A long, wrong star trail

Based on erroneous calculations performed by I. S. Bowen in 1967, some astronomers believe that the atmosphere of the Earth can cause a field of stars to rotate during a long time exposure.
11 May 2007
Eric H. Richardson

Ira Sprague Bowen (1898–1973) perfected the 5m telescope on Mount Palomar and has a much deserved reputation for meticulous work in both astrophysics and instrumentation. In 1966, after receiving the gold medal of the Royal Astronomical Society, he gave the George Darwin lecture at Burlington House (‘Future Tools of the Astronomer’). At some point during his talk, he made a few remarks 'about the ultimate limitation on the size of the field,’ showing several drawings of monochromatic star trails refracted by the Earth's atmosphere. Unfortunately, most of the drawings were incorrect.

This lecture was published in 19671 without undergoing the peer-review process, perhaps because it had been given by a prestigious invited guest lecturer. Nevertheless, this erroneous result has since been used as a standard reference by astronomers and reviewers alike. For example, in 1988, a paper published in a peer-reviewed journal2 showed a variety of star trails produced by a computer code whose accuracy had been verified by reproducing the conclusions of Bowen's lecture.

Figure 1 shows the ‘mostly incorrect’ star trails calculated by Bowen using spherical trigonometry and the corrected trails produced by ray tracing using modern optical design software. Under no circumstances, such as rotation of the detector during the time exposure, can ray tracing reproduce Bowen's results. This is because rotation would also change the trails of the North and South stars. It should be mentioned that spherical trigonometry can be used to produce the correct star trails,3 provided that the refraction vectors are always projected onto circles that include the guide star. Bowen did not do this except for the North and South stars.

Figure 1. Star trails calculated by Bowen using spherical trigonometry (left and corrected by ray tracing (right).

Figure 1 shows a 2 degree-diameter ring of six stars with a guide star in the middle at a declination of 60 degrees (Decl+60) viewed from a point on Earth at 30 degrees latitude during an 8h exposure. The points show the star positions at 1h intervals.

The most significant difference between the incorrect and correct star trails lies in the tangential separation of the images between the beginning and end of the trails of the East and West stars. This separation is about eight times longer in Bowen's drawing (see Figure 1, left) than in the correct drawing (see Figure 1, right) and is also in the opposite direction. A smaller difference can be seen in the radial spread of these trails where Bowen's result is smaller.

Bowen's error goes beyond mere academic interest. For instance, in meetings reviewing expensive new telescope projects, his findings were used to refute the claim that a telescope with an equatorial mounting represents a better design since it does not require to rotate instruments during time exposures.

The altitude-azimuth (alt-az) telescope mounting has one axle that is always vertical, the azimuth axle, and a second axle, the altitude axle, that is always horizontal. This is of some advantage in the mechanical design. If an alt-az mounting were tilted so that the vertical azimuth axle became parallel to the axis of the Earth, it would transform into a fork-type equatorial mounting, where the azimuth axle would become the polar axle and the altitude axle would be the declination axle. A mechanical design consideration for equatorial mounting is bending of its tilted axles. Although experienced mechanical engineers have offered to design modern equatorial mountings for new large telescopes, they have never been given contracts, even though the cost of such design studies represented only a small percentage of the available budgets.

The Palomar 5m telescope, which Bowen put into operation in 1948, is the largest equatorial telescope in the world. All larger optical telescopes built or planned since then feature alt-az mountings.

The equatorial telescope has a major advantage in that it can track a field of stars mostly by rotating its polar axle at the slow rate of about one revolution per day. In addition, incremental, slow changes in the rotation of the declination axle and in the elevation of the polar axle are required for perfect tracking, depending on the declination of the guide star. By comparison, the alt-az mounting would need to rotate its azimuth axle at an infinitely fast rate to track a field of stars through the zenith, that is, the part of the sky that is directly overhead. Also, its instruments would need to be rotated at the same high rate in the opposite direction. Consequently, the alt-az mounting has a small blind spot that prevents it tracking near the zenith, where stellar images are sharpest and brightest. More importantly, the main disadvantage of the alt-az design is the rapidly changing gravity vector, which causes flexures in rotating instruments at horizontal foci, even during short exposures.

Bowen's calculations were independent of the type of mounting, and he used spherical trigonometry, not ray tracing. It also seems that he performed his calculations using a simple adding machine since his notes list pages of logarithms.

The bending of light rays by atmospheric refraction makes stellar objects appear closer to the zenith. The largest refraction is close to the horizon, such as that of the sun during sunset. Refraction makes the sun appear to be slightly above the horizon when it has actually already set and is below the horizon. Differential refraction makes the sun appear oval in shape when it is actually circular, because rays from the lower part of the sun's disc are bent more than rays from the top part of the disc. Rays from objects farther from the horizon are bent much less, but the change from a circular ring of stars to an oval shape, however slight, results in significant star trails during long time exposures.

Bowen correctly calculated the amount of atmospheric refraction, but made the mistake of projecting the vectors onto polar circles on the celestial sphere instead of onto field circles that pass through the guide star in addition to opposite stars on the ring of stars. The associated spherical trigonometric equations have been published,3 comparing the incorrect with the correct equations. It should be noted that Bowen obtained a correct result for the North and South stars only because, in that case, the polar circle and the field circle happen to be identical.

An important conclusion is that a correctly tracking equatorial telescope does not need to rotate the detector during a time exposure. Another conclusion is that a lecture published by a famous scientist can be erroneous, especially if not peer-reviewed.

Eric H. Richardson
University of Victoria
Victoria, BC, Canada

Eric Harvey Richardson retired from the Dominion Astrophysical Observatory in 1991 and is now an adjunct professor of graduate studies at the University of Victoria. He also does consulting work at EHR Optical Systems. He was president of Commission 9 of the International Astronomical Union from 1979 to 1982.

1. I. S. Bowen, Future tools of the astronomer, Quart. J. Roy. Ast. Soc. 8, p. 22, 1967. 
2. J. G. Cohen, J. Cromer, Atmospheric refraction effects on the Norris and Keck multiobjects spectrographs, Pub. Astron. Soc. Pacific 100, pp. 1582-1585, 1988.
3. E. H. Richardson, Corrected calculation of star trails caused by differential atmospheric refraction, SPIE Proc. 6342, 63421N, 2006.doi:10.1117/12.692307
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