Introduction to Astrophysical Gas Dynamics

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Introduction to Astrophysical Gas Dynamics
Part 1

• Bram Achterberg
• a.achterberg_at_astro.uu.nl

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Overview

• What will we treat during this course?
• Basic equations of gas dynamics
• Equation of motion
• Mass conservation
• Equation of state

• Fundamental processes in a gas
• Steady Flows
• Self-gravitating gas
• Wave phenomena
• Shocks and Explosions
• Instabilities

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Applications

• Isothermal sphere Globular Clusters
• Solar Stellar Winds
• Blandford-Rees model for Jets
• Sound waves in an expanding Universe
• Cyclones and Jupiters Great Red Spot
• Blast waves Supernova Remnants
• Shocks and knots in Jets

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STELLAR WIND BUBBLE
Vwind 2000 km/s
Shell of swept-up interstellar gas
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Supernova Remnant Cassiopeia A
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Jet coming from a proto-star
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Central Jet in the Galaxy M87
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Numerical simulation an important tool
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P 1/(G?)1/2
Waves and oscillations sound waves, gravity
waves and asteroseismology
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P-mode oscillation of a star
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ACCRETION DISK Rotation gravity
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Classical Mechanics vs. Fluid Mechanics
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Basic Definitions
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Mass, mass-density and velocity
Mass density ?
Mass ?m in volume ?V
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Equation of Motion from Newton to
Navier-Stokes/Euler
Particle ?
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Equation of Motion from Newton to
Navier-Stokes/Euler
You have to work with a velocity field that
depends on position and time!
Particle ?
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Derivatives, derivatives
Eulerian change fixed position
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Derivatives, derivatives
Eulerian change fixed position
Lagrangian change shifting position
Shift along streamline
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Comoving derivative d/dt
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z
y
x
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Notation working with the gradient operator
Gradient operator is a machine that converts a
scalar into a vector
Related operators turn scalar into
scalar, vector into vector.
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Equation of motion for a fluid
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Equation of motion for a fluid
Non-linear term! Makes it much more difficult To
find simple solutions. Prize you pay for
working with a velocity-field
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Equation of motion for a fluid
Non-linear term! Makes it much more difficult To
find simple solutions. Prize you pay for
working with a velocity-field

• Force-density
• Can be
• internal
• pressure force
• viscosity (friction)
• self-gravity

• external

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Pressure force and thermal motions
Split velocity into the average velocity V(x,
t), and an Isotropically distributed deviation
from average, the random velocity ?(x, t)
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Acceleration of particle ?
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Acceleration of particle ?
Effect of average over many particles in small
volume
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Average equation of motion
For isotropic fluid
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Some tensor algebra
Vector
Three notations for the same animal!
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Some tensor algebra the divergence of a vector
in cartesian coordinates
Vector
Scalar
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Rank 2 Tensor
Rank 2 tensor
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Rank 2 Tensor and Tensor Divergence
Rank 2 tensor
Vector
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Special caseDyadic Tensor Direct Product of
two Vectors
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Application Pressure Force
Tensor divergence
Isotropy of Random velocities
Second term scalar x vector! This must vanish
upon averaging!!
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Application Pressure Force
Tensor divergence
Isotropy of Random velocities
Diagonal Pressure Tensor
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Pressure force, continued
Equation of motion for frictionless (ideal)
fluid
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Summary

• We know how to treat the time-derivative
• We know what the equation of motion looks like
• We know where the pressure term comes from
• We still need
• - A way to link the pressure to density and
• temperature
• - A way to calculate how the density of the
• fluid changes

