[ image of Mars ]       

The Geography of Mars

Lab

Cassini, Richer, and Picard
Experiment, 1672

Christine M. Rodrigue, Ph.D.

Department of Geography
California State University
Long Beach, CA 90840-1101
1 (562) 985-4895
rodrigue@csulb.edu
https://home.csulb.edu/~rodrigue/

Use of Parallax to Calculate Mars' Distance from Earth
and Use of Kepler's Third Law to Calculate Earth's Distance from Sun

This lab has the following objectives:
  • to let you repeat the experiment by which Jean-Dominique Cassini and his assistants, Jean Richer and Jean Picard, estimated the distance between Earth and Mars using parallax
  • to have you apply Johannes Kepler's Third Law to see how the research team used the Earth-Mars distance to estimate the Earth-Sun distance
  • to have you back-engineer what the actual angular distance between Earth and Mars must have been on that October 1st, 1672, had Cassini's team had modern equipment, given the contemporary figure for the Astronomical Unit
  • to give you an appreciation for the intellectual achievement of the Cassini team and for their meticulousness in the face of instrumental limitations
Background

Using Tycho Brahe's database, Johannes Kepler worked out three laws of planetary motion:

  1. planets follow ellipical orbits, with the sun at one of the two foci of the ellipses
    [ Kepler's First Law ]
  2. a line drawn from a planet to the sun sweeps out equal areas in equal times, so a planet has to speed up at perihelion to compensate for the shorter distance between it and the sun and slow down at aphelion to offset the longer distance
    [ Kepler's Second Law ]
  3. the cube of the semi-major axis of a planet's orbit is directly proportional to the square of its orbital period, so, if you consider two planets in orbit around the sun, the ratio of the cubes is the same as the ratio of the squares:
    [ Kepler's Third Law for two planets ]
    (where P stands for orbital Period and R stands for the semi-major Radius)

Using these laws, scientists had worked out relative distances among the various planets then known in the solar system, but they had no idea what the absolute distances were. They knew Jupiter, for example, was out there about five times as far from the sun as the earth was, but they didn't know what that meant since they didn't know how far Earth was from the sun.

That was the problem Cassini took on. As observations improved rapidly with better instrumentation, better predictions of the planets' positions could be made. It was known that Mars and Earth would be in opposition (Mars, Earth, and the sun in a straight line, with Mars and Earth on the same side of the sun) in September and October of 1672. This meant outstanding observation opportunities, because the two planets would be relatively close together. English astronomer John Flamsteed had predicted that Mars would pass in front of a particular star in the constellation Aquarius (the middle Psi star) on the first of October, 1672.

[ Mars occultation of Psi 1 ]   [ Mars occultation of Psi 2 ]
The prediction galvanized the astronomy community, and Cassini decided to exploit this once in a lifetime opportunity. With the Earth and Mars in a straight line from the sun, it would be possible to work out the Mars-Earth distance and, applying Kepler's Third Law, the Earth-Sun distance as a function of the Mars-Earth distance.

He sent a friend and collaborator, Jean Richer, down to a town called Cayenne in French Guiana on the north coast of South America. The town was 7,089 km away, which would provide a long baseline to do parallax-based observations. The baseline would not be the great circle distance of 7,089 km, though, but the chord of that distance (the distance of a tunnel going straight through the earth from Paris to Cayenne). The chord baseline distance would be roughly 6,700 km.

Richer was to make precise recordings of Mars' position at midnight on the appointed evening with respect to the Psi stars in Aquarius. At midnight that evening, Cassini and another collaborator, Jean Picard, would do the same thing from the Paris Observatory. The team had to wait nearly a year to bring their observations together and work out the trigonometry, but, even with the less precise instruments of the day, they were able to come up with an estimate for the Mars-Earth distance that was within about 7% of the modern calculation.

Interestingly enough, John Flamsteed (what is with all these guys named John or Jean in French obsessing on the same thing?) came up with another approach to parallax that didn't entail a long and arduous trip. He observed Mars at two time periods in the same night widely separated in time, so that the earth's rotation would give him the spatial separation he needed to establish his baseline. His estimate turned out pretty close to the Cassini team's estimate.

Your data and methods

Here is an idealized diagram of the parallax situation. You have a triangle of concern, the one leading from the two places on Earth and Mars, which can be subdivided into two right triangles (with the hypotenuse, opposite, and adjacent sides labelled).

[ Mars occultation of Psi 1 ]
What you need to figure out is A, the length of the adjacent side, but all you have is O or the baseline of ~3,350 km or one-half the chord distance of 6,700 km between Paris and Cayenne and the parallax angle, which is 9.5 arcseconds.

