Download Exercise 6

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).
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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”?