# Eppur si muove

We sometimes read in the history of astronomy that it was Bessel that finally proved the Earth moves around the sun with the measurement of the parallax of 61 Cygni, one of the brightest and closest stars to Earth in the Northern Hemisphere.

This is because people early on realized that if the stars are different brightnesses because they are at different distances, and if the Earth really does move around the Sun, then we should see the nearby ones appear to move with respect to the background ones annually as our light of sight towards them changes.  The effect is quite small, and it was challenging for 18th and 19th century astronomers to measure the tiny effect using only their eyes to make measurements (the biggest parallaxes are like a part in a million).

And so astronomers put a lot of effort into this, and indeed eventually Bessel pulled it off. But the clinching observational proof actually came about 100 years earlier, based on those same observations!

Astronomers were expecting to see a “reflex” motion: when the Earth moved in one direction the nearby stars would appear to slightly move in the opposite direction, so when the Earth was at one extreme of its orbit they would appear to be a bit closer to the center of Earth’s orbit (i.e. the Sun) than they should be.  Instead, astronomers kept measuring a much larger than expected motion (around 20 arcseconds instead of less than 1 arcsecond) in the wrong direction: the stars seemed to move towards a point about 90 degrees away from the Sun, towards the ecliptic.

This is what was actually happening, but it was actually pretty confusing at the time.  One way they were measuring positions was by using the zenith as an absolute direction (you can use a plumb or a liquid to determine which direction is straight up) and a telescope to see how close stars got to the zenith as they passed overhead.  So they could only measure one component of the star’s motion, so all they knew is that it was large and had the wrong phase (90 degrees from what was expected).

What was actually going on is that the stars were suffering from aberration. Imagine you have a trash can an you want to collect as much rain as possible.  If there is no wind, you should just keep the can vertical.  But, if you are on the bed of a pickup truck moving at 10 mph, then some of the rain that would have fallen into the can will hit the side instead.  To maximize rain collection, you have to tip the bucket towards the front of the truck (i.e. in the direction of the truck’s motion) to get the rain to go straight down into the can.

Wikicommons illustration of the aberration of starlight. Original here. By Brews ohare, CC BY-SA 3.0.

Similarly with telescopes: to get starlight to go straight down the optical axis of the telescope, you actually have to “lead” the motion of the Earth slightly by pointing in the direction of Earth’s motion.  This effect is equal in radians to the speed of the Earth’s orbit divided by the speed of light, or about 20″.  This is what was being observed, and this is what Bradley finally figured out in 1727.

So while hunting for the proof of Earth’s motion, astronomers actually discovered a much easier to find proof of Earth’s motion!  But it took decades to understand it.

Today, it’s tricky to repeat these measurements with modern equipment. Finding the zenith to a precision of 1″ is not something most observatories are set up to do; we almost never use high precision, absolute positions with respect the ground in astronomy any more (our instruments tend to change their pointing with temperature, humidity, and other factors, so we calibrate them on actual star positions every now and again to keep our pointing models accurate and precise).

But interestingly, there is a way most college observatories and many amateur ones can find an absolute position: star trails!  By turning off tracking and letting the stars trail, one can identify the center of the trails as the true Celestial Pole.  As the Earth goes around the Sun, one can measure star positions with respect to that and detect the aberration, thus proving the Earth orbits the Sun.

Of course, there are lots of other ways to do this: you could build an R~10,000 spectrograph and feed it light from stars on the ecliptic and measure the Doppler shift caused by Earth’s motion, or you could measure the parallax directly with differential astrometry, like Bessel did. But this is a novel solution that actually doesn’t require special equipment of good seeing, just an ordinary camera.

I was going to try this someday, and even wrote up a whole blog post about it, but finally decided that if I haven’t started by now I probably never will, so I’ve “given the idea away” in the form of a Research Note to the AAS (my sixth!).  One reason I never started is that although the equipment needed isn’t special, there are lots of complications.  One is that the Poles move on their own (due to Earth’s axial precession) so you have to remove that effect first.

As part of my plan to learn Python and Astropy I tried to make a figure showing how the apparent position of the true Celestial North Pole moves with respect to the background stars, showing the precession and the aberration.  It was surprisingly tricky!  Plotting things near coordinate singularities is not something most plotting software does well, so in the end I cheated and just plotted things as a plane chart in ecliptic coordinates and drew the Celestial coordinates in by hand.  I think it came out really well:

Figuring out how to calculate the motion of the Celestial Pole (also affected by nutation) was tricky; it turns out Astropy does not expose those functions to the user so I had to cheat.  In the end, I did it two ways that gave the same answer: I asked for the ICRS (astronomical) coordinates of the zenith at the North Pole of Earth.  Astropy converts from geocentric coordinates to barycentric coordinates by correcting for the orientation of the Earth (the axial motion) and the aberration, so this yields the green curve above.  Almost equivalently, one can ask for the Celestial Pole in CIRS coordinates (the intermediate coordinate system between the Earth and Celestrial Sphere) at many times and then ask Astropy to convert these positions to the ICRS frame.   The latter is faster.