# Keplers Third Law Of Planetary Motion Mastering Astronomy Assignment

Additional reading at www.astronomynotes.com

#### First Law

Kepler was a sophisticated mathematician, and so the advance that he made in the study of the motion of the planets was to introduce a mathematical foundation for the heliocentric model of the solar system. Where Ptolemy and Copernicus relied on assumptions, such as that the circle is a “perfect” shape and all orbits must be circular, Kepler showed that mathematically a circular orbit could not match the data for Mars, but that an *elliptical* orbit did match the data! We now refer to the following statement as Kepler’s First Law:

- The planets orbit the Sun in ellipses with the Sun at one focus (the other focus is empty).

For more information about ellipses, you can read in gory mathematical detail the page hosted at Mathworld, and there is also information on ellipses in Wikipedia.

Here is a demonstration of the classic method for drawing an ellipse:

The two thumbtacks in the image represent the two foci of the ellipse, and the string ensures that the sum of the distances from the two foci (the tacks) to the pencil is a constant. Below is another image of an ellipse with the major axis and minor axis defined:

We know that in a circle, all lines that pass through the center (diameters) are exactly equal in length. However, in an ellipse, lines that you draw through the center vary in length. The line that passes from one end to the other and includes both foci is called the **major axis**, and this is the longest distance between two points on the ellipse. The line that is perpendicular to the major axis at its center is called the **minor axis**, and it is the shortest distance between two points on the ellipse.

In the image above, the green dots are the foci (equivalent to the tacks in the photo above). The larger the distance between the foci, the larger the **eccentricity **of the ellipse. In the limiting case where the foci are on top of each other (an eccentricity of 0), the figure is actually a circle. So you can think of a circle as an ellipse of eccentricity 0. Studies have shown that astronomy textbooks introduce a misconception by showing the planets' orbits as highly eccentric in an effort to be sure to drive home the point that they are ellipses and not circles. In reality the orbits of most planets in our Solar System are very close to circular, with eccentricities of near 0 (e.g., the eccentricity of Earth's orbit is 0.0167). For an animation showing orbits with varying eccentricities, see the eccentricity diagram at "Windows to the Universe." Note that the orbit with an eccentricity of 0.2, which appears nearly circular, is similar to Mercury's, which has the *largest* eccentricity of any planet in the Solar System. The elliptical orbits diagram at "Windows to the Universe" includes an image with a direct comparison of the eccentricities of several planets, an asteroid, and a comet. Note that if you follow the Starry Night instructions on the previous page to observe the orbits of Earth and Mars from above, you can also see the shapes of these orbits and how circular they appear.

Kepler’s first law has several implications. These are:

- The distance between a planet and the Sun changes as the planet moves along its orbit.
- The Sun is offset from the center of the planet’s orbit.

#### Second Law

In their models of the Solar System, the Greeks held to the Aristotelian belief that objects in the sky moved at a constant speed in circles because that is their “natural motion.” However, Kepler’s second law (sometimes referred to as the **Law of Equal Areas**), can be used to show that the velocity of a planet changes as it moves along its orbit!

Kepler’s second law is:

- The line joining the Sun and a planet sweeps through equal areas in an equal amount of time.

The image below links to an animation that demonstrates that when a planet is near aphelion (the point furthest from the Sun, labeled with a B on the screen grab below) the line drawn between the Sun and the planet traces out a long, skinny sector between points A and B. When the planet is close to perihelion (the point closest to the Sun, labeled with a C on the screen grab below), the line drawn between the Sun and the planet traces out a shorter, fatter sector between points C and D. These slices that alternate gray and blue were drawn in such a way that the area inside each sector is the same. That is, the sector between C and D on the right contains the same amount of area as the sector between A and B on the left.

Click on this image to launch the animation in Windows Media Player. It shows a planet sweeping out equal areas in equal times.

Kepler's 2nd Law

Credit: Dr. Michael Gallis, Penn State Schuylkill

Since the areas of these two sectors are identical, then Kepler's second law says that the time it takes the planet to travel between A and B and also between C and D must be the same. If you look at the distance along the ellipse between A and B, it is shorter than the distance between C and D. Since velocity is distance divided by time, and since the distance between A and B is shorter than the distance between C and D, when you divide those distances by the same amount of time you find that:

- A planet is moving faster near perihelion and slower near aphelion.

The orbits of most planets are almost circular, with eccentricities near 0. In this case, the changes in their speed are not too large over the course of their orbit.

For those of you who teach physics, you might note that really, Kepler's second law is just another way of stating that angular momentum is conserved. That is, when the planet is near perihelion, the distance between the Sun and the planet is smaller, so it must increase its tangential velocity to conserve angular momentum, and similarly, when it is near aphelion when their separation is larger, its tangential velocity must decrease so that the total orbital angular momentum is the same as it was at perihelion.

