Entrainment of air and timekeeping of a mechanical watch at high altitude (problem attributed to A. H. Compton)

Question

S. K. Allison told¹ this Fermi problem:

During the war, Professor A. H. Compton, Enrico Fermi, and I [Allison] were traveling together to visit the Hanford Plutonium Plant in the state of Washington. The hours seemed to drag, crossing the mountains, and Enrico, who always disliked traveling, was restless and bored. After some long silences, Mr. Compton said

"Enrico, when I was in the Andes mountains on my cosmic ray trips, I noticed that at very high altitudes my watch didn't keep good time. I thought about this considerably and finally came to an explanation which satisfied me. Let's hear you discourse on this subject."

Enrico's eyes flashed. A problem! A challenge! Something to work on! Having been in several such situations before, I relaxed and prepared to enjoy the fireworks that would surely follow. He found a scrap of paper and took from his pocket the small slide rule he always carried. During the next five minutes he wrote down the mathematical equations for the entrainment of air in the balance wheel of the watch, the effect on the period of the wheel, and the change in this effect at the low pressures of high altitudes. He came out with a figure which checked accurately with Mr. Compton's memory of the deficiencies of his timepiece in the Andes. Mr. Compton acknowledged the correctness of the calculation, and I shall not forget the expression of wonder on his face.

Three years before, C. M. Cooper gave² his solution to the problem, I'm not sure I fully understood it:

In a vacuum $m_W$ is the mass of the balance wheel, the period $T$ of the watch is proportional to $m_W$; within the air a constant volume of air is attached to the balance wheel and so $T\propto m_W+m_a$ where $m_a$ is the mass of the volume of air.

We assume that the volume of air is equal to the volume of the balance wheel; we assume a 50 mm³ brass balance wheel, so $m_W=0.4$ g.

The density of air at the sea level is 1.2 kg/m³ and at 5000 m (median altitude of the Ande) is 0.7 kg/m³, so 50 mm³ of air has a mass of 60 μg at sea level and 35 μg in the Ande.

Assuming that the watch has a perfect timekeeping at sea level, $T$ in the Ande is $0.400035/0.40006=0.99994\times T$ at the sea level, so speed-up is in the order of $3600-0.99994\times3600=2/10$ second per hour.

Allison did not give the Fermi answer and I am not fully convinced of the above answer.

I had a look at The Physics Teacher, American Journal of Physics, European Journal of Physics for the "equation for the entrainment of air in the balance wheel" but with no luck... what is your solution to the problem?

¹“Enrico Fermi.” Bulletin of the Atomic Scientists 11, no. 1 (1955): 2. doi:10.1080/00963402.1955.11453543.

²Cooper, Chas. M. “Mathematics in Engineering Research.” The Mathematics Teacher 45, no. 5 (1952): 331–39. http://www.jstor.org/stable/27954041.

Answer 1

By the basics, the linear oscillator equation with liquid friction, propprtional to the velocity

$$\ddot \phi = - \omega^2 \phi -2 \ \Gamma \dot \phi$$

has solution

$$\phi(t) = A\ e^{- \Gamma t} \ \sin(\sqrt{\omega^2 -\Gamma^2} \ t)$$

The watch mechanism works by refreshing the amplitude by an angular momentum kick that replenishes the lost momentum per cycle.

So, in the linear approximation, $\Gamma$ will be reduced by the reduction of air pressure with height, increasing the effective angular frequency. But at the same time the amplitude $A$ will increase and by that, the turbulent part of the air stream, created by pumping an air stream via balance wheel and spring, will increase nonlinearly by $A^3$.

It seems to be out of reach of Fermis capabilities, to even roughly compute the combined effect wrt to the effective amplitude frequency change in the 1940ties.

The other fact: Mechanical wrist or pocket chronometers had at best a deviation of 30s per day (quartz 1s). Against what normals should one compare changes during a mountan trip? And why not about the effect in air travel with air sealed chronometers?

And finally, the deviation per day - in the first place - is a function of temperature of the spring.

Many such stories have been told during the coffee rounds after our seminars about the members of the Saints of the 1920ties. A famous anecdote precursor in optics is "when Alexander Humbold climbed the Chimborazo, the air became so thin, that he could read without glasses".

The trick of the theoretical physicists is, that, in a finite interval of time, in such difficult nonlinear problems, nobody is able to critically follow their expositions.