Definitions

Kepler's laws of planetary motion

In astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of planets in the Solar System. German mathematician and astronomer Johannes Kepler (15711630) discovered them.

Kepler studied the observations (the Rudolphine tables) of Tycho Brahe. Around 1605, Kepler found that Brahe's observations of the planets' positions followed three relatively simple mathematical laws.

Kepler's laws challenged Aristotelean and Ptolemaic astronomy and physics. His assertion that the Earth moved, his use of ellipses rather than epicycles, and his proof that the planets' speeds varied, changed astronomy and physics. Almost a century later Isaac Newton was able to deduce Kepler's laws from Newton's own laws of motion and his law of universal gravitation, using classical Euclidean geometry.

In modern times, Kepler's laws are used to calculate approximate orbits for artificial satellites, and bodies orbiting the Sun of which Kepler was unaware (such as the outer planets and smaller asteroids). They apply where any relatively small body is orbiting a larger, relatively massive body, though the effects of atmospheric drag (e.g. in a low orbit), relativity (e.g. Perihelion precession of Mercury), and other nearby bodies can make the results insufficiently accurate for a specific purpose.

Kepler's laws are formulated below using heliocentric polar coordinates. However, they can alternatively be formulated and derived using Cartesian coordinates.

First law

The first law says:"The orbit of every planet is an ellipse with the sun at one of the foci."

The mathematics of the ellipse is as follows. The equation is

$r=frac\left\{p\right\}\left\{1+epsiloncdotcosnu\right\}$
where (r,ν) are heliocentricical polar coordinates for the planet, p is the semi-latus rectum, and ε is the eccentricity, which is greater than or equal to zero, and less than one. For ν=0 the planet is at the perihelion at minimum distance:
$r_mathrm\left\{min\right\}=frac\left\{p\right\}\left\{1+epsilon\right\}$

for ν=90°: r=p, and for ν=180° the planet is at the aphelion at maximum distance:

$r_mathrm\left\{max\right\}=frac\left\{p\right\}\left\{1-epsilon\right\}$

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

$a=frac\left\{p\right\}\left\{1-epsilon^2\right\}$
The semi-minor axis is the geometric mean between rmin and rmax:
$b=frac p\left\{sqrt\left\{1-epsilon^2\right\}\right\}$
and it is also the geometric mean between the semimajor axis and the semi latus rectum:
$frac a b=frac b p$

Second law

The second law: "A line joining a planet and the sun sweeps out equal areas during equal intervals of time.

This is also known as the law of equal areas. It is a direct consequence of the law of conservation of angular momentum; see the derivation below.

Suppose a planet takes one day to travel from point A to B. The lines from the Sun to A and B, together with the planet orbit, will define a (roughly triangular) area. This same amount of area will be formed every day regardless of where in its orbit the planet is. This means that the planet moves faster when it is closer to the sun.

This is because the sun's gravity accelerates the planet as it falls toward the sun, and decelerates it on the way back out, but Kepler did not know that reason.

The two laws permitted Kepler to calculate the position, (r,ν), of the planet, based on the time since perihelion, t, and the orbital period, P. The calculation is done in four steps.

1. Compute the mean anomaly M from the formula
$M=frac\left\{2pi t\right\}\left\{P\right\}$
2. Compute the eccentric anomaly E by numerically solving Kepler's equation:
$M=E-epsiloncdotsin E$
3. Compute the true anomaly ν by the equation:
$tanfrac nu 2 = sqrt\left\{frac\left\{1+epsilon\right\}\left\{1-epsilon\right\}\right\}cdottanfrac E 2$
4. Compute the heliocentric distance r from the first law:
$r=frac p \left\{1+epsiloncdotcosnu\right\}$
The proof of this procedure is shown below.

Third law

The third law : "The squares of the orbital periods of planets are directly proportional to the cubes of the axes of the orbits." Thus, not only does the length of the orbit increase with distance, the orbital speed decreases, so that the increase of the orbital period is more than proportional.

$P^2 propto a^3$
$P$ = orbital period of planet
$a$ = semimajor axis of orbit

So the expression P2·a–3 has the same value for all planets in the Solar System as it has for Earth. When certain units are chosen, namely P is measured in sidereal years and a in astronomical units, P2·a–3 has the value 1 for all planets in the Solar System.

