Kepler 39;s Law Of Planetary Motion Pdf Download [WORK]
Gregory Muench
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:[1][2]
Newton was 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, whereas 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.[22]
A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass. This results in replacing a circular radius, r \displaystyle r , with the semi-major axis, a \displaystyle a , of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass M \displaystyle M with M + m \displaystyle M+m . However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is:
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.)
The important special case of circular orbit, ε = 0, gives θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.
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 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.
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.
2. Galileo is often credited with the early discovery of four of Jupiter's many moons. The moons orbiting Jupiter follow the same laws of motion as the planets orbiting the sun. One of the moons is called Io - its distance from Jupiter's center is 4.2 units and it orbits Jupiter in 1.8 Earth-days. Another moon is called Ganymede; it is 10.7 units from Jupiter's center. Make a prediction of the period of Ganymede using Kepler's law of harmonies.
In bounded motion, the particle has negative total energy (E < 0) and has two or more extreme points where the total energy is always equal to the potential energy of the particle, i.e., the kinetic energy of the particle becomes zero.
In unbounded motion, the particle has positive total energy (E > 0) and has a single extreme point where the total energy is always equal to the potential energy of the particle, i.e., the kinetic energy of the particle becomes zero.
2. The amount of "flattening" of the ellipse is termed theeccentricity. Thus, in the following figure the ellipses become moreeccentric from left to right. A circle may be viewed as a special case of anellipse with zero eccentricity, while as the ellipse becomes more flattened theeccentricity approaches one. Mathematically it is defined as the distance between foci dividedby the major axis length.Thus, all ellipses have eccentricities lying betweenzero and one.The orbits of the planets are ellipses but the eccentricities are so small formost of the planets thatthey look circular at first glance. For most of the planetsone must measure the geometry carefully todetermine that they are not circles, but ellipses of smalleccentricity. Pluto and Mercury are exceptions: their orbits are sufficientlyeccentric that they can be seen by inspection to not be circles. 3. The long axis of the ellipse is called the major axis, while theshort axis is called the minor axis (adjacent figure). Half of themajor axis is termed a semimajor axis. Thelength of a semimajor axis is often termed the size of the ellipse. It canbe shown that the average separation of a planet from the Sun as it goes aroundits elliptical orbit is equal to the length of the semimajor axis. Thus,by the "radius" of a planet's orbit one usually means the lengthof the semimajor axis. For a more detailed investigation of the properties ofellipses, see this ellipse applet The Laws of Planetary MotionKepler obtained Brahe's data after his death despite the attempts by Brahe'sfamily to keep the data from him in the hope of monetary gain. There is someevidence that Kepler obtained the data by less than legal means; it isfortunate for the development of modern astronomy that he was successful.Utilizing the voluminous and precise data of Brahe, Keplerwas eventually able to build on the realization that the orbits of theplanets were ellipses to formulate his Three Laws of PlanetaryMotion. Kepler's First Law:I. The orbits of the planets are ellipses, with the Sun at one focus ofthe ellipse.Kepler's First Law is illustrated in the image shown above.The Sun is not at the center of the ellipse, but is instead at one focus(generally there is nothing at the other focus of the ellipse). The planetthen follows the ellipse in its orbit, which means that the Earth-Sun distanceis constantly changing as the planet goes around its orbit. For purpose ofillustration we have shown the orbit as rather eccentric; remember that theactual orbits are much less eccentric than this.Kepler's Second Law:II. The line joining the planet to the Sun sweeps out equal areas in equaltimes as the planet travels around theellipse.Kepler's second law is illustrated in the preceding figure.The line joining the Sun and planet sweeps out equal areas inequal times, so the planet moves faster when it is nearer the Sun. Thus, a planet executes elliptical motion with constantly changing angular speed as it moves aboutits orbit. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion. Hence, by Kepler's second law, the planet moves fastest when it isnear perihelion and slowest when it is near aphelion.Kepler's Third Law:III. The ratio of the squares of the revolutionary periods for two planets is equal tothe ratio of the cubes of their semimajor axes:In this equation P represents the period of revolution (orbit) fora planet around the sun and R represents thelength of its semimajor axis. The subscripts "1" and "2" distinguishquantities for planet 1 and 2 respectively. The periods for the two planetsare assumed to be in the same time units and the lengths of the semimajor axesfor the two planets are assumed to be in the same distance units.Kepler's Third Law implies that the period for a planet to orbit the Sun increasesrapidly with the radius of its orbit. Thus, we find that Mercury, the innermostplanet, takes only 88 days to orbit the Sun but the outermost planet (Pluto)requires 248 years to do the same. Here is a java applet allowing you to investigate Kepler's Laws, and Here is an animation illustrating the actual relative periods of the inner planets.Calculations Using Kepler's Third LawA convenient unit of measurement for periods is in Earth years, and aconvenient unit of measurement for distances is the average separation of theEarth from the Sun, which is termed an astronomical unit and isabbreviated as AU.If these units are used in Kepler's 3rd Law, the denominators in the precedingequation are numerically equal to unity and it may be written in the simpleform
A quick guide to sections of "Stargazers" related to Kepler's laws can be found in the section "Kepler's Laws". In what follows, those sections will sometimes be referred to by their numbers. You can also reach the complete list of links either from "Site Map" at the top of this page or from "Back to the Home Page" at the end.
