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:

The orbit of a planet is an ellipse with the Sun at one of the two foci.
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
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:

The planetary orbit is a circle with epicycles.
The Sun is approximately at the center of the orbit.
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:

The planetary orbit is not a circle with epicycles, but an ellipse.
The Sun is not at the center but at a focal point of the elliptical orbit.
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 Mathematica|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 r_min and r_max:
The semi-minor axis b is the geometric mean between r_min and r_max:
The semi-latus rectum p is the harmonic mean between r_min and r_max:
The eccentricity ε is the coefficient of variation between r_min and r_max:
The area of the ellipse is
The special case of a circle is ε = 0, resulting in r = p = r_min = r_max = 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:

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.
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.
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.
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.
The area swept out is proportional to the overall time. This is also approximately true.
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.

In terms of elliptical parameters

In a small time the planet sweeps out a small triangle having base line and height and area, so the constant areal velocity is
The area enclosed by the elliptical orbit is. So the period satisfies
and the mean motion of the planet around the Sun
satisfies
And so,

Third law

Kepler's third law states that:

The ratio of the square of an object's orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.
Or symbolically :
Where is the object's orbital period and is the semi-major axis of its orbit.

This captures the relationship between the distance of planets from the Sun, and their orbital periods.
Kepler enunciated in 1619 this third law in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation. It was therefore known as the harmonic law. The original form of this law holds true only for planets with small eccentricities near zero.
Using Newton's law of gravitation, this relation can be found in the case of a circular orbit by setting the centripetal force equal to the gravitational force:
Then, expressing the angular velocity ω in terms of the orbital period and then rearranging, results in Kepler's third law:
A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass. This results in replacing a circular radius,, with the semi-major axis,, of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass with. However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is:
where is the mass of the Sun, is the mass of the planet, is the gravitational constant, is the orbital period and is the elliptical semi-major axis, and is the astronomical unit, the average distance from earth to the sun.

Table

The following table shows the data used by Kepler to empirically derive his law:

Planet	Mean distance to sun	Period
Mercury	0.389	87.77	7.64
Venus	0.724	224.70	7.52
Earth	1	365.25	7.50
Mars	1.524	686.95	7.50
Jupiter	5.20	4332.62	7.49
Saturn	9.510	10759.2	7.43

Kepler became aware of John Napier's recent invention of logarithms and log-log graphs before he discovered the pattern.
Upon finding this pattern Kepler wrote:
For comparison, here are modern estimates:

Planet	Semi-major axis	Period
Mercury	0.38710	87.9693	7.496
Venus	0.72333	224.7008	7.496
Earth	1	365.2564	7.496
Mars	1.52366	686.9796	7.495
Jupiter	5.20336	4332.8201	7.504
Saturn	9.53707	10775.599	7.498
Uranus	19.1913	30687.153	7.506
Neptune	30.0690	60190.03	7.504

Planetary acceleration

computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second laws.

The direction of the acceleration is towards the Sun.
The magnitude of the acceleration is inversely proportional to the square of the planet's distance from the Sun.

This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in his Principia that he considers forces from a mathematical point of view, not a physical, thereby taking an instrumentalist view. Moreover, he does not assign a cause to gravity.
Newton defined the force acting on a planet to be the product of its mass and the acceleration. So:

Every planet is attracted towards the Sun.
The force acting on a planet is directly proportional to the mass of the planet and is inversely proportional to the square of its distance from the Sun.

The Sun plays an unsymmetrical part, which is unjustified. So he assumed, in Newton's law of universal gravitation:

All bodies in the Solar System attract one another.
The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them.

As the planets have small masses compared to that of the Sun, the orbits conform approximately to Kepler's laws. Newton's model improves upon Kepler's model, and fits actual observations more accurately.
Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.

