Download pdf - Radiative Processes & Magneto Hydrodynamics (MHD)

Radiative Processes & Magneto Hydrodynamics (MHD)

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Total C.F.U. = 9 (8 CFU Lect. + 1 Ex.) 76 hr

Lectures @ AULA 1 on Mondays (14 – 16) Tuesdays ( 14 – 16) Thursdays (10 – 13)

Contact: [email protected] on tuesdays & thursdays 15:30 – 17:30

Exam: 2 dates in January – February then again from June, about 1 date/month ⇨ a short admission test will be carried out

ZEMAX


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why the admission test

18 – 29

01.01 – 12.31 2006 01.01 – 12.31 2011

fail

30


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what the admission test is

Duration: 1 h Typically: 4 questions (one/two are just statememts)Allowed consultation of Books Lecture notes Slides

possible results: Adm., Adm. w. R., Not Adm.


Overview (1)

Aims of the course:

provide a PANCHROMATIC APPROACH

to Astrophysics

provide a all (most of) the tools to understand what

we observe (radiation) in the Universe

408 MHz21 cm

rays

The Sky at various wavelengthss

Individual objects: images (1)


CONTENTS (Part – 1 – )

MHD: Statics

HD

MHD & Plasma physics

Fluid Instabilities



Radiative processes:

Continuous radiation from a charged particle (Fundamental physics of astrophysical plasmas)

0 – Black Body

1. – Bremsstrahlung

2. – Synchrotron

3. – Diffusion processes

4. – Cosmic Rays



The Interstellar medium Plasma, Neutral medium, Dust & Molecules

Atomic & molecular transitions

Einstein's Coefficients Atoms (molecules) as oscillators permitted & forbidden lines

Radiative .vs. collisional (de)excitation

The phases of the ISM – diagnostics

A case study: HI and the rotation curve in spiral galaxies


Bibliography:

Clarke & Carswell “Principles of Astrophysical Fluid Dynamics”

Rybicki & Lightman “Radiative Processes in Astrophysics”

Padmanaban “Theoretical Astrophysics” (Vol I – Astrophysical processes)

Vietri “Astrofisica delle Alte Energie”

Dopita & Sutherland “Astrophysics of the Diffuse Universe”

copy of the slides are available at http://www.ira.inaf.it/~ddallaca/PRAD.html

⇨they are just lecture notes, not sufficient for a proper study

http://www.ira.inaf.it/~ddallaca/P-RAD.html

Hydrodynamics (fluid mechanics)

Fluids are in form of either liquid or gaseous bodies.

Fluids are ideal macroscopic bodies with continuous properties.

Hydrostatics is defined by the ideal gas equation state (P,,T).

Hydrodynamics requires to know also the instantaneous velocity (P,,[T], v)v (x,y,z,t) is the velocity vector of a fluid volume element, and not that of individual fluid particles.

Description based on 1. study the fluid flowing across a given place/surface [Eulerian view] 2. study a volume element in its flow/motion [Lagrangian view]

At first the magnetic field is not considered. It will be in MHD (MagnetoHydroDynamics)

Basic mathematical tools required for understanding:

gradient of a scalar:

the vector is perpendicular to the surface where the scalar is constant

divergence of a vector

curl of a vector

2

∇ ≡ ∂∂ x

i ∂∂ y

j ∂∂z

k

∇⋅h ≡ ∂∂ x

hx∂∂ y

h y ∂∂z

hz

∇×h ≡ ∣i j k∂∂x

∂∂ y

∂∂z

hx hy hz∣

∇2 ≡ ∂2

∂ x 2 ∂2

∂ y 2 ∂2

∂z 2

Basic mathematical tools required for understanding (2):

Gauss' theorem

Stokes' theorem

∬surface V G⋅n dS = ∭Volume

∇⋅G dV

∮ G⋅d l = ∬surface∇× G dS

Mathematical tools [3]:

approximate relationships between operators and practical quantities:

U = velocity ; L = length ; T = time∂∂ x

≈ 1L

∂∂ t

≈ 1T

≈ UL

∇× ≈ 1L

∇⋅≈ 1L

∇2⋅≈ 1

L2

Astrophysical fluids

Extension of the “physical” concept.

Large scale: Gas/dust (ISM) in spiral (elliptical) galaxies

Stars in galaxies

Galaxies (& clusters) & IGM in the Universe

Small scale:Stellar winds

Accretion discs & Jets, Stars, gas/dust clouds

In general, the “gaseous” fluid is more appropriate (liquid fluid describes high pressure environments, e.g. planetary & star surfaces and interiors) since the typical densities are very small.In general the fluid is inhomogeneous (TP), according to its location

HIM , WIM, WMN, CNMThere are other compact bodies like WD & NS, where the equation of state is fundamental to define the fluid properties

Astrophysical fluids (2)

Basic concept:

Each volume element must be

small enough to preserve a single value for relevantquantities:

large enough to be statistically representative (large number of particles)

In case the fluid element has

then particles redistribute “information” and the fluid is termed collisional.All this implies that a number of statistical laws hold.

L fluid≪Lscale~qq

q=generic quantity

n Lfluid3

≫1 n=number density

L fluid≫mfp mfp=mean free path

Ideal fluid

1. no internal friction

2. changes in shape do not require work (V remains constant)

3. (incompressible)

4. continuous, microscopic volumes (dV) still contain large amounts of particles

Astrophysical examples:

the vast majority of the celestial bodies (and the space between them) can bephysically described with (magneto)hydrodynamics:

stars, plasma, stellar winds, supernovae, insterstellar gas, cosmic rays, galaxies, galactic winds, intergalactic space (e.g. hot gas in clusters of galaxies), radio sources, etc.

