Physics of the Interstellar and Intergalactic Medium

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Hilary Farmer
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1 Y4A04 Senior Sophister hysics of the Interstellar and Intergalactic edim Lectre 9: Shocks - revised Dr Graham. Harper School of hysics, TCD What a good physicist does best - Simplify eil Nebla ~8000 yr old

2 What is a -D shock? Infinite plane-parallel geometry good approximation if the radis of crvatre of the actal shock is mch greater than the thickness of the shock Shock front In nit time ( Area) x ( Area) x x x Use handot

3 Conservation Eqations: ass in the frame of the shock Conservations of mass: written for a narrow (-D) shock t ( ) ( ) t x 0 In the frame of the shock the flow is steady (time independent) ( ) t x 0 Const In passing throgh the shock mass is conserved [] Conservation Eqations: omentm in the frame of the shock Conservations of momentm: in nit time mass enters shock with momentm ( ). The mass leaves the shock with momentm ( ) the difference mst be given by the force per nit area Using Eq [] we obtain [] 3

4 4 Conservation Eqations: Energy in the frame of the shock Conservations of energy: reqires that the rate at which gas pressre does work per nit area () and the rate of flow of both internal (U) and kinetic energy (/ ) is constant across the shock Non radiating shocks: If gas acts as a perfect gas on either side of the shock then the internal energy is given by [3] U U [4] U Some detail... Sbstitte Eq [4] into Eq. [3] and reorg. [5] Use Eq. [] to divide both sides (now symmetric)

5 5 Three Jmp (J-Shock) Conditions [] [5] [] ach nmber

ach nmber Sond speed in the pstream gas c ach nmber ratio of the inflowing gas speed (as seen by the shock) to its sond speed: is the ratio of specific heats C /C - adiabatic

6 ach nmber Sond speed in the pstream gas c ach nmber ratio of the inflowing gas speed (as seen by the shock) to its sond speed: is the ratio of specific heats C /C - adiabatic index, or adiabatic exponent c c E.g., spergiant otflow moving at 5 kms - into 60K interstellar clod B star wind (600 kms - ) into WI 8000 K After some algebra... 6

7 7 Rankine-Hgoniot Relations ] [ A Relations between gas pressre, density, and velocity on either side of the shock front ] [ B

8 Strong Non-Radiating Shocks: >> [ B] 5 for 3 7 5?? Limiting vale for monatomic gas (translation) diatomic molecle (translationrotation) Can neglect pressre in the incoming flow ( ) for monatomic gas 3 4 kt µ m 3 µ m 6 k T Rest Frames: v, velocities fixed frame Observationally work in a fixed reference e.g. Spernova, ionizing star. For fast shocks we can often (bt not always) neglect v 4 v v 3 4 v v 3 3 µ m T 4 6 k 3 9 e,int 9 e 3, KE υ 3 Gas behind the shock follows in the same direction as the shock 00 kms - ~50,000 K 8

9 Entropy considerations The Rankine-Hgoniot relations do not in themselves forbid a time reversal of the shock, namely expansive shocks where sb-sonic hot gas expanding to become spersonic cool gas a rarefaction shock Entropy decreases in this process as the flow becomes more ordered nd Law of Thermodynamics forbids rarefactions shocks, bt does allows compressive shocks. Complete description is then given by the two Rankine-Hgoniot relations and the nd Law of Thermodynamics Rarefaction waves do exist (bicycle pmp) Radiating Shocks Behind the shock the kinetic energy is converted to thermal energy > 50 kms - will ionize hydrogen. lenty of electrons to excite energy levels that can radiate energy away the heated shock starts to cool Typical IS shock speeds are km s - - X-ray emitting plasma Thermally nstable t energy content energy loss rate c Λ ( Ts ) Ts 3 t c T s L c t c 3 nts n Λ 4 ( T ) s 9

10 0 Isothermal limit of a radiating shock If the cooling length is not too long then we can consider the region immediately behind the adiabatic shock and the sbseqence cooling phase as a transition zone and set in the Rankine-Hgoniot relations Now the density can contine to increase behind the shock withot limit Using the same shock-to-fixed reference frame relations ( ) v v - agnetic ields onsider a galactic magnetic field rnning parallel to the shock front It exerts a pressre on partially ionized gas he new shock mp condition for momentm conser ation becomes 8π mag B π π 8 8 B B

11 agnetic field in a -D shock The magnetic field is frozen into the partially ionized gas and the nmber of field lines is conserved across the shock. Since the compression is in -D the conservation of mass leads to Shock front x In nit time ( Area) x ( Area) B x B x B B x x Effects of agnetic Fields Typical galactic field of 3x0-6 G conseqences... Strong non-radiating shock, magnetic pressre increases by 6 giving a pressre that < /5 of the gas pressre not important However, in isothermal shocks it may be important Strong radiating shocks the compression of gas leads to the increase in B which acts against the shock limiting the shock gas density The maximm density occrs when the magnetic pressre balances the dynamic (ram) pressre B B 8π 8π max max 3 B

12 Real niverse a tad more complicated IS robes bow-shocks R Hya R Hydrae High proper motions Infrared Imaging T. Ueta If we know the stellar wind properties, we can learn abot the IS LL Ori

13 Betelgese α Ori α Orionis JAXA/Akari T. Ueta This one looks too circlar? Steady shock (not evolving) In frame of star S 0 omentm balance across the Shock, where the star-is relative velocity is IS IS IS IS W W W ( R) ( R) W IS IS W Dynamic ram pressres exceed local gas pressres Betelgese α Ori α Orionis Conservation of mass for stellar wind mass-loss υ ( ) 0 t d & 4πR dt W W Const R S Solve for R s W 4 & π I S I S dot 3x0-6 solar masses per year U IS ~ 5 km s - U W 7 km s - R s distance x angle distance00 pc, angle7 arcmin n IS cm -3 0x too big 3

ρ u = u. (1) w z will become certain time, and at a certain point in space, the value of

ρ u = u. (1) w z will become certain time, and at a certain point in space, the value of THE CONDITIONS NECESSARY FOR DISCONTINUOUS MOTION IN GASES G I Taylor Proceedings of the Royal Society A vol LXXXIV (90) pp 37-377 The possibility of the propagation of a srface of discontinity in a gas