Kepler's First Law - The Orbit of Mars
Developed by Matt Lowry, Mars Society Education Task Force
I. Introduction
In the early 1500s, the Polish astronomer Copernicus used observations and geometry to determine the distance of each planet from the center of the Sun and each planet's period of revolution around the Sun - on the bold assertion that the planets actually circled the Sun rather than the opposite. In the late 1500s, before telescopes were invented, the Danish astronomer Tycho Brahe made 20 years of extensive and accurate measurements of planets and bright stars. Near the end of his career, Tycho hired a young German mathematician, Johannes Kepler, and assigned him the task of plotting the orbit of Mars by referencing Brahe's vast store of astronomical data.
Kepler started by drawing a circle to represent the Earth's orbit (it ends up this is a decent approximation). Since Mars takes 687 Earth days to orbit the Sun once, Kepler paid attention to observations that were exactly 687 Earth days apart. In this way Mars would be in the same place in its orbit while Earth would be in a different location. Two angular readings from Mars' location from the same location on Earth 687 days apart was all that was needed - these angular readings are where the two lines of observation crossed at the same point on the orbit of Mars. Plotting many such points did not trace a circle, as Kepler had expected; rather, the orbital path of Mars was an ellipse. Kepler was the first to discover that if planets orbit the Sun, they did so in elliptical, rather than circular paths. This forms the basis of Kepler's First Law . . .
"The planets orbit the Sun in elliptical paths with the Sun at one focus of the ellipse."
In this lab you'll be duplicating the work of Johannes Kepler, and after you've drawn the elliptical orbit of Mars, you will compare the dimensions of your approximation to the actual orbit of Mars. To accomplish this task, you'll use some data from Brahe's own extensive data tables, (see table at the end of the lab).
II. Procedure
Step 1: You'll want four sheets of graph paper taped together so that you've got a roughly 14x14-inch working area.
Step 2: Mark and label a dot at the center of your map to represent the Sun. Place a compass there, and draw a 10-cm radius circle to represent the orbit of the Earth around the Sun; label this circle as "Earth's Orbit." Draw a light line from the center of the right of the paper, and mark the intersection with the Earth's orbit as 0 degrees. Mark this position of the Earth as March 21st. From now on, all of your plotting will be counter-clockwise around the circle from this reference point.
Step 3: Locate the first point in Mars' orbit (Point 1) from the data table. Do this by first marking with a protractor the position of the Earth along a circle for the date November 28, 1580. This 66.5 degrees above the 0 degree reference line. Draw and label with the date a dot to show Earth's position at this time.
Step 4: Mars at this time was in opposition - opposite to the Sun in the sky. A line from the Sun to Earth at this time extends radially outward to Mars. Draw a line from the center of your circle (the Sun) to the Earth's position at this time, and beyond the Earth through Mars. Where is Mars along this line? You'll need another sighting of Mars 687 days later when Mars is at the same place and Earth is in another.
Step 5: If you were to add 687 days to November 28, 1580, you'd get October 16, 1582. At that date Mars was measured to be lower in the sky - actually 107.0 degrees with respect to the 0 degrees reference line of March 21st. With respect to the reference line, use a protractor and a ruler and draw a line at 107.0 degrees as shown in Figure 1. Where your two lines intersect is a point along Mars' orbit.
When you finish with this step, compare your picture to the sample map on the last page of the lab; they should match up!
Step 6: With care, plot the other 13 intersections that represent point along Mars' orbit, using Tycho Brahe's data from the table.
Step 7: Connect your points, either very carefully, by freehand or with a French curve.
III. Questions & Analysis
- During what month are Mars and Earth closest to each other? When are they furthest away from each other?
- What is the conversion factor between the map distance (centimeters) and true distance (miles)? Hint: you might find your text useful for this question.
- Using a ruler, measure the furthest distance of Mars from the Sun in centimeters. Record this as the aphelion on your map. Likewise, measure and record the perihelion - the point where Mars is closest to the Sun - on your map. Using the conversion factor from #2, determine the true distance in miles for Mars during aphelion and perihelion.
- Compare your results for No. 3 to the accepted values given to you by your instructor. Calculate the percent difference between the two sets of values.
- At what point in the orbit of Mars will the planet be moving the fastest? What about the slowest? Mark these points ("fastest" & "slowest") on the map, and provide a solid physics-type explanation for your answers.
[Tycho Brahe's Data Table]
Tycho's data are grouped in 14 pairs of Mars sightings. For the first 9 pairs, the first line of data is for Mars in opposition - when Sun, Earth and Mars were on the same line - when Mars was 90 degrees to the earth horizon, directly overhead at midnight. The second line of data are positions measured 687 days later, when Mars was again in the same place in its orbit, and Earth in a different place, where a different angle was then measured. All angles given in the table read from a 0 degree reference line - the line from the Sun to Earth at the vernal equinox, March 21st. Mars at points 10-14 are non-opposition sightings. The first line of Point 10, for example, shows that when the Earth was at 277 degrees, Mars was not seen directly overhead, but at 208.5 degrees with respect to the 0 degree reference line. Then 687 days later Earth was at 235 degrees, where Mars was seen at 272.5 degrees. The data are neatly arranged for plotting - something that took Kepler years to do.
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