(Remember, for a collapsing cloud, both dR/dt and dE/dt are negative numbers.) If we solve

The average luminosity is this energy divided by the time over which it is radiated:

This is the average luminosity over the 100 year period, but the actual luminosity at the end of the period is higher, since | dR/dt | is greatest then.

Once a cloud is producing stellar luminosities by gravitational collapse, we call it a protostar. Once the cloud becomes opaque the radiation can only escape from near the surface. (When the opacity is low a photon can escape from anywhere within the volume.) Since energy escapes slowly, the temperature rises quickly. Also, a large temperature difference can exist between the center and the edge. Under these conditions, the most efficient form of energy transport from the center to just outside is by convection. This point was first realized in 1961 by the Japanese astrophysicist Chushiro Hayashi. During this stage the surface temperature stays roughly constant at about 2500 K. Since the radius is decreasing, and the temperature is approximately constant, the luminosity decreases.

During this stage the central temperature is still rising. When it is high enough, nuclear reactions start. The contraction goes on for some time in the outer parts, as the pressure builds up in the core. Eventually the pressure in the core is sufficient to halt the collapse, and the star is ready to settle into its main sequence existence.

For a protostar, the continuous spectrum peaks in the near infrared. The dust in the collapsing cloud surrounding the protostar will absorb some of the radiation. The dust will be heated, but will not be the same temperature as the star. The emission from the dust will be in the far infrared. From this we see that protostars are best observed in the infrared part of the spectrum.

15.5.2 Evolutionary tracks for protostars

When we plot an HR diagram with stars we see now, we are plotting the distribution of L and T as they are now. However, as a star evolves, its luminosity and temperature change. Therefore, its location on an HR diagram changes. If L(t) is the luminosity of a star as a function of time and T(t) is the temperature as a function of time, we can plot a series of points and connect them to follow the evolution of a star. Such a series of points is called an evolutionary track. Stars evolve so slowly compared with human lifetimes that we cannot deduce the evolutionary track by observing one star. However, by observing many stars, each at a different stage, we can infer the evolutionary tracks. (We have already used evolutionary tracks in our discussion of post main sequence evolution, in Chapters 10 and 11.)

We can also predict evolutionary tracks from theoretical models of protostars and stars. We use basic physics to calculate the physical conditions, and see how the star's radius and temperature change with time. Since the luminosity is given by L = (4vR2)(a T4), we can relate changes in R and T to changes in L and T. When we calculate model tracks, we find that the evolutionary track of a protostar depends on its mass. This is not surprising, since we have already seen that the mass determines where a star will appear on the main sequence.

Some evolutionary tracks for protostars and pre-main sequence stars are shown in Fig. 15.6. Note that the protostars appear above the main sequence. This means that for a given temperature, T, protostars are more luminous than main sequence stars of the same temperature. Protostars are also larger than main sequence stars of the same temperature. This is not surprising since protostars are still collapsing. Once the accretion phase stops, but before the main sequence is reached, we call these objects pre-main sequence stars.

Fig. 15.7 shows a model for the collapse of an interstellar cloud into a 1 M© protostar. At first the cloud is cool, and then it contracts and heats. As discussed above, the T4 increase is greater than the R2 decrease, and the luminosity of the protostar increases. The peak luminosity is reached when the temperature reaches 600 K. As the protostar becomes denser, its opacity increases. Eventually, it is harder for the radiation from the center to escape, and the luminosity begins to decrease. During this stage energy transport in

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