Keplers laws of planetary motion (KNOWLEDGE PAGE)

What does Kepler's first law of planetary motion imply?
In fact, a circle is the special case of an ellipse in which the two foci are at the same location. Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse.

Kepler's Three Laws

 

In the early 1600s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to summarize the carefully collected data of his mentor - Tycho Brahe - with three statements that described the motion of planets in a sun-centered solar system. Kepler's efforts to explain the underlying reasons for such motions are no longer accepted; nonetheless, the actual laws themselves are still considered an accurate description of the motion of any planet and any satellite.

Kepler's three laws of planetary motion can be described as follows:

  • The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
  • An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
  • The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)


 

The Law of Ellipses

Kepler's first law - sometimes referred to as the law of ellipses - explains that planets are orbiting the sun in a path described as an ellipse. An ellipse can easily be constructed using a pencil, two tacks, a string, a sheet of paper and a piece of cardboard. Tack the sheet of paper to the cardboard using the two tacks. Then tie the string into a loop and wrap the loop around the two tacks. Take your pencil and pull the string until the pencil and two tacks make a triangle (see diagram at the right). Then begin to trace out a path with the pencil, keeping the string wrapped tightly around the tacks. The resulting shape will be an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points (represented here by the tack locations) are known as the foci of the ellipse. The closer together that these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse.

 

 

The Law of Equal Areas

Kepler's second law - sometimes referred to as the law of equal areas - describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. For instance, if an imaginary line were drawn from the earth to the sun, then the area swept out by the line in every 31-day month would be the same. This is depicted in the diagram below. As can be observed in the diagram, the areas formed when the earth is closest to the sun can be approximated as a wide but short triangle; whereas the areas formed when the earth is farthest from the sun can be approximated as a narrow but long triangle. These areas are the same size. Since the base of these triangles are shortest when the earth is farthest from the sun, the earth would have to be moving more slowly in order for this imaginary area to be the same size as when the earth is closest to the sun.


 

The Law of Harmonies

Kepler's third law - sometimes referred to as the law of harmonies - compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws that describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets. As an illustration, consider the orbital period and average distance from sun (orbital radius) for Earth and mars as given in the table below.

Planet

Period
(s)

Average
Distance (m)

T2/R3
(s2/m3)

Earth

3.156 x 107 s

1.4957 x 1011

2.977 x 10-19

Mars

5.93 x 107 s

2.278 x 1011

2.975 x 10-19

 

Observe that the T2/Rratio is the same for Earth as it is for mars. In fact, if the same T2/Rratio is computed for the other planets, it can be found that this ratio is nearly the same value for all the planets (see table below). Amazingly, every planet has the same T2/R3 ratio.

Planet

Period
(yr)

Average
Distance (au)

T2/R3
(yr2/au3)

Mercury

0.241

0.39

0.98

Venus

.615

0.72

1.01

Earth

1.00

1.00

1.00

Mars

1.88

1.52

1.01

Jupiter

11.8

5.20

0.99

Saturn

29.5

9.54

1.00

Uranus

84.0

19.18

1.00

Neptune

165

30.06

1.00

Pluto

248

39.44

1.00

(NOTE: The average distance value is given in astronomical units where 1 a.u. is equal to the distance from the earth to the sun - 1.4957 x 1011 m. The orbital period is given in units of earth-years where 1 earth year is the time required for the earth to orbit the sun - 3.156 x 107 seconds. )

Kepler's third law provides an accurate description of the period and distance for a planet's orbits about the sun. Additionally, the same law that describes the T2/R3 ratio for the planets' orbits about the sun also accurately describes the T2/R3 ratio for any satellite (whether a moon or a man-made satellite) about any planet. There is something much deeper to be found in this T2/R3 ratio - something that must relate to basic fundamental principles of motion. In the next part of Lesson 4, these principles will be investigated as we draw a connection between the circular motion principles discussed in Lesson 1 and the motion of a satellite.

 

How did Newton Extend His Notion of Gravity to Explain Planetary Motion?

Newton's comparison of the acceleration of the moon to the acceleration of objects on earth allowed him to establish that the moon is held in a circular orbit by the force of gravity - a force that is inversely dependent upon the distance between the two objects' centers. Establishing gravity as the cause of the moon's orbit does not necessarily establish that gravity is the cause of the planet's orbits. How then did Newton provide credible evidence that the force of gravity is meets the centripetal force requirement for the elliptical motion of planets?

Recall from earlier in Lesson 3 that Johannes Kepler proposed three laws of planetary motion. His Law of Harmonies suggested that the ratio of the period of orbit squared (T2) to the mean radius of orbit cubed (R3) is the same value k for all the planets that orbit the sun. Known data for the orbiting planets suggested the following average ratio:

k = 2.97 x 10-19 s2/m3 = (T2)/(R3)

Newton was able to combine the law of universal gravitation with circular motion principles to show that if the force of gravity provides the centripetal force for the planets' nearly circular orbits, then a value of 2.97 x 10-19 s2/mcould be predicted for the T2/Rratio. Here is the reasoning employed by Newton:

Consider a planet with mass Mplanet to orbit in nearly circular motion about the sun of mass MSun. The net centripetal force acting upon this orbiting planet is given by the relationship

Fnet = (Mplanet * v2) / R

This net centripetal force is the result of the gravitational force that attracts the planet towards the sun, and can be represented as

Fgrav = (G* Mplanet * MSun ) / R2

Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force are equal. Thus,

(Mplanet * v2) / R = (G* Mplanet * MSun ) / R2

Since the velocity of an object in nearly circular orbit can be approximated as v = (2*pi*R) / T,

v2 = (4 * pi* R2) / T2

Substitution of the expression for v2 into the equation above yields,

(Mplanet * 4 * pi* R2) / (R • T2) = (G* Mplanet * MSun ) / R2

By cross-multiplication and simplification, the equation can be transformed into

T/ R= (Mplanet * 4 * pi2) / (G* Mplanet * MSun )

The mass of the planet can then be canceled from the numerator and the denominator of the equation's right-side, yielding

T/ R= (4 * pi2) / (G * MSun )

The right side of the above equation will be the same value for every planet regardless of the planet's mass. Subsequently, it is reasonable that the T2/R3 ratio would be the same value for all planets if the force that holds the planets in their orbits is the force of gravity. Newton's universal law of gravitation predicts results that were consistent with known planetary data and provided a theoretical explanation for Kepler's Law of Harmonies.

