Conic Sections

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## Planetary MotionThe So, for instance, the planet Jupiter might be conceived of as moving in a circle around the earth at a constant rate. This rate is measured against the field of stars. Every night Jupiter rises in the east and sets in the west together with the other heavenly bodies. On any particular night its position against the field of stars appears unchanged to the naked eye. Its position does change as time goes on, however: in general, it moves across the star field from east to west. The same is true of the other planets; Saturn is slow, while Venus and Mercury are fast. However, ancient astronomers observed that Jupiter and the other planets seem to move backward periodically, a phenomenon known as Click here for an animation. The outer circle represents the fixed sphere of the stars. Ptolemy's model was regarded as trustworthy for over a thousand years. It accurately predicted the planets' motion as far as astronomers could tell with the instruments available to them. During the Renaissance, however, Nicolaus Copernicus (1473 – 1543), a Polish mathematician and astronomer, proposed a The Copernican model is important chiefly for making possible the formulation of Kepler's Laws of Planetary Motion, which are still taught in courses on celestial mechanics today. They were developed by Johannes Kepler (1571 – 1630), a German mathematician, astronomer, and astrologer, who made use of data collected by Tycho Brahe (1546 – 1601). For us the key point is that Kepler did away with epicycles and deferents by postulating that the planets move in Here, then, are the laws: - Planets move in elliptical orbits around the sun, with the sun at one focus.
- The focal radius from the sun to the planet sweeps out equal areas over equal times.
- The square of the orbital period is proportional to the cube of the semimajor axis of the orbit.
The second law implies that each planet moves faster when it is close to the sun and slower when it is far away, as indicated on the diagram to the right. The eccentricity of the orbit is much higher than for any of the planets; comets have such orbits, however. The points indicated on the orbit are equally spaced with respect to time. Click here for an animation of the planet's motion; click here for a demonstration of the equal-area law. Each of the regions (colored green and yellow in alternation) is swept out over one-twelth of an orbital period, and they all have equal area. The third law states that, if
where Kepler's Laws describe how planets move but don't provide any rationale for this. The physical laws proposed by Isaac Newton (1642 – 1727) in his - An object not being subjected to force has constant velocity.
- The force exerted by a moving object is proportional to the rate of change of the velocity.
- Every action has an equal and opposite reaction.
His Law of Universal Gravitation is: - Two objects attract each other with a force proportional to the product of their masses and inversely proportional to the square of their distance.
These four laws can be used to justify Kepler's laws as well as Galileo's laws, if certain simplifying assumptions are made. For instance, Newton's laws imply that the sun and the earth really move around each other; but the sun is so massive compared to the earth that we can consider it as fixed in one place. We also find that hyperbolic orbits are possible, and obey the equal-area law as well. |

**Michael Ortiz, Ph.D.
Assistant Professor of Mathematics
Department of Natural and Behavioral Sciences**

Sul Ross State University Rio Grande College

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