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Kepler's Laws

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Kepler's Laws

In astronomy, Kepler's laws of planetary motion are three scientific laws describing orbital motion, originally formulated to describe the motion of planets around the Sun.

Kepler's laws are:

  1. The orbit of every planet is an ellipse with the Sun at one of the two foci.
  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 proportional to the cube of the semi-major axis of its orbit.

History

Johannes Kepler published his first two laws in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.[2] Kepler discovered his third law many years later, and it was published in 1619.[2] At the time, Kepler's laws were radical claims; the prevailing belief (particularly in epicycle-based theories) was that orbits were perfect circles. Most of the planetary orbits can be rather closely approximated as circles, so it is not immediately evident that the orbits are ellipses. 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. Kepler's laws and his analysis of the observations on which they were based 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), by asserting that the Earth orbited the Sun, proving that the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.[2]

Some eight decades later, Isaac Newton proved that relationships like Kepler's would apply exactly under certain ideal conditions that are to a good approximation fulfilled in the solar system, as consequences of Newton's own laws of motion and law of universal gravitation.[Nb 1][3] 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.[Nb 2][3] 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".[4] Together with Newton's mathematical theories, they are part of the foundation of modern astronomy and physics.[3]

First Law


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

An ellipse is a closed plane curve that resembles a stretched out circle (see the figure to the right). Note that the Sun is not at the center of the ellipse, but at one of its foci. This focal point is sometimes called the occupied focus. The other focal point, known as the empty or vacant focus, marked with a lighter dot, has no physical significance for the orbit. The center of an ellipse is the midpoint of the line segment joining its focal points. A circle is a special case of an ellipse where both focal points coincide.

The relative distance between a focal point and the center is known as the eccentricity. It can take any value greater than or equal to 0 (a circle) and less than 1 (as the eccentricity tends to 1, the ellipse tends to a parabola). The eccentricities of the planets known to Kepler varied from 0.007 (Venus) to 0.2 (Mercury). (See List of planetary objects in the Solar System for more detail).

After Kepler's death, however, bodies with highly eccentric orbits were identified, among them many comets and asteroids. Heavenly bodies such as comets with parabolic or even hyperbolic orbits are possible under the Newtonian theory and have been observed.[5]


In mathematical equations of the ellipse, using Cartesian coordinates, the ellipse has its center of symmetry as its center. This however is not helpful in studying planetary orbits, since the sun is at one focus of the ellipse, not at the center. Therefore polar coordinates rather than Cartesian coordinates are used here.

Symbolically, an ellipse can be represented in polar coordinates as:

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

where (rθ) are the polar coordinates (from the focus) for the ellipse, p is the semi-latus rectum, and ε is the eccentricity of the ellipse. Note that 0 < ε < 1 for a proper ellipse; if ε = 0, the ellipse has no eccentricity and can be analyzed directly as a case of a simple circle with the sun at the centre (see section Zero eccentricity below). For a planet orbiting the Sun, r is the distance from the Sun to the planet and θ is the angle between the planet's current position and its closest approach, with the Sun as the vertex.

At θ = 0°, perihelion, the distance is minimum

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

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

At θ = 180°, aphelion, the distance is maximum

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

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

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

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

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

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

\frac{1}{r_\min}-\frac{1}{p}=\frac{1}{p}-\frac{1}{r_\max}
pa=r_\max r_\min=b^2\,.

The eccentricity ε is the coefficient of variation between rmin and rmax:

\varepsilon=\frac{r_\mathrm{max}-r_\mathrm{min}}{r_\mathrm{max}+r_\mathrm{min}}.

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


"A line joining a planet and the Sun sweeps out equal areas during equal intervals of time."[1]

In a small time dt\, the planet sweeps out a small triangle having base line r\, and height r d\theta\,.

The area of this triangle is given by

dA=\tfrac 1 2\cdot r\cdot r d\theta

and so the constant areal velocity is
\frac{dA}{dt}=\tfrac{1}{2}r^2 \frac{d\theta}{dt}.


