Radiative Processes
in Astrophysics
9. Synchrotron
Radiation
Eline Tolstoy
http://www.astro.rug.nl/~etolstoy/astroa07/
Useful reminders
relativistic terms, and simplifications for very high
velocities are used very commonly.
when
i.e., ultra-relativistic case
binomial theorem
taylor approx.
Synchrotron Radiation
(magneto-bremsstrahlung)
Emission by ultra-relativistic electrons spiraling around magnetic field lines
Space is full of magnetic fields
typically very weak magnetic fields, but there is a plentiful
supply of relativistic electrons in low density environments
location
interstellar medium
Field strength (gauss)
10-6
stellar atmosphere
1
Supermassive Black Hole
104
White Dwarf
108
Neutron star
1012
this room
0.3
Crab Nebula
10-3
1 gauss (G) = 10-4 tesla (T)
1 tesla (T) = 1 Wb m-2
Equations of Motion
A charged particle moving in a magnetic field radiates energy. At nonrelativistic velocities, this is cyclotron radiation and at relativistic velocities
synchrotron radiation.
The relativistic form of the equation of motion of a particle in a magnetic field
is given by the Lorentz four-force:
As the force on the particle is perpendicular to the motion, the magnetic field
does no work on the particle, and so it’s speed is constant, i.e. |v| = constant.
The particle has constant speed v, but it’s direction may change. Thus:
Helical motion
r is the radius of orbit around the field lines, the ``radius of gyration'', and
! is the ``pitch angle'' or the inclination of the velocity vector to the magnetic
field lines. For motion perpendicular to the fields, ! = "/2.
The combination of circular motion and uniform motion along the field is a
helical motion of the particle
For an electron:
In ISM typical B~3 x 10-6 gauss, # = 1
Cosmic ray electrons, # = 103, $B<< 1Hz
let’s remember beaming...
Beaming means that the emitted radiation appears to be
concentrated in a narrow cone, and an observer will see radiation
from the particle only for a small fraction ~1/# of it’s orbit, which is
when the particle is moving almost directly towards the observer and
consequently there is a big doppler effect
the observer will see a pulse of radiation
confined to a time interval much
smaller than the gyration period. The
spectrum will be spread over a much
broader region than $B/2"
Since the velocity and acceleration
are perpendicular
this is an essential feature of synchrotron radiation
Radiation pulse
The leading edge of the pulse is emitted as the particle
enters the active zone (pt 1), and the trailing edge is
emitted time ~1/(#$B) later as the particle leaves the
active zone (pt 2).
1
2
the leading edge has meanwhile propagated a
distance c%t’ whereas the particle has moved v%t’
so it has almost kept up with the leading edge.
the interval between the reception of pulses is
the radiation emerges at frequency
Frequency of Gyrating Electrons
in the rest frame
At high energies, v~c, Doppler shifts (1-n·'), combined with the fact that the
vector potential A and the scalar potential ( have different retarded times at
different parts of the electron’s orbit makes the effective charge distribution
different from a simple rotating dipole, it becomes a superposition of dipole
($B), quadrapole (2$B), sextapole (3$B), etc...
Synchrotron Spectrum
If the orbit were purely circular (&="/2) then the observer would detect a series
of pulses with P=2"/$B. However since the electron’s guiding centre is
moving with velocity vcos& along the field line, and since the motion has a
component projected toward the observer v2cos2& there is a doppler compression
of the pulse period.
The pulses are spaced apart by
a distance %s:
The observed period:
Pulse width
The width of the pulse %t’ is determined
by the fraction of the gyromagnetic
period P that the electron is radiating
toward the observer.
This pulse is subject to a Doppler compression since the particle is
instantaneously moving directly toward the observer with velocity v.