Fluid statics .vs. Fluid dymanics

Fluid statics studies equilibrium

local thermodynamic conditions are all you need:

state equation (e.g. PV =nRT) alias P = f (n/V, T) = f (, T)

Dynamics instead studies/considers motions of volumes in case of motion, velocity must be taken into account, in addition,and a given quantity, like pressure is

P = f , v ,T

Fluid statics

1. Pascal Principle:In a fluid at rest, w/o any volumetric forces (e.g. gravity), P is constant within the fluid.A small force F

1 becomes strong in F

2

2. Archimede's Principle: a solid body within a fluid is subject toall the forces over its surface from the

fluid pressure. ( A.'s push up: = weightof the liquid displaced by the body)

Fluid statics (2)

3. Stevino's law:

e.g. atmospheric pressure: there is a force (gravity) acting on volume: it has a component along z [i.e. g = (0, 0, g

z )]

P z = −g zconstant

dPdz

=−g where = N = PkT

.

for an isothermal atmospheredPP

= −gkT

dz = mean molecular mass

P = P o e−g z /kT = o e−g z /kT

P o = 1.033 dyne⋅cm−2 = 1.013⋅105 N⋅m−2Pa o = 1.29⋅10−3 g⋅cm−3 = 1.29 kg⋅m−3

Fluid statics (3) air composition and

.

Fluid mechanics (1) equation of continuity (mass conservation)

mass within some given volume Vo is:

mass flowing per unit time through a surface dS is:

total mass ouflowing Vo is:

the decrease per unit time of the mass fluid in Vo is:

and equating (1) and (2)

∫V o

dV

v⋅d S = v⋅n dS [ 0 outflowing V o ]

∮ v⋅d S = ∫V o

∇ v dV 1 Gauss ' theorem

−ddt ∫V o

dV = −∫V o

∂

∂ tdV 2

−∫V o

∂

∂ tdV = ∫V o

∇ v dV

∫V o [ ∂∂ t ∇v ]dV = 0

Fluid mechanics (2) equation of continuity

the equation must hold for any volume then the integrand must vanish:

the quantity is also known as mass flux density.

If we introduce the “convective derivative”** the equation of continuity can also be written as

** it is like we concentrate out attention to a certain mass (volume) elementfollowing its motion. Such derivative represent the variation of the density ascomoving with it.

∂

∂ t ∇ v = 0

∂

∂ t ∇ v v⋅∇ = 0 Eulerian

j = vDDt

≡ ∂∂ t

v⋅∇

D

Dt ∇ v = 0 Lagrangian

Fluid mechanics (3) equation of continuity application (1) :during t the fluid moves of a certain amount x1=v1t in S1 and of x2=v2t in S2 witha mass motion of

m1 = 1 S1 x1 = 1 S1 v1 t

m2 = S2 x2 = 2 S2 v2 t

in case of mass conservation m1 = m2 namely1 S1 v1 = 2 S2 v2

i.e. S v = constantin case of incompressible fluids (= constant)

S v = constant = Q

in case of variable cross section of the tube:

i.e. velocity increases when the cross section narrows

S1

S2

v ∝1S

Fluid mechanics (4) equation of continuity application (2) : gravity plays a role!

for eq.of.c. dm1 = dm2 = dm

if = constant S1 v1 = S2 v2

there is a variation of kinetic energy

S1

S 2

z 2

z

1

z

K =12

dm v 22−v 1

2 = Lp Lg

Lg = dm g z 1−z 2

Lp = p1S1v 1dt−p2S 2 v 2dt=p1V−p 2V

12

dm v 22−v 1

2 = dm g z 1−z 2p1−p 2V but V≈

dm

12

v 12

g

p 1

gz 1 =

12

v 22

g

p2

gz 2 [Bernoulli ' s equation ]

stopheight

piezometric height

height

dm

dm

Daniel Bernoulli (17001782)

Fluid mechanics (5) equation of continuity

it follows that:

(other forms of Bernoulli's equation)

S1

S 2

z 2

z

1

z

12

v 2

g

pg

z = const

12

v 2 p g z = const

dm

dm

Fluid mechanics (6) Euler's equation Euler's equation (momentum conservation) (equivalent to F=ma)

along x, considering all the acting forces:

and we must consider the forces fx acting on the volume (i.e. gravity) and on the surface (i.e. pressure)

and from (1) and (2)

the same also along y and z; then using the vector formalism

dV x

dS

dF x = dm⋅dv x

dt= dV

dv x

dt1

dF x = dV f x[p x −p xdx ]dS = f x−dpdx dV 2

dVdv x

dt= f x−

dpdx dV i.e.

dv x

dt= 1

f x−

1

dpdx

d vdt

=[ 1f ]− 1

∇ p 3

d vdt

= [ 1f ]− 1

∇ p

Fluid mechanics (7) Euler's equation

the Lagrangian view follows the fluid element during the motionthe Eulerian view studies the velocity field

it refers to the rate of change of a given fluid volume dV as it moves about in space (which is the goal) and not the rate of change of fluid velocity at a fixed point in space.

dv over time interval dt is composed of two parts:1. change (during dt) at a fixed point in space2. difference between velocities (at the same instant) at two points dr apartwhere dr is the distance covered during dt

1.

2.

