Astrophysical fluid dynamics

Part III Mathematics 2015–16

24 lectures

gio10@cam

Introductory lecture M. W. F. 9

Professor Gordon Ogilvie DAMTP F1.02

Provisional synopsis

● Overview of astrophysical fluid dynamics and its applications● Equations of ideal gas dynamics and MHD● Physical interpretation of MHD● Conservation laws, symmetries and hyperbolic structure● Stress tensor and virial theorem● Linear waves in homogeneous media● Nonlinear waves, shocks and other discontinuities● Spherically symmetric steady flows: stellar winds and accretion● Axisymmetric rotating magnetized flows: astrophysical jets● Stellar oscillations: introduction to asteroseismology and tides● Local dispersion relation● Internal waves and instabilities in stratified rotating bodies

Practical arrangements

Lectures:● Mondays, Wednesdays and Fridays at 9:00 in MR11

Example classes (based on four example sheets):● 1. Thursday 29 October, 14:00–15:30, MR11● 2. Thursday 19 November, 14:00–15:30, MR11● 3. Thursday 3 December, 14:00–15:30, MR11● 4. Thursday 14 January, 14:00–15:30, MR15

Lecture materials, including extended notes in preparation:http://www.damtp.cam.ac.uk/user/gio10/afd.html

Revision class (based on past Tripos questions):● Thursday 12 May, 15:00–16:30, MR11

Seminars that may be of interest

DAMTP Astrophysics seminars:● Mondays at 16:00 in MR14

IoA colloquia:● Thursdays at 16:00 in the Sackler Lecture Theatre, IoA

DAMTP Fluid Mechanics seminars:● Fridays at 16:00 in MR2

All listings at http://www.talks.cam.ac.uk

● Tuesdays at 13:00 in MR14

Centre for Exoplanet Research seminars:● Wednesdays at 16:00 in the Ryle Seminar Room, Kavli Institute

Theoretical varieties of AFD

Basic models:

HDNewtonian gas dynamicsnon-relativisticcompressibleideal (inviscid, adiabatic)self-gravitatingperfect gas (usually)

MHD + magnetic fieldideal (perfectly conducting)

Theoretical varieties of AFD

Extensions (beyond this course):● Dissipative fluid (viscosity, thermal conduction, resistivity)

● Chemistry (equation of state, composition, reactions / ionization)● Radiation (various possible treatments)● Relativity

● Plasma physics / multifluid MHD / Hall effect / ambipolar diffusion

Theoretical varieties of AFD

HDMHDRHDRMHDGRHDGRMHDGRRHD...etc.

hydrodynamics

magnetohydrodynamics

radiation hydrodynamics

radiation magnetohydrodynamics

general relativistichydrodynamics

magnetohydrodynamicsgeneral relativistic

general relativisticradiation magnetohydrodynamicsGRRMHDgeneral relativisticradiation hydrodynamics

Examples of observations

Examples of numerical simulations

Useful data (in CGS units)

AU = 1.496� 1013 cm

pc = 3.086� 1018 cm

● Astronomical unit● Parsec

● Solar radius

● Newton’s constant

● Solar mass

● Boltzmann’s constant

● Speed of light● Stefan’s constant

● Solar luminosity

G = 6.674⇥ 10�8 cm3 g�1 s�2

M� = 1.989⇥ 1033 g

L� = 3.846⇥ 1033 erg s�1

R� = 6.955⇥ 1010 cm

k = 1.381⇥ 10�16 ergK�1

Joule / erg conversion: J = 107 erg

� = 5.670⇥ 10�5 erg cm�2 s�1 K�4

● Proton mass mp = 1.673⇥ 10�24 g

c = 2.998⇥ 1010 cm s�1

Some typical numbers (order-of-magnitude estimates)

● Solar-type star:⇢ ⇠ 102 g cm�3, T ⇠ 107 K

photosphere ⇢ ⇠ 10�7 g cm�3, T ⇠ 104 K

corona ⇢ ⇠ 10�15 g cm�3, T ⇠ 106 K

molecular clouds n ⇠ 103 cm�3, T ⇠ 10K

cold medium (neutral) n ⇠ 10� 100 cm�3, T ⇠ 102 K

warm medium (neutral/ionized) n ⇠ 0.1� 1 cm�3, T ⇠ 104 K

hot medium (ionized) n ⇠ 10�3 � 10�2 cm�3, T ⇠ 106 K

centre

● Interstellar medium:

⇢ n(mass density , number density )

Validity of a fluid approach

Equations of HD and MHD are derived under the assumption of smalldepartures from a local Maxwellian velocity distribution of particles

A fluid approach is valid provided that:

mean free path

mean flight time

of particlesbetween collisions

⌧ ⌧ T

� ⌧ L

Collisions tend to produce a local Maxwellian distribution, whilegradients tend to produce departures

characteristic time-scale

characteristic length-scale

of the fluid flow

Estimates: (collisional cross-section )� =1

n�, ⌧ ⇠ �

v̄, v̄ ⇠

rkT

m