- Note that addresses here are abbreviated, because you are already logged onto "Stargazers."
Thus the home page is Sintro.htm
not
"Stargazers" contains more material than can ever be covered in a regular class. Still, teachers need a wider knowledge, allowing them to pick and choose material according to circumstances, and to mention odd tidbits without detailed discussion, just to create interest. And some very lucky teachers may sometimes find in class a kid or two who really want to find out more. Such students can be directed here to satisfy their interest. This overview focuses on three items:
---what are Kepler's laws, what do they mean, and why are they important. The laws were formulated between 1609 to 16l9, and are (as usually stated):
- Planets move around the Sun in ellipses, with the Sun at one focus
- The line connecting the Sun to a planet sweeps equal areas in equal times.
- The square of the orbital period of a planet is proportional to the cube (3rd power) of the mean distance from the Sun
(also stated as-- ...of the "semi-major axis" of the orbital ellipse, half the sum of smallest and greatest distances from the Sun)
The Significance of Kepler's Laws Kepler's laws describe the motion of planets around the Sun.
Kepler knew 6 planets: Earth, Venus, Mercury, Mars, Jupiter and Saturn. The orbit of the Earth around the Sun.
This is a perspective view, the shape of
the actual orbit is very close to a circle. All these (also the Moon) move in nearly the same flat plane (section #2 in "Stargazers"). The solar system is flat like a pancake! The Earth is on the pancake, too, so we see the entire system edge-on--the entire pancake occupies one line (or maybe a narrow strip) cutting across the sky, known as the ecliptic. Every planet, the Moon and Sun too, move along or near the ecliptic. If you see a bunch of bright stars strung out in a line across the sky--with the line perhaps also including the Moon, (whose orbit is also close to that "pancake"), or the place on the horizon where the Sun had just set--you are probably seeing planets.
- Ancient astronomers believed the Earth was the center of the Universe--the stars were on a sphere rotating around it (we now know it's actually the earth that is turning) and the planets were moving on their own "crystal spheres" with variable speed. They usually moved in the same direction, but sometimes their motion reversed for a month or two, and no one knew why.
Kepler Kepler worked with Tycho Brahe, a Danish nobleman who pushed pre-telescope astronomy to its greatest precision, measuring positions of planets as accurately as the eye could make out (Brahe died in 1602 in Prague, now the Czech capital; telescopes started with Galileo around 1609). If you want to read about it, I recommend "Tycho and Kepler" by Kitty Ferguson, reviewed at or at least, read the review. Let me quote from it:
- Religious intolerance was widespread--indeed, events were moving towards the 30 years' war (1618-48), Europe's most destructive religious battle, mirrored by the civil war in Britain. Kepler was forced out of Graz, among all other employees of Protestant colleges in town, after the ruling archduke decreed they must leave the city by nightfall, that same day. It was also an era when Kepler's mother was arrested for witchcraft, when most of his numerous children died in childhood, and when Tycho's marriage was regarded as a second-rate "slegfred" union because his chosen wife was not from the nobility.
- (1)Planets move around the Sun in ellipses, with the Sun at one focus
First explain what an ellipse is: one of the "conic sections, " shapes obtaining by slicing a cone with a flat surface. A flashlight creates a cone of light: aim it at a flat wall and you get a conic section.
- Hit the wall perpendicular. The wall cuts the cone perpendicular to its axis and you get a circle of light.Slant the cone relative to the wall: an ellipse. The more you slant, the more far away the ellipse closes. The curves generated as
"conic sections"when flat
planes are cut across a cone. Finally, if the axis of the cone is parallel to the wall, the curve never closes: you get a parabola. Kepler's laws (as we now know them) allow all conic sections, and parabolas are very close to the orbits of nonperiodic comets, which start very far away. (Tilt still more and you get hyperbolas--not only don't the trajectories close, but the directions of coming and going make a definite angle).
First of all, a very famous ellipse is shown below. Its story is told in section #S7-a dafc88bca6