Experiment II-7

Determining the Orbit of Mars


Galileo discovered four moons orbiting Jupiter and challenged others to make careful measurements of their periods.  Tables of their movements were developed by Borelli (1665) and Cassini (1668).  These tables were based mainly on observations near the time when Jupiter was in opposition, (when Jupiter is opposite the Sun) because this is when Jupiter appears highest and brightest in the night sky.  The orbital periods of Jupiter's four moons were found to be 1.769 days, 3.551 days, 7.155 days, and 16.689 days.  These seemed to be very constant and predictable, just like all other heavenly motions.  Based on these figures it was possible to predict within minutes the times of eclipses and passages (the passings behind and in front of Jupiter) that would occur in future observations.

Roemer's explanation

Knowing when these passages should occur, people began to make observations at times when measurements were more difficult such as when Jupiter was nearly in conjunction (together in the sky) with the Sun, but when it was just possible to observe Jupiter just after sunset or before daybreak.  But the eclipses and passages of Jupiter's moons at conjunction, which had be predicted so precisely when Jupiter was in opposition, were consistently later than their predicted occurrence.  All the four moons seemed to be late by the same amount.  The early astronomers actually measured up to 22 minutes late, but more recent measurements have shown that the lateness is slightly less than 17 minutes.

At the subsequent oppositions six months later, all the moons were found to be back on their predicted schedule!  While mulling over this data in 1675 on a visit to Paris, the Danish astronomer Ole Roemer (1644-1710) thought of an explanation based on: sight is not instantaneous.  If light travels at a finite speed, when we see distant things we are really seeing how they were at some time in the past.  The further away from an object, the greater the time delay in our viewing.  Applying this hypothesis to the observations of Jupiter's moons, Roemer realized that when Jupiter was in conjunction with the Sun, light from Jupiter and it's moons had to travel the extra distance equal the diameter of the Earth's orbit.  Using this distance of two radii of the Earth's orbit (that is, two Astronomical Units) and the time of delay, Dutch Christiaan Huygens (1629-1695) made the first reasonable calculation of the speed of light (about 1AU / 8.5 minutes = 3.0 x 108 m/sec).

Kepler's nested solidsOther European astronomers also made important measurements and calculations.  Johann Kepler, Keppler, Khepler, Kheppler, or Keplerus (his spellings!) was conceived May 16, A.D. 1571 at 4:37 A.M. and born December 27 at 2:30 P.M. (according to him) in Weil der Stadt (Weil-the-town) in southwest Germany with the handicaps of poverty.  (In 1630 he left his own family penniless, dying while journeying alone in search of money owed him.)  His father had abandoned the family, his brother bullied him, his epileptic aunt was burned as a witch and his mother later almost met the same fate.  But Kepler received a free education because the new Lutheran church believed everyone needed to study the Bible for themselves.  Bright enough to receive advanced education designed to produce church leaders, Kepler could only find a job teaching mathematics, astronomy, and astrology.  One day in class while drawing circles just inside and outside of a triangle, he realized the geometry fixing the ratio between the two circles could be the clue to understanding the spacing of the planets.  Perhaps the spacings between the 6 planets of Copernicus (Ptolemy counted 7 including the Moon and Sun) was established by the 5 perfect flat sided solids known to Pythagoras.  Kepler's subsequent laborious effort found the sizes of the planet orbits DID match within experimental error.  Inside the orbit sphere of Saturn, Kepler calculated the largest cube that would fit.  He next calculated the largest sphere for Jupiter's orbit that would fit inside the cube then determined the ratio between the two orbits.  (Diagram above)  Saturn was the furthest away, the slowest moving, and thus the most difficult to measure  So Kepler found a 10% error acceptable.  The close or perfect fit of the other planets convinced Kepler that the world is indeed Copernican.  (Explaining the spacings of Ptolemy's 7 planets would require discovery of a sixth perfect solid.)  Publication of his Mysterium Cosmographicum in 1597 explaining the spacings of the planets established Kepler's reputation as a leading astronomer.  (Shown at right from Kepler's book is the arrangement of inner planets and larger below is the arrangement of outer planets.)

