serving pancakes to the moon

Ah total solar eclipses.  My first was somewhat East of Lewistown, Montana after driving in the dark on an icy Highway 87.  The Sun had just set on all of the horizons around me when the sonic boom arrived .  I looked up and saw a delta wing, afterburner blazing in the starry sky, streaking Eastward.  Returning to the main event I remember thinking "oh my god - the best seat in the world!" 

 

Fellow Montanan Jeri sent this after last week's eclipse note:

 

Check out yesterday (Sunday's) GF Trib. My brother-in-law is on front page flying an F106 at 1200 mph chasing the 1979 eclipse: Major Don Stevlingson. Can you imagine?

 

 Wow!

 

The Sun's diameter just happens be about 600 times that of the Moon - almost exactly how much further away the Sun is than the Moon.   The Moon's orbit is eccentric enough that sometimes it's apparent size isn't quite enough to cover the Sun and we get an annular eclipse.  If it's close enough - and our orbit around the Sun is also a bit eccentric so just how close varies - totality ranges from a very short time up to almost seven and a half minutes.  

 

You need to see a total solar eclipse at some point in your life!  If you can't make it to this one (sadly that includes me) there are a variety to chose from around the world - about two per year of varying quality.  The next big North American eclipse move in the US around Austin, Texas and arcs up to the Northeast on 8 April, 2024.  You might make hotel arrangements now.

 

Back to the post. So how do you measure the distance to the Moon?  

 

You may have done it in your first trigonometry class - it's a simple parallax calculation.   The problem is making a careful enough measurement.  And being careful you need to understand your measurement well enough to be able to estimate the error of your result.

 

 Ah data...

 

One of the attractions of physics is the core ideas and even experiments are conceptually very simple.  Execution gets a bit involved, but that's part of the fun.  You have to understand your data and what happens to it at every step.   I've been meaning to talk  about measurement and data - the provenance and handling of data and data quality.   So why not an example of how you'd measure the distance to the Moon today.

 

You've probably gone out, flashlight in hand,  on a dark Summer night.  You find your way, see the trees in a new way and perhaps see the eyes of skunk or deer.  Looking up at the stars you find yourself pointing the flashlight skyward.  Sometimes  dust or bugs in the air makes the beam visible.  Up, up and up until it disappears moving at the speed of light.  You point it at a star and wonder how long 'til the light gets there.¹   Now if you could put a mirror out there and had a good stopwatch you could measure the distance.

 

This is at the core of how a physicist (and probably many in other natural sciences) thinks - thought or gedanken experiments.  You make a concept as simple as possible so you can play with it in your mind.  You quickly learn if it's nutty or worth pursuing.  

 

This one's worth the effort.  Thanks to American and Russian space programs there are four special mirrors on the Moon.  Each is an array of corner reflectors that bounce a beam of light back in the direction it came.  The speed of light is about 186,000 miles per second and the Moon is about 240,000 miles away so a round trip is about two and a half seconds. 

 

And if it worked ideally

 

 (I love xkcd)

 

You can do this with laser, telescope and detector.  In fact kit in most University physics department would work or you could buy your own for about the price of a sports sedan.  Unfortunately it's just fun to show you can bounce light off a mirror on the Moon.  The accuracy isn't good enough to answer interesting questions like how fast is the Moon moving away from us (tides are causing it to slowly spiral out) or testing the validity of theories about gravity like General Relativity. 

 

Now you get out the chalk and work out something a bit more realisitic

 

You want to send out a boatload of photons of light.  Light travels about eight inches every nanosecond (billionth of a second), so the pulse should be short and your stop watch should be good.   A garden variety atomic clock is just fine.  And a laser comes to mind as they tend to produce very directional beams of light.  Unfortunately any beam of light diverges thanks to atmospheric turbulence.  The best you can do is a beam that spreads out at an angle of a second of arc - about three ten-thousands of a degree.  To put things in perspective think about a quarter five kilometers (three miles sort of) away or a volleyball at about 130 kilometers.  It doesn't seem like much, but the Moon is more than a bit farther out.²

 

The return trip is worse.   The retroreflectors spread light about eight times more than our atmosphere.  You want a great detector and a very bright source.  The best measurements are done using a big telescope to make the light as parallel as possible going out and to collect as much as possible coming back.  The mirror is 3.5 meters in diameter and the laser makes a short bright pulse that lasts slightly under a tenth of a billionth of a second - about the time it takes light to travel about an inch.

 

You're serving the Moon pancakes of light 3.5 meters across and three centimeters thick with a fantastic laser pointer.³

 

Now some numbers.  Due to beam spreading only one  in 30 million photons in the pancake make it to the mirror array on the Moon and about one in 30 million will make it back to the telescope.   Each pulse puts out enough photons that a handful come back. This is fewer than you'd expect from the simple model so far.  You have to consider absorption and scattering in the atmosphere and the efficiencies of the detectors and mirrors.  Accurate predictions very difficult, but we're more interested in time of flight.  All we really need to know about efficiency is do we have enough make good measurements.  

 

This is an important feature of data.  Required accuracy and error measurements depend on what you're doing.  It's good form to work out the errors for everything in case someone comes along later and wants to use what you did for something you may not have thought about. 

 

Going deeper there are some problems you must recon with if the measurement is going to be accurate to anything better than the length of a football field.  

 

° light travels more slowly in the atmosphere than it does in space.

° the density of the atmosphere drops dramatically with altitude and varies on short time scales

° the real distance to the Moon is constantly changing since the Earth and Moon are in relative motion and since the Earth rotates at about 400 meters per second at the latitude of the observatory.  Add these up and the distance between the observatory and the Moon is changing at hundreds of meters per second - and you're worried about a roughly two and a half second roundtrip.

 

See - there's a reason for starting at a simple conceptual level before diving in.  

 

There are techniques for handling these, but you need to know the errors and propagate them into your final result.  Every step along the way and the changes over time.  There is a provenance and a history associated with data that must be known or it is worthless.   Getting bitten is all too easy - every experimentalist can tell you horror stories - so you have to verify each step.  Sadly not all big data work I've seen is very carefully done and has hidden issues.

 

It is common for repeated experiments to not give the same result.  This can be a good thing as it tells you something about the two different measurements - there may be something new or perhaps a better technique.  You learn a lot from failure.  Life gets ugly when the core measurements have large errors as is common in medicine and the social sciences.  Unfortunately their culture often is less tolerant of failure and learning from failure as the natural sciences.  

 

Ah - the distance.  Measurements of well under a centimeter have been made.   Precision measurements show that the Moon is slowly becoming more distance...  3.8 centimeters a year - about an inch and a half.  Oddly enough the diameter of each retroflector in a mirror array.
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¹ I have the odd habit of telling people what star is roughly the distance it would take light to travel to Earth ..  light leaving there when they were born would be arriving about  now.   I saw a note one reader just turned 31.  Your star this time around is 61 Ursae Majoris  in the Big Dipper.  It's not terribly bright, but you could see it on a dark night.  There are probably stars closer to the date, but they're brown dwarfs that aren't quite visible given the state of infrared telescope technology.  

² Stagnant air may not be very transparent, but it can be very still an offer good "seeing" if you have enough photons.  The solar physics class labs at Caltech used to be done in the JPL parking lot.  Looking at the Sun through a smog filled sky (at least back then) was just fine as long as the sky was still.  Who would have thought that you can do serious astronomy in Pasadena during the day.

³ The laser puts out 2.3 watts - it would easily blind you coming out of the laser,  but spread over 10 square meters would be somewhat safe and on the Moon would be very weak.