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Figure 5.12 Rotational levels of NH3. Those levels lying on the lower border constitute the rotational backbone. Here, A and B are the two rotational constants, while J and K are quantum numbers that measure, respectively, the magnitude of the total angular momentum and its component along the symmetry axis. For a given value of J, the possible K-values range from J to + J. Since, according to (5.19), states with K have the same energy, it is conventional, when labeling states, to restrict K to...

Protostellar Disks

One key simplification in our analysis has been the assumption that fluid elements from the cloud envelope directly impact the protostar. However, we saw in Chapter 10 that rotation distorts the infalling trajectories. Material with sufficiently large specific angular momentum lands first in a disk and only later spirals to the stellar surface. The protostellar disk is of considerable interest itself, since it is the medium out of which planets ultimately form. We therefore want to trace the...

Hydroxyl

In quiescent clouds far from any luminous stars, it is difficult to excite the rotational states of OH. Referring again to Figure 5.18, the energy gap between the ground and first excited level of the 2n3 2 ladder has an equivalent temperature of 120 K, while the jump to the ground state of the 2nV2 ladder is 180 K. We may therefore restrict our attention to the four ground-state hyperfine transitions depicted in the figure. The Einstein A-coefficient for the strongest line at 1667 MHz (F 2+...

Torsional Alfven Waves

This form of the equation suggests another interpretation of magnetic braking. We view the twisted field itself as transporting a certain flux of angular momentum FJ along the poloidal direction. The net torque exerted by the field on our patch is then the rate at which angular momentum enters the inner face minus its efflux from the outer one. This interpretation corresponds to equation (10.40) provided we identify Remember that it is the z-component of angular momentum which is actually being...

Temperature Profile

Consider first the run of temperature in the molecular interior. Since the grain photoelectric effect relies on ultraviolet photons, it can no longer heat the gas within the molecular interior. We are left with cosmic rays as the principal heating agent. Almost all the carbon is in CO, which now becomes the major coolant. The gas temperature in this region therefore follows from where equation (7.14) gives the cosmic-ray contribution. The rising curve in Figure 8.2 is the temperature-density...

Water H2O

The water molecule has a relatively large dipole moment, almost 20 times that of CO. Here, the associated vector f is directed along the symmetry axis through the oxygen atom (Figure 5.15). Within the ground vibrational state are a large number of allowed rotational transitions at far-infrared and millimeter wavelengths. Excitation of these levels followed by prompt radiative decay provides an important cooling mechanism in shock-heated clouds, where accelerated chemical reactions produce a...

Hydrogen Ionization Rate

Determining either rCR(H2) or rCR(HI) still requires measurement of the respective ionization rates, Z(H2) or Z(HI). Here, the main impediment is our ignorance of the interstellar flux of low-energy cosmic rays. We have noted, however, that the ionizations are the first steps in a network of ion-molecule reactions that lead to the formation of more complex species. We may therefore use the observed abundances of selected molecules to infer the ionization rates indirectly. One such molecule is...

Photon Penetration

Proceeding inward, we must modify DH2 in equation (8.3) to account for the absorption of ultraviolet photons. Dust provides some of this absorption, as does the electronic excitation of H2. Thus, a hydrogen column density NH, with a molecular component NH2, corresponds to an optical depth of NH+ NH2ai(v). Here, we are neglecting dust scattering and using the fact that a grain's cross section for absorbing the far-ultraviolet photons that dissociate H2 is essentially its geometrical area. (The...

Champagne Flows

In the final, equilibrium state, nH must fall to (a0 a1)2n0H, i. e., far below the external density. It follows from equation (15.12) that the final radius Rf is Written in terms of temperature, the expansion factor is (2T1 T0)2 3, so that the radius swells by two orders of magnitude. In practice, such a large inflation never goes to completion. The expansion time may become so long that the star itself dies before equilibrium is achieved. More realistically, the ionization front reaches the...

Main Sequence Trends

Any results concerning the rotation of young stars should be viewed in context. Specifically, it helps to understand first the behavior of mature objects in this regard. Figure 16.17 displays the variation of equatorial speed with mass, or equivalently, spectral type, for single main-sequence stars in the field. Each filled circle represents an average of Veq sin i over many equal-mass stars, as obtained spectroscopically. Note that the presence of the unknown inclination angle has little...

Orion and the Pleiades

Within any young cluster, a significant spread in rotational speeds exists at all stellar masses, not just those close to solar. How the distribution of speeds varies with mass will ultimately reveal much about both magnetic braking and the internal redistribution of angular momentum. The pattern is especially well documented in the Pleiades. Figure 16.19 summarizes graphically the results of several hundred spectroscopic velocity measurements. We see immediately that most stars have relatively...

Poincare Wavre Theorem

How does rotation affect cloud morphology In the preceding section, we were able to obtain spherical structures because the supporting thermal pressure is inherently isotropic. If we now consider the cloud to be rotating about a fixed axis, the associated centrifugal force on each fluid element points away from that axis, distending the equilibrium structure accordingly. It is natural, then, to erect a cylindrical coordinate system whose z-axis lies in the direction of rotation (see Figure...

Carbon Ionization

The ultraviolet component of the radiation field is too weak at energies above 13.6 eV to significantly ionize the hydrogen or helium in molecular clouds. However, a number of heavier elements have lower ionization potentials. Of these, atomic carbon (C I) is the most abundant, with a number density relative to hydrogen of nC nH 3 x 10-4 (see Table 2.1). Any photon more energetic than 11.2 eV will eject an electron from C I. Since the kinetic energy of this electron quickly disperses to...

Centrifugal Radius

These considerations indicate that magnetic braking is both rapid and efficient. On the other hand, the braking must fail within the deeper interior of a more realistic, collapsing cloud. As the density climbs, the matter and field decouple because of the drop in ionization fraction. In addition, much of the remaining field is left behind once it is severely pinched and reconnects. Thus, the matter inside some volume of the equatorial region indeed conserves angular momentum and spins up as it...

Dusty Envelopes

The resolution of this paradox is twofold. First, the bulk of excess emission in Group I sources must not originate in a self-luminous disk, but in some more extended structure. Second, this dusty configuration must have sufficiently low column density to leave the central star optically visible. Here, we are aided by the larger intrinsic luminosities of Herbig Ae Be stars. But it is clear that the circumstellar matter cannot resemble the infalling envelope we described for spherical protostars...

Index of Astronomical Objects

Aquila, 115,433,488 Auriga, 11 Chamaeleon, 103 Crux, 119 Cygnus, 509, 545, 547 Lupus, 101 Monoceros, 63, 104 Orion, 2 Perseus, 495 Taurus, 10 Kleinman-Low (KL) Nebula, 10, 138, 226, 229, 233, 237, 455, 508, 513, 535 NGC 2023, 96 NGC 2024, 96 NGC 2068, 96 NGC 2071, 96 NGC 2244, 63, 67, 109, 385 NGC 2264, 104, 107, 109, 121, 363, 686 NGC 3603, 97, 689 NGC 7538, 97 Quintuplet Cluster, 734 R136, 417,745,750 S106, 96, 121, 257 S255 S257, 96 Large Magellanic Cloud, 97, 417, 736, 744, M31 (Andromeda...

Onset of Convection

The rising entropy profile is also of fundamental importance for convective stability. Imagine, as in Figure 11.7, that we displace a fluid element with internal density pint by a small distance Ar in the direction opposite to the local gravitational acceleration g, i. e., toward the surface of the protostar. We know from our study of hydrostatic equilibria that the pressure always declines in this direction. Hence our fluid element, if it is to maintain pressure balance with its surroundings,...

Level Populations

The high intensity of maser spots results when photons of a certain frequency stimulate a molecule to emit others of the same frequency and direction. This augmented group of photons then creates even more stimulated emission, so that the amplification is exponential. As we shall see, the entire process hinges on the fact that there are more molecules in the upper state of the transition than the lower. Recall that we have defined the excitation temperature of a two-level system through the...

Theoretical Derivation

How are we to understand this number from a theoretical perspective Species such as HCO+ are important practically, but constitute only a fraction of the cloud's ions. The majority are singly charged atoms, chiefly Na+, Mg+, Ca+, and Fe+. Most heavy elements within a dense core are actually locked up in solid grains, and only a few percent by mass exist in the gas phase. Nevertheless, the density of the gaseous component is sufficiently high that the condition of overall charge neutrality is,...

