Previous: Altitude-azimuth coordinate system

This section refers to coordinate systems designed to find objects in the night sky. Such coordinate systems must indicate the position of an object in the sky, but not its distance from Earth. Therefore, we can imagine that astronomical objects are all located on the walls of a **celestial sphere**, as the ancient Greeks imagined.

To define these coordinate systems, it is important to keep in mind that, due to the motion of Earth, objects in the sky are changing position constantly.

**Equatorial Coordinate System**: It is similar to the *Altitude-Azimuth coordinate system*, except that it based on the *celestial equator* and on the *vernal equinox*, and it is independent of the latitude and longitude of the observer. It is also independent of the annual motion of Earth around the Sun.

It consists of two coordinates:

**Declination** = δ, analogous to *latitude*, measured in degrees north (positive) or south (negative) of the *celestial equator*.

**Right ascension** = α, analogous to *longitude*, measured eastward along the *celestial equator* from the *vernal equinox*, Υ, to its intersection with the object’s **hour circle** (the circle passing through the object and the north celestial pole). It is measured in hours, minutes, and seconds (24 h = 360º).

Things affecting the *equatorial coordinate system*:

**Precession**: It is the slow change in the inclination of Earth’s rotation axis due to its non-spherical shape, and its gravitational interaction with the Sun, the Moon, and the other planets. It makes the north celestial pole make a slow circle through the heavens. Precession occurs very slowly, over a period of 25,770 years. This causes a 50.26”/year westward motion of the*vernal equinox*along the*ecliptic*(” = seconds). An additional precession caused by the planets causes also an eastward motion of the*vernal equinox*of 0.12”/year.

**Motion of the heavenly objects**: The stars (or other objects) are moving at great speed one with respect to each other. Whereas their apparent relative motion as seen from Earth is extremely slow due to the huge distances between them, it still happens.

– Calculate the effect of **precession**:

Values of α and δ need to be referred to a specific reference time, then the current values can be calculated. They are given by:

For the 1st January 1950, reference values are: *m* = 3.07327^{s}/year, and *n* = 20.0426”/year.

– Calculate effect of the **motion of the heavenly objects**:

A star’s (or any other object’s) velocity vector has two components:

**Radial velocity**,**v**_{r}, the component in the direction of the line of sight.

**Tangential/transverse velocity**,**v**_{θ}, component perpendicular to the line of sight. This component appears as a slow angular change in equatorial coordinates, known as**proper motion**,*µ*, and expressed in seconds of arc per year.

We can calculate the displacement of the star after a given time:

The angle corresponding to its change in position on the celestial sphere:

And the proper motion:

Note#1: *r* is the distance to the star from Earth.

Note#2: vectors are represented in bold letters. This is standard notation.

**Conversion from angle displacement to equatorial coordinates:**

The starting point and the final point of the displacement, combined with the north celestial pole, form a triangle projected onto a sphere. The simplest way to obtain the new *equatorial coordinates* is to assume that the change in position is so small, that we can approximate that triangle as being flat.

The final relations are:

To reach this result we had to use:

- The laws of sines, cosines for sides, and cosines for angles.
- The approximation that the triangle is flat. This means: sin
*e*~*e*and cos*e*~ 1.

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Would you please explain the term ‘first point of Aries (Y)’ or in your explanation the ‘vernal equinox’ ? Does this point supposed to be constant always?

Thanks for the question, Faisal! This is an important point that I didn’t include in the post.

The vernal equinox is one of the two times during the year where the Sun has a declination of 0º in the Sky, as seen from Earth’s equator. The other being the autumnal equinox.

The vernal equinox as well as the autumnal equinox are in fact not constant.

Now when I say that the vernal equinox is not “constant” what I mean is that it is not always pointing at the same point of the celestial sphere, it changes. In particular, according to this article, the “first point of Aries” (= vernal equinox) was pointing to the constellation Aries some 3 thousand years ago, then crossed into Pisces in about 70 BCE, and it will cross into Aquarius in the 2600 CE.

This is due to the “wobbling” of the Earth’s rotational axis with respect to the Sun. Imagine the Earth moving like a spinning top. This kind of motion is called “precession”, see wikipedia.

The period of this motion, however, is very long, of approx. 26,000 years. This would be the time it takes for the vernal equinox to go all over the 12 zodiacal constellations, so we’ll have to wait a little while to have the vernal equinox pointing to Aries again!

There are even

moremotion modes of Earth, which I didn’t go into.Hope this helped 😀