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 equal areas in equal times.

The total area enclosed by the elliptical orbit is

A=\pi ab.\,

Therefore the period

P\,

satisfies

\pi ab=P\cdot \tfrac 12r^2 \dot\theta

or

r^2\dot \theta = nab

where

\dot\theta=\frac{d\theta}{dt}

is the angular velocity, (using Newton notation for differentiation), and

n = \frac{2\pi}{P}

is the mean motion of the planet around the Sun.

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."

The third law, published by Kepler in 1619 [1] captures the relationship between the distance of planets from the Sun, and their orbital periods. Symbolically, the law can be expressed as

P^2 \propto a^3 ,

where P is the orbital period of the planet and a is the semi-major axis of the orbit.

The constant of proportionality is

\frac{P_{planet}^2}{a_{planet}^3} = \frac{P_{earth}^2}{a_{earth}^3} = 1 \frac{ \rm{yr^2} }{ \rm{AU^3} }

for a sidereal year (yr), and astronomical unit (AU). (For numeric values see List of gravitationally rounded objects of the Solar System).

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.[6] So it used to be known as the harmonic law.[7]

For circular orbits, Kepler's 3rd Law is also commonly represented as

\frac{4 \pi^2}{T^2} = \frac{G M}{R^3}

Where T is the period, G is the Gravitational constant, M is the mass of the larger body, and R is the distance between the centers of mass of the two bodies.

Generality

Godefroy Wendelin, in 1643, noted that Kepler's third law applies to the four brightest moons of Jupiter.[Nb 3]

In fact, 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).[Nb 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 mass as much as 1/1047.3486 (Jupiter) and 1/3497.898 (Saturn) of the solar mass,[8] 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.[Nb 5] 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, which assumed circular orbits. If the eccentricity of a planetary orbit is zero, then Kepler's laws state:

  1. The planetary orbit is a circle
  2. The Sun is in the center
  3. The speed of the planet in the orbit is constant
  4. 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 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:

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

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 equator 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

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

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.

Planetary acceleration

Relation to Newton's laws

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

  1. The direction of the acceleration is towards the Sun.
  2. 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:

  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 from the Sun.

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

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

As the planets have small masses compared to that of the Sun, the orbits conform 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.

Acceleration vector

From the heliocentric point of view consider the vector to the planet \mathbf{r} = r \hat{\mathbf{r}} where r is the distance to the planet and the direction \hat {\mathbf{r}} is a unit vector. When the planet moves the direction vector \hat {\mathbf{r}} changes:

\frac{d\hat{\mathbf{r}}}{dt}=\dot{\hat{\mathbf{r}}} = \dot\theta \hat{\boldsymbol\theta},\qquad \dot{\hat{\boldsymbol\theta}} = -\dot\theta \hat{\mathbf{r}}

where \scriptstyle \hat{\boldsymbol\theta} is the unit vector orthogonal to \scriptstyle \hat{\mathbf{r}} and pointing in the direction of rotation, and \scriptstyle \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:

\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}}. So

\ddot{\mathbf{r}} = a_r \hat{\boldsymbol{r}}+a_\theta\hat{\boldsymbol{\theta}}

where the radial acceleration is

a_r=\ddot{r} - r\dot{\theta}^2

and the tangential acceleration is

a_\theta=r\ddot{\theta} + 2\dot{r} \dot{\theta}.

The inverse square law

Kepler's second law implies that the areal velocity \tfrac 1 2 r^2 \dot \theta is a constant of motion. The tangential acceleration a_\theta is zero by Kepler's second law:

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

So the acceleration of a planet obeying Kepler's second law is directed exactly towards the sun.

Kepler's first law implies that the area enclosed by the orbit is \pi ab, where a is the semi-major axis and b is the semi-minor axis of the ellipse. Therefore the period P satisfies \pi ab=\tfrac 1 2 r^2\dot \theta P or

r^2\dot \theta = nab

where

n = \frac{2\pi}{P}

is the mean motion of the planet around the sun.

