# Why Cepheids have Period Luminosity relation?

According to my astronomy teacher, Cepheids is a type of variable stars that has Luminosity-Period relationship:

$$M \propto log(T)$$ , where $$T$$ is the pulsation period of Cepheids.

But I have a question regarding this, there are a lot of variable stars in the universe, why only Cepheids shows a Luminosity-Period relationship while other variables don't show it, does it have something to do with the properties of Cepheids?

$$Tρ = \sqrt{C}$$ , where

Where $$T$$ is the pulsation period and ρ is the density of the star., but I do not see how this helps to explain my question.

I also found a pdf on the internet that explains why variables have luminosity period relation, but I have difficulties to explain that only Cepheids and some type of variables have luminosity period relation.

• Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. Commented Jul 4 at 3:33
• It is a class paper from Princeton, I can't really find the author. Commented Jul 4 at 3:48
• It is actually possible to find the author. The URL of that resource contains the string "~gk", and the string "A403" The A403 refers to the name of the course: Stars and Star Formation, Spring 2011, and "~gk" is the princeton astrophysics department abbreviation for the person who was in charge of that course. From the Faculty members overview: Gilian Knapp, emerita professor. Given that the resource is part of old course notes: it seems unlikely that Princeton astrophysics will keep that material available forever. Commented Jul 4 at 16:43

The timescale of any large scale radial variation in a star will, even on a dimensional basis, be proportional to $$(G\bar{\rho})^{-1/2}$$, where $$\bar{\rho}$$ is the average density. Thus $$T \propto M^{-1/2}R^{3/2}\ .$$

Cepheid variables have, very roughly the same mass and surface temperature, but a big range of radii. Since luminosity is $$\propto R^2$$, then that gives $$T \propto L^{3/4}$$ and a period-luminosity relationship.

The pulsations occur at a particular range of temperatures and radii in the Hertzsprung-Russell diagram called the instability strip. Thus is where the conditions are suitable for a subphotospheric layer of ionised helium. This has the property of increasing its opacity when compressed and heated and this is what drives the pulsations.

Period-luminosity relationships are not unique to Cepheids - they are also present in RR Lyrae variables and several other classes of pulsator.

Of course, to get a period-luminosity relationship, you need the pulsators to have a range of radii whilst other physical variables are more narrowly constrained. Or, if on the main sequence, a range of masses. But if the pulsation mechanism demands a fixed temperature then that isn't always possible.

• Could you expand on why ionized helium shell with create the pulsation? If a star is variable because of its temperature or radius is changing, that means it has a luminosity period relation? Commented Jul 4 at 13:15
• @Polaris5744 I got from your question that you understood the kappa mechanism but wanted to know why there was a period-luminosity relationship. Commented Jul 5 at 7:33

There is some material by Bill Janesh, Case Western Reserve University

Pulsating variable stars

Bill Janesh points out that in order to have a pulsation there must be a process going on that in both directions is overshooting an equilibrium point.

The overshooting is most difficult to understand/model, it would seem.
Light and heat tend to come into an equilibrium state; that's why it is possible to obtain an exact measurement of a black body spectrum. If something repeats well then you can measure it well. Thermal equilibrium between heat and light repeats well.

However, in the case of variables with a regular pulsation cycle something is thwarting the normal tendency to equilibrium state.

With Cepheid variability:
We have that there is a phase where internal absorption of light exceeds the amount of emission, so that heat accumulates. At some point that accumulation of heat rolls over, and a phase ensues where emission exceeds absorption.

This cyclic unbalance of absorption and emission is attributed to difference of opacity of states of ionization of Helium.

It seems to me that the opacity story is incomplete. If a material becomes more opaque it heats up, and immediately it will start emitting more light, due to its higher temperature.

But what you need is something that causes a lag, some form of hysteresis

Hysteresis effects are difficult to model.

Incidentally, until now I had failed to appreciate that Cepheid variability is a very transient state. A state of Cepheid variability is a state that pretty much every star will enter one or more times during the star's overall lifetime, and it will last for only a very short amount of time (compared to the overall luminous lifetime of the star.

It would seem that in order for Cepheid variablity to occur a lot of transient properties have to line up in a very specific way.

It would seem: a highly specific set of conditions must line up for a star to enter a phase of Cepheid variabiity. It seems that the combination of conditions is so specific that stars in Cepheid phase are lookalikes. That would go towards explaining why there is a strong correlation between period of the Cepheid cycle, and luminosity of the star.

• "very little information out there about the physics/chemistry of pulsating stars." - huh? This has been a major research area in stellar astrophysics for many decades. Commented Jul 5 at 9:36
• @AndersSandberg I have removed the remarks suggesting little information is available. The sources I encountered seemed to focus on the minimum of data necessary to use Cepheid variables for distance assessment, with very little attention for the underlying physics/chemistry. Apparently the sources I encountered are not representive. I should have refrained from statement about availability of information. Commented Jul 7 at 5:16