Sommers-Bausch Observatory - University of Colorado


Celestial Coordinates


This essay is excerpted from the Sommers-Bausch Observatory's "APAS 1010 Laboratories - Introduction to Astronomy" lab manual, 1996, by Keith Gleason.


GEOGRAPHIC COORDINATES

The Earth's geographic coodinate system is familiar to everyone - the north and south poles are defined by the Earth's axis of rotation; equidistant between them is the equator. North-south latitude is measured in degrees from the equator, ranging from -90 degrees at the south pole, 0 degrees at the equator, to +90 degrees at the north pole. East-west distances are also measured in degrees, but there is no "naturally-defined" starting point - all longitudes are equivalent to all others. Humanity has arbitrarily defined the prime meridian (0 degrees longitude) to be that of the Royal Observatory at Greenwich, England (alternately called the Greenwich meridian).

Each degree of a 360 degree circle can be further subdivided into 60 equal minutes of arc ('), and each arc-minute may be divided into 60 seconds of arc ("). The 24-inch telescope at Sommers-Bausch Observatory is located at a latitude of 40 d 0'13" North of the equator and at a longitude 105 d 15'45" West of the Greenwich meridian.


ALT-AZIMUTH COORDINATES

The alt-azimuth (altitude - azimuth) coordinate system, also called the horizon system, is a useful and convenient system for pointing out a celestial object.

One first specifies the azimuth angle, which is the compass heading towards the horizon point lying directly below the object. Azimuth angles are measured eastwardly from North (0 deg azimuth) to East (90 deg), South (180 deg), West (270 deg), and back to North again (360 deg = 0 deg). The four principle directions are called the cardinal points.

Next, the altitude is measured in degrees upward from the horizon to the object. The point directly overhead at 90 deg altitude is called the zenith. The nadir is "down", or opposite the zenith. We sometimes use zenith distance instead of altitude, which is 90 deg minus the altitude.

Every observer on Earth has his own separate alt-azimuth system; thus, the coordinates of the same object will differ for two different observers. Furthermore, because the Earth rotates, the altitude and azimuth of an object are constantly changing with time as seen from a given location. Hence, this system can identify celestial objects at a given time and location, but is not useful for specifying their permanent (more or less) direction in space.

In order to specify a direction by angular measure, you need to know just how "big" angles are. Here's a convenient "yardstick" to use that you carry with you at all times: the hand, held at arm's length, is a convenient tool for estimating angles subtended at the eye:


EQUATORIAL COORDINATES

Standing outside on a clear night, it appears that the sky is a giant celestial sphere of indefinite radius with us at its center, and upon which stars are affixed to its inner surface. It is extremely useful for us to treat this imaginary sphere as an actual, tangible surface, and to attach a coordinate system to it.

The system used is based on an extension of the Earth's axis of rotation, hence the name equatorial coordinate system. If we extend the Earth's axis outward into space, its intersection with the celestial sphere defines the north and south celestial poles; equidistant between them, and lying directly over the Earth's equator, is the celestial equator. Measurement of "celestial latitude" is given the name declination (DEC), but is otherwise identical to the measurement of latitude on the Earth: the declination at the celestial equator is 0 deg and extends to 90 deg at the celestial poles.

The east-west measure is called right ascension (RA) rather than "celestial longitude", and differs from geographic longitude in two respects. First, the longitude lines, or hour circles, remain fixed with respect to the sky and do not rotate with the Earth. Second, the right ascension circle is divided into time units of 24 hours rather than in degrees; each hour of angle is equivalent to 15 deg of arc. The following conversions are useful:


                24 h  =  360 d         1 m  =  15'
                 1 h  =  15 d          4 s  =  1' 
                 4 m  =  1 d           1 s  =  15"

The Earth orbits the Sun in a plane called the ecliptic. From our vantage point, however, it appears that the Sun circles us once a year in that same plane; hence, the ecliptic may be alternately defined as "the apparent path of the Sun on the celestial sphere".

The Earth's equator is tilted 23.5 deg from the plane of its orbital motion, or in terms of the celestial sphere, the ecliptic is inclined 23.5 deg from the celestial equator. The ecliptic crosses the equator at two points; the first, called the vernal (spring) equinox, is crossed by the Sun moving from south to north on about March 21st, and sets the moment when spring begins. The second crossing is from north to south, and marks the autumnal equinox six months later. Halfway between these two points, the ecliptic rises to its maximum declination of +23.5 deg (summer solstice), or drops to a minimum declination of -23.5 deg (winter solstice).

