# Kepler's laws of planetary motion: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

Figure 1: Illustration of Kepler's three laws with two planetary orbits. (1) The orbits are ellipses, with focal points ƒ1 and ƒ2 for the first planetƒ1 and ƒ3 for the second planet. The Sun is placed in focal point ƒ1. (2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2. (3) The total orbit times for planet 1 and planet 2 have a ratio a13/2 : a23/2.

In astronomy, Kepler's laws give an approximate description of the motion of planets around the Sun.

Kepler's laws are:

1. The orbit of every planet is an ellipse with the Sun at a focus.
2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.[1]
3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

## History

Kepler's laws were discovered empirically, around 1605, by Johannes Kepler who found them by analyzing the astronomical observations of Tycho Brahe.[2] Almost a century later, Isaac Newton proved that relationships like Kepler's would apply exactly under certain ideal conditions approximately fulfilled in the solar system, as consequences of Newton's own laws of motion and law of universal gravitation, using classical Euclidean geometry.[3][4]

Because of the nonzero planetary masses and resulting perturbations, Kepler's laws apply only approximately and not exactly to the motions in the solar system.[5][3]

Voltaire's Eléments de la philosophie de Newton (Elements of Newton's Philosophy) was in 1738 the first publication to call them "laws".[6]

Kepler's laws and his analysis of the observations on which they were based, the assertion that the Earth orbited the Sun, proof that the planets' speeds varied, and use of elliptical orbits rather than circular orbits with epicycles—challenged the long-accepted geocentric models of Aristotle and Ptolemy, and generally supported the heliocentric theory of Nicolaus Copernicus (although Kepler's ellipses likewise did away with Copernicus's circular orbits and epicycles).[2] Together with Newton's mathematical theories, they are part of the foundation of modern astronomy and physics. [3] after Isaac Newton showed[4] that relationships of similar form would apply exactly as consequences of the laws of motion and gravitation under certain ideal conditions (not exactly fulfilled in the the solar system).

## Generality

These laws approximately describe the motion of any two bodies in orbit around each other. (The statement in the first law about the focus becomes closer to exact as one of the masses becomes closer to zero mass. Where there are more than two masses, all of the statements in the laws become closer to exact as all except one of the masses become closer to zero mass and as the perturbations then also tend towards zero).[4] The masses of the two bodies can be nearly equal, e.g. CharonPluto (~1:10), in a small proportion, e.g. MoonEarth (~1:100), or in a great proportion, e.g. MercurySun (~1:10,000,000).

In all cases of two-body motion, rotation is about the barycenter of the two bodies, with neither one having its 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, the barycenter may be deep within the larger object, close to its center of mass. In such a case it may require sophisticated precision measurements to detect the separation of the barycenter from the center of mass of the larger object. But in the case of the planets orbiting the Sun, the largest of them are in mass as much as 1/1047.3486 (Jupiter) and 1/3497.898 (Saturn) of the solar mass,[7] and so it has long been known that the solar system barycenter can sometimes be outside the body of the Sun, up to about a solar diameter from its center.[8] Thus Kepler's first law, though not far off as an approximation, does not quite accurately describe the orbits of the planets around the Sun under classical physics.

Since Kepler stated his results with reference to the Sun and the planets, and did not know of their wider applicability, this article also discusses them with reference to the Sun and its planets.

At the time, Kepler's laws were radical claims; the prevailing belief (particularly in epicycle-based theories) was that orbits should be based on perfect circles. Kepler's observation was significant support for the Copernican view of the Universe, and still has relevance in a modern context. A circle is a form of ellipse, and most of the planets follow orbits of low eccentricity, which can be rather closely approximated as circles, so it is not immediately evident that the orbits are elliptical. Detailed calculations for 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 too. That allows also for highly eccentric orbits (like very long stretched out circles). Bodies with highly eccentric orbits have been identified, among them the comets and many asteroids, discovered after Kepler's time. The dwarf planet Pluto was discovered as late as 1929, the delay mostly due to its small size, far distance, and optical faintness. Heavenly bodies such as comets with parabolic or even hyperbolic orbits are possible under the Newtonian theory and have been observed.[9]

To understand the second law 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. So 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 equivalent to the fact that the force perpendicular to the radius vector is zero. The "areal velocity" is proportional to angular momentum, and so for the same reasons, Kepler's second law is also in effect a statement of the conservation of angular momentum.

