Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
A Basic Course onSupernova Remnants
• Lecture #1– How do they look and how are observed?– Hydrodynamic evolution on shell-type SNRs
• Lecture #2– Microphysics in SNRs - shock acceleration– Non-thermal emission from SNRs
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Basic concepts of shocks• Quantities conserved across
the shock discontinuity– Mass– Momentum– Energy
• For a strong shock, i.e.the jump conditions are:
• Compression ratio (r=u1/u2):– 4, for a non relativistic fluid– 7, for a relativistic one
12
1122
22 pVpV 1
21112
2222 2/2/ wVVwVV
1122 VV
shock111 ,, up 222 ,, up
V
2111 Vp
21121212 1
2;
1
1;
1
1VpVV
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
More complex than this
• Collisionless shocks– Coulomb equilibration scale (order of parsecs)
But shocks are much sharper than that
– Even tiny magnetic fields are more effective(gyration radius)
– Free to escape along the field lines? Not in the presence fluctuations(e.g. MHD waves)
pcskm000,1cm1
4.24
1sh
1
30
2/5
Vn
T
TL
p
eeq
1
3 μG10km/s10km000,10U
BU
m
m
eB
mcr
pG
2
res||
2
/
cyclesafterradians1
1in/,If
BBr
N
rBBBB
G
G
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Thermal and non-thermal particles
• Naif view– Electrons & ions are shocked
independently
– Similar Vth, i.e. Te~(me/mp)Tp
• Anomalous electron heating, mediated by MHD waves?(Cargill & Papadopoulos 1988, + … )
• Possibly observed? (Ghavamian et a. 2007)
Using Balmer line profile,
Te & Tp derived independently
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Even more striking, evidence fornon-thermal, relativistic particles
• Radio synchrotron emission n SNRs• And even in X-rays, in a few of them
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
shock
X
flow speed
(in the shock reference frame)
Diffusive shock acceleration
• Fermi acceleration– Converging flows– Particle diffusion
(How possible, in acollisionless plasma?)
• Scattering on MHD waves
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
A test particle approach (Bell 1978) • Collision against a (N.R.) moving wall:
• Momentum after N cycles:
i.e.
213
2uu
v
pp
R
NRvv+2U = v(1+2U/v)
U
pp(1+2U/c)
0
1
21
3
21)( p
v
uuNp
N
i i
N
i ivuu
p
Np
121
0
1
3
2)(ln
(averaged over directions)
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Probability of having N cycles
• Return probability
• Probability of N cycles
222 2
12
vudvvu x
u
v
x
222 2
1
2
uvdvuv x
v
u
x
222 2
12
uvdvuv x
u
v
x
2
2
2ret /1
/1
vu
vu
flow
flowP
down
up
2
1 2
2
/1
/1
N
i i
i
vu
vuNP
N
i i
N
i i
i
vu
vu
vuNP
12
1 2
2 14
/1
/1ln2ln
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Compare the two formulas
from which
and finally the distribution
For r=4, σ=2. Spectral index 0.5 (as in radio!)
Diffusivity is fundamental for the process to take place, but does not appear explicitly
N
i ivuu
p
Np
121
0
1
3
2)(ln
N
i i
N
i i
i
vu
vu
vuNP
12
1 2
2 14
/1
/1ln2ln
pppp
Ppf rruuuu )1/()2()/()2( 2121)(
)/(3
0021
2
212
..ln3
lnuuu
p
ppPei
p
p
uu
upP
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
The convection-diffusion eq. • A different approach to the problem
• Heuristic explanation:– Advected flow– Diffusive flow
– Diffusion in momentum space
provided that
),()(
3
1),(),(),()( pxpf
px
xu
x
pxfpxpxfxu
x
),()( pxfxu
x
pxfpx
),(
),(
),(),()(
3
1),(
)(
3
1pxfp
ppxpf
x
xu
ppxpf
px
xu
px
xup
)(
3
1
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Solving the equation
• Boundary conditions
• Velocity profile:• Integrate between x=+∞ and x=- ∞
(now x has disappeared)
• Solution of linear equation:
),()(
3
1),(),(),()( pxpf
px
xu
x
pxfpxpxfxu
x
0;:0at
0;:at
22
11
x
fffx
x
fffx
xuux
u 21:jump
)()()( 21231
1122 ppfuupfupfu p
Bppfppdppfp
p
)()2()( 11
2
0
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
A cosmic-ray precursor
• In the unshocked medium
• Accelerated particle may reach, in front of the shock, a distance
Any effect on the pre-shock fluid ?
),()(
3
1),(),(),()( pxpf
px
xu
x
pxfpxpxfxu
x
0),(
),(),()(
x
pxfpxpxfxu
x
)(
),(
xu
px
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Dimensional quantities• Parallel mean free path
• Diffusion coefficient
• Perpendicular diffusion
(can be much lower than the parallel one)
Gr || 1;/)( -2res BBp
1
72
μG10pc10
B
m
m
eB
mcr
pG
eBmcv 3/3/ 3||||
2||
2|||| 1//1/ Gr
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Characteristic times• Acceleration time• Age• Synchrotron losses
• Loss-dominated regimenaturally locatedin the X-ray range
Independent of B strength
p
p p
pd
uuuu 02
2
1
1
21acc
3 22acc u
c
u
2/12/3
2
31
2
2
3
2/12/3
2
21
2
2
acc
mc
eB
e
mc
mc
eB
e
mc
mc
eB
u
c
mc
eB
u
c
syn
1
2
12
2cutoff km/s1000keV1.0
uu
e
mc
Diffusion must be efficientalso upstream !!