#### Third Law

Kepler had all of Tycho’s data on the planets, so he was able to determine how long each planet took to complete one orbit around the Sun. This is usually referred to as the **period **of an orbit. Kepler noted that the closer a planet was to the Sun, the faster it orbited the Sun. He was the first scientist to study the planets from the perspective that the Sun influenced their orbits. That is, unlike Ptolemy and Copernicus, who both assumed that the planet's “natural motion” was to move at constant speeds along circular paths, Kepler believed that the Sun exerted some kind of force on the planets to push them along their orbits, and because of this, the closer they are to the Sun, the faster they should move.

Kepler studied the periods of the planets and their distance from the Sun, and proved the following mathematical relationship, which is Kepler’s Third Law:

- The square of the period of a planet’s orbit (P) is directly proportional to the cube of the semimajor axis (a) of its elliptical path.

What this means mathematically is that if the square of the period of an object doubles, then the cube of its semimajor axis must also double. The proportionality sign in the above equation means that:

where k is a constant number. If we divide both sides of the equation by , we see that:

This means that for every planet in our solar system, the ratio of their period squared to their semimajor axis cubed is the same constant value, so this means that:

We know that the period of the Earth is 1 year. At the time of Kepler, they did not know the distances to the planets, but we can just assign the semimajor axis of the Earth to a unit we call the Astronomical Unit (AU). That is, without knowing how big an AU is, we just set . If you plug 1 year and 1 AU into the equation above, you see that:

So for every planet, if P is expressed in years and a is expressed in AU. So if you want to calculate how far Saturn is from the Sun in AU, all you need to know is its period. For Saturn, this is approximately 29 years. So:

So Saturn is 9.4 times further from the Sun than the Earth is from the Sun!

For a more historical approach, see in particular the articles *Astronomia nova* and *Epitome Astronomiae Copernicanae*.

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

- The orbit of a planet is an ellipse with the Sun at one of the two foci.
- A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
^{[1]} - The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Most planetary orbits are nearly circular, and careful observation and calculation are required in order to establish that they are not perfectly circular. Calculations of the orbit of Mars, whose published values are somewhat suspect,^{[2]} indicated an elliptical orbit. From this, Johannes Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits.

Kepler's work (published between 1609 and 1619) improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.^{[3]}

Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System to a good approximation, as a consequence of his own laws of motion and law of universal gravitation.

## Comparison to Copernicus[edit]

Kepler's laws improve the model of Copernicus. If the eccentricities of the planetary orbits are taken as zero, then Kepler basically agrees with Copernicus:

- The planetary orbit is a circle
- The Sun is at the center of the orbit
- The speed of the planet in the orbit is constant

The eccentricities of the orbits of those planets known to Copernicus and Kepler are small, so the foregoing rules give good approximations of planetary motion; but Kepler's laws fit the observations better than Copernicus'.

Kepler's corrections 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*of the elliptical orbit. - Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the
*area speed*is constant.

The 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 through 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

which is close to the correct value (0.016710219) (see Earth's orbit). The calculation is correct when perihelion, the date the Earth is closest to the Sun, falls on a solstice. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.

## Nomenclature[edit]

It took nearly two centuries for the current formulation of Kepler's work to take on its settled form. Voltaire's *Eléments de la philosophie de Newton* (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws".^{[4]}^{[5]} The *Biographical Encyclopedia of Astronomers* in its article on Kepler (p. 620) states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande.^{[6]} It was the exposition of Robert Small, in *An account of the astronomical discoveries of Kepler* (1804) that made up the set of three laws, by adding in the third.^{[7]} Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.^{[5]}^{[8]}

Further, the current usage of "Kepler's Second Law" is something of a misnomer. Kepler had two versions, related in a qualitative sense: the "distance law" and the "area law". The "area law" is what became the Second Law in the set of three; but Kepler did himself not privilege it in that way.^{[9]}

## History[edit]

Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.^{[10]}^{[3]}^{[11]} Kepler's third law was published in 1619.^{[12]}^{[3]} Notably, Kepler had believed in the Copernican model of the solar system, which called for circular orbits, but could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit (Mars coincidentally having the highest eccentricity of all planets except Mercury^{[13]}). His first law reflected this discovery.

Kepler in 1621 and Godefroy Wendelin in 1643 noted that Kepler's third law applies to the four brightest moons of Jupiter.^{[Nb 1]} The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664, but by 1670 his *Philosophical Transactions* were in its favour. As the century proceeded it became more widely accepted.^{[14]} The reception in Germany changed noticeably between 1688, the year in which Newton's *Principia* was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published.^{[15]}

Newton is credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law; while the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.^{[16]}

## Formulary[edit]

The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.

### First law of Kepler[edit]

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

Mathematically, an ellipse can be represented by the formula:

where is the semi-latus rectum, *ε* is the eccentricity of the ellipse, *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. So (*r*, *θ*) are polar coordinates.

