Физика космических лучей / HEA12

6Shock Waves in Astrophysics

Figure 4: Supernova shock waves - Example: SNR related to most recent supernova in our Galaxy: G1.9+0.3. Orange: X-rays (Chandra 2007), blue: Radio (VLA 1985). The di erence in size between the two images gives clear evidence for expansion, allowing the time since the original supernova explosion (about 140 years)

to be estimated [Credits: CXC/NASA].

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Shock=discontinuity moving through medium at speed larger than speed of sound (upstream)"

Figure 5: Non-relativistic shock wave in the reference frame of the un-shocked medium, v1 = 0 (lab.frame, left) and in reference frame where the surface of the discontinuity is at rest, vs = 0 (shock rest frame, right). The shock advances into the un-shocked medium at speed vs. In rest frame of the shock, upstream medium approaches it at speed v1 = vs. The shocked uid moves away from the shock front at speed v2 = 1v1= 2. The shocked uid thus approaches the un-shocked uid at speed vrel = v1 v2.

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7Fluid Dynamics

Conservation laws for a given volume xed in space:

)"Euler equations of uid dynamics (inviscid ows!)":

Mass per unit volume (continuity):

@@t + r ( ~v) = 0 with uid density, ~v uid velocity.

Momentum per unit volume:

d~vdt = @~v@t + (~v r)~v = rP

with F =external force (density)= 0, and rP =force due to pressure gradient.

Energy per unit volume:

with u = U internal energy per unit volume.

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8One-dimensional Shock & Rakine-Hugoniot Jump Conditions

Consider 1D (plane) steady shock, without magnetic and gravitational eld: Apply conservation laws to reference frame in which discontinuity is at rest:

Mass/Continuity:

Momentum (identity: r(~v ~v) = 2(~v r)~v and with continuity eq.):

where red expressions ="Rankine-Hugoniot jump conditions" are obtained by integrating over discontinuity of the shock front.

) Three equations for three unknowns!

Noting that U = P=[ ( 1)] from ideal gas law, with U internal energy per unit mass.

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9Solutions of conservations laws (plane steady shock)

= Cp=Cv ratio of speci c heats ( = 5=3 for mono-atomic gas).

2.Mach number Mi := vi=cs, i.e. M1 = ratio of shock speed to upstream sound speed, ) Pi + ivi2 = Pi + i c2i (vi2=c2i ) = Pi(1 + Mi2) etc.

Solutions in terms of upstream Mach number:

Density and velocity discontinuity:

Together, these imply temperature discontinuity (with T / P= ):

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for mono-atomic gas = 5=3, thus

r = (8=3) : (2=3) = 4

Density increase:

2 = r 1 = 4 1

Velocity decrease:

v2 = v1=r = v1=4

Temperature increase (M1 1), noting c2s = (P= ) = kT=mp:

(with P = nRT=V , P= = kT=m, k, Boltzmann constant, m mass of single particle):

i.e., for v1 = 1000 km/s, T2 107 K.

) Shocks convert bulk kinetic energy of upstream medium to thermal energy downstream.

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10Collisionless shocks

Rankine-Hugoniot conditions give macroscopic description of shock.

Shock microphysics (lab.): atomic collisions produce heat and increase downstream temperature, lower bulk velocity downstream.

Shock microphysics (space): In ISM, density too low, collisional mean free path much larger than system size ) collisionless shocks.

Energy exchange through electromagnetic interactions (wave-particle interactions/re ection). Typical shock scale is thermal particle Larmor radius (a few 100 to 1000 km).

Particles carrying more energy ("injection problem") have larger Larmor radius, and see shock as discontinuity.

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11 Di usive shock acceleration

Principle:

Particle directions randomized/isotropized by MHD waves upstream and downstream of shock (e cient pitch-angle scattering).

Waves generated:

{downstream by shock-induced turbulence (clumpy ISM, instabilities...).

{upstream by energetic particles themselves (cosmic-ray streaming).

In whichever frame, an energetic particle approaching the shock experiences a magnetic wall with velocity v = jv1 v2j = v1 (1 1r ).

"Collisions" are always head-on ) 1st order Fermi acceleration.

Particles have a nite probability to escape downstream.

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Figure 6: Di usive shock acceleration: Energetic particles get isotropized in the downstream and upstream rest frame, respectively, by scattering o waves quasi-embedded in the background plasma (2nd order Fermi e ects assumed being negligible). The situation is symmetrical: On each crossing of the shock front, they essentially experience head-on collisions with v = jv1 v2j, leading to 1st order Fermi acceleration.

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12 Escape probability

Energetic particles have nite probability to escape towards downstream:

Flux of energetic particles (density n0) crossing planar shock surface in one direction (1 ! 2; only those with projected cos < 0 are crossing):

Since shock is assumed to be non-relativistic, only small fraction of energetic particle is lost per cycle!

If we inject N0 particles at time t = 0, there are N0 (1 Pesc)n particles left after cycle n.

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