In SI units: $frac\left\{P^\left\{2\right\}\right\}\left\{a^\left\{3\right\}\right\} = 3.00times 10^\left\{-19\right\} frac\left\{s^\left\{2\right\}\right\}\left\{m^\left\{3\right\}\right\} pm 0.7%,$.

The law, when applied to circular orbits where the acceleration is proportional to a·P−2, shows that the acceleration is proportional to a·a−3 = a−2, in accordance with Newton's law of gravitation.

The general equation, which was derived from Newton's law of gravity, is

$left\left(\left\{frac\left\{P\right\}\left\{2pi\right\}\right\}right\right)^2 = \left\{a^3 over G \left(M+m\right)\right\},$
where $G$ is the gravitational constant, $M$ is the mass of the sun, and $m$ is the mass of planet. The latter appears in the equation since the equation of motion involves the reduced mass. Note that P is time per orbit and P/2π is time per radian.

See the actual figures: attributes of major planets.

This law is also known as the harmonic law.

Position as a function of time

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 from center to perihelion, the semimajor axis,
$varepsilon=\left\{|cs|over a\right\},$ the eccentricity,
$b=asqrt\left\{1-varepsilon^2\right\},$ the semiminor axis,
$r=|sp| ,$ the distance from sun to planet.
and the angle
$nu=angle zsp,$ the planet as seen from the sun, the true anomaly.

The problem is to compute the polar coordinates (r,ν) of the planet from the time since perihelion, t.

It is solved in steps. Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:

• x is the projection of the planet to the auxiliary circle; then the area $|zsx|=frac a b cdot|zsp|$
• y is a point on the auxiliary circle such that the area $|zcy|=|zsx|$

and

$M=angle zcy$, y as seen from the centre, the mean anomaly.

The area of the circular sector $|zcy| = frac\left\{a^2 M\right\}2$, and the area swept since perihelion,

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

The mean anomaly M is first computed. The goal is to compute the true anomaly ν. The function ν=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 ν from the eccentric anomaly E. Here are the details.
$|zcy|=|zsx|=|zcx|-|scx|$

$frac\left\{a^2 M\right\}2=frac\left\{a^2 E\right\}2-frac \left\{avarepsiloncdot asin E\right\}2$
Division by a²/2 gives Kepler's equation
$M=E-varepsiloncdotsin E$.
The catch is that Kepler's equation cannot be rearranged to isolate E. The function E=f(M) is not an elementary formula, but Kepler's equation is solved either iteratively by a root-finding algorithm or, as derived in the article on eccentric anomaly, by an infinite series.

Having computed the eccentric anomaly E from Kepler's equation, the next step is to calculate the true anomaly ν from the eccentric anomaly E.

Note from the geometry of the problem that

$acdotcos E=acdotvarepsilon+rcdotcos nu.$
Dividing by a and inserting from Kepler's first law
$frac r a =frac\left\{1-varepsilon^2\right\}\left\{1+varepsiloncdotcos nu\right\}$
to get
$cos E$
=varepsilon+frac{1-varepsilon^2}{1+varepsiloncdotcos nu}cdotcos nu =frac{varepsiloncdot(1+varepsiloncdotcos nu)+(1-varepsilon^2)cdotcos nu}{1+varepsiloncdotcos nu} =frac{varepsilon +cos nu}{1+varepsiloncdotcos nu}. The result is a usable relationship between the eccentric anomaly E and the true anomaly ν.

A computationally more convenient form follows by substituting into the trigonometric identity:

$tan^2frac\left\{x\right\}\left\{2\right\}=frac\left\{1-cos x\right\}\left\{1+cos x\right\}.$
Get
$tan^2frac\left\{E\right\}\left\{2\right\}$
=frac{1-cos E}{1+cos E} =frac{1-frac{varepsilon+cos nu}{1+varepsiloncdotcos nu}}{1+frac{varepsilon+cos nu}{1+varepsiloncdotcos nu}} =frac{(1+varepsiloncdotcos nu)-(varepsilon+cos nu)}{(1+varepsiloncdotcos nu)+(varepsilon+cos nu)}

frac{1-varepsilon}{1+varepsilon}cdotfrac{1-cos nu}{1+cos nu}

frac{1-varepsilon}{1+varepsilon}cdottan^2frac{nu}{2}. Multiplying by (1+ε)/(1−ε) and taking the square root gives the result
$tanfrac nu2=sqrtfrac\left\{1+varepsilon\right\}\left\{1-varepsilon\right\}cdottanfrac E2.$

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

One could even develop a series computing ν directly from M.

