Spectroscopic Parallax

The discovery of a relationship between the spectral class of a star and its absolute magnitude was a tremendous moment for astronomers. Now, if an astronomer could deduce the spectral class of a particular star, the next step would be to compare that spectral type on the HR Diagram to its absolute magnitude. Then it is a simple matter of applying the magnitude-distance formula to the star to determine where it is.

A simplified version of the HR Diagram is seen above. It can be readily seen that there is a relationship between a star's spectral class and both its luminosity and its absolute magnitude. Because the parallax measurements of stars within a 500 light year radius from our Sun were so accurate, we were able to test whether the relationship between their spectral class and absolute magnitudes was precise enough for our study, and it turned out to be so. Now, if we can take a photograph of a distant star and determine its spectral class, we have a means of determining its absolute magnitude. If we know the absolute magnitude, we can visually determine its apparent magnitude and utilize the magnitude-distance formula to measure that star's distance.

Magnitude-distance formula

This formula relates the apparent magnitude mv, the absolute magnitude Mv, and the distance in parsecs.

mv-Mv = -5 + 5(log10(d)

rearranging the formula yields:

d = 10^{(mv-Mv+5)/5}

once again, distance is measure in parsecs (3.26 light years in one parsec).

Three examples of using the magnitude-distance formula are given below:


SPECTROSCOPIC PARALLAX EXAMPLES

So, the technique of "spectroscopic parallax" uses the relationship between a star's Absolute Magnitude (Mv) and spectral class (as learned from the HR Diagram). The first step is to get the spectral "signature" of a particular star of interest. Let's say that you find a nice yellow star like the Sun in some evening constellation. Tau Ceti is just such a candidate, and it in the constellation Cetus, but is is not very bright. With a spectral classification of G8, it shows an Absolute Magnitude (Mv) on the HR Diagram of We can know the absolute magnitude from the HR diagram of 5.7. Actual photometric measurements of this star give an Apparent Magnitude (mv) of 3.5. Now we just pllug in the numbers to get the distance:

d = 10^{(3.5 - 5.7 + 5)/5} ... which is 10^.56 ... which is 3.63 parsecs. Multiply 3.63 parsecs times 3.26 light years per parsec, gives a distance of 11.84 light years to this star.

The HR Diagram is even useful for large stars like Deneb. This is the bright star in the constellation Cygnus, and Deneb is a Yellow Super Giant. With a spectral classification of A2, you need to look at the Super Giant clump near the top of the HR Diagram to find the Absolute Magnitude (Mv) of -7.1. (I am using exact values, but you can use estimates for later exercises since it is the principle that I want you to know). Photometric measurements of Deneb give an Apparent Magnitude (mv) of 1.26. Now just plug in the numbers:

d = 10^{1.26-(-7.1) + 5}/5} ... which is 10^13.36/5 ... which is 10^2.67 ... which is 469.9 parsecs ... which becomes 1531 light years!

Thus, astronomers who use Spectroscopic Parallax are not really measuring any parallax angles, but using properties of spectral classification and absolute magnitude to determine the distance to stars. The fact that this method matched well with stars whose actual parallax angles are known lends strength to the use of this method.

The use of spectroscopic parallax helped astronomers map the stars in the Milky Way and thus give us a 3D map of our galaxy without sending a satellite up into the space above the plane of the Milky Way (a journey requiring a ludicrous number of years).

To measure the distance to other galaxies precisely, we need to know the spectra of individual stars within them, and this becomes incredibly difficult for most galaxies because our telescopes have trouble isolating individual stars in distant galaxies. Henrietta Leavitt discovered an interesting group of stars which changed in brightness on a repeated basis and gave us a tool to measure things that are even farther away.

To learn about the use of Standard Candles, please move on now to Cepheid Variables, or return to either Measuring Star Distances, or go to the Syllabus .


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