Kepler's laws of planetary motion

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

Kepler's laws are: #The orbit of every planet is an ellipse with the Sun at one of the two foci. #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.

Kepler's laws are strictly only valid for a lone (not affected by the gravity of other planets) zero-mass object orbiting the Sun; a physical impossibility. Nevertheless, Kepler's laws form a useful starting point to calculating the orbits of planets that do not deviate too much from these restrictions.

Isaac Newton solidified Kepler's laws by showing that they were a natural consequence of his inverse square law of gravity with the limits set in the previous paragraph. Further, Newton extended Kepler's laws in a number of important ways such as allowing the calculation of orbits around other celestial bodies.

The past Johannes Kepler published his first two laws in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler did not discover his third law until many years later, and it was published in 1619. 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. Voltaire's Eléments de la philosophie de Newton (Elements of Newton's Philosophy) was in 1738 the first publication to call Kepler's Laws "laws". Together with Newton's mathematical theories, they are part of the foundation of modern astronomy and physics.

First Law

:"The orbit of every planet is an ellipse with the Sun at one of the two foci."

An ellipse is a particular class of mathematical shapes that resemble a stretched out circle. (See the figure to the right.) Note as well that the Sun is not at the center of the ellipse but is at one of the focal points. The other focal point is marked with a lighter dot but is a point that has no physical significance for the orbit. Ellipses have two focal points neither of which are in the center of the ellipse (except for the one special case of the ellipse being a circle). Circles are a special case of an ellipse that are not stretched out and in which both focal points coincide at the center.

How stretched out that ellipse is from a perfect circle is known as its eccentricity; a parameter that varies from 0 (a simple circle) to 1 (an ellipse that is so stretched out that it is a straight line back and forth between the two focal points). The eccentricities of the planets known to Kepler varies from 0.007 (Venus) to 0.2 (Mercury). (See List of planetary objects in the Solar System for more detail.) Yet even the orbit of Mercury is not that far off of round.

After Kepler, though, bodies with highly eccentric orbits have been identified, among them many comets and asteroids. 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.

Symbolically an ellipse can be represented in polar coordinates as: :$r=\backslash frac\{p\}\{1+\backslash varepsilon\backslash ,\; \backslash cos\backslash theta\},$ where (r, θ) are the polar coordinates (from the focus) for the ellipse, p is the semi-latus rectum, and ε is the eccentricity of the ellipse. For a planet orbiting the Sun then r is the distance from the Sun to the planet and θ is the angle with its vertex at the Sun from the location where the planet is closest to the Sun.

At θ = 0°, perihelion, the distance is minimum :$r\_\backslash mathrm\{min\}=\backslash frac\{p\}\{1+\backslash varepsilon\}.$

At θ = 90° and at θ = 270°, the distance is $\backslash ,\; p.$

At θ = 180°, aphelion, the distance is maximum :$r\_\backslash mathrm\{max\}=\backslash frac\{p\}\{1-\backslash varepsilon\}.$

The semi-major axis a is the arithmetic mean between r_min and r_max: :$\backslash ,r\_\backslash max\; -\; a=a-r\_\backslash min$ so :$a=\backslash frac\{p\}\{1-\backslash varepsilon^2\}.$

The semi-minor axis b is the geometric mean between r_min and r_max: :$\backslash frac\{r\_\backslash max\}\; b\; =\backslash frac\; b\{r\_\backslash min\}$ so :$b=\backslash frac\; p\{\backslash sqrt\{1-\backslash varepsilon^2\}\}.$

The semi-latus rectum p is the harmonic mean between r_min and r_max: :$\backslash frac\{1\}\{r\_\backslash min\}-\backslash frac\{1\}\{p\}=\backslash frac\{1\}\{p\}-\backslash frac\{1\}\{r\_\backslash max\}.$

The eccentricity ε is the coefficient of variation between r_min and r_max: :$\backslash varepsilon=\backslash frac\{r\_\backslash mathrm\{max\}-r\_\backslash mathrm\{min\}\}\{r\_\backslash mathrm\{max\}+r\_\backslash mathrm\{min\}\}.$

The area of the ellipse is :$A=\backslash pi\; a\; b\backslash ,.$ The special case of a circle is ε = 0, resulting in r = p = r_min = r_max = a = b and A = π r².

Second law

:"A line joining a planet and the Sun sweeps out equal areas during equal intervals of time." 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.

