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This is a basic question, but I may as well ask it. I had always thought that the core of a main-sequence star is defined as the part hot enough for nuclear fusion. Some dictionaries seem to agree with this.

However, many books say that protostars and pre-main-sequence stars also have cores, though they are not undergoing hydrogen fusion.

So what exactly is the definition of a stellar core? If it doesn't necessarily have to be hot enough for fusion, what separates it from the rest of the star?

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I think that there isn't a strict answer to this question. However, I believe the answer is that there's a difference between the core of a hydrogen-burning star and the core of a protostar or star-forming, gas cloud.

For a hydrogen-burning star, the core, as you say, is the region of the star where fusion is taking place. This is surrounded by the radiative and convective zones (which one comes first depends on the mass of the star). For true stars, the concept of the core is well defined.

For things like protostars and star-forming gas clouds, the concept of the core is less well defined and more a nomenclature for a region of the object rather than a strict definition. I don't think it is correct to assume that because one is talking about the core of a protostar, that they are referring to the core as it is defined for a hydrogen-burning star.

So what does it mean to be a core in a protostar or star-forming gas cloud? I will first point out that protostars can still be fusing. In the pre-main sequence stage, most stars will be fusing deuterium and we can consider the core to be the region where this is occurring. A protostar does not become a fully fledged star until it begins hydrogen burning, but deuterium fusion requires lower temperatures and pressures than the full proton-proton chain of hydrogen fusion (of which deuterium fusion is a part) and so can occur in protostars.

We might also distinguish a core from outer regions based on a density or compositional change. If we plot up something like the density (or really any property that changes with radius), we are likely to see a drastic jump signifying some sort of boundary. Cores for star-forming regions may be defined by highly dense regions, high temperature regions, or regions "where the physics changes" because we've passed some critical threshold at some radius. These cores are less well defined and can't be likened to stellar cores very well.

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You are right; when referring to stars such as the Sun, the core is usually defined as the volume in the centre where fusion is taking place. In the case of larger stars with fusion occurring in multiple layers, the core is still taken to be the area in the middle where nuclear fusion is producing the heaviest elements.

With protostars or pre-main-sequence stars, I haven't found any articles recently that specifically refer to cores as part of the star's structure. It is possible they are using the word 'core' to just mean 'the centre'. If you're able to provide links to where you have seen these terms used, we might be able to offer some context.

It is possible that bodies not hot enough to begin hydrogen fusion, such as brown dwarves, may have layered and differentiated structures akin to Jupiter, in which case, the 'core' is the solid or near-solid region in the centre where accreted heavier elements eventually accumulate. The core's boundary would then be taken as the region where one would transition from heavy elements to the compressed metallic hydrogen that surrounds it.

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