The explorations of space end on a note of uncertainty... we measure shadows and we search among ghostly errors of measurement for landmarks that are scarcely more substantial.
--Edwin Hubble The Realm of the Nebula
Trigonometric parallax works only for the stars closest to us, not only in our own galaxy but pretty much in our own backyard. Spectroscopic parallax allows us to go much further, out to any star cluster for which we can graph the main sequence. However, the red main sequence stars are generally very faint making it nearly impossible to resolve them even in the galaxies closest to us. Thus we must use more indirect means to determine how far away other galaxies are.
One of the most powerful tools involves the use of Cepheid Variable Stars.
The first Cepheid variable was discovered in 1784 by Edward Pigott and John Goodricke; the latter was a deaf mute who had, rather unusually for the time, received a good education and had become enamored of astronomy. The two men were particularly good at spotting changes in the brightnesses of stars and found, along with Cepheids, many classes of variable stars. To a large extent, such variables were initially treated as curiosities, their true nature unknown. With time, virtually all variable stars have proven themselves to be particularly useful stars. Eclipsing binaries reveal stellar masses; novae tell us about how mass can be transferred between stars; supernovae reveal details of the end stages of very massive stars; Mira variables reflect the radical changes happening deep within red giants. And virtually all stars have proven to be useful distance indicators. Cepheids in particular are luminous (they correspond to supergiant stages of the lives of fairly high mass stars) and their brightness varies characteristically, making them easy to find even in very distant galaxies.
It was not until the early 20th century that the importance of Cepheid variables was fully recognized. Henreitta Leavitt, hired by the Harvard Astronomy Department to search thousands of photographic plates for variable stars, soon developed a technique for finding them quickly using a combination of positive and negative images from plates taken on different dates. She placed the plates together, matching up several of the brightest stars. The black stars on the negative would then line up with the white stars on the positive making everything grey, except where the variables left either a white or black ring. While searching images of the Small Magellanic Cloud (SMC), she found more than 2000 variable stars, all by eye, all from photographic plates, and all before any electronic computers or measuring devices! The light curve of a variable star is a plot of the star’s brightness over time. Leavitt was able to plot the light curves of the variable stars she found and classify them, and made one of the most important discoveries in astronomy. As she found more and more SMC Cepheids she noticed a trend in their behavior: the brighter Cepheids seemed to always have the longer periods. She was eventually able to determine a mathematical relationship between the two properties, subsequently known as the period-luminosity (PL) relationship:
log L = 1.5 log P + 1.3,
where L is the star’s average luminosity in solar units, and P is the variable’s period in days. Cepheids can have periods ranging from 3 to 100 days. Her results were published under the name of the observatory director, Edward Pickering, in 1912. At the time, women could not publish in Harvard Observatory publications even though Pickering asked to have the paper published in Leavitt’s name. Four years after her death, Professor Mittag-Leffler nominated her for the Nobel prize in astrophysics. Unfortunately, the prize is never given posthumously.
With the period from the light curve, one can use the PL relationship to determine the luminosity of the star. With the luminosity, determining the distance becomes a simple matter of using either the inverse-square law or the distance modulus: the same method used for spectroscopic parallax and standard candles! Which method you use depends on the data you have: if you have luminosity and fluxes (apparent brightness) you use the inverse-square law and if you have magnitudes you use the distance modulus. Both are described below.
The flux and luminosity are related by the inverse square law:
where L is the luminosity, f is the flux (apparent brightness) in and d is the distance. Any set of units can be used, though the flux must have the same unit of energy per second as the luminosity, and the same unit of length as the distance. Solving for the distance gives
Historically, astronomers used magnitudes to talk about the brightness of stars. The first magnitude system was developed by the Greek astronomer Hipparchus, who divided the stars up into 6 groups, calling the brightest group the first magnitude, the second brightest the second magnitude, etc. Astronomers have refined the system, and now include negatives and numbers larger than 6. For example, the Sun has an apparent magnitude (m) of -26, Sirius a -1.42, and Hubble can see down to about 30.
Since the human eye responds to light on a logarithmic scale (i.e. a light that looks about twice as bright as another light actually puts out about 10 times more photons), the magnitude system is also logarithmic: a difference of 2.5 in magnitude corresponds to a factor of 10 in actual brightness. This leads to a slightly crazy way of talking about the brightness of stars, but also makes some things easier. For example, the PL relation for magnitudes is
M= - 3.8 log P + 1.5
which is slightly simpler than the one above. Astronomers can also define the distance modulus using magnitudes.
The apparent magnitude m is how bright a star appears, so it corresponds to the flux. Note however that the numbers are in reverse order and a difference of 2.5 in m is equal to a factor of 10 in f, so it is not a simple matter to convert between the two. Similarly, the absolute magnitude (M) corresponds to the luminosity, but again a bigger number is dimmer and a difference of 2.5 is equal to a factor of 10 in luminosity. The absolute magnitude is defined as the magnitude the star would appear to be if it were 10 pc away, so for any star that is actually 10 pc away, m and M are equal.
By converting flux and luminosity into apparent and absolute magnitude, astronomers are able to derive the distance modulus:
where d is the distance to the star in parsecs. Solving this for d gives
Since there is more error involved in the determination of M than in m, astronomers frequently refer to the distance modulus rather than the actual distance. That also means astronomers can use a simple difference instead of calculating out the distance! This is the technique Edwin Hubble used to show that the "spiral nebulae" were in fact galaxies outside our own, and eventually enabled him to develope his law for the expanding universe.
In this lab you will have to identify several cepheids from a field of stars. Your GSI will then provide you with data for those stars so you can generate their light curves and find the distance to the galaxy they are in. This is best done in groups of 3 - 4
magnitude short version
Last modified: 11/30/07 by SAM Original magnitude version by MM