from Mysterium Cosmographicum
Planet Solid spacer Copernican spacing Kepler's model % error
cube 0.635 0.577 10%
tetrahedron 0.333 0.333 0%!!!
dodecahedron 0.757 0.795 5%
icosahedron 0.795 0.795 0%!!!
octahedron 0.702 0.707 3%

Aristocrat Tycho Brahe (1546-1607) lived a luxurious life after his father died saving the king of Denmark from drownding in a mote.  While at university he noted a predicted conjunction was days late and decided more accurate predictions would require more accurate observations and measurements.  So with sinecures from the King, he built the world's largest observing tools which he calibrated to the limits perceivable by human vision.  One of his early accomplishments was to observe a nova, a star never before seen that seemed to appear in the presumably unchanging sky.  After the king's death and the royal financing ended, Brahe obtained employment as the royal mathematician and astronomer to the last Holy Roman Emperor in Prague.  Brahe hoped to establish a compromise system in which the Sun and Moon orbited the Earth, while all other planets orbited the Sun.  Realizing he needed an assistant for the tedious calculations, he hired Kepler.

Kepler's laws

Kepler was assigned to calculate the orbit of Mars since Brahe found it most poorly fit predictions.  Kepler, claiming he could do the problem in 8 days, took 8 years.  After a year, Brahe died, perhaps due to over indulgence, Kepler stole Brahe's journals, and himself obtained the job as royal mathematician and astronomer.  Brahe had made 10 precise sightings of Mars and Kepler made two more.  These sightings did not fit the Copernican system, nor did adding an equant help.  In frustration, Kepler plotted what he called a clean orbit then noted Mars changes speed.  Drawing out rays from the Sun every 1°, Kepler noticed that the radius of Mars sweeps out equal areas in equal times.  It moves more rapidly when closer to the sun, then slower when further away.  After rejecting several kinds of orbits that closely fit the sightings because he had confidence in the precision and accuracy of Brahe's measurements, Kepler found the path of Mars was actually an ellipse with the Sun at one focus!  The universal belief was false that the perfect heavens required circular motion.

Note several lessons in the history:


Accurately observing the position of an object like Mars in the sky only provides direction but not its distance.  This information alone could not establish the location of Mars.  Since ancient times everyone presumed the stars were fixed to the celestial sphere.  Copernicus had claimed that the celestial sphere was a great distance away.  But Mars, like all planets, was thought to be considerably closer and continuously moving.

Mars parallax

According to the Copernican system, the Earth is moving in a circular orbit providing moving vantage points.  So a second observation at a different time provides a different view of Mars.  But Mars is also moving, making a second random viewing of no value for determining its location.  Ancient astronomers had noted that Mars falls behind the stars, moving even more slowly than the Sun so that it is passed by the Sun 37 times in 79 years.  But according to Copernicus, IF the Earth and Mars are orbiting the central Sun at the same rate, like cars travelling the same speed together around a speedway, Mars would NEVER APPEAR to be passed by the Sun.  So during one Earth orbit taking one year, Mars makes a hidden common orbit around the Sun.  But Mars is actually slower, being passed by the Earth 37 times in 79 years.  So according to Copernicus, Mars actually completes only 79 - 37 = 42 orbits in 79 years.  If this is correct, Mars will complete one orbit and be back to the same actual location in the sky every 1.88 years = 687 days.  If our second observation of Mars is 687 days after the first, we will have two perspectives of Mars in the same location allowing us to determine one point on Mars' orbit using parallax.

