Acceleration mechanisms Stochastic: - collisions among particles/clouds Systematic: - H field compression + scattering/diffusion - shocks - e-m processes (e.g. Low frequency-large amplitude waves in pulsar magnetosphere) Daniele Dallacasa – Radiative Processes and MHD Section -5 : Acceleration Mechanisms
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Acceleration mechanisms - INAFddallaca/P-Rad_5.pdf · Fermi's Acceleration: basic process Proposed by E. Fermi in 1954. Let's consider a charged particle, given that electrons are
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Acceleration mechanisms
Stochastic: collisions among particles/clouds
Systematic: H field compression + scattering/diffusion shocks
Proposed by E. Fermi in 1954.Let's consider a charged particle, given that electrons are known to achieve very high energies (e.g. cosmic rays, synchrotron radiation).
Electrostatic fields cannot survive given the enormous conducibility. There is a magnetic field in the cloud only. However, when the charge moves into the moving cloud (seen from the observer's frame) it ''feels'' an electric field as well
is the field produced by the moving cloud, which ismoving at a speed u while the electron moves at v
v u
After a scalar product with the charge velocity we get
(in fact the second term is zero!)
This means that the energy of the electron change in casethe Lorentz force is active, and this requires that the magnetizedcloud is in motion
Type I collision
Type II collision
v uu
(Type II)
(Type I)
Fermi's Acceleration: basic process 3
Elastic collision between a cloud and a particle with |v | ≫ | u |: conservation of E and p
In terms of particle energy (')
f I =vul
f II =v−ul
⟨
t⟩F
= f I I f IIII = 4uv
2ul
=8vl uv
2
=F
t = o etF F ≈
lv2u2
F ≈ 1011 yr
Fermi's Acceleration: basic process 4
Type I interactions happen more often than Type II
A factor 2 is more appropriate than 8 (valid for head on collisions only)If we integrate in time we obtain:
once the particle is accelerated at the required velocity, then should be able to leave the region where acceleration takes place. Namely the confining time c should be of the order of (or slightly larger than) the acceleration time F
For initial velocities of ~ 10 km/s and the cloud size and number density (distance 10100 pc) relativistic velocities are achieved at
F ≈ 105 yr
k = o k
Nk = No pk
ln Nk/No
ln k/o=
ln p
ln = m
Nk= N
ok
o m
N d = cost −1md
Fermi's Acceleration: spectral shape
In SNR, “clouds” have higher velocities (thousands km/s) and l is small (0.1 pc) and
Spectrum of Fermi's acceleration processes:
k = # of collisions = energy in(de)crease per collisionp = probability to remain within the acceleration region
i.e. power – law energy distribution
Crab nebula (1054)
Observations of radio supernovae
The observations shown aside (images on thesame scale!) require that efficient particleacceleration takes place on time scales as shortas a few weeks
unperturbedv=0
v2
v1accelerating particle
(v)
Shock waves and Fermi's collisions(1)
if a perturbation moves at a speed exceeding cs , a discontinuity is created
region of particles to be accelerated is moving at strong shock is moving at
if particles cross several times the shock front before they gain enough energy to leave the acceleration region
cs=kTmH
sound speed for anidealgas noH field
v2=3/4v1
v2v1
Shock waves and Fermi's collisions (2)
the combination of the two velocities allow a particle to have Fermi – Itype collisions in a row (and rebounds with unperturbed clouds at rest).
the occurrence between collisions is
and the energy gain in time is
v2=3 /4v1
unperturbedv=0
v2
v1accelerating particle
(v)
1=0
2≈2v2v=32v1v
f≈ v2l
ddt
≈ 32v1vv2l
≈ 34v1l =
F
c ≈l
v1confining time
=1F
c
Shock waves and Fermi's collisions (3): an example. Cas A
2720 Jy at 1 GHz. The SN occurred at a distance of approximately 11,000 ly away. The expanding cloud ofmaterial left over from the supernova is now approximately 10 ly across. Despite its radio brilliance, however, it is extremely faint optically, and is only visible on longexposure photographs. It is believed that first light from the stellar explosion reached Earth approximately 300 years ago but there are no historical records of any sightings of the progenitor supernova, probably due to interstellar dust absorbing optical wavelength radiation before it reached Earth (although it is possible that it was recorded as a sixthmagnitude star by John Flamsteed on August 16, 1680
It is known that the expansion shell has a temperature of around 50 million degrees Fahrenheit (30 megakelvins), and is travelling at more than ten million miles per hour (4 Mm/s).
