speed along its entire orbit.

## 4. Can Kepler's third law be applied to satellites as well as planets?

Yes, Kepler's third law, also known as the law of harmonies, can be applied to any two objects that are orbiting around each other. This includes satellites orbiting around a planet or a moon orbiting around a planet. The law states that the ratio of the cube of the semi-major axis of an orbit to the square of the orbital period is the same for all objects orbiting the same central body.

## 5. How does the mass and distance of a planet affect its orbit according to Kepler's laws?

According to Kepler's laws, the mass and distance of a planet do not affect the shape of its orbit. The first law states that all planets move in elliptical orbits around the sun, regardless of their mass or distance. The second law states that a planet's speed in its orbit is determined by its distance from the sun, not its mass. And the third law states that the ratio of the semi-major axis to the orbital period is the same for all objects, regardless of their mass or distance from the central body.

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