Kepler's laws
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In astronomy, Kepler's three laws of planetary motion are:
- "The orbit of every planet is an ellipse with the sun at a focus."
- "A line joining a planet and the sun sweeps out equal areas during equal intervals of time."
- "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."
These three mathematical laws were discovered by German mathematician and astronomer Johannes Kepler (1571–1630), and used by him to describe the motion of planets in the Solar System. They describe the motion of any two bodies orbiting each other.
Kepler worked from the observations of Tycho Brahe, which he published as the Rudolphine tables. Around 1605, Kepler found that Brahe's observations of the planets' positions followed these relatively simple mathematical laws.
Kepler's laws challenged Aristotelean and Ptolemaic astronomy and physics. His assertion that the Earth moved, his use of ellipses rather than epicycles, and his proof that the planets' speeds varied, changed astronomy and physics. Almost a century later Isaac Newton was able to derive Kepler's laws from Newton's own laws of motion and his law of universal gravitation, using classical Euclidean geometry.
In modern times, Kepler's laws are used to calculate approximate orbits for artificial satellites, and bodies orbiting the Sun of which Kepler was unaware (such as the outer planets and smaller asteroids). They apply where any relatively small body is orbiting a larger, relatively massive body, though the effects of atmospheric drag (e.g. in a low orbit), relativity (e.g. Perihelion precession of Mercury), and other nearby bodies can make the results insufficiently accurate for a specific purpose.
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Introduction to the three laws
Generality
These laws describe the motion of any two bodies in orbit around each other. The masses of the two bodies can be nearly equal, e.g. Charon—Pluto (~1:10), in a small proportion, e.g. Moon—Earth (~1:100), or in a great proportion, e.g. Mercury—Sun (~1:10,000,000).
In all cases both the bodies orbit around the common center of mass, the barycenter, with neither one having their center of mass exactly at one focus of an ellipse. However, both orbits are ellipses with one focus at the barycenter. When the ratio of masses is large, i.e. with planets orbiting the Sun, the barycenter is deep within the larger object close to its center of mass. In this case it requires sophisticated precise measurements to detect the separation of the barycenter from the center of mass of the larger object. Thus Kepler's first law accurately describes the orbits of the planets around the Sun.
Since Kepler stated these laws as they apply to the Sun and the planets, and did not know of their generality, this article discusses these laws as they apply to the sun and its planets.
First law
- "The orbit of every planet is an ellipse with the sun at a focus."
Symbolically:
where (r, θ) are heliocentric polar coordinates for the planet, p is the semi-latus rectum, and ε is the eccentricity.
At the time, this was a radical claim; the prevailing belief (particularly in epicycle-based theories) was that orbits should be based on perfect circles. This observation was very significant at the time as it supported the Copernican view of the Universe. This does not mean it loses relevance in a more modern context. A circle is just one form of an ellipse, but most of the planets follow an orbit of low eccentricity, meaning that they can be crudely approximated as circles. So it is not evident from the orbit of the planets that the orbits are indeed elliptic. However, Kepler's calculations proved they were, which also allowed for other heavenly bodies farther away from the Sun with highly eccentric orbits (like very long stretched out circles). These other heavenly bodies indeed have been identified as the numerous comets or asteroids by astronomers after Kepler's time. The dwarf planet Pluto was discovered as late as 1930, the delay mostly due to its small size and its highly elongated orbit compared to the other planets. Nevertheless, heavenly bodies such as comets with parabolic or even hyperbolic orbits are possible under the Newtonian theory and have been observed.
Second Law
- "A line joining a planet and the sun sweeps out equal areas during equal intervals of time."
Symbolically:
where is the "areal velocity".
This is also known as the law of equal areas. To understand this let us suppose a planet takes one day to travel from point A to point B. The lines from the Sun to points A and B, together with the planet orbit, will define an (roughly triangular) area. This same area will be covered every day regardless of where in its orbit the planet is. Now as the first law states that the planet follows an ellipse, the planet is at different distances from the Sun at different parts in its orbit. This leads to the conclusion that the planet has to move faster when it is closer to the sun so that it sweeps an equal area.
Kepler's second law is an additional observation on top of his first law. It is equivalent to the fact that the net tangential force involved in an elliptical orbit, as per his first law, is zero. The "areal velocity" is proportional to angular momentum, and so for the same reasons, Kepler's second law is also a statement of the conservation of angular momentum.
Third Law
- "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."
Symbolically:
where P is the orbital period of planet and a is the semimajor axis of the orbit.
The proportionality constant is the same for any planet around the sun.
.
For example, suppose planet A is four times as far from the sun as planet B. Then planet A must traverse four times the distance of Planet B each orbit, and moreover it turns out that planet A travels at half the speed of planet B. In total it takes 4×2=8 times as long for planet A to travel an orbit, in agreement with the law (82=43).
This law used to be known as the harmonic law.
