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Subject: Re: Table A4 + elevation?
From: George Huxtable (george@XXX.XXX)
Date: Fri May 02 2003 - 11:57:03 EDT
Doug Royer is having a lot of success with his altitudes. Certainly much
better than is needed for the purpose of correcting a lunar!
Now I think he has realised the high accuracy obtainable from a stable
platform on-land, using a reflector which avoids all the uncertainties
contributed by the horizon. And he is keen to eliminate any remaining
sources of error.
This is where it starts to get hard, trying to whittle down fractions of an
arc-minute.
We may be able to contribute better if he provides us with his raw
observations, with precise details of the corrections that he has applied.
Combined Errors.
Let's tick off some of the possible sources of error.
First, errors in the predicted position. Read the paragraph in the Nautical
Almanac about accuracy on page 261. The almanac dec and GHA are given to
the nearest 0.1 arcminutes (a bit more error than that, for Sun and Moon).
Then the interpolation tables (into which you can enter time only to the
nearest second) give an increment and a v-correction or d-correction to add
on, and each of these is given only to the nearest 0.1 arcmin. Then, if
it's a star, SHA must be added, and that's only given to the nearest 0.1.
So there are all these possible small errors, most limited to ±.05 arcmin,
and by chance, some working one way, some the other, they will tend to
cancel to some extent. But also, by chance, occasionally they will happen
to combine all in the same direction, to produce quite a significant error.
Sometimes it will move the dec (or the RA) up, sometimes down.
If the predictions of the celestial bodies are provided by computer, rather
than from tables, then if the job is done well those errors can become
negligible.
Because the Earth's rotation (on which the tables are based) isn't entirely
constant, or predictable, GMT diverges slowly from atomic time,and this is
made up by occasional leap-second jumps to bring broadcast time close to
GMT. But they never quite correspond, and the almanac on page 254 states
that they can diverge by 0.9 sec. So unless you know what that divergence
is at the time, your watch may differ from almanac-GMT by up to 0.9 sec, no
matter how accurate a timekeeper it is. This affects GHAs, but not decs.
Next, there are errors in the data reduction procedure. If a tabular method
is used, you can only enter data to the nearest 0.1 min, and the resulting
calculated-altitude is given only to the nearest 0.1, so again another
possible error of ±.05. If reduction is done with a computer or calculator,
such errors can be almost eliminated.
Then there are errors in the observation, which are under the control of
the observer, to some extent, and he has to estimate what they might be.
Allowing for index error is important, and the final tweak of the knob
should always be in the same direction for index checks as for each
observation. A good sextant might have its calibration known to 0.2 min.
Then there are the corrections that have to be made to the observed
altitude, each one being to the nearest 0.1 arcmin, so ±.05 arcmin. These
are index error, combined refraction/parallax/semidiam, and perhaps an
extra correction from table A4 for non-standard temperature and pressure.
Now we have to take the difference between the calculated altitude and the
(corrected) observed altitude, to get the error in the position line. The
various errors we have considered can sometimes push one of these up and
pull the other down, in which case their combined effect is to increase the
overall error in the position line further still. At other times, quite by
chance, they will instead tend to cancel.
I think you can see that with all these accumulated small errors, with the
best will in the world, it 's quite easy for an overall error of 0.4 arcmin
to result in a position line.
Nautical Almanac table A4.
Doug says he has used this table to correct for non-standard refractions,
but I am not sure how he has done so, at his height which (I think I
remember) he quoted as 2100 ft.
Vaguely remembering (or maybe misremembering?) the density (relative to
water) of air as 1/830 and of Mercury as 13.6, it seems to me that at
Doug's height of 2100 ft. the atmospheric pressure will be reduced by 7.5%
below what it is at ground level. (Somebody please check!) If that's
correct, then the local pressure will reduce to 935 millibars, compared
with its "standard" sea-level value of 1010. That takes it well outside the
realms of the diagram that goes with table A4, which goes no further down
than 970.
A mercury barometer on Doug's wall would probably give a trustworthy
measure of local pressure, but an aneroid usually would not. These are
usually adjusted to give a pressure corresponding to that at sea-level (or
roughly so). Indeed, that has to be done, or else, when in place up a
mountain, the pointer would end up way off scale.
I'm a bit surprised, then, that Doug quotes a local pressure of 970 mb,
unless there was a rather intense local anticyclone at that time. Not that
it will make much difference to his overall results, except for bodies at
very low altitudes, with a lot of refraction.
George.
>2nd correction: wrote down the wrong gmt in the letter but used the correct
>gmt in the calculations.040416 thru 041822.
>
>-----Original Message-----
>From: Royer, Doug [mailto:doug.royer@XXX.XXX]
>Sent: Thursday, May 01, 2003 17:13
>To: NAVIGATION-L@XXX.XXX
>Subject: Re: Table A4 + elevation?
>
>
>Correction: 051113 gmt 05-01 Polaris Hs = 32*-23.3'
>
>-----Original Message-----
>From: Royer, Doug [mailto:doug.royer@XXX.XXX]
>Sent: Thursday, May 01, 2003 12:59
>To: NAVIGATION-L@XXX.XXX
>Subject: Table A4 + elevation?
>
>
>On 04-26 I completed 6 sets of reflected Sun altitudes during the day.I'm at
>2100 ft. elevation.Temps. were 60-68*F while doing this.Sextant is 0.1' off
>arc.After all corrections were made the standard error was about 0.1' of arc
>for each set.I used Mr. Hebard's advice about elevation and Table A4.A4
>corrections as he stated;between + 0.1- 0.3 for the temp. and elevation.Each
>set of sightings had 3 shots and times to average against a calculated Hc
>slope from Ho-229. All's well.Outstanding final position.Checked against the
>software I was only off the above error. I didn't save the data.
>04-30 I tried 3 sets of reflected star altitudes.Temp. was a steady 43*F
>throughout.All averaged as above.After all corrections were made,the LOPs
>drawn and checked against the software, the error turned out to be + 0.8' of
>arc for each group.This was a constant for all this nights sights.
>Here is the data from 04-30.Ic = + 0.1'. A4 cor. = 0.0' (column H. 970 mb
>and 43*F). A2 cor. = -1.4' to -1.5'. zt = 2100 hrs. 04-30
>050416 gmt 05-01 Spica Hs = 32*43.2'
>051113 gmt 05-01 Polaris Hs = 32*42.7'
>051822 gmt 05-01 Procyon Hs = 34*42.7'
>The resulting error put the final LOP pos. less than 1/4 mi. from where I
>was.All is real well!! However,usually I have errors around 0.1' of arc when
>averageing.That would have placed the LOP pos. about a mile away.I can live
>with either of these final positions but was wondering why this error
>outside the usual error.
>The reflections in the horizon and scope were clear and bright so I feal the
>error isn' t in this area.Each seperate time was timed to the closest
>secound.I don't think it was a time error.
>I'm wondering about the temp. and the altitude as the cause.All the errors
>are the same for each averaged sight.Do any of you guys have an idea on
>this?
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contact George Huxtable by email at george@XXX.XXX by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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