## Appendix F Astronomical coordinates and timekeeping

Coordinate systems

When we want to locate a star, or any other astronomical object, we only need to specify its direction. We don't need its distance. We therefore need only two coordinates, two angles, to locate an astronomical object. Sometimes, it is convenient to think (as the ancients did) of the stars as being painted on the inside of a sphere, the celestial sphere. Just as we can locate any place on the surface of Earth with two coordinates, latitude and longitude, we need two coordinates to locate an object on the celestial sphere.

We choose coordinate systems for convenience in a particular application. In general, to set up a coordinate system we first identify an equator and then choose coordinates that correspond to latitude and longitude.

A convenient system for any particular observer is the horizon system. The horizon becomes the equivalent of the equator in that system. The angle around the horizon, measured from north, through east, south and west, is the azimuth. The angle above the horizon is called elevation. Instead of elevation, we can use the zenith distance, which is the angle from the zenith (overhead) to the object. From their definitions, we can see that the sum of the zenith distance and the elevation is always 90°. The azimuth ranges from 0° to 360°, and the elevation from —90° to 90° (with negative elevations being below the horizon). The problem with this system is that, as the Earth rotates, the azimuths and elevations of the stars change in a complicated way. It would not be very useful to prepare a catalog of stars, just giving their azimuths and elevations.

One solution is to use a coordinate system based on the projection of the Earth's equator onto the celestial sphere, the celestial equator. Such a system is called an equatorial coordinate system. The angle above or below the celestial equator is called the declination, and it is designated by the symbol "<5". It ranges from —90° to 90°. As the Earth rotates, the declinations of objects don't change. Also, the declination doesn't change as observers move around the Earth. There are two ways to measure the other coordinate.

(1) We can measure the angle, going westward, from the observer's meridian. This is called the hour angle, H. We can measure it from 0° to 360°, but it is convenient to express it in units of time, from 0 to 24 hr. As the Earth rotates, the hour angle of each object changes, but in a simple way, increasing by one hour for each hour that the Earth rotates. In addition, at any instant, observers at different longitudes will measure different hour angles for the same object. (The hour angles differ by the difference in longitudes.) (2) To achieve a coordinate system that is the same for all observers and doesn't change with time, we must fix that coordinate system with respect to a location on the celestial sphere. We choose as our reference point one of the two intersections of the celestial equator and the ecliptic (the Sun's path around the sky). These two points are called equinoxes. (When the Sun is at either equinox, all observers on Earth have a 12 hr day and a 12 hr night. This occurs on the first day of spring and the first day of fall.) We choose the vernal equinox, the point where the Sun is on the first day of spring, as our starting point for the coordinate, right ascension, designated by the symbol a. The right ascension is measured from 0 to 24 hr, increasing from west to east. That is, objects with higher right ascensions cross an observer's meridian later than objects with lower right ascensions.

One effect of the Earth's precession is to move the equinoxes by about 50 arc sec per year. Thus, the origin of our coordinate system is drifting. Therefore, in compiling a catalog of objects, the time, or epoch, at which the coordinates apply must be specified. The epoch is usually put in parentheses after the a and <5, for example, a(2000) and 5(2000). An observer can then calculate where the objects will be on the date they are to be observed (a relatively simple calculation). To keep things simple, we generally agree on standard epochs, and keep our catalogs on a common standard. Catalogs that are just coming out use the standard epoch 2000. By changing the standard epoch every 50 years, we don't have to change too often, and we are able to keep the catalog coordinates reasonably close to the actual coordinates. (In 50 years, the origin will have moved by less than one degree.)

Other coordinate systems are useful for studying particular sets of objects. For example, in studying the Solar System, ecliptic coordinates are useful. In this system, the ecliptic latitude ft is measured above or below the ecliptic (from —90° to 90°), and the ecliptic longitude A is measured around the ecliptic, starting at the vernal equinox and increasing eastward (from 0° to 360°).

