天体物理流体动力学基础

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"Principles of Astrophysical Fluid Dynamics" 是一本专门为第三年大学学生编写的教材，涵盖了天体物理流体力学的基本原理和方法。这本书旨在介绍处理天体物理学中高度压缩流体所需的概念和技巧，同时也可以作为高级本科和研究生的参考书。作者们在书中通过基本方程的推导，辅以直观的理解性文字，强调了依赖流体力学过程的可观测天文现象。书中包含超过50个练习题，适用于剑桥大学天体物理学最后一年的本科生和刚开始的研究生。 **流体力学基础** 流体力学是研究流体及其运动规律的科学，它在理解宇宙中的许多基本过程中起着至关重要的作用。天体物理流体力学特别关注的是在极端条件下，如高密度、高速度或强重力环境下的流体行为。这本书从基础出发，介绍了流体动力学的关键方程，包括牛顿第二定律、连续性方程、纳维-斯托克斯方程以及能量方程。 **天体物理应用** 在天文学中，流体力学的应用广泛，例如： 1. **恒星结构**：流体动力学帮助我们理解恒星内部的压力平衡、温度分布以及核反应如何维持其稳定性。 2. **超新星爆炸波**：当恒星死亡并爆炸时，产生的冲击波是流体动力学研究的重要对象，这些爆炸对星系内的元素分布有深远影响。 3. **吸积盘**：行星、恒星或黑洞周围的吸积盘，其物质如何在重力作用下向中心物体移动，也是流体力学研究的重点。 **教学方法与受众** 本书基于剑桥大学多年的课程经验编写，适合有基础物理学背景的学生学习。作者通过解释和推导基本方程，同时提供直观的解释，使学生能够更好地理解和应用这些概念。此外，大量的习题设计有助于巩固学生的理论知识，并提升他们解决实际问题的能力。 **作者背景** 作者Cathie Clarke是剑桥大学的理论天体物理学教授，并在克莱尔学院担任天体物理学的研究主任。她开发了这个天体物理流体力学课程，旨在提供一个全面的教育框架，让学生能够深入理解和应用流体力学原理到天文学的实际问题中。《天体物理流体力学原理》不仅是一本深入浅出的教学书籍，也是研究天体物理现象的宝贵参考资料，它连接了理论与观测，为理解宇宙提供了强大的工具。

4 Introduction to concepts

surrounding the correct equation of state for hydrogen under extreme

conditions of density and pressure in the relevant temperature range.

The above thumbnail sketch has stressed the diversity of fluids in

the Universe and has perhaps suggested that their study will involve

consideration of much complex microphysics. One of the beauties of

fluid dynamics is however the way that the microphysical complexity

is all contained in a single parameter – the equation of state. Once

armed with an equation of state, the fluid dynamicist can simply insert

this into the general equations of fluid dynamics. This is not to suggest

that the outcome of this exercise is necessarily simple, but there is a

pleasing generality that runs through the subject. We will, for example,

be able to deduce the structure of white dwarfs and neutron stars

as readily as of stars composed of conventional gas, simply because

they each have well-defined equations of state and in each case we

can consider the stars to be in a state of equilibrium. However, this

discussion anticipates much of the contents of later chapters. Before

we can embark on deriving the fluid equations, it is necessary that we

now introduce some important fluid dynamical concepts.

1.2 The concept of a ‘ﬂuid element’

The fluid equations are based on the concept of a ‘fluid element’, i.e.

a region over which we can define local variables (such as density,

temperature, etc.). The size of the region is assumed to be such that:

(i) It is small enough that we can ignore systematic variations across

it for any variable we are interested in, i.e. the region size 

region

much smaller than a scale length for change of any relevant variable q

(where a scale length is the scale over which q varies by order unity).



region

 

scale

∼

q

 (1.1)

(ii) It is large enough that the element contains sufficient particles

for us to ignore fluctuations due to the finite number of particles

(‘discreteness noise’). Thus if the number density of particles per unit

volume is n, we require that

n

region

 1 (1.2)

The above two criteria must be met by any system that is describable

as a fluid. In addition, collisional fluids must satisfy the following

criterion:

1.3 Formulation of the ﬂuid equations 5

(iii) The fluid element is large enough that the constituent particles

‘know’ about local conditions through colliding with each other, so if

the mean free path is , we require that



region

  (1.3)

What do we mean by a collisional fluid? The essential point is

that if the particles in a fluid interact with each other (which does not

necessarily imply that they physically collide), then they will attain a

distribution of, say, particle speed that maximises the entropy of the

system at that temperature. (For this discussion we can simply regard

the entropy of a system as being a measure of the number of microscop-

ically distinct configurations that correspond to a given macroscopic

property of the fluid locally.) Therefore a collisional fluid at a given

temperature and density will have a well-defined distribution of particle

speeds in the local rest frame, and hence a corresponding pressure. Thus

one can derive an equation of state (relationship between density, tem-

perature and pressure) for a collisional fluid. Almost all the fluids we

consider in this book, be they ideal gases or degenerate, are collisional

fluids with corresponding equations of state. However, as noted above,

we can also consider systems of orbiting stars in a galaxy, or grains in

Saturn’s rings, as being fluids, even though the particles now do not

interact sufficiently frequently to satisfy criterion (1.3) above. In this

case, the distribution of particle speeds locally does not correspond to

an entropy maximum but instead depends on the initial distribution of

particle speeds. The fact that one cannot write down an equation of state

for collisionless systems means that it is a hard problem, whose solu-

tion is non-unique, to find, for example, the structure of stellar orbits

in a galaxy in equilibrium. In this book we will avoid this difficulty

by considering collisional fluids (i.e. conventional liquids and gases).