The third law, published by Kepler in 1619 [1] captures the relationship between the distance of planets from the Sun, and their orbital periods. For example, suppose planet A is 4 times as far from the Sun as planet B. Then planet A must traverse 4 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 order to maintain equilibrium with the reduced gravitational centripetal force due to being 4 times further from the Sun. 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 third law used to be known as the harmonic law,[10] because Kepler enunciated it 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.[11]

This third law currently receives additional attention as it can be used to estimate exoplanetary orbital radius as to know the distance of planets to its respective central star, and help to decide if this distance is inside habitable zone of the corresponding star.[12] The concrete, generic formula, would be:

d[AU]3 = P[years]2 * M[SolarMasses].

## Zero eccentricity

Kepler's laws refine upon the model of Copernicus. If the eccentricity of a planetary orbit is zero, then Kepler's laws state:

1. The planetary orbit is a circle with the Sun in the center.
2. The speed of the planet in the orbit is constant
3. The square of the sidereal period is proportionate to the cube of the distance from the Sun.

Actually the eccentricities of the orbits of the six planets known to Copernicus and Kepler are quite small, so this gives excellent approximations to the planetary motions, but Kepler's laws give even better fit to the observations.

Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly. Kepler's corrections to the Copernican model are not at all obvious:

1. The planetary orbit is not a circle, but an ellipse, and the Sun is not in the center of the orbit, but in a focal point.
2. Neither the speed nor the angular speed of the planet in the orbit is constant, but the area speed is constant.
3. The square of the sidereal period is proportionate to the cube of the mean between the maximum and minimum distances from the Sun.

The time from the March equinox to the September equinox is around 186 days, while the time from the September equinox to the March equinox is only around 179 days. This elementary observation shows that the eccentricity of the orbit of the earth is not exactly zero. The intersection between the plane of the equator and the plane of the ecliptic cuts the orbit into two parts having areas in the proportion 186 to 179, while a diameter cuts the orbit into equal parts. So the eccentricity of the orbit of the Earth is approximately

$\varepsilon\approx\frac \pi 4 \frac {186-179}{186+179}\approx 0.015$

which is close to the correct value. (See Earth's orbit).

## Nonzero planetary mass

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

1. From the second law alone it follows that the acceleration is directed towards the Sun.
2. From the two laws together it follows that the magnitude of the acceleration is in inverse proportion to the square of the distance from the Sun.

This suggests that the Sun may be the physical cause of the acceleration of planets.

Newton defined the force on a planet to be the product of its mass and the acceleration. So:

1. Every planet is attracted towards the Sun.
2. The force on a planet is in direct proportion to the mass of the planet and in inverse proportion to the square of the distance between the planet and the Sun.

Here the Sun plays an unsymmetrical part which is unjustified. Newton generalized Kepler's laws by assuming that

1. All bodies in the solar system attract one another with a force of gravitation.
2. The force between two bodies is in direct proportion to 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 to Kepler's laws approximately. Newton's model improves Kepler's model and gives better fit to the observations.

Deviations from Kepler's laws due to attraction from other planets are called perturbations.

The proportionality constant in Kepler's third law is related to the masses according to the following expression:[13]

${a^3 \over \left({{P}/{2\pi}}\right)^2}=G (M+m),$

where a is the length of the semi-major axis of the orbit, P is time per orbit and P/2π is time per radian. G is the gravitational constant, M is the mass of the Sun, and m is the mass of the planet.

The difference between Kepler's constant for Jupiter and that for Earth, due to the mass of Jupiter, is approximately a tenth of a percent. (See data tabulated at Planet attributes).

## Mathematics of the three laws

### First law

Figure 2: Kepler's first law placing the Sun at the focus of an elliptical orbit
"The orbit of every planet is an ellipse with the Sun at a focus."

Symbolically:

$r=\frac{p}{1+\varepsilon\, \cos\theta},$

where (rθ) are heliocentric polar coordinates for the planet, p is the semi-latus rectum, and ε is the eccentricity.

Figure 4: Heliocentric coordinate system (r, θ) for ellipse. Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = rmin and for θ = 180°, r = rmax.

