Where can I get software to predict the azimuth angle of Polaris to better than 1 arcsecond accuracy?

There are many factors that enter into predicting the position of a star from the ground at this level of accuracy. Most commercially available software will not work. There are many effects that have to be accounted for:

Position of the star
Precession of earth axis
Nutation of earth axis
Chandler Wobble of earth axis
Stellar Aberration
Proper Motion of Polaris
Location of observer on earth
Time of observation
Atmospheric refraction
General relativistic effects

Location of Star First, you have to obtain the Right Ascension and Declination of Polaris computed for a reference epoch. The Smithsonian Astrophysical Observatory star catalog gives star positions only to an accuracy of +/- 0.8 arcseconds which is not adequate. The Hubble Guide Star Catalog gives star positions to an accuracy of +/- 0.5 arcseconds which is slightly better but still not good enough for any one specific star such as Polaris. The 1996 edition of the US Nautical Almanac available from the US Naval Observatory, lists the position of Polaris to a precision of about 6 arcseconds using the FK4 astrometric system developed for epoch 1930.0 and presessed to the current date.

The 'FK5 Catalog' lists the computed positions for some 5000 stars to an accuracy of +/- 0.02 arcseconds which is in the right ball park. This catalog is an enhanced version of the FK4 astrometric reference catalog created for epoch 1930, but including a much more sophisticated model of the earth's motion. The Hipparcos Input Catalog, which is based on the FK5 positions, gives the J2000.0 position of Polaris to an accuracy of 0.03 arcseconds as: RA = 2h 31m 48.704s and Dec = +89d 15' 50.72" and indicates a proper motion of: 15mu(ra)cos(dec) = +0.038 arcseconds/year and mu(dec) = -0.026 arcseconds/year. The yearly publication Apparent Places of the Stars for 1996 is also based on FK5 and gives the position of Polaris on February 8, 1996 as: RA = 2h 28m 0.61s, Dec = +89d 15' 5.49". Positions are given every day of the year and include aberration, precession and major long term nutation corrections.

Proper motion The next correction is for the proper motion of Polaris which is about 0.046 arcseconds per year from the Standard Astrometric Epoch of your table. Once you have added this, it is such a small increment that you do not have to recompute it for a few years. We will ignore it here.

Relativistic effects You might also consider adding a correction to account for the fact that we are sitting at the bottom of a gravitational well, and that incoming light rays will be slightly bent. For Polaris, at sea level, this amounts to only about 0.002 arcseconds or so, and you can ignore it. For astrometric satellites such as Hipparcos, however, this effect has to be added.

Precession Next, you have to correct this position for the precession of the earth to translate it from the date of the catalog position, to the date when you made the observation. This amounts to about 10 arcseconds per year for Polaris as the earth's polar axis moves along a 25,800 year circle in the sky with a diameter of some 47 degrees.

Chandler Wobble Then there is the correction for the wobble of the earth's axis, known as the Chandler Wobble, the amplitude of which can vary from 0.1 to as much as 0.7 arcseconds. The earth is not a perfect sphere, but slightly flattened, and it rotates about a slightly different axis than the mean figure of the earth. Also, earthquakes cause mass near the surface to get moved around which changes the moment of inertia of the earth. These effects cause the direction of the pole to trace a roughly circular path about the nominal north pole direction on a roughly 14 month (428 day) basis, however the mean amplitude varies in a very complex pattern from cycle to cycle.

Nutation An even slower effect is the nutation of the polar axis on a roughly 19 year cycle due to the changing lunar gravitational attractions upon the earth's equatorial bulge. There are several components to this effect. The Lunar component amplitude is +/-9 arcseconds towards the ecliptic pole with a period of 18.6 years. The solar component is +/- 1.2 arcseconds over 0.5 years; There is a 'fortnightly nutation' of +/- 0.1 arcseconds per 5 days; there is also a seasonal variation caused by the movement of airmasses of +/- 0.18 arcseconds per year.

Observer location Next, you need to know your latitude and longitude on the earth to an accuracy better than one second of arc which, given the earth's polar radius of 6356.779 kilometers, requires a positional accuracy of less than 30 meters. This is available from USGS topological maps of the viewing location, or from the Global Positioning System of satellites, with the help of a transponder. You can practically buy this at Radio Shack!

Local time Next, you have to determine where on its diurnal circuit around the north celestial pole it is. Polaris moves in a circle about 1 degree in radius every 24 hours which is 1 arcsecond every 8 seconds of clock time. This means you need fairly precise timekeeping using the WWV timesignals. The local time is then converted into Universal Time, and from the time of the year and the longitude of the observer, you determine the Local Sidereal Time which tells you which Right Ascension is on the north-south meridian at the time of the observation. For this you need to know the observing date to convert from Ut to Local Sidereal Time.

Next, you use the Right Ascension, Declination for the date of the observation, together with the Local Sidereal Time, to compute a first approximation to Polaris's Azimuth and Elevation. This is a trivial computation using two simple trigonometric formulae.

Atmospheric refraction Now, you have to correct the elevation angle for atmospheric refraction which for sightings of Polaris at elevation angles below 80 degrees will exceed 1 arcsecond. This correction, however, only affects the elevation angle and not the azimuth angle so in principle we don't need to worry about it if we only want azimuth angles. It makes the elevation of the star look higher than it actually is. Unless you are an eskimo, most observations are made at earth latitudes near 30 - 40 degrees so that Polaris will be at elevation angles between 30 - 40 degrees where atmospheric refraction will easily exceed 1 arcsecond. There are standard formulae which can make this correction, but it changes with atmospheric humidity and pressure.

Stellar aberration The next correction we have to make is for stellar aberration. Because the earth moves at about 30 kilometers per second in its orbit, star images make a 20 arcsecond loop about their true positions in the sky. The geometry of this effect is that stars near the North and south poles of the earth's orbit in the ecliptic plane execute circular motions, and stars in the plane of the ecliptic move from side to side along the ecliptic plane. We need to compute the ecliptic coordinates of Polaris, and from the date of the observation, determine which direction the offset due to aberration will be from the astrometrically determined positions of Polaris's TRUE position. In principle, if the observations are made from an earth satellite, a similar effect will be present at a level of 5 arcseconds for satellites with orbital velocities of 8 kilometers/sec. Under these conditions, of course, no correction is needed for atmospheric refraction!

Once we have determined the true Right Ascension and Declination of Polaris at the moment of observation, and have added corrections for aberration and atmospheric refraction. we now have the APPARENT position of Polaris for the day, date and time of the observation, and from a specific earth latitude and longitude. All these steps are relatively straight forward and can even be represented by a mathematical formula, but there would be several steps needed to do the calculation. Any table would be VERY lengthy because it would have to have 8 seconds in time resolution for every day of the year, for every 30 meters on the surface of the earth! Each year there would be a new table to account for the new nutation constants and the conversion of UT to Local Sidereal Time.

Copyright 1997 Dr. Sten Odenwald
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