NavList:

Thank you Frank for your feedback .

In my initial reply , I mentioned "We know that averaging just 2 Heights over a 5 minute time period yields a maximum error of 0.1' in the worst case.

Accordingly, with only its quadratic error increase being significant here, averaging just 2 heights over a 10 minute period - which is the case here for Venus - yields a maximum error of 0.4' ."

I was surprised that nobody raised an eye-brow here since this "rule of thumb" does not seem to be widely known.

Its use is fairly infrequent. Rather than a "need to know" item, it is just a "nice to know" one.

Nonetheless, let's get at it !

You will find in the enclosed document the Mathematics enabling to compute how much over a time period the observed heights do depart from an actual linear increase or decrease.

In my initial reply I indicated that :

The "No Running Fix" Averaging values yields a 00:35 position at N34°58.7' / W067°07.3'

The "Full Running Fix" yields a 00:35 position at N34°58.7' / W067°07.5' , as marginally more reliable. By the way, I am using Venus Center of light coordinates accurate to +/- 4".

From the enclosed document, you can easily compute that over such a +/- 10 minute UT interval, the heights actual maximal departure from a straight line is 0.15', which is well under the "worst case maximum error" boundary stated then as 0.4' .

If we then replace our earlier [straight line] mean value applicable to Venus, i.e. 00:35:00 Venus at 21°22.25' by its quadratic correction of 0.15' hereabove, we are to solve the following slightly modified system :

00:35:00 [unchanged] Polaris at 34°27.0' , and

00:35:00 modified Venus at 21°22.25' + 0.15' = 21°22.4' , and :

lo and behold : the resulting Fix becomes N34°58.5'  /  W067°07.5' which is now identical to the earlier full running fix.

Magic, isn't it ?

The earlier quoted "rule of thumb" here-above actually needs to be slightly corrected into "averaging just 2 Heights over a 4 minute time period yields a maximum error of 0.1' in the worst cases".

Last word about this rule of thumb. It is actually derived from 3 independent variables (Latitude, Declination and Local Hour Angle), and its discussion may become rather complicated. I can't retrieve my full study here but this "Rule of Thumb" is a quite solid one.

Hence for the 10 minute period of our exercise, which is 2.5 times wider, the expected maximum error is slightly in excess of 0.6'.

What are such extreme cases ? Extreme cases do occur when both Observer and Body are near the Equator with the Body close from its Local Apparent Noon.

Since in this example, we are quite far from such an "extreme case" configuration, it explains why the actual departure of heights vs. UT's is only 0.15' (vs. 0.6') and this value certainly can be regarded as negligible here since both "Full Running Fix" Venus LOP's fall about 1.7 NM apart.

Now this 4 minute Time Interval is a very interesting one on a practical standpoint.

On a steady deck and with reasonable training you can fit 5 successive sextant observations on one same body into such an interval even if you are alone. These 5 observations enable you to divide errors by 2. It is the best "efficiency / toil" ratio you can think of. You can absolutely use the average value of each series of 5 observations in this 4 minute time interval without degrading your end result.

If you ever thought of "improving" such efficiency - say : divide error by 3 - you would need 9 observations.

You even want to divide you accidental errors by 4 ? Then shoot 16 observations in a row. And now you are to face many issues : the actual heights curve is no longer linear, chances are that you will incorrectly compute your average UT and Height. And last but not least, you have already used a quite significant part of your precious observation time window on just one body ...

Best Regards to all,

Antoine M. Couëtte