At θ = 0°, the minimum distance is

$r_\mathrm{min}=\frac{p}{1+\varepsilon}.$

At θ = 90°, the distance is $\, p.$

At θ = 180°, the maximum distance is

$r_\mathrm{max}=\frac{p}{1-\varepsilon}.$

The semi-major axis is the arithmetic mean between rmin and rmax:

$a= \frac{r_{\rm min}+r_{\rm max}}{2} =\frac{p}{1-\varepsilon^2}.$

The semi-minor axis is the geometric mean between rmin and rmax:

$b=\sqrt{r_{\rm min}r_{\rm max}} =\frac p{\sqrt{1-\varepsilon^2}}.$

The semi-latus rectum is the harmonic mean between rmin and rmax:

$\frac{1}{r_{\rm min}}-\frac{1}{p}=\frac{1}{p}-\frac{1}{r_{\rm max}}.$

The area of the ellipse is

$A=\pi a b\,.$

The special case of a circle is ε = 0, resulting in

r = p = rmin = rmax = a = b

and

A = π r2.

### Second law

Figure 3: Illustration of Kepler's second law. The planet moves faster near the Sun, so the same area is swept out in a given time as at larger distances, where the planet moves more slowly.
"The line joining a planet and the Sun sweeps out equal areas during equal intervals of time."[1]

Symbolically:

$\frac{d}{dt}\left(\frac{1}{2}r^2 \dot\theta\right) = 0,$

where $\tfrac{1}{2}r^2 \dot\theta$ is the "areal velocity".

This is also known as the law of equal areas. It also applies for a parabolic trajectory and a hyperbolic trajectory and a radial trajectory.

### 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:

${P^2} \propto {a^3}$

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.

$\frac{P_{\rm planet}^2}{a_{\rm planet}^3} = \frac{P_{\rm earth}^2}{a_{\rm earth}^3}.$

So the constant is 1 (sidereal year)2(astronomical unit)−3 or 2.97472505×10−21 s2m−3. See the actual figures: attributes of major planets.

## Position as a function of time

Kepler used these three laws for computing 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 (r,θ) to a planetary position as a function of the time t since perihelion, and the orbital period P, is the following four steps.

1. Compute the mean anomaly M from the formula
$M=\frac{2\pi t}{P}$
2. Compute the eccentric anomaly E by solving Kepler's equation:
$\ M=E-\varepsilon\cdot\sin E$
3. Compute the true anomaly θ by the equation:
$\tan\frac \theta 2 = \sqrt{\frac{1+\varepsilon}{1-\varepsilon}}\cdot\tan\frac E 2$
4. Compute the heliocentric distance r from the first law:
$r=\frac p {1+\varepsilon\cdot\cos\theta}$

The important special case of circular orbit, ε = 0, gives simply θ = E = M.

The proof of this procedure is shown below.

### Proof

FIgure 5: Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeled S and the planet P. The auxiliary circle is an aid to calculation. Line xd is perpendicular to the base and through the planet P. The shaded sectors are arranged to have equal areas by positioning of point y.

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

s the Sun (at one focus of ellipse);
z the perihelion
c the center of the ellipse
p the planet

and

$\ a=|cz|,$ distance between center and perihelion, the semimajor axis,
$\ \varepsilon={|cs|\over a},$ the eccentricity,
$\ b=a\sqrt{1-\varepsilon^2},$ the semiminor axis,
$\ r=|sp| ,$ the distance between Sun and planet.
$\theta=\angle zsp,$ the direction to the planet as seen from the Sun, the true anomaly.

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

$\ x,$ the projection of the planet to the auxiliary circle
$\ y,$ the point on the circle such that the sector areas |zcy| and |zsx| are equal,
$M=\angle zcy,$ the mean anomaly.

The sector areas are related by $|zsp|=\frac b a \cdot|zsx|.$

The circular sector area $\ |zcy| = \frac{a^2 M}2.$

The area swept since perihelion,

$|zsp|=\frac b a \cdot|zsx|=\frac b a \cdot|zcy|=\frac b a\cdot\frac{a^2 M}2$$= \frac {a b M}{2}$ ,

is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.

