5.3: Evolution onto the Main Sequence

a. Problems concerning the Formation of Stars

Since we began this discussion with the assumption that stars form by contraction from the interstellar medium, honesty requires that we describe several forces that mitigate against that contraction. For a star to form by gravitational contraction from the interstellar medium, all sources of energy which support the initial cloud must be dominated by gravity. For the typical interstellar cloud with sufficient mass to become a star, we shall see that not only is this not true of the collective sources of energy, but also it is not true of them individually.

The Internal Thermal Energy From the Virial theorem as derived in Chapter 1 [equation \ref{1.2.34}], the internal kinetic energy of the gas of the cloud must be less than one-half the gravitational energy in order for the moment of inertia to show any accelerative contraction. Thus for a uniform density gas at a certain temperature T, the mass must be confined inside a sphere of a certain radius R_c. That radius can be found from \[2\left(\frac{3 \rho k T}{2 \mu m_h}\right)\left(\frac{4 \pi R_c^3}{3}\right) \leq \frac{G M^2}{R_c}\label{5.2.1}\]

or \[R_c \leq \frac{G M \mu m_h}{3 k T} \approx \frac{0.25\left(M / M_{\odot}\right)}{T}\quad \mathrm{pc}\label{5.2.2}\]

This distance is sometimes known as the Jeans length, for it is the distance below which a gas cloud becomes gravitationally unstable to small fluctuations in density. For a solar mass of material with a typical interstellar temperature of 50 K, the cloud would have to be smaller than about 5×10^-3 pc with a mean density of about 10⁸ particles per cubic centimeter. This is many orders of magnitude greater than that found in the typical interstellar cloud, so it would seem unlikely that such stars should form.

The Rotational Energy The Virial theorem can also be used to determine the effects of rotation on a collapsing cloud. Again, from Chapter 1, the rotational kinetic energy must be less than one-half the gravitational potential energy in order for the cloud to collapse. So \[2\left(\frac{1}{2} \mathrm{I} \omega^2\right) \leq \frac{G M^2}{R_c}\label{5.2.3}\]

which for a sphere of uniform density and constant angular velocity gives \[R_c \leq\left(\frac{5 G M}{2 \omega^2}\right)^{1 / 3}\label{5.2.4}\]

The differential rotation of the galaxy implies that there must be a shear or velocity gradient which would impart a certain amount of rotation to any dynamical entity forming from the interstellar medium. For an Oort constant, A = 16 km/s/kpc, this implies that \[R_c \leq 0.9\left(\frac{M}{M_{\odot}}\right)^{1 / 3} \quad \mathrm{pc}\label{5.2.5}\]

Thus, it would seem that to quell rotation, the initial mass of the sun must have been confined within a sphere of about 0.7 pc.

Magnetic Energy A similar argument concerning the magnetic energy density M, where \[M=\frac{H^2}{8 \pi} \frac{4 \pi R_c^3}{3}\label{5.2.6}\]

can be made by appealing to the Virial theorem with the result that \[R_c \leq 0.37\left(\frac{M}{M_{\odot} H_{\mu g}}\right)^{1 / 2}\quad\mathrm{pc}\label{5.2.7}\]

For a value of the ambient interstellar galactic magnetic field of 5 microgauss we get \[R_c \leq 0.17\left(\frac{M}{M_{\odot}}\right)^{1 / 2}\quad\mathrm{pc}\label{5.2.8}\]

How are we to reconcile these impediments to gravitational contraction with the fact that stars exits? One can use the rotational and magnetic energies against one another. A moderate magnetic field of a spinning object will cause a great deal of angular momentum per unit mass to be lost by a star through the centripetal acceleration of a stellar wind. The resulting spin-down of the star will weaken the internal sources of the stellar magnetic field itself. Observations of extremely slow rotation among the magnetic A_p seem to suggest that this mechanism actually occurs. Clouds can be cooled by the formation of dust grains and molecules, as long as the material is shielded from the light of stars by other parts of the cloud. The high densities and low temperatures observed for some molecular clouds imply that this cooling, too, is occurring in the interstellar medium. However, unless some sort of phase transition occurs in the material, the thermal cooling time is so long that it is unlikely that the cloud will remain undisturbed for a sufficient time for the Jeans' condition to be reached. Thus it seems unlikely that the Jeans' condition can be met for low-mass clouds.

