Q:

Is an interstellar clock orbitally possible?

A Kardashev Type III civilisation has need of a long-scale timepiece, and as such has arranged a series of planets around a black hole such that their orbital periods can be used to tell the time in the same way an analogue clock would be on Earth (don't worry about how they know where 12 o' clock is).
However there is an issue: This clock is meant to last a long time (on the order of tens of billions of years), and the orbits of the planets must remain precisely calibrated to their original orbital periods. Another slight wrinkle is that this race plans to leave the universe (long story. Blame a space wizard) for a little while and so won't be able to actively modify any of the orbits while they're away. A drift of 1 part in 1014 (the same as the TAI) is permissible.
Given that this civilisation is capable of carting around black holes and planets (potentially to an extragalactic location if needed), is this 'orbital clock' possible? If not, what is the maximum accuracy one could hope to achieve (given that there are multiple planets in this clock)?
Please note: The hard science tag really matters here. Equations to prove yes or no are a must.

A:

Your main problem is that the planets will comprise a chaotic system. This is why the stability of the Solar System is so difficult (possibly impossible) to determine. Our best models are valid for perhaps $\sim10^8$ years - at best (see Laskar et al. (2004)). This is the Lyapunov time, over which orbits are definitely chaotic (the 50 million years quoted there is an extreme lower estimate). Determining that requires calculating the system's Lyapunov exponent, which is not easy and which I will not do here. The two are related, though, by the equation
$$|\delta Z(t)|=e^{\lambda t}|\delta Z_0|$$
for Lyapunov exponent $\lambda$, time $t$, and separation $Z(t)$.
If we assume that this system has a similar Lyapunov exponent, then over even $\sim10^9$ years, the system is chaotic, and the clock becomes essentially useless.

Another issue that makes the idea of a central clock - or an extragalactic clock - a bit of a pain in general is time dilation.
Some postulates:

Being a Type III civilization, these beings have control of the entire galaxy and likely inhabit it. They therefore need the clock to work in all parts of it.
The galaxy has a non-uniform density, and therefore a non-uniform potential. Even treating it as a large disk and neglecting individual effects of bodies, there will be gravitational time dilation.
The civilization will survive for a long period of time, and therefore any discrepancies must be tiny; otherwise, the effects will snowball.

I'll take the calculations from John Rennie's answer here. If we assume a central potential of $\Phi(r=0)=6.4\times10^{11}\text{ J kg}^{-1}$, and time dilation of the form
$$\Delta t_0=\Delta t_{\infty}\sqrt{1-\frac{2\Delta\Phi}{c^2}},\quad\Delta t_{\infty}\approx\Delta t_{\text{edge}}$$
where $\Delta t_{\text{edge}}$ is a time interval at the edge of the galaxy, then we find that
$$\Delta t_{\text{edge}}-\Delta t_0\sim7\times10^{-7}\Delta t_{\text{edge}}$$
That causes a discrepancy much greater than one part in $10^{14}$ - not from inaccuracies in the clock, but just because time will tick differently at different points in the galaxy. This clock cannot be used throughout the galaxy - or even in a small portion of it. It would be much better to use different clocks in different regions. And if you put it outside the galaxy, you can't even use it in the majority of galactic locations.
Yes, you could make corrections depending where you are in the galaxy, but the equation given above - and in the linked answer - is only an approximation. It takes some more complicated computations to correctly figure out gravitational time dilation in the galaxy to enough precision, and frankly, there's a point at which it's just not worth it.

