Gas Dynamics
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Appendix A Gas Dynamics The nature of stars is complex and involves almost every aspect of modern physics. In this respect the historical fact that it took mankind about half a century to understand stellar structure and evolution (see Sect. 2.2.3) seems quite a compliment to researchers. Though this statement reflects the advances in the first half of the 20th century it has to be admitted that much of stellar physics still needs to be understood, even now in the first years of the 21st century. For example, many definitions and principles important to the physics of mature stars (i.e., stars that are already engaged in their own nuclear energy production) are also relevant to the understanding of stellar formation. Though not designed as a substitute for a textbook about stellar physics, the following sections may introduce or remind the reader of some of the very basic but most useful physical concepts. It is also noted that these concepts are merely reviewed, not presented in a consistent pedagogic manner. The physics of clouds and stars is ruled by the laws of thermodynamics and follows principles of ideal, adiabatic, and polytropic gases. Derivatives in gas laws are in many ways critical in order to express stability conditions for contracting and expanding gas clouds. It is crucial to properly define gaseous matter. In the strictest sense a monatomic ideal gas is an ensemble of the same type of particles confined to a specific volume. The only particle–particle interactions are fully elastic collisions. In this configuration it is the number of particles and the available number of degrees of freedom that are relevant. An ensemble of molecules of the same type can thus be treated as a monatomic gas with all its internal degrees of freedom due to modes of excitation. Ionized gases or plasmas have electrostatic interactions and are discussed later. N.S. Schulz, The Formation and Early Evolution of Stars, Astronomy and Astrophysics 371 Library, DOI 10.1007/978-3-642-23926-7, © Springer-Verlag Berlin Heidelberg 2012 372 A Gas Dynamics A.1 Temperature Scales One might think that a temperature is straightforward to define as it is an everyday experience. For example, temperature is felt outside the house, inside at the fireplace or by drinking a cup of hot chocolate. However, most of what is experienced is actually a temperature difference and commonly applied scales are relative. In order to obtain an absolute temperature one needs to invoke statistical physics. Temperature cannot be assigned as a property of isolated particles as it always depends on an entire ensemble of particles in a specific configuration described by its equation of state. In an ideal gas, for example, temperature T is defined in conjunction with an ensemble of non-interacting particles exerting pressure P in a well-defined volume V . Kinetic temperature is a statistical quantity and a measure for internal energy U . In a monatomic gas with Ntot particles the temperature relates to the internal energy as: D 3 U 2 NtotkT (A.1) where k D 1:381 1016 erg K1. The Celsius scale defines its zero point at the freezing point of water and its scale by assigning the boiling point to 100. Lord Kelvin in the mid-1800s developed a temperature scale which sets the zero point to the point at which the pressure of all dilute gases extrapolates to zero from the triple point of water. This scale defines a thermodynamic temperature and relates to the Celsius scale as: o T D TK D TC C 273:15 (A.2) It is important to realize that it is impossible to cool a gas down to the zero point (Nernst Theorem, 1926) of Kelvin’s scale. In fact, given the presence of the 3 K background radiation that exists throughout the Universe, the lowest temperatures of the order of nano-Kelvins are only achieved in the laboratory. The coldest known places in the Universe are within our Galaxy, deeply embedded in molecular clouds – the very places where stars are born. Cores of Bok Globules can be as cold as a few K. On the other hand, temperatures in other regions within the Galaxymayevenriseto1014 K. Objects this hot are associated with very late evolutionary stages such as pulsars and -ray sources. Fig. A.1 illustrates examples of various temperature regimes as we know them today. The scale in the form of a thermometer highlights the range specifically related to early stellar evolution: from 1 K to 100 MK. The conversion relations between the