NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Greg Rudzinski
Date: 2013 May 19, 18:36 -0700
As David has pointed out my position of 40° N 82° W is not a complete solution to this puzzle so I will modify my answer to be a great circle track line of equal star altitudes between the late twilight position of 40° N 82° W and the early twilight position of 42° 4' N 92° 38' W. This gives a 498 NM great circle track line where at 2:00:00 UT 5/19/2013 the star pair of Procyon and Capella will have equal altitudes during the evening twilight period. This assumes that a sextant is to be used normally with a natural horizon while upon a body of water. If a bubble octant is used then the entire evening is available for this puzzle.
Greg Rudzinski
Re: An equal altitude puzzle
From: David Fleming
Date: 2013 May 19, 15:53 -0700
Given two sights we get a two point fix which we resolve by what is close to where we think we are.
Two equal sights without knowing the altitude can not yield a point solution. Greg's answer maybe possible but they do not constitute the complete solution.
As both I and Peter Hakel pointed out, the position loci is an arc, part of a great circle. It is a great circle passing through the midpoints of the P->C arcs and perpendicular to the great circle through the two stars.
See attached illustration to follow this reasoning.
The great circle containing the stars defines an axis of symmetry which passes through the second great circle that contains the solution loci. Where that axis intersects the position GC is 90° from each star. Move 1° on the position loci GC closer to the stars from the 90° point and the arc between those points defines altitudes greater than 1°.
The attached illustration further shows how the Sun 6° or more below horizon restricts the position loci.
We are left with the sawtooth labeled part of the arc as a possible position.
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