Kepler's laws of planetary motion


In astronomy, Kepler's laws of planetary motion give good approximations for the orbits of planets around the Sun. They were published by Johannes Kepler from 1608-1621 in three works Astronomia nova, ''Harmonice Mundi and Epitome Astronomiae Copernicanae''. The laws were based on Kepler's concept of solar fibrils adapted to the accurate astronomical data of Tycho Brahe. These laws replaced the circular orbits and epicycles of Copernicus's heliostatic model of the planets with a heliocentric model that described elliptical orbits with planetary velocities that vary accordingly. The three laws state that:
  1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
  2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.
The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law establishes that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the longer its orbital period.
Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation.

Comparison to Copernicus

's laws improved the model of Copernicus. According to Copernicus:
  1. The planetary orbit is a circle with epicycles.
  2. The Sun is approximately at the center of the orbit.
  3. The speed of the planet in the main orbit is constant.
Despite being correct in saying that the planets revolved around the Sun, Copernicus was incorrect in defining their orbits. Introducing physical explanations for movement in space beyond just geometry, Kepler correctly defined the orbit of planets as follows:
  1. The planetary orbit is not a circle with epicycles, but an ellipse.
  2. The Sun is not at the center but at a focal point of the elliptical orbit.
  3. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed is constant.
The eccentricity of the orbit of the Earth makes the time from the March equinox to the September equinox, around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the Sun parallel to the equator of the Earth cuts the orbit into two parts with areas in a 186 to 179 ratio. The areas swept out in an ellipse will differ by twice the area of a triangle with the focus and the two minor axis points as vertices. This triangle has area where is the semi-minor axis length and is the linear eccentricity. Thus the difference in areas is Since the eccentricity is given by we have where is the difference in swept areas. Since the area of an ellipse is we have so the eccentricity of the orbit of the Earth is approximately
which still a factor of two off from the correct value. The accuracy of this calculation requires that the two dates chosen be along the elliptical orbit's minor axis and that the midpoints of each half be along the major axis. As the two dates chosen here are equinoxes, this will be correct when perihelion, the date the Earth is closest to the Sun, falls on a solstice. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22, but this difference is actually bigger than our difference in inter-equinox times, which explains our substantial error.

History

Kepler's laws were developed based on a physical theory of planetary motion in which the Sun emitted magnetic fibrils which pulled the planets into orbits. The fibrils were somewhat elastic allowing non-circular motion driven by the inertia of the planets.
In Astronomia nova, Kepler articulated his first law, showing that Mars' orbit is elliptical,
having found them by analyzing the astronomical observations of Tycho Brahe. Kepler had believed in the Copernican model of the Solar System, which called for circular orbits, but he could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit – Mars coincidentally having the highest eccentricity of all planets except Mercury. His first law reflected this discovery.
In his Astronomia nova, Kepler did not present his second law in its modern form. He did that only in his Epitome Astronomiae Copernicanae of 1621.
Kepler had two versions of the second law, related in a qualitative sense: the first "distance law" and later the "area law". The distance form was only correct for orbits that were almost circular, but the area form was correct for all elliptical orbit. The "area law" is what became the second law in the set of three. This law had little impact on astronomy because calculations of planetary positions using the law were approximate and time consuming. The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664, but by 1670 his Philosophical Transactions were in its favour. As the century proceeded it became more widely accepted.
Kepler's third law was published in 1619 in his Harmonice Mundi. In 1621, Kepler noted that his third law applies to the four brightest moons of Jupiter. Godefroy Wendelin, the first well-known astronomer to adopt Kepler's laws, gave a detailed account of the third law in 1652.
Kepler's work had little initial impact. His work was as strong defense of Copernicanism which had fallen out of fashion in part because of opposition by Tycho Brahe. In 1627 Kepler published the Rudolphine Tables containing many accurate astronomical observations accumulated by Brahe. The breadth and accuracy of the tables allowed astronomers to compare Kepler's formula to good quality data. At first these difficult calculations were off putting, but once undertaken more astronomers became convinced of Kepler's approach.
The reception in Germany changed noticeably between 1688, the year in which Newton's Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published.
Newton understood that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law, whereas the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.

As three laws

It took nearly two centuries for the current formulation of Kepler's work to take on its settled form. Voltaire's Eléments de la philosophie de Newton of 1738 was the first publication to use the terminology of "laws". The Biographical Encyclopedia of Astronomers in its article on Kepler states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande. It was the exposition of Robert Small, in An account of the astronomical discoveries of Kepler that made up the set of three laws, by adding in the third. Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.

Formulary

The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.

First law

Kepler's first law states that:
The orbit of every planet is an ellipse with the sun at one of the two foci.

Mathematically, an ellipse can be represented by the formula:
where is the semi-latus rectum, ε is the eccentricity of the ellipse, r is the distance from the Sun to the planet, and θ is the angle to the planet's current position from its closest approach, as seen from the Sun. So are polar coordinates.
For an ellipse 0 < ε < 1 ; in the limiting case ε = 0, the orbit is a circle with the Sun at the centre.
At θ = 0°, perihelion, the distance is minimum
At θ = 90° and at θ = 270° the distance is equal to.
At θ = 180°, aphelion, the distance is maximum
The semi-major axis a is the arithmetic mean between rmin and rmax:
The semi-minor axis b is the geometric mean between rmin and rmax:
The semi-latus rectum p is the harmonic mean between rmin and rmax:
The eccentricity ε is the coefficient of variation between rmin and rmax:
The area of the ellipse is
The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = πr2.

Second law

Kepler's second law states that:
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the Sun, then slower when farther from the Sun. Kepler's second law states that the blue sector has constant area.

History and proofs

Kepler notably arrived at this law through assumptions that were either only approximately true or outright false and can be outlined as follows:
  1. Planets are pushed around the Sun by a force from the Sun. This false assumption relies on incorrect Aristotelian physics that an object needs to be pushed to maintain motion.
  2. The propelling force from the Sun is inversely proportional to the distance from the Sun. Kepler reasoned this, believing that gravity spreading in three dimensions would be a waste, since the planets inhabited a plane. Thus, an inverse instead of the inverse square law.
  3. Because Kepler believed that force would be proportional to velocity, it followed from statements #1 and #2 that velocity would be inverse to the distance from the sun. That force is proportional to velocity is an incorrect tenet of Aristotelian physics, but the errors of assumption in statements #2 and #3 essentially cancel, so that it is approximately true that velocity is inverse to the distance from the sun.
  4. Since velocity is inverse to time, the distance from the sun would be proportional to the time to cover a small piece of the orbit. This is approximately true for elliptical orbits.
  5. The area swept out is proportional to the overall time. This is also approximately true.
  6. The orbits of a planet are circular.
Nevertheless, the result of the second law is exactly true, as it is logically equivalent to the conservation of angular momentum, which is true for any body experiencing a radially symmetric force. A correct proof can be shown through this. Since the cross product of two vectors gives the area of a parallelogram possessing sides of those vectors, the triangular area dA swept out in a short period of time is given by half the cross product of the r and dx vectors, for some short piece of the orbit, dx.
for a small piece of the orbit dx and time to cover it dt.
Thus
Since the final expression is proportional to the total angular momentum, Kepler's equal area law will hold for any system that conserves angular momentum. Since any radial force will produce no torque on the planet's motion, angular momentum will be conserved.