[Eternal Student (OP)]

Hi.

    I'm looking for references to explain why the Luminosity of a Cepheid has the relationship it has with the period of pulsation.

   Wikipedia has plenty of talk about the data and the fact there is such a relationship but only limited waffle about the why.

   Some on-line reference would be great, although I think I've got a very old copy of   "An introduction to modern astrophysics", Carroll and Ostlie where I can find it.   I'll have a glance at that if no one has a better suggestion.

.... Actually I've had 10 minutes to look through Carroll and Ostlie before finishing writing this post.   They seem to be explaining the Luminosity change by temperature variations.... oh and pages of stuff... and  Helium ionisation gets a mention...  and more pages of stuff...

   Well,  before I spend hours ploughing through Carroll and Ostlie, I'd still be keen to seek alternative references.

Best Wishes.

[Origin]

That's a good question.  Last I knew there was not a very good answer as to what causes that relationship.  Maybe someone here knows of some recent theories.

[evan_au]

There are several groups of variable stars with slightly different period-brightness relationship:
- Type I Cepheids are 4-20 times more massive than the Sun; since luminosity increases as the 4th power of the mass, these are very bright stars indeed.
- Type II Cepheids are around half the mass of the Sun, and very old stars (judging by the very low levels of elements beyond Helium).

In both cases, the ionisation of Helium is thought to be a driving factor:
- When Helium is fully ionised, it is more opaque; when it is only singly ionised, it is more transparent
- When Helium is transparent, it lets out light from the surface of the star, and the outer atmosphere of the star cools and shrinks.
- As it compressed, the outer atmosphere heats up, and more Helium becomes doubly ionised. This has two effects: (1) from outside the star, the light is attenuated, ie the star appears dimmer, and (2) The atmosphere heats up and starts to expand
- As the atmosphere expands, it cools, and the Helium becomes more transparent (ie the star looks brighter from a distance)
- The cycle then repeats

The variable behavior is just a phase that stars pass through. Massive stars like Type I Cepheids burn their fuel quickly, and are likely to explode in a supernova.

See: https://en.wikipedia.org/wiki/Cepheid_variable#Pulsation_model

[Eternal Student (OP)]

Hi.

Thanks @Origin and @evan_au

See: https://en.wikipedia.org/wiki/Cepheid_variable#Pulsation_model

   I have already done that.
   The ionisation of Helium explains why the brightness pulsates.   At least it explains the pulsation if the Helium ionisation layer is right out on the surface of the Cepeheid star where it can act like varible light shade. 
    What is missing completely is an explanation of why the average luminosity is as large as it is.   You need something linking the avergage luminosity to the period of pulsation.   Why is a rapidly pulsing star of overall lower absolute magnitude?

    I've only just started to glance at the Carroll and Ostlie text mentioned.   They've been explaining the total luminosity as a function of temperature of the surface and the total amount of surface.  The total amount of surface doesn't seem to be too important, there's only a small % variation anyway.  It seems that the surface temperature is what shows the greatest variation.
   Now they do mention Helium ionisation but the situation is complex since they suggest that depending on the size and temperature of the star the single-to-double Helium ionisation layer isn't always at the surface of the star, sometimes it's much deeper within it.   The Opacity of the Helium ionisation layer should have no effect on the amount of light emitted by the surface if it's actually underneath the surface.
    I don't know, haven't finished reading and my copy of the book is years out of date anyway.   Maybe it is the latest opinion that there is always a suitable Helium ionisation layer on the outer surface of the star.

Best Wishes.

LATE EDITING:   This is too many words.  Let's try to shrink it or summarise it:
The Helium ionisation might explain why the star pulsates in brightness.   However, it does nothing to link the average luminosity with the period of that pulsation.

[evan_au]

This researcher used open-source MESA software to model the behaviour of stars which enter the Cepheid variable phase. He tabulates temperature, radial velocity, etc for several stars he modeled.
http://www.aoc.nrao.edu/events/nmsymposium/2019/contributions/talks/guzik.pdf

Some of his diagrams suggest that an individual star can start oscillating due to one effect, and stop. Then later in it's lifetime, it can start oscillating due to a different effect and stop, potentially entering and exiting this behaviour several times before the star eventually dies.

[Eternal Student (OP)]

Hi.

Thanks @evan_au .    I've looked through those slides (presumably they were over-head slides).

Some of his diagrams suggest that an individual star can start oscillating due to one effect, and stop. Then later in it's lifetime, it can start oscillating due to a different effect and stop, potentially entering and exiting this behaviour several times before the star eventually dies.

   The slides are a little hard to follow without any audio description but I think I can see what you're saying.   Some of their/his stellar evolution lines cross the instability strip a couple of times (maybe more).  It's hard to follow the evolution once there are several lines on the diagrams (but I'll take your word on it).
    The slide about V1334 Cygnus showed a change from an overtone mode of pulsation to the fundamental mode, which was more obvious.  It's hard to guess what effect caused that, the expansion / contraction may still have been driven at the same frequency (based on what is in his auxiliary slides) and that drive may have been caused by the Helium ionisation layer without significant change - but something about the star changed as time progressed (density or something in the outer layers).
   [NOTE:  Just my best guesses from what I can see in the slides.   Its biased by, and based on, the stuff mentioned in Carrol and Ostlie's book where they describe the Helium III ionisation layer as being somewhere within the star and the main piston driving the expansion and contraction of the star].

He tabulates temperature, radial velocity, etc for several stars he modeled.

   If I've followed the diagrams correctly, on slide page 14 the effective temperature of a variable is shown changing from 4600 → 5200 → back to 4600K  in a time period of about 5.4 days (matching the period of brightness pulsation).   That's consistent with what the Carroll and Ostlie book is saying:   Changes in Luminosity seem to be explained just by changes in surface temperature  (not by the variable opacity of the Helium ionisation layer and any assumption that this would be in the surface layers where it can act as a variable light shade).

Other interesting results from the slides
    Plenty of (minor but sometimes not with overlapping error bars) disagreements between models he refers to as being "in the literature",   the models produced by the MESA software    and   real data from observation.

Thanks very much indeed for finding that information.  I've enjoyed looking through it.

Best Wishes.

[evan_au]

Changes in Luminosity seem to be explained just by changes in surface temperature, not by the variable opacity of the Helium ionisation layer

The visible light we see is from the surface of the star. Depending on pressure vs temperature vs altitude, I could imagine:
- The Helium "switch" could be beneath this surface layer of (mainly) hydrogen, cutting off the surface layer from the underlying heat source, causing the surface temperature to fluctuate. What we see is mostly the changing temperature of Hydrogen.
- The Helium "switch" could be above this surface layer of (mainly) hydrogen, cutting off our view of the surface layer except when the helium layer is particularly transparent, causing the surface temperature to fluctuate. What we see is mostly the changing temperature of Helium.