Cosmology - Cepheid Variables

For a very long time, nobody considered galaxies to be present outside our Milky Way. In 1924, Edwin Hubble detected Cepheid’s in the Andromeda Nebula and estimated their distance. He concluded that these "Spiral Nebulae" were in fact other galaxies and not a part of our Milky Way. Hence, he established that M31 (Andromeda Galaxy) is an Island Universe. This was the birth of Extragalactic Astronomy.

Cepheid’s show a periodic dip in their brightness. Observations show that the period between successive dips called the period of pulsations is related to luminosity. So, they can be used as distance indicators. The main sequence stars like the Sun are in Hydrostatic Equilibrium and they burn hydrogen in their core. After hydrogen is fully burned, the stars move towards the Red Giant phase and try to regain their equilibrium.

Cepheid Stars are post Main Sequence stars that are transiting from the Main Sequence stars to the Red Giants.

Classification of Cepheids

There are 3 broad classes of these pulsating variable stars −

• Type-I Cepheids (or Classical Cepheids) − period of 30-100 days.

• Type-II Cepheids (or W Virginis Stars) − period of 1-50 days.

• RR Lyrae Stars − period of 0.1-1 day.

At that time, Hubble was not aware of this classification of variable stars. That is why there was an overestimation of the Hubble constant, because of which he estimated a lower age of our universe. So, the recession velocity was also overestimated. In Cepheid’s, the disturbances propagate radially outward from the centre of the star till the new equilibrium is attained.

Relation between Brightness and Pulsation Period

Let us now try to understand the physical basis of the fact that higher pulsation period implies more brightness. Consider a star of luminosity L and mass M.

We know that −

$$L \propto M^\alpha$$

where α = 3 to 4 for low mass stars.

From the Stefan Boltzmann Law, we know that −

$$L \propto R^2 T^4$$

If R is the radius and $c_s$ is the speed of sound, then the period of pulsation P can be written as −

$$P = R/c_s$$

But the speed of sound through any medium can be expressed in terms of temperature as −

$$c_s = \sqrt{\frac{\gamma P}{\rho}}$$

Here, γ is 1 for isothermal cases.

For an ideal gas, P = nkT, where k is the Boltzmann Constant. So, we can write −

$$P = \frac{\rho kT}{m}$$

where $\rho$ is the density and m is the mass of a proton.

Therefore, period is given by −

$$P \cong \frac{Rm^{\frac{1}{2}}}{(kT)^{{\frac{1}{2}}}}$$

Virial Theorem states that for a stable, self-gravitating, spherical distribution of equal mass objects (like stars, galaxies), the total kinetic energy k of the object is equal to minus half the total gravitational potential energy u, i.e.,

$$u = -2k$$

Let us assume that virial theorem holds true for these variable stars. If we consider a proton right on the surface of the star, then from the virial theorem we can say −

$$\frac{GMm}{R} = mv^2$$

From Maxwell distribution,

$$v = \sqrt{\frac{3kT}{2}}$$

Therefore, period −

$$P \sim \frac{RR^{\frac{1}{2}}}{(GM)^{\frac{1}{2}}}$$

which implies

$$P \propto \frac{R^{\frac{3}{2}}}{M^{\frac{1}{2}}}$$

We know that – $M \propto L^{1/\alpha}$

Also $R \propto L^{1/2}$

So, for β > 0, we finally get – $P \propto L^\beta$

Points to Remember

• Cepheid Stars are post Main Sequence stars that are transiting from the Main Sequence stars to Red Giants.

• Cepheid’s are of 3 types: Type-I, Type-II, RR-Lyrae in decreasing order of pulsating period.

• Pulsating period of Cepheid is directly proportional to its brightness (luminosity).