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About three weeks ago, we were discussing how celestial observations
could be used in navigating a spacecraft. I had proposed that the only
viable approach would be a three-dimensional analogue of the horizontal
sextant angles technique used in coastal navigation. Frank Reed, in a
typically sharp response, wrote (in part):

 > I can measure
 > the angles between various extremely distant objects (the "fixed" stars)
 > and various nearby objects whose positions are known for every instant
 > of time from almanac data. To make it specific, suppose I measure the
 > angle between the star Antares and the Earth's near limb. I then add the
 > Earth's semidiameter (which I can measure or more likely calculate from
 > an assumed position). Now I have the center-to-center angle between the
 > Earth and Antares. Let's suppose it's 120 degrees (an observer on the
 > Earth at sea level directly beneath my location would measure an
 > altitude of 30 degrees). Instead of a circle of position, this sight
 > gives us a "cone" of position. The cone's vertex is at the Earth's
 > center and it extends out to infinity as a two-dimensional surface (in
 > 3d space). You're somewhere on that cone.


It has taken me a while to make the time to understand Frank's view of
the geometry and longer to find the time to respond.

As best as I can now understand it, the "cone" that Frank postulates
assumes that the light from Antares arrives as parallel rays. To an
observer near the Earth (meaning anywhere within the orbit of Mars and
likely well outside it) that is obviously true, with a precision higher
than the observer's observations would likely be. However, long before
infinity was reached (which it can never be!), an observer travelling
outward on that cone would receive light rays from Antares that were not
   remotely parallel to those reaching Earth. Hence, the locus of the
observed 120-degree angle is not a cone.

I suggest that it is, instead, exactly what I suggested it was back on
May 2nd: an analogue to a horizontal sextant angle. Specifically, I
suggest that the locus is the shape swept out by an arc of a circle,
passing through Earth and Antares, when that arc is rotated around the
chord Earth-Antares. Close to the Earth, such a locus would approximate
to the cone suggested by Frank but the approximation would break down at
greater distances.


A small point, of exactly zero practical relevance. Still, it is best
that this list's archives not contain uncontested errors.


Trevor Kenchington



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Trevor J. Kenchington PhD                         Gadus@iStar.ca
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