Wikipedia:Reference desk/Archives/Mathematics/2020 February 6

= February 6 =

How close to the pole does a 21st century constellation border line segment need to be for the northernmost or southernmost point to avoid being an endpoint?
Without the help of crossing the 6 or 18 hour meridians, which guarantees this even near the equator. Sagittarian Milky Way (talk) 21:35, 6 February 2020 (UTC)


 * If the interior of a border segment of constant right ascension contains a celestial pole, that pole is its northern-/southernmost point, but not an endpoint. Then the distance of the segment to the pole is zero. None of the IAU designated constellations has such a segment. Otherwise, one endpoint is northernmost and the other is southernmost. For border segments of constant declination, all points are equally north (or south), so all their points, including the endpoints, are both northernmost and southernmost. The 6 and 18 hour meridians play no role in this. Or have I misunderstood the question? --Lambiam 05:26, 7 February 2020 (UTC)


 * They usually slant after 1875.0 AD making one end northernmost and the other southernmost (because of precession) however Ursa Minor has segments like that where the celestial pole is moving directly at or away from the interior of an ex-parallel of latitude line segment. I am wrong about the 6 and 18 hour meridians though, this is where many points reach minimum or maximum latitude every 26 millennia but even though the ends of a short piece of parallel crossing those meridians move in opposite directions (on the latitude scale) nothing special happens, the line segment is still tilted. Sagittarian Milky Way (talk) 08:37, 7 February 2020 (UTC)


 * So you are not talking about the current epoch but also far into the future (but presumably not indefinitely far; in 2 billion years the Moon's stabilizing effect on the Earth's axial tilt will have ceased). But how do you define north and south? With respect to a moving celestial pole, or with respect to a fixed equatorial coordinate system? And which epoch do you use in defining the orientation of the system? Where does 1875 AD come from? --Lambiam 14:29, 7 February 2020 (UTC)


 * Epoch in the 21st century, a rough range of bounds on what is the declination of the least polar segment whose declination maxima or minima is internal is fine. Least polar meaning closest to the 21st century equator. 1875.0 is the deciyear they are referring to in constellation boundary definitions. i.e. the southern border of Scorpius is minus 45.5 degrees everywhere but only in 1875.0. Sagittarian Milky Way (talk) 00:00, 8 February 2020 (UTC)


 * All things celestial are moving and wobbling, and there is an embarrassment of coordinate systems one can choose from. Let us fix one: that in which the constellation boundaries are neatly “horizontal” and “vertical”: the celestial coordinates for B1875.0. To avoid all ambiguity, let us call these the F-coordinates (F for Fixed), which determine two F-poles and an F-equator. For an arbitrary epoch E, we have, similarly, a system of E-coordinates. This system varies in time; fixing it by putting E := B1875.0 gives the F-coordinates. You did not explicitly answer my question how to define north and south, but I understand this is w.r.t. the E-equator. We may then equally well use the distances to the E-poles. Disregarding the motion of the ecliptic and nutation, the E-poles describe small circles on the F-sphere, with an angular radius of about 23.4°. Did I understand correctly that the centres of these circles are the ecliptic poles, and that their E-coordinates are independent of E? Then they are 23.43929° away from the F-poles. I assume that this is the same as the “about 23.4°” of the aforementioned radius, since the F-poles must lie on these circles. (I did not find more precise values than “about 23.4°”.) Now take a “horizontal” arc on the F-sphere, with endpoints P and Q that are less than 180° apart in terms of rotation around the F-axis. The two F-meridians (half great circles connecting the F-poles) that go through P and Q form the boundaries of a segment of the F-sphere containing the arc PQ. Call that segment S. Consider the great circle C through the two F-poles that also goes, at epoch E, through the two E-poles. If one of the E-poles falls inside segment S, circle C intersects the arc PQ in an interior point, and then that point is extremal in distance to one of the E-poles. If the north E-pole is inside S, the intersection point is northernmost; otherwise the south E-pole is inside S and it is southernmost. Otherwise, at least one of the endpoints P and Q is extremal. This has nothing to do with how close to the E-pole that arc is. It can be arbitrarily close (a minute distance east or west from segment S). --Lambiam 17:30, 9 February 2020 (UTC)
 * You are correct, the center of precession is the ecliptic pole: while most galaxies tour every meridian and ~47 degrees of declination in 26,000 years, ecliptic pole galaxies are stuck at the same coordinates, ignoring each galaxy's unique space velocity, relativistic aberration and 0.005 degrees of year-scale wobble (which constellation borders and paper atlas books all ignore) and the ~2 degrees of dekamillennium-scale wobble (which paper atlases account for but only shrinks the ~23.4 degrees by 0.03 degrees from 1875 to 2101, I assume 0.03 degrees has little effect on which meridians PQ must avoid or cross). Sagittarian Milky Way (talk) 00:33, 10 February 2020 (UTC)