Cepheid variable stars - candles in the universe
Did you know that the Sun is a variable star? The changes in brightness are not great, but they are measurable. As we know, the Sun varies luminosity over an 11 year cycle and this phenomenon is driven by certain inner processes. The difference between oscillations is one thousandth of the magnitude.
Before we continue with our subject, let briefly repeat basic terms. While brightness is our individual sense on quantity of light, luminosity (L) is related physical quantity representing the total energy of a star radiation. The apparent magnitude (m) of a celestial body is a measure of its brightness as seen on the Earth. The absolute magnitude (M) is defined as an apparent magnitude an object would have if it were 10 parsecs away from the Earth. Additional explanation of these notions can be found in this brief introduction to astronomy.
A variable star is a star which change its brightness either because of
- variation in luminosity or because
- its light is blocked before reaching an observer.
The first type of stars is called intrinsic while another one is extrinsic variable stars. Pulsating variables are of special importance among intrinsic variable stars. These are stars which change its magnitude as a consequence of alternating of star's radius i.e. volume. Eruptive variables change brightness by mass ejection of a star, flashes or similar processes. Cataclysmic stars pass an explosive change, like novae and supernovae. Extrinsic variable stars can be divided into two groups, eclipsing binaries and rotating variables.
There is no star with perfectly steady brightness. First off all, the higher volume of a star the larger oscillations in its brightness. Also, there is change in brightness during very long periods of time because of stellar evolution. There is relatively small number of stars counted as variable in the overall population of stars, around 1%.
One of the most known variable star is δ Cephei, very bright star in the constellation Cepheus. It varies from magnitude 3.6 to 4.3. The period is 5 days, 8 hours and 48 minutes. Its variability is first noticed and described in detail by John Goodricke in 18. century. δ Cephei's variability is caused by pulsation, which is also a case with many other stars. δ Cephei is the prototype of whole class of stars, named δ Cepheids or Classical Cepheids. In our galaxy, there is several hundreds such stars. For example, Polaris is a Cepheid.
One may assume that the brightness of these stars will be the largest when the surface (or volume) is the biggest. This would be true if the star had the same surface temperature all the time. However, pulsation process is very complex and this is not the case. Thus, brightness is the largest when the temperature is the highest.
Three types of pulsating variables shows almost linear relationship between luminosity and period. These are RR Lyrae, W Virginis and δ Cepheids, as Figure 3 presents. The graph M – log (P) is almost a line, i.e.
Discovery of galaxies
In spite of the fact that the universe is full of galaxies, we became aware of this in the 20th century. An important role in this, Edwin Hubble's discovery, played Cepheids.
Finding Cepheid variables in several spiral nebulae, including Triangulum Galaxy, Edwin Hubble estimate its distance and realized that those nebulae are galaxies.
where a and b are certain constants. Thankfully to this relation we have a powerful method for distance determination. It is possible to determine distances of the objects which contain a Cepheids or they are in their neighbourhood. Once we know period P, the average absolute magnitude follows directly from the equation above. Finally, the relation M=m+5-5log(r) leads to the distance r.
The advantage of this method is that the Cepheids are very bright stars and Cepheids serve as standard candles for determination of distances in the universe. The size of Milky Way and the distance from some neighbouring galaxies is determined by this method.
When Cepheids were observed in spiral nebulae, Edwin Hubble has shown in 1923 that those nebulae are actually spiral galaxies. Perhaps it sounds unbelievable, but we were not aware that the universe is full of galaxies until early 20th century. In the time when Edwin Hubble started with his observation at Mount Wilson in California in 1919, it was believed that the universe is consisted entirely of the Milky Way Galaxy. Habble observed Cepheid variables in a few spiral nebulae inducing Triangulum Nebula, which can be seen at Figure 1. His observation and calculation in period 1922-1923 has shown that nebulae are objects outside our own galaxy. Then it became clear that those nebulae are galaxies and that the universe is full of galaxies.
Types of pulsating stars
Cepheids can be divided into two classes, younger δ Cepheids and older W Virginis. As we already mentioned these two groups, together with RR Lyrae have similar relationship between luminosity and brightness.
RR Lyrae and W Virginis are situated in globular clusters and galactic halo whereas Classical Cepheids around the galactic plane. These are very bright and giant stars. β Cepheids have very short periods, typically few hours.
Variable star type
As the name says, prototype of this class is β Cep. Mira variables are named after the Mira star (Omicron Ceti). These are red giant star with huge brightness amplitude, typically 3 to 7 magnitudes.
Furthermore, there are some other interesting types of pulsating variable stars.
Video: The animation of Chi Cygni, a 408-day pulsating star. This star is a variable star of the Mira type in the constelation Cygnus. It changes its brightness from 5th to 13th magnitude. It is around 345 light years far away. Its variability is discovered by the astronomer Gottfried Kirch in 1686.
Animation of Chi Cygni, a 408-day pulsating star
Case study: Variable star as a pendulum
In spite of the complexity of the processes that cause pulsation, the form of oscillations is subjected to a simple rule. Particles on a star surface oscillate with the same frequency as it would be the simple pendulum of length equal to the radius of star, in the gravitational field as on the surface of the star. This fact provides us to calculate the density of a star.
We start with the relation for the period P of simple pendulum. Then we use defining relations for gravitational acceleration, volume V of a ball and for density ρ (Figure 5). Finally, resulting equation gives the average density of a star, assuming we know related period. Although this model is very simple, obtained results are slightly different to exhaustive theoretical calculation.
For example, having known the period of δ Cep it folllows directily from the relation that related density is 0.75kg/m3.
Cepheid variables - summary
- Delta Cephei is a very luminous variable star in the Cephei constellation, that periodically changes its brightness. The period of Delta Cephei is 5 days, 8 hours and 48 minutes.
- Later, many other stars with the same characteristics has been observed and nowadays we know several thousands these stars in our galaxy. They are named Cepheids, or Cepheid variables, after the first observed star.
- Cepheids are usually giant and supergiant stars with temperatures between 5000 and 7500. Its typical mass is around 5 mass of Sun.
- More precisely, Cepheids can be divided into subclasses representing different masses, ages and evolution.
- Thankfully to relation between luminosity and period within Cepheids, these stars provide a method for a distance determination (for details see the chapter “Period-luminosity relationship”).
- Examples of Cepheids: Polaris (Alpha Ursae Minoris) with period of 4 days with a small variation of magnitude, Eta Aquilae with period of 7.1 days and magnitude 3.5-4.4, ...
Quiz, check your knowledge!
Are you looking for a hobby? Observation of variable stars can be more than a hobby! It is one of very few areas where amateurs can achieve scientific discovery.
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