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Wobbly Earth imparts apparent star motion

Jan 1, 2003

When navigating with the 57 navigational stars, the first thing one notices on a starry night is that they appear to move across the sky. But this is only apparent motion; it is Earth that is moving, not the stars. For all practical purposes, the stars are fixed. Were Earth frozen in space, the stars would not appear to move on any time scale perceptible to the naked eye. Now, glance at the sidereal hour angle (SHA) and declination of a star on the daily pages of the Nautical Almanac. One will see that its position changes over the course of a month or so. For example, the SHA and declination of Procyon are 245° 16.9' and N 5° 14.9' in January 1991, but 245° 17.0' and N5° 14.8' in March 1991. What causes this slow drift in the position of the stars?

The rotation of Earth increases the GHA of a star about 15° per hour and leaves the declination constant. The Nautical Almanac factors this change in GHA out of the stellar positions and into the GHA of Aries. Thus, when one looks at the SHA of a star on the daily pages, the position does not include the rotation of Earth. The apparent change in the position of the stars in the Nautical Almanac is due to more subtle effects. There are, in fact, several at work: precession, nutation, and annual aberration. These also affect the sun, moon, and planets, but the orbital motions of these bodies swamp these small drifts and make them difficult to discern.

Precession, discovered by Hipparchus in 100 BC, is a very slow drift in the direction of Earth's rotational axis, caused by the gravitational attraction of the sun and the moon on Earth's equatorial bulge. Precession is familiar in a spinning top whose axis drifts slowly (compared to the spin rate) in a circle around the vertical. For Earth, this drift has a period of about 26,000 years. The drift moves the axis of rotation in roughly a 47° circle. (The tilt of Earth's rotational axis does change slightly; the tilt is currently 23.5 degrees, but was 24.5 degrees in 7,530 BC, and will be about 22.5 degrees in 12,030 AD.)

The most striking result of precession is that Earth's pole has not always been underneath the star Polaris. Further, it will not remain under Polaris for long (on a 26,000-year time scale, that is). Polaris will be at its closet approach to the celestial pole in about the year 2100 when it will be within half a degree. However, in about 12,000 years, the star Vega will be the north star. In the northern hemisphere, we are fortunate to live in a time when there is a bright star near the north celestial polethe southern hemisphere is not currently so lucky. Precession causes the celestial sphere to drift about 0.8 minutes of arc per year.

The discovery of precession was a remarkable achievement in 100 BC. Since the processional drift is quite slow, the discovery depended on accurate observations of the celestial sphere recorded 150 years before the actual discovery. The correct explanation for this observed change in stellar positions was not known for many centuries, however. It was Isaac Newton who correctly explained the phenomenon.

Nutation is a more rapid and variable effect than precession. It is a small wobble in the drift of Earth's rotational axis superimposed on precession. Nutation is caused primarily by variations in the direction and distance of the Moon. The time scales for the wobble vary considerably, but the dominant component has a period of 18.6 years. Interestingly, the most rapidly varying components have periods of 10 days or less, though these cause a navigationally insignificant effect.

Unlike precession, nutation affects the tilt of Earth's rotational axis on a relatively short time scale. The tilt can vary as much as 0.3 minutes of arc, altering, for example, the maximum declination of the sun from year to year and even from solstice to solstice. In June 1991 the maximum declination listed in the Nautical Almanac was N 23° 26.6', but in December it was only S 23° 26.4'.

The sun's maximum declination is a good way to examine nutation in the Nautical Almanac since the declination of the sun at the solstice is equal to the tilt of the axis (at least to the accuracy used in the Nautical Almanac). In addition, the sun's maximum declination is unaffected by annual aberration (discussed below) and so is a "pure" measure of nutation. The plot of the sun's maximum declination (north and south) in an accompanying diagram shows that nutation is a complicated phenomenon, but its effect on position is generally less than 0.4 minutes of arc. It was first discovered in 1748 by the British astronomer James Bradley (1692-1762) who found small changes in the declination of the star y Draconis. He measured the declination from the transit altitude of the star. This technique is just the opposite of measuring one's latitude by observing a transit. When one measures one's latitude, the known declination of a star or, more commonly, the sun is used. Bradley knew his latitude, and employed this to measure the declination of Draconis. Finally, the motion of Earth around the Sun combined with the finite speed of light causes an effect known as annual aberration. This effect is perhaps the most interesting since it can be observed entirely in the course of a single year. Light travels very fast, but not infinitely fast. When incoming light is observed from a star, the direction from which it appears to come is influenced by the speed of Earth's orbit about the Sun. This is the same effect that alters the apparent direction of falling rain when one runs through it. Rain falling from directly overhead will appear to come from a direction in front of a runner. Similarly, the light from a star appears to come from a direction inclined toward the direction of Earth's motion. Over the course of a year, the direction of the earth's motion about the Sun varies as Earth moves in its nearly circular orbit. This causes an apparent drift of the stars in very small ellipses about their average positions. The size of this drift is at most 0.4 minutes of arc.

The SHA and declination of the stars should drift around the small ellipse over the course of a year, returning to almost the same values. However, precession and nutation alter this cyclical motion. An accompanying diagram shows a plot of the position (declination and SHA) of the star Procyon in 1990 and 1991; look for the cyclical variation due to annual aberration superimposed on the more steady precessional and nutational drift. The shape of the ellipse depends on the star's position relative to the ecliptic (the path in the sky traced out by the sun's motion during a year). For a body on the ecliptic (like the sun), the ellipse collapses to a line and the position shifts along the ecliptic. Thus, the declination of the sun at the solstices is unaffected by annual aberration. For a body directly above or below the plane of the ecliptic, the ellipse becomes a circle.

Like precession, annual aberration is a cyclic phenomenon. However, it is 26,000 times more rapid. Thus, it cannot be detected by simply observing the celestial sphere for hundreds of years. On this time scale, precession is a large effect and can be detected without precise instruments, but aberration never changes the position of the stars by more than 0.4 minutes of arc. No matter how long the stars are observed, an instrument capable of measuring positions to one tenth of a minute of arc is required to detect aberration. Thus, it was not discovered until 1728, again by Bradley. His observations were correctly interpreted as an effect of the finite speed of light, since the Dutch astronomer Roemer had established that light traveled with finite velocity in 1675. Bradley's observations were the first direct proof of Earth's motion.

As an exercise, glance through the pages of the Nautical Almanac to convince oneself that the size of the yearly changes in SHA and declination are about 0.4 to 0.8 minutes of arc. This is the size of the drifts due to precession, nutation, and annual aberration. For stars with high declinations, the SHA will appear at first glance to change more than this. Navigators know that the distance between the meridians of longitude on Earth becomes smaller and smaller as they approach the poles until the meridians finally come together. The same effect causes the changes in the position of the celestial sphere to be more pronounced in the SHA than in the declination. At high declination (latitude), a short east-west distance translates to a large change in SHA (longitude).

This has been a brief introduction to the most important effects that influence the apparent motion of the "fixed" stars used in navigation. While the apparent motion of the stars due to the rotation of Earth is the most obvious, there are several smaller effects that must be accounted for in the construction of the Nautical Almanac. In total, the positions of the stars can drift by as much as 0.8 minutes of arc per year, a naviga-tionally significant change.

Thomas R. Metcalf is an assistant astronomer at the University of Hawaii and lives in Kaneohe on Oahu.


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