It should be stressed that a fluid element is purely a conceptual

entity – 

region

does not enter into the fluid equations. However, if a system

is to be described by the fluid equations, its properties must be such that

there are values of 

region

that simultaneously satisfy the conditions above.

1.3 Formulation of the ﬂuid equations

There are two (sensible) approaches to formulating the equations for

mass density, momentum and energy in a fluid:

(i) Eulerian description

Consider a small volume at a fixed spatial position. The fluid flows

through the volume with physical variables specified as functions of

time and the (fixed) position of the small volume: density  = rt,

6 Introduction to concepts

temperature T = Trt, etc. The change of any measurable quantity

as a function of time at that position is /t of the quantity, evaluated

at the fixed position.

(ii) Lagrangian description

In this approach one chooses a particular fluid element and examines

the change in variables (density, temperature, etc.) in that particular

element. So the (spatial) reference system is comoving with the fluid.

Thus one might examine the behaviour of  = at, where a is a

label for a particular fluid element – which might be the coordinates at

a chosen initial time, for example. The time derivative is now a partial

one at a fixed a (i.e. for a fixed bit of the fluid), and the rate of change

with respect to time for a fixed element is denoted D/Dt.

In the Lagrangian description, position is not an independent vari-

able but instead r = rat. The Lagrangian description refers to the

world as seen by an observer riding on a fluid element (e.g. a river

viewed from a boat adrift on it); the Eulerian one refers to the world

as seen at a fixed spatial position (e.g. a river viewed from the bank).

The Lagrangian approach is useful if the path of an individual

element is important, e.g. when a particular element has some property

which distinguishes it from all the others. Usually this is not the case;

however, one can think of particular instances where it is important in

astronomy (e.g. tracing the trajectory of a parcel of gas that has been

enriched by heavy elements as it is ejected into the interstellar medium

by a supernova). The Eulerian approach is usually more useful if the

motion of individual fluid elements is not of interest. It is particularly

useful for steady flows, i.e. those where the quantities at a given

position do not change. Then /t of all variables = 0 everywhere.

Steady flows have no special properties in Lagrangian descriptions

since in a steady flow an individual element still changes its properties

in general as it goes from place to place.

This conceptual split between the Eulerian and Lagrangian formu-

lations translates into two entirely distinct ways of simulating fluid

dynamical problems on a computer. Eulerian codes set up a grid of

fixed boxes and compute the changes of all variables in each box

as the flow evolves. Lagrangian codes instead set up an ensemble of

particles which represent fluid elements and follow the trajectories of

the particles in the flow. There are advantages and disadvantages to

each approach which are being much discussed at present in relation

to simulating the formation of stars and galaxies, two highly topi-

cal problems in astrophysical fluid dynamics. Eulerian codes in their

simplest form have the disadvantage that you have to decide at the

beginning of the calculation how you are going to distribute your grid

cells, i.e. in what regions of the flow do you want fine gridding (high

1.4 Relation between the Eulerian and Lagrangian descriptions 7

resolution). However, as the flow evolves you might need high reso-

lution in a region that you would not have predicted at the outset of

the calculation, unless your problem has very regular symmetry. The

Lagrangian approach circumvents this difficulty since high resolution

will automatically be achieved in high density regions of the flow

where lots of particles end up. However, the problem with using par-

ticles to model a continuous fluid is then how to compute the density

at each point (number of particles divided by some sampling volume

is an obvious approach, but then the density calculated might fluctuate

unphysically as particles enter and leave the sampling volume). Much

work has gone into devising codes that minimise the inherent noisi-

ness of particle based methods, particularly a class of codes known as

Smoothed Particle Hydrodynamics (SPH). In the meantime, Eulerian

codes are becoming more cunning through developing the capability of

reconfiguring the grid automatically during the calculation in order to

achieve high resolution where it is needed (Adaptive Mesh Refinement

methods: AMR). There is a large and evolving literature on computing

methods for fluid problems – see e.g. the Von Karman Institute Lecture

Series Monograph: Computational Fluid Dynamics.

1.4 Relation between the Eulerian and

Lagrangian descriptions

Consider any quantity Q (which may be a scalar or a vector) in a fluid

element which is at position r at time t. At time t +t the element is

at r +r, and then, straight from the definition

= lim

t→0



Qr +rt+t −Qrt

t



 (1.4)

The numerator is

Qr +rt+t −Qrt=Qrt+t −Qrt

+Qr +rt+t −Qrt+t (1.5)

which is, to first order in r, t,

Qrt

t

t +r · Qrt+t (1.6)

and so, expanding the second term,

Qrt

t

t +r ·



Qrt+

 Q

t

t ···



 (1.7)

The r ·

 Q

t

t term is of second order in small quantities, so, in the

limit as t and r → 0,

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