Prof. dr. A. Achterberg, Astronomical Dept Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit Gas Dynamics, Lecture 5 (The Solar Wind; Waves: theoretical introduction) see: www.astro.ru.nl/~achterb/

Theoretical results for steady flow that we will use today (from last Lecture): Bernoulli’s Law:

Solar wind velocity as measured by Ulysses satellite

The Parker Model Assumptions: The wind is steady and adiabatic The flow is spherically symmetric Neglect effect of magnetic fields and rotation star

There must be a sonic radius where flow speed = sound speed

Basic Equations: steady, spherically symmetric flow Conservation of mass in steady flow Bernoulli: conservation of energy Entropy is constant: Adiabatic Flow Gravitational potential of a single star

Mass conservation: continuity equation Steady flow in radial direction:

General approach: use “constants of motion” : stream lines are KNOWN!

General approach: use “constants of motion” : stream lines are KNOWN!

Aim: to convert all the equations into a single equation for the velocity V(r): Step 1: calculate density change

Step 2: Calculate velocity change

Step 3: combine velocity and density results: Adiabatic sound speed

Step 4: covert result into a differential equation

Parker’s equation for a spherical stellar wind: Special velocity: sound speed (“Mach One”) Special radius: critical radius

Mathematical Interlude: singularities in differential equations (1)

Mathematical Interlude: singularities in differential equations (2a) 1. Problems arise if solution curve passes through point where That is the same as saying:

Mathematical Interlude: singularities in differential equations (2b) 1. Problems arise if solution curve passes through point where 2. Slope of curve changes sign if solution curve passes through a point where ; if B is monotonic, THEN Y(x) has a SINGLE maximum or a minimum!

Mathematical Interlude: singularities in differential equations (3) SPECIAL CASE: CRITICAL POINT SOLUTION THROUGH ONLY IN THIS CASE IS A MONOTONIC SOLUTION Y(x) POSSIBLE!

Mathematical Interlude: singularities in differential equations (4) Formal solution near critical point :

Mathematical Interlude: singularities in differential equations (5) Formal solution near critical point :

Solution space Parker Eqn: diagram

Solution space for Parker’s Equation Accelerating wind solution: V > 0 and dV/dr > 0! Solution should remain regular at all radii!

Solution space for Parker’s Equation Critical Point Condition:

Wind and Breeze Solutions Special case: Isothermal Wind with constant temperature

Accretion Solution

Bondi Accretion Critical Point Condition:

Isothermal Bondi Accretion

Similar flows: Laval Nozzle (jet engines)

Basic equations:

Similar flows: (2) 2. Astrophysical jets:

Stellar Winds and Jets: similarities and differences Steady flow Steady flow Large opening angle Small opening angle Parker-equation Parker-type equation Flow geometry known Pressure known

Waves

Simple (linear) waves Properties: Small perturbations of velocity, density and pressure Periodic behavior (“sines and cosines”) in space and time No effect of boundary conditions

Small amplitude waves: 1. fundamental approach Wave amplitude is small: position of a fluid element can be ‘decomposed’ as: unperturbed position small displacement

Small amplitude waves: 2. The plane wave assumption Small displacement exhibits a sinusoidal behavior in space and time: plane wave representation Phase factor Complex amplitude vector

Small amplitude waves: 3. Phase, wave vector, wavelength and frequency Alternative formulation of plane waves: angular frequency wave period wave vector wavelength

Waves, wavelength and the wave vector

Small amplitude waves 4. what does ‘small amplitude’ mean? 4. Wave amplitude is small in the following sense: |a| is much smaller than the wavelength λ; |a| is much smaller than gradient scale of the flow; Density and pressure and temperature variations remain small:

Mathematical technique: Perturbation analysis: Expand fundamental equations in displacement ξ(x,t); Neglect all terms of order ξ2 and higher! Express density and pressure variations in terms of ξ(x,t); Neglect all terms of order ξ2 and higher! Find equation of motion for ξ(x,t) where only terms linear in ξ(x,t) appear; - Substitute plane wave assumption.

