A Modest Proposal

* Texted Ray over the weekend about the possibility of designing a scintillator-based Geiger counter substitute that used the iPhone as a user interface. We talked about it on the phone for a little while this morning.

* Ray asked me to write up a paragraph or so of text for the proposal based on the recent results on the GPS timing system. Here's what I've got so far, about a page. Asked Ray for some feedback on what I should cut.

We recently completed a series of tests of the DeLorme GPS2058 module which we are using in our demonstration system for purposes of assigning absolute time stamps to individual shower detection events. We characterized the time stability of the module by comparing its output against a high-precision 10.0000 MHz OCXO (oven-controlled crystal oscillator) having 10 ppb (parts per billion) frequency stability, used as a frequency reference, and plotted the relative Allan deviation (a measure of frequency instability) between these two timing sources (below left). After correcting for the expected quantization error (due to the 10 Msps sampling rate) of ±50 ns, we found that the fine-grained Allan deviation (between adjacent 1-second frequency averaging windows) was 1.67 × 10⁻⁷ (in dimensionless units), which, at the 10 MHz OCXO frequency, translates into ±1.67 Hz of relative frequency instability, or in other words, ±167 ns RMS deviation from the mean in the time of arrival of individual PPS (pulse per second) timing pulses received from the GPS module. This then implies a comparable lower bound to the uncertainty of absolute time measurements based directly on the GPS PPS edges. We could improve the accuracy somewhat by averaging the GPS time readings over longer intervals, but (given the 10 MHz OCXO frequency) not to better than a ±25 ns lower bound that would be set by quantization error even if we used dual-edge triggered flip-flops in our input capture circuit to achieve 20 Msps. But without using differential GPS or military-grade access codes for improved accuracy, the GPS network does not offer users any better time resolution than around that level in any case.

If we assume, for the time being, an uncertainty of ∆t = ±160 ns for the absolute time measurements at a given detector site, then this limits the angular resolution for determinations of the bearing of a large-scale (atmospheric or interstellar) cosmic-ray shower detected at multiple sites. For two detectors separated by a straight baseline of length b < 7,926 km (diameter of Earth), and for an interstellar shower coming from a heading on the celestial sphere at an angle θ away from the baseline axis, the uncertainty in the measurement of θ is given by |∆θ| = c∆t/b (sin θ)⁻¹, in the approximation where
c∆t << b, and where the time reference at the other detector is assumed to be perfectly accurate. The same formula also applies to atmospheric showers for baselines that are less than that of a typical shower diameter at ground level (a few km). In the figure at above right, we plot this quantity for a variety of baseline lengths and source elevations (relative to the baseline axis). So for example, for two sites that are spaced 1,000 km apart, the angle θ for an interstellar shower coming from a direction nearly orthogonal to the baseline axis could be determined to within about 10 seconds of arc once the coincidence between the two detection events (occurring ~1 ms apart, or less) has been identified.

OK, now done a shorter (1 paragraph) description:

Our demonstration system for timestamping cosmic-ray shower events utilizes a DeLorme GPS2058 module for (once per second) absolute time synchronization, together with a Connor-Winfield 10.0000 MHz OCXO (oven-controlled crystal oscillator) with 10 ppb frequency stability for more fine-grained, higher-precision time measurements. We recently completed a series of tests on these components for purposes of verifying the absolute accuracy that we can expect to obtain for comparison of event times between detector sites; our empirical results and analysis will be described in detail in an IEEE journal article (currently in preparation). Our measurements of relative Allan deviation (frequency instability) are shown in the figure at left below. From this data, we inferred that there is an estimated ±167 ns RMS jitter in the time of arrival of individual timing pulses from the GPS unit, which is likely traceable to the low-offset phase noise of the TCXO (temperature-compensated crystal oscillator) which is used for timing purposes inside the GPS module. Given the implied time uncertainty of |∆t| ≈ 160 ns, we can calculate the uncertainty |∆θ| = c|∆t|/b (sin θ)⁻¹ in the inferred angle θ between a shower's heading and the baseline axis between two detector sites that are a distance b apart (below, right). So for example, given two detector sites spaced 1,000 km apart, the off-axis angle θ for an (interstellar) shower coming from a direction nearly orthogonal to the baseline axis could be determined to within about 10 seconds of arc, once the coincidence between the two detection events has been identified.

Also digging up some other graphs to post on project page... OK, all of the recent graphs (and some explanation discussion) are now posted at http://www.neutralino.org/projects/cosmici#TOC-Recent-Results.