Stellar measurement

Distances
The oldest method of measuring the distance from our solar system to a distant star is the parallax method. To use this method, astronomers measure the right ascension on the sky of the star at two times of the year, half a year apart. The two measurements will differ by a small angle with respect to the most distant stars in that region of the sky. Exactly half this angle is the parallax angle, having symbol p. This is the angle that the star makes with the sun and the position of the earth at a right angle with that star. The distance s of the star, in astronomical units (AU), is:

$$\,\!s = \cot p$$

In the range of the very small angles typically encountered, the cotangent of the angle measure (in radians) is very nearly equal to the reciprocal, and thus:

$$\,\!s \approx \frac {180 \times 3600}{p \times \pi}$$

where p is measured in seconds of arc.

The cotangent of one second (1/3600 of a degree) of arc is approximately 206,264.81. No parallax angle for any star will be larger than one second. Therefore astronomers initially defined a unit of stellar distance, the parsec (symbol pc), from this relationship. One parsec is the distance corresponding to a parallax angle of one second of arc. Hence:

$$1 pc \approx 206,264.81 AU$$

or about 3.3 light years.

However, the error of measurement of parallax angle is 0.005 arc seconds, and beyond a distance of 100 parsecs, this error becomes significant. 700 stars are near enough to measure their distances directly by using parallax. To measure distances further out than this, astronomers typically use absolute and relative magnitudes, or they apply Hubble's Law to the star's estimated redshift.

Positions and movements
The most common system for describing the position of a star in the sky is the equatorial system. This system uses two coordinates:
 * 1) Right ascension on the sky, or the number of hours required for the earth to rotate before an observer can see the star at its highest point in the sky. The zero for right ascension is midnight on the day of the vernal equinox.
 * 2) Declination, or the north-south angle between the star and the celestial equator.

All stars move, but the most distant stars are considered "fixed" because their motion would be undetectable. The proper motion (symbol m) of any star is the angular velocity of its position across the sky. This describes the motion at right angles to the line of sight of the observer. To convert this to actual tangential velocity, multiply the tangent of this angular velocity by the star's distance.

The motion in line of sight, or radial velocity, is currently determined from spectral shift.

Magnitudes
The visual magnitude system is defined as follows: a star of any given magnitude is about 2.512 times as bright as is a star of the next magnitude. Hipparchus devised the magnitude system, and Ptolemy refined it further. By convention, an arbitrary sample of the twenty brightest stars that they could observe were assigned to the first magnitude, and the stars that they could barely observe were assigned to the sixth. Sixth-magnitude stars are actually 100 times less bright than first-magnitude stars. Magnitude levels between these extremes are assigned on a logarithmic scale. Thus, given two stars of brightness l1 and l2, their magnitude difference (V2 - V1) relates to their respective brightnesses in this way:

$$\,\!V_2 - V_1 = 2.5 \times \log \frac{l_1}{l_2}$$

The absolute magnitude of any star is the visual magnitude that it would have if it were ten parsecs distant. To convert apparent magnitude V to actual magnitude M, use this formula:

$$\,\!M = V + 5 \times \log \frac{s_0}{s}$$

where s0 is the standard distance. This distance is ten parsecs, or about 2,062,650 AU.

Brightness declines with the square of distance, and squares correspond to doubling of logarithms. One must then multiply that result by 2.5 to stay within the magnitude scale.

Colors and spectra
The color of a star is objectively quantifiable. To determine color, astronomers view the star through a variety of colored filters and compute color indices as the differences in apparent magnitudes through the various filters. Stellar colors vary, in order from the coolest to the hottest, from red to yellow to white to blue-white to blue or violet. This is the same gamut of colors that a black body shows as its temperature rises.

In addition, each star has a unique spectrum, which depends on the gases and other elements that it contains, and their distribution. A spectrum can serve two purposes:
 * 1) It can serve as a unique signature for the star, to distinguish it from other stars.
 * 2) It can provide information on the star's radial velocity vis-à-vis the earth.

To accomplish the latter, astronomers note the placement of various lines in the spectrum and then determine the star's likely constituent elements from the spacing of those lines. Lines that are out of place are shifted, either toward the blue or toward the red. Nearly all stellar spectra are shifted toward the red; this redshift indicates a recession, either of the star or of the part of space where the star resides.