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Kepler's laws #1
Kepler's 1st. Law - The Orbit Law
The orbit of a planet is in the shape of an ellipse, with the parent star at one focus.
Kepler's 2nd. Law - The Area Law
A planet moves such that an imaginary line between it and the parent star sweeps out
equal areas in equal times.
Kepler's 3rd. Law - The Period Law
The square of the orbital period of a planet is proportional to the cube of the mean
distance to its parent star.
T is the orbital period
rM is the mean distance between the planet and the parent star
Kepler's 3rd law derived from Newton's Law of Gravitation
The centripetal force F keeping a mass m in orbit is given by:
The angular velocity ω is the angle (in radians) traced out when the mass travels v metres in one second. Stating this definition in an equation and making v the subject:
Substitung for v into our equation for centripetal force:
NB The centripetal force is provided by gravity.
For a star mass Ms and planet mass m,
Equating equations (i and (ii , (force of gravity = centripetal force)
The period T of the orbital motion is the circumference (in radians) divided by the angular velocity. Making ω the subject of the equation:
and substituting for ω into equation (iii
We obtain the expression:
Now making T 2 the subject:
If we now remove the constants G MS π by making the equation a proportionality:
Hence Kepler's 3rd Law is consistent with Newton's Law of Gravitation.
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