It is clear from equation \ref{5.2.2} that \(\mathrm{R_c \sim \sqrt{T / \rho}}\), so for a given temperature the Jeans' length increases with decreasing density. However, the Jean's mass increases as the cube of the Jeans' length. Thus, for a cloud of typical interstellar density to collapse, it must be of the order of 10⁴ M_⊙. It is thought that the contraction of these large clouds creates the conditions enabling smaller condensations within them to form protostars. The pressure that the large contracting cloud exerts on smaller internal perturbations of greater density may squeeze them down to within the Jeans' length after which these internal condensations unstably contract to form the protostars of moderate mass. These are some of the arguments used to establish the conditions for gravitational contraction upon which all stellar formation depends, and since stars do form, something of this sort must happen.

b. Contraction out of the Interstellar Medium

Since we have given some justification for the assumption that stars will form out of clouds of interstellar matter which have become unstable to gravitational collapse, let us consider the future of such a cloud.

Homologous Collapse For simplicity, consider the cloud to be spherical and of uniform density. The equation of motion for a unit mass of material somewhere within the cloud is \[\frac{d^2 r}{d t^2}=-\frac{G M(r)}{r^2}\label{5.2.9}\]

If we assume that the material at the center doesn't move [that is, v(0) = 0], then the first integral of the equations of motion yields \[v^2=\frac{8 \pi G \rho_0}{3} \int_0^r r d r=\frac{4 \pi G \rho_0}{3} r^2\label{5.2.10}\]

or \[\mathbf{v} \propto \mathbf{r}\label{5.2.11}\]

This says that at any time the velocity of collapse is proportional to the radial coordinate. This is a self-similar velocity law like the Hubble law for the expansion of the universe, only in reverse. Thus, at any instant the cloud will look similar to the cloud at any other point in time, only smaller and with a higher density ρ₀. Thus, the density will remain constant throughout the cloud but steadily increase with time. Since the velocity is proportional to r, the collapse is homologous and we obtain Lane's law of Chapter 2 [equation \ref{2.3.9}] which completely specifies the internal structure throughout the collapse.

One should not be left with the impression that this homologous collapse is uniform in time. It is not; rather, it proceeds in an accelerative fashion, resulting in a rapid compaction of the cloud. When the density increases to the point that internal collisions between particles produce a pressure sufficient to oppose gravity, the equations of motion become more complicated. Some of the energy produced by the collapse leaks away in the form of radiation from the surface of the cloud and a temperature gradient is established. These processes destroy the self-similar, or homologous, nature of the collapse, and so we must include them in the equations of motion.

However simple and appealing this solution may be, it is a bit of a swindle. The mathematics is correct, and the assumption that v(0) = 0 may be quite reasonable. However, it is unlikely that most clouds are spherically symmetric and of uniform density. Density fluctuations must exist and without the pressures of hydrostatic equilibrium to oppose the central force of gravity, there is no a priori reason to assume spherical symmetry. Normally this would seem like unnecessary quibbling with an otherwise elegant solution. Unfortunately, these perturbations are amplified by the collapse itself and destroy any possibility of the cloud maintaining a uniform density.

Non-Homologous Collapse Let us consider the same equations of motion as before so that the first integral is given by \[\int_{v_i^2}^{v^2} d v^2=2 \int_{r_i}^r \frac{4 \pi}{3} G \rho_0 r d r=2 \int_{r_i}^r \frac{G M(r) d r}{r^2}\label{5.2.12}\]

Now we wish to follow the history of a point within which the mass is constant so that M(r) is constant and \[v^2(R)-v_i^2=2 G M(R)\left(\frac{1}{R}-\frac{1}{R_i}\right)\label{5.2.13}\]

The variable R has replaced r, and this introduces a minus sign into equation \ref{5.2.13}, since for a collapse dr = -dR. Equation \ref{5.2.13} is just the energy integral, so there are no surprises here. Now we change variables so that \[x=\frac{R}{R_i} \quad \alpha=\frac{2 G M\left(R_i\right)}{R_i^3} \quad u=\frac{v}{R_i}\label{5.2.14}\]