As with longitude, there is no obvious starting point for right ascension, so astronomers have assigned one: the point of the vernal equinox. Starting from the vernal equinox, right ascension increases in an eastwardly direction until it returns to the vernal equinox again at 24 h = 0 h.

The Earth precesses, or wobbles on its axis, once every 26,000 years. Unfortunately, this means that the Sun crosses the celestial equator at a slightly different point every year, so that our "fixed" starting point changes slowly - about 40 arc-seconds per year. Although small, the shift is cumulative, so that it is important when referring to the right ascension and declination of an object to also specify the epoch, or year in which the coordinates are valid.


TIME AND HOUR ANGLE

The fundamental purpose of all timekeeping is, very simply, to enable us to keep track of certain objects in the sky. Our foremost interest, of course, is with the location of the Sun, which is the basis for the various types of solar time by which we schedule our lives.

Time is determined by the hour angle of the celestial object of interest, which is the angular distance from the observer's meridian (north-south line passing overhead) to the object, measured in time units east or west along the equatorial grid. The hour angle is negative if we measure from the meridian eastward to the object, and positive if the object is west of the meridian.

For example, our local apparent solar time is is determined by the hour angle of the Sun, which tells us how long it has been since the Sun was last on the meridian (positive hour angle), or how long we must wait until noon occurs again (negative hour angle).

If solar time gives us the hour angle of the Sun, then sideral time (literally, "star time") must be related to the hour angles of the stars: the general expression for sidereal time is

Sidereal Time = Right Ascension + Hour Angle

which holds true for any object or point on the celestial sphere. It's important to realize that if the hour angle is negative, we add this negative number, which is equivalent to subtracting the positive number.

For example, the vernal equinox is defined to have a right ascension of 0 hours; thus the equation becomes

Sidereal time = Hour angle of the vernal equinox

Another special case is that for an object on the meridian, for which the hour angle is zero by definition. Hence the equation states that

Sidereal time = Right ascension crossing the meridian

Your current sidereal time, coupled with a knowledge of your latitude, uniquely defines the appearance of the celestial sphere; furthermore, if you know any two of the variables in the expression ST = RA + HA , you can determine the third.

The following illustration shows the appearance of the southern sky as seen from Boulder at a particular instant in time. Note how the sky serves as a clock - except that the clockface (celestial sphere) moves while the clock "hand" (meridian) stays fixed. The clockface numbering increases towards the east, while the sky rotates towards the west; hence, sidereal time always increases, just as we would expect. Since the left side of the ST equation increases with time, then so must the right side; thus, if we follow an object at a given right ascension (such Saturn or Uranus), its hour angle must constantly increase (or become less negative).


SOLAR VERSUS SIDEREAL TIME

Every year the Earth actually makes 366 1/4 complete rotations with respect to the stars (sidereal days). Each day the Earth also revolves about 1 degree about the Sun, so that after one year, it has "unwound" one of those rotations with respect to the Sun; on the average, we observe 365 1/4 solar passages across the meridian (solar days) in a year. Since both sidereal and solar time use 24-hour days, the two clocks must run at different rates. The following compares (approximate) time measures in each system:


        SOLAR       SIDEREAL            SOLAR       SIDEREAL
      365.25 days  366.25 days         24 hours    24h 3m 56s
      1 day        1.00274 d           23h 56m 4s  24 hours
      0.99727 d    1 day               6 minutes   6m 1s

The difference between solar and sidereal time is one way of expressing the fact that we observe different stars in the evening sky during the course of a year. The easiest way to predict what the sky will look like (i.e., determine the sidereal time) at a given date and time is to use a planisphere, or star wheel. However, it is possible to estimate the sidereal time to within a half-hour or so with just a little mental arithmetic.

At noontime on the date of the vernal equinox, the solar time is 12h (since we begin our solar day at midnight) while the sidereal time is 0h (since the Sun is at 0h RA, and is on our meridian). Hence, the two clocks are exactly 12 hours out of syncronization (for the moment, we will ignore the complication of "daylight savings"). Six months later, on the date of the autumnal equinox (about September 22) the two clocks will agree exactly for a brief instant before beginning to drift apart, with sidereal time gaining about 1 second every six minutes.

