Kepler's laws of planetary motion

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In astronomy, Kepler's laws of planetary motion are three scientific laws describing motion of planets around the Sun.

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 planet and ƒ1 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.

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[edit]

Johannes Kepler published his first two laws in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.[2] Kepler's third law was published in 1619.[2]

Kepler's laws challenged the long-accepted geocentric models of Aristotle and Ptolemy, and followed the heliocentric theory of Nicolaus Copernicus 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]

Most planetary orbits are almost circles, so it is not obvious that they are actually 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 in 1622 and Godefroy Wendelin in 1643 noted that Kepler's third law applies to the four brightest moons of Jupiter.[Nb 1]

Isaac Newton proved in 1687 that relationships like Kepler's would apply in the solar system to a good approximation, as consequences of Newton's own laws of motion and law of universal gravitation.

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".[3]

Together with Newton's theories, they are part of the foundation of modern astronomy and physics.[4]

First Law[edit]

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

Mathematically, an ellipse can be represented by the formula:

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

where (rθ) are polar coordinates, p is the semi-latus rectum, and ε is the eccentricity of the ellipse.

Note that 0 < ε < 1 for an ellipse; in the limiting case ε = 0, the orbit is a circle with the sun at the centre (see section Zero eccentricity below).

For a planet r is the distance from the Sun to the planet, and θ is the angle to the planet's current position from its closest approach, as seen from the Sun.

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

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[edit]

"A line joining a planet and the Sun sweeps out equal areas during equal intervals of time."[1]
The same blue area is swept out in a given time. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity

In a small time dt\, the planet sweeps out a small triangle having base line r\, and height r d\theta\, and area 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}. The planet moves faster when it is closer to the Sun.

The area enclosed by the elliptical orbit is \pi ab.\, So the period P\, satisfies

P\cdot \tfrac 12r^2 \frac{d\theta}{dt}=\pi a b

and the mean motion of the planet around the Sun n = {2\pi}/P satisfies

r^2{d\theta} = a b n dt .

Third law[edit]

"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.

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

Mathematically, the law says that the expression  n^2 a^3 has the same value for all the planets in the solar system.

Zero eccentricity[edit]

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 plane throught the sun parallel to the equator of the earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately

\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[edit]

A sudden sunward velocity change is applied to a planet. Then the areas of the triangles defined by the path of the planet for fixed time intervals will be equal. (Click on image for a detailed description.)

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.

Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.

Acceleration vector[edit]

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[edit]

Kepler's laws say that

r^2\dot \theta = nab

is constant.

The tangential acceleration a_\theta is zero:

\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 laws is directed towards 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\,

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 include the Keplerian motions, as shown, but they also include motions where the orbit is a hyperbola or parabola or a straight line. See Kepler orbit.

Newton's law of gravitation[edit]

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 m is the mass of the planet and \alpha has the same value for all the planets. According to Newton's third Law, the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun, M. So

\alpha = GM

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_{{Planet},{Sun}}^2}\hat{\mathbf{r}}_{{Planet},{Sun}}

which is the acceleration of the Kepler motion.

Position as a function of time [edit]

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, 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[edit]

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=n t,

where n is the mean motion.

Eccentric anomaly, E[edit]

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.[7] 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[edit]

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[edit]

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[edit]

Notes[edit]

  1. ^ Godefroy Wendelin wrote a letter to Giovanni Battista Riccioli about the relationship between the distances of the Jovian moons from Jupiter and the periods of their orbits, showing that the periods and distances conformed to Kepler's third law. See: Joanne Baptista Riccioli, Almagestum novum … (Bologna (Bononiae), (Italy): Victor Benati, 1651), volume 1, page 492 Scholia III. In the margin beside the relevant paragraph is printed: Vendelini ingeniosa speculatio circa motus & intervalla satellitum Jovis. (The clever Wendelin's speculation about the movement and distances of Jupiter's satellites.)
    In 1622, Johannes Kepler had noted that Jupiter's moons obey (approximately) his third law in his Epitome Astronomiae Copernicanae [Epitome of Copernican Astronomy] (Linz (“Lentiis ad Danubium“), (Austria): Johann Planck, 1622), book 4, part 2, page 554.

References[edit]

  1. ^ a b Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law", Wolfram Demonstrations Project. Retrieved December 27, 2009.
  2. ^ a b c Holton, Gerald James; Brush, Stephen G. (2001). Physics, the Human Adventure: From Copernicus to Einstein and Beyond (3rd paperback ed.). Piscataway, NJ: Rutgers University Press. pp. 40–41. ISBN 0-8135-2908-5. Retrieved December 27, 2009. 
  3. ^ 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. 
  4. ^ 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.).
  5. ^ Burtt, Edwin. The Metaphysical Foundations of Modern Physical Science. p. 52.
  6. ^ Gerald James Holton, Stephen G. Brush (2001). Physics, the Human Adventure. Rutgers University Press. p. 45. ISBN 0-8135-2908-5. 
  7. ^ MÜLLER, M (1995). "EQUATION OF TIME -- PROBLEM IN ASTRONOMY". Acta Physica Polonica A. Retrieved 23 February 2013. 

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