HISTORY 135C

Department of History
University of California, Irvine
 Instructor:    Dr. Barbara J. Becker

The Relative Distances of Mars, Venus, and Mercury from Earth

DETERMINING THE RELATIVE DISTANCE TO AN OUTER PLANET: MARS

When Copernicus put Earth into motion around the Sun, he made us all astronomical surveyors and gave us a long baseline from which to triangulate the relative distances of the planets. He reasoned that an earthbound observer could determine the relative distance to an outer planet -- like Mars -- by (1) observing and recording its apparent location with respect to the background stars on one occasion, and then (2) sighting the planet a second time when it had completed one trip around the Sun. How did Copernicus know when to make this second observation? You can use Copernicus's method and recent information about the position of Mars as seen against the background stars to determine its relative distance from the Sun. To simplify this task, first sight Mars on the date of its opposition. [When an outer planet is at opposition, it is exactly opposite the Sun in Earth's sky.  That means it will rise at sunset, be located directly overhead at midnight, and set at sunrise.] Because of this unique alignment, you can easily draw in the line of sight to Mars whenever it is in opposition as shown in the following diagram: Figure 1.  Mars at opposition. There's no way to know exactly where Mars is located along this line of sight.  To pinpoint Mars's true position, you will need to sight it again, this time from a different point of view. But Mars is in motion -- even Ptolemaic astronomers would agree with that!  You can't conduct a successful triangulation if the distant object is in a different place each time you sight it! Fortunately, Mars returns to this same heliocentric location one martian year, or 687 days, later. Figure 2.  Mars, one martian year later. Where is the Earth located at this time?  In 687 days, Earth will have circled the Sun once and be well on its way toward completing a second orbit. Figure 3.  Earth, one martian year later. You won't see Mars directly overhead at midnight.  To sight Mars, you'll have to look to the eastern horizon near dawn. The angular distance of a planet from the Sun is called its elongation.  Mars will be located somewhere along your new line of sight.  But where? Figure 4.  Sighting Mars. Like a surveyor who has sighted a distant object from two ends of a long baseline, you can now determine the location of Mars by noting where the two lines of sight to Mars intersect. Figure 5.  Pinpointing Mars's position. Using geometry, you can now determine the distance of Mars from the Sun relative to the Earth-Sun distance (one astronomical unit, or 1 AU).  In fact, you can easily approximate Mars's distance from a simple scale drawing: Figure 6.  Determining Mars's relative distance to the Sun. Lucky for you, since 1980 I've been keeping a record of the positions of Mars and Earth when Mars is (1) in opposition and (2) 687 days after opposition.  This record can be found in Table I.

TABLE I.  Positions of Earth and Mars (1980-2003)

Date	Earth's Position	Elongation of Mars
24 Feb 1980	156°	opposition
11 Jan 1982	112°	100°
30 Mar 1982	192°	opposition
15 Feb 1984	149°	99°
10 May 1984	231°	opposition
28 Mar 1986	187°	97°
10 Jul 1986	288°	opposition
27 May 1988	245°	93°
28 Sep 1988	5°	opposition
16 Aug 1990	326°	91°
27 Nov 1990	65°	opposition
10 Oct 1992	13°	93°
7 Jan 1992	108°	opposition
24 Nov 1993	59°	99°
12 Feb 1995	143°	opposition
31 Dec 1996	101°	101°
17 Mar 1997	177°	opposition
2 Feb 1999	135°	100°
24 Apr 1999	214°	opposition
8 Mar 2001	169°	98°
13 Jun 2001	262°	opposition
1 May 2003	218°	95°

Click here to find a printable worksheet like the one shown below. Sample worksheet. On the worksheet, the solid line circle depicts Earth's orbit.  The marks surrounding the circle represent divisions of 5° to locate Earth's position in its orbit.  The dot in the center represents the sun.  The line which intersects the circle at 0° represents the vernal equinox. Copernicus believed the orbits of all the planets were circular.  For him, one set of measurements was sufficient to determine the dimensions of Mars's orbit. Johannes Kepler had access to several sets of these measurements, thanks to the meticulous record-keeping of Danish astronomer Tycho Brahe.  Using these data, Kepler discovered that Mars's orbit was not circular. Use the data for other oppositions of Mars to determine its position in other years.  Describe the orbit of Mars that you have plotted.  How does Mars's distance from the Sun vary?  When do you think Mars moves fastest in its orbit?  When is it closest to Earth? Look up Mars's distance from the Sun.  How do your values compare?

METHOD FOR DETERMINING THE RELATIVE DISTANCE TO THE INNER PLANETS: VENUS AND MERCURY

Ancient observers knew that Mercury and Venus never strayed very far from the Sun. Figure 7.  As seen from Earth, Venus never strays far from the Sun on the sky. But these early astronomers were hard pressed to explain why this happened.  No obvious links connected the motions of the Sun, Moon, and planets. Some speculated that Mercury and Venus might actually travel around the Sun while the Sun orbited the Earth.  Others suggested that Mercury and Venus were somehow tethered to the Sun and thus followed along as it moved. Some astronomers placed Venus closer to the Sun, while others reasoned that rank belonged to Mercury.  There was no conclusive evidence to guide opinion on these matters. Nicholas Copernicus (1473-1543) tried a completely different arrangement.  He placed the Earth in motion around the Sun at a distance farther than both Mercury and Venus. When either of these planets reached its greatest angular separation from the Sun -- its greatest elongation -- Copernicus reasoned (based on simple geometric principles) that an earthbound observer's line of sight was simply a tangent to that planet's orbital path.  Knowing this, he was able to determine the planet's relative distance from the Sun. Using the same worksheet on which you located the orbital distance of Mars: put the Earth on the point where its orbit crosses the vernal equinox imagine that you have just observed Venus at its greatest elongation from the Sun, which is 46° place your protractor along the vernal equinox centered on the Earth measure a 46° angle draw in this sighting line so that it cuts across the Earth's orbital path; the portion of this line that lies inside the Earth's orbit is called a chord Figure 8. bisect this chord Figure 9. draw a line from the Sun to the chord's center Figure 10. the point where these two lines meet marks Venus's location along your line of sight measure the length of the line between Venus and the Sun calculate the ratio of this length to the radius of Earth's orbital path in your drawing This is the distance of Venus from the Sun measured in astronomical units.   use a compass to draw in the orbit of Venus Figure 11. Repeat this procedure to find the orbital distance of Mercury in astronomical units. [NOTE:  Mercury's greatest elongation is 27°.]