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Generating a Lunar Distance example (with some simplifications to make things 
easier/more understandable/not much less accurate.  If this offends you, 
please stop reading now!)


Go to:

http://aa.usno.navy.mil/data/docs/celnavtable.php

Pick an appropriate date, time, and location (i.e. the moon is up and situated 
reasonably).  Enter your selections.  For example:

Date and time of observation:

Year: 2009     Month: August     Day:   1

Hour:  2  Minute:  47  Second: 45   UT 

Assumed position:

Enter best-estimate sea level coordinates.

Latitude:
X north    south  38  o  53  '  

Longitude:
  east  X  west   77  o  01  '   

You'll get a table of data.

We will use Jupiter as our body.  Just to get rolling and to make a point, 
we'll first calculate the true Lunar Distance (center to center).  First 
we'll do it in celestial coordinates:

True LD = asin (sin Jup Dec *sin Moon Dec + cos Jup dec * cos Moon Dec * cos (Moon GHA - Jup GHA)

= 68 deg 20.9 min

Second we'll do it in local alt-az coordinates using calculated altitudes, Hc:

True LD = asin (sin Jup Alt *sin Moon Alt + cos Jup Alt * cos Moon Alt * cos (Moon Az - Jup Az)

= 68 deg 22.0 min

Whoa! They should be the same!  (Note the parallelism in the equations.  Same 
would be seen in the Great Circle Distance equation.  All the same triangle.)

Problem is that the USNO gives only 0.1 deg (6 min) resolution in Az.  Fine 
for plotting and star ID; not good enough for LD calculation.  Luckily, we 
have all we need to do our own calculation of Az.

Moon Az = 180 + ATAN2(COS Moon LHA * SIN Lat - TAN Moon Dec * COS Lat, SIN Moon LHA )

(Careful with the order of Atan2 arguments, it varies!  Above is MS EXCEL 
order.)  ((so why atan2, not atan?  Because it keeps things in the correct 
quadrant with no fuss!))

where, Moon LHA = Moon GHA + Long (-West) = 21.028

Moon Az = 200.270 deg

Likewise, for Jupiter,  Jup Az = 129.990 deg

Using these values for Az in our alt-az LD equation, we get:

True LD = 68 deg 20.9 min

Well, that's a relief.  It agrees, as it must!  (The bodies are in the same 
places with the same separation whatever coordinate system we might choose.  
Since separation is a relative quantity, it must be the same.)

Now that we know how to calculate LD in local alt-az coordinates, we are ready 
to calculate the observed LD.  (Which is what we need to generate an 
example.)

We just use the above equation replacing calculated altitudes, Hc with 
observed altitudes Ho.  (Assume IC=0, Dip=0, or already corrected out.)  In 
the USNO notation, Ho = Hc - Sum of Altitude Corrections.  BUT, since we 
observe separation from near or far limb of the Moon, not the Lower Limb,  we 
don't use the SD part of Moon correction.  So what we'll calculate is 
separation to the center, adding or subtracting the SD for far or near limb 
respectively.

Moon Ho = 20 deg 48.6 min (to LL, to use with Frank) or 21 deg 3.5 min (to 
center, to use in Observed LD calculation)

Jup Ho = 18 deg 2.0 min (to LL, since USNO includes a Jupiter SD correction.  
Some, Frank for instance, assume we have observed center.  Difference is .4) 
so, use 18 deg 2.4 min (to center, for Frank observed altitude AND Observed 
LD calculation)  ((although body altitude has small effect))

Observed LD = Moon SD + asin (sin Jup Ho *sin Moon Ho + cos Jup Ho * cos Moon Ho * cos (Moon Az - Jup Az)

= 68 deg 44.8 min

That's it. Done!

It you paste the below URL, you'll get Frank's result for our example.  Quite close.



http://www.historicalatlas.com/lunars/lunars_pre.asp?LatDeg=38&LatMin=53&LatName=N&LonDeg=77&LonMin=1&LonName=W&IC=&temp=50&press=29.80&hteye=0&BodySel=Jupiter&BodyAltDeg=18&BodyAltMin=2.4&BodyLimb=LL&MoonAltDeg=20&MoonAltMin=48.6&MoonLimb=LL&GMTmonth=August&GMTday=1&GMTyear=2009&GMTHrs=2&GMTMin=47&GMTsec=45&gmtsel=Greenwich+Mean+Time&LunarDeg=68&LunarMin=44.8&LunarName=Far&obcheck=on&flatcheck=on

(We have ignored flattening and oblatness.  In the case of the Sun, the Moon SD discussion applies.)

In the attached spreadsheet, enter the green from USNO (or elsewhere) and use red.

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