This third law currently receives additional attention as it can be used to estimate the distance from an exoplanet to its central star, and help to decide if this distance is inside the habitable zone of that star.

Symbolically: :$P^2\; \backslash propto\; a^3\; \backslash ,$ 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.

:$\backslash frac\{P\_\{\backslash rm\; planet\}^2\}\{a\_\{\backslash rm\; planet\}^3\}\; =\; \backslash frac\{P\_\{\backslash rm\; earth\}^2\}\{a\_\{\backslash rm\; earth\}^3\}.$

So the constant is 1 (sidereal year)²(astronomical unit)⁻³ or 2.97472505×10⁻¹⁹ s²m⁻³. See the actual figures: attributes of major planets.

For general primary bodies, the proportionality constant can be calculated as follows:

:$\backslash frac\; \{P^2\}\; \{a^3\}\; =\; \backslash frac\; \{4\; \backslash pi^2\}\; \{M\; G\},$

where M is the mass of the primary body (assumed to be much larger than that of the secondary,) and G is the gravitational constant.

See also scaling in gravity.

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 exactitude 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 exactitude as all except one of the masses become closer to zero mass and as the perturbations then also tend towards zero). 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. 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.

Zero eccentricity

Kepler's laws refine the model of Copernicus. If the eccentricity of a planetary orbit is zero, then Kepler's laws state: #The planetary orbit is a circle #The Sun is in the center #The speed of the planet in the orbit is constant #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: #The planetary orbit is not a circle, but an ellipse #The Sun is not at the center but at a focal point #Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed is constant. #The square of the sidereal period is proportionate to the cube of the mean between the maximum and minimum distances from the Sun.

Estimating the eccentricity of earth orbit

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 equator 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 :$\backslash varepsilon\backslash approx\backslash frac\; \backslash pi\; 4\; \backslash frac\; \{186-179\}\{186+179\}\backslash approx\; 0.015$ close to the correct value (0.016710219). (See Earth's orbit).

The accuracy of this approximation depends on the date of the perihelion, the date that the Earth is closest to the Sun. When the perihelion occurs on an equinox, the equation above gives a value of zero. When the date is on a solstice, the value will be at a maximum. The current perihelion, near January 4, is fairly close to the Winter Solstice on December 21 or 22.

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. #The direction of the acceleration is towards the Sun. #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. (See Newton's laws of motion). So: #Every planet is attracted towards the Sun. #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 from the Sun.

Here the Sun plays an unsymmetrical part which is unjustified. So he assumed 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 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. See two-body problem.

The deviation of the motion of a planet from Kepler's laws due to attraction from other planets is called a perturbation.

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

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

The proof of this procedure is shown below.

Mean anomaly

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 :$\backslash \; a=|cz|,$ distance between center and perihelion, the semimajor axis, :$\backslash \; \backslash varepsilon=\{|cs|\backslash over\; a\},$ the eccentricity, :$\backslash \; b=a\backslash sqrt\{1-\backslash varepsilon^2\},$ the semiminor axis, :$\backslash \; r=|sp|\; ,$ the distance between Sun and planet. :$\backslash theta=\backslash angle\; zsp,$ 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 :$\backslash \; x,$ the projection of the planet to the auxiliary circle :$\backslash \; y,$ the point on the circle such that the sector areas |zcy| and |zsx| are equal, :$M=\backslash angle\; zcy,$ the mean anomaly.

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

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

The area swept since perihelion, :$|zsp|=\backslash frac\; b\; a\; \backslash cdot|zsx|=\backslash frac\; b\; a\; \backslash cdot|zcy|=\backslash frac\; b\; a\backslash cdot\backslash frac\{a^2\; M\}2\; =\; \backslash frac\; \{a\; b\; M\}\{2\},$ is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t. :$M=\{2\; \backslash pi\; t\; \backslash over\; P\},$ where P is the orbital period.

Eccentric anomaly

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 :$E=\backslash angle\; zcx$, 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. :$\backslash \; |zcy|=|zsx|=|zcx|-|scx|$

:$\backslash frac\{a^2\; M\}2=\backslash frac\{a^2\; E\}2-\backslash frac\; \{a\backslash varepsilon\backslash cdot\; a\backslash sin\; E\}2$ Division by a²/2 gives Kepler's equation :$M=E-\backslash varepsilon\backslash cdot\backslash sin\; E.$

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. Series are also used; they must be truncated, reducing accuracy.