If we take the initial velocity of the cloud to be zero and the initial value of x to be 1, then equation \ref{5.2.13} becomes \[u^2=\left(\frac{d x}{d t}\right)^2=\frac{\alpha(1-x)}{x} \quad 0 \leq x \leq 1\label{5.2.15}\]

which can be integrated over time to give \[t(x)=\alpha^{-1 / 2}\left[\frac{\pi}{2}-\operatorname{Sin}^{-1} \sqrt{x}+\left(x-x^2\right)^{1 / 2}\right]\label{5.2.16}\]

Now a can be related to the mean density, so that we can rewrite equation \ref{5.2.16} as \[t\left(\frac{R}{R_i}\right)=\left[\frac{3}{8 \pi G \overline{\rho_i\left(R_i\right)}}\right]^{1 / 2}\left\{\frac{\pi}{2}-\operatorname{Sin}^{-1}\left(\frac{R}{R_i}\right)^{1 / 2}+\left[\frac{R}{R_i}+\left(\frac{R}{R_i}\right)^2\right]^{1 / 2}\right\}\label{5.2.17}\]

If \(\mathrm{\langle\rho_i(R_i)}\rangle\) is constant and not a function of \(\mathrm{R_i}\), then we recover the homologous contraction which is clearly not uniform in time. However, if the initial mean density is a decreasing function of \(\mathrm{R_i}\), then the collapse time of a sphere of material \(\mathrm{M(R_i)}\) is an increasing function of \(\mathrm{R_i}\). This means that initial concentrations of material will become more concentrated and any inhomogeneities in the density will grow unstably with time.

This is essentially the result found by Larson¹ in 1969. If the cloud is gravitationally confined within a sphere of the Jeans' length, the cloud will experience rapid core collapse until it becomes optically thick. If the outer regions contain dust, they will absorb the radiation produced by the core contraction and reradiate it in the infrared part of the spectrum. After the initial free-fall collapse of a 1M_⊙ cloud, the inner core will be about 5 AU surrounded by an outer envelope about 20000 AU When the core temperature reaches about 2000 K, the H₂ molecules dissociate, thereby absorbing a significant amount of the internal energy. The loss of this energy initiates a second core collapse of about 10 percent of the mass with the remainder following as a "heavy rain". After a time, sufficient matter has rained out of the cloud, and the cloud becomes relatively transparent to radiation and falls freely to the surface, producing a fully convective star. While this scenario seems relatively secure for low mass stars (i.e., around 1M_⊙), difficulties are encountered with the more massive stars. Opacities in the range of 1500 to 3000 K make the evolutionary tracks somewhat uncertain. Indeed, there are some indications that massive stars follow a more homologous and orderly contraction to the state where they become fully convective.

Although this is the prevailing picture for the early phases of the evolution for low-mass protostars, there are some difficulties with it. Such stars would be shielded from observation by the in-falling rain of material until quite late in their formation. Since the entire configuration including the rain is hardly in a state of hydrostatic equilibrium, the arguments given below would not pertain until quite late in the star's formation, by which time the star may well have reached the main sequence. There seems to be little support in observation for this point of view, and the entire subject is still somewhat controversial.

Michael Disney² has pointed out that the details of the collapse from the interstellar medium depend critically on the ratio of the sound travel time to the free-fall time in the contracting protostar. Although this ratio is typically unity [equations \ref{3.2.4}, \ref{3.2.6}, and \ref{3.2.9}], small departures from unity appear to matter. The free-fall time is basically the time during which the collapse takes place, and the sound travel time is the time required for the interior to sense the effects of pressure disturbances initiated at the boundary. Thus, if \(\tau_{\mathrm{s}} / \tau_{\mathrm{f}}>1\), the interior tends to be unaffected by the boundary pressure during the collapse. Any external pressure will then tend to compress the matter in the outer part of the collapsing cloud without affecting the interior regions, removing any density gradients that may exist in the perturbation and forcing the collapse to be more nearly homologous. This would reduce the effect of the rain and cause the protostar to collapse more as a unit. Any initial velocity resulting from of the homologous collapse of the large cloud will only exacerbate the situation by significantly shortening the time required for the collapse. Thus the initial phases of star formation remain in some doubt and probably depend critically on the circumstances surrounding the initial conditions of the collapse of the larger cloud.

c. Contraction onto the Main Sequence

Once the protostar has become opaque to radiation, the energy liberated by the gravitational collapse of the cloud cannot escape to interstellar space. The collapse will slow down dramatically and the future contraction will be limited by the star's ability to transport and radiate the energy away into space. Initially, it was thought that such stars would be in radiative equilibrium and that the future of the star would be dictated by the process of radiative diffusion in the central regions of the star. Indeed, for most stars this is true for the phases just prior to nuclear ignition. However, Hayashi³ showed that there would be a period after the central regions became opaque to radiation during which the star would be in convective equilibrium.

