What is an angle?

When studying atmospheric optics, and astronomy in particular, it is customary to think of the surroundings (sky, soil, etc.) being a giant sphere with directions. Directions on this celestial sphere are given by angles. An angle can be thought of as a measure of rotational distance.

Example: when you turn up your stereo, you rotate the volume knob over a certain angle. You can't say that you rotate it over a certain distance, because with rotations, distance becomes a function of the distance to the point of rotation itself.

Rotations in science are measured in radians, where 6.28 radians are a full circle (2pi); more common is the degree, where 360 degrees are a full circle.

It is very important to understand the concept of angles in order to understand the science of atmospheric optics, especially halo phenomena, which are always measured in degrees.

Note: the degree measuring angular distance is unrelated to degrees of temperature, freedom, education, deterioration, etc., although the word has the same meaning: degree as in the amount of rotation.

The vectors (arrows) are at angles of 60° to each other, as the full circle measures 360° and there are six such equal angles.

The celestial sphere

The celestial sphere is an imaginary sphere with unspecified radius, centered on the observer (you). It moves while you move, and consequently, the direction-vectors of things around you (trees, cats, clouds, the sun) move over the sphere (if they don't already to so by their own movement). Some things are really far away, like the sun, and appear to be fixed to a certain direction.

This may sound all very trivial. But try to think in this abstract way, if you want to study atmospheric optics. All halos, rainbows, sky polarizations etc. etc. are given by directions in the sky or on the ground, or better put, directions on this celestial sphere, where the sky and ground are part of. The idea of directions is certainly not trivial!

Example: this is why a rainbow seems fixed in space in a rainstorm, even though the rainstorm may move. The rainbow is seen in a certain direction on the sphere, which only depends on your position and the position of the sun in the sky. If you move, the rainbow will move with you, and if the rainbow stays long enough in the sky for the sun to move appreciably, it will move according to the sun's position, to stay centered around the antisolar point (which is below the horizon). If there is no rainfall at the rainbow's required position, you will not see the rainbow.

Spherical coordinate system To describe the actual position of an object in three-dimensional space, you have to know three numbers. Many of us are used to rectangular coordinates, but in atmospheric optics and astronomy the coordinates of an object in the sky are more conveniently given by: the elevation angle1, the azimuthal angle and the distance. Elevation angle: angle of the object above the horizon (also referred to as altitude). Objects below the horizon have this angle negative. The range is -90° to +90°. Azimuthal angle: ''horizontal'' angle of the object, measured from a certain reference point on the horizontal plane (usually this is true north, although sometimes the point on the horizon directly below the sun (above if the sun has set) is used). The range is 0° to +360° (360° and 0° are coincident points). Distance (or, better put, radius): the distance the object is away from you, as measured along the direction given by the azimuthal and elevation angle. This reference system is called a spherical coordinate system and is of crucial importance describing positions of atmospheric halos and celestial bodies in the sky.

The celestial sphere as used frequently in atmospheric optics to describe the directions of light sources, halos and other phenomena.

The figure above shows this imaginary celestial sphere with reference points, arcs and planes drawn into it. We have:

• Observer: this is you, at the center of the sphere, making an observation.

• Horizontal plane: this plane is almost coplanar with the horizon when an observer is earthbound. It has an elevation angle of 0°.

• Wind directions: north (0° azimuth), east (90° azimuth), south (180° azimuth) and west (270° azimuth) define four reference points on the observer's horizon.

• Zenith: the point directly above the observer, with elevation angle +90° (azimuthal angle is undefined for this point).

• Nadir: the point directly below the observer, with elevation angle -90° (azimuthal angle is undefined for this point).

• Solar point: the point on the celestial sphere where the sun is.

• Antisolar point: the point on the celestial sphere directly opposite to the sun (angular distance of 180°). Also called the opposition point.

• Subsolar point: the point below the sun, given by the solar azimuthal angle and negative solar elevation angle.

• Anthelic point: the point opposite to the sun but at the same elevation angle (i.e. 180° difference in azimuth).

It is important to realize that always two of the latter four points lie above the horizontal plane. Never more, and never less (except when the sun is on the horizontal plane, in which case the four points form coincident pairs). But, more than two of these points may still lie above the observer's horizon! If the observer is on a high mountain, or observing from space, earth's horizon does not coincide with the horizontal plane anymore, since the earth is curved. So, there is a difference between horizon and horizontal plane, which becomes more pronounced if you observe from a higher altitude.

The solar, subsolar, antisolar and anthelic points may lie above or below the horizontal plane, depending on the elevation of the sun. At night, the subsolar and antisolar points lie above this plane. For example, gegenschein, also called the opposition light, is centered on the antisolar point and is visible in dark nights. Other example: during a lunar eclipse, the moon is very near or on the antisolar point, since this is where earth's shadow cone is directed to. Hence, a lunar eclipse is only visible when the sun is at or below the horizon for an earthbound observer, and is always full.²

1. In mathematics and physics, the zenith point actually has a polar angle of 0° and the nadir 180°; but in atmospheric optics and astronomy it is more useful to use the elevation angle instead of the polar angle. Both measure the same.

2. Due to atmospheric refraction of light, which bends light rays downward about 0.5° for objects on the horizon, it is possible for an earthbound observer to look beyond the horizon and observe both the sun and the eclipsed moon at the same time.