Variable Stars in M15
One of the most common types of variable stars—those stars that fluctuate in brightness over regular time intervals—is the RR Lyrae class of variables. Most of the RR Lyrae stars are characterized by periodic changes in brightness with cycles less than 24 hours long. In this exercise we will use the regular variation in brightness of six RR Lyrae stars to determine the distance to the globular cluster in which those stars are located. This technique of determining distance by the use of a standard candle and an equation called the distance modulus is one of the most powerful tools in the astronomical arsenal.
Photo courtesy NOAO/AURA/NSF
By studying eight photographic negatives of the globular cluster M15 (shown at right), we will be able to compare the six RR Lyrae stars to other non-varying stars in the same photograph whose apparent magnitudes are well known. By carefully comparing each RR Lyrae star to a marked non-varying star that is brighter than the RR Lyrae star and to another non-varying star that is dimmer than the variable, we can interpolate the value of the RR Lyrae star's apparent magnitude (m). Eventually, a calculation using the accepted absolute magnitude (M) of all RR Lyrae stars will allow us to determine the distance to the cluster.
Several key points are important to remember when working on this exercise:
- The photographs are negatives. This means a darker stellar image actually corresponds to a brighter star. In other words, the darker the star's image, the smaller its magnitude.
- The RR Lyrae stars are labeled A through F and are the same in each photograph. They may change brightness but the stars are the same stars. They simply vary in brightness on different cycles.
- The non-varying stars used for comparison with the RR Lyrae stars are labeled with numbers (for example, 156). That number is the star's apparent magnitude without the decimal point. It doesn't take much of a stretch of the imagination to realize that a decimal point on a photographic negative looks amazingly like a star. Therefore, in our previous example, that star's apparent magnitude is m = +15.6 .
- When trying to figure out that apparent magnitude of the RR Lyrae stars, try to use stars nearby on the same photograph for your comparison. Nearby stellar images are less likely to suffer from errors in photographic development or changes in the clarity of the atmosphere.
- On some photographs, the non-varying stars are not labeled. Remember that the apparent magnitudes of these stars are the same even though the numbers are missing. Simply find the unmarked stars and copy their apparent magnitudes to the unlabelled photographs.
- DO NOT use the stellar images of the non-varying stars on a nearby, labeled photograph to estimate the apparent magnitudes of the RR Lyrae stars on the unlabeled photo. The development procedure and changes in cloudy cover can cause enormous errors to occur when you compare stellar images from different photographs. For example, the best thing to do is copy the numbers from photo 468 to photo 465 before comparing the RR Lyrae stars in 465 to the non-varying stars.
Procedure
Look at photo 468 (the second photo from the left in the top row).
Note that RR Lyrae star C is slightly below the center of M15 and surrounded by non-varying stars of apparent magnitudes 150, 154, 157, 160, and 164. After a careful examination of the labeled stars and star C, we see that RR Lyrae star C is slightly darker than the star labeled 154 but not as dark as star 150. That means its apparent magnitude is between +15.0 and +15.4 . Since it is closer in appearance to the star at m = +15.4 but only slightly darker (brighter), estimate the apparent magnitude of star C as m = +15.3 .
Repeat this procedure with the other variable stars in photograph 468 and then continue on to the other photographs. Eventually you will have a table containing eight different values of apparent magnitude for each of the six RR Lyrae stars (that's six measurements on each of the eight photographs).
After all of the estimated magnitudes are determined, calculate an average apparent magnitude for each of the six RR Lyrae stars. This yields the average value for each of the stars. Now, we can assume RR Lyrae stars in general have the same average apparent magnitude. To find that value, average the six average magnitudes. In other words, average the average magnitudes of stars A through F. The result will be a single average of the average magnitudes which will be denoted m.
Distance Determination
To determine the distance to M15, we will use the accepted result that M = + 0.5 for RR Lyrae stars in general. So, for the distance modulus
M = m + 5 – 5logd
we can insert + 0.5 and your value of the average apparent magnitudes for M and m respectively.
Example Calculation
If m = +15.4, then the distance modulus can be written as
and solved as follows
0.5 = +15.4 + 5 – 5logd
0.5 = 20.4 – 5logd
5logd = 19.9
logd = 19.9/5 = 3.98
Now, we can make use of a little-remembered law of logrithms usually taught in pre-calculus that states 10logx = x. Our equation then becomes
10logd = d = 103.98
or
d = 9,550 pc
Since 1 parsec = 3.26 light-years, the distance to M15 is also d = 31,100 light-years.