∂ v∂ t

∣ x , y ,z=constant dt

dx∂ v∂x

dy∂ v∂ y

dz∂v∂z

= d r⋅∇ v


let's divide by dt

and then substituing in (3) we get the Euler's equation (1755) also knownas momentum conservation equation:

includes all the external (non HD) forces. In a gravitational field alone

d v =∂v∂ t

dt d r⋅∇ v

d vdt

=∂ v∂ t

v⋅∇ v

∂ v∂ t

v⋅∇ v = [ 1f ]− 1

∇ p

D vDt

= [ 1f ]− 1

∇ p

f = − ∇

Leonhard Euler (17071783)


In case also friction (depending on v2) and other nonconservative forces are considered, new coefficients and ' are introduced:

if we forget magnetic forces

which is known as NavierStokes equation and v = is termed kinematic viscosity;

the quantity

represents the balance between inertia and viscosity and then between turbulence (low viscosities, high R ) and laminar motion (high low R ).

we will come back to this again ...

∂ v∂ t

v⋅∇ v =D vDt

= −1∇ p− ∇

∇ 2v

'[ ... H ...]

∂ v∂ t

v⋅∇ v =D vDt

= −1∇ p− ∇

∇

2v

R=v⋅∇v

∇2 v ' ....

Reynolds ' numberClaude L. Navier1785 – 1836

George Stokes 1819 – 1903

Fluid mechanics (9bis) Euler's equation

R=v⋅∇v

∇2 v ' ....Reynolds ' number

Fluid mechanics (9ter) Euler's equation

Pext

Fluid mechanics (9quater) ram pressure

For a given surface, the fluid contributes with an isotropic pressure (thermodynamics!) arisingfrom the random (thermal) motions of the particles (Pfluid)

In case the fluid is in motion, at a speed v, there is an additional term v 2, arising from thebulk motion of the fluid.

This is known as ram pressure, and rules the motion of a fluid within another fluid. The surface separating two fluids reaches the pressure equilibrium:

There are several examples of ram pressure in astrophysics.

sect. 2.4, pp 1719 Clarke & Carswell

Pfluid v

P fluid ⇐⇒ P ext v 2

Fluid mechanics (10) equation of energy

It is the application of the 1st law of thermodynamics to fluid mechanics: u= internal energy per unit volume h= p + u = enthalpy per unit volume (H=pV+ U)

the conservation of energy can be written as

Energy flux vector

to see the meaning of this equation let's integrate over some volume V

amount of energy flowing out the volume V per unit time

∂∂t

12v 2u =− ∇ [v v 2

2h ] = − ∇ [v v 2

2up ]

∂∂t∫ 1

2v 2u dV = −∫ ∇ [v v 2

2h ]dV

∂∂ t∫ 1

2v 2u dV = −∮ [v v 2

2h ]dS

Fluid mechanics (11) equation of energy

Other expressions of the energy equation (also in nonconservative form) maybe given:

we consider and w the energy and the enthalpy per unit mass (instead ofthe values u and h per unit volume)

conversely, if V* 1/ is the specific volume and P* the external pressure(modulus) and Q* the heat source

further expressions include also entropy.

∂∂t

12v 2=−∇ [v v

2

2w ]

D

Dt P ∗ DV ∗

Dt= Q ∗

Fluid mechanics (12) Summary

1. Equation of continuity

is scalar and contains and v, namely 4 variables2. Euler's (NavierStokes') equation

is a vector relation and adds the variable p3. Equation of energy

is scalar and introduces one new variable (h or u)

therefore we have 5 independent equations with 6 variables and one more relation is required to solve the full set.

In general the fluid state equation is required to fix this problem.

∂

∂ t ∇ v v⋅∇ = 0

∂ v∂ t

v⋅∇ v =D vDt

= −1∇ p− ∇

∇ 2v

D

Dt P ∗ DV ∗

Dt= Q ∗

Fluid mechanics (12bis) Applications

the “millennium simulation” (Springel et al. 2005) www.mpagarching.mpg.de/galform/millennium/

(M)hydrocodes:ENZO – EulerianGADGET Lagrangian

Fluid mechanics (12ter) Applications

Fluid mechanics (13) Vortices

Let's go back to NavierStokes' equation

It can be rewritten by using the vector identity

Let's consider the rotational of such relation (rotational of gradients are 0):

let's also introduce

12∇ v 2 = v× ∇×v v⋅∇ v

∂ v∂ t

v⋅∇ v = −1∇ p − ∇

∇2 v

∂ v∂ t

= −12∇ v 2v× ∇×v −

1∇ p − ∇

∇ 2v

∂ ∇×v

∂ t= ∇×[ v× ∇×v ]

∇2 ∇×v

∂

∂ t= ∇× v×

∇

2

2 = ∇×v

Fluid mechanics (14) Vorticity

The quantity is known as vorticity and is similar to an angular velocityIf we apply the Stokes' theorem to a circumference with radius r :

but the left side is then

12∮

v⋅d l =12∫∫ ∇×v ⋅n dS=∫∫ ⋅n dS≈ r 2

r v v = r

Fluid mechanics (15) Vortices

If viscosity is negligible ( =0)

Then...the flux of across any given surface in motion with the fluid is constant

Kelvin's theorem:“In an inviscid, barotropic flow with conservative body forces, the circulationaround a closed curve moving with the fluid remains constant with time"

Similar to H field in a plasmaThe structure of the velocity field is bound to the fluid element.

If such element changes shape during its motion, then also the velocity field is modified accordingly

Consequences....In an ideal fluid (viscosity is 0) vortices cannot be either created or disrupted

Viscosity is necessary for creation/disruption of vortices

∂

∂ t= ∇×v×

William Thomson,Lord Kelvin, (1824 – 1907)

Hydrostatic equilibrium (1)

It represents a condition where time derivatives are 0 (i.e. equilibrium) and thefluid does not move (v = 0 i.e. hydrostatic).

Mass and energy conservation equations are identically satisfied. In case of an inviscid fluid, the momentum conservation becomes

AIM: in case we know the barotropic equation of state [i.e. p = p ()] it is possible to solve for the pressure / density distribution everywhere [p = p (r) and/or r.

