ASTRO 801
Planets, Stars, Galaxies, and the Universe

Spectroscopic Parallax

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Because cluster stars are so homogeneous in their properties, we can often measure a few stars in the cluster and then generalize the result and apply it to the whole cluster. For example, you can measure the age of the cluster by estimating the age of the Main Sequence Turn-Off, as we just saw. Similarly, you can measure the distance to the cluster if you can find some technique to measure the distance to any single star. Since all measurements that you make have some error associated with them, though, if you measure the distance to one star, your estimate of its distance may be off by as much as 10% (the typical level of precision for many techniques). If you can measure many stars in the cluster and get many estimates of the distance, you can get a more precise estimate of the true distance by taking the average of all of the individual distance measurements. Even if all of the individual estimates are off by 10%, the measurement error associated with any one star becomes less important.

Astronomers often use a method of fitting the HR diagrams of clusters to a standard HR diagram in order to measure the clusters' distances more precisely, since this technique uses all of the stars to get a distance measurement. Here is another place where astronomical jargon can be confusing. Recall that the first successful method for measuring the distance to stars was trigonometric parallax. Because of this, the word parallax became used interchangeably with distance measurement. So, even though the term parallax should only refer to the method for measuring the apparent shift of a star, it is not used in that way only. This method of measuring the colors and brightnesses of many stars and comparing them to the colors and luminosities of a known set of stars is referred to as spectroscopic parallax.

You can create a “standard” HR diagram in a few ways. For example, you can theoretically calculate how bright and how hot the stars should be using mathematical models and plot those in a luminosity/temperature version of the HR diagram (an isochrone, just like we discussed previously). Also, if you can measure the distance to many nearby stars by the method of trigonometric parallax, you can convert the apparent brightness measurements for these stars into luminosities. Either way, you have created an HR diagram that shows luminosity on the y-axis. Now, if you measure the apparent brightness and color for many stars in your star cluster, you can plot these stars on the same diagram as your “standard” stars. Because all of the stars in the cluster are the same distance away from us, all of them will have an equal displacement along the y-axis. That is, the cluster's Main Sequence will just appear to be shifted vertically in the HR diagram from the standard stars, because the luminosity of a star drops off due to the inverse-square law for light. If you measure how much fainter you have to shift the entire set of standard stars so that they overlap the Main Sequence of the cluster, you can estimate how far away the cluster is using the relationship:

Fcluster = Lstandard / 4πR2

Click on the forward arrow in the animation below to see this demonstrated.

In this animation, we first see an HR diagram for a cluster plotted with luminosity on the y-axis. A blue region is fitted to the Main Sequence of this cluster, providing a calibrated Main Sequence with a known luminosity for all colors. Next, we see a new HR diagram for a nearby cluster with apparent brightness plotted on the y-axis. The calibrated Main Sequence is too bright, but if we shift it down, we can match it to the Main Sequence of the nearby cluster. This gives us an estimate of how much fainter the cluster is compared to the calibrator, which gives an estimate of distance. Lastly, we see the HR diagram for a more distant cluster. It is fainter yet, since it is more distant. However, by just offsetting the calibrated Main Sequence to fainter magnitudes, it matches the Main Sequence of the distant cluster, too, allowing us to measure its distance.

Credit: Penn State Astronomy & Astrophysics

You could apply this same method just using the measurements for one star (that is, measure Fstar and estimate Lstar from a standard Main Sequence), but since you are lining up the entire Main Sequence of the cluster with the standard HR diagram, you are using many hundreds or thousands of stars to calculate the distance, and the final result is much more precise.