A.   GENERAL CONSIDERATIONS

• The life cycle of stars is literally the pulse of the universe. We see the universe largely via the UV/optical/infrared radiation produced by nuclear fusion within stable, self-luminous massive spheres, as they evolve through cosmic time.

• Stars form in dense (largely molecular) regions of the interstellar medium, evolve according to their initial mass and composition, and die either explosively, when their nuclear fuel is exhausted, or quietly, as a compact, slowly cooling white dwarf "star".

• Within dense clouds of gas & dust, stars form within a mass range, ~100 > M/M(sun) > 0.08 . The upper limit appears to be set by the pressure of radiation from these extremely luminous stars which inhibits additional accretion of mass onto the newly formed high-mass star. The lower mass limit is set by the conditions necessary for proton fusion to occur in the core. Objects of lower mass, in which the core pressure, density & temperature are insufficient to "ignite" fusion, are known as brown dwarfs.

• Gravity is the dominant force which drives star formation. Dense regions of the ISM collapse under their own self-gravity, limited by the heating which occurs as mass falls within the gravitational potential of the growing protostars.

• As the mass density increases, energy can be removed from the collapsing protostellar cloud largely by infrared radiation; continuous emission from heated dust grains, and line emission from various molecular transitions. This radiation increases with density.

B.   STAR-FORMING REGIONS

Stars form within dense regions of the ISM. Lets consider the issue of a gas+dust cloud:

What conditions induce a cloud to initiate a collapse ?   Which processes hinder that collapse ?
What is the typical size of a star-forming region ripe for collapse ?

The essence of the problem is to understand large region, 0.1 to 1 parsec in size, with a density of ~10⁶ atoms per cm³ can collapse to a size scale ~ 1 million times smaller.

i)   THE VIRIAL THEOREM

Interstellar clouds must first reach marginal equilibrium, otherwise their internal random motions would disperse rather than collapse the cloud. The virial theorem relates the kinetic and potential energy of a gravitating system.

The virial theorem simply states that, for a system of self-gravitating particles (like an ideal gas, or even a cluster of stars) that is neither contracting nor expanding (ie, that is in equilibrium):

2 < K >   =   − < U >

where K = kinetic energy, and U = potential energy. For gravitationally-bound systems in equilibrium, the total energy of the system equals 1/2 of the time-averaged potential energy:

2 < E >total   =   < U >

where   < E >_total   =   < K > + < U >


ii)   COLLAPSE OF A PROTOCLOUD

For a cloud of ideal gas:

< E >total   =   N(3/2kT)   »   Nkt

where N is the total number of particles in the cloud.

also,

< U >grav   »   − GM2/R

where R is the size of the collapsing cloud.

Note also that, for a spherical cloud with constant density:

< U >grav   =   − 3GM2 / (5R)

(which is close to our approximation for U above).

Assume that the cloud is a molecular hydrogen gas cloud:

N = M / (2m_H)

Plugging into the virial theorem:

Þ   2 (NkT)   =   - GM² / R

Þ   MkT / m_H   =   − GM² / R

For a spherical cloud,

M   =   (Volume_sphere × r)   =   (4pR³ / 3) r

And so:

Þ (4pR³ rkT) / (3m_H)   =   − G(4 pR³)² / (3R)

R » 107 (T / r ) 1/2   meters

This physical scale is known as the Jean's Length

Thus, to make R as small as possible, the r needs to be as large as possible, and T as small as possible:

⇒   A cloud that is dense and very cold makes the best site for star formation.


iii)   ANGULAR MOMENTUM PROBLEM

Since all protostellar clouds are thought to have some rotation, we have to consider angular momentum when thinking about the process of star formation.

Recall conservation of angular momentum, and prove that the "surface" of a solar-like star would have an impossible velocity, if the protostellar cloud starts out with an initial radius of 0.1 pc, and the outer parts of that cloud rotate with a modest speed of only 1 km/s.



Answer here!

The essence of the problems: The angular momentum present in an contracting IS cloud has to somehow be removed before this cloud can collapse to a protostar. Otherwise the cloud will assume a disk-like shape, and remain stable against further collapse.

Magnetic fields have been invoked as a mechanism to diffuse the angular momentum by channeling mass through stellar wind outflows.

Now, within the Galaxy, all mass (gas, dust, stars) within the disk rotates about the Galactic center. The smaller the area from which a cloud forms the better the forming a star, since the angular momentum of the protostellar cloud will be minimized.




iv)   MOLECULAR CLOUDS

• Opaque (optically thick, A_v > 10-20 mags), dense, cold clouds of molecular hydrogen (H₂). The high opacity shields the inner H₂ from UV photodissociation. Molecular clouds are surrounded by shells of neutral atomic HI.

• T » 10K, r ³ 10⁹ molecules / m³. How does this density compare to the gas density in this room ?

• Dust grains are an important component of molecular clouds. H₂ molecules form on the grain surfaces, and the dust also shields molecules from stellar UV. IRAS observations during the 1980s confirmed this model showing that the 100mm flux, produced by dust grains, closely traces emission from the CO molecule.
  

