Period-Luminosity Relation for Variable Stars

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Harlow ShapleyIn the 1540's Nicolaus Copernicus removed the Earth from the center of the universe. He put the Sun at the center. Copernicus' view held up against the observational evidence for hundreds of years. In the 1910's the Sun was removed from the center of the universe and relegated to a typical patch in the galactic disk far from the center of the Galaxy. Harlow Shapley (lived 1885--1972) made this discovery by determining the distances to very old star clusters. He used the inverse square law of light brightness on a particular type of variable star in those old star clusters.

Some stars are very useful for finding distances to clusters and to other galaxies because they have a known luminosity that is large, so they can be seen from great distances away. Bright objects of a known luminosity are called standard candles (though, in our modern day we should perhaps call them "standard bulbs"). Standard candle objects are used to measure large distances. The particular standard candle stars Shapley used are in the last stages of their life and pulsate by changing size. They are trying to re-establish hydrostatic equilibrium but the thermal pressure is out of sync with the gravitational compression. The expanding star overshoots the equilibrium point. Then gravity catches up and contracts the star. But gravity contracts the star beyond the equilibrium point. The thermal pressure increases too much and the cycle continues.

changing size and temperature of Cepheid and RR-Lyrae variables


In 1912 Henrietta Leavitt (lived 1868--1921) published the results of her study of variable stars in the Large and Small Magellanic Clouds. These are two small satellite galaxies orbiting the Milky Way. The linear size of each Magellanic Cloud is much smaller than its distance from us. Therefore, to a good approximation all of the stars in each galaxy are at the same distance from us. Leavitt found a very useful relationship for a certain type of variable star called a Cepheid variable (after the prototype in the constellation Cepheus; see stage 6 of a star's life cycle). The fainter Cepheids in the Magellanic Clouds have shorter periods. Because all of the Cepheids in a Magellanic Cloud are at the same distance from us, Leavitt reasoned that the more luminous Cepheids pulsated more slowly. This is the period-luminosity relation. Leavitt did not know the distances to the Magellanic Clouds, so she could not tell what the actual value of the luminosity part of the relation was.

Astronomers had to wait a few years for Harlow Shapley to calibrate Leavitt's relation using Cepheids in our galaxy for which the distances could be determined. In the calibration process Shapley put actual values to the luminosity part of the period-luminosity relation. With a calibrated period-luminosity relation astronomers could use Cepheid variables as standard candles to determine the distances to distant clusters and even other galaxies.

Cepheids have pulsation periods of 1 to 50 days. In the 1950's astronomers found that there are two types of Cepheids:

Because the luminosity of Cepheids can be easily found from the pulsation period, they are very useful in finding distances to the star clusters or galaxies in which they reside. By comparing a Cepheid's apparent brightness with its luminosity, you can determine the star's distance from the inverse square law of light brightness. The inverse square law of light brightness says the distance to the Cepheid = (calibration distance) × Sqrt[(calibration brightness)/(apparent brightness)]. Recall that brightnesses are specified in the magnitude system, so the calibration brightness (absolute magnitude) is the brightness you would measure if the Cepheid was at the calibration distance of 10 parsecs (33 light years). In some cases the calibration distance may be the already-known distance to another Cepheid with the same period you are interested in. As described below, Cepheid variable stars are a crucial link in setting the scale of the universe.

deriving the distance from the flux

Early measurements of the distances to galaxies did not take into account the two types of Cepheids and astronomers underestimated the distances to the galaxies. Edwin Hubble measured the distance to the Andromeda Galaxy in 1923 using the period-luminosity relation for Type II Cepheids. He found it was about 900,000 light years away. However, the Cepheids he observed were Type I (classical) Cepheids that are about four times more luminous. Later, when the distinction was made between the two types, the distance to the Andromeda Galaxy was increased by about two times to about 2.3 million light years. Recent studies using various types of objects and techniques have given a larger distance of between 2.5 to 3 million light years to the Andromeda Galaxy (a measurement using eclipsing binaries gives a distance of 2.52 million light years; another measurement using red giants gives a distance of 2.56 million light years; another measurement using Cepheids gives 2.9 million light years; and measurements using RR-Lyrae give 2.87 to 3.00 million light years).

more luminous variables have longer pulsation periods

more luminous variables have longer pulsation periods

RR Lyrae

Another type of pulsating star similar to the Cepheids are the RR Lyrae variable stars (named after the prototype star RR Lyrae). They are smaller than Cepheids and, therefore, have shorter periods and lower luminosities. They pulsate with a period between 5 and 15 hours (Cepheid pulsation periods are greater than 24 hours). Low-mass stars will go through a RR Lyrae pulsation stage while the high-mass stars will go through a Cepheid stage. Because low-mass stars live longer than high-mass stars, the Cepheid stars as a group are younger than the RR Lyrae stars.

RR Lyrae are found in old star clusters called globular clusters and in the stellar halo part of our galaxy. All of the RR Lyrae stars in a cluster have the same average apparent magnitude. In different clusters, the average apparent magnitude was different. This is because all RR Lyrae have about the same average absolute magnitude (=+0.6, or 49 solar luminosities). If the cluster is more distant from us, the RR Lyrae in it will have greater apparent magnitudes (remember fainter objects have greater magnitudes!).

RR Lyrae stars can be used as standard candles to measure distances out to about 760,000 parsecs (about 2.5 million light years). The more luminous Cepheid variables can be used to measure distances out to 40 million parsecs (about 130 million light years). These distances are many thousands of times greater than the distances to the nearest stars found with the trigonometric parallax method. The method of standard candles (inverse square law) provides a crucial link between the geometric methods of trigonometric parallax and the method of the Hubble-Lemaître Law for very far away galaxies. (The Hubble-Lemaître Law is explained further later.) In fact, this link between the parallax and the Hubble-Lemaître Law was so crucial that the diameter of the Hubble Space Telescope's mirror was primarily determined by how large a mirror (its resolving power and light gathering power) would be needed to pick out Cepheids in the other galaxies and the Cepheid distance measurement of 18 galaxies was one of the three Key Projects for the Hubble Space Telescope during its first decade of operation (see also). All of the pretty pictures of other objects during that time were just an extra bonus.

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last updated: May 28, 2019

Is this page a copy of Strobel's Astronomy Notes?

Author of original content: Nick Strobel