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Re: Exercise Lunar Distance with Mercury
Date: 2009 Sep 22, 23:49 -0400

```Hi Jeremy

Since Mercury is an inferior planet and very near to the Sun, Mercury always
shows a phase when we can see it.

Did you see the phase of Mercury when performing the observation?  What power scope?

When aligning the limb of Mercury to the limb of the Moon, were both curved
limbs in alignment or were caused to estimate the tangency, because the
phases of both objects were out of alignment.

Just curious!

Best Regards

________________________________________
From: navlist@fer3.com [navlist@fer3.com] On Behalf Of
antoine.m.couette@club-internet.fr [antoine.m.couette@club-internet.fr]
Sent: Sunday, September 20, 2009 12:48 PM
To: NavList@fer3.com
Subject: [NavList 9816] Re: Exercise Lunar Distance with Mercury

La Bruy�re, Sep 20, 2009

Dear Jeremy,

Thank you for interesting examples to work ! First time I ever heard of a Moon-Mercury Lunar.

This is a typical case when we can work on averaged values both for Sextant Distances anf for UT values.

Therefore, I have worked all results from the following values :

- Mean Sextant Distance = 41�05'58 (41 Degrees 05 Minutes 58/100 of a Minute,
further down shortened to the following format : 41D05.58M). This value has
been corrected for your Index error of 0.8' (remove, i.e. substract the error
, i.e. remove 0.8' ) or your Index correction of -0.8' (add the correction,
readings is that values(possibly) there is a "inaccurate" result somewhere
for either Observation 2, 3 or 4 since observed values should all increase as
times increase, and
- Mean UT tagged to the value herabove : 16h47m15.0s

I have therefore (re)computed heights since they are required, even to a
rather good accuracy (probably better than what is usually mentioned on
Navlist) if you want to make accurate Distance clearing and take in account
Body Shape distortion due to differential refraction (probably not
significant in this current case since Moon is fairly high in the sky, but
this may become a quite significant effect when observed bodies are (much)
lower).

For all subsequent altitudes computations I have assumed that your height of
eye is 106 ft (rings a bell ???? . this seems a value you often use) and that
both Temperature and Pressure are standard.

NOTE 1 : I am hereby publishing a number of digits which sometimes do not have
any physical meaning (such as heights to 0.001 '), or Celestial Bodies
Coordinates to better to 1 arc second while I have computed then to an
accuracy of 3 arc second only here.

SINCE YOU have NOT "TIME TAGGED" YOUR (GPS?) POSITION, I HAVE MADE TWO ASSUMPTIONS :

CASE ONE : FIRST ASSUMPTION
First assumption (the most probable one I think ... ??? ) : Your indicated
(GPS ?) position is for 17:00 UT, in which case your 16h47m15.0s position is
S3629.7E2624.5 , then at which time, you get  sextant observed altitudes of
14�48'559 (14D48.559M) for Mercury and 55�06'118 (55D06.118M) for the Moon
lower limb.

CASE TWO : SECOND ASSUMPTION
Second assumption (less likely case ??? ) is that your indicated (GPS ?)
position for 16h47m15.0s is exactly the one you are mentioning, i.e.
S3631.1E02620.8, then at which time, you get  sextant observed altitudes of
14�51'328 (14D51.328M) for Mercury and 55�08'565 (55D08.565M) for the Moon
lower limb.

NOTE 2 : Shortly simplified, it can be stated that a Lunar can be seen as a
set of four data : choose any two and the other two result of your choice,
sometimes in a unique way, sometimes not. The four data considered hereafter
are as follows:

(1)     UT of Sextant Distance,
(2)     Sextant Distance,
(3)     Position at UT Sextant distance, and
(4)     Heights observed
I will not extensively cover all cases hereunder, but only the few one interesting for immediate use.

NOTE 3 : All computations results presented hereafter have been corrected for
all known effects of parallax, Earth oblateness, augmented semi-diameter,
refraction and body shape distortion due to differential refraction, horizon
depression, and in case of Planets for phase correction as follows : I am
assuming that you "swing-kiss" the center of gravity of the Planet light over
the Moon Limb.

NOW, HERE ARE MY RESULTS

Since they have not been published in this thread so far, the following data pertain to Mercury :

- Distance to Earth is .907240981 UA, giving an Equatorial parallax close to 9.69 ",
- Visual magnitude is 0.33 ,
- Semi-diameter is 3.707 arcseconds, and
- Corrections for phase angle are -1.605 arcseconds for RA and +0.851
arcseconds for Declination, both reckoned from body center of gravity towards
center of light.

CASE ONE : Your assumed position at 16h47m15s0 is assumed to be S3629.7E02624.5

CASE ONE Subcase ONE

If we assume that Both (1) and (3) just defined immediately above show no
error at all, then the Observed Sextant distance is equal to 41�05'127
(41D05.127M), in which case we can compare it with your sextant observed
value corrected for Index error of 41�05'58 (41D05.58M). You had an average
observation value error of 0.45 arc minute, which is already quite good.
