Kepler's Laws

Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky.

1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.

2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times.

3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

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

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The Law of Orbits

All planets move in elliptical orbits, with the sun at one focus.

This is one of Kepler's laws. The elliptical shape of the orbit is a result of the inverse square force of gravity. The eccentricity of the ellipse is greatly exaggerated here.

Describing an ellipse
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Orbit Eccentricity

The eccentricity of an ellipse can be defined as the ratio of the distance
between the foci to the major axis of the ellipse. The eccentricity is zero for a circle. Of the planetary orbits, only Pluto has a large eccentricity.
Eccentricity examples


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Examples of Ellipse Eccentricity

Planetary orbit eccentricities
Mercury .206
Venus .0068
Earth .0167
Mars .0934
Jupiter .0485
Saturn .0556
Uranus .0472
Neptune .0086
Pluto .25

Outer planets


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The Law of Areas

A line that connects a planet to the sun sweeps out equal areas in equal times.

This is one of Kepler's laws.This empirical law discovered by Kepler arises from conservation of angular momentum. When the planet is closer to the sun, it moves faster, sweeping through a longer path in a given time.
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The Law of Periods

The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

This is one of Kepler's laws.This law arises from the law of gravitation. Newton first formulated the law of gravitation from Kepler's 3rd law.

Kepler's Law of Periods in the above form is an approximation that serves well for the orbits of the planets because the Sun's mass is so dominant. But more precisely the law should be written

In this more rigorous form it is useful for calculation of the orbital period of moons or other binary orbits like those of binary stars.

Table of data
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Data: Law of Periods

Data confirming Kepler's Law of Periods comes from measurements of the motion of the planets.

Planet
Semimajor
axis
(1010m)
Period
T (y)
T2/a3
(10-34y2/m3)
Mercury 5.79 0.241 2.99
Venus 10.8 0.615 3.00
Earth 15.0 1 2.96
Mars 22.8 1.88 2.98
Jupiter 77.8 11.9 3.01
Saturn 143 29.5 2.98
Uranus 287 84 2.98
Neptune 450 165 2.99
Pluto 590 248 2.99

The quantity T2/a3 depends upon the sum of the masses of the Sun and the planet, but since the mass of the Sun is so great, adding the mass of the planet makes very little difference.

Data from Halliday, Resnick, Walker, Fundamentals of Physics 4th Ed Extended. Table 15-3

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