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 magnitudedistance 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 magnitudedistance
formula to measure that star's distance.
Magnitudedistance formula
This formula relates the apparent magnitude mv, the absolute magnitude Mv,
and the distance in parsecs.
mvMv = 5 + 5(log10(d)
rearranging the formula yields:
d = 10^{(mvMv+5)/5}
once again, distance is measure in parsecs (3.26 light years in one parsec).
Three examples of using the magnitudedistance 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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