M

where m is the mean molecular weight of the material in the cloud; mH is the mass of a

Figure 2.1. Molecular cloud Barnard 68 observed by the ESO Very Large Telescope at a range of visible and near-infrared wavelengths between 0.44 ^m and 2.16 ^m. Credit European Southern Observatory.

hydrogen atom; N is the total number of molecules in the cloud; and kB is the Boltzmann constant.

The gravitational binding energy of a cloud of radius R is given by

GM2 - r and hence an interstellar cloud should collapse if EG > ET, or

Assuming that the cloud has a uniform density p, its mean radius will be

Substituting for R in Equation (2.8), and rearranging, we obtain Jeans' expression for the minimum mass of cloud of temperature T and density p that will collapse

Substituting a temperature of 10 K, and a density of 10~14 kg m 3, we find that Mj « 1029 kg, or approximately 0.1 M0 (where M0 is the mass of the Sun), and from Equation (2.9), R = 1.4 x 1014 m (or 0.015 light-years, or -1,000 AU). Hence, according to Jeans' theory, the denser parts of the ISM should be unstable to Jean's collapse. However, this theory ignores effects such as magnetic fields and gas flow in the clouds that oppose the collapse. In reality it thus appears that in most cases some sort of external compression is also required to initiate collapse, such as collision between two clouds, impact of a shock wave from a nearby exploding star, or the action of the spiral density wave that periodically sweeps through the galaxy (Jones, 2007).

Once the whole cloud starts to contract, the denser parts of the cloud contract more quickly and thus the cloud quickly fragments, with each cloud fragment condensing to form its own star system. Thus new stars seem to form in clusters, as is observed.

2.3.2 Formation and evolution of circumstellar disks

As each cloud fragment collapses, gravitational binding energy is released as thermal energy. In the initial stages the temperature rise is small since the opacity of the nebula is very thin and thus this energy efficiently radiates away. However, the center of the nebula is predicted to collapse more quickly than the edge, and once its density increases to the point where it starts to become opaque, the temperature rise is rapid. Although this rise in temperature tends to slow the rate of collapse, the temperature of the central "proto-star" still continues to rise until the temperature of the core is sufficient to initiate fusion.

The whole cloud fragment will have some net rotation. Hence, as the cloud collapses the inner parts will begin to rotate more rapidly by conservation of angular momentum. Material on or near the net rotation axis will fall freely towards the center whereas the infall elsewhere is moderated by centrifugal force. Hence, a circumstellar disk forms in the plane perpendicular to the rotation axis. Approximately 50% of young stellar objects (YSOs) that have been observed to date are surrounded by disk-like or ring-like structures (Andre and Montmerle, 1994; Drouart et al., 1999). There are four classes of these:

(1) Class 0. The youngest class of YSO with a large mass of circumstellar material (0.5 M0 or more) and a lifetime of approximately 10,000 yr (M0 is the current mass of the Sun).

(2) Class I. Mass -0.1 M0, lifetime around 100,000 yr, extending up to few 1,000 AU across.

(3) Class II. Mass -0.01 M0, lifetime around 1 Myr and optically thick at 10 ^m with excess thermal emission.

(4) Class III. Optically thin at 10 ^m, and radially more compact at around 100 AU.

These observations are consistent with the evolutionary scenario where a massive envelope (Class 0) rapidly collapses to form a proto-star in a timescale of only

Emission-line composite image Continuum image i_i

Figure 2.2. Hubble Space Telescope image of a young circumstellar disk (Orion 114—426) in the Orion Nebula (after McCaughrean and O'Dell, 1996). The left-hand panel shows an emission-line composite, made by combining data from three narrow-band filters centered on bright emission lines from the nebula (namely, [O iii], Ha and [N ii]). The strong emission lines provide a bright background which reveals the circumstellar disks as silhouettes around their young stars. The right-hand panel shows the corresponding continuum image taken through the medium bandwidth F547M filter. The central star shows up most clearly in the right-hand panel, and a faint reflection nebula is also seen above and below the plane of the silhouette disk. Courtesy of Mark McCaughrean and the Astronomical Journal.