There are several possible applications:

slabs of stratified (isothermal) fluids

(isothermal) spherical distribution

0 = − ∇ −1∇ p namely

1∇ p = − ∇

z

Hydrostatic equilibrium (2)

For an isothermal distribution the hydrostatic equilibrium can be written in (polar)spherical coordinates, and then

i.e. surfaces with constant p, coincide

In general, the equation of state

is known as polytrope, and can be interpreted as a power law.

More will be given in “Stellar Astrophysics”.

Details in Clarke & Churchwell, Chap. 5

∇ p = − ∇ becomesdpdr

= −d

dr

p = K 1

1n

Fluid mechanics (16) sound waves

Let's consider a compressible fluid at rest (vo = 0), with initial pressure po

and density o in which gravity and magnetic forces can be neglected.Let's generate a small 1D perturbation along the x axis so that

let's substitute these relations in the 1D eq. of motion

x ,t = o x ,t = o 'p x ,t = po p x ,t = pop 'v x ,t = v o v x ,t = v ' .

∂v∂ t

v∂v∂x

= −1∂p∂x

−1∂ f∂x

∂v '∂ t

v '∂v '∂ x

= −1

o '∂ p '∂ x

∂v '∂ t

1o

∂ p '∂ x

= 0


and also in the 1D equation of continuity

we are left with

then let's derive the first equation wrt x nd the second wrt t and subtractingthe second we get

∂v '∂ t

1o

∂p '∂x

= 0

∂

∂ t

∂ v∂x

= 0

∂∂ t

o ' o ' ∂v '∂x

= 0 [v '∂ '∂ t

] neglected

∂ '∂ t

o∂v '∂ x

= 0

1o

∂ '∂ t

∂v '∂ x

= 0

1o

∂ '∂ t

∂v '∂x

= 0


we can consider that all motions in an ideal fluid are adiabatic to the lowestorder of approximation

and substitute in the previous equation to get

which is the expression of a wave propagating along the x axis with velocity

solutions are n the form

∂ p∂

o

∂2 '∂x 2 −

∂2 '∂ t 2 = 0

p ' = p = ∂p∂

o

= ∂ p∂ o

'

∂2 p '∂x 2 −

∂2 '

∂ t 2 = 0

' = f x − c s t g x c s t

c s = ∂ p∂

o

1/2


is the sound speed in an ideal gas;

in case of an adiabatic perturbation (wait few slides.....)

in case of isothermal perturbation

In general a fluid in motion is described by M=v/cs , known as Mach number,

and in case either M < 1 or M > 1 the motion is termed sub/super – sonic

cs = ∂p∂

o

1/2

P cs

2 = P=

−1 = kT

c s2 = P

= kT

[ is mean molecular atomic mass ]

Ernst Mach(1838 – 1916)


we can be written releasing the 1D starting condition as (3D)

whose solutions have the following expression:

waves are longitudinal (displacement along the direction of propagation)

complete with p. 386387 Padmanabhan

∂ p∂

o

∂2 '∂x 2 −

∂2 '∂ t 2 = 0

0 =∂2 '∂ t 2 − c s ∇

2 '

'0

= Ae i k⋅x± t where

2 = k 2 cs2 ; A≪1


Going back to 1D expression, the solutions have the form

If the perturbation can be assumed to be adiabatic (like in most cases foran ideal gas) pV = cost namely p=p

o (

o) then (p )

( = atomic/molecular mass)

In case of an isothermal perturbation (then p )

For Earth atmosphere:

p ' = p = f p x − c s t g p x c s t ' = = f x − c s t g x c s t

cs = ∂p∂

o

1/2

= [ po

o −1]

1 /2

= p

1/2

= −1 1 /2= kT

1/2

c s = ∂ p∂

o

1/2

= p

1/ 2

= kT

1 /2

po = 1.013250⋅105 N m−2 ; o = 1.2928 kg m−3

k=1.38⋅10−23 J K−1 ; ~ 0.029kg m−3 ; N 6.022⋅1023mol−1 : = 1.4 .

Fluid mechanics (22) incompressible fluids

A fluid is considered incompressible when (space and time variations are negligible)

neglecting all forces except pressure, let's consider

and using Euler's equation in its linear approximation

implying 2v 2 = p and considering p p/2 we get p v 2which can be entered in the expression

≪ 1

≃ ∂∂p p =1cs

2 p

∂ v∂ t

v⋅∇ v = −1∇ p

v

v 2

L = 2v 2

L ≈ −1

pL

≃1cs

2 p =1cs

2 v 2

≃v 2

cs2 = M 2

Fluid mechanics (23) incompressible fluids

Then for having an uncompressible fluid, it is necessary that its motionis subsonic, M 2 ≪ 1.

In all cases when namely ( v 0.4 c

s )

the fluid can be considered incompressible.

Alternatively taking

the velocity along the x axis is

and it cames out that the pressure is

from the def of sound speed

and finally

= f x−c s t

v ' =∂

∂ t= f '

p ' = −∂

∂ t= c s

'

p ' = c s2

v ' = c s '

Fluid mechanics (23bis) De Lavalle Nozzle

A consequence of mass conservationin a steady adiabatic flow. The fluid is compressible!

details on Padmanaban p. 389

no details, comments on the figure

v S = constantd v v

= −dSS

[omissis ]dSS

= −[1−v 2

cs2 ]dv

v

Gustaf de Lavalle(1845 – 1913)

Fluid mechanics (24) Shock waves

Shock waves are not reversible discontinuities in the properties of a fluidpropagating within that fluid.