Hayashi Evolutionary Tracks In Chapter 4 we found that once convection is established, it is incredibly efficient at transporting energy. Thus, as long as there are no sources of energy other than gravitation, the future contraction will be limited by the star's ability to radiate energy into space rather than by its ability to transport energy to the surface. We have also learned that the structure of a fully convective star will essentially be that of a polytrope of index n = 1.5. We may combine these two properties of the star to approximately trace the path it must take on the Hertzsprung-Russell diagram.

With gravitation as the only source of energy and the contraction taking place on a time scale much longer than the dynamical time, the Virial theorem allows one-half of the change in gravitational energy to appear as the luminosity and be radiated away into space. The other half will go into the internal energy of the star increasing the internal temperature. Thus, \[L=\frac{1}{2} \frac{d\left(G M^2 / R\right)}{d t}=-\frac{\frac{1}{2} G M^2}{R^2} \frac{d R}{d t}\label{5.2.18}\]

Since the luminosity is positive, dR/dt must be negative which ensures that the star will contract. Since the luminosity is related to the surface parameters by \[L=4 \pi R_*^2 \sigma T_e^4\label{5.2.19}\]

the change in the luminosity with respect to the radius will be \[\frac{d L}{d R_*}=\frac{4 L}{T_e} \frac{d T_e}{d R_*}+\frac{2 L}{R_*}\label{5.2.20}\]

Equation \ref{5.2.19} is essentially a definition of what we mean by the effective temperature. As long as the star remains in convective equilibrium, it will be a polytrope and the contraction will be a self-similar, and thus homologous, contraction.

Since the rate of stellar collapse is dictated by the photosphere's ability to radiate energy, we should expect the photospheric conditions to dictate the details of the collapse. Indeed, as we shall see in the last half of this book, the eigenvalues that determine the structure of a stellar atmosphere are the surface gravity and the effective temperature. So as long as the stellar luminosity is determined solely by the change in gravity, and the energy loss is dictated by the atmosphere, we might expect that the independent variable T_e to remain unchanged. However, it is necessary to show that such a sequence of models actually forms an evolutionary sequence. The extent to which this will be true depends on the radiative efficiency of the photosphere. This is largely determined by the opacity. At low temperatures the opacity will increase rapidly with temperature owing to the ionization of hydrogen. This implies that any homological increase of the polytropic boundary temperature at the base of the atmosphere will be met by an increase in the radiative opacity and a steepening of the resultant radiative gradient. This increase in the radiative opacity also forces the radiating surface farther away from the inner boundary, causing the effective temperature to remain unchanged. A much more sophisticated argument demonstrating this is given by Cox and Giuli⁴.

The star can effectively be viewed as a polytrope wrapped in a radiative blanket, with the changing size of the polytrope being dictated by the leakage through the blanket. The blanket is endowed with a positive feedback mechanism through its radiative opacity, so that the effective temperature remains essentially constant. The validity of this argument rests on the ability of convection to deliver the energy generated by the gravitational contraction efficiently to the photosphere to be radiated away. With this assumption, we should expect the effective temperature to remain very nearly constant as the star contracts. Thus dT_e/dR_* in equation \ref{5.2.20} will be approximately zero, and we expect the star to move vertically down the Hertzsprung-Russell (H-R) diagram with the luminosity changing roughly as R_*² until the internal conditions within the star change. Thus for the Hayashi tracks \[\frac{d T_e}{d R_*}=\frac{d T_e}{d L}=0 \quad \frac{d \ln L}{d \ln R_*}=+2\label{5.2.21}\]

While the location of a specific track will depend on the atomic physics of the photosphere, the relative location of these tracks for stars of differing mass will be determined by the fact that the underlying star is a polytrope of index n = 3/2. From the polytropic mass-radius relation developed in Chapter 2 [equation \ref{2.4.21}] we see that \[\mathrm{M}^{1 / 3} \mathrm{R}=\text { constant }\label{5.2.22}\]

and that \[\frac{d \ln R_*}{d \ln M}=-\frac{1}{3}\label{5.2.23}\]

