shifts left on the HR diagram, towards higher temperatures, as mass increases. Although this model is extremely crude, these qualitative observations are fully supported by numerical simulations. At high temperatures, the atmosphere's opacity begins to be dominated by Kramers' opacity law instead of the H- ion, with a=1 and b=-4.5 In that case, A=0.2 in our crude model, far higher than 0.05, and the star is no longer on the Hayashi track. In Stellar Interiors, Hansen, Kawaler, and Trimble go through a similar derivation without neglecting multiplicative constants, and arrived at: where is the molecular weight per particle. The authors note that the coefficient of 2600K is too low—it should be around 4000K—but this equation nevertheless shows that temperature is nearly independent of luminosity. Numerical results The diagram at the top of this article shows numerically computed stellar evolution tracks for various masses. The vertical portions of each track is the Hayashi track. The endpoints of each track lie on the main sequence. The horizontal segments for higher-mass stars show the Henyey track. It is approximately true that: . The diagram to the right shows how Hayashi tracks change with changes in chemical composition. Z is the star's metallicity, the mass fraction not accounted for by hydrogen or helium. For any given hydrogen mass fraction, increasing Z leads to increasing molecular weight. The dependence of temperature on molecular weight is extremely steep—it is approximately . Decreasing Z by a factor of 10 shifts the track right, changing by about 0.05. Chemical composition affects the Hayashi track in a few ways. The track depends strongly on the atmosphere's opacity, and this opacity is dominated by the H- ion. The abundance of the H- ion is proportional to the density of free electrons, which, in turn, is higher if there are more metals because metals are easier to ionize than hydrogen or helium. Observational status Observational evidence of the Hayashi track comes from color-magnitude plots—the observational equivalent of HR diagrams—of young star clusters. For Hayashi, NGC 2264 provided the first evidence of a population of contracting stars. In 2012, data from NGC 2264 was re-analyzed to account for dust reddening and extinction. The resulting color-magnitude plot is shown at right. In the upper diagram, the isochrones are curves along which stars of a certain age are expected to lie, assuming that all stars evolve along the Hayashi track. An isochrone is created