## Spectroscopic Parallax

For the nearest stars, the change in the position of the Earth in its orbit results in a tiny shift in the position of the nearby star relative to the distant background stars. This shift is called the trigonometric parallax. You can see the effect by holding your thumb up at arms length, closing your left eye, and lining up your thumb with something across the room. Now, alternate back and forth between having your right eye open and your left eye open and you’ll see the position of your thumb shift relative to an object further away. Move your thumb closer, and the shift is larger. That is the essence of trigonometric parallax.

Trigonometric Parallax

The distance to the star in parsecs (1 pc = 3.26 ly) is just

Now, a star’s distance, apparent brightness, and “true” (or intrinsic) brightness are related in the following way:

M = m + 5 (1 – log d)

where M = the absolute magnitude of the star

and m = the apparent magnitude of the star

and d = the distance to the star in parsecs

The absolute magnitude is the apparent magnitude the star would have if it were at a distance of 10 parsecs. Looking at it another way, the absolute magnitude is a proxy for the intrinsic brightness. The apparent magnitude is the star’s apparent brightness (as seen from Earth).

While the above equation is highly useful for general purpose calculations, to get the most accurate values astronomers must take into account atmospheric and interstellar extinction. And, anytime we deal with a star’s luminosity and its apparent brightness at some distance, d , we must specify the photometric system and optical filter that is being used. Or, less commonly (for practical reasons), we specify that the star’s luminosity and apparent brightness is to include all wavelengths of the electromagnetic spectrum, thus bolometric magnitudes are to be used.

Spectroscopic parallax is a bit of a misnomer, but here’s how it works for approximating the distance to main-sequence stars that are too far away to exhibit a measurable, reasonably certain, trigonometric parallax: measure the apparent magnitude of the star, and then using its spectrum to find its position on the H-R diagram, read off its absolute magnitude. Using your measured apparent magnitude and the star’s estimated absolute magnitude, you can solve for d the distance in the above equation.

Hertzsprung–Russell (H-R) diagram

The star’s color (the x-axis on the H-R diagram) is easy to measure, but a deeper analysis of the spectral lines is needed to determine whether the star is a main-sequence, giant, or supergiant star (or something else).

## Apparent Magnitude, Absolute Magnitude, and Distance

A simple equation relates apparent magnitude, absolute magnitude, and distance.  Know any two, and you can calculate the third.

Known: Apparent Magnitude (m), Absolute Magnitude (M)
Unknown: Distance (d), in parsecs Known: Apparent Magnitude (m), Distance (d)
Unknown: Absolute Magnitude (M) Known: Distance (d), Absolute Magnitude (M)
Unknown: Apparent Magnitude (m) If this were a perfect universe, the known quantities could always be measured as precisely as one desires.  But, of course, that isn’t the case.

Apparent Magnitude – if the observations are made from the surface of the Earth, atmospheric reddening and extinction (atmospheric r/e) must be taken into account to determine the apparent magnitude above the Earth’s atmosphere.  Even above the Earth’s atmosphere, cosmic reddening and extinction (cosmic r/e) must also be quantified.  Both atmospheric r/e and cosmic r/e1 cause the observed apparent magnitude to appear fainter than it otherwise would be, and bluer wavelengths are more severely affected than redder wavelengths.  The net result is to make objects appear fainter and redder than they would be if there were a perfect vacuum between source and observer.

Absolute Magnitude – is a measure of the intrinsic brightness of a celestial object, and this can only be measured indirectly for objects outside of our solar system.

Distance – is difficult to measure for objects outside of our solar system.  Trigonometric parallax gives the most accurate results for nearby stars, but uncertainty increases rapidly with increasing distance.

Apparent magnitude is the only one of these quantities that is a direct instrumental measurement: absolute magnitude and distance are determined indirectly and thus are subject to greater uncertainty.

`1Atmospheric reddening and extinction (atmospheric r/e) is traditionally called atmospheric extinction, and cosmic reddening and extinction (cosmic r/e) is traditionally called interstellar reddening.  Since in both cases light is both reddened and diminished in intensity, and because "cosmic" encompasses both interstellar and intergalactic matter between source and observer, I suggest here that atmospheric r/e and cosmic r/e might be an improvement in terminology.`