## 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).

## Sagittarius Time Machine

The bright stars that outline our constellations beckon to us from a remarkably wide range of distances. Many of these stars are super-luminous giant stars and hot blue dwarf stars. More typical stars like our Sun—and the even more abundant red dwarf stars—are much too faint to see with our unaided eyes, unless they are only a few light years away. Thus many of the stars we see when we look up at the night sky are the intrinsically brightest ones, the “whales among the fishes.”

Trigonometric parallax directly provides us with the best estimate of the distance to each of these stars (provided they are not more than a few hundred light years away), and once you know the distance, it is easy to calculate when the light you are seeing tonight left each one of them. It is enjoyable to contemplate what was going on in Earth history when each star’s light began its long journey across interstellar space, the tiniest fraction of which is reaching your eyes as you look up on a clear night.

This article is the next in a series featuring the major stars of a prominent constellation. We turn now towards Sagittarius, which is currently crossing the celestial meridian at the end of evening twilight.

Below you will find a chart showing the constellation Sagittarius and the bright stars that define its outline. The official IAU-approved star names are listed, where available, or the Bayer designation. There’s a printer-friendly PDF version of this chart at the bottom of this article. There’s room for you to write in the year when the light we are currently receiving left the photosphere of each star, using the provided table (which is updated automatically to today’s date).

The table below contains all the relevant information. There are three tabs: Parallax, Distance, and Time. The first three columns of each tab show the star name, the Bayer designation, and the spectral type and luminosity class listed in SIMBAD.

On the Parallax tab, the parallax in millarcseconds (mas) is listed in column D, along with the uncertainty in the parallax in column E, and the year the parallax was published in column F. All are from SIMBAD. I will update these values as new results become available, but please post a comment here if you find anything that is not current, or is incorrect.

On the Distance tab, the parallax and parallax uncertainty for each star is used to calculate the range of possible distances to the star (in light years) in columns D through F. The nominal value given in column E is our current “best guess” for the distance to the star.

On the Time tab, the range of distances from the Distance tab are used to determine the range of years when the light we are seeing at this point in time would have left the star. The earliest year (given the uncertainty in parallax) is shown in column D, the most likely year in column E, and the latest year (given the uncertainty in parallax) in column F.

Here’s a printer-friendly PDF version of the Sagittarius chart where after printing you can enter the nominal year from column E of the Time tab next to the name for each star. The year values on the Time tab will update automatically to reference the current date.

## Scorpius Time Machine

The bright stars that outline our constellations beckon to us from a remarkably wide range of distances. Many of these stars are super-luminous giant stars and hot blue dwarf stars. More typical stars like our Sun—and the even more abundant red dwarf stars—are much too faint to see with our unaided eyes, unless they are only a few light years away. Thus many of the stars we see when we look up at the night sky are the intrinsically brightest ones, the “whales among the fishes.”

Trigonometric parallax directly provides us with the best estimate of the distance to each of these stars (provided they are not more than a few hundred light years away), and once you know the distance, it is easy to calculate when the light you are seeing tonight left each one of them. It is enjoyable to contemplate what was going on in Earth history when each star’s light began its long journey across interstellar space, the tiniest fraction of which is reaching your eyes as you look up on a clear night.

This article is the first in a series featuring the major stars of a prominent constellation. We turn now towards Scorpius, which is currently crossing the celestial meridian at the end of evening twilight.

Below you will find a chart showing the constellation Scorpius and the bright stars that define its outline. The official IAU-approved star names are listed, where available, or the Bayer designation. There’s a printer-friendly PDF version of this chart at the bottom of this article. There’s room for you to write in the year when the light we are currently receiving left the photosphere of each star, using the provided table (which is updated automatically to today’s date).

The table below contains all the relevant information. There are three tabs: Parallax, Distance, and Time. The first three columns of each tab show the star name, the Bayer designation, and the spectral type and luminosity class listed in SIMBAD.

On the Parallax tab, the parallax in millarcseconds (mas) is listed in column D, along with the uncertainty in the parallax in column E, and the year the parallax was published in column F. All are from SIMBAD. I will update these values as new results become available, but please post a comment here if you find anything that is not current, or is incorrect.

On the Distance tab, the parallax and parallax uncertainty for each star is used to calculate the range of possible distances to the star (in light years) in columns D through F. The nominal value given in column E is our current “best guess” for the distance to the star.

On the Time tab, the range of distances from the Distance tab are used to determine the range of years when the light we are seeing at this point in time would have left the star. The earliest year (given the uncertainty in parallax) is shown in column D, the most likely year in column E, and the latest year (given the uncertainty in parallax) in column F.

Here’s a printer-friendly PDF version of the Scorpius chart where after printing you can enter the nominal year from column E of the Time tab next to the name for each star. The year values on the Time tab will update automatically to reference the current date.