First, divide 9.5 by 3,600 (60 seconds times 60 minutes) to get the parallax angle in degrees. What would that (extremely tiny number) be?

1.  _________________________

You're trying to imagine creating an insanely skinny angle between the adjacent side and the hypotenuse and then resting the adjacent side along the straight line from Mars to Earth. If you imagine sliding a ruler along the length of the adjacent side at right angles to it, you're trying to move that ruler just until until the open end of the angle (between H and A) is 3,350 km wide (the width of the O or opposite side in your triangle, now formed by your imaginary ruler). How far down the A side do you have to slide the O ruler?

From trigonometry come some useful ratios: SOH, CAH, TOA (sine=opposite/hypotenuse; cosine=alternative/hypotenuse; tangent=opposite/alternate). We don't care about the hypotenuse, so that gets rid of the sine and the cosine. We're interested in figuring out the length of the alternate side from knowing the opposite side (3,350 km), so that leaves us with the tangent of that very skinny triangle. You need to convert the angle of that skinny triangle from degrees into tangent form. You can do that with a calculator (tan function) or, in Open/LibreOffice Calc, which would be =tan(radians(9.5/3600)) or =tan(radians(whichever cell you put the degrees calculation in)). So, what would that tangent be?

2.  _________________________

So, now you have two of the three elements of TOA (tangent = opposite divided by alternate, T = O/A): the tangent and the opposite (3,350 km). But what we need is the length of the alternate side. So, let's algebraically re-arrange the formula to A = O/T. Go on ahead and divide O or 3,350 by that insanely small tangent you just calculated. The answer for A is the Earth- Mars distance.

3.  _________________________

Now that you have the distance between Mars and Earth at opposition on 1 October 1672, you can apply Kepler's Third Law to figure out the Cassini team's estimate of the Astronomical Unit, or the distance between the Earth and Sun. Remember that the ratio formed by the squared periods of the two planets' years or periods is the same as the ratio formed by the cubes of their distances from the sun. So, let's calculate the ratio of the squared periods. Earth's year is 365.25 Earth days long; Mars' year is 686.98 Earth days long (or 668.60 sols/martian days long). What's the square of Mars' revolutionary period?

4.  _________________________

And what's the square of Earth's period?

5.  _________________________

Divide Mars' squared period by Earth's squared period to get that ratio:

6.  _________________________

That ratio is the same as the ratio of the cubed radiuses or distances from the sun to each planet. So, let's now take the cube root of that ratio between the squared periods.

With a calculator, you'd enter the ratio and then the x√Y key and then the number 3, then the equals sign. In a spreadsheet, you would create a reference to the cell where you calculated that ratio. I'll call that cell C6 here (use whichever cell you used to calculate that ratio!). In another cell, you'd type =C6^1/3 or =c6^(1/3) (in a spreadsheet, you raise the number to the 1/3rd power, which is the same as taking the cube root, which spreadsheets don't do directly).

So, whichever way you did it, what would that cube root be?

7.  __________________________

Now, subtract 1 from cube root of the ratio of the squared Mars and Earth years. That is?
8.  _________________________

Divide the Earth-Mars distance by this answer. That is roughly the Earth-Sun distance that Cassini, Richer, and Picard figured out, or the Astronomical Unit. What is their estimate of the au?

9.  _________________________

Now, this estimate just blew the astronomical community away. Thanks to Kepler, they knew the relative distances of the various solar system bodies in the newly accepted Copernican heliocentric model, but no-one knew what the actual distance of a single au was. Once the Cassini team (and, independently, Flamsteed) had that number down, and it was so huge, the true scale of the solar system dawned on everyone. The solar system was vaster than anyone had imagined, a realization almost as drastic as the Copernican rearrangement of everything around the sun.

Given the instruments of the day, imprecision in the distance between Paris and Cayenne, trying to get two teams observing at exactly midnight their time, and the slight movement of Mars between the midnight of Paris and the later midnight of Cayenne, there would have to be large uncertainty bars around these estimates, but, even so, they got very close to the modern value for the au, within about 7%, just from sheer logic, imagination, and meticulous record-keeping.

You might optionally try redoing the whole process with contemporary figures:

  • Mars parallax: 8.794143 arcseconds
  • one half Paris-Cayenne chord distance 3,340 km

10. _________________________ (optional extra credit)
The current estimate of the astronomical unit is 149,597,870.691 km.

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This document is maintained by Dr. Rodrigue
First placed on the web: 02/04/12
Last updated: 02/06/18