Jse

This limiting luminosity is thus proportional to the mass of the star. Beyond this value, the radiation pressure predominates, and the star becomes unstable. It should be mentioned that the Eddington limit applies to massive objects, and not to solartype stars.

During the course of its evolution towards the Main Sequence, the star maintains an essentially constant luminosity and surface temperature. From the equations just given, it may be deduced that the luminosity is proportional to the cube of the mass, and the temperature is proportional to the ratio of the mass of the star to its radius. Observations have allowed the value of the exponent to be refined:

where LQ and MQ are the luminosity and mass of the Sun.

Fig. 5.17 Evolution of protostars on the Hertzsprung-Russell Diagram, from the collapse of the molecular cloud to the Main Sequence. The Hayashi Limit, at a constant temperature of about 4000 K, corresponds to an instability region. The star cannot cross this limit, and loses luminosity, at a constant temperature, before joining the Main Sequence (After Acker, 2005)

The method of energy transport depends on the mass of the star. For stars with masses comparable to that of the Sun, radiative equilibrium predominates, whereas convective equilibrium prevails in low-mass stars (< 0.3 Ms). The outer convective zones reduce as the mass of the star increases.

Let us return to consideration of the evolution of stars after their T-Tauri phase. We have seen that this phase is marked by an extremely high luminosity and a relatively thick disk (a Class II object, see Fig. 5.13). One star, FU Orionis, observed by Herbig, allows us to describe the stage following the T-Tauri stage, and which is the last stage of star formation (a Class III object, see Fig. 5.13). The FU-Ori phase is characterized by an even more violent eruptive phase, during which the star ejects a large fraction (or even almost all) of the mass of the disk. During phases II and III, the surface temperatures of these objects increases to about 4000 K, then their luminosity decreases before they reach the Main Sequence (Fig. 5.17).

The lifetimes of protoplanetary disks is estimated to be some ten million years (see Fig. 5.13), as derived from observations: young stars surrounded by disks all have ages less than ten million years. This observational fact translates into an extremely powerful constraint on models for planetary formation within these disks.

Fig. 5.17 Evolution of protostars on the Hertzsprung-Russell Diagram, from the collapse of the molecular cloud to the Main Sequence. The Hayashi Limit, at a constant temperature of about 4000 K, corresponds to an instability region. The star cannot cross this limit, and loses luminosity, at a constant temperature, before joining the Main Sequence (After Acker, 2005)

5.2.7 The Structure of Protoplanetary Disks

As a working hypothesis, we assume a viscous disk, externally isolated (this is obviously an approximation, because accretion is also generally present). The evolution of this disk may be described by assuming Keplerian motions around the central star, also assuming that radial pressure gradients are absent. We may then write:

where a is the angular velocity and j the kinetic moment. M* is the mass of the central star, and r the distance from the star. The vertical structure of the disk may be calculated (where the z-axis is perpendicular to the plane of the disk) by assuming that the gas is isothermal and in hydrostatic equilibrium along that axis. We then find the following relationship between the scale height, h, and the distance from the star:

where vs and vk are, respectively, the velocity of sound and the Keplerian velocity at distance r from the star. It will be seen that the thickness, h, of the disk increases with distance from the star, which agrees with observations (Figs. 5.9 and 5.16). Because the disk is thin, the Keplerian velocity is far greater than the velocity of sound:

Dynamical equations describing the evolution of a thin disk must take viscosity into account. This may be expressed as a couple t; the latter is proportional to the coefficient of viscosity, to the density, and to the distance from the star. By taking conservation of angular momentum into account, it may be shown that, in the inner part of the disk, the radial velocity, vr, of the gas is negative (i.e., directed towards the centre). There is therefore a transfer of material inwards. The mass-loss rate may be expressed as follows:

The equations just discussed describe the case of a disk with no gain of material from the outside. In the case of a stationary accretion disk, we make the assumption of a constant rate of accretion of interstellar material falling onto the disk, independent of time and of distance from the star, and expressed as a fraction of a stellar mass. The surface density is a function of the accretion rate and the viscosity, v.

The viscosity in a flow of gas with a velocity gradient is caused by an exchange of kinetic moment along the axis that is perpendicular to the motion (i.e., the effect of friction). If one only takes internal friction into account in estimating the viscosity of the disk, it seems that the dynamical effect on the disk is small. Nevertheless, a realistic calculation requires that turbulence in the disk must be taken into account. Turbulence is revealed by the presence of violent, chaotic motion, active a various spatial scales, and capable of causing significant exchange of kinetic moment. Large-scale eddies may be described as having a diameter similar to the thickness, h, of the disk, and a velocity equal to the speed of sound, vs. In the absence of turbulence, the coefficient of viscosity v may be written:

where u is the average kinetic moment, and l the mean free path. When turbulence is present, the viscosity coefficient needs to be modified to take account of the effects of the eddies. Empirically, we may write:

where the coefficient a is less than 1. On a similar empirical basis, one generally use a value of a that is close to 0.01. Disks modelled in this manner are known as 'a-disks'. Attempts have been made to estimate the density of the surface of the disk as a function of distance r. By assuming that Vs2 is proportional to the temperature T and to r3/4, it is found that the surface density also decreases as r3/4.