In this experiment we wish to make key observations about Mars' direction in the sky, then use those measurements to plot the famous orbit of Mars and determine its parameters.


sky camera Below is a portion of a wide angle photograph of the March 21, 1931 night sky taken with a wide angle camera (shown at right) at the Harvard College Observatory.  A search was made of the Observatory's vast collection of routinely made photographic plates for pairs of photographs showing Mars 687 days apart.  Mars is back at the same location but the Earth is 43 days shy of completing two orbits in that time period.  So knowing where the Earth is on each date and knowing the direction in the sky towards Mar provides two perspectives allowing us to use parallax to plot the location of Mars.  On this and the next two screens are 8 pairs of such sky photographs and accompany sky maps of those regions of the sky.
  1. Print a copy of each of the 16 photographs of the sky at the same scale (pixels per inch).  (You will have more data than Kepler!)  Because the night sky is dark, this would consume a lot of black printer ink.  So to make this experiment more environmentally friendly (and less expensive requiring less ink), all sky photographs have been inverted so white stars will look black, and the expansive black sky will appear white.
  2. Print at the same scale each of the 16 portions of the red sky map on the most transparent paper you have available for your printer.  You could purchase special transparent plastic used for printing presentations for overhead projectors, however that is not essential.  It is possible to see enough through most printer paper if you hold the photograph and map in front of a bright light or window.
  3. Look on each sky chart for several stars in a pattern (such as the big dipper).  Look an the photograph with the same code letter seeking to find the same pattern in the brightest stars.  Align the star map with the stars in the photograph.
  4. Look for a bright (black) object on the photograph that does not appear on the map.  Since Mars moves compared to the stars, it can not have a fixed location on the star map.  Use the ecliptic (annual path of the sun) and the scale (degrees from the Vernal Equinox) to determine and record the longitude (angular direction) to Mars.
Mars A plate star map A
  1. View and print 15 additional sky photographs and star maps.

With these 16 measurements, we are now ready to plot the orbit of Mars and analyze its shape.

  1. Using a large piece of paper place a dot for the Sun near the center and draw a circle to represent the Earth's orbit around the Sun with a radius that extends about half way to the edges of the paper.  (If you previously plotted the orbit of the Earth via Experiment II-6, that provides a superior start.)
  2. Draw a ray out from the sun representing the location of the Earth on March 21, the Vernal Equinox.  The direction TOWARDS the Sun on this date is traditionally assigned 0° longitude.
  3. Using a protractor to measure angles clockwise, and rays out from the Sun, determine the location of the Earth on the dates of each photographic plate.  Since the Earth orbits 360° around the Sun in 365 days, we could determine the following longitudes base on approximately 1.0°/day.
Longitude of the Sun viewed from Earth on Dates of Photographs
Plate Date Longitude          Plate Date Longitude
A March 21, 1931 B February 5, 1933 316°
C April 20, 1933 29° D March 8, 1935 347°
E May 26, 1935 65° F April 12, 1937 21°
G September 26, 1939 184° H August 4, 1941 131°
I November 22, 1941 239° J October 11, 1943 198°
K January 21, 1944 300° L December 9, 1945 255°
M March 19, 1946 358° N February 3, 1948 314°
O April 4, 1948 13° P February 21, 1950 332°
  1. At each Earth position, accurately draw a ray parallel to the 0° line constructed in step 7.
  2. Continuing to measure clockwise, draw a ray in the direction of Mars as determined from the location of Mars when each photograph is overlayed with the appropriate star map.
  3. Place a dot for the location of Mars where rays intersect for each pair of photographs 687 days apart.
  4. Attempt to find the best off centered circle to pass closest to the 8 Mars locations.  Mark the center of this circle.
  5. Recalling Kepler's claim that the Sun is at one focus of it's orbit ellipse, measure the distance between the Sun and the circle center, then locate the second empty focus an equal distance from the circle center but opposite the Sun.
  6. Construct an ellipse by tying a string into a tight loop around pins located at the two foci and a pencil held at one location of Mars.  Holding the loop of string taut, construct the ellipse.  Adjust the ellipse as necessary to fit the 8 locations of Mars.  Check measurements for any points that don't come close to fitting the ellipse.
  7. Measure the aphelion and perihelion distances, take the average to determine average radius, a.  Measure the distance from the circle center to either focus, c. Divide c/a to determine the eccentricity of the ellipse.
Do you concur with Kepler's conclusions?

Optional Experiment

Use the photographs and star maps to measure the latitude of Mars on each date.  Use this information to calculate the tilt of Mar's orbit compared to that of the Earth.



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created 19 April 2003
latest revision 16 May 2010
by D Trapp
Mac made