Lab
RR Lyrae Stars and the Distance to the Globular Cluster M4 (This lab requires on-line measurements.)

Objective

To determine the distance to the globular cluster, Messier 4 (M4), by using observations of an RR Lyrae star and the mean absolute magnitude for RR Lyrae variables; to compare advantages and disadvantages of observing RR Lyrae stars to Cepheid variables; to summarize what we learn about the Milky Way when we learn the distances to globular clusters.

Introduction

As stars evolve, their atmospheres become unstable and the star becomes intrinsically variable. Some stars vary erratically, some semi-regularly, and some regularly. Two special classes of regular variable stars have become the keys to determining distances within our galaxy and the distances to neighboring galaxies. The Cepheid variables are evolved young, massive stars and lie within the crowded spiral arms of a galaxy. Since they are a stage in the life of a massive star, there are relatively few of them in a galaxy. Cepheid variables have periods that range from a few days to a few hundred days. The other class of variable stars, the ones we are concerned with here, are RR Lyrae variables (named after the prototype star RR in the constellation of Lyra). RR Lyrae stars are evolved old, low-mass stars, and can be seen in the uncrowded halos of galaxies, especially in globular clusters. They are a stage in the evolution of a lower-mass star, and therefore are generally more numerous than Cepheid variables. A single globular cluster may have dozens of RR Lyrae stars within its population of stars. Periods of RR Lyrae stars are typically 0.3 to 1 day, making it possible to see one or more periods (cycles) in a single night of observations.

Astronomers have observed thousands of Cepheid and RR Lyrae variables. As might be expected from the types of stars that become Cepheid variables, these stars are very luminous, with luminosities ranging from 100 to over 10,000 times that of the Sun. Through calibration of the RR Lyrae stars, astronomers have found that these stars are much less luminous--on average only 40-50 times as luminous as our Sun--than the Cepheids. The RR Lyrae stars have a mean absolute magnitude (M) of 0.75. Even though they are not very luminous, RR Lyrae variables serve an important function. If we can determine a mean apparent magnitude (m) for an individual star in a globular cluster, we can calculate the distance to the star and thus the globular cluster by using the magnitude equation:

In this exercise, we will use this calibration and the magnitude equation to determine the distance to the globular cluster, M4 (fourth object in the Messier Catalog of star clusters and nebulae). At 7000 light years distance (2150 pc), M4 is the nearest globular cluster, as seen from its large apparent size. It contains more than 100,000 stars, including many RR Lyrae variables. M4 lies in a direction close to the huge Rho Ophiuchi dust complex and therefore suffers substantial dust obscuration. From our point of view, this dust dims the stars in the globular cluster by about 1 magnitude.

Procedure

Here is a full-field view of M4 and an enlargement of its northeastern part (PSS #863E). One of the stars in M4, No. 42 in a catalog of the variables found there, is identified. You will also find a series of 20 pictures of #42 and its neighboring stars. These 20 pictures were taken over the course of 12 hours one (winter?) night. The star #42 is easy to spot as it is obviously changing size (and thus luminosity) between frames, but if you do not see it right away, it is at the center of each photograph.

To determine the magnitude of a star, astronomers choose a number of "standard" stars within each frame and calibrate the variable star against these standards. We will do a similar process here, using just one image of the series, frame 1, to do so. Within this frame are six standard stars. They are identified in frame No. 1 shown below. Using the calibration of standard stars, we find a "diameter-versus-apparent-magnitude" correlation by assuming the two are linearly related. This means that the brighter the star looks to us, the larger the size it is on the image. Once we have this relationship, we simply measure the diameter of Star #42 in each of the 20 frames and determine the corresponding apparent magnitude from the graph. Since the change in the star takes place as time passes, we then graph apparent magnitude versus time to see the change in its brightness.

Exercise

What you will turn in: 1) your table of apparent magnitude versus diameter for the standard stars; 2) the graph of apparent magnitude versus measured diameter; 3) your completed table on the apparent magnitude versus time for Star #42, 4) the graph of apparent magnitude of star #42 versus time; and 5) the answers to the "calculations" and "questions" sections.

At the left is a reproduction of image #1 of the starfield showing the standard stars and their calibrated magnitudes. Click on the image ON-LINE and you will get an enlarged interactive version set up in a program that allows you to measure the width of the standard stars (indicated by their apparent magnitudes being written next to them) in pixels. Follow the directions as given on the web page with the image.

Once you have calculated the diamters of the standard stars, plot apparent magnitude versus diameter (in pixels) on the graph (Apparent Magnitude vs Measured Diameter in Pixels). You will then have a graph of the apparent magnitude of each standard star (y axis) as a function of its measured diameter in pixels (x axis). Draw a best-fit STRAIGHT line through the points. This graph will be used to derive the magnitude of variable star No. 42 in each of the 20 frames.

Measure the diameter, in pixels, of star #42 (the center star) in each frame shown in the series of 20 pictures. . The frames have been randomized to avoid biased measurements (good scientific technique). Take the series of 20 measurements and write the diameters in column 3 of Table 2. When finished with these measurements, have your partner take the other series of 20 measurements and write the diameters in column 4 of Table 2. Average the two diameters for each frame and write the averages in column 5. Using your calibration-of-apparent-magnitude-versus-diameter graph from the standard stars, interpolate the apparent magnitude for the variable star for each frame and record in column 6. Plot apparent magnitude vs. time (where the time is given in the fraction of the day since the observations started) on a apparent magnitude of star 42 versus time graph.

First, "connect-the-dots" and then, when you have an idea of what the "light curve" looks like, draw a smooth curve through the data points.

REMINDER IF YOU ARE USING YOUR OWN GRAPH PAPER: the graph must have the smallest magnitude value (for brightest) at the top of the "y" axis and the largest magnitude value (for dimmest) at the bottom of the "y" axis.

Calculations and Questions

See "calculations" and "questions" PDF file.