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Re: Lunars - Oblateness Correction
From: Andr�s Ruiz
Date: 2008 Jul 15, 08:14 +0200

The Oblateness is a negative correction:

(ref: Corrections for sextant altitude, at my web page)
For the Moon, the Oblateness of the Earth will be taken into account:
OB = 0.0032*(SIN(2B)*COS(z)*SIN(H)-SQ(SIN(B))*COS(H)) [�]
Where:
*   B: latitude of the observer
*   z: azimuth of the Moon
Approximate values are sufficient for the calculation.
At mid-latitudes and for altitudes of the Moon below 60�, a simple approximation is made:
OB = - 0.0017 * cos H

The correction for parallax is:
PA = HP * COS( H ) + OB

Flattening: f=1-b/a
For WGS84 f = 1.0/298.257223563
HP*f = 0.0032 aprox 1/300

In Lunars this equation is approximate to avoid the use of the azimuth, removing the 3rd order terms by:
PA = HP * COS( H ) * (1-(sin(B))^2/300)

Am I wrong? This is the formula that I use in lunar calculations.
Or is PA = HP * COS( H ) * (1+(sin(B))^2/300)

Andr�s

-----Mensaje original-----
De: NavList@fer3.com [mailto:NavList@fer3.com] En nombre de frankreed@HistoricalAtlas.net
Enviado el: domingo, 13 de julio de 2008 3:02
Para: NavList@fer3.com
Asunto: [NavList 5817] Lunars - Oblateness Correction

Since this has come up in a several recent messages, I thought I would spell
out the way it's handled in my online lunars calculator.

The approach is straight out of Chauvenet. What we call "oblateness" is the
slight flattening of the Earth at the poles relative to the equator
converting it into an "oblate spheroid" rather than a perfect sphere. The
degree of flattening is approximately one part in 300 (1/297 is closer but
not significantly so for these purposes). Chauvenet usually refers to this
property as the "compression" of the Earth's spherical shape.

There are two pieces to the correction. First, you correct the HP taken from
the almanac (this is Chauvenet's "Table XIII" correction):
HP=HP0*(1+(sin(Lat))^2/300).
This is a small correction. On average, the error from ignoring it is
roughly 0.05 minutes of arc in the clearing process which corresponds to an
error in longitude of about 1.5' or about 1 nautical mile on average. The
correction is larger at higher latitudes, but of course this has a smaller
impact on the position fix since the longitude lines converge.

Second, you correct the lunar distance (after clearing it, but it doesn't
really matter whether you do it at the beginning instead) with a small
increment:
inc_LD=sin(Lat)*[(HP/150)*(sin(Dec2)/sin(LD)-sin(Dec1)/tan(LD))]
where Dec1 is the declination of the Moon and Dec2 that of the Sun or other
body. The error in longitude, converted to nautical miles, that would result
from ignoring this second small correction would be, on average, about equal
to
(1.35 n.m.)*sin(2*Lat).
By the way, Chauvenet has a factor of "A" in his equation and tells the
reader to look it up in a small table giving log(A) as a function of
latitude. If you're trying to figure it out from that table, bear in mind
that the table actually shows log(A)+10 which was normal back then. The
small variation of A with latitude is not at all important. Chauvenet had
the clever idea that most of this small correction could be included in the
lunar distance tables in the almanac. Then the correction would have been
simply
inc_LD=sin(Lat)*x
where x is equal to everything in the square brackets above. I have
tables on my web site. Anybody want it?

PLEASE NOTE: these corrections have been included in the calculations of the
lunar distance clearing tool on my web site for over three years. You can
directly assess the significance of the total oblateness correction by
selecting "Ignore Oblateness" in the "Options" section. If you're trying to
get an exact assessment of your skill or your sextant's arc error, there's
no reason to turn off the oblateness calculation. But for historical
NAVIGATIONAL calculations or just for general understanding, there may be
times when you want to turn off the oblateness correction.

Incidentally, there's no real reason to bother with the remaining details of
Chauvenet's own, rather idiosyncratic method of clearing lunars. It was
rarely used, and it offers no really significant advantage. But the general
discussion in his book is definitely worth reading.

-FER
www.HistoricalAtlas.com/lunars

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