$M={2 \pi t \over P},$

where P is the orbital period.

The mean anomaly M is first 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

$E=\angle zcx$, 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.

$\ |zcy|=|zsx|=|zcx|-|scx|$
$\frac{a^2 M}2=\frac{a^2 E}2-\frac {a\varepsilon\cdot a\sin E}2$

Division by a2/2 gives Kepler's equation

$M=E-\varepsilon\cdot\sin E.$

This equation gives M as a function of E, determining E for a given M is known as the inverse problem. There are several equations that can be used, but they all have an infinite number of terms. In practice these equations must be truncated which reduces their accuracy. Numerical solvers and iterative algorithms are more commonly used.

Having computed the eccentric anomaly E the next step is to calculate the true anomaly θ from the eccentric anomaly E.

Note from the figure that

$\overrightarrow{cd}=\overrightarrow{cs}+\overrightarrow{sd}$

so that

$a\cdot\cos E=a\cdot\varepsilon+r\cdot\cos \theta.$

Dividing by a and inserting from Kepler's first law

$\ \frac r a =\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta}$

to get

$\cos E =\varepsilon+\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta}\cdot\cos \theta$$=\frac{\varepsilon\cdot(1+\varepsilon\cdot\cos \theta)+(1-\varepsilon^2)\cdot\cos \theta}{1+\varepsilon\cdot\cos \theta}$$=\frac{\varepsilon +\cos \theta}{1+\varepsilon\cdot\cos \theta}.$

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:

$\tan^2\frac{x}{2}=\frac{1-\cos x}{1+\cos x}.$

Get

$\tan^2\frac{E}{2} =\frac{1-\cos E}{1+\cos E}$$=\frac{1-\frac{\varepsilon+\cos \theta}{1+\varepsilon\cdot\cos \theta}}{1+\frac{\varepsilon+\cos \theta}{1+\varepsilon\cdot\cos \theta}}$$=\frac{(1+\varepsilon\cdot\cos \theta)-(\varepsilon+\cos \theta)}{(1+\varepsilon\cdot\cos \theta)+(\varepsilon+\cos \theta)}$$=\frac{1-\varepsilon}{1+\varepsilon}\cdot\frac{1-\cos \theta}{1+\cos \theta}=\frac{1-\varepsilon}{1+\varepsilon}\cdot\tan^2\frac{\theta}{2}.$

Multiplying by (1+ε)/(1−ε) and taking the square root gives the result

$\tan\frac \theta2=\sqrt\frac{1+\varepsilon}{1-\varepsilon}\cdot\tan\frac E2.$

We have now completed the third step in the connection between time and position in the orbit.

One could even develop a series computing θ directly from M. [2]

The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:

$\ r=a\cdot\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta}.$

## Derivation from Newton's laws of motion and Newton's law of gravitation

Kepler's laws are concerned with the motion of the planets around the Sun. Newton's laws of motion in general are concerned with the motion of objects subject to impressed forces. Newton's law of universal gravitation describes how masses attract each other through the force of gravity. Using the law of gravitation to determine the impressed forces in Newton's laws of motion enables the calculation of planetary orbits, as discussed below.

In the special case where there are only two particles, the motion of the bodies is the exactly soluble two-body problem, of which an approximate example is the motion of a planet around the Sun according to Kepler's laws, as shown below. The trajectory of the lighter particle may also be a parabola or a hyperbola or a straight line.

In the case of a single planet orbiting its Sun, Newton's laws imply elliptical motion. The focus of the ellipse is at the center of mass of the Sun and the planet (the barycenter), rather than located at the center of the Sun itself. The period of the orbit depends a little on the mass of the planet. In the realistic case of many planets, the interaction from other planets modifies the orbit of any one planet. Even in this more complex situation, the language of Kepler's laws applies as the complicated orbits are described as simple Kepler orbits with slowly varying orbital elements. See also Kepler problem in general relativity.

