Positional Astronomy

It is better, of course, to know useless things than to know nothing. -Seneca


This page collects my notes on the calculation and manipulation of astronomical coordinates.

A conundrum in writing a discussion like this is that it is hard to know where to start. In explaining one thing, you want to talk about something you have yet to define or to use terms you have yet to describe. I have done my best to avoid this, though I may have erred in places. Sometimes I use a term before defining it, and quickly follow up with some clarification.

Almost every topic being discussed has fine points and nuances that are not being discussed in depth. Some limitation of this sort is essential to allow the reader to get an overall perspective of the topics, which is the purpose of this discourse.

The Earth - coordinates on the globe

In performing astronomical calculations, it is vital to know where you are on the earths surface. The usual way of specifying this is via latitude and longitude, which should be reasonably familiar to anyone who has spent any time with a globe (model of the earth).

The line of zero longitude passes through both poles and through the Royal Observatory at Greenwich, England. In general, longitude is given as positive or negative values with magnitudes less than 180 degrees. Positive values are east of Greenwich, and negative values are west of Greenwich. Longitude in North America is measured from east to west in units of degrees. Since North Americans often think as if the rest of the world does not exist, they often will speak of a "longitude of 119 degrees, while they really should say either 119 degrees west, or negative 119 degrees longitude.

Latitude is measured from zero at the equator to 90 degrees at either pole. Latitudes north of the equator are positive or "north" latitudes; latitudes south of the equator are negative or "south" latitudes.

The pair of numbers, longitude and latitude, specify the location of a site on the surface of the earth. The coordinates of Tucson, Arizona are (longitude, latitude) -110.93 and +32.12.

The celestial pole

The earth of course rotates on an axis that emerges at the north and south poles. If this axis is projected into the sky, it defines the north and south celestial poles, and these are used as reference points for the coordinate system used to describe the position of the stars. The north celestial pole is fairly close to what we presently call the north star (polaris). It actually is not that close, it is about 0.7 degrees away; farther away on the sky than the moon is wide.

Measuring angles

Most folks use degrees to measure angles. Astronomers do too, sometimes. They also use hours, which can lead to some confusion. As every schoolboy knows, there are 360 degrees in a circle. When hours are used to measure angles, there are 24 hours in a circle. An angle in hours can be simply converted to degrees by multiplying by 15. Correspondingly time minutes and seconds differ from angular minutes and seconds by the same factor of 15. To attempt to avoid confusion, time minutes are just called "minutes" whereas angular minutes are called "arc-minutes". Similarly, time seconds are just called "seconds" whereas angular seconds are called "arc-seconds". There are 60 seconds in a minute, and 60 minutes in a hour. Similarly there are 60 arc-seconds in a arc-minute, and 60 arc-minutes in a degree.

Radians are also used to measure angles, but always in hidden places within software. There are 2 pi radians in a circle. Nobody who doesn't deserve to be severely punished ever lets radians escape from inside a computer program.

The stars - RA and Dec

The position of stars in the sky are specified by two values, in much the same spirit as longitude and latitude. The analog of longitude is Right Ascension, often abreviated "RA". The analog of latitude is declination, sometimes shortened to "dec". Declination is very much like latitude in that it is specified in degrees north or south of the celestial equator. RA is different than longitude, first of all because it is specified in hours rather than degrees, and secondly because it is measured continuously towards the east.

RA and dec serve to specify the position of objects on the celestial sphere. The celestial poles are projections of the earths axes of rotation. The celestial equator is the projection of the plane perpendicular to that rotation axis. The zero point for RA is known as "the first point in Aries", which is the place in the Sky where the sun crosses the celestial equator during the Spring Equinox. If you are new to all of this, it is perhaps best to accept this as some arbitrarily chosen point in the sky and to ignore the details for a while.

It is worth noting briefly a couple of fine points with regard to stellar coordinates. The first is that the stars themselves are in motion. For many purposes (and many stars) this can be ignored, but this motion is significant for stars fairly close to the earth. The change of the coordinates of a star with respect to time is called "proper motion" and has been determined and recorded for stars for which it is important.

The second point worth noting is that the reference points for the entire system of stellar coordinates is in motion (relative to the earth). This is because the earths rotation axis is moving relative to a system of coordinates fixed among the stars. This motion is called precession. The rotation axis of the earth sweeps out a cone with angular radius of nearly 23.5 degrees every 26,000 years. Precise astronomical calculations must account for precession, and stellar coordinates are given for a specific moment in time (most commonly the year 2000, specified as J2000), and may be adjusted, if desired, for the current time.

Polaris has an RA of 2:31:49.09456 hours and a declination of +89:15:50.7923 degrees in the J2000 Epoch. Notice the sexigesimal notation where RA is given in hours, minutes, and seconds and declination is given in degrees, minutes, and seconds. Also notice that RA minutes and seconds are units of time, different from the angular minutes and seconds given for declination.


