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 suncentered 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:
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.
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 31day 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.
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 
Average 
T2/R3 
Earth 
3.156 x 107 s 
1.4957 x 1011 
2.977 x 1019 
Mars 
5.93 x 107 s 
2.278 x 1011 
2.975 x 1019 
Observe that the T2/R3 ratio is the same for Earth as it is for mars. In fact, if the same T2/R3 ratio 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 
Average 
T2/R3 
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 
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(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 earthyears 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 manmade 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 1019 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 1019 s2/m3 could be predicted for the T2/R3 ratio. 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) / RThis 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 ) / R2Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force are equal. Thus,
(Mplanet * v2) / R = (G* Mplanet * MSun ) / R2Since the velocity of an object in nearly circular orbit can be approximated as v = (2*pi*R) / T,
v2 = (4 * pi2 * R2) / T2Substitution of the expression for v2 into the equation above yields,
(Mplanet * 4 * pi2 * R2) / (R • T2) = (G* Mplanet * MSun ) / R2By crossmultiplication and simplification, the equation can be transformed into
T2 / R3 = (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 rightside, yielding
T2 / R3 = (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.

Johannes Kepler discovered a simple relationship between the average distance of a planet from the Sun (called its semimajor 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 semimajor axis to the third power equals the period squared:
A^{3} = P^{2} 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: M_{1} + M_{2} = A^{3} / P^{2} 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 10^{33} grams, or 1.99 X 10^{30} kilograms. The semimajor 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: M_{1} + M_{2} = V^{3}P^{3} / 2^{3}(pi)^{3}P^{2} M_{1} + M_{2} = V^{3}P / 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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