Imagine a furnace that floats freely in space and is enveloped in a massive cloud of gas. You are in control of the fuel supply to the furnace. At the lowest setting, the heat it produces is radiatively conducted away through the gas cloud and radiated from its surface. Then at some critical setting, some part of the cloud becomes convective - in places cells of hot gas rise, just as hot air rises from a radiator, and are replaced by downward moving cells of cold gas. As you turn up the fuel supply more, the convectively stirred region grows steadily in radial extent. Eventually, the cloud becomes fully convective.
Consider the location, at each stage in the experiment, of the cloud in a plot of luminosity L versus surface temperature T. Initially there is not much heat to radiate, so both L and T are small. Then as L rises, T tends to rise also. But it turns out that once the cloud becomes fully convective, T becomes very nearly independent of L; at this point the cloud swells sufficiently rapidly with increasing L that the additional luminosity can be radiated from the surface without an appreciable change in temperature. Hayashi (1961) discovered this surprising behavior of fully convective stars, and their locus in the (T, L) plane is called the Hayashi line in his honor.
Figure 1 Left panel: the evolution of stars massive enough to burn H on the MS before they reach the MS. Each curve is labeled with the object’s mass in solar units. At first these objects are fully convective and descend almost vertically along the Hayashi track. Stars with masses ℳ ≳ 0.3 ℳ⊙ eventually cease to be fully convective and move sharply leftwards towards the ZAMS. The positions reached by stars after 1, 10, and 100 Myr from birth are marked with dots. Right panel: the equivalent figure for low-mass objects. At first the descent of these objects is slowed by the combustion of Li and D. Once an object has exhausted its D, it moves more rapidly towards lower temperatures and luminosities. [From data published in D’Antona & Mazzitelli (1994)]
