Kepler’s Laws

Johannes Kepler in his working with data together with Tycho Brahe, put forth three laws which described the motion of the planets across the sky.

The laws are:

1. The Law of Orbits:

   1. The planets move in ellipses with the Sun at one focus

   You can prove this by measuring the diameter of a projected image of the Sun at different times of the year. (NB although the Earth does move in an elliptical orbit this orbit is every nearly circular. The only planet with a markedly elliptical orbit is Pluto.)

   Distance from the Earth to the Sun at its closest (perihelion)           = 1.471×10¹¹ m

   Distance from the Earth to the Sun when at its furthest (aphelion)   = 1.521×10¹¹ m

2. The Law of Areas:

   A line drawn from the planet to the Sun sweeps out equal areas in equal times.

   This means that bodies move faster when closer to the Sun – the best example of this being the long period comets. They spend only a few months close to the Sun before moving off into deep space – not returning for maybe many hundreds of years.

3. The Law of Periods:

   The square of the period of any planet is directly proportional to the cube of the semi-major axis of its orbit.The Earth is 150×10⁹ m from the Sun and it takes one year (3.16×10⁷ s) to orbit the Sun.

   Therefore Kepler’s constant (r³/T²) for the Solar System is (150×10⁹)³/(3.16×10⁷)² = 3.37×10¹⁸m³s^-2.

Kepler’s laws were derived for orbits around the sun, but they apply to satellite orbits as well.

GENERAL EVALUATION (POST YOUR ANSWERS USING THE QUESTION BOX BELOW FOR EVALUATION AND DISCUSSION):

1. State the mathematical expression of Isaac Newton’s law of gravitation.
2. What is the mathematical relationship between gravitational intensity and gravitational potential?
3. State the Kepler’s laws.
4. Considering a rocket launched from the earth’s surface, show that the velocity of escape V₀is  given by:  V0=2gR−−−−√
5. Estimate the possible gravitational force between the earth and the moon at 4 × 10⁸m apart if  their masses are 6 ×10²⁴kg and 7 ×10²²