In the ideal case: LTE is supposed to hold everywhere except in the shock front (x=0) a stationary regime is considered (t=0)

There is a search for relationships between perturbed and unperturbed parameters)

Fluid mechanics (25) shock waves

a shock is formed when a perturbation is moving with a velocity larger thanthe sound speed pushing/compressing the fluid encountered during its motion

also adiabatic sound waves can grow to shock waves [c

s is higher where is higher, leaving the

denser regions progressively move at higher velocities (c

s continuously grows!) until a

jump is formed ]

the shock compresses, heats and drags the shocked material

piston(v

t)

x

shock

downstream upstream(perturbed fluid) (unperturbed fluid v

1=0)

2,p

2,2,T2

,p1,,T1

v2

vsh


it is possible to define a (comoving) surface where physical quantities abruptly change;

in that RF all the HD equation must hold[in the simplest case, they have a stationary form (/t =0), and happen along agiven direction, i.e. x]

∂v x

∂x= 0

∂

∂xv x

2p = 0 ; ∂

∂ xv x v y = 0 ; ∂

∂xv x v z = 0

∂

∂ x[v x w

v 2

2] = 0


Case 1: Any mass can't cross the surface density can't be 0, consequently all velocities are 0; from the second equation

and then only can be discountinuous (tangential discontinuity)

it is possible to show that such tangential discontinuities are unstable (grow).In this case the two fluids will be mixed by turbulence

side view / top view

Case 2: In general (= all the other cases) the mass flux across the surfaceis not 0 and are not 0 anymore while become continuous.v 1x ,v 2x v y ,v z

1 v 1x = 2v 2x = 0∂ p∂x

= 0

, v y ,v z

x x xx x xx x x


Let's choose a RF on the shock front: unperturbed material falls with v'1 = v

sh

The shocked matter is dropped behind at v'2 = v

sh v

2

An external observer measures v1= 0 , v

2 and v

sh > v

2

Lets write the conservation laws for HD fluids on the discontinuity surface:

where and w=+p/are the energy and is the enthalpy per unit mass.

1v ' 1=2v ' 2 i.e.1

2=

v ' 2

v ' 1

1v ' 12p 1=2v ' 2

2 p 2 i.e. 1v ' 12−2v ' 2

2 = p 2−p1

1v ' 1v ' 1

2

21

p1

1w 1

= 2v ' 2 v ' 2

2

22

p2

2w 2

PierreHenry Hugoniot(1851 – 1887)


then p2 and v'

2 can be obtained; finally after some algebra we get

and then 2 and v'

2 are

known as HugoniotRankine relationships in the shock RF.From the first equation

in the observer's frame, with M ≫ 1:

2

1=

1M 2

2 − 1M 2 =v ' 1

v ' 2

p 2

p 1

=2M 2−−1

1

v ' 2

v ' 1

=2 − 1M 2

1M 2 { =14

if monoatomic gas , M ∞

= 1 if M 1 }v 2 = v sh − v ' 2 =

34

v sh

William Rankine(1820 – 1872)


Strong (adiabatic) shock waves : in case of M ≫ 1

since PV T then P / T and we can constrain the jump in temperature:

The temperature (and pressure) of the shocked material can growwithout limitations.

v ' 1

v ' 2

=2

1=

1M 2

2 − 1M 2 ≃ 1 − 1

p2

p1

=2M 2−−1

1≃

2 1

⋅M 2

T 2

T 1

=p2

p1

⋅1

2≃

2 1

⋅M 2⋅−11

=2−1

12 ⋅M 2


The shock converts kinetic to internal (thermal) energy

Also entropy is discontinuous on the shock surface s2s1 and then s2 > s1 while it is preserved upstream and downstream. The entropy increase is originated by collisions among fluid particles

Thickness: is of the order of the mean free path ; from the relation n =1

=1/n

where n=numeric density; =cross section of the process

In summary, knowing the parameters of the unperturbed medium (pedex 1) it is therefore possible to derive those of the shocked material

For instance: electronic radius 5.3 x 10 11 m, nISM ~ 1 cm 3, 1 pc =3.09 x 10 16m, UA = 1.5 10 11 m

Fluid mechanics (31bis) shock waves

The shock converts kinetic to internal (thermal) energy

The case of SNR: ~ 1 – 10 cm – 3 kT~ 1000 K; = m

H ~ 1.7 x 10 – 27 kg ;

cs = (kT/)1/2

vSN

~ 104 km/s;

The star envelope expands at M » 1

Kepler SNR 1604

Fluid mechanics (31bis) shock waves

Consequences on and on the ambient medium (bubbles!)

NGC6946

Fluid mechanics (32bis) astrophysical shock waves


Isothermal shock waves:

the gas is capable to rapidly radiate the thermal energy acquired crossing the shock surface (the cooling time is extremely short) then T

2=T

1

it is equivalent the have =1 in all the earlier equations.

the fluid, then, can be highly compressed, and this is true if the energytransferred by the shock to the ambient gas is (~) immediately dissipated(e.g. via radiation)

details: either chap 7.2 cowie & carswell or chap 8.11 Padmanaban

2

1= M 2

p 2

p1

= M 2

rT

T1T2

isothermal

coolinglength

adiabatic

Fluid mechanics (33) Fermi acceleration and shock waves

In presence of (strong) shock waves, acceleration may produce relativisticcharged particles:

if l = mean free path between 2 collisions;f = v/2 l is the frequency of interactions

application to SNR: vsh

~10000 km/s; l = 0.001 pc ; F ~ 106 s 0.1 yr

to have electrons with 1 GeV, about 15 F

are requiredto be considered: oblique shocks [transparencies]relativistic Fermi acceleration [idem]

(SN)

*

vsh

v1=0

v2

vv 2 =34

v sh

2 = 2v 2

v =

32

v sh

v

d

dt= 2⋅f = 3

4

v sh

l = FPOSTPONED

Fluid mechanics (34) MagnetoHydroDynamics (MHD)