Emission-line composite image Continuum image i_i

Figure 2.2. Hubble Space Telescope image of a young circumstellar disk (Orion 114—426) in the Orion Nebula (after McCaughrean and O'Dell, 1996). The left-hand panel shows an emission-line composite, made by combining data from three narrow-band filters centered on bright emission lines from the nebula (namely, [O iii], Ha and [N ii]). The strong emission lines provide a bright background which reveals the circumstellar disks as silhouettes around their young stars. The right-hand panel shows the corresponding continuum image taken through the medium bandwidth F547M filter. The central star shows up most clearly in the right-hand panel, and a faint reflection nebula is also seen above and below the plane of the silhouette disk. Courtesy of Mark McCaughrean and the Astronomical Journal.

100,000 years, together with an extended disk of mass between 0.01 M0 and 0.1 M0 (Class I and Class II), which evolves over a timescale of 1 Myr to 10 Myr into a less massive disk with lower density (Class III). The upper bound of estimated disk masses is consistent with the theoretical upper mass of ^0.3 M0 derived from stability arguments of Shu et al. (1990). Particularly clear examples of these circumstellar disks are seen in the Orion Nebula. Here, several such nebulae are seen in front of a glowing background interstellar cloud, which is lit by the light of stars already created (Figure 2.2) (McCaughrean and O'Dell, 1996; McCaughrean et al., 1998). As can be seen the circumstellar disk is of the order of 1,000 AU across. This formation scenario requires a mechanism for disk dispersal on timescales of 1 Myr to 10 Myr. Solar mass stars cannot easily blow away dense circumstellar disks and thus accretion onto the star seems more likely. However, in order for this accretion to proceed some way is needed of losing or redistributing angular momentum.

By conservation of angular momentum, one would expect the nebula to rotate fastest towards the center and thus the proto-Sun should initially have been rotating very rapidly. However, the Sun now contains only 1% of the solar system angular momentum (0.5% in spin, and 0.5% in orbital rotation of Sun about the solar system barycenter, which is just outside the Sun) with the bulk (85%) now contained in the orbital angular momenta of Jupiter and Saturn. Hence, to explain the current state of the solar system a means is needed whereby the circumstellar disk was accreted or dissipated and where most of the proto-Sun's angular momentum was lost. Two key processes have been identified.

(1) Turbulence. Conditions in the early circumstellar disk are likely to have been very turbulent. This turbulence would have led to a net transfer of mass outward in the outer part of the disk, and inwards towards the proto-Sun nearer the center. Associated with this mass flow would have been a net transfer of angular momentum from the proto-Sun to the disk. The transfer of angular momentum would have caused the disk to spread farther and farther out into space, reducing its density. In the inner disk, the mass loss would have been towards the proto-Sun and also to space via outflow along the rotation axis, in collimated jets which are observed in proto-star nebulae, as shown in Figure 2.3 (McCaughrean et al. 1994). These polar jets are probably collimated both by the dense circumstellar disk itself and by the magnetic field of the protostar. They are seen to be episodic in nature, which is consistent with them being fed by turbulent infall of the inner circumstellar disk. However, while this bipolar outflow would have carried off as much as 10% of the nebula mass, it would not have accounted for much of the angular momentum.

(2) T-Tauri phase. The Sun is currently about half-way though its main sequence (Lewis, 1995). Just prior to the main sequence phase, a period called the T-Tauri phase (named after a star currently observed in the constellation of Taurus) occurs, which is marked by a considerable outflow of gas in the solar wind, and also high UV radiation. For a star with the mass of the Sun, this period would have lasted about 10 Myr and would have led to the Sun losing about 10% of its mass, and a significant proportion of its angular momentum. As we shall see later, the high solar wind associated with this phase would have swept away any remaining fragments of the solar nebula, and thus the timing has profound implications on the formation of giant planet atmospheres.

