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|This article has statements which are anachronisms.|
Most planetary orbits are almost circles, so it is not apparent that they are actually ellipses. Calculations of the orbit of the planet Mars first indicated to Kepler its elliptical shape, and he inferred that other heavenly bodies, including those farther away from the Sun, have elliptical orbits also. Kepler's work broadly followed the heliocentric theory of Nicolaus Copernicus by asserting that the Earth orbited the Sun. It innovated in explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.
Isaac Newton showed in 1687 that relationships like Kepler's would apply in the solar system to a good approximation, as consequences of his own laws of motion and law of universal gravitation. Together with Newton's theories, Kepler's laws became part of the foundation of modern astronomy and physics.
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 (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws". The Biographical Encyclopedia of Astronomers in its article on Kepler (p. 620) 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 (1804) 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.
Further, the current usage of "Kepler's Second Law" is something of a misnomer. Kepler had two versions of it, related in a qualitative sense, the "distance law" and the "area law". The "area law" is what became the Second Law in the set of three; but Kepler did himself not privilege it in that way.
Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler's third law was published in 1619.
Kepler in 1622 and Godefroy Wendelin in 1643 noted that Kepler's third law applies to the four brightest moons of Jupiter.[Nb 1] The second law ("area law" form) was contested by Nicolaus Mercator in a book from 1664; but by 1670 he was publishing in its favour in Philosophical Transactions, and as the century proceeded it became more widely accepted. 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.
Mathematically, an ellipse can be represented by the formula:
Note that 0 < ε < 1 for an ellipse; in the limiting case ε = 0, the orbit is a circle with the sun at the centre (see section Zero eccentricity below).
For a planet 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.
At θ = 0°, perihelion, the distance is minimum
At θ = 90° and at θ = 270°, the distance is equal to the semi-latus rectum.
At θ = 180°, aphelion, the distance is maximum
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 = π r2.
|Semi-major axis||Orbital eccentricity||Interfocal distance||Location of f2|
|Jupiter||7,783 mn km||0.048386||75 mn km||Roughly half-way between Mercury's (58 mn km) and Venus's (108 mn km) orbital distances.|
|Saturn||14,267 mn km||0.053862||154 mn km||Roughly at Earth orbit distance (150 mn km).|
This law relates radial distance and angular velocity in eliptical orbits. That is, in a perfectly circular orbit, the orbital radius of the satellite would be constant and therefore so would be its observed angular velocity. In elliptical orbits, the orbital radius of the satellite will vary and therefore so will its angular velocity. This is shown in the above animation where the satellite travels "faster" (greater angular velocity) when closer to the parent object, then "slower" (less angular velocity) at a more distant radius. The result is that the blue sectors are shorter but wider when close to the body, then longer but narrower at a greater distance. Kepler's 2nd law states that for a given elliptical orbit, any two sectors of equal time duration will have the same area. This implies that radial distance and angular velocity have an inversely proportional relationship in a given orbit; angular velocity is minimum at apoapsis and maximum at periapsis. The constant of proportionality is the rate at which area in the ellipse is covered.
In a small time the planet sweeps out a small triangle (or, more precisely, a sector) having base line and height and area and so the constant areal velocity is
The planet moves faster when it is closer to the Sun.
The area enclosed by the elliptical orbit is So the period satisfies
and the mean motion of the planet around the Sun satisfies
|Perihelion distance||Perihelion speed||Aphelion distance||Aphelion speed||Sector area swept|
in one second
|Mercury||46.0 mn km||59,000 m/s||69.8 mn km||38,900 m/s||1.3565×1015 m2|
for both perihelion and aphelion
|Sedna||11,423 bn km||4,640 m/s||~ 140,000 bn km||~ 430 m/s||3.01×1016 m2 (based on perihelion measurements)|
2.65×1016 m2 (based on aphelion measurements)
|“||The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.||”|
The third law, published by Kepler in 1619  captures the relationship between the distance of planets from the Sun, and their orbital periods.
Kepler enunciated 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. So it used to be known as the harmonic law.
Mathematically, the law says that the expression has the same value for all the planets in the solar system.
|Period & Semi-major axis|
(in relation to Earth)
|Period 2 : Semi-major axis 3|
(in relation to Earth)
|Period & Semi-major axis|
(in real terms)
|Period 2 : Semi-major axis 3|
(in real terms)
|Venus||0.615 yr, 0.723 AU||0.3785 : 0.3785||19.4 mn secs, 1.082×1011 m||3.769×1014 : 1.267×1033||1 : 3.362×1018|
|Neptune||164.8 yrs, 30.1 AU||27160 : 27280||5.2 bn secs, 4.498×1012 m||2.704×1019 : 9.103×1037||1 : 3.366×1018|
The modern formulation with the constant evaluated reads:
In the full formulation under Newton's laws of motion, should be replaced by , where is the mass of the orbiting body. Consequently, the proportionality constant is not truly the same for each planet. Nevertheless, for all planets in our solar system such that variations in the proportionality constant are negligible.
Kepler's laws refine the model of Copernicus, which assumed circular orbits. If the eccentricity of a planetary orbit is zero, then Kepler's laws state:
Actually, the eccentricities of the orbits of the six planets known to Copernicus and Kepler are quite small, so the rules above give excellent approximations of planetary motion, but Kepler's laws fit observations even better.
Kepler's corrections to the Copernican model are not at all obvious:
The nonzero 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, so the eccentricity of the orbit of the Earth is approximately
which is close to the correct value (0.016710219). (See Earth's orbit). The calculation is correct when the perihelion, the date that the Earth is closest to the Sun, is on a solstice. The current perihelion, near January 4, is fairly close to the solstice on December 21 or 22.
This suggests that the Sun may be the physical cause of the acceleration of planets.
Here the Sun plays an unsymmetrical part, which is unjustified. So he assumed Newton's law of universal gravitation:
As the planets have small masses compared to that of the Sun, the orbits conform to Kepler's laws approximately. Newton's model improves upon Kepler's model and fits actual observations more accurately. (See two-body problem).
A deviation in the motion of a planet from Kepler's laws due to the gravity of other planets is called a perturbation.
Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.
where is the unit vector orthogonal to and pointing in the direction of rotation, and is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.
So differentiating the position vector twice to obtain the velocity and the acceleration vectors:
where the radial acceleration is
and the transversal acceleration is
Kepler's laws say that
The transversal acceleration is zero:
So the acceleration of a planet obeying Kepler's laws 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
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
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.
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. See Kepler orbit.
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 world, besides i itself.
In the special case where there are only two bodies in the world, 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 world 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
The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the time t since perihelion, is the following four steps:
The important special case of circular orbit, ε = 0, gives θ = E = M. 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.
The Keplerian problem assumes an elliptical orbit and the four points:
The problem is to compute the polar coordinates (r,θ) 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 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.
When the mean anomaly M is computed, the goal is to compute the true anomaly θ. The function θ=f(M) is, however, not elementary. Kepler's solution is to use
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.
Note from the figure that
Dividing by and inserting from Kepler's first law
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:
Multiplying by 1+ε gives the result
This is the third step in the connection between time and position in the orbit.
The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law: