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I was thinking about the possibility of determining what phase of evolution a variable star is in when it is located in the instability strip (say, a Cepheid object). Can you differentiate whether it is evolving toward the blue (in HR diagram) or back toward the giant branch?

I was looking at a figure recently in Carroll and Ostlie's Intro to Astrophysics text, which brought this to mind. It shows on the HR diagram where Cepheids lie, which is right in this strip.

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You can, in fact, differentiate between the crossings of the instability strip in certain types of variable stars. There have been quite a few evolutionary models devoted to this, and many, as well as scores of observations, show that the pulsational period $P$ of a Cepheid variable changes during each crossing. Look at, for instance, the period changes of 60 galactic Cepheids, shown in Figure 2 of Pietrukowicz:

enter image description here

The first and third crossings occur when the star is evolving redwards (towards the right of the HR diagram), and show positive $\dot{P}$. The second crossing occurs when the stars are evolving bluewards (towards the left of the HR diagram), and show negative $\dot{P}$.

Models and observations by Neilson et al. (2014) show the same thing (see Figure 1):

enter image description here

The initial large period change indicates the first crossing, the smaller negative period change indicates the second, and the third less drastic increase indicates the third.

Similar indicators should be present in further crossings (fourth and fifth crossings in the later stages of core and shell helium burnings), and while the amplitude of $\dot{P}/P^2$ would likely be different, the same general positive/negative trends should be present. For more information, see Turner & Berdnikov (2004).

There doesn't seem to be as much information on period changes in non-Cepheid variable stars, although Smith (2013) states that most Mira variables show only minute period changes, if any, and RR Lyrae variables may show period increases as they evolve redwards across the instability strip toward the end of their lives, as is the case with Cepheids. This, by the way, does not fully agree with evolutionary theory.

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