The radial acceleration a_r is

a_r = \ddot r - r \dot \theta^2= \ddot r - r \left(\frac{nab}{r^2}

\right)^2= \ddot r -\frac{n^2a^2b^2}{r^3}. Kepler's first law states that the orbit is described by the equation:

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

Differentiating with respect to time

-\frac{p\dot r}{r^2} = -\varepsilon \sin \theta \,\dot \theta

or

p\dot r = nab\,\varepsilon\sin \theta.

Differentiating once more

p\ddot r =nab \varepsilon \cos \theta \,\dot \theta

=nab \varepsilon \cos \theta \,\frac{nab}{r^2} =\frac{n^2a^2b^2}{r^2}\varepsilon \cos \theta . The radial acceleration a_r satisfies

p a_r = \frac{n^2 a^2b^2}{r^2}\varepsilon \cos \theta - p\frac{n^2 a^2b^2}{r^3}

= \frac{n^2a^2b^2}{r^2}\left(\varepsilon \cos \theta - \frac{p}{r}\right). Substituting the equation of the ellipse gives

p a_r = \frac{n^2a^2b^2}{r^2}\left(\frac p r - 1 - \frac p r\right)= -\frac{n^2a^2}{r^2}b^2.

The relation b^2=pa gives the simple final result

a_r=-\frac{n^2a^3}{r^2}.

This means that the acceleration vector \mathbf{\ddot r} of any planet obeying Kepler's first and second law satisfies the inverse square law

\mathbf{\ddot r} = - \frac{\alpha}{r^2}\hat{\mathbf{r}}

where

\alpha = n^2 a^3=\frac{4\pi^2 a^3}{P^2}\,

is a constant, and \hat{\mathbf r} is the unit vector pointing from the Sun towards the planet, and r\, is the distance between the planet and the Sun.

According to Kepler's third law, \alpha 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 includes 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.

Newton's law of gravitation

By Newton's second law, the gravitational force that acts on the planet is:

\mathbf{F} = m \mathbf{\ddot r} = - \frac{m \alpha}{r^2}\hat{\mathbf{r}}

where \alpha only depends on properties of the Sun. According to Newton's third Law, the Sun is also pulled towards 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 the equation for gravitational force should be

\mathbf{F} = - \frac{GMm}{r^2}\hat{\mathbf{r}}

where G is a universal constant. This is Newton's law of universal gravitation.

The acceleration of solar system body i is, according to Newton's laws:

\mathbf{\ddot r_i} = G\sum_{j\ne i} \frac{m_j}{r_{ij}^2}\hat{\mathbf{r}}_{ij}

where m_j is the mass of body j, r_{ij} is the distance between body i and body j, \hat{\mathbf{r}}_{ij} is the unit vector from body i pointing 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, Planet and Sun, the acceleration becomes

\mathbf{\ddot r}_{Planet} = G\frac{m_{Sun}}{r_^2}\hat{\mathbf{r}}_

which is the acceleration of the Kepler motion.

Position as a function of time

Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.

The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the time t since perihelion, and the mean motion n = 2π/P, is the following four steps:

1. Compute the mean anomaly
M=nt
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. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.

The proof of this procedure is shown below.

Mean anomaly, M

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

s the Sun (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=n t,

where n is the mean motion.

Eccentric anomaly, E

When the mean anomaly M is computed, the goal is to compute the true anomaly θ. The function θ=f(M) is, however, not elementary.[9] 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 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.

True anomaly, θ

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.

Distance, r

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}.

See also

Notes

References

Bibliography

  • 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, 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 [2]
  • Equant compared to Kepler: interactive model [3]
  • Kepler's Third Law:interactive model [4]
  • Solar System Simulator (Interactive Applet)
  • Kepler and His Laws, educational web pages by David P. Stern

Template:Johannes Kepler

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