Physics & Astronomy Modules

# Celestial Coodinates

## Mapping the sky

Challenged to map the entire universe you might begin by establishing a reference system within which to locate everthing as you progressively add entries to the catalog. The potential frames of reference are not unique, independent or absolute, and contemporary physics informs us that any system we create is relative to some other. It follows that there are frames of reference attached to Earth, the Sun, the center of mass of the solar system, the solar system itself, our galaxy, and even a mean of all observable galaxies. The most widely used, and the basis for comprehensive modern catalogs of celestial objects, is a system in space that is oriented by the directions of Earth's axis and one in its equatorial plane, in the same way we find longitude and latitude. Since you may already know that Earth's rotation axis is not fixed in direction, you probably are thinking that such a system could not be ideal because it must change continously. Indeed, that's correct, and the solution is to accept the fact, and identify particular times or epochs at which to define the system. For example, the one in current use is for the orientation of the Earth at the beginning of year 2000. A sphere surrounding the Earth at that moment, its equator being an extension of Earth's equator, and its poles extensions of Earth's axis would look like this.

## Celestial coordinates

The plane of Earth's orbit represents for us the plane of the solar system, and in our sky we see its direction traced out by the path of the Sun over a year with respect to the stars. On this figure it is the yellow plane and its intersection with the celestial sphere is a great circle called the ecliptic. Earth's axis is inclined from the normal to the plane of its orbit by the mean obliquity of the ecliptic $\epsilon = 23^\circ\,26^\prime\,21.45^{\prime\prime} - 46.815^{\prime\prime} T \, -0.0006^{\prime\prime}T^2 +0.0081^{\prime\prime}T^3$ where $T$ is Julian centuries from 2000.0. The angle decreases slowly on a time scale of centuries and is usually desecribed as approximately 23.5 degrees, as shown in this graphic.

The Sun progresses along the ecliptic over the course a year and as seen from the northern hemisphere is below the projection of Earth's equator on the sky for half a year and above it for the other half. The crossing points are the equinoxes named for the beginning of the northern hemisphere seasons. The spring or vernal equinox in the sky is also an event when the Sun is at that point, within a day of March 21. That direction in the sky is reference direction, along with the direction of Earth's axis, that fixes the celestial coordinates for epoch 2000.0.

Ecliptic is the great circle arc on the celestial sphere that is the intersection of Earth's orbital plane with the sky. It is also the annual path of the Sun on the sky as seen from Earth's center.

Equator is the great circle arc on the celestial sphere that is the intersection of Earth's equatorial plane with the sky.

Celestial poles are the intersections of Earth's axis with the celestial sphere. Because of the rotation of the Earth, as seen from its surface the sphere appears to rotate around these points. In the northern hemisphere the star Polaris is only about $44^\prime$ (44 arcminutes or about 0.75 degree) from the north celestial pole at this time.

Declination is measured in degrees north (+) or south (-) from the celestial equator. Analogous to latitude on Earth, a star with declination 0 degrees is on the celestial equator, and a star with declination 45 degrees is half way from the equator to the north celestial pole.

Right Ascension is measured like a clock on the sky in hours toward the east from a reference celestial meridian. This meridian is set by convention at a point where the plane of Earth's orbit around the Sun intersects with the plane of its equator projected on the sky. There are two such points, and the one we use for the zero point of Right Ascension is where the Sun passes from below the equator to above the equator as seen from the center of the Earth. Since this happens in a northern hemisphere spring when days and nights have equal duration, it is the point that marks the vernal "equinox". Right ascension is analogous to longitude on Earth which is fixed with respect to Earth's prime meridian through Greenwich, England.

Zenith is the direction normal to Earth's local surface looking straight up. Its complement is the nadir, the direction directly under your feet.

Elevation also sometimes confusingly called "altitude" is the angle from the horizon the object. An elevation of 90 degrees puts it overhead.

Azimuth is the angle around in the horizontal sense, measured from due north. A azimuth of 0 degrees is north, 180 degrees is south, 90 degrees is east, and 270 degrees is west. Elevation and azimuth determine the direction to an object in the local sky at that moment. Because of Earth's rotation, these coordinates are constantly changing and an Earth-based telescope must continually update the direction in which it is pointed, its elevation and azimuth pointing, to track a celestial object.

Meridian is a great circle arc from pole to pole. The local meridian goes through the zenith and the passing of a star across the meridian marks its highest elevation in the sky at that site.

Hour angle is the time elapsed after an object passes through the local meridian. At 0h the object is on the meridian, and when the hour angle (usually "HA") is positive the object passed the meridian that number of hours earlier. A negative HA means the object has not yet reached the meridian.

Earth's axis slowly precesses in space, approximately rolling around a perpendicular to the plane of its orbit, creating a virtual cone in space. The sense of rotation seen from above looking down on it is clockwise, opposite the sense of Earth's daily rotation. The precession period is 26,000 years, over which the point of the vernal equinox moves around the celestial equator one full turn, or it makes 1/26,000 of a turn per year. That's more than 1 degree per century. The places in the sky marking the north and south celestial poles will trace out a full circle over 26,000 years, centered on the normal to the plane of the solar system and with an angle of about 23.5 degrees to it that is the tip of Earth's axis from the perpendicular to its orbital plane.

If we look up at the ecliptic pole, the circle over this 26,000 years is progressed in a counter clockwise sense. A bright star we might call the "north" or "pole" star marking this point will change with 26,000 year cycle.

Consequently the celestial coordinates of any star will depend on the year the reference for the equinox is set. It may be convenient to set a specific year close to the one for which measurements are made for high precision data, or to refer positions to a standard epoch. Currently this is year 2000, which replaced 1950. When giving coordinates, it is essential to state the epoch.

## Orbits, Ephemerides, and Proper Motion

Everything is in motion, and the Sun's motion through space, as well as the motion of stars or any object we look at, alters where it will appear in the sky.

The objects in our solar system, Earth included, move on paths determined by gravity and the laws of motion. If we know where they are now and how they are moving at this moment, then from their mutual gravitational interactions we can compute where they will be and how they will be moving at any time past or future. The parameters that enable that modeling are called the orbital elements, and they are known for all identified bodies in the solar system, natural and man-made. Given the orbital elements we can predict events, and timing is termed an "ephemeris". From these we know where the Earth will be at any time, how it will be oriented in space, and where any object of interest will be too. The calculation of the direction to look, that is its the instantaneous celestial coordinates, is a matter of vector algebra. It is complex for objects which are nearby and have a fast apparent motion on the sky, while for distant objects it is straightforward because the motion on the sky is nearly linear in space and time. If we take the Earth's motion out of the problem of where to look at a distant object typically beyond the solar system, then we would need to know the intrinsic motion of that object due to its velocity in space, and how far it is from us. The resulting motion on the sky each year is its "proper motion".

The small yearly proper motion of our nearest stellar neighbor, Proxima Centauri, is $-3.781^{\prime\prime}$ in right ascension, and $+0.770^{\prime\prime}$ in declination is discernable with respect to more distant stars over a few years. However, since all other stars are more distant than Proxima, the proper motion of stars is often disregarded over short time intervals and we may speak of them as "fixed", but they are not.