Observed position is S3629.7E02624.5 since (3) is forced into the
computations.

CASE ONE Subcase TWO

If we then assume that both (2) and (3) show no error at all, then you get a
UT error equal to 1m32.8s, which gives an observation UT of 16h48m47,8s and
obviously an observed position of S3629.7E02624.5 since (3) is still forced
into the computations.

CASE ONE Subcase THREE (a)

If we assume that all (2) and (3) show no error at all, then we get a UT error
of 0m53.2 s, a UT Time of 16h48m08s2 and an observed position of
S3632.0E02610.8 .

CASE ONE Subcase THREE (b)

We can notice that the following data also fulfill (2) and (3) if considered
as exact : UT error of 0m54.1s with a 16h48m09.1s UT position at
S1935.0E02805.3

CASE ONE Subcase FOUR

An interesting one here : what if we assume that both (1) and (2) show no
error ? I have not seen anything published for this case.

Well, we can start with a simple case : a grazing Moon occultation with a
pinpoint body (star, planet or asteroid) is fairly simple to visualize,
likewise if and when we consider a pinpoint light source with Moon.
Its becomes a funnier story when the second body has a non zero apparent
diameter : Mister Sun to name it !!! Any ideas here ???

*******

CASE TWO : Your assumed position at 16h47m15s0 is assumed to be S3631.0E02620.8

CASE TWO Subcase ONE

If we assume that Both (1) and (3) just defined immediately above show no
error at all, then the Observed Sextant distance is equal to 41�05'179
(41D05.179M), in which case we can compare it with your sextant observed
value corrected for Index error of 41�05'58 (41D05.58M). You had an average
observation value error of 0.40 arc minute, which is good. Observed position
is S3631.0E02620.8 since (3) is forced into the computations.

CASE TWO Subcase TWO

If we then assume that both (2) and (3) show no error at all, then you get a
UT error equal to 1m22.1s, which gives an observation UT of 16h48m37,1s and
obviously an observed position of S3631.0E02620.8 since (3) is still forced
into the computations.

CASE TWO Subcase THREE (a)

If we assume that all (2) and (3) show no error at all, then we get a UT error
of 0m47.0s, a UT Time of 16h48m02s0 and an observed position of
S3633.0E02608.7 .

CASE TWO Subcase THREE (b)

We can notice that the following data also fulfill (2) and (3) if considered
as exact : UT error of 0m47.9s with a 16h48m02.9s position of S1933.3E02804.0

CASE TWO Subcase FOUR

Already covered in CASE ONE Subcase FOUR hereabove.

*******

NOTE 4 ( Last Note) : Planetary data computations are based on the following theories :
VSOP82 and NGT : These data handed over to me by my late good friend Dr Pierre
Bretagnon of the Bureau des Longitudes in Paris. Dr Pierre Bretagnon "merged"
both VSOP82 and NGT theories into the VSOP87 theory. So I am essentially
getting the same results as VSOP 87. As stated earlier, Moon and Mercury
apparent coordinates have been computed to an accuracy of 3 arcseconds here.
Mercury phase correction on both Right Ascension and Declination are computed
to an accuracy better than 1/100 th of an arcsecond.

For the Moon I am using the ELP 2000-85 theory from M. Jean and Mrs Michelle
Chapront from the CNRS in Paris and who are working in Bureau des Longitudes,
a copy of ELP-2000 85 was handed over to me by Mrs Michelle Chapront.

For Precession I am using Dr Pierre Bretagnon's Precession values to their
full accuracy as derived from NGT. Their current precision better than 1/1000
arcsecond and will degrade to about 1/10 arc second after 1000 years
before/after 2000.0 .

For Nutation I am using here J. Wahr's Theory (Wahr, J. 1981) to its full
accuracy. Its current accuracy is  5/100 arc second and it is degrading to
about 1 arc second after 1000 years before/after 2000.0 . Although not
ourstandingly accurate by to-day standards, this Theory had the great merit
of being the first one used by the IAU which incorporated a elastic Earth
model.

For Stars Apparent computation positions, I am using the FK6 positions when
available, and the Hipparcos Position by default of FK6 published values. It
enables to compute stars apparent positions - including gravitational light
deflection - to 1/1000 arc second (when compared with documents using the J.
Wahr's Theory such as the AA for a few years starting with the 1984 volume).

For the TT-UT values I am using the information given in :

eclipse.gsfc.nasa.gov/SEcat5/deltatpoly.html

which I consider being the best single comprehensive source both for recent
and ancient historical times (Thanks to M. Morrison and al)
In a few words, the accuracies I am using are definitely above the required
standards for CelNav with a conventional Sextant.

*******

Comments on the hereabove (all or any part) are welcome. I may not have the
immediate time to reply since I am in "standby" and might fly again (to the
USA ??) to-night or to-morrow morning.

******

Till then, Thank again Jeremy and

Best Regards to you all

Enjoy !!!!!!!

Antoine M. "Kermit" COUETTE
N4641.753W00054.875
Hopefully, no typos hereabove.

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