Acceleration vector

From the heliocentric point of view consider the vector to the planet where is the distance to the planet and is a unit vector pointing towards the planet.
where is the unit vector whose direction is 90 degrees counterclockwise of, and is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.
Differentiate the position vector twice to obtain the velocity vector and the acceleration vector:
So
where the radial acceleration is
and the transversal acceleration is

Inverse square law

Kepler's second law says that is constant.
The transversal acceleration is zero:
So the acceleration of a planet obeying Kepler's second law is directed towards the Sun.
The radial acceleration is
Kepler's first law states that the orbit is described by the equation:
Differentiating with respect to time
or
Differentiating once more
The radial acceleration satisfies
Substituting the equation of the ellipse gives
The relation gives the simple final result
This means that the acceleration vector of any planet obeying Kepler's first and second law satisfies the inverse square law
where
is a constant, and is the unit vector pointing from the Sun towards the planet, and is the distance between the planet and the Sun.
Since mean motion where is the period, according to Kepler's third law, has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire Solar System.
The inverse square law is a differential equation. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a hyperbola or parabola or a straight line.

Newton's law of gravitation

By Newton's second law, the gravitational force that acts on the planet is:
where is the mass of the planet and has the same value for all planets in the Solar System. According to Newton's third law, the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun,. So
where is the gravitational constant.
The acceleration of Solar System body number i is, according to Newton's laws:
where is the mass of body j, is the distance between body i and body j, is the unit vector from body i towards body j, and the vector summation is over all bodies in the Solar System, besides i itself.
In the special case where there are only two bodies in the Solar System, Earth and Sun, the acceleration becomes
which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.
If the two bodies in the Solar System are Moon and Earth the acceleration of the Moon becomes
So in this approximation, the Moon moves around the Earth according to Kepler's laws.
In the three-body case the accelerations are
These accelerations are not those of Kepler orbits, and the three-body problem is complicated. But Keplerian approximation is the basis for perturbation calculations.

Position as a function of time

Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.
The procedure for calculating the heliocentric polar coordinates of a planet as a function of the time t since perihelion, is the following five steps:

Compute the mean motion, where P is the period.
Compute the mean anomaly, where t is the time since perihelion.
Compute the eccentric anomaly E by solving Kepler's equation: where is the eccentricity.
Compute the true anomaly θ by solving the equation:
Compute the heliocentric distance r: where is the semimajor axis.

The position polar coordinates can now be written as a Cartesian vector and the Cartesian velocity vector can then be calculated as, where is the standard gravitational parameter.
The important special case of circular orbit, ε = 0, gives. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.
The proof of this procedure is shown below.

Mean anomaly, ''M''

The Keplerian problem assumes an elliptical orbit and the four points:

s the Sun ;
z the perihelion
c the center of the ellipse
p the planet

and

distance between center and perihelion, the semimajor axis,
the eccentricity,
the semiminor axis,
the distance between Sun and planet.
the direction to the planet as seen from the Sun, the true anomaly.

The problem is to compute the polar coordinates of the planet from the time since perihelion, t.
It is solved in steps. Kepler considered the circle with the major axis as a diameter, and

the projection of the planet to the auxiliary circle
the point on the circle such that the sector areas |zcy| and |zsx| are equal,
the mean anomaly.

The sector areas are related by
The circular sector area
The area swept since perihelion,
is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.
where n is the mean motion.

Eccentric anomaly, ''E''

When the mean anomaly M is computed, the goal is to compute the true anomaly θ. The function θ = f is, however, not elementary. Kepler's solution is to use
x as seen from the centre, the eccentric anomaly
as an intermediate variable, and first compute E as a function of M by solving Kepler's equation below, and then compute the true anomaly θ from the eccentric anomaly E. Here are the details.
Division by a²/2 gives Kepler's equation
This equation gives M as a function of E. Determining E for a given M is the inverse problem. Iterative numerical algorithms are commonly used.
Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ.
But note: Cartesian position coordinates with reference to the center of ellipse are
With reference to the Sun =, r =
True anomaly would be arctan, magnitude of r would be.

True anomaly, ''θ''

Note from the figure that
so that
Dividing by and inserting from Kepler's first law
to get
The result is a usable relationship between the eccentric anomaly E and the true anomaly θ.
A computationally more convenient form follows by substituting into the trigonometric identity:
Get
Multiplying by 1 + ε gives the result
This is the third step in the connection between time and position in the orbit.

Distance, ''r''

The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:
Using the relation above between θ and E the final equation for the distance r is:

Explanatory notes

General bibliography

Kepler's life is summarized on pp. 523–627 and Book Five of his magnum opus, Harmonice Mundi, is reprinted on:
A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example:.