Download Gas Dynamics, Lecture 2 (Mass conservation, EOS) see: www

Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit
-Equation of motion;
-Relation between pressure
and thermal velocity dispersion;
-Form of the pressure force
Each degree of freedom carries an energy
1
2
kbT
Point particles with mass m:
1
2
m  x2  12 m  y2  12 m  z2  12 kbT


2
kbT

3
m

 kbT
 P
 
3
 m
2

  nkbT

R T
P

kb
R 
 universal gas constant;
mH
m
=
 mass in units of mass hydrogen atom.
mH
Adiabatic change: no energy is irreversibly
lost from the system, or gained by the system
dU  PdV = 0
Adiabatic change: no energy is irreversibly
lost from the system, or gained by the system
dU  PdV = 0
Change in internal
energy U
Work done by pressure forces
in volume change dV
Thermal
energy density:
Wth  n

1
2
m
2

kinetic energy of
thermal motion
Pressure:
R T 2
P
 3 Wth

R T
 nkbT 

3
2
3
2
Thermal equilibrium:
Adiabatic change:
Thermal equilibrium:
Adiabatic change:
Product rule for ‘d’-operator:
(just like differentiation!)
Adiabatic pressure change:
For small volume:
mass conservation!
Polytropic gas law:
Ideal gas law:

P  K  

 1

T

K
'


R T 

P
 
Thermal energy density:
P
R T
Wth 

  1 (  1) 
Polytropic index
mono-atomic gas:

5
,  =1:
3
ISOTHERMAL
2D-example:
A fluid filament is deformed
and stretched by the flow;
Its area changes, but the
mass contained in the
filament can NOT change
So: the mass density must
change in response to
the flow!
right boundary box:
M in   ( x, t )V ( x, t ) t
left boundary box:
M out   ( x  x, t )V ( x  x, t ) t
d  M 
  M  
 t  M in  M out
dt
=  ( x, t )V ( x, t ) t   ( x  x, t )V ( x  x, t ) t

  t x  V 
x
d  M 

  
 t   t x   =   t x  V 
dt
x
 t 

 
  V   0
t x
 


  Vx    Vy    Vz   0
t x
y
z

 Ñ   V   0
t
Velocity at each point
equals fluid velocity:
Definition of tangent
vector
Velocity at each point
equals fluid velocity:
Definition of tangent
vector:
Equation of motion
of
tangent vector:
Volume: definition
A = X , B = Y, C = Z
The vectors A, B and C are carried along by the flow!
Volume: definition
A = X , B = Y, C = Z
Volume: definition
A = X , B = Y, C = Z
Special choice:
orthogonal triad
General
volume-change
law
Special choice:
Orthonormal triad
General
Volume-change
law
Volume
change
Mass conservation:  V = constant
Volume
change
Mass conservation:  V = constant
Comoving
derivative
Divergence product rule
&
(Self-)gravity
Self-gravity and Poisson’s equation
Potential: two
contributions!
Poisson equation for
potential associated
with self-gravity:
Laplace operator
Application: The Isothermal Sphere
as a
Globular Cluster Model
All motion is
‘thermal’ motion!
Pressure force is
balanced by gravity
Typical stellar orbits
N-particle simulation (Simon Portugies-Zwart, Leiden)
The Isothermal Sphere: assumptions
Mass density:
 ( r )  n( r ) m
Stellar Temperature:
kbT
  x2   y2   z2   2
m
Stellar Pressure:
P( r )  n( r ) kbT   ( r ) 2
Governing Equations:
Equation of Motion: no
bulk motion, only pressure!
Hydrostatic Equilibrium!
r
Density law and Poisson’s Equation
Hydrostatic Eq.
Exponential density law
‘Down to Earth’ Analogy:
the Isothermal Atmosphere
Low density &
low pressure
g  Ñ    geˆz  ( z )  gz
z
Constant
temperature
 dP  ˆ R T d  ˆ
ÑP
ez
 ez 
 dz
 dz 
Force balance:
High density &
high pressure
Earth’s surface: z = 0
 RT d

0  Ñ P   g =  
  g  eˆz
  dz

‘Down to Earth’ Analogy:
the Isothermal Atmosphere
 RT d

0  Ñ P   g =  
  g  eˆz
  dz

Earth’s surface: z = 0
‘Down to Earth’ Analogy:
the Isothermal Atmosphere
 RT d

0  Ñ P   g =  
  g  eˆz
  dz

Set to zero!
1 d
g


 dz
RT
 ( z )   (0) exp( z / H )
 ( z )   (0) exp(
P( z )  P(0) exp( z / H )
Earth’s surface: z = 0
 gz
RT
)   (0) exp(
  (0) exp(  z / H ) , H 
RT
g
 ( z )
RT
)
Density law and Poisson’s Equation
Hydrostatic Eq.
Exponential density law
Poisson Eqn.
Spherically symmetric
Laplace Operator
Density law and Poisson’s Equation
Hydrostatic Eq.
Exponential density law
Poisson Eqn.
Spherically symmetric
Laplace Operator
Scale Transformation
Density law and Poisson’s Equation
Hydrostatic Eq.
Exponential density law
Poisson Eqn.
Spherically symmetric
Laplace Operator
Scale Transformation
WHAT HAVE WE LEARNED SO FAR…..
Introduction dimensionless
(scaled) variables
Single equation describes
all isothermal spheres!
Solution:
For   r / rK

 2 4 
   0 1   
6 45 


1: 
2 4

   6  120
For   r / rK

2 0
2
  2 
2

2

Gr

1: 
2
   log   
 

 2

Solution:
For   r / rK
1:
 2 
0 dr 4 r  2 Gr 2 
r
M (r )
2
 80 rK2 r
2 2 r
G
What’s the use of scaling with rK ?
All ‘thermally relaxed’ clusters look the same!
Tidal Radius
Galactic tidal force ~ self-gravity
GM cl
  GM gal
 rt

2
rt
R 
R2

 M cl
rt  
 2M
gal

 2GM gal rt

3
R

r
t
1/ 2



R
  
M cl  2.5 10 

 5 km/s 
6
M cl  8 0 rK2 rt
3
 R 


 10 kpc 
3/ 2
M