Equations \ref{5.2.19}, and \ref{5.2.20} also imply that \[\frac{d L}{d M}=\frac{2 L}{R_*} \frac{d R_*}{d M}+\frac{4 L}{T_e} \frac{d T_e}{d M}\label{5.3.24}\]

If we inquire as to the spacing of the vertical Hayashi tracks in the H-R diagram, then we can look for the effective temperatures for stars of different mass but at the same luminosity. Thus, we can take the left-hand side of equation \ref{5.2.24} to be zero and combine the right-hand side with equation \ref{5.2.23} to get \[\frac{d \ln T_e}{d \ln M}=+\frac{1}{6}\label{5.2.25}\]

This extremely weak dependence of the effective temperature on mass means that we should expect all the Hayashi tracks for the majority of main sequence stars to be bunched on the right side of the H-R diagram. Since the star is assumed to be radiating as a blackbody of a given T_e and is in convective equilibrium, no other stellar configuration could lose its energy more efficiently. Thus no stars should lie to the right of the Hayashi track of the appropriate mass on the H-R diagram; this is known as the Hayashi zone of avoidance.

We may use arguments like these to describe the path of the star on the Hertzsprung-Russell diagram followed by a gravitationally contracting fully convective star (see Figure 5.1). As we suggested, this contraction will continue until conditions in the interior change as a result of continued contraction.

As the star moves down the Hayashi track, the internal temperature will increase in a homologous fashion so that T = µM/R. Hence we could expect the adiabatic gradient \(\nabla \mathrm{T}_{\mathrm{ad}}=\mu \mathrm{M} / \mathrm{R}^2\). However, the radiative gradient is \[\frac{d T}{d r}=-\frac{3 \kappa \rho L(r)}{16 \pi a c T^3 r^2} \sim \frac{\kappa\left(M / R^3\right) L}{(\mu M / R)^3 R^2} \sim \frac{\kappa L}{\mu^3 M^2 R^2}/label{5.2.26}\]

To find the homological behavior of the radiative opacity, we may use the approximate formulas [equation \ref{4.1.19}] for Kramer’s-like opacity. Making use of the homology transformations for p and T we can calculate the ratio of the adiabatic to radiative gradient ℜ as \[\Re \sim \frac{M^{s-n+3}}{L\left(R^{s-3 n}\right)}\label{5.2.27}\]

As the star contracts down the Hayashi track, ℜ will steadily increase. At some point, depending on the dominant source of opacity, the adiabatic gradient will exceed the radiative gradient, and convection will cease. This will not happen globally all at once; rather, a radiative core will form that propagates outward until the entire star is radiative. At that point the mode of collapse will change because the primary barrier to energy loss will move from the photosphere to the interior and the diffusion of radiant energy. Since all the models on the Hayashi tracks are convective polytropes, we might expect this point to happen at the same value of ℜ for stars of differing mass. If this is the case, then, remembering that for stars on the Hayashi tracks L . R², we may use equation \ref{5.2.27} to find that the locus of points of constant ℜ lies along a line such that \[\frac{d \ln L}{d \ln M}=2\left(\frac{s-n+3}{s-3 n+2}\right)\label{5.2.28}\]

Figure 5.1 shows the schematic tracks for fully convective stars and radiative stars on their way to the main sequence. The low dependence of the convective tracks on mass implies that most contracting stars will occupy a rather narrow band on the right hand side of the H-R diagram. The line of constant radius clearly indicates that stars on the Henyey tracks continue to contract. The dashed lines indicate the transition from convective to radiative equilibrium for differing opacity laws. The solid curves represent the computed evolutionary tracks for two stars of differing mass⁵.

Remembering that for electron scattering n = s = 0, while for Kramers' opacity s = 7/2 and n is 1 or 0.75, depending on the relative dominance of free-free to bound-free opacity, we can obtain the appropriate mass luminosity law for the dominant source of opacity at the point of transition from convective to radiative equilibrium. Combining this with equation \ref{5.2.25}, we find that the locus of points in the H-R diagram will be described by \[\frac{d \ln L}{d \ln T_e}= \begin{cases}+18 & \text { electron scattering } \\ +21.24 & \text { Schwarzschild opacity for bound-bound opacity } \\ +26.4 & \text { Kramers' opacity for free-free opacity }\end{cases}\label{5.2.29}\]

For the very massive stars, radiation pressure may play an important role toward the end of the Hayashi contraction phase, so that the onset of radiative equilibrium occurs sooner, increasing the value on the right-hand side of equation \ref(\{5.2.28} slightly. But for stars with a mass less than about 3M_⊙ equations \rel{5.2.25}, and \rel{5.2.29} will describe their relative position on the H-R diagram with some accuracy.