A coordinate system that is useful for studying galactic structure is the galactic coordinate system. (We touched on this briefly in Chapter 16.) The galactic latitude b is measured above and below the galactic plane. The galactic longitude / is measured from the galactic center, increasing in the same direction as the right ascension.

Once an object is located in one coordinate system, those coordinates can be transformed into any of the other systems. The equations for those transformations are beyond the scope of this brief summary, but the calculations can be done on a hand calculator, and certainly by a computer that would be involved in pointing a large telescope.

### Timekeeping

It is natural to use the Earth's rotation as a basis for timekeeping. We can keep track of the Earth's rotation by noting the motion of the stars. For each rotation of the Earth the stars make a full circle in the sky. We can therefore measure time by choosing a star, or other point in the sky, and seeing the fraction of the daily circle that it has made.

We choose the reference point to be the vernal equinox. We measure a time, called local sidereal time (LST), by the progress of the vernal equinox. When the vernal equinox is at an observer's meridian, the LST is zero for that observer. One hour later, the LST is 1 hr; in addition, the hour angle of the vernal equinox is 1 hr. This means that the LST is simply the hour angle of the vernal equinox. When the LST time is 1 hr, objects with a right ascension of 1 hr will be on the meridian. This means that the LST is also equal to the right ascension of the object that happens to be on the meridian. Observers at two different points on Earth will have different LSTs. The LSTs will differ by the longitude difference between the two observers.

Most of our civil timekeeping is referenced to the Sun. We could define a local solar time, based on the hour angle of the Sun. This is what a sundial measures, but this is not very useful for civil systems. For uniformity, civil systems utilize time zones. This means that the Sun can be as much as a half an hour ahead or behind your local time, even more for some very wide time zones.

As the Earth moves around the Sun, the Sun appears projected against a changing background of stars (progressing through the constellations of the zodiac). This means that the right ascension of the Sun increases by an average of 3m56.56s per day. The time for the Sun to go from one passage of your meridian to the next is this much longer than the time for a star to go from one passage to the next. Therefore, a day by the Sun, a solar day, is longer, by this amount, than a day according to the stars, a sidereal day.

Another problem with solar time is that the right ascension of the Sun does not change smoothly. This is the result of two effects. (1) The Earth's orbit is elliptical, so the Earth moves faster when it is closer to the Sun, and slower when it is farther away. This variable speed is mirrored in the apparent motion of the Sun against the background of stars. (2) The Sun moves along the ecliptic, which makes a 23.5° angle with the celestial equator. Therefore, the right ascension and declination of the Sun are both changing. Even if the Sun were to move along the ecliptic at a constant rate, its right ascension would change at a variable rate. Because of these two effects, we define a fictitious object, called the mean Sun, which moves along the celestial equator at a uniform rate. Time kept by the mean Sun is called the mean solar time, and it is the time that would be kept by a clock. The relationship between the mean Sun and the real Sun is given by a quantity called the equation of time. This quantity is depicted graphically by the distorted figure "8" that appears in the empty areas of some globes. This is called an analemma.

To the extent that the solar time is used in astronomical timekeeping, it is usually universal time (UT), which is the mean solar time at Greenwich, England. It is often useful to convert from UT to LST for any observer. This is done with the aid of a publication, such as the Astronomical Almanac, published by the US Naval Observatory. The Almanac gives, for each date, the LST at Greenwich at 0h UT. To this, we add the UT of interest, multiplied by a factor to account for the difference between sidereal and solar times. This gives the LST at Greenwich at the UT of interest. We then subtract the longitude of the observer L. We can write this as

Where LST(0,0) is the LST at Greenwich at 0h UT.

All of this is complicated by the effects of precession, and the wobble of the Earth, known as nutation. While true sidereal time is the actual hour angle of the vernal equinox, our sidereal clocks really keep a mean sidereal time. There is also a problem in the definition of a year. A sidereal year is the time for the Sun to return to the same place with respect to the fixed background of stars. We could use this definition, but after a long time the precession will cause the seasons to occur in different months. For this reason, we use a tropical year, defined as the time it takes for the Sun to travel from