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Connection with thermodynamics Ideal Gas Law
Isotropic gas of point particles in
Thermodynamic Equilibrium
Temperature is defined in terms of kinetic energy
of the thermal motions!
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Connection with thermodynamics Ideal Gas Law
Isotropic gas of point particles in
Thermodynamic Equilibrium
Ideal Gas Law in terms of temperature T and
number-density n (? nm ?nmH , R kb / mH)
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Summary
Comoving derivative
Equation of motion
Pressure
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Density Changes and Mass Conservation
Two-dimensional example A fluid filament is
deformed and stretched by the flow Its area
changes, but the mass contained in the filament
can NOT change
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Density Changes and Mass Conservation
Two-dimensional 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!
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Curves and volumes carried by flow
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Velocity at each point equals fluid velocity
Definition of tangent vector
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Velocity at each point equals fluid velocity
Definition of tangent vector
Equation of motion of tangent vector
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Volume definition
A ?X , B ?Y, C ?Z
The vectors A, B and C are carried along by the
flow!
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Volume definition
A ?X , B ?Y, C ?Z
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Volume definition
A ?X , B ?Y, C ?Z
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Special choice orthogonal triad
General volume-change law
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Special choice Orthonormal triad
General Volume-change law
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Mass conservation and the continuity equation
Volume change
Mass conservation ? ?V constant
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Mass conservation and the continuity equation
Volume change
Comoving derivative
Mass conservation ? ?V constant
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The continuity equation the behaviour of the
mass-density
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The continuity equation the behaviour of the
mass-density
Divergence chain rule
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The continuity equation the behaviour of the
mass-density
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What have we learnt so far?

• The meaning of the time-derivative
• Equation of motion for a frictionless fluid
• The origin and interpretation of the pressure
• term
• The response of the mass-density
• to the flow

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What have we learnt so far?

• The meaning of the time-derivative
• Equation of motion for a frictionless fluid
• The origin and interpretation of the pressure
• term
• The response of the mass-density
• to the flow

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What have we learnt so far?

• The meaning of the time-derivative
• Equation of motion for a frictionless fluid
• The origin and interpretation of the pressure
• term
• The response of the mass-density
• to the flow

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What have we learnt so far?

• The meaning of the time-derivative
• Equation of motion for a frictionless fluid
• The origin and interpretation of the pressure
• term
• The response of the mass-density
• to the flow

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Thermodynamics
Force balance
pressure times piston area
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Thermodynamics
Force balance
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First law of thermodynamics
Change in internal energy heat added by
external sources - work done by gas
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Entropy A measure of disorder
Second Thermodynamic law
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The Adiabatic Gas Law the behaviour of pressure
Thermodynamics
Special case adiabatic change
U internal energy, T temperature, S
entropy and ? volume
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The Adiabatic Gas Law the behaviour of pressure
Thermodynamics
Special case adiabatic change
Gas of point particles of mass m
Internal energy
Pressure
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Thermal equilibrium
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Thermal equilibrium
Adiabatic change
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Thermal equilibrium
Adiabatic change
Chain rule for d-operator
(just like differentiation!)
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Adiabatic Gas Law a polytropic relation
Adiabatic pressure change
For small volume mass conservation!
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Specific Heat and Entropy
Specific Volume contains unit mass
Thermodynamics of unit mass
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Specific Heat and Entropy
Specific Volume contains unit mass
Thermodynamics of a unit mass
Specific energy e and pressure P
Specific heat coeff. at constant volume

• is kept constant!
• d(1/?) 0

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Specific Heat and Entropy
Specific Volume contains unit mass
Thermodynamics of a unit mass
Specific energy e and pressure P
Specific heat coeff. at constant volume
Specific heat coeff. at constant pressure dP 0
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Thermodynamic law for a unit mass, rewritten in
terms of specific heat coefficients
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Definition specific entropy s

• is the specific heat ratio
• 5/3 for ideal gas of point particles

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Definition specific entropy s
Case of constant entropy (adiabatic gas) ds 0
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(Self-)gravity
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Self-gravity and Poissons equation
Potential two contributions!
Poisson equation for the potential
associated with self-gravity
Accretion flow around Massive Black Hole
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Self-gravity and Poissons equation
Potential two contributions!
Poisson equation for Potential associated with
self-gravity
Laplace operator
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Summary Equations describing ideal(self-)gravita
ting fluid
Equation of Motion
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Summary Equations describing ideal(self-)gravita
ting fluid
Equation of Motion
Continuity Equation behavior of mass-density
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Summary Equations describing ideal(self-)gravita
ting fluid
Equation of Motion
Continuity Equation behavior of mass-density
Ideal gas law Adiabatic law Behavior of
pressure and temperature
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Summary Equations describing ideal(self-)gravita
ting fluid
Equation of Motion
Continuity Equation behavior of mass-density
Ideal gas law Adiabatic law Behavior of
pressure and temperature
Poissons equation self-gravity