Spherical Astronomy without Trig

© 1996 by Robert C. Moler

Spherical astronomy concerns the directions of celestial objects, and uses the concept of the celestial sphere. There is of course the heavy use of spherical trigonometry in most calculations. Those of us who sweated through plane trigonometry in highschool probably just touched spherical trig. However, there are some valuable spherical astronomy relationships that can be calculated without heavy math.

The sky appears as a dome over the earth. If one includes what is below the horizon, it becomes a sphere, what we call the celestial sphere. The earth too is a sphere (or nearly so) which we can imagine is located at the center of the celestial sphere. Spherical astronomy is concerned with angles. A complete circle contains 360 degrees. The sperical analog of the straight line is the great circle, which if it were part of a plane, would cut the sphere in half. It is the shortest spherical distance between two points.

We locate positions on the earth by their longitude, or east-west position and latitude or north-south position. Latitude is fairly straight forward in that the 180 degree half circle from the north to the south pole is measured from +90º to -90º. The great circle equidistant from the poles at 0º is the equator. Longitude is much more arbitrary. There is no natural east-west starting point, so when a method for determining longitude was discovered, each country used its capitol city or the observatory at that city as longitude 0. When everyone got together to standardize on a single longitude 0, that of England's Greenwich Observatory was chosen. Lines of equal longitude are great circles that intersect at the poles and are called meridians. Lines of equal latitude are sometimes called parallels and are all small circles except for the equator.

Celestial coordinates are a projection of the earth's coordinates onto the celestial sphere. The names have been changed to confuse astronomy 101 students. Earthly latitude is declination on the celestial sphere. Longitude becomes right ascension, and to be even more confusing is measured in hours rather than degrees.

The celestial sphere from our location appears tilted, because we are at some distance from the north pole. For the purposes of this article I will assume Traverse City's latitude is 44 degrees. It is actually closer to 45° , but since 45 is half of 90, it isn't as good for illustration purposes.

The diagram right shows the earth within the celestial sphere. If Traverse City at 44°  north longitude, is at the top of the earth the horizontal ellipse is the horizon, a great circle. The celestial equator is at 90°  from the earth's axis. The star symbol represents Polaris the north pole star, which really isn't exactly at the pole, but we will assume here that it is. The angle Polaris makes with the horizon is L the same as the location's latitude. The great circle line from the north compass point through the celestial pole and the zenith and down to the south compass point is called the meridian, as if it were our own personal meridian of longitude passing overhead. So for us L is 44°. C is the angle from the celestial pole to the zenith. It is called the co-latitude and is 90° - 44°  = 46°. As can be seen by the diagram the altitude (angle with respect to the horizon) of the celestial equator as it crosses the meridian is equal to the co-latitude.

The celestial equator is special in that it intersects the horizon at the east and west compass points.

The sun appears to move on a great circle that is tilted to the celestial equator by 23.5°. It is not shown in the diagram to reduce clutter. However On March 20th the sun is on one of the intersection points with the celestial equator called the vernal equinox. So it rises due east reaches 46°   altitude at local noon, and sets due west. On June 20th the sun is farthest north at +23.5°   declination; an event called the summer solstice. At local noon on that day it will be 46 + 23.5 or 69.5°   altitude. On September 22nd, autumnal equinox, the sun will be back on the celestial equator. On December 21st, the winter solstice, the sun will be the farthest south and its altitude at local noon will be 46 - 23.5 or 22.5°, quite a change from June.

At every location on the earth, with the exception of the equator, there is a section of the sky which is always visible, and another section of sky that is always hidden. These are the circumpolar areas, where stars never rise or set, but seem to move around and around the pole of the sky. Since the pole is at an altitude of one's latitude above the horizon, it makes sense that the circumpolar areas are within one's latitude in degrees from the pole, or they have a declination above one's co-longitude. Thus taking our example of Traverse City's latitude of 44°  (It's actually 44°  45'), everything in the sky with a declination of 90 - 44 = 46°  never sets. Conversely anything with a southern declination from -46°  to -90°  never rises for us in Traverse City. These areas are marked CP in the diagram at left.

If you were transported to the north or south pole of the earth, then all the stars would be circumpolar. Any star you saw would always be up. Any star below the horizon would never rise.

Let's take a look at some first magnitude stars that are or nearly are circumpolar here in Michigan.. Capella in the constellation Auriga the charioteer has a declination of +46°  Making it circumpolar for latitudes greater than 44°  north. That is north of a line from Ludington to Standish. Deneb in the constellation Cygnus the swan has a declination of 45º 16' which makes it circumpolar for locations north of 44°  44'. Compare that with Traverse City's actual latitude.

Capella is a winter star, and I remember spotting it as it began to rise in the north northeast while driving home after midnight from summer viewing nights at the Lanphier Observatory in Glen Arbor. It's a rather stark reminder that summers are short, and winters long here.

Now let's look at locations at opposite parts of the celestial sphere. When a planet is opposite the sun in the sky, it is said to be in opposition from the sun. This is also true of the full moon. The planets and moon stay near the ecliptic, the great circle that is the apparent path of the sun in the sky. When a body is at opposition from the sun it will rise when the sun sets, and set when the sun rises. The body at opposition from the sun also has the opposite declination than the sun. The full moon in summer has a southern declination. Therefore the summer full moon lies low in the south, where the sun was 6 months earlier. The winter full moon rises high, where the summer sun was.

Now let's look at effects that center on the rising and setting of bodies that are located near the equinox points of the sky, the points where the ecliptic crosses the celestial equator. Let's concentrate on rising phenomena in the diagrams at right. If the full moon is opposite the sun in the sky, and its travel is along the ecliptic; We can see why the Harvest Moon rises at much less than the 50 minutes later rise time per day average. Remember that the Harvest Moon is the nearest full moon to the Autumnal Equinox (That's where the sun is). So the moon is near the Vernal Equinox point, and its motion is at a shallow angle to the horizon. In the calendar check out the full moon rising intervals near the vernal and autumnal equinoxes.

Questions? Send Email to me at bob@bjmoler.org

Updated: 08/11/00