Henyey Evolutionary Tracks After sufficient time has passed for the adiabatic gradient to exceed the radiative gradient, convection ceases and the main barrier to energy loss is no longer the ability of the photosphere to radiate energy into space. Rather the radiative opacity of the core slows the leakage of energy generated by gravitational contraction, and the atmosphere no longer provides the primary barrier to the loss of energy. Further contraction now proceeds on the Kelvin-Helmholtz time scale. As the star continues to shine, the gravitational energy continues to become more negative, and to balance it, in accord with the Virial theorem, the internal energy continues to rise. This results in a slow but steady increase in the temperature gradient which results in a steady increase in the luminosity as the radiative flux increases. This increased luminosity combined with the ever-declining radius produces a sharply rising surface temperature as the photosphere attempts to accommodate the increased luminosity. This will yield tracks on the H-R diagram which move sharply to the left while rising slightly (see Figure 5.1). For the reasons mentioned above, the beginning of these tracks will be along a series of points which move upward and to the left for stars of greater mass.

We may quantify this by asking how the luminosity changes in time. We differentiate equation \ref{5.2.18} and obtain \[\frac{d L}{d t}=-\frac{1}{2} \frac{d^2 \Omega}{d t^2}=-\frac{a G M^2}{2 R^2}\left[-\frac{2}{R}\left(\frac{d R}{d t}\right)^2+\frac{d^2 R}{d t^2}\right]\label{5.2.30}\]

The parameter a is simply a measure of the central condensation of the model, which we require to be independent of time. This requirement is satisfied if the contraction is homologous. If we further invoke the Virial theorem and require that the contraction proceed so as to keep the second derivative of the moment of inertia equal to zero, then \[\frac{d^2 I}{d t^2}=\frac{d^2}{d t^2}\left(\alpha M R^2\right)=0 \Rightarrow\left(\frac{d R}{d t}\right)^2+R \frac{d^2 R}{d t^2}=0\label{5.2.31}\]

Using this to replace the second derivative in equation \ref{5.2.30}, we get \[\frac{d \ln L}{d \ln R}=-3\label{5.2.32}\]

Multiplying equation \ref{5.2.24} by (R/L)(dM/dR) and combining with the above result, we get \[\frac{d \ln T_e}{d \ln R}=-\frac{5}{4} \quad \frac{d \ln L}{d \ln T_e}=+\frac{12}{5}\label{5.2.33}\]

Thus we can expect the star to move upward and to the left on the H-R diagram with a slope of 2.4. This path will eventually carry it to the main sequence where nuclear burning will set in as a consequence of the steady increase in the central temperature. Since the effect of onset of nuclear burning will be similar for a wide range of main sequence stars, we can expect the luminosity distribution of the Henyey tracks with mass to be reflected on the main sequence. As a result, we might expect that equation \ref{5.2.28} will reflect the main sequence Mass-Luminosity relation. Indeed, Harris, Strand, and Worley⁶ give empirical values of 2.76 for the exponent on the mass of the mass-luminosity relation for the lower main sequence and 4 for the upper main sequence. The values obtained from equation \ref{5.2.28} corresponding to different types of opacity are \[\frac{d \ln L}{d \ln M}= \begin{cases}+3 & \text { electron scattering } \\ +3.54 & \text { Schwarzschild's opacity } \\ +4.4 & \text { Kramers' opacity }\end{cases}\label{5.2.34}\]

Although the proper evolutionary tracks for a star contracting to the main sequence require more exact modeling than can be done with polytropes, the overall effects can be estimated by considering sequences of equilibrium configurations linked by a description of those physical processes which limit the energy flow from the star. To dramatize this, we included in Figure 5.1 some calculated evolutionary tracks for two stars. The salient features of the pre-main sequence evolution are reasonably described by the curves in spite of the crude assumptions involved.