Time is perhaps the most vital and important aspect of astronomical calculations. Civil time (also referred to as mean solar time) is what everyone is familiar with, being displayed on watches, cell phones, and computers. Sidereal time is what is important for astronomical calculations, and converting from civil time to sidereal time is a vital, complex, and interesting task. It is possible to talk about sidereal time at some specific location, such as the sidereal time at Greenwich. In general though, the only sidereal time of interest is the local sidereal time (LST), which is the sidereal time at the current location. The local sidereal time gives the RA of the object directly overhead at any moment. Unlike civil time, sidereal time ignores time zones and changes continuously with changes in longitude.

It is convenient to think of sidereal time as tracking the motion of the stars overhead. The local sidereal time gives the hour angle (see below) of the zero point for RA (the infamous "first point in Aries"). Of course what sidereal time really does is to track the rotation of the earth with respect to a reference frame fixed with respect to the stars (insofar as we can talk about such a thing). I say this because I expect some dunce to send me an email and point out that it is the earth that is rotating, and not the stars. It is really a question of choosing our frame of reference, and it is usually most convenient to fix our coordinate frame at the current location on earth and talk about the stars rotating, so I go ahead and do just that.

A sidereal day is shorter than a solar day (a solar day being what civil time deals with). This is rooted in the fact that the earth orbits the sun. Each night at the same solar time, we find ourselves looking at a slightly different view of the sky (being that much further along on our trip around the sun). If we were to go outside and carefully view the sky the same time on two consecutive nights, we would find that the stars had moved by about 4 minutes. After a year had gone by, we see the same view we had begun with. 4 minutes is 24 hours divided by 365. A sidereal day is 23.93447 solar hours.

Zenith and Meridian

The point in the sky directly overhead is refered to as the zenith. The line from the north pole through the zenith, and beyond is the meridian. Note that the meridian is a line of constant RA, but an ever changing value of RA as the stars move in the sky.

Hour Angle

Once we have chosen an object and specified a local sidereal time, it is convenient to talk about the hour angle of that object. The hour angle is generally given in time units (hours), not degrees. The hour angle indicates how far the object is from the meridian. Another way to think about it is that if the hour angle is positive, that is how long it has been since the object was on the meridian. If the hour angle is negative, that is how long we must wait until the object crosses the meridian. So, the hour angle is zero for an object on the meridian, positive for an object west of the meridian, and negative for an object east of the meridian.
Notice that hour angle and RA have opposite sign. HA increases for objects further west, whereas RA increases for objects further east.

Altitude and Azimuth

Another coordinate system, which is somewhat awkward for performing calcuations, but convenient for building large telescopes, is the system of altitude and azimuth. This system is centered on an observer and specifies a point in the sky from the observers point of view. Azimuth is the direction: East, West, North, or South, given as an angle from north and through the east. North is 0, east is 90, south is 180, and west is 270. Altitude is the angle from the ideal horizon (the true horizon is often cluttered with mountain, trees, and buildings) and up to the zenith. An object on the horizon is at an altitude of 0, and an object at the zenith is at an altitude of 90. Altitude is synonymous with elevation, and the terms are used interchangeably.

Given the RA and Dec coordinates of an object, the observers location in longitude and latitude, and the local sidereal time, it is possible to calculate the altitude and azimuth of an object.
This link gives details on the math required to do this.

Some telescope are built with altazimuth mounts. They have two axes of motion (azimuth and altitude) and these axes can be moved in such a way as to follow objects moving in the sky.

Parallactic Angle

Note that the word "parallactic" is somewhat of a misnomer, there is no relationship to parallax.

Care must be taken not to confuse this with the position angle.

To describe the parallactic angle we need to discuss two lines (actually great circles) in the sky. One is the line of constant RA which passes through the celestial pole and the object. The other is the line that passes through the zenith and the object. Notice that the second line is dynamic - as the object moves from east to west across the sky, this line constantly moves.

The parallactic angle is the angle between the constant RA line (circle) containing the object and the line (circle) containing both the object and the zenith. Notice that there is a mathematical singularity if the object passes through the zenith. This is resolved by defining the parallactic angle as zero when the object crosses the meridian. The parallactic angle of an object is constantly changing as the object moves from east to west.

The parallactic angle is important for imaging objects with a telescope with an altazimuth mount. If the telescope rotates the image it produces in synchronization with the parallactic angle, the image will be stationary on a detector.

Position Angle

Many objects in the sky (namely stars) have perfect circular symmetry and there is no top or bottom or orientation that we care about. Other objects, like planets, galaxies, and such (extended objects) have a definite top and bottom and it is desirable to specify their angular orientation in some way. This is done using the position angle.

Care must be taken not to confuse this with the parallactic angle. Position angle was first defined to specify the orientation of visual binary stars. Again, two lines (circles) on the sky must be defined. The first is the line of constant RA from the primary star to the north celestial pole. The second is the line from the primary star to the secondary star. It is positive from north through the east. It is the angle between the line from the primary star If we consider an extended object such as a planet or galaxy (as described above), we can call the

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Tom's home page / tom@mmto.org