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.

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.

Examples of Ellipse Eccentricity

Planetary orbit eccentricities Mercury 水星 .206 Venus 金星 .0068 Earth 接地 .0167 Mars 火星 .0934 Jupiter 木星 .0485 Saturn 土星 .0556 Uranus 天王星 .0472 Neptune 海王星 .0086 Pluto 冥王星 .25 Outer planets 外行星 行星轨道偏心率

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. 这是开普勒定律之一。这个由开普勒发现的实证定律源于角动量守恒。当行星离太阳更近时，它运动得更快，扫过更长的路径，而在给定的时间内。

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. 这是开普勒定律之一。该定律源于万有引力定律。牛顿首次从开普勒第三定律出发制定了万有引力定律。

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.

Data: Law of Periods

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

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