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Kepler's First Law:The orbit of a planet about the Sun is an ellipse with the Sun at one focus.Figure 1 shows a picture of an ellipse. It is constructed by specifying two focus points, F1 and F2, of the ellipse. All points on the ellipse, such as P have the property that the sum of the distance between P and F1 and the distance between P and F2 is a constant. The dimension of an ellipse is often described by giving its major axis and minor axis. In descriptions of orbits in the solar system, however, it is more common to use the semi-major axis to describe the size of the orbit, and the eccentricity of the ellipse to describe its shape. The eccentricity is given by the ratio of the distance between the two focus points to the length of the major axis of the ellipse. The periapsis, or the shortest distance between the orbiting body and the central mass, is determined by the product of the semi-major axis and the complement of the eccentriciy (1 - e): if the body is orbiting the sun, this is the perihelion, symbolized by q): q = a (1 - e). A circle is a special case of an ellipse, with an eccentricity of 0, or so that q = a.                                                                           Kepler's Second Law:A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.Figure 2 illustrates Kepler's Second Law. Consider the line between the Sun and point A on the elliptical orbit. After a certain amount of time, the planet will have moved along the orbit to point B, and the line between the Sun and the planet will have swept over the cross hatched area in the figure. Kepler's Second Law states that for any two positions of the planet along the orbit that are separated by the same amount of time, the area swept out in this manner will be the same. Thus, suppose that it takes the planet the same amount of time to go between positions C and D as it did for the planet to go between positions A and B. Kepler's Second Law then tells us that the second cross hatched area between C, D, and the Sun will be the same as the cross hatched area between A, B, and the Sun.Kepler's Second Law is valuable because it gives a quantitative statement about how fast the object will be moving at any point in its orbit. Note that when the planet is closest to the Sun, at perihelion, Kepler's Second Law says that it will be moving the fastest. When the planet is most distant from the Sun, at aphelion, it will be moving the slowest.                               Kepler's Third Law:The squares of the sidereal periods of the planets are proportional to the cubes of their semimajor axes.We have defined the semimajor axis of the orbit above, in our discussion of Kepler's First Law. The sidereal period of a planet's orbit is the time that it takes a planet to complete one orbit around the Sun. Kepler discovered a quantitative relationship between these two properties of the orbit. If P is the period of the orbit, measured in years, and a is the semimajor axis of the orbit, measured in Astronomical Units, thenP2 = a3
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