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

True anomaly

Note from the figure that :$\backslash overrightarrow\{cd\}=\backslash overrightarrow\{cs\}+\backslash overrightarrow\{sd\}$ so that :$a\backslash cdot\backslash cos\; E=a\backslash cdot\backslash varepsilon+r\backslash cdot\backslash cos\; \backslash theta.$ Dividing by $a$ and inserting from Kepler's first law :$\backslash \; \backslash frac\; r\; a\; =\backslash frac\{1-\backslash varepsilon^2\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}$ to get :$\backslash cos\; E\; =\backslash varepsilon+\backslash frac\{1-\backslash varepsilon^2\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}\backslash cdot\backslash cos\; \backslash theta$ $=\backslash frac\{\backslash varepsilon\backslash cdot(1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta)+(1-\backslash varepsilon^2)\backslash cdot\backslash cos\; \backslash theta\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}$ $=\backslash frac\{\backslash varepsilon\; +\backslash cos\; \backslash theta\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}.$ 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: :$\backslash tan^2\backslash frac\{x\}\{2\}=\backslash frac\{1-\backslash cos\; x\}\{1+\backslash cos\; x\}.$ Get :$\backslash tan^2\backslash frac\{E\}\{2\}\; =\backslash frac\{1-\backslash cos\; E\}\{1+\backslash cos\; E\}$ $=\backslash frac\{1-\backslash frac\{\backslash varepsilon+\backslash cos\; \backslash theta\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}\}\{1+\backslash frac\{\backslash varepsilon+\backslash cos\; \backslash theta\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}\}$ $=\backslash frac\{(1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta)-(\backslash varepsilon+\backslash cos\; \backslash theta)\}\{(1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta)+(\backslash varepsilon+\backslash cos\; \backslash theta)\}$ $=\backslash frac\{1-\backslash varepsilon\}\{1+\backslash varepsilon\}\backslash cdot\backslash frac\{1-\backslash cos\; \backslash theta\}\{1+\backslash cos\; \backslash theta\}=\backslash frac\{1-\backslash varepsilon\}\{1+\backslash varepsilon\}\backslash cdot\backslash tan^2\backslash frac\{\backslash theta\}\{2\}.$ Multiplying by (1+ε)/(1−ε) and taking the square root gives the result :$\backslash tan\backslash frac\; \backslash theta2=\backslash sqrt\backslash frac\{1+\backslash varepsilon\}\{1-\backslash varepsilon\}\backslash cdot\backslash tan\backslash frac\; E2.$

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.

Distance

The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law: :$\backslash \; r=a\backslash cdot\backslash frac\{1-\backslash varepsilon^2\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}.$

Derivation from Newton's laws

Kepler's laws are concerned with the motion of the planets around the Sun. They are expressed as equations connecting the coordinates of the planet, and the time variable, with the parameters describing the position, size and shape of the orbit, the socalled orbital elements.

Newton's laws of motion are concerned with the motion of objects subject to impressed forces. Newton's law of universal gravitation specifies these forces. Together these laws constitute differential equations satisfied by planetary motions. Solving these equations constitute the n-body problem.

The solutions to the two-body problem, where there are only two particles involved, say, the sun and one planet, can be expressed analytically. These solutions include the elliptical Kepler orbits, but motions along other conic section (parabolas, hyperbolas and straight lines) also satisfy Newton's differential equations.

The solutions deviate from Kepler's laws in that #the focus of the conic section is at the center of mass of the two bodies, rather than at the center of the Sun itself. #the period of the orbit depends a little on the mass of the planet.

The language of Kepler's laws also applies when the motion of a planet is affected by the attraction from the other planets, as the orbits are described as Kepler orbits with slowly varying orbital elements and in the case of the two-body problem in general relativity.

The derivations below involve the art of solving differential equations. The derivations below use heliocentric polar coordinates, see Figure 4. Kepler's second law is derived first, as the derivation of the first law depends on the derivation of the second law.

They can also be formulated and derived using Cartesian coordinates.

