Detecting mercury in PPM concentrations

One problem with me and my brain trust is that we are all a bit old. When it comes to trying to detect low concentrations of mercury (i.e. contact with a bowl from two thousand years ago,) the best we can think of is gas chromatography. This is about one step up from swabbing with reagents and hoping to see a color change.

A little back and forth with an actual archeologist reveals that the current art is portable x-ray fluorescence.  The instrument is the size of a hand-drill, and looks a bit like a Star Trek phasor. Some more back and forth with an XRF/pottery expert confirms this is the right tool. Now I just need to find one in New York.

P.S. These blog posts are lagging realtime by a couple of weeks. I'm trying to keep the chronology roughly straight rather than posting up-to-the minute results.

Molcajetes

Those are molcajetes: traditional mortars (the actual bowls) and pestles (the clubs.) They are typically made of basalt and used to grind chilies, etc, to make salsa.

Unfortunately, although "molcajete" means "grinding bowl" in the Nahuatl language, it also seems to be the term of choice for all legged shallow pottery bowls from Mesoamerica, even if the bowl is not a grinding bowl. This annoys my wife: over the second course at Le Bernardin, she mentions that people need to get an ethnoarcheologist to show the bowl around Oaxaca villages, asking people what is really called.

I'm pretty sure the pottery bowl I have was never designed for grinding food. In fact, I'm pretty sure almost none of the pottery molcajetes were designed for grinding. Naturally, I am pointed to a counterexample:

But most are not grinding bowls. Here's candidate number 3:

It's sitting in the Walters Art Museum in Baltimore.

Oh, and those bulbous legs are called "mammiform." Which just means breast-shaped. That concludes today's lesson.

That bowl in the Metropolitan Museum of Art?

I almost forgot. The day after I got back from Belize, I took a taxi up to the Met, paid my $25 suggested donation, and snapped a few pictures of the bowl in question. Apologies for the quality of the image - all I had on me was my ancient iPhone.

Anyway, the inside of the bowl is a nice, smooth curved surface. Almost too nice to get a real sense of depth and the overall curvature.

Torsion and Rotation

I haven't done any physics in a while, but a bowl suspended from a twisted length of thread shouldn't be too hard.

A twisted thread provides the force that spins the bowl. It should obey Hooke's law: . This just says the torque (twisting force) is opposite to the direction of the twist, and equal to the amount of twist (theta) scaled by a constant (k) that is a property of the thread. Interestingly, the force does not depend on the force the bowl is exerting on the thread (i.e. the weight of the bowl,) nor does it depend on the length of the thread.

Given the torque, we can derive the angular acceleration using Newton's laws:   This says the acceleration equals the torque divided by the moment of inertia of the bowl. I estimated the moment of inertia as 2 kg at 10 cm from the center of the bowl, or .02 kg-m^2. Of course, it's really not a constant, because it will increase as the mercury rotates and moves outwards from the center.

Integrating the acceleration over time gives the angular velocity. The plot above gives a sense of how the velocity of the bowl changes over time. Note that the shape of the curves should be reflective of reality, but the actual values (e.g. revolutions per minutes) should not be trusted at all: they are based on a complete guess for the term k / I. Unluckily, I can't find values for k for types of twine online. Luckily, if we actually put a bowl on a string, it should be easy to experimentally derive this term for the system.

Anyway, the system does spend a good portion of the time at a relatively constant velocity (the flatish peak in the velocity plots.) That's good, but I have an idea for how to get a much flatter, longer  peak in the rotational velocity...

I buy a bowl

The Guardian article on Teotihuacan creates a a lot of low level noise on the internet.  An article on Slashdot generates over a hundred comments and probably hundreds of thousands of page views.

The statistician in me worries that even an outre hypothesis (e.g 1 in 10000) exposed to 100,000 people may result in someone else exploring the same idea and buying up any candidate precursor bowls that can be found on-line. Because I am eventually going to want to test candidate bowls for mercury residue, I don't want the risk of them all vanishing into unknown private hands.

I decide to buy one bowl online. The bowl pictured above is the one I choose. My reasoning is:

1. It's got a good functional feel, no decoration, bulbous legs and slots like the Met bowl.
2. It's a bit of a mess, definitely not a museum quality piece because the legs don't match. This reduces the chance it is an outright fake.
3. It's been it the USA for decades, so the ethical issues with buying artifacts are reduced. I figure I can give it back to the country of origin once I'm done (if the country can even be identified) for further ethical transgression minimization.
4. It's not too expensive.

This plan does not sit well with my wife. She's ok with, say, private ownership of a Crimea war Minie ball, but not with a nondescript classic Greek potshard. I'm in her immoral camp for pursuing this.

Here's another view of the bowl:

Note how the nubs on the other two legs are horizontal rather than vertical. And they have no slots. And the foot pads are substantially larger.

A bowl suspended by forty feet of string?

Why not four feet? Or four hundred feet of string?

The longer the string, the better: a longer string can take more winding up, and so impart more spin to the bowl for a longer period of time. However, to prevent the wind from blowing the apparatus around, it needs to be inside a building or in a shaft in the earth, or similar.

Here's Scientific American on the shaft at Teotihuacan (it's near the center of this post's image, off a little towards the lower right.)

The tunnel itself was discovered when a heavy rainstorm exposed a shaft that led to a spot about halfway down its length. The shaft’s purpose remains a mystery but scientists believe the tunnel had a ceremonial purpose, and it is possible that the shaft was used for astronomical purposes.

That shaft was 14 meters (about 40 feet) deep. A number of other astronomical holes or shafts of about the same depth seem to have existed.  El Caracol looks to be about that height. I haven't found any location with a much deeper shaft, so 40 feet seems the best guess.

Assuming the user of the hypothetical telescope will be situated near the focal length of the telescope, then the minimum focal length is .2 meters (with the viewer near the bowl,) while the maximum focal length would be approximately the length of the string (with the viewer near the suspension point of the apparatus.)

In the long focal length case, the bowl would rotate at around sqrt(447/14) = 5.6 revolutions per minute.

Potttery and Paraboloids

Here's a bowl from Cenote. The picture comes from Arlen Chase's PhD thesis. I've aligned the image, added a grid, and plotted a parabola.

The fit is good out to a radius of 9 centimeters. That doesn't prove much, because pretty much any smoothly curved bowl will fit some parabola for some internal radius.

Working on the assumption of a 9cm radius of pool of mercury, with a y height of 1cm at that radius, the equation of the parabola would be approximately:

y = x*x / 81

a = 1/81

a = 1/(4f)

4f = 81

f is approximately 20 centimeters.

Given focal length of .2 meters, we can derive the required rotational velocity:

f * rpm * rpm = 447

rpm = sqrt(447/.2) = sqrt(2235) = 47

So, the bowl would be rotating at around 47 revolutions per minute.

These numbers are problematic for my theory. The focal length is too short assuming the goal is to reflect the image via a small mirror above the center of the bowl, then into some type of eyepiece that must be 9cm from the center axis (otherwise the viewer's head would obstruct the incoming light.) Secondly, 47 rpm is really fast: even with a forty foot string suspending the bowl, that would be 5 twists per foot of twine to get even five minutes of rotation. At 47 rpm, a forty foot string would be vibrating quite intensely, probably enough to ruin the optical surface of the mercury.

Of course, lower rotational speeds would be reasonable. The mercury would still form a parabola, it would just be deeper in the center of the bowl, and thus a less efficient use of the metal. The more scarce the mercury was, the more we would expect the rotational speed to conform to the parabola of the bowl.

I ran though the math for other similar bowls in the Cenote paper. They all produced numbers in the 40-70 rpm range.