Magnetic fields become important in case of substantial electron (ion) densityi.e. T > 104 K [~106 109 K are often typically found in astrophysics]

an ionized fluid is known as plasma which can be treated as and ideal fluid when the linear size of a phenomenon exceeds the Debye length [1. random kinetic energy of particles is much larger than the electrostatic potential energy between two particles 2. the E field of a single particle is nulled by the field of its neighboring charges with opposite sign 3. charge density

e 0 ]

An astrophysical plasma must satisfy:

1. fluid velocity is small (v ≪ c, and all the terms v/c 2 can be neglected)2. electric conductivity is large ( )3. linear scale ≫

D

4. collisions are ruling the plasma physics (vc > v

L > v

phenomenon)

i.e. dimensions of the fluid exceed the Larmor radius which is larger than the mean free path between collisions

D ≡ kT4ne e2

1 /2

= 6.9cm To K

1 /2

ne

cm−3 −1/2

Peter Debye 18841966

L(cm) T (K)solar corona 11 6 16 18 5HII 18 4 13 28 5WD 8 6 16 12 8NS 6 9 22 14 13

20 2 11 28 623 8 19 44 7

Lab. Plasma 2 4 13 3 3

(1/sec) (sec) H (Gauss)

ISGasIGGas

=ne e2

2me

v=

108 T 3 /2

gFF

sec−1

H =4c 2 L2 sec

Fluid mechanics (35) MHD

Typical orders of magnitude (numbers should be interpreted as 10 x)

where (H) is a timescale for “diffusion” of a magnetic field H, wait for a few slides!


Maxwell equations and their HD approximation;

Let's consider the divergence of the third equation:

that can interpreted as charge conservation

add 7.17 e 7.18 dispense: decadimento esponenziale e diffusione della carica

∇⋅H = 0∇⋅E = 4e

∇× H = 4c

j 1c∂ E∂ t

∇×E = −1c∂ H∂ t

0 = 4c

∇⋅j 1c∂∂ t

∇⋅E = 0 = 4c

∇⋅j 1c∂∂ t

4e

0 =∂e

∂ t∇⋅e v

corrente di spostamento

James Clerk Maxwell1831 – 1879


in the same equation compare the two contributions:

in (nearly) all astrophysical cases, given that and L are generally large.

Consequence:

taking the divergence: (Ohm's law)

neutral plasma!

4c

j ⇔1c

∂ E∂ t

Ohm ' s law E =j

orders of mag1c

∂ E∂ t

≈ 1c

∂j∂ t

≈ 1c

jT

≈ 1c

j vL

it turns out that4c

∣j∣ ≫ 1c

j vL

∇× H = 4c

j 1c∂ E∂ t

≈ 4c

j

∇⋅∇× H = 4c

∇⋅j = 0 ∇⋅E = 0

e = 0


Fields in observer's (E, H ) and rest (E', H' ) frames of a plasma moving at v≪c

take the first equation:

E ' = E vc× H

H ' = H −vc×E

j ' = j − e v ≈ j

' e=e −1c2 v⋅j ≈ e ≈ 0

E = E ' −vc× H =

j

−vc× H

= c4

∇× H −vc× H

≈ c4

HL−

vc

H = −vc

H 1− c 2

4v L = −vc

H 1−1Rm

Lorentz force on a moving charge

Hendrik Lorentz 1853 – 1928


Magnetic Reynolds' number

is generally true in astrophysics:in most cases R

m is about 10 6 or even much larger

consequence: (... in general ...)

astrophysical electric fields are extremely weak ( 0) and can be neglected; induced fields from moving charges are generally negligible (except....).

Furthermore....

Rm = 4v Lc 2 ≫ 1

c 2

4= m magnetic viscosity

Rm = v Lm


Summary of MHD approximations:

∇⋅H = 0∇⋅E = 4e ≈ 0

∇× H = 4c

j 1c∂ E∂ t

≈ 4c

j

∇×E = −1c∂ H∂ t

E ' = E uc× H ≈ 0

H ' = H −uc× E ≈ H

j ' = j − e v ≈ j = E vc×H

∂ H∂ t

= −c ∇×E

j ' = j − e v = E ' ≈ j = E vc×H

¿∂ H∂t

= −c ∇× j

−vc×H j =

c4

∇× H

∂ H∂ t

= −c ∇× c

4∇× H −

vc× H

≈ ∇×v× H −c 2

4∇× ∇× H ∇× ∇×H = ∇ ∇⋅H − ∇2 H

= ∇×v× H c 2

4∇

2 H = ∇× v× H m ∇2 H

≃v H

L

c 2

4

HL2

=v H

L 1 1

Rm ∂ H∂ t

≈ vHL 1

1Rm


Maxwell equations and their HD approximation;

Fluid mechanics (41bis) MHD

There are two limiting cases:

1. H field diffusion (v = 0)

i.e. the field decays and fades away into the ambient mediumexercise: compute in a number of astrophysical cases: ISM (HII region, bulge corona), WD,...

