If a satellite is orbiting on a circle around a main body, at the time of an occultation this orbit is projected in the Bessel plane over a band which "axis" is the central line of occultation.
If the halfwidth of this band is 1 , the probability that the center of the satellite will pass between the distances r' and r" is given by the formula :
If you will take r' and r" on different sides of the central line, take r' < 0 . Please note that I don't deal with the tiny changes in the projected orbit during the few minutes of the observation. I also neglect the possibility, when the observer is almost in the plane of the orbit, that the satellite could be hidden either in front or behind the main body. Remember that if you want to compute with this formula, you must be in radian mode. To compute in arc degrees, the constant
p should be replaced by 180 . Let's say now that the projected width of the orbit is 2a , that the satellite radius is s , and that the observer is at the distance d from the central line. The probability that the shadow will pass over the observer is :If d < a - s :
If a - s < d < a + s :
If d > a + s the probability is (of course) 0 .
Let us take as an example the case of Nephthys which was occulting TYC 1385-00019-1 on 2003 april 28th evening. According to the data from IMCCE team (including Jerome Berthier and Pascal Descamps), let us suppose that the suspected satellite has a diameter 2s = 8km , with a halfwidth of projected orbit a = 110km.
For d being the distance of the observer from the central line, we have :
d km proba
0 23/1000
50 27/1000
80 34/1000
90 40/1000
100 57/1000
103 69/1000
104 76/1000
105 87/1000
106 122/1000
107 114/1000
108 106/1000
109 96/1000
110 86/1000
111 75/1000
112 61/1000
113 43/1000
113.5 30/1000
113.9 14/1000
113.99 4/1000
114 0/1000
Thus the probability to see a satellite rises sharply when the observer is near an edge of the band swept by the projected orbit. [Note that the picture is even more contrasted if the orbit is elliptic. There is no longer symmetry, the central line of the main body shadow is no longer the "axis" of the band. The probability is highest near the farthest edge of the band swept by the projected elliptical orbit].
The problem is that the uncertainty on central line is large : where one must go and find a place where the probability to come back from the hunt not empty handed is the best (no more than one chance out of ten, yet) ? Worse : when the latest observations make suspect that there is something wrong either in the orbit or in the star position ? And worse : when we don't know much about the orbit [as almost every time !] ?
But the lesson is : if you can't move, and if you are not clouded, take a look : the prospects could be not so bad !
Astrofriendly, Raymond Dusser. 2003.05.01