In order to use this scheme- Kepler had to have a series of Martian ob.pdf
- 1. In order to use this scheme, Kepler had to have a series of Martian observations 687 days apart, and here Tycho's otserving books provided a veritable gold mine. We can follos this procedure using the same Tyehonic observations that Kepler used; these are found in the following table. For this exercise, graph poper will be used to make it easier to liae up the peotractor. Find the center of the araph paner (the Sgn) and draw a straight ling from the center to the risht hand rdae, parallel to the urid lines. The end of this line represents the basic 00 directice ia space. the direction of the Sun as seen from the Earth at the time of the vemal equinox. Heliocentric longitude is measured from this linc. Draw a circle of 4 cantimeters radius te represent the F.arth's orbit. (Alitough the orbits of Mars and the Earth are actually ellipses, we will see that the circles are pretry good approximations,) Using the protractor centered on the Sun and the vereal cquines as the startine line lav off ceunterclockwise the heliocentric longitudes of the Farth for each date in the table, and label the date. Next with the Earth as the center and with 0 in the direction of and along a line parallel to the vernal squinas. mark off ceuntereleckwise the directions for the scesentric longifudes of Mars, For each pair of observations, the position of Mars in its orbit lies at the imfersection of the lines drawn from both Earth positions through the corresponding goocentric leagitudes of Mars. The nexi step is to determine the center of the circle that will approximate the orbit of Mars. What is needed is the center of the circle which can be determined by taking the perpendicular bisector of three chords of the circle. This method will be discused in class. Dae to constraction errors, these three lines may not all incrsect in one point, so you may have to establish the "best center". The distance from the center to one of Mars' positions will give you the radius of the eircle. Draw the circle representine the orbit of Mars The line from the Sun through the eenter of Mars orbit will lie on the major axis of the oebit of Mars. Derw in the maior avis so. that it wasses threash the Smn. the center of Mars" orbit, and intersets the erbit of Mars. lahel the perihelion and aphelion positions of Mars. Determine the distance Mars is from the Son using the scals that one astronamical unit equals 5 ecatimeters. The eceestricity (c) of the orbit is the ratio of the distanse betwoen the Sun and the center of the orbit for Mars and the semi-major axis of the orbic, measured int the same usits. Determine the eccentrisity (c). PURPOSE: To deternine the orbit of Mars from a limined set of observations. PROCEDURE: Using some of the oboervations of Mans made by Tycho and the method developed by Kepler, an approsimute erbit for the planet Mars will be determined. From our earthbound viewpoint, the eccentricity of Mars ocbit provides the most interesting aspest of the planef's motion. Thes relatively large eecentricity makes the elliptical shape of Mars' orbil eomparatively casy to stady. As lohannes Kepler stid, "to diseover the secret of the cosmos, we mast use the motion of Mars; oherwise it will nemain etemally hidden". As Mars and the Earit move in their otbits, they come close to each other about every two years. Because Mar' path around the Sen is moch more off-center than ours, some oppositions briag Mars much nearer to Earth than ethery. The very closest approaches occur approximately 15 years apart, at so-called perihelic egpositions. Thus, it was at a particularly chose epposition in 1877 that Asaph Ilall discovered the satellites of Mars and Giovanni Schiaperelli first observed the canali on its surface. At one of the next favorable elose approachex, in 1894. Percival Lowell male a big splach with his theory of intelligent life on Mars, and again in 1907 the "Mars furor" roumed kriefly. The next favorable perihelic opposition is in 2018 . Alhough the Martian oebit has an appeciakle eccentricity, it is a mistake to think of its shape as obvious ellipie; rather, it looks very mach like a circle that is off-centered from the Sun. We will investigate this feahare of Mars' oefit, cmploying seme of the same data from Tycho Beahe that Kepier used in the carly 16005 x , and we shall follow the method used by Kepler. Kepler knew that it took 687 days for Mars to
- 2. complete one helioccatric circuit. On the other hand, the liarth would retum to the same place in its ewn orbit every ycar- 365 days for one circuit, 730 days for two. Hence, if Mars is observed at a particular place, he knew that it would be back at the same spot in 687 dayx, whereas the Earth wovld not quite have completed its second revolution; therefore Mars would be sighed foven a different vantage poiat. This clever triangulation technique is seen in the diagram below. In order to use this seheme, Kepler had to have a scries of Martian observativas 687 days apart, and here Tycho's observing books provided a veritable sold mime. We can follow this procedure using the same Tychonic observations that Kepler used these are found in the following table.