For an ellipse 0 < *ε* < 1 ; in the limiting case *ε* = 0, the orbit is a circle with the sun at the centre (i.e. where there is zero eccentricity).

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

At *θ* = 90° and at *θ* = 270° the distance is equal to .

At *θ* = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)

The semi-major axis*a* is the arithmetic mean between *r*_{min} and *r*_{max}:

The semi-minor axis*b* is the geometric mean between *r*_{min} and *r*_{max}:

The semi-latus rectum*p* is the harmonic mean between *r*_{min} and *r*_{max}:

The eccentricity*ε* is the coefficient of variation between *r*_{min} and *r*_{max}:

The area of the ellipse is

The special case of a circle is *ε* = 0, resulting in *r* = *p* = *r*_{min} = *r*_{max} = *a* = *b* and *A* = *πr*^{2}.

### Second law of Kepler[edit]

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

^{[1]}

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the sun, then slower when farther from the sun. Kepler's second law states that the blue sector has constant area.

In a small time the planet sweeps out a small triangle having base line and height and area and so the constant areal velocity is

The area enclosed by the elliptical orbit is So the period satisfies

and the mean motion of the planet around the Sun

satisfies

### Third law of Kepler[edit]

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

This captures the relationship between the distance of planets from the Sun, and their orbital periods.

Kepler enunciated in 1619^{[12]} 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.^{[17]} So it was known as the *harmonic law*.^{[18]}

Using Newton's Law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the centripetal force equal to the gravitational force:

Then, expressing the angular velocity in terms of the orbital period and then rearranging, we find Kepler's Third Law:

A more detailed derivation can be done with general elliptical orbits as well as the center of mass. This results in replacing a circular radius, , with the elliptical semi-major axis, , as well as replacing the large mass with . However, with planet masses being so much smaller than the sun, this correction is often ignored. The full corresponding formula is:

where is the mass of the sun, is the mass of the planet, and is the gravitational constant, is the orbital period and is the elliptical semi-major axis.

The following table shows the data used by Kepler to empirically derive his law:

Planet | Mean distance to sun [AU] | Period [days] | |
---|---|---|---|

Mercury | 0.389 | 87.77 | 7.64 |

Venus | 0.724 | 224.70 | 7.52 |

Earth | 1 | 365.25 | 7.50 |

Mars | 1.524 | 686.95 | 7.50 |

Jupiter | 5.2 | 4332.62 | 7.49 |

Saturn | 9.510 | 10759.2 | 7.43 |

Upon finding this pattern Kepler wrote:^{[19]}

"I first believed I was dreaming...But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance."

For comparison, here are modern estimates:

Planet | Semi-major axis [AU] | Period [days] | |
---|---|---|---|

Mercury | 0.38710 | 87.9693 | 7.496 |

Venus | 0.72333 | 224.7008 | 7.496 |

Earth | 1 | 365.2564 | 7.496 |

Mars | 1.52366 | 686.9796 | 7.495 |

Jupiter | 5.20336 | 4332.8201 | 7.504 |

Saturn | 9.53707 | 10775.599 | 7.498 |

Uranus | 19.1913 | 30687.153 | 7.506 |

Neptune | 30.0690 | 60190.03 | 7.504 |

## Planetary acceleration[edit]

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 inversely proportional to the square of the planet's distance from the Sun (the*inverse square law*).

This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in his Principia that he considers forces from a mathematical point of view, not a physical, thereby taking an instrumentalist view.^{[20]} Moreover, he does not assign a cause to gravity.^{[21]}

Newton defined the force acting 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 acting on a planet is directly proportional to the mass of the planet and is inversely proportional to the square of its distance from the Sun.

The Sun plays an unsymmetrical part, which is unjustified. So he assumed, in 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 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 approximately to Kepler's laws. Newton's model improves upon Kepler's model, and fits actual observations more accurately (see two-body problem).

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

### Acceleration vector[edit]

See also: Polar coordinate § Vector calculus, and Mechanics of planar particle motion

From the heliocentric point of view consider the vector to the planet where is the distance to the planet and is a unit vector pointing towards the planet.

where is the unit vector whose direction is 90 degrees counterclockwise of , and is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.

Differentiate the position vector twice to obtain the velocity vector and the acceleration vector:

(1) The orbits are ellipses, with focal points

*F*

_{1}and

*F*

_{2}for the first planet and

*F*

_{1}and

*F*

_{3}for the second planet. The Sun is placed in focal point

*F*

_{1}.

(2) The two shaded sectors

*A*

_{1}and

*A*

_{2}have the same surface area and the time for planet 1 to cover segment

*A*

_{1}is equal to the time to cover segment

*A*

_{2}.

(3) The total orbit times for planet 1 and planet 2 have a ratio (

*a*

_{1}/

*a*

_{2})

^{3/2}.

*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*=

*r*

_{min}and for θ = 180°,

*r*=

*r*

_{max}.

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