Orbits of Planets and Satellites

Derive Kepler’s Third Law

1. Equate the equations for centripetal force and gravitational force
2. Simplify for v squared
3. Substitute v for 2πr/T
4. Then rearrange for r cubed over T squared

Derive the straight line equation for T and r

How do you get to it?

1. Start from Kepler’s third law
2. Take logs of both sides
3. Split the logs on the LHS
4. Rearrange for T as the subject
5. Divide both sides by 2

log(T) = 1.5log(r) - 0.5log(GM/4π^2)

What information can we get from the gradient and from the y-intercept of the logarithmic graph?

gradient - proves Kepler’s law

y-intercept - can be used to calculate the mass of source mass

Define escape velocity

The minimum speed an object must travel at to escape the gravitational field at the surface of a mass

What are the features of a satellite in a geostationary orbit?

* orbits directly above the equator
* has an orbital period of 24 hours

What is common to all satellites in geostationary orbits and why?

They all have the same orbital radius because orbital radius depends on mass of the earth and the time period not mass of the satellite

What is a polar orbit?

a (low earth) orbit that always passes over the North and South poles

What is a synchronous orbit?

When the orbital period of the satellite is equal to the rotational period of the object that it is orbiting

What is a geosynchronous orbit?

an 24 hour orbit that is similar to geostationary but is tilted away from the equator

it appears to move across the sky

How do you calculate the velocity of a satellite in orbit?

From equating centripetal force to gravitational force and rearranging for v

How do you calculate the total energy of a satellite in orbit? What is it?

Add the satellite’s kinetic energy and potential energy

\-(GMm)/2r

How do you calculate the escape velocity of a satellite in orbit?

equate kinetic energy and gravitational potential energy then rearrange for v

where r is the radius of the satellite’s orbit (distance from centre of source mass to satellite)

What are geostationary satellites useful for and why?

sending TV and telephone signals because they are always above the same point on the Earth, plane of aerials or transmitters don’t need to be alterred

What is a low earth orbit?

a satellite that orbits the Earth at a low altitude (up to 2000km) so it appears to move aorund the sky

What are low-orbit satellites useful for and why?

* monitoring the weather, making observations about unreachable places
* because they are closer and so require less powerful transmitters