Acceleration vector

From the heliocentric point of view consider the vector to the planet $\backslash mathbf\{r\}\; =\; r\; \backslash hat\{\backslash mathbf\{r\}\}$ where $r$ is the distance to the planet and the direction $\backslash hat\; \{\backslash mathbf\{r\}\}$ is a unit vector. When the planet moves the direction vector $\backslash hat\; \{\backslash mathbf\{r\}\}$ changes: :$\backslash frac\{d\backslash hat\{\backslash mathbf\{r\}\}\}\{dt\}=\backslash dot\backslash hat\{\backslash mathbf\{r\}\}\; =\; \backslash dot\backslash theta\; \backslash hat\{\backslash boldsymbol\backslash theta\},\; \backslash qquad\; \backslash dot\backslash hat\{\backslash boldsymbol\backslash theta\}\; =\; -\backslash dot\backslash theta\; \backslash hat\{\backslash mathbf\{r\}\}$ where $\backslash scriptstyle\; \backslash hat\{\backslash boldsymbol\backslash theta\}$ is the tangential (azimuthal) unit vector orthogonal to $\backslash scriptstyle\; \backslash hat\{\backslash mathbf\{r\}\}$ and pointing in the direction of rotation, and $\backslash scriptstyle\; \backslash theta$ 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: :$\backslash dot\backslash mathbf\{r\}\; =\backslash dot\; r\; \backslash hat\backslash mathbf\{r\}\; +\; r\; \backslash dot\backslash hat\backslash mathbf\{r\}\; =\backslash dot\; r\; \backslash hat\{\backslash mathbf\{r\}\}\; +\; r\; \backslash dot\backslash theta\; \backslash hat\{\backslash boldsymbol\backslash theta\},$ :$\backslash ddot\backslash mathbf\{r\}\; =\; (\backslash ddot\; r\; \backslash hat\{\backslash mathbf\{r\}\}\; +\backslash dot\; r\; \backslash dot\backslash hat\{\backslash mathbf\{r\}\}\; )\; +\; (\backslash dot\; r\backslash dot\backslash theta\; \backslash hat\{\backslash boldsymbol\backslash theta\}\; +\; r\backslash ddot\backslash theta\; \backslash hat\{\backslash boldsymbol\backslash theta\}\; +\; r\backslash dot\backslash theta\; \backslash dot\backslash hat\{\backslash boldsymbol\backslash theta\})\; =\; (\backslash ddot\; r\; -\; r\backslash dot\backslash theta^2)\; \backslash hat\{\backslash mathbf\{r\}\}\; +\; (r\backslash ddot\backslash theta\; +\; 2\backslash dot\; r\; \backslash dot\backslash theta)\; \backslash hat\{\backslash boldsymbol\backslash theta\}.$

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

Equations of motion

Assume that the planet is so much lighter than the Sun that the acceleration of the Sun can be neglected.

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 ($\backslash scriptstyle\; m$) times the acceleration vector of the planet ($\backslash scriptstyle\; \backslash ddot\backslash mathbf\{r\}$) equals the mass of the Sun ($\backslash scriptstyle\; M$) times the mass of the planet ($\backslash scriptstyle\; m$), divided by the square of the distance ($\backslash scriptstyle\; r$), times minus the radial unit vector ($\backslash scriptstyle\backslash hat\{\backslash mathbf\{r\}\}$), times a constant of proportionality ($\backslash scriptstyle\; G$). This is written:

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

Dividing by $\backslash scriptstyle\; m$ and inserting the acceleration vector gives the vector equation of motion :$(\backslash ddot\; r\; -\; r\backslash dot\backslash theta^2)\; \backslash hat\{\backslash mathbf\{r\}\}\; +\; (r\backslash ddot\backslash theta\; +\; 2\backslash dot\; r\; \backslash dot\backslash theta)\; \backslash hat\{\backslash boldsymbol\backslash theta\}=\; -GMr^\{-2\}\backslash hat\{\backslash mathbf\{r\}\}$

Equating components, we get the two ordinary differential equations of motion, one for the acceleration in the $\backslash scriptstyle\; \backslash hat\{\backslash mathbf\; r\}$ direction, the radial acceleration :$\backslash ddot\; r\; -\; r\backslash dot\backslash theta^2\; =\; -GMr^\{-2\},$ and one for the acceleration in the $\backslash scriptstyle\; \backslash hat\{\backslash boldsymbol\backslash theta\}$ direction, the tangential or azimuthal acceleration: :$r\backslash ddot\backslash theta\; +\; 2\backslash dot\; r\backslash dot\backslash theta\; =\; 0.$