While Kepler's laws are expressed either in geometrical language, or as equations connecting the coordinates of the planet and the time variable with the orbital elements, Newton's second law is a differential equation. So the derivations below involve the art of solving differential equations. Kepler's second law is derived first, as the derivation of the first law depends on the derivation of the second law. The derivations that follow use heliocentric polar coordinates, that is, polar coordinates with the Sun as the origin. See Figure 4. However, they can alternatively be formulated and derived using Cartesian coordinates.[14][15]

### Equations of motion

Assume that the planet is so much lighter than the Sun that the acceleration of the Sun can be neglected. In other words, the barycenter is approximated as the center of the Sun. Introduce the polar coordinate system in the plane of the orbit, with radial coordinate from the Sun's center, r and angle from some arbitrary starting direction θ.

Newton's law of gravitation says that "every object in the universe attracts every other object along a line of the centers of the objects, proportional to each object's mass, and inversely proportional to the square of the distance between the objects," and his second law of motion says that "the mass times the acceleration is equal to the force." So the mass of the planet times the acceleration vector of the planet equals the mass of the Sun times the mass of the planet, divided by the square of the distance, times minus the radial unit vector $\hat{\mathbf{r}}$, times a constant of proportionality. This is written:

$m\ddot\mathbf{r} = -\frac{G M m}{r^2} \hat{\mathbf{r}}$

where a dot on top of the variable signifies differentiation with respect to time, the second dot indicates the second derivative, and the position vector $\mathbf{r} = r \hat{\mathbf{r}}$.

In polar coordinates, as discussed in Vector calculus and Polar coordinates in an inertial frame of reference

$\dot\hat{\mathbf{r}} = \dot\theta \hat{\boldsymbol\theta}, \qquad \dot\hat{\boldsymbol\theta} = -\dot\theta \hat{\mathbf{r}}$

where $\hat{\boldsymbol\theta}$ is the tangential (azimuthal) unit vector orthogonal to $\hat{\mathbf{r}}$ and pointing in the direction of rotation, and θ is the polar angle.

So differentiating the position vector $\mathbf{r} = r \hat{\mathbf{r}}$ twice to obtain the velocity and the acceleration vectors:

$\dot\mathbf{r} =\dot r \hat\mathbf{r} + r \dot\hat\mathbf{r} =\dot r \hat{\mathbf{r}} + r \dot\theta \hat{\boldsymbol\theta},$
$\ddot\mathbf{r} = (\ddot r \hat{\mathbf{r}} +\dot r \dot\hat{\mathbf{r}} ) + (\dot r\dot\theta \hat{\boldsymbol\theta} + r\ddot\theta \hat{\boldsymbol\theta} + r\dot\theta \dot\hat{\boldsymbol\theta}) = (\ddot r - r\dot\theta^2) \hat{\mathbf{r}} + (r\ddot\theta + 2\dot r \dot\theta) \hat{\boldsymbol\theta}.$

Note that for constant distance, $\ r$, the planet is subject to the centripetal acceleration, $r\dot\theta^2$, and for constant angular speed, $\dot\theta$, the planet is subject to the Coriolis acceleration, $2\dot r \dot\theta$.[16]

Inserting the acceleration vector into Newton's laws, and dividing by m, gives the vector equation of motion

$(\ddot r - r\dot\theta^2) \hat{\mathbf{r}} + (r\ddot\theta + 2\dot r \dot\theta) \hat{\boldsymbol\theta}= -GMr^{-2}\hat{\mathbf{r}}$

Equating components, we get the two ordinary differential equations of motion, one for the acceleration in the $\hat{\mathbf r}$ direction, the radial acceleration

$\ddot r - r\dot\theta^2 = -GMr^{-2},$

and one for the acceleration in the $\hat{\boldsymbol\theta}$ direction, the tangential or azimuthal acceleration:

$r\ddot\theta + 2\dot r\dot\theta = 0.$

### Deriving Kepler's second law

Only the tangential acceleration equation is needed to derive Kepler's second law.

The magnitude of the specific angular momentum

$\ell = r^2 \dot \theta$

is a constant of motion, even if both the distance $\ r$, and the angular speed $\dot\theta$, and the tangential velocity $r \dot \theta$, vary, because

$\frac{d\ell}{dt} =\frac{d(r^2 \dot \theta)}{dt} = r^2 \ddot \theta+2r \dot r\dot \theta=r(r \ddot \theta+2\dot r \dot \theta)=0$

where the expression in the last parentheses vanishes due to the tangential acceleration equation.

