Calibration of a clock

Question

I was inspired by this interesting question on this forum:

How do I measure an earth year without a clock?

Say you're stranded on an alien planet without any significant tools. How would you recreate the SI system of units with some accuracy?

The meter is fairly easy within less than 1% since most people know their height to within a cm or less than an inch.

How about the second? Given the unreliability of biological clocks, I'd have to measure it. But I can't use a pendulum because I don't know $g$. I can't build a mechanical mass-spring system because I don't know the stiffness of materials (in general). To measure stiffness I'd have to re-create the SI Newton, which is a derived unit from seconds.

1. How could you re-create the SI second? The least technically complicated answer the better (pendulums beat RC circuits which beat atomic clocks and radioactivity measurements).

2. Is the only way to re-create the second is to measure $c$ or build an RC circuit (basically the same things since I can use $\varepsilon_0 =\frac {1}{\mu_0 c^2}$)?

3. If point 2 is correct, how would you measure $c$ or build an RC circuit in the most simple manner?

Myself, I believe that (1) is not possible with mechanical systems, because (2) is true for classical mechanics so I'll have to build an optical interferometer, a radio/receiver to measure c, or an RC circuit. For point (3) given copper wire and iron (which are thousand year old technology) it's possible to build a magnetic amplifier radio and try to measure the time to bounce off an obstacle.

Thoughts?

EDIT: This is different from If time standard clocks and any memories about the time standard are destroyed, can we recover the time standard again?. That question assumes I'm still on Earth, and I'm possibly a member of civilization. Then I can re-create the meter fairly accurately from it's original definition as a fraction of the circumference of the Earth, and for the second I can build a pendulum of period 1s, since g has not changed.

The challenge here is, given a facts that a reasonably educated person would know (their height, the value of c and how it relates to other values. I know some basic astronomical facts but don't know the names and position of all the uncatalogued stars), excellent physics knowledge (i.e. I understand special relativity, lie grouops, ect) and perfect mathematical ability (I can derive any known mathematical result I need) can I recreate the SI system within a reasonable margin of error?

Also, people have asked if they can assume certain things (i.e. can I see Jupiter, does the planet have an earth). Assume away, you'd be answering interesting specific cases and exploring other techniques. However, the more general the answer, the better.

Finally, if there are any doubts read the original question at the very top. I'm ultimately trying to get a satisfying answer for that scenario.

Answer 1

You've stated that you'd recreate an SI length unit $\text{m}$ (meter) from knowledge of your own height. So you've got a reasonably accurate ruler.

Create a small angle pendulum with length $L$.

Use this clock to measure the speed of light (in vacuum). Call this $c_p$ (measured with the planet's pendulum period).

The ratio of $c$ (measured in SI units and remembered by you, despite being marooned) and $c_p$ measured with the planet's pendulum is:

$\Large{\frac{c_p}{c}=\sqrt{\frac{g_p}{g}}}$, with $g$ acceleration of Earth gravity (remembered!) and $g_p$ acceleration of planet's gravity.

Calculate $g_p$ and use it to convert 'planet seconds' to 'Earth seconds'.

Bonus: Use your ruler to create an accurate container of $1 dm^3$ (1 liter): filled with water that's $1\text{ kg}$.

Answer 2

If your hypothetical stranded astronaut is able to use her own head-to-sole height as a length reference, I would expect her to count seconds by muttering "mississippi one, mississippi two, mississippi three" the way she has been doing since playground days.

If your astronaut is a musician she might recall a piece of music for which she knows the metronome marking.

The human brain is an excellent clock, especially with training.

Answer 3

I think we can divide most potential solutions into 3 broad categories depending on what sort of reference is used:

1. Some property inherent to your body or brain: As mentioned in rob's answer, the most obvious is probably to use one of several second counting methods, or a song, drum beat or similar that you have experience performing at a fixed pace. Some people might have a speech or a lecture that they have given many times and know precisely how long time it takes. Although rare, some people have perfect pitch, which might for example be used in some kind of Doppler shift experiment to determine a speed. The accuracy that can be achieved with these methods probably varies a lot depending on experience and aptitude. As a test, I tried counting 100 seconds two times using 1001, 1002, 1003, etc. The first time it took 92 seconds, the second time it took 106 seconds, so I didn't make it within 5% but others may.
2. A fundamental physical constant: If you can measure a physical constant, such as $c$, $G$ or $h$, and you know its value in SI units you can get the length of a second from that. All methods I can think of would require building some fairly complicated things (considering you're literally starting from scratch) and/or some very time consuming astronomical observations (probably takes more than an Earth year). There's also the problem that most people don't know physics and even fewer know the numerical values of physical constants in SI units, except maybe for $c$.
3. A material property: This is similar to the physical constant category but instead a known material property is used. This assumes that you have access to the same or similar materials as on Earth and that their properties are either constant or have some dependence you can account for. For example, if you could produce some pure metals for which you know the Young's modulus and do some good metal working you could produce a spring for which you could calculate the spring constant. Then you could calculate the oscillating frequency for a known weight hanging from it and use that as a time keeper. This might be an easier object to produce than what would be needed for some schemes to determine $c$ but on the other hand, the physics/engineering knowledge needed is probably more rare. Another solution might be to use knowledge of the speed of sound, assuming the atmosphere has a similar composition as on Earth. If the temperature differs a lot from the reference temperature you might compensate for it using your own estimate of the temperature and the fact that the speed of sound is proportional to the square root of the temperature. One could for instance build some form of simple wind instruments whose frequency is proportional to the speed of sound divided by the length of the resonator. If two similar instruments were played simultaneously and the length of the resonator could be manually changed in at least one of them, they could first be tuned to the same frequency and then the length could be changed to produce a distinctive beat effect. The beat frequency $f$ should be given by $f=v\cdot\Delta L/L^2$, where $v$ is the speed of sound, $L$ is the length of the resonator and $\Delta L$ is the change in $L$. To be able to get a small $f$ with a relatively large $\Delta L$ one would want $L$ to be as large as possible.

In the end, it seems like most methods that I can think of are dependent on some pre-learned knowledge or skill that far from everyone possesses and/or requires building some non-trivial stuff. Which method is most feasible probably depends on the person and on what sort of resources are available at the location. The best thing would probably be if several methods could be employed to independently determine the length of a second and hopefully at least some of them would give a similar answer.