Putting it together
Spectrum
to get the spectrum we just take the Fourier transform of the pulse train
bandwidth
high harmonic of gyro-frequency
i.e. 1012 th harmonic
for #~104
observable radio spectrum of cosmic ray electrons
relativistic motion has boosted the frequency by factor 108
Total Power Radiated
acceleration is perpendicular to the velocity (a|| = 0)
for an isotropic distribution of velocities it is necessary to average over all
angles for a given speed ', given ! is the pitch-angle between field & velocity:
Synchrotron Loss Time (cooling):
Electrons in a plasma emitting synchrotron radiation cool down. The time
scale for this to occur is given by the energy of the electrons divided by the rate
at which they are radiating away their energy. The energy E = #mc2 so
this sets an upper limit to the electron energy as a function of time since the
electrons were injected. Even if the electrons were infinitely energetic they will
have cooled to
after time t, and electrons of lower initial energy will have E < Emax
the half-life of a synchrotron emitting electron
typical cooling times
Typical
B
(gauss)
tcool
10-6
1010yrs
1
5days
104
10-3sec
white dwarf
108
10-11sec
neutron star
1012
10-19sec
location
interstellar
medium
stellar
atmosphere
Super-massive
black hole
Power Law Energy Distribution
In a wide range of astrophysical applications, the energy spectrum of relativistic
electrons is a power-law as might be produced by a stochastic acceleration
mechanism.
A good example is the Fermi mechanism which operates in supernovae remnants:
electrons scatter off turbulent magnetic ``bubbles’’ and are pushed towards
equipartition but before they can achieve statistical equilibrium they escape the
remnant around energies of 1015 eV. The resulting energy distribution:
Where p is the spectral index (~2.5 for cosmic rays). To compute the emissivity
or the emission coefficient we assume (1) uniform magnetic field (2) power law
energy distribution (3) isotropic velocities
Given the frequency spectrum for electrons of a given energy:
Tricky integral
Get a good approximation by assuming that all the electrons radiate at their
critical frequency, )c. Then, per unit solid angle:
Now substitute for E and dE in terms of ) an d).
After some reduction, one finds
This formula is approximate, but it differs from the exact expression by a
numerical factor, of order unity. In particular it has the correct spectral index
! = (1*p)/2. For cosmic ray electrons p~2.5, thus !=*0.75.
Radiation losses by the high energy particles will lead to an abrupt cutoff in the
spectrum no matter how high the upper limit E2 is
the spectrum
we can derive a lot about a spectrum simply using the fact that the electric field
is a function of & only through #& (beaming & ~ 1/#)
where & is the polar angle about the direction of motion (beam)
where t is the time measured in the observers frame, and the relation between &
and t is:
thus the time dependence of the E-field is:
we don’t yet know the constant of proportionality, which may depend on
any physical parameters (except t) - but we can still derive the general
dependence of the spectrum on $.
the fourier transform of the E-field is:
using definition (from part I)
definition - energy/unit freq/ unit
solid angle
can show that:
integrating over solid angle and
dividing by orbital period (both
independent of frequency)
where F is a dimensionless function, and C a constant of proportionality,
and T is the pulse duration.
can now evaluate C by comparing the total power evaluated by the
integral over $ to the previous result for P
we do not know what
is until we specify F(x), but we can
assume it is a non-dimensional arbitrary value, and still determine C.
from
and
thus, for high relativistic case
by each electron is
the power per unit freq emitted
Spectral Index
power-law electron distribution
no factor
(except in
)
often spectrum can be assumed to be a power law (for a frequency range).
in this case, define the spectral index as the constant, s:
e.g., Rayleigh-Jeans part of black-body has s = -2
can hold for the particle distribution law of relativistic electrons
often the number density of particles with energies between E and E+dE
(or and
) can be expressed:
the total power radiated by per unit volume per unit freq by such a
distribution is given by the integral over
times the single
particle radiation formula over all energies
change variables
and note
the limits on the integral correspond to the limits
on . However if the limits are sufficiently wide
and the integral is approx. constant, and so
and depend
and
this means that the spectral index s is related to the particle distribution
index p, by
when electrons are moving at velocities close to the speed
of light two effects alter the nature of the radiation
1. the radiation is beamed
an electron moving with Lorentz factor towards an
observer emits radiation into a cone, of opening angle
which means an observer will only see
radiation from a small portion of the orbit when the
cone is pointed towards us - a pulse of radiation which
becomes shorter for more energetic electrons.
2. the pulse is foreshortened
for an electron moving at v~c a photon emitted at the
end of the pulse almost catches up with the photon from
the start.