The Cepheid Yardstick

The Cepheid Yardstick The Cepheid Yardstick

Introduction

At the age of nineteen, John Goodricke, an eighteenth century English astronomer observed that the star delta-Cephei brightened and dimmed in approximately 5-day cycles.

When you spend a few days in a dark place with clear skies, you can observe the same cycles that Goodricke did in 1784. delta-Cephei is a star in the constellation Cepheus, near the north star. By keeping a careful record of its brightness relative to nearby stars over the course of a few nights, you can trace out a curve like the one shown in the figure.

Since Goodricke's time, astronomers have discovered and catalogued thousands of stars whose brightness, or luminosity, like delta-Cephei's, undergoes cycles with periods of a few days. Such stars, called Cepheid variables, play an important role in determining the distances to nearby galaxies. This is because of a relationship, first noted by the American astronomer Henrietta Leavitt in 1912, between a Cepheid's period and its intrinsic brightness. If we know how much light a Cepheid in a distant galaxy gives off, we can calculate how far away it must be for it to appear as faint to us as it does. This lab explores this relationship and its application in detail.

We cannot directly measure the distances to stars, and stars in other galaxies are much too far away for parallax measurements. We can, however, measure the periods of any Cepheid variable stars we can find in them. By assuming that these Cepheids obey the same period - absolute luminosity relation that their friends nearby have been observed to follow, we can compute how far away a galaxy must be in order that its Cepheids have their observed apparent luminosities. The distances to galaxies as far away as several million light years have been measured in this fashion.

Cepheids in the SMC

In 1522, a rag-tag crew of 21 sailed into the Spanish harbour of Seville. They were all that remained of Ferdinand Magellan's crew of 270 that had set sail from that same port three years before. Magellan and most of his crew hadn't survived the voyage, but those who had had sailed 'round the world. Among the stories they brought back to Europe was one of two fuzzy patches of starlight visible in the southern hemisphere's sky, which came to be known as the large and small Magellanic clouds.

Today we know the Magellanic ``clouds'' to be nearby galaxies. Some of the stars in them are Cepheid variables whose periods and apparent luminosities have been measured. The data for nine Cepheids in the Small Magellanic Cloud or SMC are collected in the accompanying table.

Star	Period (days)	App. Mag.	log(Period)
HV1871	1.2413	17.21	0.0934
HV1907	1.6433	16.96	0.216
HV11114	2.7120	16.54	0.433
HV2015	2.8742	16.47	0.459
HV1906	3.0655	16.31	0.486
HV11216	3.1148	16.31	0.493
HV11113	3.2139	16.56	0.507
HV212	3.9014	15.89	0.591
HV11112	6.6931	15.69	0.826

Make a graph of apparent magnitude vs. log(Period) on a piece of graph paper. Draw a line through the data which represents approximately the period-luminosity relationship you observe.

What, qualitatively, is the relationship between a Cepheid variable's period and its luminosity?
Since all these Cepheids are the same distance away from us, their relative apparent magnitudes are the same as their relative absolute magnitudes. For this reason, when Henrietta Leavitt first made this plot, she realized that it suggested an approximate relationship between a Cepheid's period and its absolute magnitudes, and conjectured that this approximate relationship holds for all Cepheids. Suppose, for example, there were a Cepheid in the SMC with the same period as delta-Cephei, 5.3663 days (log(5.3663)=0.730). Read off from your graph what you would expect its apparent magnitude to be. You may indicate a range of reasonable values by including an uncertainty (i.e, +/-) in your answer.
```
        m(SMC-Ceph) = 
```
The observed apparent magnitude of delta-Cephei is
```
        m(delta-Ceph) = 4.0
```
much brighter than the apparent magnitudes of Cepheids in the SMC. This suggests that the SMC is much farther away than delta-Cephei. The distance to delta-Cephei has been determined using a variety of independent methods to be about
```
	D(delta-Ceph) = 850 +/- 80 ly = 265 +/- 25 pc
```
Using the Magnitude equation:
```
	M = m + 5 - 5log(d)
```
where d is in parsecs, compute the absolute magnitude(M) of delta-Cephei.
```
        M(delta-Ceph) = 
```
The period-luminosity relation tells us that this is also the absolute magnitude of the (fictitious) SMC Cepheid with the same period as delta-Cephei, whose apparent magnitude you already estimated. Using the magnitude equation again, in this form:
d=10^(m-M+5)/5
compute the distance to the SMC.
```
         Distance to SMC = 
```
Give your answer in parsecs and in light-years (1 parsec=3.2 light years). The SMC is the second CLOSEST galaxy to the Milky Way (just slightly farther away than the Large Magellanic Cloud).
The accepted distance to the SMC is 0.06 Mpc. Discuss briefly the possible source of errors in your calculated distance.