Can you use pulsar observations to determine absolute time? How long can you go without anything else?

Question

In this old answer of Steve Allen's, he quotes this nice passage

Imagine for a moment what would happen if, just as a practical joke, someone found a way to stop all atomic clocks, just for a short time. This would cause such a tremendous disturbance in world affairs that the loss of TAI would be a totally insignificant matter! Furthermore, when it came to setting it up again, the phase of TAI could be retrieved to within a few tenths of a microsecond by observations of rapidly rotating pulsars...

-- Claude Audoin & Bernard Guinot, p. 252, sec. 7.3.1 of "The Measurement of Time: Time, Frequency and the Atomic Clock", Cambridge University Press, 2001

I'd like to explore this scenario in a bit more detail. Specifically, suppose that our practical joker (let's call him Richard for definiteness's sake) managed to stop all running time standards for a well-defined time $\tau$ and then set them running again. Richard then challenges us to determine $\tau$. To keep things simple, Richard has agreed to provide, if we ask him nicely, an upper bound $T$ on $|\tau|$ - but, again, he challenges us to accept the largest, loosest bound that we can.

As Audoin and Guinot point out, you can use pulsar observations to constrain $\tau$. Suppose you can observe a pulsar that oscillates regularly at a period $T_1$ of 10s. Then, if you know that $|\tau|<T_1/2$, you can match the phase of the observed oscillations with the historical record, you can determine $\tau$ to the same precision to which you know $T_1$.

If $\tau$ is not smaller than $T_1$, of course, this won't work, as you won't know how many periods will have elapsed. However, if you have more than one pulsar, you can significantly extend the range of $\tau$ that you'd be able to pin down. Indeed, if pulsar 2 has a period $T_2$ of 11s, then it goes in and out of phase with pulsar 1 over a period of 110s, so we can recover $\tau$ if we're guaranteed that $|\tau|<$55s. Similarly, if you add a third pulsar with period 13s, the range goes up to (13x11x10)/2 s=715s - much longer than the individual periods of the three pulsars.

In real life, of course, we know a good many pulsars, so you could do quite a bit with their signals to extend the range of $\tau$s that we can recover from. (On the other hand, they do oscillate very rapidly, so even if the skip is thousands of periods long you're only on the few-second regime. But then their periods can be known to high accuracy, which lengthens the period of collective oscillations between two or more pulsars - i.e. it lengthens their effective LCM once you factor in uncertainties in the periods.) Also, as Chris White points out, pulsars can have glitches, which poses a new problem of its own. (But then, you'd need to have glitches in a substantial fraction of known pulsars to really compromise your ability to recover.) And presumably there's more things I haven't considered (what are they?).

So, my question is: using currently known data on observed pulsars and their periods, what's the longest $\boldsymbol\tau$ from which we could recover? Is it long enough that we could use longer-period sources like the period decay of neutron binaries to get further fixes on $\tau$? With what sort of accuracy could we recover $\tau$?

If this has already been explored in the literature then I'm happy to take a reference, but otherwise I'm looking for detailed and in-depth treatments that teach me lots of physics. I'll chip in some rep carrots to spice things up in a few days.

Also, to be clear - answers need not exclusively use pulsars. If there is some other method that can constrain $\tau$, using either other types of astronomical sources or using earthbound phenomena and experiments, then I'll be happy to learn about those!

Answer 1

One piece of physics that you've missed is the most pulsars spin down due to the emission of magnetic dipole radiation. For instance, the crab pulsar has a period of 33.5028 (plus a few more sig figs) milliseconds, but slows down by 38 nanoseconds per day. Furthermore, the size of several more increasing order derivatives is known accurately.

So in principle, just measuring the period of the crab pulsar gives you an estimate of $\tau$. Once I locate the uncertainty in dp/dt I'll report on what $\tau$ that would limit you to.

There are of course glitches in the timing of some pulsars (including the Crab), which would lead to errors if the pulsar was not being continuously monitored. So you do need to observe multiple pulsars. However, this is done right now. There are currently "pulsar clocks" in operation that keep time with the best atomic clocks. The International Pulsar Timing Array uses monitoring observations of about 30 pulsars to look for timing irregularities caused by gravitational waves. So I think that even if you stopped all pulsar observations for a time $\tau$, they are predictable enough (on human timescales) to recover what the time is to better accuracy than an atomic clock; indeed the claim has been made that these pulsar arrays reveal problems in the time frame set by atomic clocks and that only the pulsar array timing should be used for GW detection.

EDIT: The Crab pulsar ephemeris is updated monthly by Jodrell Bank. e.g. for February 2015 the pulse frequency is $\nu_0=29.6684986853 (3)$ s$^{-1}$ (the number in brackets is the uncertainty in the last sig. fig. The rate of change of frequency is $\dot{\nu}=-369636.52 (0.35) \times 10^{-15} $ s$^{-2}$.

So let's say the atomic clocks stopped at $t=0$ for this ephemeris and stay off for at time $\tau$. The frequency of the crab pulsar will be $\nu = \nu_0 +\tau \dot{\nu}$ and the phase is $$ \phi = 2\pi \int_{0}^{\tau} (\nu_0 + \dot{\nu} t)\ dt $$

If $\tau < (\Delta \nu_0 )/(\Delta \dot{\nu})$ ($\simeq$ 10 days for the Crab) then the uncertainty in the first term on the RHS dominates term dominates $$ \frac{\Delta \tau}{\tau} \simeq \frac{\Delta \nu_0}{\nu_0},$$ which is $\Delta \tau \simeq 10^{-12}\tau$ for the Crab pulsar. So, accurate to a microsecond over 10 days (assuming that the phase can be ascertained accurately). The problem is, as you point out in your question, if you lose the monitoring you don't know how many cycles have occurred - so the time is known to a microsecond, but is ambiguous by multiples of the rotation period of 33 milli-seconds!

Just measuring the pulsar frequency might be useful in this regard and I'm currently struggling to work that out...