  In clockwise direction, maps of the 60mm, 100mm, 21 cm, & CO emission from a region along the galactic plane (For additional details, see Heyer & Tereby ApJ 502, 265 (1998)

  Note that the agreement between the CO and the IRAS fluxes is not extremely tight. However, these data provide evidence suggesting a correlation between the amount of dust and molecular hydrogen:

  

• Molecular hydrogen, H₂, is difficult to observe directly in the ISM, as there are no emission/absorption lines in the visible or centimeter radio bands. H₂ is a diatomic molecule, and has vibrational bands in the near-IR, and some electronic transitions in the far-UV, near 1000 A. The Far Ultraviolet Spectroscopic Explorer (FUSE), orbited in 1999, has been used to observe the UV lines of this molecule.
  

• To determine the physical conditions in star forming regions, we must observe tracers of H₂, other molecular species whose density tracks the density of the far more prevelant H₂. The CO molecule, observed at 2.6 mm (and other mm & sumbillimeter wavelengths, is the primary tracer.
  

  False color map of CO emission at 2.6 mm looking at the galactic mid-plane.
  Source: UMass FCRAO Galactic Plane Survey

• For every 10⁴ H₂ molecules, there is about one CO molecule. Other molecular tracers are OH, CS, C₃H₂, H₂O, and formaldehyde.
  

C.   TIMESCALE FOR COLLAPSE

As the collapse progresses, the cloud center becomes more dense, causing the collape time near the center to decrease, relative to the more diffuse edges. This process continues in a non-linear fashion, with the core collapsing at a faster rate until the outer envelope ends up collapsing in "free-fall" towards the central dense concentration:

One can estimate the timescale for a "test" particle, located in the diffuse outer envelope, to collapse into the central region:

Assume that the test particle is in an orbit about the center, such that its semimajor axis = a = R / 2.

Using Kepler's 3rd Law:

But m_test is much smaller that that of the central condensation, M, thus:

Plugging in for M, assuming a central mass density r₀, and also plugging in for a, we get that:

A more exact computation (using integration, see text on P-22 for details) gives the same answer.

Thus:

where t_ff is expressed in units of seconds, when r₀ is expressed in units of kg/m³.

t_ff is the "free-fall" timescale for collapse of an interstellar cloud.

An example: What would be the free-fall time for a giant molecular cloud whose central density r ₀ is 2 x 10^-13 kg / m³ ?

Answer is here

D.   FLAVORS OF YOUNG STARS

Indentification and analysis of protostars and pre- main sequence stars is a major focus of current astrophysics. New tools, such as the high angular resolution provided by HST, and the infrared capabilities of the Spitzer Space Telescope have opened new windows for the exploration of star formation.
Astonomers have labeled different flavors of stars still in the formation process:

i)   YSOs - Young Stellar Objects, or "Herbig Haro Objects"


HST images of Herbig-Haro objects. For a full-resolution image click here

Located within or near dusty molecular clouds, YSO's have red optical colors, and are often appeared to possess visible gas clouds connected to the star. YSOs are low mass stars "approaching" the main sequence. Millimeter wave radio data discovered powerful bipolar molecular outflows from many YSO's, believed to part of the process for removing angular momentum from the contracting protstar gas cloud. HST has produced spectacular images of optical counterparts to these outflows, and the circumstellar disks which surround the YSOs.



Propagating knots within the jets. For a full-resolution image click here



And a more detailed diagram:



Find out more at this HST page, and at the UMass Herbig-Haro Object Catalog.

ii)   OB ASSOCIATIONS - Loose groupings of O and B stars

iii)   T-TAURI STARS - Pre-main sequence stars (ranging from 0.5 - 3 M ) which are highly variable in their luminosity, and often exhibit mass flows of 10^-7 to 10^-8 M per year. P-Cygni profiles are the observational evidence of these mass flows:





iv)   LBV's (Luminous Blue Variables) - Very massive (£ 60 M) stars have strong P-Cygni profiles, indicating strong stellar winds during "eruptions". P-Cygni and Eta Carinae (mass   ~120 M) are classic examples, having had eruptions in 1600 and 1837-1860, respectively.



Read more about this fascinating object here

(E)   THE EVOLUTIONARY PATH OF A YSO ON THE HR DIAGRAM

i)   POINT #1 :

As the effective temperature steadily increases in the protostellar cloud, the opacity of the outer layers is dominated by the H^- ion (the extra electrons come from partial ionization of heavier elements - hydrogen remains largely neutral).

The large opacity causes the envelope of the contracting protostar to become convective (sometimes all the way down to the stellar core).

This deep convection limits the evolutionary path in the HR diagram to a vertical line, called the Hayashi track.


ii)   POINT #2 :

As the luminosity decreases, due to the collapse, making the radiating surface area smaller, the effective temperature slightly increases.


iii)   POINT #3 :

Here, the temperature in the core increases to the point that the core material becomes ionized sufficiently for energy to be transported by radiation again (as opposed to convection). When a radiative core develops, the luminosity increases. At the same time, the core temperature becomes high enough to ignite some nuclear reactions (depending on the mass of the protostar). Deuterium will fuse before individual protons, so this process ignites at lower temperatures. This proecess causes a significant increase in the temperature of the star.


iv)   POINT #4:

Once the core temperature and density is sufficient to ignite hydrogen fusion, the star becomes a zero-age main sequence (ZAMS) star, with the core region transporting energy via radiation. The timescale for "settling" onto the main sequence strongly depends on the star's final mass - massive stars quickly reach the MS (less than a few 10⁵ years), while low-mass M dwarfs can take over 100 million years to reach a final stable configuration.