Observations of YSOs suggest that the Sun had accreted most of its mass very rapidly within about 100,000 yr of the start of collapse. At this stage the Sun was surrounded by a circumstellar disk. The inner part of the disk would have been very hot due to a combination of opacity, turbulent frictional heating, and solar luminosity. Temperatures out to approximately 1 AU probably exceeded 2,000 K, evaporating almost every solid constituent. Farther out, temperatures reduced with distance from the Sun and thus the "condensation line'' (i.e., the distance from the Sun that different minerals and ices would have condensed from the nebula) occurred at varying distances from the Sun, with more volatile materials condensing farther from the Sun than less volatile materials. As time progressed and the disk cooled and spread out, these condensation lines would have moved inwards towards the Sun at a rate governed by both the radiative heat loss to space, and the frictional heating generated by turbulence. Not all parts of the disk may have been dominated by turbulence, however. In the outer disk, beyond perhaps 50 AU, the nebula density is predicted to have been so low that turbulence would not have been able to play such an important role. In this low-turbulence region, material would rotate about

Figure 2.3. Wide-field medium-resolution near-IR (molecular hydrogen line at 2.122 ^m) image of HH212. Data taken in December 1996 using the Calar Alto 3.5 m telescope in Spain. The image shows the bi-polar jet from the formation of a star near the center of the frame and hence the circumstellar disk appears to be almost edge-on. Material can be seen to be ejected in pulses and there is a clear symmetry between the pulses in the upper and lower jets. The episodic nature of the outflow points to the turbulent nature with which material falls from the circumstellar disk onto the central star. Courtesy of Mark McCaughrean.

Figure 2.3. Wide-field medium-resolution near-IR (molecular hydrogen line at 2.122 ^m) image of HH212. Data taken in December 1996 using the Calar Alto 3.5 m telescope in Spain. The image shows the bi-polar jet from the formation of a star near the center of the frame and hence the circumstellar disk appears to be almost edge-on. Material can be seen to be ejected in pulses and there is a clear symmetry between the pulses in the upper and lower jets. The episodic nature of the outflow points to the turbulent nature with which material falls from the circumstellar disk onto the central star. Courtesy of Mark McCaughrean.

Figure 2.4. Variation of nebula pressure in the circumstellar disk above the ecliptic plane.

the protostar with a period governed by Kepler's orbital laws and thus this part of the nebula is called the Keplerian disk. Such two-component circumstellar disks have been observed (Drouart et al., 1999; Guilloteau et al., 1997). Hence, while we expect the material in the inner turbulent disk to have undergone significant mixing and thermal reprocessing, material in the outer Keplerian disk will have been largely unmixed and unprocessed from its pre-solar form.

Modeling of the evolution of circumstellar disks is usually done with the alpha disk model of Shakura and Sunyaev (1973). In such models the surface density of the disk, £(r, t) (where r is the distance from the star and t is the time) is solved, where it is assumed that turbulence alone redistributes mass and angular momentum. The level of turbulence is prescribed by a single, dimensionless scaling parameter a and values of between 10-2 and 10-3 yield disk lifetimes consistent with the inferred lifetimes of circumstellar disks (Hartmann et al., 1998; Hueso and Guillot, 2003). More advanced models (e.g., Huré et al., 2001) use the beta parameterization of turbulence of Richard and Zahn (1999). While the structure of the gaseous part of the nebula is determined via hydrostatic equilibrium and centrifugal forces in the radial direction, the structure in the vertical direction (i.e., in the direction parallel to the rotation axis; Figure 2.4) is determined by hydrostatic forces only. Collapse in this direction stops when the pressure gradient force equals the gravitational force: that is,

GMkZ

where r is again the distance from the proto-star; z is the height of a parcel of gas above the disk plane; Ms is the mass of the proto-star; and d is the angular elevation of the parcel above the disk plane as seen from the protostar. Assuming hydrostatic equilibrium, the change in pressure between heights z and z + dz above the disk plane is given by dP = -Pgzdz = (- H)^) dz (2.12)

where ^ is the molecular weight of the gas; and R is the gas constant, and hence in equilibrium

where H is the scale height given by

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