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Antoine Couëtte wrote:
> Further to results last obtained on Greg's second Lunar dated Aug 14th, 
2012, (i.e. the one with Lunar Distance = 2°05'8 See hereunder), I need to 
take back some comments on the comparison made yesterday between own results 
and Frank's calculator results obtained on this specific Lunar 
> 
> 
> With NO phase correction, if I enter my value of 2°05'980 into Frank's 
Calculator, I am getting a grade of Error in Lunar 0', Error in Longitude 
0°01'5.

My program allows only the center, near limb, or far limb of the target 
body to be observed. It expects that the user will observe a limb 
instead of the center if the target planet appears non-circular. In 
fact, it analyzes the illumination angles and recommends the correct 
limbs for altitude and lunar distance. That's very nice when the choice 
is not obvious. But as we shall see, the program needs help if the 
observer prefers to shoot the center of light.

In this run I use the far limb in order to get the semidiameter of 
Venus. (If the center of Venus is selected, semidiameter is not needed 
and not displayed.) Then I will adjust the computed lunar distance to 
the center of Venus, and to the center of light.


time:
2012-08-13 13:43:56.000 UTC
2012-08-13 13:45:03.184 TT (Gregorian)
JD 2456152.5 0.572954 (TT)
66.777 seconds delta T

estimated position:
+34°10.400' -119°13.800' north lat, east lon
2 meters above ellipsoid

atmosphere:
19° C (66° F) at observer
1012.5 mb (29.90" Hg) altimeter setting
1012.3 mb (29.89" Hg) pressure at observer

Moon to Venus separation angle:
    2°21.135' computed, center to center, unrefracted
       0.069' refraction
    2°21.066' center to center, refracted
      15.174' Moon near limb refracted semidiameter
       0.200' target far limb refracted semidiameter
    2°06.092' Moon near limb to Venus far limb
    2°05.800' observed angle
-  0°00.292' observed - computed

unrefacted separation angle rate:
-19.3" per minute (topocentric)
79% of total angular velocity

geocentric coordinates (true equator and equinox,
DE421 ephemeris, IAU 2006 precession, 2000A nutation):
  6h11m27.228s +20°49'04.1"  Moon RA and dec.
14.998' apparent semidiameter
  6h23m24.748s +19°55'58.0"  Venus RA and dec.
0.201' semidiameter

  3h16m36.972s  local apparent sidereal time

geocentric separation angle and rate:
    2°56.337' center to center
-27.77" per minute
84% of total angular velocity

illumination conditions:
75.6° 4.5° Sun unrefracted az, el
149.9° zenith angle, Moon to Sun
131.8° Moon phase angle
150.0° zenith angle, Venus to Sun
91.0° Venus phase angle

zenith angles:
165.5° Moon to Venus
346.1° Venus to Moon

recommended limbs:
Use Moon lower limb.
Use Venus lower limb.
Use Moon near limb.
Use Venus far limb.


My programe says (observed - computed) lunar distance = -.292'. But 
remember, that assumes an observation of the far limb of Venus. 
Refracted semidiameter = .200', so, if we assume an observation of the 
center of Venus, (observed - computed) = -.292 + .200 = -.092'.

In reality, the center of light was observed, so the -.092' error 
requires a phase correction.

> With Phase correction, if I enter my value of 2°05'896 into Frank's 
Calculator, I am getting a grade of Error in Lunar 0', error in Longitude 
0°00'5.

I have never applied correction for phase angle, and did not find much 
information or formulas. A 1976 paper by Safronov begins with some 
classic formulas, then presents a new model - which is too elaborate for me.


http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1978SvA....22...78S&db_key=AST&page_ind=0&data_type=GIF&type=SCREEN_VIEW&classic=YES

In the Explanatory Supplement to the Astronomical Almanac is a 
polynomial for Mercury and Venus phase corrections. It was derived 
during the creation of the DE200 ephemeris. Rewritten for phase angle in 
degrees (vs. quadrants) and better computational efficiency, it is:

-.03 + (4.33e-3 + (5.80e-5 - 3.57e-7 * i) * i) * i

That is the correction in semidiameters. E.g., at 45° phase angle, the 
center of light is .25 semidiameters from the center of the body. At 
phase angle 0 (fully illuminated) the correction is an obviously wrong 
-.03, but the small error is trivial since Mercury and Venus are not 
observable at phase angles near 0 or 180.

The 91.0° Venus phase angle and that formula give a correction of .575 * 
.200' = .115'. My program says the far limb of Venus is the correct one 
for a limb shot. Therefore, the correction is away from the Moon - but 
not directly away! As already noted:

150.0° zenith angle, Venus to Sun
346.1° zenith angle, Venus to Moon

Zenith angle is the "azimuth" from one body to another on the celestial 
sphere. The zenith is "north". E.g. if the Moon is exactly above Venus, 
the zenith angle from Venus to the Moon is zero. The angle increases 
counterclockwise. This is actually the same sense as azimuth, though it 
seems backward because you're looking out from inside the celestial sphere.

In this case, the direction of the Sun from Venus, with respect to the 
direction of the Moon, is 150.0 - 346.1 = 163.9°. The correction to the 
observed lunar distance is the cosine of that angle, times the .115' 
phase correction computed above. So add the -.111' to the observed lunar 
distance (center of light) to get the estimated center of Venus observation.

Recall that I found the observed - computed lunar distance = -.092', if 
computed for the center of Venus. If computed for the center of light, 
that becomes -.203'.

That assumes the correction affects only the distance, not the angles. 
But they do change, especially that close to the Moon. Unfortunately, I 
don't have time to do a better job. I can only hope that the small 
semidiameter of Venus prevents any serious error. Also, as Safronov says 
in his paper, there's significant disagreement among the standard formulas.

One last thought: as dear old George used to say, a single observation 
says little about accuracy. You could have been lucky, or unlucky. Even 
a run of several observations could have systematic errors in common.

-- 

   

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