Hydroxyl OH

We saw in Chapter 3 how molecular clouds are threaded by the magnetic field lines that permeate interstellar space. Compression of this field creates an effective pressure that partially supports clouds against gravitational collapse. To measure B accurately, a molecular probe should have a relatively large magnetic moment. Especially important in this regard are molecules with one unpaired electron and therefore a nonzero electronic angular momentum. Classified chemically as free radicals,...

Hydrogen H2

We now consider a few of the species that have been especially fruitful in the study of molecular clouds. Table 5.1 lists abundances and important transitions for all the molecules discussed in this chapter, as well as some others of interest. The sixth column gives the energy difference between the upper and lower states. This energy is expressed as an equivalent temperature by using the Boltzmann constant kB. Also given is the Einstein coefficient Aul, i. e., the probability per time of...

Magnetic Flux Freezing

We have seen that neither their internal, thermal pressure nor centrifugal force are strong enough to support most clouds against collapse. We therefore turn our attention to the interstellar magnetic field. A promising indication that magnetic forces are generally adequate for the task comes from a simple virial theorem analysis. As we noted in Chapter 3, the typical fields in giant complexes are sufficiently strong that the terms M and W in equation (3.18) are comparable in magnitude. It is...

Bubbles and Shells

An individual supernova remnant typically spans a few tens of parsecs and cools significantly after 106 yr or so. On the other hand, there are much larger regions in our Galaxy, with diameters ranging from 102 to 103 pc, that display a similar morphology. That is, they consist of a hot, tenuous interior surrounded by a thick surface layer of relatively cold and dense gas. Judging from the measured velocity of the outer component, expansion has been proceeding for some 107 yr. The process must...

Dust Envelope and Opacity

The gas raining down on the protostar originates much farther away, in the outer envelope. This is the infalling region where, as we noted in 10.2, the gas temperature rises sluggishly with density as a result of efficient cooling by dust. Despite the nomenclature, we recall that the matter here does not fall until it is inside the rarefaction wave gradually spreading throughout the cloud. Most of this expanding volume is nearly transparent to the protostellar radiation. However, as the...

Continuum Veiling

Veiling Optical Infrared

The question of what creates non-chromospheric Ha is a difficult one that is far from resolved. Before delving into the matter, let us first complete our description of the emission characteristics of T Tauri stars generally. Lines in the ultraviolet region of the spectrum also attest to elevated activity. This radiation was first extensively studied using the International Ultraviolet Explorer (IUE) satellite, launched in 1978. The spectrum of classical T Tauri stars from 1100 to 3100 A...

The Stellar Birthline

Throughout this book, we have frequently utilized the birthline as a theoretical construct within the HR diagram. We presented the curve as the locus from which pre-main-sequence tracks descend, as well as the fiducial zero-point for stellar ages. But what is the motivation, from an observational viewpoint, for introducing this concept And once the idea is accepted, with what precision can we pinpoint the curve's location, from either observation or theory We saw in previous chapters that the...

Ammonia NH3

Here, IA, , and IC are the moments of inertia about the principal axes of rotation, while JA, Jb , and JC are the corresponding projections of the total angular momentum vector J. The Figure 5.10 Molecular structure of NH3, showing the three principal axes of rotation. Figure 5.11 Torque-free motion of NH3. The molecule rotates about the axes perpendicular to the plane of hydrogen atoms, with associated angular momentum JA. The latter vector in turn precesses about the axis lying along the...

Interstellar Radiation

A second important heating agent for molecular clouds is the diffuse radiation field that permeates interstellar space. We need to understand in detail how these photons impinging on the gas create thermal energy. We should also look to stars embedded within the clouds as additional heating sources. Frequency log v (Hz) Figure 7.4 Mean intensity of the interstellar radiation field, expressed as a function of frequency. Frequency log v (Hz) Figure 7.4 Mean intensity of the interstellar radiation...

Deuterium Thermostat

Once deuterium ignites near the center of a low-mass protostar, the induced convection quickly spreads. Soon, the whole interior is unstable, apart from a thin, outer settling region of negligible mass. The open circle in Figure 11.6 marks the onset of full convection. By this point, deuterium burning has significantly increased the protostar's radius. The degree of swelling depends somewhat on the assumed accretion rate. Figure 11.8a displays R (M*) for three different M-values. The increase...