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Astronomy 100 Names: Exercise 6: Stellar distance and age determination methods Parallax Parallax is the measurement of the apparent motion of stars due to our viewing them at different points in Earth’s orbit. Further stars exhibit little parallax; it is difficult to measure parallax beyond about 300 light years. However, these far stars then serve as “background” for the closer stars that do exhibit measureable parallax. The parallax of a given star is half of the angular distance it seems to move over a six-month period. 1. Why a six-month period? 2. Examine the three sheets (one paper and two transparencies) that show a oneyear period sequence of astrophotographs of the globular cluster Palomar 3. If parallax is measured over a six-month period, why do you need a full year of photographs? 3. Cunningly overlay the transparencies on the paper so that the starfields coincide (that is, most of the stars are lined up). There should be four objects that have “moved”. How many of these are stars? 4. Of the stars that “move”, determine their parallaxes in arc-seconds. Use the scale given in the photographs. 5. The formula relating distance and parallax is d = 1/p, where d is the distance to the star in parsecs (pc) and p is the parallax of the star measured in arc-seconds. 1 parsec is about 3.26 light years. Determine the distances to the stars in question 4. Extra credit — what is the limit of distance using the parallax method using these photographs? Justify your answer! Using the index to figure out distance and age – the method of standard candles Astronomers, including Edwin Hubble, began to use the Hertzsprung-Russell diagram as a powerful tool to determine stellar properties. Initially, the method of standard candles for individual stars was applied to whole clusters within the Milky Way. You will recreate this distance determination. You should have three sheets of paper with the B-V versus magnitude plots of NGC 3292, 47 Tucanae and NGC 362/Small Magellanic Cloud. You should also have a transparency of the B-V versus magnitude plot for nearby stars. 6. To determine the age of a cluster: • Pick a cluster’s graph. Align the transparency such that the axes of the two graphs coincide. • This is somewhat tricky: while keeping the axes pointed in the right direction (in other words, without rotating the sheet), slide the transparency over the other graph until the pattern of points on the transparency nearly or exactly matches the pattern of points on the underlying graph (in other words, such that the slope of a best fit line in both graphs are coincident). • Notice that the y-axis is now offset. For example, the “10” on the transparency is no longer on the same level as the “10” on the underlying sheet. Calculate the offset (this is easy). Notice that this offset is the difference on the two graphs between m, the magnitude of the cluster’s stars, and Mv, the magnitude of nearby stars. Write this offset, called the distance modulus, into the table below. Cluster NGC 3293 47 Tucanae NGC 362 Small Magellanic Cloud Distance modulus Distance (pc) Repeat the process for all four clusters. 7. Now use the graph below to translate the distance modulus to an actual distance in parsecs, and fill in the last column on the table. 8. This process of distance determination is called main-sequence fitting. Why does this work? Equivalent question: Since this is the method of “standard candles”, the name implies that there is a standard “wattage” light that is being used; what. physically, is the “standard candle”? 9. The relative age of the stars within a cluster can be determined by the spectral type. For instance, consider the NGC 3293 cluster. What spectral type (using the Cannon OBAFGKML stars) seem to abound in this cluster? So is this a relatively old or relatively young cluster? 10. Order the four clusters in terms of increasing apparent age. Determining distances using a different standard candle – the Cepheid variable star Henrietta Leavitt at Harvard University, in the early part of the 20th century, studied a class of variable stars called the Cepheids. These stars vary in brightness in a cyclical pattern, and are bright enough to be seen within another galaxy. Leavitt determined that absolute magnitude (M) of a Cepheid variable was mathematically related to the period (P, measured in days) of its brightness cycle: M = – 1.35 – 2.78 log10 P The equation above is a much more refined version than the one Leavitt determined but nevertheless contains the same relationship between magnitude and period. As mentioned, one can observe Cepheid variables in other (nearby) galaxies because they are intrinsically quite bright. One such galaxy is M100, and the following graph shows the plot on apparent brightness (m) versus time (t). (courtesy European Space Agency) 11. a. Estimate the period of one cycle of Cepheid 1 (in days). b. Using the Leavitt equation, calculate the absolute magnitude (M) of Cepheid 1. To figure out the distance to Cepheid 1, we need to calculate a distance modulus, like in main sequence fitting part of this exercise. 12. a. Calculate the distance modulus, m – M. b. Determine the distance to Cepheid 1 by converting the distance modulus into an actual distance. Can you use the graphs in the previous section (the main sequence fitting section) to determine this distance? Why not? c. Then it’s back to a formula; the graphs in the previous section are based on the following equation: D = 10 ((m−M ) + 5 ) / 5 where the distance D is in parsecs (pc). Calculate the distance to Cepheid 1 and convert the distance into megaparsecs (Mpc). € 13. The distance to M100 using other methods (the Hubble method, which we will cover later) was determined to be 17.1 ± 1.8 Mpc. How does your value compare (calculate how many percent off, and whether you overestimated or underestimated the distance)? Is the Cepheid variable a reasonable “standard candle”?