The Geography of Mars

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

• 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

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

   

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

   

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:

   

   (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.

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).

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.  _________________________

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.  _________________________

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.  _________________________

5.  _________________________

6.  _________________________

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.  __________________________

8.  _________________________

9.  _________________________

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)