A false color image composited of data from three sources. Red is infrared data from the Spitzer Space Telescope,orange is visible data from the Hubble SpaceT, and blue and green are data from the Chandra Xray Observatory
Accelerazione di Betatrone:Sites like “magnetic traps” there is transfer of energy ||H is compressed || H expands ||
no net energy change,
unless..... random interactions redistribute the energy increase at (H+H) to other particles
HH
Ht
1 2 3
HH
Ht
1 2 3
{P ∥
2 =13P1
2
P ⊥
2=23P1
2}At location (2){
P ∥2 =
13P1
2
P ⊥2 =
23P1
2HHH
Ptot2 =P2
2=13P1212 HH
H }At location (1+)
HH
Ht
1 2 3
At location (2+){P ∥
2=
13P2
2=
19P1
212 HHH
P ⊥
2=
23P2
2=
29P1
212 HHH
Ptot2=P2
2=13P1
212 HHH }
{P ∥
2=
13P22=
19P1212 HH
H P ⊥
2=
23P22=
29P1
212 HHH H
HH
Ptot2=P3
2=19P1
212 HHH 12 H
HH }At location (3 )
In 3 again the particle is at the same initial H field (as in 1)and the total momentum change is
and in terms of energy:
p2 = p32 − p1
2 =29p12 HH H H
H 2
≈29p12 H
H 2
= p2/2m
≈29HH
ddt
≈
tH≈
29
dln H
dt
time for CRs
Em wave propagation in a plasma
Let's consider an astrophysical plasma, composed by ionized gas which is, however, neutral as a whole.Maxwell equations are defined for vacuum, but can be adapted toa plasma if we consider charge and current densities
e and j
Let's consider the dielectric constant:
Pulsation of incoming radiation
Pulsation of bound electrons
Free electron number densityBound electron number density
Pulsation of free electrons (=0)
In the radio domain, ω < ω
i and
can be neglected
plasma frequency
We define the refraction index:
where the plasma frequency vp has been defined
only waves with v > vp can travel across the region, whilethose with v < vp are reflected (nr becomes imaginary)
Below the plasma cutoff frequency there is no propagation
In the ionosphere: ne ~ 106 cm3 implies vp ~ 107 Hz
In the interstellar medium: ne ~ 10 3 10 cm 3 implies v
p ~ 3 ∙ 102 3 ∙103 Hz
nr≡ r ≃ 1−4e2
me
ne
2=1− p
2
p= e2ne
me
= 9.1⋅103 ne Hz
Wave propagation
the em wave travels with group velocity
The time necessary to travel from A to B at a given frequency is
A L B
(for v >> vp )
In case it is possible to detect this effect in a given particular case, then
it becomes feasible to directly measure ne along the LOS
Dispersion Measure
Observing at two different frequencies, the arrival time will be different!
Dispersion Measure a direct measurement
The slope of the pulse arrivaltime .vs. frequency provides a measure of
Distances, however, are difficultto determine, except in a few lucky cases, like globular clusters
D.M. = ∫0
Lne dl
Faraday Rotation
Propagation effect arising from an “external” magneticfield H which causes an anisotropic transmission.Let's consider what happens along the field direction:
Let's us assume that the propagating em wave is polarized and sinusoidalas a superposition between a LCP and a RCP components.
where + is for RCP and – is for LCP. The dielectric constant is no longer a scalar and becomes a tensor: the “two” modes have different refraction index
nr R ,L = 1−p
2
11±p /cos
where is the angle between the direction of em wave propagation and H
Michael Faraday (17911867)
E t = E o e−i t 1±2
Faraday Rotation
Along the field direction
and then in equation ** we get as a solution
which provides a dielectric constant
and therefore the propagation speed of the two orthogonal modes are different, originating a shift in their relative phase, which implies a rotation of the polarization vector
Bo = Bo 3
v t = −ie
me ±H E t
R ,L = 1−p
2
±H
How to measure RM:
Polarization sensitive observations provide the measurement of m and at various discrete frequencies (wavelengths).
●Plot 2 and ( n) ●get the slope of the best fit line●obtain RM
blue points represents the observations, while red points are for the nambiguities
slope provides RM
In case the “Faraday screen” is spatially resolved, the net effect is justa rotation of the linear polarization vector in a given direction
The rotation measure determines the magnetic field along the LOS weighted on the electron density ne and if also the D.M. is available
(radians)
⟨H ∥ ⟩ ∝R.M.D.M.
∝∫neH ∥ dl
∫nedl
Examples of RM and FR
RM needs various frequencies to be measured
Depolarization may take place
RM may be very different in various locations ofthe same radio source > local to the rsource
Examples of RM and FR (2)
Stratified RM structure (implications on H and ne)