scales are: T E D kT D 8:61712 108 keV ŒK 12:3985 E D keV (A.3) ŒA˚ A.1 Temperature Scales 373 100 billion 10 million 0.001 Gamma– rays 10 billion 1 million 0.01 1 billion 100,000 0.1 X–rays Radio 100 million 10,000 1 10 million 1,000 10 Ultra voilet 1 million 100 100 – 100,000 10 1,000 Optical 10,000 1 10,000 Infra– 1,000 0.1 100,000 red mm Sub 100 0.01 1 Million – 10 0.001 10 Million 1 0.0001 100 Million Kelvin eV Angstrom Fig. A.1 The temperature scale in the Universe spans over ten orders of magnitude ranging from the coldest cores of molecular clouds to hot vicinities of black holes. The scale highlights the range to be found in early stellar evolution, which roughly spans from 10 K (e.g., Barnard 68) to 100 MK (i.e. outburst and jet in XZ Tauri and HH30, respectively). Examples of temperatures in- between are ionized hydrogen clouds (e.g., NGC 5146, Cocoon Nebula), the surface temperatures of stars (e.g., our Sun in visible light), plasma temperatures in stellar coronal loops (e.g., our Sun in UV light), magnetized stars (i.e., hot massive stars at the core of the Orion Nebula). The hottest temperatures are usually found at later stages of stellar evolution in supernovas or the vicinity of degenerate matter (e.g., magnetars). Credits for insets: NASA/ESA/ISAS; R. Mallozzi, Burrows et al. [387], Bally et al. [1159], Schulz et al. [532] For young stars the highest temperatures (of the order of 100 MK) observed occur during giant X-ray flares that usually last for hours or sometimes a few days, and jets. The coolest places (with 1 to 100 K) are molecular clouds. Ionized giant hydrogen clouds usually have temperatures around 1,000 K. The temperatures of stellar photospheres range between 3,000 and 50,000 K. In stellar coronae the plasma reaches 10 MK, almost as high as in stellar cores where nuclear fusion requires temperatures of about 15 MK. The temperature range involved in stellar formation and evolution thus spans many orders of magnitudes. In stellar physics the high temperature is only topped by temperatures of shocks in the early phases of a supernova, the death of a massive star, or when in the vicinity of gravitational powerhouses like neutron stars and black holes. 374 A Gas Dynamics A.2 The Adiabatic Index The first relation one wants to know about a gaseous cloud is its equation of state, which is solely based on the first law of thermodynamics and represents conservation of energy: dQ D dU C dW (A.4) where the total amount of energy absorbed or produced dQ is the sum of the change in internal energy dU and the work done by the system dW: In an ideal gas where work dW D P dV directly relates to expansion or compression and thus a change in volume dV against a uniform pressure P , the equation of state is: PV D NmRT D nkT (A.5) 7 1 1 where R D 8:3143435 10 erg mole K , Nm is the number of moles, and n is the number of particles per cm3. The physics behind this equation of state, however, is better perceived by looking at various derivatives under constant conditions of involved quantities. For example, the amount of heat necessary to raise the temperature by one degree is expressed by the heat capacities: Â Ã Â Ã dQ dQ Cv D and Cp D (A.6) dT V dT P where d=dT denotes the differentiation with respect to temperature and the indices P and V indicate constant pressure or volume. The ratio of the two heat capacities: D CP =CV (A.7) is called the adiabatic index and has a value of 5/3 or 7/5 depending on whether the gas is monatomic or diatomic. For polyatomic gases the ratio would be near 4/3 (i.e., if the gas contains significant amounts of elements other than H and He or a mix of atoms, molecules, and ions). The more internal degrees of freedom to store energy that exist, the more CP is reduced, allowing the index to approach unity. A mix of neutral hydrogen with a fraction of ionized hydrogen is in this respect no longer strictly monatomic, because energy exchange between neutrals and ions is different. Similarly, mixes of H and He and their ions are to be treated as polyatomic if the ionization fractions are large. Deviations from the ideal gas assumption scale with n2=V 2, which, however, in all phases of stellar formation is a very small number. Thus the ideal gas assumption is quite valid throughout stellar evolution. A very important aspect with respect to idealized gas clouds is the case in which the radiated heat is small. For many gas clouds it is a good approximation to assume that no heat is exchanged with its surroundings.