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

$r=acdotfrac\left\{1-varepsilon^2\right\}\left\{1+varepsiloncdotcos nu\right\}.$

Derivation from Newton's laws

Kepler's laws are about the motion of the planets around the sun, while Newton's laws more generally are about the motion of point particles attracting each other by the force of gravitation.

In the special case where there are only two particles, and one of them is much lighter than the other, and the distance between the particles remains limited, then the lighter particle moves around the heavy particle as a planet around the Sun according to Kepler's laws, as shown below. Newton's laws however also admit other solutions, where the trajectory of the lighter particle is a parabola or a hyperbola. These solutions show that there is a limitation to the applicability of Kepler's first law, which states that the trajectory will always be an ellipse.

In any two-body system (the two body problem), the objects always orbit around their common center of mass (barycenter). When one object is very heavy compared to the other, the common center of mass is simply assumed to be the same as the center of mass of the heavy object (which is nearly correct). However, Kepler's laws can be used even in the case of two objects of similar mass, by decomposing the motion of each body as an ellipse around the barycenter.

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.

Deriving Kepler's second law

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, times a constant of proportionality. This is written:

$mcdotddotmathbf\left\{r\right\} = frac\left\{Mcdot m\right\}\left\{r^2\right\}cdot\left(-hat\left\{mathbf\left\{r\right\}\right\}\right)cdot G$

where a dot on top of the variable signifies differentiation with respect to time, and the second dot indicates the second derivative.

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

$dothat\left\{mathbf\left\{r\right\}\right\} = dotnu hat\left\{boldsymbolnu\right\}$
where $hat\left\{boldsymbolnu\right\}$ is the tangential unit vector, and
$dothat\left\{boldsymbolnu\right\} = -dotnu hat\left\{mathbf\left\{r\right\}\right\}.$

So the position vector

$mathbf\left\{r\right\} = r hat\left\{mathbf\left\{r\right\}\right\}$
is differentiated twice to give the velocity vector and the acceleration vector
$dotmathbf\left\{r\right\} =dot r hatmathbf\left\{r\right\} + r dothatmathbf\left\{r\right\}$
=dot r hat{mathbf{r}} + r dotnu hat{boldsymbolnu},
$ddotmathbf\left\{r\right\}$
= (ddot r hat{mathbf{r}} +dot r dothat{mathbf{r}} ) + (dot rdotnu hat{boldsymbolnu} + rddotnu hat{boldsymbolnu} + rdotnu dothat{boldsymbolnu}) = (ddot r - rdotnu^2) hat{mathbf{r}} + (rddotnu + 2dot r dotnu) hat{boldsymbolnu}.

Note that for constant distance, $r$, the planet is subject to the centripetal acceleration, $rdotnu^2$, and for constant angular speed, $dotnu$, the planet is subject to the coriolis acceleration, $2dot r dotnu$.

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

$\left(ddot r - rdotnu^2\right) hat\left\{mathbf\left\{r\right\}\right\} + \left(rddotnu + 2dot r dotnu\right) hat\left\{boldsymbolnu\right\}= -GMr^\left\{-2\right\}hat\left\{mathbf\left\{r\right\}\right\}$

Equating component, we get the two ordinary differential equations of motion, one for the radial acceleration and one for the tangential acceleration:

$ddot r - rdotnu^2 = -GMr^\left\{-2\right\},$
$rddotnu + 2dot rdotnu = 0.$

In order to derive Kepler's second law only the tangential acceleration equation is needed. Divide it by $r dotnu:$

$frac\left\{ddotnu\right\}\left\{dotnu\right\} +2frac\left\{dot r\right\}\left\{r\right\}=0$
and integrate:
$logdotnu +2log r = logell,$
where $logell$ is a constant of integration, and exponentiate:
$r^2dot nu =ell .$
This says that the specific angular momentum $r^2 dot nu$ is a constant of motion, even if both the distance $r$ and the angular speed $dotnu$ vary.

The area swept out from time t1 to time t2,

$int_\left\{t_1\right\}^\left\{t_2\right\}frac 1 2 cdot basecdot d\left(height\right) = int_\left\{t_1\right\}^\left\{t_2\right\}frac 1 2 cdot rcdot rdot nu dt=frac 1 2 cdotell cdot\left(t_2-t_1\right)$
depends only on the duration t2t1. This is Kepler's second law.

