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Why Helium-burning stars are found in a horizontal branch on the H-R diagram? Why do they have roughly the same luminosity?

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This is explained in the Wikipedia article

Stars on the horizontal branch all have very similar core masses, following the helium flash. This means that they have very similar luminosities, and on a Hertzsprung–Russell diagram plotted by visual magnitude the branch is horizontal.

To expand a little. In stars of a certain mass range, helium builds up in the core until it reaches a specific mass, at which point the "Helium flash" occurs and burning of helium to carbon and oxygen starts throughout the core. When things settle down, helium burning is going on in the core, which is more or less the same size, independently of the original mass of the star. Since this is the main power source of these stars, they all have about the same luminosity.

The variation across the branch comes from how much remaining gas there is outside the helium-burning shell -- more gas means a larger cooler star radiating the same total amount of energy

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    $\begingroup$ Several important corrections: the helium flash does not fuse most of the helium, only a tiny fraction. So a horizontal branch star has a helium-fusing core (not a helium fusing shell). It is true that these cores start out with similar masses, and light diffusion through that core sets the luminosity, so that's why they are all similar luminosity. The amount of hydrogen does set the size and surface temperature, and because it is puffed out a bit by a hydrogen burning shell, the light that has diffused out of the core passes through the envelope without altering the luminosity much. $\endgroup$ – Ken G Apr 2 '18 at 13:31
  • $\begingroup$ Thanks. I had misunderstood about the helium flash. Will edit when I have a minute $\endgroup$ – Steve Linton Apr 2 '18 at 15:09
  • $\begingroup$ @SteveLinton "When things settle down, helium burning is going on in the core, which is more or less the same size, independently of the original mass of the star." I don't understand this. Why the core should be more or less the same size? $\endgroup$ – velut luna Apr 7 '18 at 4:38
  • $\begingroup$ Because that's the size at which helium burning starts $\endgroup$ – Steve Linton Apr 8 '18 at 20:10
  • $\begingroup$ @SteveLinton Then why doesn't similar argument hold for Hydrogen fusion (Main sequence star)? That is, to say that hydrogen fusion starts when the core size is more or less the same, so that all main sequence stars have roughly the same luminosity? $\endgroup$ – velut luna Apr 9 '18 at 17:09

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