Perturbation analysis: simple mechanical example Small-amplitude motion; Valid in the vicinity of an equilibrium position;

Perturbation analysis: fundamental equations Equilibrium position:

Perturbation analysis: motion near x = 0 Taylor expansion near x = 0: General case:

Perturbation analysis: like a harmonic oscillator Equation of motion near x=0: “spring constant”

Perturbation analysis: fundamental solutions

Mathematical preliminaries Aim: 1) Construct a generally valid method for perturbation analysis in fluids or gases; 2) Express all perturbations in terms of the displacement vector (x,t) and its derivatives. In the end you should only see a linear equation with things like:

Who measures what variation in a wave Who measures what variation in a wave? Lagrangian and Eulerian variations Two fundamental types of observer in fluid mechanics: Observer fixed to coordinate system measures the Eulerian perturbation: Observer moving with the flow measures the Lagrangian perturbation

Lagrangian labels: useful mathematical concept are carried along by the flow

Lagrangian labels: useful mathematical concept Conventional choice: position x0 of a fluid-element at some fixed reference time t0 As always:

Re-interpretation of time-derivatives: At a fixed position Comoving with the flow

Re-interpretation of time-derivatives + Re-interpretation of perturbations: At a fixed position Comoving with the flow Lagrangian and Eulerian perturbations: Lagrangian: Eulerian:

Important consequence: Commutation Relations for derivatives! All at a fixed position in the coordinate grid Moving with the flow

Relation between Lagrangian and Eulerian perturbations Stay at old position! Follow the fluid to new position! Unpertubed value at old position!

Relation between Lagrangian and Eulerian perturbations (2)

Relation between Lagrangian and Eulerian perturbations (3)

Relation between Lagrangian and Eulerian perturbations (4)

Final result for small perturbations: Small change induced by ξ in Q at fixed position Effect of position shift ξ

Almost trivial example of these rules (1):

Almost trivial example of these rules (2): Formal calculation:

Perturbation analysis: general approach (example: sound waves)

Application: velocity perturbation due to small-amplitude wave (1) Commutation Rules

Application: velocity perturbation due to small-amplitude wave (2) Commutation Rules Definition of the comoving derivative (V = unpertubed velocity!):

Eulerian and Lagrangian velocity perturbations: General relation between the two kinds of perturbations:

Summary: velocity perturbations (1) Special simple case: stationary unperturbed fluid that has V = 0:

Summary: velocity perturbations (2) Another special case: unperturbed fluid has uniform velocity V ≠ 0:

Density perturbation: 1D case Mass conservation:

Density perturbation (2)

Density perturbation (2)

Density perturbation (3)

Density perturbation (4)

Generalization results from 1D to 3D: One dimension: Three dimensions:

Pressure perturbation: Adiabatic flow: entropy conservation if you move with the flow From general relation between Lagrangian and Eulerian perturbations ΔP and δP:

Summary: changes in fluid quantities induced by a small fluid displacement ξ(x,t):

Perturbation analysis: general approach

Linear sound waves in a homogeneous, stationary gas Main assumptions: Unperturbed gas is uniform: no gradients in density, pressure or temperature (P = ρ = 0); Unperturbed gas is stationary: without the presence of waves the velocity vanishes (V = 0); The velocity, density and pressure perturbations associated with the waves are small

Immediate consequence: perturbations are “simple”: Velocity associated with the wave: Density perturbation associated with the wave Pressure perturbation associated with the wave:

Immediate consequence: perturbations are “simple”: Velocity associated with the wave: Density perturbation associated with the wave Pressure perturbation associated with the wave: This is KINEMATICS, not DYNAMICS!

Aim: to derive the DYNAMICS of the problem! To derive a linear equation of motion for the displacement vector (x,t) by linearizing the equation of motion for the gas. Method: Take the Lagrangian variation of the equation of motion.

Perturbing the Equation of Motion To find the equation of motion governing small perturbations you have to perturb the equation of motion!

Unperturbed gas is uniform and at rest: Apply a small displacement

Unperturbed gas is uniform and at rest: Apply a small displacement Because the unperturbed state is so simple, the linear perturbations in density, pressure and velocity are also simple!

Effect of linear perturbations on the equation of motion: fluid acceleration Use commutation rules again:

Effect of linear perturbations on the equation of motion: pressure force Use commutation rules again: I have used:

Effect of linear perturbations on the equation of motion What do we know at this point:

Finally: the equation of motion for ξ(x,t):

Grand finale: an equation for plane sound waves!

Grand finale: an equation for plane sound waves!