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Brad Morris has been delving somewhat deeper into the Moon's ecliptic 
latitude and longitude.

He has found his way to daily polynomials of the Moon's position, which 
allow Right Ascension (RA) and declination to be calculated to very high 
precision (microdegrees, it seems, from the number of decimal places in the 
given constants.

Then he has calculated ecliptic (= celestial) latitude and longitude of the 
Moon, using the Meeus formulae I provided in [7272] as follows-

==============================
If you have dec and RA of a body, and know the tilt  at that date (which
varies very slightly year on year, currently 23.438�) it's straightforward
to calculate the ecliptic latitude, from-
sin lat = sin dec cos tilt - cos dec sin tilt sin RA
and the ecliptic longitude, from -
tan long = ( sin RA cos tilt + tan dec sin tilt ) / cos RA

These came from Meeus, Astronomical Algorithms (which everyone who tills
such ground should own) equations 13.2 and 13.1.
================================

A problem he has found is that there's an ambiguity in the solution for 
obtaining long using tan long. This is because there are two possible 
angles, in the range 0� to 360�, which have the same tangent. Those angles 
are 180� apart, and the calculator / computer has no immediate way of 
knowing which is the right one (it will always choose the result to be in 
the range -90� to +90�): but we do. The long and the RA always have to be in 
the same quadrant; indeed, they are never more than a few degrees different, 
which becomes obvious if you sketch out the geometry of the two arcs 
involved. So if, AND ONLY IF, the RA is outside the range -90� to +90�, 
should you adjust the arctan result by 180�.

So the modified expression Brad provided, as-
Ecliptic Long = 180 + atan(((sin(RA)cos(ObliquityEcliptic) + 
tan(dec)sin(ObliquityEcliptic))/cos(RA))
will give wrong answers if RA is between -90� and +90�. It should NOT be 
used!

But if, instead of using the atan (arc tangent) function, he uses the 
alternative version that's available on most computers and many calculators, 
usually labelled ATAN2 or something similar, that's designed to put the 
result in the right quadrant by looking at the signs of numerator and 
denominator separately. He may need to do a bit of fiddling of the 
expression to get there, but it ought to work.

Brad pointed out-

"You can find the low precision values of the Ecliptic Latitude and 
Longitude of the Moon in the Astronomical Almanac.  The published values are 
to two decimal places of degrees."

Thanks. I hadn't realised that. I have some Astronomical Almanacs on the 
shelves, from many years back, and there they are. But only to hudredths of 
a degree, so not really adequate for many purposes, such as lunar distance 
and occultations. And only for one moment in each day, so not really usable.

If the daily polynomials are all that he needs, then fine. But one problen 
in using that method is that he needs to have access to a website that 
provides that information, and the one he quoted only does so to the end of 
2009. What if he wanted details to predict an occultation next year, say, or 
to back-predict one made by Shackleton?

Well, it's quite possible to roll-your-own Moon position predictions, if a 
precision of a millidegree, or so, will suffice. Like most things, it's all 
in Meeus, Astronomical Algorithms. There's a list of 60 or so terms for Moon 
ecliptic longitude, about 30 for latitude. Not too hard to set up for a 
computer; then simply give it a date and time, in the past or in the future, 
and it will do the rest. Biggest unknown is in future delta-T.

If I could do it on a programmable portable calculator, 20 years ago (which 
still works), Brad should be able to do it on a PC or a Mac.

If a more precise predictions than a millidegree are required, more detailed 
periodic terms are called for, and more are presumably available since 
Meeus' publication, which dates to 1991. He stated then, that many hundreds 
of such terms had been calculated by the Chapront pair, and they published 
"Lunat tables and programs from 4000BC to AB 8000" in 1991 (I haven't seen 
it). The terms converge only slowly, so each factor-of-10 increase in 
precision will call for many additional such terms.

I tried Googling "Moon longitude", and found, quickly,

"NASA - Solar and Lunar Coordinates
This theory contains a total of 37862 periodic terms, namely 20560 for the 
Moon's longitude, 7684 for the latitude, and 9618 for the distance to Earth. 
..."

That looked promising, with rather more terms than anyone on Navlist might 
call for, but the more important ones could presumably be useful.

However, I got a message back that "Internet Explorer cannot display the 
webpage". Anyone know why?

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

========================================

----- Original Message ----- 
From: 
To: 
Sent: Saturday, February 07, 2009 11:56 PM
Subject: [NavList 7296] Re: Fix by Occultations


|
| Hi George
|
| You can find the low precision values of the Ecliptic Latitude and 
Longitude of the Moon in the Astronomical Almanac.  The published values are 
to two decimal places of degrees.
|
| If you go to
| http://asa.usno.navy.mil/SecD/LunarPoly.html
| you will find the co-efficients to the Lunar Polynomials.
|
| When you use them, you will derive the high precision RA, declination and 
HP.  The results can be converted (as you point out) to Ecliptic Latitude 
and Longitude.
|
| One issue I found, however, is that I had to add 180 degrees the ATAN 
function provided to obtain correlation.  That is, the equation reads
|
| Ecliptic Long = 180 + atan(((sin(RA)cos(ObliquityEcliptic) + 
tan(dec)sin(ObliquityEcliptic))/cos(RA))
|
| EXAMPLE
| 7June09
|
| Polynomial Coefficients
| RA
| 245.4302085
| 13.3954668
|  0.0795003
| -0.0392653
| -0.0030594
|  0.0007151
|
| Dec
| -25.6866823
| -1.3663138
|  0.6207190
|  0.0099947
| -0.0035461
| -0.0000342
|
| HP
| 0.91121504
| -0.00532634
| 0.00057368
| 0.00001426
| 0.00000293
|
| YIELDS (at midnight, we can easily shift the time)
| RA 245.4302085
| Dec -25.6866823
| HP 0.91121504
|
| CONVERTING TO ECLIPTIC LAT LONG
| Elip Lat -4.110005927
| Elip Long 247.9338333
|
|
| The Astronomical Almanac 2009 publishes the 0h TT data as
|
| RA 16h 21m 43.25s
| Dec -25d 41m 12.1s
| HP 54m 40.37s
|
| Ecliptic Lat -4.11
| Ecliptic Long 247.93
|
| It seems to be worth while to go through the polynomial expansion, 
particularly if we intend to include the Limb Effects (mountains) like Frank 
suggests.
|
| Best Regards
| Brad
|
|
|
| |
|
| 


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