∂ H∂ t

= ∇×v× H c 2

4∇ 2 H =

= ∇×v× H m ∇2 H ≃

v HL

c 2

4

HL2

∂ H∂ t

=c 2

4∇ 2 H ≈ −

c 2

4

HL2 since ∇×v× H = 0

H ≃ H o e−t / =L2 4

c 2


2. H frozen in matter (v 0 ; ) then the second term goes to 0

Let's consider a closed curve defining a surface S1 moving to another closed surfacedefining a surface S2 (different from S1) at a subsequent time dt.The variation of the flux of H across each surface is:

∂ H∂ t

= ∇×v× H

S2

S1

ddt∬ H⋅n dS ≃

≃ 1dt

[∬S 2

H tdt ⋅n dS 2−∬S 1

H t ⋅n dS1]

≃ 1dt

[∬S 2

H t ⋅n dS 2∬S∂ H∂ t

⋅n dS−∬S1

H t ⋅n dS1]

≃ ∬S∂ H∂ t

⋅n dS 1dt

[∬S 2

H t ⋅n dS 2−∬S 1

H t ⋅n dS1]

if S t = S 1 S 2 S 3

≃ ∬S∂ H∂ t

⋅n dS 1dt

[∬S t

H t ⋅n dS t−∬S 3

H t ⋅n dS3]

S3


This implies the magnetic flux conservation

v

∬S∂ H∂ t

⋅ndS 1dt

[∬S t

H t ⋅n dS t − ∬S 3

H t ⋅n dS3] =

= ∬S∂ H∂ t

⋅n dS 1dt

[∭V∇⋅H t ⋅n dV − ∬S3

H t ⋅n dS3] =

but n⋅dS∣S 3= d l ×v dt

= ∬S∂ H∂ t

⋅n dS − 1dt

∮S3

H t ⋅ d l ×v dt =

= ∬S∂ H∂ t

⋅n dS − ∮S 3

H t ⋅ d l ×v =

= ∬S∂ H∂ t

⋅n dS − ∮S 3

v× H t ⋅d l =

= ∬S∂ H∂ t

⋅n dS − ∬S∇×v× H ⋅n dS =

= ∬S[∂ H∂ t

− ∇×v× H ]⋅n dS =

= ∬ c 2

4∇

2 H dS 0 [∞]

S2

S1

S3

dl

Fluid mechanics (43bis) MHD

Physical interpretation:

in full analogy to the Kelvin's theory (vortex generation / conservation)the magnetic field lines cannot be crossed by matter in motion but suchmatter can stretch and bend the field lines

In case the plasma expands, the field strength decreases and the other way round. It is therefore possible to create large fields in case of substantial compression of the plasma.

There are many astrophysical examples. WD, NS, IS & IG plasma, etc.

v

S2

S1

S3

dl

Crab nebula (radio waves)

Fluid mechanics (43ter) MHD

Plasma have an additional pressure arising from the H field: depending on their balance with the dynamical contributions, the fate of the plasma willbe determined

v

S2

S1

S3

dl

P v 2

H 2

8field lines will be moderately bent :

the field will confine the plasma

P v 2≫

H 2

8matter motion will sweep magnetic field lines

Magnetic pressure

Fluid mechanics (45) magnetic energy

Let's consider the magnetic energy, defined as

and the time variation of the magnetic energy is

W H = ∭V

H 2

8dV

∂W H

∂ t= 1

4∭ H⋅∂ H

∂ t dV =

= 14

∭ H⋅ ∇×v×H c 2

4∇2 H dV =

= 14 ∭ H⋅∇×v× H dV∭

c 2

4H⋅∇2 H dV

∇× ∇× H = ∇ ∇⋅H − ∇2 H

a⋅ ∇⋅b = ∇⋅b×a b⋅∇×a

Fluid mechanics (46) magnetic energy

the previous relationship becomes

which in turn becomes (lots of algebra skipped!).....

which measures the variation of magnetic energy in a time unit as aconsequence of the work done on a unit volume by the magnetic force

Field amplification via magnetic dynamo takes place when kinetic(mechanic) energy is converted to magnetic energy

14 ∭H⋅∇×v×H dV∭

c2

4H⋅∇2 H dV

∂W H

∂t= 1

4∫V

∇⋅[v×H × Hv× H ⋅∇× H ]dV =

= −1c ∫V

v⋅j× H dV j = c4

∇×H

F mag ≈ j ×H

Fluid mechanics (47) magnetic forces

Magnetic forces:

with the vector identity

but this operator represents H times the derivative along the field line

and can be interpreted as a resistive term to variations to changesto the local magnetic field geometry

F H =j× Hc

=1

4 ∇×H × H

F H =1

14

H⋅∇ H −1∇ H 2

8

∇× H × H = H⋅∇ H −12∇H 2

isotropic H pressure

tension along field lines

H⋅∇ = H ∂∂s


Going to the magnetic tension:

The magnetic field therefore produces two forces perpendicular to the fluidvelocity and to the local direction of the H field itself:1. directed towards C, aims to shorten the field line (like a guitar cord!)2. opposing to gradient of the magnetic pressure along field lines

.C

R

Hĥ

ň

ĥs

14

H⋅∇ H =1

4∇ H H

=1

4H ∂

∂sH h =

H 2

4∂ h∂s

h

8∂H 2

∂s

1. perpendicular to local H (ĥ) direction

if R is the curvature radius of a field line it can be

written as H2/4Rthis term is 0 for a uniform H

2. along local H (ĥ) directionsignificant only in case of substantial

variation of the H field intensity;identical to [but with opposite sign (but computed along

the field line) this term is 0 for a uniform H

stress tensor


Going one further step back:

these two terms produce an anisotropic mechanical effect, depending on H geometry; limiting cases:

1. Uniform H field, tension is zero

2. Disordered H field on scales much smaller than the plasma:both terms play a role and in general we end out with a isotropic pressure

[examples: compression l along H and perpendicularly to H ]

F H =1

14

H⋅∇ H −1∇ H 2

8 tension along

field linesisotropic H pressure

13

H 2

8

Fluid mechanics (50) Alfvén waves

Waves in a magnetic field: also when = const (incompressible ) and = 0 there could be transversewaves as a consequence of H 2/4allowing this effect since this termrepresent an oscillator.

