NavList:

In order answer the question of the likelihood of getting 0.5nm accuracy from a series of Sun sights around LAN I set up a Monte Carlo simulation of 10,000 trials in Excel using the 14 observation times listed in the earlier post. The true altitude was calculated in the usual way, taking into account the d = -0.9’ per hour change in declination of the Sun, and a normally distributed random error was added to each value. The standard deviation was 0.26’ which is the standard deviation intercepts in the actual observations when the first (outlier) observation is excluded. The maximum was found by least squares fit to a parabola. The results are shown in the attached histogram. The x-axis units are seconds with t = 0 representing LAN. The offset of the peak the distribution arises from the Sun’s changing declination ( https://NavList.net/imgx/v32n1-6.pdf ) and amounts to 9.6 seconds. I had ignored this effect in when calculating longitude previously but since we are approaching the equinox d is getting close to its maximum. Including it chances things a bit.

Error in longitude including all sights: 2.7nm
Error in longitude excluding 1st sight (outlier): 1.3nm

The results of the Monte Carlo can be used to produce a probability distribution for the error in determining the time of maximum altitude and hence the error in longitude. The plots are attached. There was about a 10% chance of determining longitude with an error of 0.5nm or less, even odds of better than 1.75nm and 95% probability of doing better than 3.5nm. These figures will depend on the number and spacing of the sights and the standard deviation of the altitude measurements.

Robin Stuart

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