For every month since the last fall equinox, sidereal time gains 2 hours over solar time. We simply count the number of elapsed months, multiply by 2, and add the time to our watch (converting to a 24-hour system as needed). If daylight savings is in effect, we subtract 1 hour from the result to get the sidereal time.

For example, suppose we wish to estimate the sidereal time at 10:50 p.m. Mountain Daylight Time on August the 19th. About 11 months have elapsed since fall began, so sidereal time is ahead of standard solar time by 22 hours - or 21 hours ahead of daylight savings time. Equivalently, we can say that sidereal time lags behind daylight time by 3 hours. 10:50 p.m. on our watch is 22h 50 m on a 24-hour clock, so the sidereal time is 3 hours less: ST = 19h 50m (approximately).


ENVISIONING THE CELESTIAL SPHERE

With time and practice, you will begin to "see" the imaginary grid lines of the alt-azimuth and equatorial coordinate systems in the sky. Such an ability is very useful in planning observing sessions and in understanding the apparent motions of the sky. To help you in this quest, we've included four scenes of the celestial sphere showing both alt-azimuth and equatorial coordinates. Each view is from the same location (Boulder) and at the same time and date used above (10:50 pm MDT on August 19th, 1993). As we comment on each, we'll mention some important relationships between the coordinate systems and the observer's latitude.

Looking North

From Boulder, the altitude of the north celestial pole directly above the North cardinal point is 40 deg, exactly equal to Boulder's latitude. This is true for all observing locations:

Altitude of the pole = Latitude of observer

The +50 deg declination circle just touches our northern horizon. Any star more northerly than this will be circumpolar - that is, it will never set below the horizon.

Declination of northern circumpolar stars > 90 deg - Latitude

Most of the Big Dipper is circumpolar. The two pointer stars of the dipper are useful in finding Polaris, which lies only about 1/2 degree from the north celestial pole. Because these two stars always point towards the pole, they must both lie approximately on the same hour circle, or equivalently, both must have approximately the same right ascension (11 hours RA).

Looking South

If you were standing at the north pole, the celestial equator would coincide with your local horizon. As you travel the 50 degrees southward to Boulder, the celestial equator will appear to tilt up by an identical angle; that is, the altitude of the celestial equator above the South cardinal point is 50 deg from the latitude of Boulder. Your local meridian is the line passing directly overhead from the north to south celestial poles, and hence coincides with 180 degrees azimuth. Generally speaking, then,

Altitude of the intersection of the celestial equator with the meridian = 90 deg - Latitude

Since the celestial equator is 50 deg above our southern horizon, any star with a declination less than -50 deg is circumpolar around the south pole, and will never be seen from Boulder.

Declination of southern circumpolar stars < Latitude - 90 deg

The sidereal time (right ascension on the meridian) is 19h 43m - only 7 minutes different from our estimate.

Looking East

The celestial equator meets the observer's local horizon exactly at an azimuth of 90 deg; this is always true, regardless of the observer's latitude:

The celestial equator always intersects the east and west cardinal points

At the intersection point, the celestial equator makes an angle of 50 deg with the local horizon; in general,

The intersection angle between celestial equator and horizon = 90 deg - Latitude

Our view of the eastern horizon at this particular time includes the Great Square of Pegasus, which is useful for locating the point of the vernal equinox. The two easternmost stars of the Great Square both lie on approximately 0 hours of right ascension. The vernal equinox lies in the constellation of Pisces about 15 degrees south of the Square.

Looking Up

From a latitude of 40 deg, an object with a declination of +40 deg will, at some point in time during the day or night, pass directly overhead through the zenith. In general

Declination at zenith = Latitude of observer

The 24 Ephemeris Stars in the SBO Catalog of Astronomical Objects have Object Numbers ranging from #401 to #424. Each of these moderately-bright stars passes near the zenith (within 10 deg or so) over the course of 24 hours. At any time, the ephemeris star nearest the zenith will usually be the star whose last two digits equals the sidereal time (rounded to the nearest hour). For example, at the current sidereal time (19h 43m), the zenith ephemeris star is #420 (d Cygni).

At this time of the year (late summer) and at this time of night (mid-evening), the three prominent stars of the Summer Triangle are high in the sky: Vega (in Lyra the lyre), Deneb (at the tail of Cygnus the swan), and Altair (in Aquila the eagle). However, the Summer Triangle is not only high in the sky in summer, but at any period during the year when the sidereal time equals roughly 20 hours: just after sunset in October, just before sunrise in May, and even around noontime in January (though it won't be visible because of the Sun).