Deriving Kepler's first law

The expression
$p=ell ^2 G^\left\{-1\right\}M^\left\{-1\right\}$
has the dimension of length and is used to make the equations of motion dimensionless. We define
$u =pr^\left\{-1\right\}$
and get
$-GMr^\left\{-2\right\}=-ell^2 p^\left\{-3\right\}u^\left\{2\right\}$
and
$dot nu =ell r^\left\{-2\right\}=ell p^\left\{-2\right\}u^2.$

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

$dot X=frac \left\{dX\right\}\left\{dt\right\}=frac \left\{dX\right\}\left\{dnu\right\}cdotfrac \left\{dnu\right\}\left\{dt\right\}=frac \left\{dX\right\}\left\{d nu\right\}cdot dotnu=frac \left\{dX\right\}\left\{d nu\right\}cdotell p^\left\{-2\right\}u^2.$

Differentiate

$r =pu^\left\{-1\right\}$
twice:
$dot r = frac\left\{d\left(pu^\left\{-1\right\}\right)\right\}\left\{dnu\right\}cdotell p^\left\{-2\right\}u^\left\{2\right\} = -pu^\left\{-2\right\}frac\left\{du\right\}\left\{dnu\right\}cdotell p^\left\{-2\right\}u^\left\{2\right\}= -ell p^\left\{-1\right\}frac\left\{du\right\}\left\{dnu\right\}$
$ddot r = frac\left\{ddot r\right\}\left\{dnu\right\}cdotell p^\left\{-2\right\}u^\left\{2\right\}= frac\left\{d\right\}\left\{dnu\right\}\left(-ell p^\left\{-1\right\} frac\left\{du\right\}\left\{dnu\right\}\right)cdotell p^\left\{-2\right\}u^\left\{2\right\}= -ell^2 p^\left\{-3\right\}u^\left\{2\right\}frac\left\{d^2 u\right\}\left\{dnu^2\right\}$
Substitute into the radial equation of motion
$ddot r - rdotnu^2 = -GMr^\left\{-2\right\}$
and get
$\left(-ell^2 p^\left\{-3\right\}u^2frac\left\{d^2u\right\}\left\{dnu^2\right\}\right) - \left(pu^\left\{-1\right\}\right)\left(ell p^\left\{-2\right\}u^2\right)^2 = -ell ^2 p^\left\{-3\right\} u^2$
Divide by $-ell^2 p^\left\{-3\right\}u^2$ to get a simple non-homogeneous linear differential equation for the orbit of the planet:

$frac\left\{d^2u\right\}\left\{dnu^2\right\} + u = 1 .$

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
$frac\left\{d^2u\right\}\left\{dnu^2\right\} + u = 0$
These solutions are
$u = epsiloncdotcos\left(nu-nu_0\right)$
where $epsilon$ and $nu_0,$ are arbitrary constants of integration. So the result is
$u = 1+ epsiloncdotcos\left(nu-nu_0\right)$

Choosing the axis of the coordinate system such that $nu_0=0$, and inserting $u=pr^\left\{-1\right\}$, gives:

$pr^\left\{-1 \right\} = 1+ epsiloncdotcosnu .$

If $epsilon<1 ,$ this is Kepler's first law.

Deriving Kepler's third law

The area of the planetary orbit ellipse is
$A=pi a b=pi a\left(asqrt\left\{1-epsilon^2\right\}\right)=pi a^2sqrt\left\{1-epsilon^2\right\},$
The area speed of the radius vector sweeping the orbit area is
$dot A=ell/ 2 ,$
where
$ell^2=pGM=a\left(1-epsilon^2\right)GM ,.$
The period of the orbit is
$P=frac A dot A =frac \left\{pi a^2sqrt\left\{1-epsilon^2\right\}\right\}\left\{ell/2\right\}=2pifrac \left\{a^2sqrt\left\{1-epsilon^2\right\}\right\}\left\{ell\right\},$
satisfying
$left\left(frac P\left\{2 pi\right\}right\right)^2=left\left(frac \left\{a^2sqrt\left\{1-epsilon^2\right\}\right\}\left\{ell\right\}right\right)^2=frac \left\{a^4\left(1-epsilon^2\right)\right\}\left\{ell^2\right\}=frac \left\{a^4\left(1-epsilon^2\right)\right\}\left\{a\left(1-epsilon^2\right)GM \right\}=frac\left\{a^3\right\}\left\{GM\right\},$
which is Kepler's third law.

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 .
• Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN-10 0-521-57597-4