Planets rotate around the Sun because of the curve in the space they should follow a circular path and the distance between planet and Sun should be at a distance. Given the fact that earth has a elliptical orbit around the sun, and the distance between Earth and Sun varies according to position of the earTH
 

Kepler's laws of planetary motion. ... Three lawsdevised by Johannes Kepler to define the mechanics of planetary motion. The first law states that planets move in an elliptical orbit, with the Sun being one focus of the ellipse.Image result for kepler's 1st law definition

Kepler's three laws of planetary motion can be stated as follows: (1) All planets move about the Sun in elliptical orbits, having the Sun as one of the foci. (2) A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time.
In fact, a circle is the special case of an ellipse in which the two foci are at the same location. 
Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse.
orbit: 
Kepler's second law of planetary motion. Kepler's second law of planetary motion describes the speed of a planet traveling in an elliptical orbit around the sun. It states that a line between the sun and the planet sweeps equal areas in equal times.
Kepler's 3rd Law: P2 = a. 3Kepler's 3rd law is a mathematical formula. It means that if you know the period of a planet's orbit (P = how long it takes the planet to go around the Sun), then you can determine that planet's distance from the Sun (a = the semimajor axis of the planet's orbit).
NEWTON'S VERSION OF KEPLER'S THIRD LAW. Johannes Kepler discovered a simple relationship between the average distance of a planet from the Sun (called its semi-major axis, A, measured in Astronomical Units) and the amount of time it takes a planet to orbit the Sun once (called its orbital period, P, measured in years).
 

NEWTON'S VERSION OF KEPLER'S THIRD LAW
Johannes Kepler discovered a simple relationship between the average distance of a planet from the Sun (called its semi-major axis, A, measured in Astronomical Units) and the amount of time it takes a planet to orbit the Sun once (called its orbital period, P, measured in years). For objects orbiting the Sun, the semi-major axis to the third power equals the period squared:

A3 = P2

There were two problems with this relation. First, Kepler did not know how it worked, he just knew it did. Second, the relation does not work for objects which are not orbiting the Sun, for example, the Moon orbiting the Earth. Isaac Newton solved both these problems with his Theory of Gravity, and discovered that the masses of the orbiting bodies also play a part. Newton developed a more general form of what was called Kepler's Third Law that could apply to any two objects orbiting a common center of mass. This is called Newton's Version of Kepler's Third Law:

M1 + M2 = A3 / P2

Special units must be used to make this equation work. If the data are not given in the proper units, they must be converted.

The masses must be measured in solar masses, where one solar mass is 1.99 X 1033 grams, or 1.99 X 1030 kilograms.

The semi-major axis must be measured in Astronomical Units, where 1 AU is 149,600,000 kilometers, or 93,000,000 miles.

The orbital period must be measured in years, where 1 year is 365.25 days.

This relation has many uses: determining the mass of a planet by looking at its moon(s), studying binary star systems, even determining the mass of the Galaxy!

There is a problem, however, with the way the equation is written above. Often, we are not able to determine to a high degree of accuracy the average distance between, say, two binary stars. We must use a modified version of NVK3L for very distant objects.

To achieve this modification, we must first introduce an equation for velocity, how fast an object is traveling. Everybody who has driven a car has encountered the formula for velocity. The speedometer on a car measures velocity in miles per hour, or kilometers per hour. Now miles or kilometers are ways of measuring distance, hours are what we use to measure time, and "per" is a word signaling division. Therefore, the formula for velocity is

Velocity = Distance traveled / Time to travel

How does this relate to NVK3L? Remember that our real problem is often that we do not know the average distance between the two objects that are orbiting each other. Many times, we can only clearly see one of the objects that is orbiting!  But velocity is something we can measure, as long as we can see one of the partners, using the Doppler Effect.

Technically what we are measuring is the orbital velocity of the visible partner, which can be related to the distance traveled by the visible partner in its orbit and the time it takes the visible partner to orbit once.  That time is simply  the orbital period P, which is generally easy to observe.  What we usually don't know is the distance traveled around the orbit by the visible partner, called the circumference of the orbit.  This circumference is related to the average distance, A, by the formula

Circumference = C = 2 (pi) A

So the velocity equation becomes

Velocity = V = C / P = 2 (pi) A / P

Remember that we can compute velocity using  the Doppler Effect.  We can observe the orbital period easily.  It is the value of A that is typically very hard to find.  So we turn the equation above around, and solve for A:

A = V P / 2 (pi)

We can now take this value of A and plug it in to Newton's Version of Kepler's Third Law to get an equation involving knowable things, like V and P:

M1 + M2 = V3P3 / 23(pi)3P2

M1 + M2 = V3P / 8(pi)3

What this equation is basically telling us is, the more mass there is in a system, the faster the components of that system are moving as they orbit each other.  We shall not use this more complicated version of NVK3L for homework calculations, but we will use the concept in our discussion of black holes.
 

 

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