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Hello,

I tried to work out the position from the photo taken in Southern Arizona (Frank published it under Antoine's Mercury lunar topic's subject name here).

The task was quite challenging. Not theoretically (because all the steps, that should be done, are clear). But when I started to solve the problem practically, several unexpected problems arose.

It seems obvious that the first choise for scaling the distances in the picture is the diameter of the Moon.  As the altitude of the Moon is quite low (we can see landscape in the photo), even the augmentation of SD is not a big deal. But problem is that:

1) the resolution of the photo is not sufficient to measure the SD value more accurate than some +/-0,05’;

2) the lunar distancies are much larger than SD. This means that if my SD estimate is off by only 0,05’, at lunar distance 10° this will accumulate an error of 2’ in lunar distance. And this is very significant error for this “Fix by Lunars” method.

3) the images of the stars are dim, creating some additional uncertainty in distances;

4) I am not sure, how large are the dissortions in the photo. Can I trust the angular measurements 5°…10° appart?

 I tried to get some positon lines based on the best estimate of the Moon’s SD, but I failed. The position lines where scattered over a wide area. Choosing some Assumed position, I tried to calculate the distances between some stars and compare with those in the image. But the star distances were off signifcantly. Then I abbandoned this approach.

Next idea was to find some distance between the two stars, that could be used as the scaling benchmark. At the same time this distance should not be affected by change of the altitude by several degrees (because of uncertainty of altitudes). It means that the two stars should be almost at the same height and at the same time be at the distance which is comparable with other lunar distances.

Fortunately there are two stars that answer these criteria: stars HIP79374 and HIP77859. In the attached Fig.1. they are marked as A and B. The trees in the photo reveal that the horizon is not very far bellow the bottom edge of the photo. So we can assume that the altitude of the two stars is not less then about 6 degrees. I calculated the distance between the stars at this altitude thus providing the scaling factor for the photo. Also I rechecked how the distance is changed by error in altitude of several degrees. It was quite small. At last I had some more reliable angle in this picture.

Then I measured 5 lunar distances relative to: 1) Mercury; 2) Star A; 3) Star B; 4) Star C; and 5) Star D.  

After plotting the positions lines I rechecked some star/star distances from several best estimated positions. They were within 1’. It prooved that there is no hope to get closer than this value. Therefore I plotted error bands, assuming error in lunar distances 1’.

The results are shown in the attached Fig.2…Fig.4.

Additionally I used 2 position lines that crosses the lunar position lines closer to 90 degrees:

1) position line from minimal altitude of the Moon (we can see in the photo that this value can not be less then about 7 degrees;

2) position line from the Sun with altitude 12 degrees below horizon (nautical twilight).      

The area of possible position is shown with yellow colored background. As we can see the error ellipse is situated in NW SE direction with the center close to 32.5°N 110.5°W.

But if the angular dissortions in the photo are larger, the position could be even out of the borders I have shown. 

Lesson learnt from this: don’t expect always get position lines crossed at angle close to the angle as the lunar distances are positioned in the sky. As we can see the position line from star’s C lunar and the position line from Mercury lunar are crossing only at angle 19° on map, despite the angle in the sky is about 77°.

I made additional simulation to check the crossing angle on map when one star is directly above the Moon and other to the left, thus forming angle close to 90 in the sky (and the Moon’s altitude is about 10 degrees). Fig.5. shows the arrangement of stars Rasalhague and Zubenelgenubi relative to the Moon and horizon. But in Fig.6. we can see the position lines on map. The lines (or more precisely — curves) cross at angle that is about a half of the angle we see in the sky. Here our position is not favorable for most effective crossing angle. Apparently the coresponding cone of position cuts the surface of the Earth forming sharp turns.

I wonder: are there any simple rules that allow to evaluate the most effective position line crossing angle? Maybe when the Moon is low in the sky, it is better when it is not far from meridian? (the expected accuracy and impact of the altitude of the Moon and star's positions is regulary discused in Frank's posts, therefore I am not mentioning this aspect).

Modris Fersters