A. What Is a Star and Where Do They Come From?

You may wonder how we decide between various objects such as stars and planets. Here are some possible statements that may seem reasonable to distinguish them:

• Planets orbit around stars (wrong! [movie])
• Stars shine by their own light, planets reflect light (not entirely correct)
• Stars are much bigger than planets (some "stars" [brown dwarfs] are barely larger than Jupiter)

In fact, none of these properties is sufficiently unique to use to classify when an object is a star or planet. The single distinguishing factor that defines a star is the occurrence of nuclear fusion. Nuclear fusion is the fusing (combining) of two atomic nuclei to form a heavier atomic nucleus, giving off a tremendous amount of energy. This is the same process that occurs in the hydrogen bomb (H-bomb), and requires tremendous pressure and temperature. In an H-bomb, the release of energy cannot be contained, resulting in an explosion, but in the center of a star the gravitational force due to the outer layers of the star keep the energy from escaping immediately, so the fusion process continues steadily over billions or even 10's of billions of years. An sufficiently large object that is too small to sustain nuclear burning of any kind is called a planet, but if the object is large enough that nuclear burning of some type occurs, it is called a star. Brown dwarfs, intermediate between a planet and a true star, do have nuclear burning, but of deuterium rather than hydrogen.

To make a star, then, we have to collect enough mass in a small enough region of space that the temperature and pressure grows high enough to begin nuclear fusion. The material to make a star comes from the gas and dust of the interstellar medium (ISM). It is gravity that provides the force needed to keep the object in one piece and cause it to shrink over time until it "ignites." An object that has enough mass to be a star, and is on its way to becoming dense enough and hot enough for nuclear fusion, is called a protostar. It is not yet a star, but given enough time it will become one.

Planetary systems are born during or just after the period of collapse of the protostar, and it is this process that we will examine in detail in this lecture.

B. Basic Picture of the Collapse

Clouds of gas and dust collapse and form Young Stellar Objects (YSOs), sometimes visible and sometimes enshrouded in dust. The relevant physics of the collapse includes

• Size scale for collapse (Jeans Criterion): L ~ 10⁷ ( T / r )^1/2 ~ 0.1 pc for T = 10 K, r = 10^-15 kg m^-3
  This estimate comes from applying the virial theorem , which if you will recall says that half the potential energy difference between two equilibrium states of a system is available to be (and must be) radiated away, and the other half goes into internal heating of the system. The thermal energy is
  E_thermal = NkT
  while the potential energy of a spherical distribution of mass is
  U = - GMm/R = GM ²/L
  where we equate the radius R with the length scale L for the system. We can roughly relate the total number of particles in the cloud, N, with the mass of the cloud by considering that all of the particles are all hydrogen molecules (the cloud is very cold, so we do not consider individual atoms): N = M / 2m_H. Then equating E = -U /2, (from the Virial Theorem) we have kT /2 m_H ~ GM / L. Writing this in terms of the density of the cloud, r = M /(4/3pL³ ), we have the final result L ~ ( kT / m_H Gr)^1/2 = 10⁷ ( T / r )^1/2 . The text goes through a somewhat more careful argument to give the Jeans Criterion: R_J = (15kT / 4pGmm_Hr)^1/2, but this differs by less than 10% from the simpler expression above.
• Time scale for (free-fall) collapse: t_ff = 6.4 x 10⁴ r^-1/2 ~ 10⁵ yr for r = 10^-15 kg m^-3
  Free fall means that the particles are in orbit around their common center of mass, and do not collide, but fall freely. We can estimate how long it will take for a particle to fall from the edge of the cloud by taking 1/2 of its Keplerian orbital period, assuming that it falls directly in, so that the semi-major axis of the orbit ellipse is a = r /2, where r is the radius of the cloud. From Kepler's third law,
  P = [(4p² / GM) a³ ]^1/2 = 2 t_ff .
  Again relating the mass to the density, as we did above, and using this value for the semi-major axis, we have:
  t_ff = (3p / 32Gr)^1/2 = 6.4 x 10⁴ r ^-1/2
  Of course, the collapse will only proceed until collisions do become important. Then the collapse will slow, and the object will begin to heat up. We will look at this in more detail shortly.
• Angular momentum: m_i v_i r_i = m_f v_f r_f => v_f = v_i (r_i / r_f ) = 1 m/s (10¹⁵ m/10⁹ m) ~ 1000 km s^-1
  Conservation of angular momentum, the same law that causes a skater to spin faster as they pull their arms closer to their body, causes the collapsing cloud to spin faster as it gets smaller. Even for very slowly spinning clouds initially, the vast change in radius as the cloud collapses to a stellar-sized object leaves the object with enormous spin velocity. We can estimate this final velocity very simply by equating the initial angular momentum (L_i = m_i v_i r_i) to the final angular momentum as above. Thus, the object spins up by a factor of 1 million as its radius drops by the same factor.
• Magnetic fields: conservation of magnetic flux (magnetic field strength / unit area) F = pr²B => B_f = B_i (r_i / r_f )² ~ 100 T for B_i = 0.1 nT.
  Under the conditions of the collapsing cloud, another quantity is conserved--the magnetic flux. The increase in magnetic flux is even greater than that for spin, because it depends on the square of the radius.

Thus, we can expect a cloud of typical size 0.1 pc to collapse over perhaps 1 million years, and when it reaches stellar size it should be spinning very rapidly and be highly magnetized. This is exactly what is observed, except that the spin rate and magnetic field strength are somewhat less than predicted by these simple arguments. Our best understanding of the reason for the discrepancy is that the strong magnetic fields themselves slow down the rotation rate by coupling the spinning core to the outer disk, and dynamical processes cause the proto-stellar object to shed some of its mass and much of its magnetic field. Observational Evidence

• O,B Associations
• stellar disks
• disks in Orion Nebula (OMC-1)
• protoplanetary disks
  • Fomalhaut
  • AU Microscopii
  • Beta Pictoris
  • Fomalhaut Again

Difficulty of making these observations: The following three images show the star field around Fomalhaut, a zoomed view of the star field with an apparent circular gap in stars, and the same view with an actual image taken from the Deep Sky Survey (DSS). You can see why there are no stars cataloged near Fomalhaut--the extreme brightness of Fomalhaut makes seeing any stars nearby completely impossible. Imaging trying to see the protoplanetary disk--it requires clever techniques of occulting (covering up) the star in order to make the ring visible. Fomalhaut overview, Fomalhaut zoom, Fomalhaut DSS Image