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
SN 1006 spectrum• Rather standard ( -0.6) power-law
spectrum in radio(-0.5 for a classical strong shock)
• Synchrotron X-rays below radio extrapolation
Common effect in SNRs (Reynolds and Keohane 1999)
• Electron energy distribution:
• Fit power-law + cutoff to spectrum:
“Rolloff frequency”
)/exp()( maxEEEEN se
))/(exp()( 2/1rolloff S
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Measures of rolloff frequency
• SN 1006 (Rothenflug et al 2004)
• Azimuthal depencence of the break
Truly loss limited? Changes in tacc? Varying η?
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Very sharp limbs in SN 1006
ASCA
Chandra
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
B from limb sharpness
Profiles of resolved non-thermal X-rayfilaments in the NE shell of SN 1006
(Bamba et al 2004)
Length scales 1” (0.01 pc) upstream 20” (0.19 pc) downstream
Consistent withB ~ 30 μG
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
A diagnostic diagram• Acceleration time
tacc = 270 yr• Derivation of
the diffusioncoefficients:u=8.9 1024 cm2s-1
d=4.2 1025 cm2s-1 (Us=2900 km s-1)to compare withBohm=(Emaxc/eB)/3
rolloff
tsync> tacc
> Bohm
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Acceleration times & energies
• (Theoretical) need for large fields
• The case of a perpendicular field
• BUT how to inject particles?(mean free path has tobe comparable with the shock width)
1
2
2
2acc
mc
eB
u
c
u
2||
2|||| 1//1/ Gr
1
2
2
22acc 1
mc
eB
u
c
u
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Not just test particles ?• (Indirect) evidences that cosmic-ray
component is dynamically relevant (ions)– Large magnetic field
• If synchrotron-losses regime• If interpretation of narrow filaments is correct
– Deviations from predicted fluid behaviour• RS closer to FS• Too low post-shock (ion) temperature
– Effects of a shock precursor
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Indirect tests on the CRs• Some “model-dependent” side effects of efficient
particle acceleration• Forward and reverse shock are closer, as effect of
the energy sink• HD instabilities behavior depends on the value of eff
(Decourchelle et al 2000)
(Blondin and Ellison 2001)
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
SNR 1E 0102.2-7219• Very young and bright SNR in the SMC• Expansion velocity (6000 km s-1, if linear expansion)
measured in optical (OIII spectra) and inX-rays (proper motions)
• Electron temperature~ 0.4-1.0 keV, whileexpected ion T ~ 45 keV
• Very small Te/Ti, or Timuch less than expected?Missing energy in CRs?
(Hughes et al 2000, Gaetz et al 2000)
Optical
Radio
X-rays
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Gamma-ray emissionA definitive way to measure the field?
• Measurement of gamma-ray emission, produced by the same electrons that emit X-ray synchrotron, would allow one to determine the value of B.
SynchrotronIC
Radio X-ray γ-ray
νFν
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
• On the other hand, there is another mechanism giving Gamma-ray emission– accelerated ions– p-p collisions– pion production– pion decay (gamma)
• Lower limit for B
• Need for “targets”
(molecular cloud?) • Efficiency in in accelerating ions?
(The origin of Cosmic rays)
(Ellison et al 2000)
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
A self-regulating model• If acceleration is efficient, cosmic-ray
precursor upstream• Generation of MHD waves, by streaming
instabilities• Turbulent amplification of upstream field• Effects on the diffusion coefficient• A smaller diffusion coefficient makes
further acceleration more efficientCLOSING THE LOOP
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Shock modificationDynamical effects of the
accelerated particles ontothe shock structure
(Drury and Voelk 1981)
•Intrinsically non linear
•Shock precursor
•Discontinuity (subshock)
•Larger overall compression factor
•Accelerated particle distribution is no longer a power-law
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Deviations from Power-Law• In modified shocks,
acc. particles withdifferent energiessee different shockcompression factors.Higher energy Longer mean free path Larger compress.factor Harder spectrum
• Concavity in particledistribution.
(also for electrons)
Standard PL
Thermal
Blasi Solution
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
The injection of electrons ?• Theory predicts (~ high) values of the
efficiency of shock acceleration of ions.• Little is known for electrons• Main uncertainty is about the injection
process for electrons– Shock thickness determined by the mfp of
ions (scattering on magnetic turbulence)– Electrons, if with lower T, have shorter mfps– Therefore for them more difficult to be
injected into the acceleration process
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
Optical emission in SN1006• “Pure Balmer” emission
in SN 1006
• Here metal lines are missing (while they dominate in recombination spectra)– Extremely metal deficient ?
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
“Non-radiative” emission• Emission from a radiative shock:
– Plasma is heated and strongly ionized– Then it efficiently cools and recombines– Lines from ions at various ionization levels
• In a “non-radiative” shock:– Cooling times much longer than SNR age– Once a species is ionized, recombination is
a very slow process
• WHY BALMER LINES ARE PRESENT ?
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
The role of neutral H• Scenario: shock in a partially neutral gas• Neutrals, not affected by the magnetic
field, freely enter the downstream region• Neutrals are subject to:
– Ionization (rad + coll) [LOST]– Excitation (rad + coll) Balmer narrow– Charge exchange (in excited lev.)Balmer broad
(Chevalier & Raymond 1978, Chevalier, Kirshner and Raymond 1980)
•Charge-exchange cross section is larger at lower vrel
•Fast neutral component more prominent in slower shocks
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
H-alpha profiles
(Hester, Raymond and Blair 1994)
(Kirshner, Winkler and Chevalier 1987)
Cygnus Loop
•FWHM of broad component (Ti !!)
•FWHM of narrow component
• (T 40,000 K – why not fully ionized?)
MEASURABLE QUANTITIES
•Intensity ratio
•Displacement (not if edge-on)
Rino Bandiera, Arcetri Obs., Firenze, Italy A Basic Course on SNRs
THE END