Star Formation and the Fragmentation of Giant Molecular Clouds


Introduction:

The question of our own origins has fascinated humans from the dawn of civilization to the present time.  In more modern times, though, our thoughts have turned not only to the origin of our species, but also to the origins of the environment in which we live, and in turn to the origins of the universe itself.  Stars are a key building block in the galaxy and hence the universe, and as such, their formation from gaseous clouds, specifically giant molecular clouds, can provide truly valuable insight into the inner workings of the galaxy.
 
Giant molecular clouds consist primarily of molecular Hydrogen (H2) and are usually located in the disk of the galaxy.  These clouds fragment into dense cores, which in turn condense into stars.  The specific mechanisms of the cloud fragmentation key to stellar formation are not yet completely understood by scientists.  However, research indicates that fragmentation is dependent upon a number of complex factors, including the mass of the cloud, formation of molecules within the cloud, as well as magnetic fields.  This paper seeks to describe what is known about the fragmentation of giant molecular clouds in the early stages of star formation.  The paper is divided into two sections, the first providing a general overview of the early stages of star formation, and the second dealing more specifically with fragmentation. 

 

Part I:  The Basics of Star Formation:Horsehead Nebula


Giant Molecular Clouds and Dense Cores

Stars form from large regions of molecular hydrogen (H2) gas in the galaxy.  These regions typically have radii on the order of 10 pc, and masses on the order of 105 Msun.  The picture on the right is of a dark cloud in Orion known as the Horsehead Nebula taken from the Anglo-Australian Observatory (Carkner  A Star is Born).  Thus the giant molecular clouds, as they are often referred to, are relatively dense, having number densities on the order of 103 cm-3 and generally possess some form of internal structure consisting of dense cores (the locations of star formation) and other structures.  The formation of these structures is a subject not yet well understood by astronomers, yet it is related to self-gravitation, magnetic fields, turbulence, and pressure gradients within the clouds and will be discussed further in Part II.

The Jean's Mass - Gravitational Collapse

It is in these cores that star formation begins.  If a core holds enough mass, the force inward due to gravity will overcome the force outward due to gas pressure, and the cloud will thus begin to collapse.  The critical mass is determined by balancing the two forces and solving for the mass.  This is known as the Jean’s mass and is defined as:

where k and G are the Boltzman and gravitational constants respectively, and r0 is the initial mass density of the core.  Due to the dependence on one over the square root of density, a higher initial density will result in a lower Jean's mass; a more dense cloud will condense will a lower mass.  Given average parameters for a high-mass dense core of nH2 ~ 104 cm-3 and T ~ 10 K, the Jean’s mass can be calculated to be on the order of 1034 g or 10 MsunClick here to see detailed calculation.  A typical mass for this class of core is 40 Msun; hence dense cores are gravitationally unstable.
 
An important thing to remember is that the dense core must be transparent to its own radiation.  This allows the radiation energy to escape out into the surrounding medium so that the radiation pressure does not impede gravitational collapse.  However, as the cloud collapses, the density rises, as does the optical depth of the cloud.  This means the collapsing core becomes opaque to its radiation.  This mechanism causes the cloud to stop collapsing, and the star is born.

The Accretion Disk

The picture below shows stages of the accretion of gas of a protobinary system (Bate 2000).
Accretian


When a collapsing cloud (or any collapsing fluid) has an initial non-zero angular momentum it will collapse to an accretion disk surrounding a central mass.  This can be demonstrated mathematically by simply adding a centripetal acceleration term to the equation of motion:

where Mr represents the mass contained in radius r, and w the angular velocity.  Conservation of angular momentum thus shows that the collapse will stop in the plane perpendicular to the axis of rotation (when the centripetal and gravitational forces balance).  The resultant disk will then accrete material onto the protester.

 

Part II:  FragmentationFragmentation

A problem arises in the above theory of star formation when one considers the giant molecular cloud as a whole.  The mass of an average GMC is on the order of 105 Msun, the temperature ~10 K, and the number density 30 cm-3 (Cabrit 1994).  Again referring to the formula for the Jean’s mass, one finds that the molecular cloud mass is supercritical: Mj = 133 MsunClick for detailed calculation.  The cloud should just condense into a giant star of 105 Msun.  However, stars of this mass have never been observed.  Instead of collapsing as a whole, the clouds fragment into smaller cores (as described above), and it is these cores which collapse into protostars.    The  to the left shows a possible sequence of fragmentation (Bate 2000).

Why Fragment?

 

As stated above, the exact mechanisms of fragmentation are not completely understood as of yet.  It is, however, proposed that the origin of fragmentation “probably involves a complex interplay between turbulence, magnetic fields, pressure gradients, and self-gravity” (Cabrit 1994).  None of these factors by itself can account for the observed hierarchical structure of the giant molecular clouds; however, they each logically play some role.  V.C. Reddish lists and describes in detail a number of historically attempted solutions to the problem of fragmentation in Stellar Formation of 1978.  An overview of this will be presented here.

One possible solution to the fragmentation question is statistically random fragmentation by volume.  This approach yields a solution for the number of fragments per unit volume as:

for N0 being the total number of fragments, V0 the average volume of a fragment, K0[y] the Bessel function, and a = V/V0.  When a uniform density is assumed, n[V] is proportional to the number of fragments by mass, as density is proportional to mass by unit volume.

Unfortunately, the results from this approach do not match the experimental data.  Adjustments to this theory that resulted in closer observational agreement were made Kruszewski.  A minimum possible mass per fragment was considered.  This mass was determined through the Jean’s criterion, as any smaller mass would not collapse.  This resulted in a mass distribution much closer to that experimentally observed (yet still off by an order of magnitude), with the frequency distribution of masses:

 A second approach which yielded predictions yet closer to observations relied on random fragmentation according to mass.  This theory relied on an assumption of a power function relationship between the frequency distribution and the mass of a fragment:

 


A detailed study adjusting the parameters gave results agreeing with observation.  However, there is physical justification for the theory, and hence its validity cannot be determined.

 Another approach considers a series of fragmentations, each leaving behind equal amounts of mass at each step:

Nn Mn = constant

Again, though, this approach finds results close to, but not in complete agreement with observation. 

Molecule formation can also be considered as a possible cause for fragmentation.  This approach suggests that the formation of H2 molecules in a cloud leads to the clouds fragmentation.  Here the assumption that molecules form on interstellar grains is necessary.  The rate of molecule formation on a grain is given by

 

where E is the absorption energy of atoms on the grains.  When molecules begin to form on the grains, they begin to cool; this leads to an increased formation rate by the above equation.

Any variations in density within a cloud will result in variations in the optical depth throughout the cloud.  This in turn leads to a celularization of molecular regions: areas of longer optical depth will block external radiation from reaching other areas, creating cells in which radiation cannot reach.  Inside these cells the temperature will continue to drop, leading to a drop in the internal pressure.  The cell then condenses to reach pressure equilibrium, and in doing so the cellular temperature begins to rise.  This chain reaction leads to “inevitable” fragmentation.

Reddish’s work, being over two decades old, obviously cannot consider newer developments and theories on fragmentation.  Among these is the consideration of the role of magnetic fields in giant molecular cloud evolution.  In poorly ionized regions subjected to magnetic fields, ambipolar diffusion takes hold.  Here, electrically neutral particles will drift towards the center of a region due to self-gravitation.  However, any ionized particles will remain held to the magnetic field lines, effectively reducing the mass to flux ratio and increasing the instability of the region.  This effect is obviously related to the ionization rate: where ionization is lowest, the ambipolar time scale is closest to the freefall time.  This accounts for the observation that condensation generally occurs within regions optically thick to most external radiation - where only cosmic ray ionization is able to occur (Cabrit 1994).  Unfortunately, these last two solutions are unable to fully describe the phenomenon of hierarchical structure in giant molecular clouds.  Instead, they need some structure to already be in place before they can describe the resulting fragmentation.

Conclusion:


Whatever the cause of fragmentation, it is highly important in the formation of stars.  Some mechanism dependent on a number of environmental factors allows for the giant molecular clouds to partition themselves into regions which will condense and form stars.  As of yet no theory has been able to fully describe and agree with observations: theories dependent on random statistical fragmentations give solutions differing by orders of magnitude from what is observed, and theories based on inherent physical properties cannot fully describe the necessary hierarchy of structure.  These theories are still being studied and new theories are developing to describe this mechanism, which is the key starting point in the cycle of stellar evolution.

 

References:

  1. Bate, Matthew.  Star Formationhttp://www.astro.ex.ac.uk/people/mbate/Research/SF.html 14 November 2000.
  2. Cabrit, Sylvie: “Molecular Clouds and Star Evolution” in Star Formation and Techniques in Infrared and mm-Wave Astronomy.  T.P. Ray & S.V.W. Beckwith eds. Springer-Verlag, Berlin 1994, 1-48.
  3. Carrol, Bradley and Ostlie, Dale.  Modern Astrophysics.  Addison-Wesley, Reading, MA, 1996.
  4. Reddish, V.C.: “Stellar Formation.  Pergamon, Elmsford, NY, 1978.
  5. Carkner, Lee.  A Star is Born.  http://www.astro.psu.edu/users/carkner/ttauri/star1.html