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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 complimenttoresearchers.Thoughthisstatementreflectstheadvances
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,manydefinitions andprinciplesimportanttothephysics
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
sametypeofparticles 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.
A.1 Temperature Scales
One might think that a temperature is straightforward to define as it is an
everydayexperience.Forexample,temperatureisfelt outsidethe house, inside
at the fireplace or by drinking a cup of hot chocolate. However, most of what is
258 A Gas Dynamics
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:
U = 3N kT (A.1)
2 tot
−16 −1
where k =1.381×10 erg K .
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:
T =T =T +273.15o (A.2)
K C
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 pres-
ence 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, temper-
14
atures in other regions within the Galaxy may even rise to 10 K. Objects
this hot are associated with very late evolutionary stages such as pulsars and
γ-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:
−8 T
E=kT=8.61712×10 [K]keV
E=12.3985keV (A.3)
˚
λ[A]
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. Ion-
ized giant hydrogen clouds usually have temperatures around 1,000 K. The
A.1 Temperature Scales 259
Gamma
0.001
100 billion 10 million
rays
1 million 0.01
10 billion
0.1
100,000
1 billion
Xrays
1
10,000
100 million
10
10 million
1,000
Ultra
1 million 100 100
violet
100,000 1,000
10
Optical
10,000 10,000
1
Infra
100,000 red
1,000 0.1
Sub
100 0.01 1 Million
mm
10 Million
10 0.001
Radio
100 Million
1 0.0001
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
andHH30,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
UVlight), 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. [136], Bally et al. [52], Schulz et
al. [761].
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 tem-
peratures 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.
260 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=dU+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 = PdV directly relates to expansion or compression
and thus a change in volume dV against a uniform pressure P, the equation
of state is:
PV=NmRT=nkT (A.5)
7 −1 −1
whereR=8.3143435×10 ergmole K ,Nmisthenumberofmoles,andn
3
is the number of particles per cm . 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:
C =dQ andC =dQ (A.6)
v dT p dT
V 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:
γ =C /C (A.7)
P V
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 inter-
nal degrees of freedom to store energy that exist, the more C is reduced, al-
P
lowing 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 frac-
tions are large. Deviations from the ideal gas assumption scale with n2/V2,
which, however, in all phases of stellar formation is a very small number. Thus
the ideal gas assumption is quite valid throughout stellar evolution.
Averyimportant 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 approxima-
tion to assume that no heat is exchanged with its surroundings. The changes
in P,T,andV in the adiabatic case are then:
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