The area swept out from time t1 to time t2,

$\ \int_{t_1}^{t_2}\frac 1 2 \cdot {\rm base}\cdot d({\rm height}) = \int_{t_1}^{t_2}\frac 1 2 \cdot r\cdot r\dot \theta dt=\frac 1 2 \cdot\ell \cdot(t_2-t_1)$

depends only on the duration t2t1. This is Kepler's second law.

### Deriving Kepler's first law

To derive Kepler's first law, define:

$\ u =pr^{-1}\,$

where the constant

$p=\ell ^2 G^{-1}M^{-1}\,$

has the dimension of length. Then

$GMr^{-2}=\ell^2 p^{-3}u^{2}$

and

$\ \dot \theta =\ell r^{-2}=\ell p^{-2}u^2.$

Differentiation with respect to time is transformed into differentiation with respect to angle:

$\ \dot X=\frac {dX}{dt}=\frac {dX}{d\theta}\cdot\frac {d\theta}{dt}=\frac {dX}{d \theta}\cdot \dot\theta=\frac {dX}{d \theta}\cdot\ell p^{-2}u^2.$

Differentiate

$\ r =pu^{-1}$

twice:

$\dot r = \frac{d(pu^{-1})}{d\theta}\cdot\ell p^{-2}u^{2} = -pu^{-2}\frac{du}{d\theta}\cdot\ell p^{-2}u^{2}= -\ell p^{-1}\frac{du}{d\theta}$
$\ddot r = \frac{d\dot r}{d\theta}\cdot\ell p^{-2}u^{2} = \frac{d}{d\theta}\left(-\ell p^{-1} \frac{du}{d\theta}\right)\cdot\ell p^{-2}u^{2} = -\ell^2 p^{-3}u^{2}\frac{d^2 u}{d\theta^2}$

Substitute into the radial equation of motion

$\ddot r - r\dot\theta^2 = -GMr^{-2}$

and get

$\left(-\ell^2 p^{-3}u^2\frac{d^2u}{d\theta^2}\right) - (pu^{-1})(\ell p^{-2}u^2)^2 = -\ell ^2 p^{-3} u^2$

Divide by $-\ell^2 p^{-3}u^2$ to get a simple non-homogeneous linear differential equation for the orbit of the planet:

$\frac{d^2u}{d\theta^2} + u = 1 .$

An obvious solution to this equation is the circular orbit

$\ u = 1.$

Other solutions are obtained by adding solutions to the homogeneous linear differential equation with constant coefficients

$\frac{d^2u}{d\theta^2} + u = 0$

These solutions are

$\ u = \varepsilon\cdot\cos(\theta-\theta_0)$

where $\ \varepsilon$ and $\theta_0\,$ are arbitrary constants of integration. So the result is

$\ u = 1+ \varepsilon\cdot\cos(\theta-\theta_0)$

Choosing the axis of the coordinate system such that $\ \theta_0=0$, and inserting $\ u=pr^{-1}$, gives:

$\ pr^{-1 } = 1+ \varepsilon\cdot\cos\theta .$

If $\ |{ \varepsilon}|<1 ,$ this is Kepler's first law.

### Deriving Kepler's third law

In the special case of circular orbits, which are ellipses with zero eccentricity, the relation between the radius a of the orbit and its period P can be derived relatively easily. The centripetal force of circular motion is proportional to a/P2, and it is provided by the gravitational force, which is proportional to 1/a2. Hence,

$P^2 \propto a^3$

which is Kepler's third law for the special case.

In the general case of elliptical orbits, the derivation is more complicated.

The area of the planetary orbit ellipse is

$A=\pi a b=\pi a(a\sqrt{1-\varepsilon^2})=\pi a^2\sqrt{1-\varepsilon^2}\, .$

The area speed of the radius vector sweeping the orbit area is

$\dot A=\ell/ 2 \,$

where

$\ell^2=pGM=a(1-\varepsilon^2)GM \,.$

The period of the orbit is

$P=\frac A \dot A =\frac {\pi a^2\sqrt{1-\varepsilon^2}}{\ell/2}=2\pi\frac {a^2\sqrt{1-\varepsilon^2}}{\ell}\,$

