Astronomy 162: Professor Barbara Ryden

Wednesday, January 29

PULSATING STARS

``But I am constant as the Northern Star
Of whose true-fixed and resting quality
There is no fellow in the firmament.''
- William Shakespeare,
Julius Caesar, 3, 1

Key Concepts

Some giants and supergiants vary in luminosity because they pulsate in and out.
Cepheid stars and RR Lyrae star are examples of pulsating variable stars.
The greater the luminosity of a Cepheid star, the longer its period of oscillation.

(1) Variable stars have luminosities which increase and decrease with a regular period.

Main sequence stars have a nearly constant luminosity. (During the next five billion years, the Sun's luminosity will double; that's an increase of only 0.02% every million years.)

By contrast, some giants and supergiants have luminosities which regularly increase and decrease. The period of the luminosity fluctuations (that is, the time between peaks in the brightness) can range from a few hours to a few years.

(2) Cepheid stars and RR Lyrae stars are variable because they pulsate in and out.

The two most interesting types of variable star are Cepheid variables and RR Lyrae variables.

Cepheid variables are named after the star Delta Cephei (the fourth brightest star in the constellation Cepheus). The luminosity of Delta Cephei varies by a factor of two, with a period of 5 days. Polaris, the Northern Star, is also a Cepheid. (Thus, when Shakespeare has Julius Caesar say ``I am constant as the Northern Star'', astronomically knowledgeable playgoers may savor a double irony.)

Cepheid variables have the following properties:

Period = 2 to 60 days
Average Temperature = about 6000 Kelvin
Average Luminosity = 300 to 40,000 L_sun

The fluctuations about the average luminosity can be large (as in the case of Delta Cephei) or small (as in the case of Polaris, where the fluctuations are too small to be detected with the naked eye).

RR Lyrae variables are named after the star RR Lyrae, in the constellation Lyra.
RR Lyrae stars have the following properties:

Period = 4 hours to 1 day
Average Temperature = about 7000 Kelvin
Average Luminosity = about 80 L_sun

RR Lyrae variables thus have shorter periods and lower luminosities than Cepheid variables.

The variation in the luminosity of RR Lyrae and Cepheid stars results from the fact that they pulsate in and out. The radius of a Cepheid can vary by as much as 10 or 20 percent. (Remember that a giant or supergiant star such as a Cepheid or RR Lyrae variable has a small dense core and a large, low-density envelope. In the case of a Cepheid or RR Lyrae variable, it is only the envelope which expands and contracts. The core remains constant in size, and continues to produce energy at a steady rate.) As the envelope expands and contracts, its surface temperature varies by as much as 1000 degrees Kelvin.

Because the radius R and surface temperature T change, so does the luminosity L. Remember,
L = 4 pi R² sigma T⁴

But why does the star pulsate at all??

The textbook has a rather involved description involving the properties of a layer of ionized helium in the star's envelope. A simpler, stripped-down explanation goes something like this:

When a Cepheid is compressed, it becomes opaque.
Photons are trapped inside, heating the gas and increasing its pressure.
The high-pressure gas expands, becoming transparent.
Photons escape, the gas cools, the pressure drops.
As the pressure drops, the Cepheid is compressed by gravity.

This cycle repeats as long as the Cepheid (or the RR Lyrae, which pulsates by the same mechanism) is in the instability strip, the region of the H-R diagram where stars are unstable to pulsation.

(3) The greater the luminosity of a Cepheid star, the longer its period of oscillation.

It was discovered empirically in 1912, by the Harvard astronomer Henrietta Leavitt, that a Cepheid's period of pulsation is linked to its luminosity. (This was long before anyone knew WHY Cepheids varied in luminosity.) Miss Leavitt was studying Cepheid variables in the Large Magellanic Cloud (a nearby galaxy) when she discovered that the brightest Cepheid variables in the galaxy had the longest period of pulsation. This relation holds true in our own galaxy too, and in all other galaxies where Cepheid variables have been detected.

High Luminosity, Long Period.

For instance, a very luminous Cepheid, with L = 40,000 L_sun, has a very long period, P = 60 days. On the other hand, the dimmest Cepheids, with L = 300 L_sun, have short periods, P = 2 days. A plot of the period-luminosity relation for Cepheid variables (as well as for RR Lyrae variables) is shown below:

(A complicating factor: There are two types of Cepheids. In addition to the standard Type I Cepheids that I've been discussing, regions of the universe which have few heavy elements give rise to a different species of Cepheid: Type II. Type II Cepheids, because they have a different chemical composition, have a different period-luminosity relationship.)

The Period-Luminosity relation is useful because it lets you measure the distance to Cepheid stars which are quite far away.

Measure the period P of a Cepheid variable.
From the period-luminosity relation, determine the luminosity L.
Measure the apparent brightness b.
Compute the distance from the equation: L = 4 pi d² b

Because Cepheids are so luminous, they can be seen to large distances. Parallaxes can be used to determine distances to 500 parsecs or so. However, the period-luminosity relation can be used to determine the distance of a Cepheid about 40 MILLION parsecs away. This lets us find the distance to all the galaxies in our neighborhood. (At least all the galaxies which contain Cepheid variables - Cepheid stars are rare, and must be hunted for carefully.)

The galaxy M100 is 17 million parsecs (56 million light years) away -- 25 times farther than the Andromeda galaxy. The distance to M100 was determined from the bright Cepheid stars within the galaxy, which are visible using the Space Telescope. Click on the image of M100 below, and you will see six snapshots of an extremely luminous Cepheid in that galaxy, varying with a period of 51 days.
(Image credit: J. Trauger, [Jet Propulsion Laboratory], & NASA)

Cepheid variables are a vital tool for measuring the distances of relatively nearby galaxies (out to about 40 million parsecs, or 130 million light years). RR Lyrae stars can also be used to measure distances, but since they are less luminous, they can only be seen to smaller distances.

Prof. Barbara Ryden (ryden@astronomy.ohio-state.edu)

Updated: 2003 Jan 28