What is Moving?

Don't be confused by the discussion of motion on the HR diagram. We don't mean by this that a star is literally moving in space (which is a separate issue that has nothing to do with the HR diagram). We mean instead that the point corresponding to the star on the HR diagram is moving. That is, motion on the HR diagram is shorthand for a change in the relation between surface temperature and luminosity for the star.

Motion on the HR Diagram

As protostars contract to the main sequence, both their luminosities and their surface temperatures change. Therefore, we may envision them following a path in time on the HR diagram. Such a path on the HR diagram for a star or a protostar is called an evolutionary track. As we shall see, the entire life history of a star, from birth to death, may be exhibited by plotting evolutionary tracks for the star on an HR diagram.
Hayashi Tracks for Protostars
Collapsing protostars are expected to be fully convective (see the right panel). The Japanese astronomer C. Hayashi showed that this has two important implications for protostar evolution.

  • Contracting protostars must initially follow an almost vertical path downward on the HR diagram. These paths on the HR diagram for collapsing protostars are called Hayashi tracks.
  • There is a region to the right of the HR diagram where it is not possible for stable protostars to exist. This region is called the Hayashi forbidden zone.

    • The following figure illustrates the Hayashi forbidden zone and a Hayashi track for a star of mass similar to that of the Sun.

    The general path that is followed is that initially the protostar descends almost vertically along the boundary of the forbidden zone. The protostar will continue to follow this almost vertical path so long as it remains fully convective.

    Technically Speaking: Kramer's Law

    The transition from the Hayashi track to more horizontal motion on the HR diagram depends on the development of a radiative core, and that depends crucially on the opacity in the center of the star. The opacity K is given approximately by Kramer's law:

    K = aD / T 3.5

    where a is a constant, D is the density, and T is the temperature. Since this depends inversely on a large power of T, the opacity goes down rapidly with increasing temperature. It is this strong temperature dependence of the opacity that can cause the star to develop a radiative core and leave the vertical Hayashi track.

    Development of a Radiative Core
    But as the protostar contracts, its center becomes hotter and this lowers the opacity to photons in the core. As a consequence, a region in the center of the star will eventually cease to be convective because radiation can now transport the energy outward at a sufficient rate to prevent convection (see the discussion of opacities in the adjacent box and recall the general discussion of stellar energy transport in Chapter 18).
    A Star Is Born
    With further contraction, the radiative core grows in size relative to the convective region overlying it. When the star develops a radiative core, it stops following the vertical Hayashi track and begins to drift upward and to the left on the HR diagram (that is, its luminosity and surface temperature increase). Finally, when the temperature and density in the core reach the critical values, hydrogen fusion turns on and the track bends over vertically and stops as the star stabilizes on the main sequence in hydrostatic equilibrium.
    Tracks for Stars of Different Mass
    How does this sequence of events depend on the mass and the composition of the star? There is a weak dependence on composition because this can influence the opacities, but we will not discuss it in detail. The dependence on mass is much more dramatic and is summarized in the following figure, which shows the protostar evolutionary track corresponding to stars of various main-sequence masses. The numbers marked along each track give the time in years to reach that point. We shall discuss the time to collapse to the main sequence in more detail shortly.

    Notice that large mass stars move almost completely horizontally to the main sequence, very low mass stars move almost completely vertically to the main sequence, and the time to collapse to the main sequence is much shorter for more massive stars. For example, a 100 solar mass star collapses almost horizontally on the HR diagram to the main sequence in only about 10,000 years, but a 0.1 solar mass stars takes 100 million years to follow its vertical HR path to the main sequence. This animation illustrates the evolutionary tracks for contraction of protostars to the main sequence.

    Understanding the Dependence on Mass
    This dependence of the protostar evolutionary tracks on mass can be understood through the following observations.

    1. The Hayashi tracks have only a weak dependence on mass, with increasing mass shifting the vertical Hayashi track slightly to the left in the HR diagram.
    2. The time for transition from fully convective to a radiative core has a strong dependence on mass, since in massive protostars the center heats up more rapidly and therefore becomes radiative more quickly.
    3. Massive stars and protostars evolve more rapidly than less massive ones.

    From these observations we may conclude that for the most massive stars the protostar develops a radiative core very quickly and the track to the main sequence is almost completely horizontal and very fast, while for the lowest mass stars the center never becomes radiative and the collapse to the main sequence is essentially completely vertical along the Hayashi track and very slow. Only intermediate mass stars show the more complex behavior of a long vertical Hayashi track followed by substantial horizontal motion once the core becomes radiative, and their time to collapse to the main sequence is between that for massive and very light stars.