How exactly was Giovanni Cassini able to measure the distance to Mars?

Question

Recently learned that Cassini was able to calculate the distance to Mars quite well using parallax in 1672. I was surprised, since even at opposition of Mars, the parallax (with respect to the Earth's Center) is about 30 sec arc. While I know that a few decades earlier, accuracy of 2 min arc was considered the best in the world. (Later edit: During these "few decades", the telescope emerged and enhanced dramatically the resolution.)

In my research I did find this answer and the refereed document. I am not repeating the question, my question is exactly about the technicalities of the operation. Cassini instead of using Earth's center and several measurements across the day, used two locations at the surface of the Earth 12,000 km apart. That indeed was helpful as it would make the parallax about 45 arc sec (in the referred doc the figure is 20 sec arc - not sure why.). I would say that this would have required a setting (equipment, etc.) that would allow an accuracy of about 10 arc sec or even less to make this parallax calculation legit. So my first question is (*):

What equipment did Cassini use and what was the accuracy?

Now, there is another problem that is addressed in the document, which is basically making the observation at the same time. From my calculation Mars can move (with respect to fixed stars) maybe about 2.5 arc sec in one minute of time (maybe I'm wrong though, please correct me if I am). Hence I would like to the observation at the different locations to be performed within in 3 minutes span at max. I don't think that such accuracy should be too hard to achieve even in 1671 using conventional methods like of specific star at specific azimuth; hence I was surprised again to read in the document some interesting method of time-synchronization:

In addition, the time at which Mars is measured must be precisely synched between Paris and Guiana. This is because the Earth is spinning and the angle is changing throughout the night. While it may not look with your naked eye that the planets are moving, with a strong telescope this motion becomes increasingly rapid. Cassini and Richter used the orbits of Jupiter's moons to sync the clocks in Paris and Guiana.

This leads to my second question:

How can we achieve time synchronization (with resolution of couple of minutes) using Jupiter's moons?

(*) Later edit: thanks to answer I now realized the the figure of 12,000 km I've adapted from the document is not correct. the value is about half. This makes 20 arcsec closer to the truth - and makes it even harder in terms of required resolution. It also raises another question of why this operation was necessary in the first place - as the relative advantage of not using the Earth's center parallax (i.e., measure in the same location but several hours apart) is not that big; maybe the willingness to avoid any calculated (rather than observed) figure. (though in that time the pace of Mars was settled quite well.

Answer 1

This is my first post on Stack Exchange so bear with me, I'm new to this.

I've been looking into the Cassini / Richter measurement for quite some time, and I think I have something to add here.

Using Stellarium, I can see that Mars would have been at opposition in about September of 1672. From Paris, the declination measured -8.48.37, and from South America, the declination measured -8.48.22 (RA was basically the same). This means the actual figure was about 15 arc seconds of separation. This was a relatively close opposition of the two planets.

The distance between Paris and French Guiana is 7200km over land. Divide this by 111km per degree to get an Earth central angle of 65 degrees. Then, use the radius of the earth to solve the triangle for the baseline between the two locations. You should get about 6800km.

Cassini and Richter used "air telescopes" to make the measurement. Basically, it's a lens in a bucket (with the bottom cut out), on a pole, with a 210 rope extending to the focal point. At the focal point, they either placed a secondary lens (eyepiece) or projected the image on paper (I've been trying to replicate this unsuccessfully with a 12 inch diameter lens, so I'm guessing they used an eyepiece).

The image scale at this focal length is about 3.2 arcseconds per mm. So Mars would have appeared HUGE in the FOV, or about 8mm wide if they successfully projected an image onto paper.

Cassini and Richter each plotted Mars on a star map as accurately as they could. Then when Richter arrived back in Paris, they placed the two Mars positions on the same map. Then, it's simply a matter of taking a ruler and measuring the distance between the two positions. Remember, the scale of the map was probably 3.2arcseconds/mm. The two positions of Mars would have been about 4.7mm apart on the map, but with the uncertainty in the measurement, they probably measured something closer to 6mm. Multiply this by the image scale of 3.2arcseconds/mm, and there you have your 20 arc seconds.

To get the distance to mars, you first convert this to radians: 20 * (1/3600) * (Pi/180) = 0.000097 radians. Because this is a VERY small angle, you can use the small angle approximation, in which case distance = baseline over the angle:

Therefore distance = 6800 / 0.000097 = 70 million KM.

Using Kepler's third law, p^2=a^3 (basically Time^2 = Distance^3 ), we know that Mars's orbit it 1.52 times the radius of Earth's orbit. You can use the duration of each planet's year to get the radius of each planets orbit.

To answer your questions about time synchronization, I'm guessing they made the measurement a few times, with a several time syncs such as "Start your clock at the ellipse of IO". Star your second observation at the occultation of Ganymede" or some such scenario, which would have been planned out before Richter's trip across the ocean.

Sources/Links: https://www.mccarthyobservatory.org/pdfs/pm020102.pdf http://tonic.physics.sunysb.edu/~dteaney/F12_mystery/lectures/l6notes.pdf

I used Stellarium to determine the approximate date of opposition (a simple way to approximate opposition is to scroll through time until the Sun and Mars are sitting opposite each other on the horizon). I couldn't find any exact dates for this observation on the internet, a trip to my local (or university) library might be in order...