In general if we consider a small perturbation ( with the addition of B= B0z) the final solution must satisfy

These are known as Alfvén waves (MHD waves) which propagate along thefield lines with a typical velocity v

A which can be defined as follows:

v A = H 2

41

=H

4

2 − v a2 k ∥

2 [4−2c s2v a

2 k 2c s2 v a

2 k 2k ∥2 ] = 0

Fluid mechanics (51) Alfvén waves

Therefore Alfvén waves are the only way for propagating perturbations inan incompressible, non viscous, magnetized plasma.

In case the fluid is also compressible there are also longitudinal waves.

The general case is

1. perpendicularly to local H direction ( =0 ) we have sonic waves propagating with Cs

2. along field lines we have magnetosonic waves propagating with

cms = c s2v A

2

v 2=

c s2 v a

2

2 [1 ± 1−4cos2

c s2v a

2

c s2v a

22 ]

Fluid mechanics (52) Alfvén and shock waves

In presence of a shock and assuming that there is a H field parallel to theshock surface, the momentum conservation becomes:

which gives (after some algebra)

therefore in case of an adiabatic shock (cs2 = (p/) the H field can be

compressed, i.e. amplified, by a factor of 4.

1v ' 12p 1

H 12

8= 2v ' 2

2 p2 H 2

2

8

H 2

H 1

=2

1

Fluid mechanics (53) Alfvén and shock waves

In case of isothermal shock T2=T

1 , c

s2 = (p/) then

and the eq. of momentum conservation

and reintroducing the Alfvén speed, in case of vA >> c

s1 ,c

s2 and 2>1

2

1= 2

v 1

v A1

= K

cam

pi H

ast

rofis

ici &

effe

tto d

inam

o 7.

4.8

p1 = 1c s12 ; p 2 = 2 c s2

2 with v 2p=const

1v ' 121cs1

2 H1

8= 2 v ' 2

2 2c s22

H 2

8= 2v ' 2

2 2 c s22 2

1 2 H 1

8

Fluid mechanics (44) H flux conservation

Magnetized fluid motion: the NavierStokes equation in case of a H fieldwe consider internal dynamic (but not magnetic) viscosity

taking a linear approximation of this equation:

MHD systems with the same combination of M, F, R, K have the same behaviour; self – similarmotions – astrophysical conditions can be reproduced in lab if they have the same numerical set

∂ v∂ t

= −v⋅∇v −1∇ p − ∇

∇

2v j× Hc

∂v∂t

≃v 2

L−

PL

− g

vL2

H 2

4L

=v 2

L −1 −Pv 2

−gLv 2

v L

H 2

4v 2 =

v 2

L − 1 −1

M 2 −1F

1R

1

K 2 R =

v L

Reynolds K =vv a

=v

H 2/ 4Karman

M =vc s

=v

p /Mach F =

v 2

g LFroude

Fluid mechanics (55) Fluid (in)stabilities

– a stable (steady state, i. e. / t=0) configuration can be interesting in astrophysics since it is more likely to be found than unstable configurations. It can be considered persistent.

– perturbations are either diminished or there is the possibility of oscillations or waves about the equilibrium.

Fluid mechanics (55bis) Fluid instabilities

Instabilities in many different forms in (M)hydrodynamical systems

– equilibrium configuration which undergoes to a small perturbation studied in its linear approximation.– the perturbation is described by plane waves – dispersion relation is obtained – imaginary frequencies pinpoint the onset of instabilities

A fluid in motion relative to another body (fluid) may alter its own configuration if small perturbations grow rather than damping out

common astrophysical instabilities:

RayleighTaylor “mushroom” cloudsKelvinHelmholtz shear flow rollupRichtmyerMeshkov shock interface instabilityConvective buoyant overturnThermal runaway coolingJeans gravitational collapse

e i t − k⋅x

k

Fluid mechanics (56) Fluid instabilities

RayleighTaylor instability

Two incompressible, inmiscible, fluids with a sinusoidal perturbation at the interface:

– fluids at rest; 1 > 2 ; constant (in time) gravitational field

– unstable configuration in absence of surface tension

– bubbles (fingers) of less dens fluid rise to compensate bubbles (fingers) of heavier fluid dripping down

P1,

1

P2,

↓g

↕ p(x)(x)

2= k 2 g z 1 ∗

2−1

2is negative if 21

∣dTdz ∣ 1− 1

Tp ∣dp

dz∣ Schwarzschild stability criterion


RayleighTaylor instability

↓g


KelvinHelmoltz instability

Two incompressible, inmiscible, fluids in relative motion at a speed v, with a sinusoidal perturbationat the interface:

– Linearized Euler equation becomes

–

v1, P

1,

1

v2 = 0

P2,

↕ p(x)(x)

∂ v∂ t

= −∇ p

then take the divergence

∇ 2 P=0 ∇⋅v = 0

p = f z e [i kx− t ] leads t o ∂2 f z

∂z 2− k 2f

z = 0

solutions : f z ~ e±kz f z ~ ekzruled out diverges at large z

above the boundary surface : P1 ~ ekz e [i kx− t ]

∂v z

∂ tv

∂v z

∂ x=−

k P 2

2z

Fluid mechanics (59) Fluid instabilities clouds over mt. Duval (Australia)

Atmospheric layers in Saturn


Jeans instability

Arises when gas pressure is unable to overcome selfgravity:let's consider 1D adiabatic perturbation to a compressible gas at rest withdensity 0 and pressure P 0:

the linearized equations in presence of selfgravity only are:

taking the partial derivative wrt time of the continuity equation and and wrt x of Euler eq.

[to be completed + omissis]

[end of MHD slides]

Fluid mechanics (99)

[end of. HD & MHD]