HCOL 196, April 22, 2011

We postponed the presentations until Wednesday the 27th and Friday the 29th, so that we would not interfere with people’s holiday plans. Each team knows when they are scheduled to present. It is important for people to arrive on time for presentations, because the time available is limited. Please, everyone, make every effort to be on time.

On Friday I talked about work that I’ve done using Bayesian methods. I talked about a particular kind of star, Cepheid variables, which are very important in astronomy. These stars are fairly bright, and they pulsate regularly.

In 1908, Henrietta Leavitt discovered that the intrinsic luminosity of these stars depends on their pulsation periods: The longer the period, the more luminous the star is. This is called the period-luminosity relationship. Of course, as the star gets farther away, it will become fainter to us, but the intrinsic luminosity doesn’t depend on the distance of the star. This gives us a way to measure the distance to one of these stars, if we know its intrinsic luminosity: Measure its apparent luminosity, which decreases as the square of the distance to the star (the “inverse square law”), and calculate the distance from the intrinsic and apparent luminosities.

To do this, we need to calibrate the period-luminosity relationship. That is, even though we know from Leavitt’s work that there is a relationship, we do not have the tools to convert a period into an intrinsic luminosity. Leavitt was looking at these stars in objects that had a number of them (the nearby galaxies known as the Magellanic clouds), so that all the stars were at about the same distance from us. Since they were all at the same distance, the period-luminosity relationship showed up as a relationship between the periods and the apparent luminosities of these stars. But no one knew the distance to the Magellanic clouds, so this was missing a crucial piece of information, that is, the distance to at least one Cepheid variable.

The direct method of measuring the distance to a star is to observe it as the Earth goes around the Sun. Because of the Earth’s motion, the direction to the star changes slightly. If we can measure this very small change in angle, we can determine the distance to the star by trigonometry, as shown in the figure:

The trouble is that the angles are very small, less than 1 second of arc even for the nearest star (Proxima Centauri). When we started our work on the Hubble, the errors of parallaxes were about ±0.01 second of arc, which means that we could not measure distances to stars more than about 100 parsecs, or 326 light years away. Unfortunately, the nearest Cepheid is almost three times farther away than that, so another method was needed. (Since then, we have measured the distance to this star to quite high precision, but we’d like to have other, independent measures). But, as you can see from the picture of our galaxy, most stars are just too far away.

The pulsations affect both the size of the star (which expands and contracts) and the surface temperature of the star (when gases expand, they cool, and vice versa). We can measure both the apparent brightness and the temperature of the star as it expands and contracts, and that allows us to measure the angular change in the size of the star quite accurately (see “measure as angle” in the diagram below). At the same time, we can measure the linear changes in the size of the star (in meters, for example) by observing the change in the wavelengths of spectral lines emitted by the star, through the Doppler shift (the same principle that police use to detect speeders). The angular change and the linear change are related to the distance of the star, so from this information we can calculate the distance to the star (in the equations in the diagram I’ve converted to compatible units, this is just for illustration).

Our problem was fairly complicated, since the light curves of Cepheids are not nice smooth waves, but they rise slowly and fall more rapidly. We can approximate the complicated actual curve by adding up nice smooth waves of different frequencies, with appropriate amplitudes. Estimating those amplitudes is at the heart of the Bayesian program that we developed. Our results were consistent with the results of other methods.

So, with the period-luminosity relationship calibrated, astronomers are able to measure the distances to many distant objects that contain Cepheid variables (mostly galaxies). I hasten to point out that many astronomers have worked on this problem over the decades, and our work is intended to augment what others have done.

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