The Hayashi Track

When a star first forms, it is powered not by nuclear fusion but simply by gravity. It shrinks, which causes a release of gravitational energy. This tends to heat it, which slows its shrinking. But it releases energy in the form of light, which tends to cool it down. Then it can keep shrinking. So you need math to figure out exactly what happens.

In 1961, Chushiro Hayashi figured it out. Stars between one tenth and twice the mass of the Sun begin life by shrinking while staying at the same temperature! As a result their overall luminosity goes down. This is called the ‘Hayashi track’, and you can see it happening in the vertical blue lines here:

Later, the star’s surface cools down—but for some reason the star expands enough to get brighter! Then we say it’s on the ‘Henyey track’. This change happens sooner for more massive stars. You can see it happening when the blue lines bend to the right and go back up.

The red lines say how old a star is. You can see that our Sun followed the Hayashi track for about 10 million years. In the process, its luminosity decreased by a factor of 10.

The region to the right of the blue lines is called the ‘forbidden zone’. It’s impossible for stars to stay here for long, because they’re highly convective: hot gas from below rises to the surface, quickly cooling these stars. When stars are first born they start out here—but they very quickly move to the Hayashi track.

For more, try these:

• Wikipedia, Hayashi track.

• Wikipedia, Henyey track.

The boundary of the forbidden zone is called the ‘Hayashi limit’, and you can see a derivation of it here:

• Wikipedia, Hayashi limit.

I haven’t developed enough intution for stellar mechanics to explain these things well. As usual, it seems you need to learn the mathematical models, then think about them long enough until they become intuitive. It’s particularly important to get a good sense of the two mechanisms whereby energy moves through a star—radiation and convection—and when one or the other is dominant. But I don’t understand that well! So instead of trying to explain why young stars work the way they do, let me quote Wikipedia, which is pretty readable here:

Stars form when small regions of a giant molecular cloud collapse under their own gravity, becoming protostars. The collapse releases gravitational energy, which heats up the protostar. This process occurs on the free fall timescale, which is roughly 100,000 years for solar-mass protostars, and ends when the protostar reaches approximately 4000 K. This is known as the Hayashi boundary, and at this point, the protostar is on the Hayashi track. At this point, they are known as T Tauri stars and continue to contract, but much more slowly. As they contract, they decrease in luminosity because less surface area becomes available for emitting light. The Hayashi track gives the resulting change in temperature, which will be minimal compared to the change in luminosity because the Hayashi track is nearly vertical. In other words, on the HR diagram, a T Tauri star starts out on the Hayashi track with a high luminosity and moves downward along the track as time passes.

The Hayashi track describes a fully convective star. This is a good approximation for very young pre-main-sequence stars because they are still cool and highly opaque, so that radiative transport is insufficient to carry away the generated energy and convection must occur. Stars less massive than 0.5 M☉ remain fully convective, and therefore remain on the Hayashi track, throughout their pre-main-sequence stage, joining the main sequence at the bottom of the Hayashi track. Stars heavier than 0.5 M_☉ have higher interior temperatures, which decreases their central opacity and allows radiation to carry away large amounts of energy. This allows a radiative zone to develop around the star’s core. The star is then no longer on the Hayashi track, and experiences a period of rapidly increasing temperature at nearly constant luminosity. This is called the Henyey track, and ends when temperatures are high enough to ignite hydrogen fusion in the core. The star is then on the main sequence.

Lower-mass stars follow the Hayashi track until the track intersects with the main sequence, at which point hydrogen fusion begins and the star follows the main sequence. Even lower-mass ‘stars’ never achieve the conditions necessary to fuse hydrogen and become brown dwarfs.

Why does having a higher interior temperature decrease a star’s central opacity? I would have thought more ionization increased the opacity.

By the way, the image in this post came from Steven W. Stahler and Francesco Palla’s paper “The formation of stars”, and it was republished here:

• Francesco Palla, 1961–2011: Fifty years of Hayashi tracks, First Stars IV: From Hayashi to the Future, AIP Conference Proceedings, Vol. 1480, 2012, pp. 22–29.