In the case of the primordial solar nebula, the evolution of temperature and pressure as a function of distance from the Sun has been calculated, based on the a-disk theory, as a function of a and of the accretion rate. Wood and Morfill (1988) carried out this modelling, expressing the opacity of the disk k as k = K0 T2 (5.21)

where k = 10-6cm.g-1 K-2 for temperatures between 160 and 1600 K (corresponding to absorption by metals and silicates) and k = 2 x 10-4cm.g-1 K-2 for T < 160 K (absorption by ices). From the equations for the conservation of mass, of momentum, and energy, the authors obtained the following expressions for grains predominantly consisting of water ice:

T (K) = 54600 a-1/3. M1/2. |2/3 .k^/3. r-3/2 (5.22)

P(bar) = 1.7710-7a-5/6. M3/4. |2/3.k02/3. r-9/4 (5.23)

In fact, the relationship to r varies as a function of the composition of the grains. It is close to r-1 for material that is predominantly silicates or metals, which is in agreement with the relationship that has been measured for the inner Solar System (Dubrulle, 1993).

Figure 5.18 shows the distribution of T and P as a function of distance from the centre r, for a = 10-2 and an accretion rate that varies between 10-6 and 10-9 solar masses per year. The total mass of the disk (in solar masses Mq ) may be expressed (Wood and Morfill, 1988) as:

where R is the maximum size of the disk, in AU. For a = 10-3, Md varies from 0.03 to 0.003 Ms for an accretion rate that varies between 10-6 and 10-9 MQ/yr. For a = 10-3, the value of Md varies between 0.13 (10-6 Mq/yr) and 10-9(10-9 MQ/yr).

More recently, more elaborate models have been developed, notably by Papaloizou and Terquem (1999), and which have used a more accurate expression for the opacity, taking account of grains, molecules, atoms, and ions. The pressure curves are in good agreement. The temperatures are, overall, in good agreement

Fig. 5.18 Evolution of temperature and pressure in the equatorial plane of a disk, according to the model by Wood and Morfill (1988). The horizontal plateau in the temperature curve corresponds to a transitional region where the opacity has values intermediate between those for ices and that for refractory materials (After Wood and Morfill, 1988)

Fig. 5.18 Evolution of temperature and pressure in the equatorial plane of a disk, according to the model by Wood and Morfill (1988). The horizontal plateau in the temperature curve corresponds to a transitional region where the opacity has values intermediate between those for ices and that for refractory materials (After Wood and Morfill, 1988)

Fig. 5.19 Evolution of the temperature and pressure profiles as a function of time within the primitive solar nebula (After Hersant etal.,2001)

Fig. 5.19 Evolution of the temperature and pressure profiles as a function of time within the primitive solar nebula (After Hersant etal.,2001)

0 10 20 30 40 50 60 70 80 Radius (AU)

with the results shown in Fig. 5.19 for T < 160K, but differ significantly at higher temperatures (Wood, 2000).

In the case of the solar nebula, the evolution of the pressure and temperature curves as a function of time has been modelled by Hersant et al. (2001), assuming a monotone decrease in the accretion rate with time. Figure 5.19 shows the evolution of T and P over time, and where the origin for time corresponding to the moment when the Sun attained its current mass. Knowledge of the radial distribution of temperature and pressure are crucial factors in the construction of models of planetary formation, because they determine the sequence in which solids condense.

5.2.8 Composition of the Gas and Dust

The gas content of protoplanetary disks is dominated by molecular hydrogen. H2 is detected from its quadripolar transitions, particularly in the near infrared (at 2.1 |m), and in the intermediate infrared (S(1) at 17 |m, and S(0) at 27 |m). It may be noted that hydrogen's presence is only detectable at temperatures of several hundred K. The tracer that is universally used is CO, which is observed at its millimetric and submillimetric rotational transitions. The very high spectral resolutions (R > 106) and spatial resolutions (< 1 arcsecond) that are reached by millimetric interferometers, in conjunction with the possibility of observing several transitions, enable the gas to be traced in the disk with spatial resolutions of several tens of AU, and also allow its velocity and temperature fields to be determined. The 13CO(2—1) line at 220 GHz, which has relatively low opacity, enables the median plane of the disk to be probed, while the 12CO(2—1) line at 230 GHz probes the outer surface of the disk, at about three scale heights. It appears that the temperature in the median plane is less than that at the outer surface (d'Alessio, 1999), as a result of the mechanism described earlier (5.2.5, Fig. 5.16). The temperature in the median plane (about 15 K) may lead to the partial condensation of CO.

Numerous other ions and molecules (or both) have been detected in protostellar disks: CS and H2CO (see Fig. 5.2), HCN, HNC, C2H, HCO+, DCO+, etc. (Dutrey et al., 2006).

The dust in protostellar disk exhibits the infrared spectroscopic signature of the silicates (amorphous or crystalline), carbon (either amorphous or in the form of PAH), oxides, or ices (H2O) that form the mantle of the grains. The SWS spectrometer on the ISO satellite enabled these components to be positively identified. In particular, it revealed the similarity between the spectrum of Comet Hale-Bopp and that of certain disks surrounding young objects (such as HD 100546; see Fig. 5.20).

It should be noted that the typical size of the grains in protostellar disks is larger than one micrometre, which is comparable with the size of interplanetary dust, and significantly larger than interstellar grains. This property may be the sign of an accretion process in protoplanetary disks.

The slope of the spectrum of disks in the millimetre region may be used to determine the structure of the grains. The coefficient of k0 may be described by

Fig. 5.20 Comparison of the SWS-ISO spectra of Comet Hale-Bopp (dashed curve) with that of the young object HD 100546. The similarities between the spectral signatures is notable. They are attributed to forsterite (the crystalline silicate Mg2SiO4), to amorphous olivine, to FeO, and to various forms of PAH (After Malfait et al., 1998)

5 Star Formation and Protoplanetary Disks C/1995 O1(Hale Bopp) vs. HD 100546

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