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My understanding is that giant stars are formed when they leave the main sequence and begin to burn elements heavier than hydrogen. Can some stars, however, be born as giants or do all stars start off as main sequence dwarfs?

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It depends what you mean by "born". If by "born" you mean commence nuclear fusion of hydrogen, then all stars are "born" on the main sequence as hydrogen burning stars and there aren't any exceptions, but massive main sequence stars can be as large as evolved red giants. However, if by "born" you mean the earliest stages of star formation then yes, stars that are contracting towards the main sequence and are yet to fuse hydrogen in their cores are big and red and have similar radii, temperatures and luminosities to red giants.

Details

Not all giant stars have evolved from the main sequence. Some main sequence stars are small (G, K and M dwarf stars), but more massive main sequence stars are much bigger and can approach the size of cooler "red giants" that have evolved off the core hydrogen burning main sequence.

The reason for this is that main sequence stellar luminosity depends strongly on mass as something like $L \propto M^{4}$, while the surface temperature goes as something like $T_{\rm eff} \propto M^{1/2}$, but the luminosity of a star is controlled by both its radius and its effective surface temperature as $L \propto R^2 T_{\rm eff}^4$. Thus the radius of a main sequence star varies as something like $R \propto M$. Plotting lines of constant radius on a Hertzsprung-Russell diagram makes this clearer (see below).

HR diagram with lines of constant radius

The diagram shows that the hottest main sequence stars (with masses $>30M_\odot$) have radii that are comparable with red giants like Arcturus or Aldebaran. Such massive stars are extremely short-lived (and hence rare) and have a very short (few million years or less) main sequence phase. An example would be Theta Orionis C in the Orion Nebula cluster, which is an extremely young O-type main sequence star.

What is not shown on this diagram though is stars that are in the pre main-sequence phase. These stars are not yet burning hydrogen in their cores but occupy the same region in the Hertzsprung-Russell diagram as red giants and hence have similar radii to red giants. They are contracting towards the main sequence on timescales of millions of years and can be found in star forming regions - see the schematic below, showing of the track of a pre-main sequence star as it evolves towards the main sequence.

Pre main sequence stars

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  • $\begingroup$ For anyone else curious about the S-shaped kink near the end of the pre-main sequence evolution, it's apparently the Henyey track. $\endgroup$ Mar 26, 2023 at 20:45
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You seem to be mixing up red giants, a late stage in the life cycle of most stars, and giant stars in general, which are any stars with particularly large radius and luminosity.

Stars larger than about a quarter of the sun's mass will greatly expand and cool toward the end of their life, forming red giants with a small, dense core and fusion occurring in shells around the core. Not all red giants have this origin, and not all giants are red giants. Rigel, for example, is a blue supergiant, Betelgeuse is a red supergiant, and neither was ever a main sequence dwarf.

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    $\begingroup$ This seems rather wrong to me. Both Rigel and Betelgeuse started as main sequence stars, which means they were dwarfs, by definition. (Are you suggesting neither star ever fused hydrogen in their cores?) $\endgroup$ Mar 25, 2023 at 3:49
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The answer literally depends on which direction you look. Some "dwarfs" born on the main sequence could be as big as "giants" that have evolved off it.

Wikipedia reports that the main sequence, into which all stars are born, is called "dwarfs", but these "dwarfs" merge with the evolved "giants" in their position on the Hertsprung-Russel Diagram and their physical characteristics:

Main-sequence stars are called dwarf stars,[1][2] but this terminology is partly historical and can be somewhat confusing. For the cooler stars, dwarfs such as red dwarfs, orange dwarfs, and yellow dwarfs are indeed much smaller and dimmer than other stars of those colors. However, for hotter blue and white stars, the difference in size and brightness between so-called "dwarf" stars that are on the main sequence and so-called "giant" stars that are not, becomes smaller. For the hottest stars the difference is not directly observable and for these stars, the terms "dwarf" and "giant" refer to differences in spectral lines which indicate whether a star is on or off the main sequence. Nevertheless, very hot main-sequence stars are still sometimes called dwarfs, even though they have roughly the same size and brightness as the "giant" stars of that temperature.[3]

We see this merging in the H-R diagram reproduced below from the article. The white and bluish stars at the upper end of the S-shaped main sequence curve are actually brighter as well as hotter than the collection of "giant" stars seen to the right. Astrophysicists use spectroscopic information, not size or brightness, to distinguish such main-sequence "dwarfs" from evolved "giants".

enter image description here

Source

Cited References

  1. Harding E. Smith (21 April 1999). "The Hertzsprung-Russell Diagram". Gene Smith's Astronomy Tutorial. Center for Astrophysics & Space Sciences, University of California, San Diego. Retrieved 2009-10-29.

  2. Richard Powell (2006). "The Hertzsprung Russell Diagram". An Atlas of the Universe. Retrieved 2009-10-29.

  3. Moore, Patrick (2006). The Amateur Astronomer. Springer. ISBN 978-1-85233-878-7.

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    $\begingroup$ Having read about the possibility of some Gen 1 stars being > 10,000M⊙ I'm wondering if they will ever be added to this diagram... $\endgroup$
    – Michael
    Mar 26, 2023 at 20:34
  • $\begingroup$ A close look reveals that the upper left end on the main sequence curve reveals a few points marked with >10000 sun luminosities (masses are not plotted on this diagram). They would already be pretty big. $\endgroup$ Mar 26, 2023 at 20:38

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