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

:$\backslash ell\; =\; r^2\; \backslash dot\; \backslash theta\; \backslash ,$

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

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

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

The area swept out from time t₁ to time t₂,

:$\backslash \; \backslash int\_\{t\_1\}^\{t\_2\}\backslash frac\; 1\; 2\; \backslash cdot\; \{\backslash rm\; base\}\backslash cdot\; d(\{\backslash rm\; height\})\; =\; \backslash int\_\{t\_1\}^\{t\_2\}\backslash frac\; 1\; 2\; \backslash cdot\; r\backslash cdot\; r\backslash dot\; \backslash theta\; \backslash ,\; dt\; =\; \backslash frac\; 1\; 2\; \backslash cdot\backslash ell\; \backslash cdot(t\_2-t\_1)$

depends only on the duration t₂ − t₁. This is Kepler's second law.

Deriving Kepler's first law

To derive Kepler's first law, define:

:$\backslash \; u\; =pr^\{-1\}\backslash ,$

where the constant

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

has the dimension of length. Then

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

and

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

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

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

Differentiate

:$\backslash \; r\; =pu^\{-1\}$

twice:

:$\backslash dot\; r\; =\; \backslash frac\{d(pu^\{-1\})\}\{d\backslash theta\}\backslash cdot\backslash ell\; p^\{-2\}u^\{2\}\; =\; -pu^\{-2\}\backslash frac\{du\}\{d\backslash theta\}\backslash cdot\backslash ell\; p^\{-2\}u^\{2\}=\; -\backslash ell\; p^\{-1\}\backslash frac\{du\}\{d\backslash theta\}$

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

Substitute into the radial equation of motion

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

and get

:$-\backslash ell^2\; p^\{-3\}u^2\backslash frac\{d^2u\}\{d\backslash theta^2\}\; -\; (pu^\{-1\})(\backslash ell\; p^\{-2\}u^2)^2\; =\; -\backslash ell\; ^2\; p^\{-3\}\; u^2$

Divide by the right hand side to get a simple non-homogeneous linear differential equation for the orbit of the planet:

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

An obvious solution to this equation is the circular orbit

:$\backslash \; u\; =\; 1.$

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

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

These solutions are

:$\backslash \; u\; =\; \backslash varepsilon\backslash cdot\backslash cos(\backslash theta-\backslash theta\_0)$

where $\backslash scriptstyle\; \backslash varepsilon\; \backslash ,$ and $\backslash scriptstyle\; \backslash theta\_0\backslash ,$ are arbitrary constants of integration. So the result is

:$\backslash \; u\; =\; 1+\; \backslash varepsilon\backslash cdot\backslash cos(\backslash theta-\backslash theta\_0)$

Choosing the axis of the coordinate system such that $\backslash scriptstyle\; \backslash ,\; \backslash theta\_0=0$, and inserting $\backslash scriptstyle\; u=pr^\{-1\}$, gives:

:$\backslash \; pr^\{-1\; \}\; =\; 1+\; \backslash varepsilon\backslash cdot\backslash cos\backslash theta\; ,$

:$\backslash \; r\; =\; \backslash frac\{p\}\{1+\; \backslash varepsilon\backslash cdot\backslash cos\backslash theta\}\; .$

If this is the equation of an ellipse and illustrates 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,

:$P^2\; \backslash propto\; a^3$

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

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

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

:$\backslash dot\; A=\backslash ell/\; 2\; \backslash ,$

where

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

The period of the orbit is

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

satisfying

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

implying Kepler's third law

:$P^2\; \backslash propto\; a^3.\; \backslash ,$

See also

Kepler orbit

Kepler problem

Kepler's equation

Circular motion

Gravity

Two-body problem

Free-fall time

Laplace–Runge–Lenz vector

Notes

References

Kepler's life is summarized on pages 523–627 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 .

Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN 0-521-57597-4

V.I. Arnold, Mathematical Methods of Classical Mechanics, Chapter 2. Springer 1989, ISBN 0-387-96890-3

External links

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.

Audio - Cain/Gay (2010) Astronomy Cast Johannes Kepler and His Laws of Planetary Motion

University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion

Equant compared to Kepler: interactive model

Kepler's Third Law:interactive model

Solar System Simulator (Interactive Applet)

Kepler and His Laws, educational web pages by David P. Stern

Category:Johannes Kepler Category:Celestial mechanics Category:Equations