Clock Period-Temperature Correlation

The relationship between the average clock period and the average temperature of the CP-4 sensor is shown in Figure 3 for data taken during an interval approximately 75 days long, from 1988 July 20 to 1988 October 5. Corresponding average periods and temperatures were computed over intervals as closely matched as possible. The clock period ranges from 8 s minus 15 µs to 8 s plus 6 µs, while the CP-4 temperature ranges from 13.5° to 20°C. It is clear that the relationship is far from linear, having a sharp maximum near 18°C. The relationship also exhibits a larger scatter than that expected from the error bars assigned on the basis of a 0.5 ms uncertainty for each comparison of the satellite clock with the KSC clock.

Some of the scatter is due to uncertainties in the determination of the temperature. For instance, there may be a systematic error because the CP-4 sensor is not inside the DP and hence does not give the exact temperature of the clock circuit. It is possible that a linear combination of several sensors would better reflect the temperature of the clock circuit than the CP-4 sensor itself, and thereby reduce the scatter seen in Figure 3. To keep this analysis simple, we used only that portion of the data within which the clock period-temperature relationship is approximately linear - the region where the CP-4 temperature is less than 18°C and clock period is less than 8 s. Initially the regression was performed on all six temperatures mentioned in Section 2b (plus a constant temperature to allow for a nonzero intercept), but it was found that BP-3 and CP-3 had coefficients close to zero. They were therefore deleted, and the regression was repeated on the other four temperatures, again including a constant temperature to allow for nonzero intercept.

The coefficients from this regression are shown in Table 1, together with normalized coefficients which sum to unity and therefore represent an approximation to a synthesized temperature of the clock. The rms scatter in the clock period for this regression is 0.886 µs, compared to 1.176 µs for the regression on CP-4 and a constant temperature. The minimum rms scatter, based on just the 0.5 ms precision of the clock comparison, is roughly 0.5 µs. The correlation between the clock period and the combination temperature is shown in Figure 4a. The correlation is tighter than it is for CP-4 alone, not only at low temperatures where the regression was performed, but also at high temperatures as well.

The next part of this investigation is to determine whether the clock period-temperature relationship can be used as a practical tool for recovering the local clock rate, and thence the local clock phase by integration. The relationship shown in Figure 4a is sharply peaked, and cannot be fit with any obvious function, such as a low-degree polynomial. We therefore decided to approximate the relationship numerically by forming normal points at a close enough spacing that intermediate values can then be obtained by linear interpolation. The normal points were formed by averaging the clock period and the combined temperature in 0.5° steps, spaced every 0.25°. This overlapping of bins makes a smoother function than would independent bins spaced every 0.5°. The resulting relationship is shown in Figure 4b.

As a test of the reliability of our conversion of temperature into clock period provided by Figure 4b, we compared the synthesized periods with the measured periods for the same 20-day interval shown in Figures 1 and 2. This interval includes a passage back and forth across the maximum in the relationship. The measured periods are identical to those shown in Figure 1, while the synthesized periods are obtained by transforming the temperature data to clock period using the relationship shown in Figure 4b. This comparison is presented in Figure 5, with the measured clock periods indicated by diagonal crosses and the synthesized periods by a continuous line. For comparison over a single satellite orbit between passes over the KSC ground station (diagonal cross with error bars), there is only one point which deviates by as much as 2 sigma. The measured periods across sets of remote passes of about 16 hours duration (diagonal crosses with negligible error bars) are also close to the average synthesized periods over corresponding intervals. Thus the comparison between measured and synthesized periods is generally pretty good.