We’ll be covering:
- Circular Orbits
- Kepler’s 3rd Law of Planetary Motion
- Types of Orbit
We’ve already seen how circular motion works.
Let’s talk about circular orbits caused by gravitational force.
What are the components of circular orbits?
|Centre of Circular Motion||Centre of mass of larger mass (e.g. planet)|
|Radius, r||Radius of planet + altitude of satellite
r = R + h
|Period, T||Time taken to complete one revolution.
You can use these values to calculate:
|Centripetal Force, Fc||Gravitational Force
F = GMm/r2
|Centripetal Acceleration, ac||Gravitational Field Strength
a = g = GM/r2
How are these all related?
|Fc = Fg
|Linear speed = circumference of orbit/period
v = 2πr/T
|mv2/r = GMm/r2
v2 = GM/rSince g = GM/r2,
|From relation 1:
m(2πr/T)2/r = GMm/r2Rearranging:
|From Relation 3:
For different objects orbiting the same body (e.g. planets in a solar system), M is the same (mass of the Sun).Thus, it can be concluded that:
T2 ∝ r3
“The square of the period is proportional to the cube of the orbital radius”
This is Kepler’s 3rd Law of Planetary Motion.
Let’s talk about a few types of Earth orbit:
|Low Earth Orbit||Satellites orbiting close to the vicinity of Earth.
Their altitude is relatively small enough, such that r ≈ R.Thus, by calculation:
|Geostationary Orbit||Satellites in equatorial orbits which the same period of rotation as the Earth.
Thus, they are stationary in the sky to an observer on the ground.
T = TEarth = 24 hours
This information can be used to calculate other aspects of motion: