Prestellar Phase

Let us consider a cloud in the blue sky of the Earth. Why does this cloud not pack together to form a small star in the air? The answer lies in the Jeans criterion, which says that in order a cloud, interstellar or whatever, to gravitationally contract and form a star, its self-gravity must win over the internal pressure forces which resist to contraction.

If an interstellar cloud starts collapsing, its gravity becomes stronger and stronger. However, remarkably the pressure in the cloud does not vary, since the temperature remains constant because the dust grains are able to radiate the whole potential energy liberated by contraction. This means that smaller parts of the cloud can further contract on their own. This is the process of fragmentation leading to the formation of star clusters and associations.

18.1 Overview and Signatures of Star Formation

The star formation process represents a density jump by a factor of about 1023. The typical density of the interstellar medium is about 10~23 g cm~3, while the average solar density is 1.4 g cm~3. Major changes in the matter properties occur during such a density change. It is meaningful to distinguish three phases in star formation:

- The pre-stellar phase: it covers the contraction and fragmentation phases of an interstellar cloud under its gravitation. This phase is essentially isothermal due to the efficient cooling by dust grains.

- The proto-stellar phase: it concerns the evolution of the fragment up to the stage where the growth of internal pressure in the central core stops the fast contraction and fragmentation. The central core reaches hydrostatic equilibrium and evolves nearly adiabatically.

- The pre-main sequence phase: it is the phase of the evolution of the central object from the Hayashi line up to the zero-age main sequence (ZAMS).

*This chapter may form the matter of a basic introductory course.

A. Maeder, Physics, Formation and Evolution of Rotating Stars, Astronomy and 475

Astrophysics Library, DOI 10.1007/978-3-540-76949-1-18, "

© Springer-Verlag Berlin Heidelberg 2009

The physical conditions, timescales and observational properties are different in these three phases.

Figure 18.1 shows the timescales and some properties of stars reaching a mass M at the end of the star formation process. Up to an age of (1-2) x 105 yr, most stars experience accretion and are surrounded by a rotating accretion disk. Protostars are generally still embedded in a cocoon. The low-mass protostars show strong colli-mated jets, while massive protostars excite an ultra-compact HII region (UCHII) which further expand.

Massive stars show large bipolar outflows, thus a fraction of the infalling mass returns to the interstellar medium. The disks around massive protostars dissipate rapidly as a result of their high luminosities, which also further produce extended HII regions. Massive stars are still embedded in their accreting material, when they experiences H ignition. For lower mass stars, the disks last for a large part of the pre-MS phase.

The range of sizes and masses of the star-forming clouds is quite large:

- Small star-forming zones: with sizes of about 10 pc and masses of a few 102 M0 (cf. Taurus region).

- Large star-forming regions: of the order of 100 pc and 104 M0 (cf. the Orion region which contains about 10 O-type stars).

- Giant HII regions (GHII): they are equivalent to 100 times Orion, with masses up to a few 106 M0. The 30 Dor nebula in the Large Magellanic Cloud is a magnificent example.

100 M/M0 10

Fig. 18.1 Timescales and properties of stars reaching a final mass M during their formation. The time zero is counted since the formation of a hydrostatic core. Initially, accretion and massive disks maintain the star inside a cocoon. For intermediate masses, the disks survive up to the beginning of the MS phase, while for lower masses the disks disappear before the end of the pre-MS phase. For massive stars, the accretion phase and disks (if any) are present even after the star has reached the ZAMS. TAMS indicates the end of the MS phase, WR means Wolf-Rayet stars. BH, NS and WD mean black holes, neutron stars and white dwarfs. Adapted from Yorke [627]

Fig. 18.1 Timescales and properties of stars reaching a final mass M during their formation. The time zero is counted since the formation of a hydrostatic core. Initially, accretion and massive disks maintain the star inside a cocoon. For intermediate masses, the disks survive up to the beginning of the MS phase, while for lower masses the disks disappear before the end of the pre-MS phase. For massive stars, the accretion phase and disks (if any) are present even after the star has reached the ZAMS. TAMS indicates the end of the MS phase, WR means Wolf-Rayet stars. BH, NS and WD mean black holes, neutron stars and white dwarfs. Adapted from Yorke [627]

- Starbursts: they are equivalent to upto 104-105 times Orion and mostly result from interactions of galaxies.

Observations of star formation in the Galaxy and in external galaxies are performed by various techniques, which provide different signatures:

- Large molecular clouds are observable in radio emission. The main molecule H2 has no easy observable line. However, the molecule CO has an important line at X = 2.6 mm, which permits the radio cartography of cold molecular clouds. A number ratio CO/H2 ~ 6 x 10~5 is usually adopted to estimate the cloud mass. Likely this ratio changes with the content in heavy elements of the region or galaxy studied.

- The grains in contracting clouds are a strong IR source, as well as the grains heated by the radiation of newly formed O stars.

- Dark clouds are observable in visible light: the Coal Sack as well as dark regions in Cygnus are beautiful examples. The Bok globules are dark spots visible on the bright background illuminated by young stars.

- Nebular emission lines in HII regions are a signature of recent star formation proportional to the number of ionizing O-type stars.

- The broad resonance UV lines, as observed by the Hubble Space Telescope, with often P-Cygni profiles, are a signature of massive star formation visible in the integrated spectrum of galaxies.

- The emission lines of WR stars, discernible in the integrated spectra of galaxies, provide a signature of star formation in the distant universe.

18.2 The Beginning of Cloud Contraction

The onset of the collapse of an interstellar cloud depends on the initial conditions, in particular on the temperature T and density q in the cloud. In general, these conditions are not the same in a large cloud, which may give birth to a star cluster, or in a cloud fragment which will form a star. Turbulence and magnetic field also influence the collapse (Sect. 18.3).

18.2.1 The Jeans Criterion

An interstellar cloud starts contracting, when gravity forces overcome the forces due to the gradient of internal pressure, the cloud becomes gravitationally unstable. The Jeans criterion determines the conditions for the onset of contraction. Let us consider an isothermal sphere of mass M, radius R and average temperature T. Let Pequ be the ambient pressure with which the cloud is in equilibrium. The Virial theorem (Sect. 1.3) states that

where Ec is the kinetic energy, Q the potential energy, Cv the specific heat by unit of mass at constant volume. For a monatomic and perfect gas of mean molecular weight j, Cv = cv = (3/2)k/(pmu). The density distribution in the initial configuration is not necessarily constant. For a polytrope (Sect. 24.5) of index n, one has Q = -[3/(5 - n)] (GM/R2), so that q = 3/(5 - n). For a constant density, n = 0 and q = 3/5, while for a more centrally condensed object, q is larger. The equilibrium pressure at the surface of a cloud varies with its parameters M, T and R,

CvMT qGM2 2nR3 - 4 nR4

The behavior of Pequ as a function of radius is sketched in Fig. 18.2. For small values of R, the negative term in R-4 dominates, while for larger R the positive contribution in R-3 is winning. Pequ reaches a maximum at Rj, then it decreases ^ 0 as R further increases. For fixed values of M, T and q, the derivative dP/dR = 0 defines the value Rj of the radius of the maximum pressure that the cloud in equilibrium can sustain,

The stability depends on the value of the actual radius R with respect to Rj.

- For R < RJ: a reduction of radius R produces a reduction of Requ. Thus, if the cloud was initially in equilibrium with its surrounding medium, after a small decrease of R the sustainable pressure becomes smaller than the actual pressure

Fig. 18.2 Schematic variation of the equilibrium pressure Pequ as a function of radius R for a spherical body of mass M and temperature T in equilibrium

and the cloud starts contracting. This leads to a smaller value of R, which in turn makes a further reduction of the sustainable pressure, etc. The situation is unstable and the cloud collapses. - For R > Rj: an increase of R makes a reduction of Pequ. If the cloud was initially in equilibrium with the ambient medium, the external pressure is now too strong and the cloud contracts recovering its initial size.

The critical radius RJ can be expressed as a function of the average T and q in the cloud. Writing the mass M of the cloud at the critical limit MJ = (4k/3)qRj, one eliminates MJ from (18.3) and gets

For given T and q, configurations with a radius smaller than RJ starts collapsing. With radius (18.4), we can write the mass MJ, called the Jeans mass, above which a cloud of given T and q becomes gravitationally unstable,

The higher the temperature, the larger the mass for initiating the collapse, since gravitation has to overcome a larger internal gas pressure. At a given T, a higher density favors collapse. We also see that if the initial cloud has a more peaked density distribution, i.e., a larger q , the Jeans mass MJ would be smaller for the same average T and q. This means that the collapse is initiated more easily. Numerically, for a homogeneous density q = 3/5 and with the first two terms on the right of (18.5), one gets a coefficient 3.548 (i.e., about 2 a/tc, which is also found sometimes!). For neutral atomic gas with solar composition, with a mean molecular weight ^ = 0.77, we get

where T is in K and q is in g cm-3. Thus, the masses of collapsing clouds are rather large: for dense clouds with T = 10 K and q = 10-22 g cm-3, MJ « 367 M0. Since only clouds with masses much larger than current stellar masses can start contraction, we see that a process of fragmentation is necessary to form stars. The Jeans mass is a fundamental parameter determining the beginning of cloud collapse, either for galaxy or star formation.

18.2.2 Various Expressions of the Jeans Criterion

Several accurate and inaccurate expressions of the Jeans mass are found in literature:

1. If one ignores the external pressure in (18.1), the Jeans radius in Eq. (18.3) is a factor of 2 smaller. The numerical coefficient in the expression of the Jeans mass (18.5) would be 27/(%/2 • 4) instead of 27/16. However, this solution is physically less satisfactory.

2. Let us derive the Jeans criterion by simple considerations on the free-fall timescale tff (Sect. 18.2.4) and on the sound-crossing timescale tcs, which are


for a medium of density q, size X and sound velocity cs. The time tcs is the propagation time of acoustic waves in the medium, as such it also characterizes the action of pressure forces. One has two cases:

- If tcs < tf: pressure forces are acting faster than dynamical perturbations, thus the system is in hydrostatic equilibrium.

- If tcs > iff: gravitational collapse occurs faster than pressure adjustments and there is little opposite effect from thermal pressure.

In view of (18.7), contraction occurs if the size of the collapsing region is smaller than the Jeans length XJ defined by tcs = iff, i.e.,

This expression of the Jeans length is approximate (cf. 18.16). 3. The Jeans length can also be obtained from the linear perturbations of a 1D infinite medium. One starts from the continuity, Euler and Poisson equations (Sect. 1.1) and considers small perturbations p1, qi, v and gi around a non-perturbed solution with P = const. and q = const. and zero velocity v. The three equations become to the first order, f + QV' V1 = £ = - (t),d ^ + -• (18'9)

The perturbations are adiabatic and the sound velocity is given by (C.26) cS = (dp/do)ad. We take the divergence of the perturbed Euler equation

dt Q

As the spatial and time variations of the perturbations are not correlated, we have V • (dv1/dt) = (d/dt)(V • v1) and with (18.10)


This equation admits as a solution a plane wave of the form

and the resulting dispersion equation is


the frequency is imaginary, i.e., there is an exponential decay or growth of the perturbation wave, for example, the density grows exponentially. This occurs for characteristic lengths larger than the jeans wavelength

which is to be compared to (18.8). Some authors [47] take for the Jeans mass, the mass in a sphere of radius At/2 (case 1), others authors [229] take MJ = q Aj3 (case 2). In the first case, one has

while in case 2, one has a factor of n3/2. The sound velocity is cs = \J~Y (C.27), where y is the ratio of the specific heats, y = cP/cV. For the isothermal sound speed, y equals unity (e.g., 18.45) and we get for case 1,

The numerical coefficient in (18.18) is 2.916 in case 1, instead of 5.568 in case 2. These values are to be compared to 3.548 in (18.5). The differences between the various numerical factors are not too critical in view of the uncertainties concerning the velocities. We prefer expression (18.5) since it does not impose an infinite medium and allows some external pressure, which is not the case for the derivation of (18.16) from linear perturbations.

18.2.3 Initializing the Cloud Collapse

Over a large interval of densities from q « 10~23 to q « 10-13 g cm-3, the collapsing cloud remains isothermal (Sect. 18.4). An interstellar cloud with an actual mass M < MJ will not start gravitational contraction, unless some external effect compresses the gas. If this happens, the increase of density q produces a decrease of Mj and thus the actual cloud mass M may happen to be larger than the corresponding theoretical value MJ. When this occurs, collapse is initiated. Several mechanisms are able to produce the necessary density increase to initiate cloud contraction.

- Contagious star formation: if star formation starts in a galaxy rich in gas, the process of star formation may propagate through the galaxy like a forest fire. Shocks due to the ionization fronts around newborn massive stars hit the neighboring gas clouds and produce density enhancements. Shocks are also produced by supernova explosions. The large association of Scorpius-Centaurus is a magnificent example of sequential star formation. One observes from one side of the association to the other an age sequence, the oldest cluster with an age of a few 107 yr is the most scattered, then there is a young dense cluster with an age of a few 106 yr and finally the youngest objects of about 105 yr are compact IR sources.

- Density wave in spiral galaxies: the density waves associated to the arms of spiral galaxies produce a local compression of the gas, which initiates star formation. This is why young clusters are in spiral arms.

- Galaxy interactions: the collisions and interactions of galaxies produce gas compression and are responsible for intense star formation in starburst galaxies. On the average, observations in the distant universe show that star formation was on the average more active in the past.

- Cloud collision: Cloud-cloud collisions were the first mechanism proposed for star formation; however, this process plays a modest role.

18.2.4 The Timescale

The dynamical timescale or free-fall timescale tf characterizes the changes of mechanical equilibrium in a gravitational configuration. This timescale is generally much shorter than the thermal timescale. It plays an essential role in star formation and characterizes any dynamical event, for example the core collapse in supernova explosions. To derive tf, we start from (1.14)

r dP GMr

4 nr2

dMr 4nr4 '

If the cloud is isothermal (Sect. 18.4), there is no pressure gradient and the equation becomes simply r = -OM • (18.20)

We suppose that the whole mass M of the cloud participates to the collapse and the integration of this equation (after multiplying it by r) gives

Two sets of initial conditions are particularly interesting: Timescale for the Growth of Density

We consider a cloud with an infinite extension and no initial motion, i.e., at t = 0, one has ri = ^ and ri = 0. By expressing the mass in terms of the mean density g and radius r at the time considered, we get from (18.21) r2 = (8 nOg r2)/3. This gives


For contraction, we choose the sign minus. Since the mass remains constant, we have 3 (dr/r) = —(dg/g). The characteristic time tff of the free fall is

-dt 1 446 sec tff = g -= = , numerically tff =-1—, (18.23)

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