satisfying

$\left(\frac P{2 \pi}\right)^2=\left( \frac {a^2\sqrt{1-\varepsilon^2}}{\ell}\right)^2 = \frac {a^4(1-\varepsilon^2)}{\ell^2} = \frac {a^4(1-\varepsilon^2)}{a(1-\varepsilon^2)GM }=\frac{a^3}{GM}\,$

implying Kepler's third law

$P^2 \propto a^3 \ .$

## Notes

1. ^ a b Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law", Wolfram Demonstrations Project. Retrieved December 27, 2009.
2. ^ a b Holton, Gerald James; Brush, Stephen G. (2001). Physics, the Human Adventure: From Copernicus to Einstein and Beyond. Rutgers University Press. pp. 40–41. ISBN 0-8135-2908-5. Retrieved December 27, 2009.
3. ^ a b c See also G E Smith, "Newton's Philosophiae Naturalis Principia Mathematica", especially the section Historical context ... in The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.).
4. ^ a b c Newton's showing, in the 'Principia', that the two-body problem with centripetal forces results in motion in one of the conic sections, is concluded at Book 1, Proposition 13, Corollary 1. His consideration of the effects of perturbations in a multi-body situation starts at Book 1, Proposition 65, including a limit argument that the error in the (Keplerian) approximation of ellipses and equal areas would tend to zero if the relevant planetary masses would tend to zero and with them the planetary mutual perturbations (Proposition 65, Case 1). He discusses the extent of the perturbations in the real solar system in Book 3, Proposition 13.
5. ^ Kepler "for the first time revealed" a "real approximation to the true kinematical relations [motions] of the solar system", see page 1 in H C Plummer (1918), An introductory treatise on dynamical astronomy, Cambridge, 1918.
6. ^ Wilson, Curtis (May 1994). "Kepler's Laws, So-Called". HAD News (Washington, DC: Historical Astronomy Division, American Astronomical Society) (31): 1-2. Retrieved December 27, 2009.
7. ^ Astronomical Almanac for 2008, page K7.
8. ^ The fact was already stated by Newton ('Principia', Book 3, Proposition 12).
9. ^
10. ^ Gerald James Holton, Stephen G. Brush (2001). Physics, the Human Adventure. Rutgers University Press. p. 45. ISBN 0813529085.
11. ^ Burtt, Edwin. The Metaphysical Foundations of Modern Physical Science. p. 52.
13. ^ Gerald James Holton, Stephen G. Brush (2001). Physics, the Human Adventure. Rutgers University Press. p. 136. ISBN 0813529085.
14. ^ Hyman, Andrew. "A Simple Cartesian Treatment of Planetary Motion", European Journal of Physics, Vol. 14, pp. 145–147 (1993).
15. ^ Further derivations are given in E T Whittaker, Analytical Dynamics (2nd edn. Cambridge, 1917), see pages 86 onwards.
16. ^ Although this term is called the "Coriolis acceleration", or the "Coriolis force per unit mass", it should be noted that the term "Coriolis force" as used in meteorology, for example, refers to something different: namely the force, similar in mathematical form, but caused by rotation of a frame of reference. Of course, in the example here of planetary motion, the entire analysis takes place in a stationary, inertial frame, so there is no force present related to rotation of a frame of reference.

## References

• Kepler's life is summarized on pages 627–623 and Book Five of his magnum opus, Harmonice Mundi (harmonies of the world), is reprinted on pages 635–732 of On the Shoulders of Giants: The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4
• A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pages 161–164 of Meriam, J. L. (1966, 1971), Dynamics, 2nd ed., New York: John Wiley, ISBN 0-471-59601-9 .
• Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN 0-521-57597-4

• B.Surendranath Reddy; animation of Kepler's laws: applet
• Crowell, Benjamin, Conservation Laws, http://www.lightandmatter.com/area1book2.html, an online book that gives a proof of the first law without the use of calculus. (see section 5.2, p. 112)
• David McNamara and Gianfranco Vidali, Kepler's Second Law - Java Interactive Tutorial, http://www.phy.syr.edu/courses/java/mc_html/kepler.html, an interactive Java applet that aids in the understanding of Kepler's Second Law.
• University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion [3]
• Equant compared to Kepler: interactive model [4]
• Kepler's Third Law:interactive model [5]