Eline Tolstoyhttp://www.astro.rug.nl/~etolstoy/astroa07/

9. SynchrotronRadiation

Radiative Processesin Astrophysics

relativistic terms, and simplifications for very highvelocities are used very commonly.

taylor approx.

binomial theorem

wheni.e., ultra-relativistic case

Useful reminders

Synchrotron Radiation (magneto-bremsstrahlung)

Emission by ultra-relativistic electrons spiraling around magnetic field lines

Space is full of magnetic fields

10-3

0.3

1012

108

104

1

10-6

Field strength (gauss)location

interstellar medium

stellar atmosphere

Supermassive Black Hole

White Dwarf

Neutron star

this room

Crab Nebula

1 gauss (G) = 10-4 tesla (T)1 tesla (T) = 1 Wb m-2

typically very weak magnetic fields, but there is a plentifulsupply of relativistic electrons in low density environments

Equations of MotionA charged particle moving in a magnetic field radiates energy. At non-relativistic velocities, this is cyclotron radiation and at relativistic velocitiessynchrotron radiation.The relativistic form of the equation of motion of a particle in a magnetic fieldis given by the Lorentz four-force:

As the force on the particle is perpendicular to the motion, the magnetic fielddoes 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 magneticfield lines. For motion perpendicular to the fields, ! = "/2.

The combination of circular motion and uniform motion along the field is ahelical motion of the particle

For an electron:

In ISM typical B~3 x 10-6 gauss, # = 1

Cosmic ray electrons, # = 103, $B<< 1Hz

Since the velocity and accelerationare perpendicular

Beaming means that the emitted radiation appears to beconcentrated in a narrow cone, and an observer will see radiationfrom the particle only for a small fraction ~1/# of it’s orbit, which iswhen the particle is moving almost directly towards the observer andconsequently there is a big doppler effect

this is an essential feature of synchrotron radiation

the observer will see a pulse of radiationconfined to a time interval muchsmaller than the gyration period. Thespectrum will be spread over a muchbroader region than $B/2"

let’s remember beaming...

1 2 the leading edge has meanwhile propagated adistance 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 leading edge of the pulse is emitted as the particleenters the active zone (pt 1), and the trailing edge isemitted time ~1/(#$B) later as the particle leaves theactive zone (pt 2).

Radiation pulse

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 thevector potential A and the scalar potential ( have different retarded times atdifferent parts of the electron’s orbit makes the effective charge distributiondifferent 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 seriesof pulses with P=2"/$B. However since the electron’s guiding centre ismoving with velocity vcos& along the field line, and since the motion has acomponent projected toward the observer v2cos2& there is a doppler compressionof the pulse period.The pulses are spaced apart bya distance %s:

The observed period:

Pulse widthThe width of the pulse %t’ is determinedby the fraction of the gyromagneticperiod P that the electron is radiatingtoward the observer.

This pulse is subject to a Doppler compression since the particle isinstantaneously moving directly toward the observer with velocity v.

Putting it together

to get the spectrum we just take the Fourier transform of the pulse train

observable radio spectrum of cosmic ray electrons

relativistic motion has boosted the frequency by factor 108

high harmonic of gyro-frequency

i.e. 1012 th harmonicfor #~104

bandwidth

Spectrum

Total Power Radiated

acceleration is perpendicular to the velocity (a|| = 0)

for an isotropic distribution of velocities it is necessary to average over allangles for a given speed ', given ! is the pitch-angle between field & velocity:

this sets an upper limit to the electron energy as a function of time since theelectrons were injected. Even if the electrons were infinitely energetic they willhave cooled to

after time t, and electrons of lower initial energy will have E < Emax

Electrons in a plasma emitting synchrotron radiation cool down. The timescale for this to occur is given by the energy of the electrons divided by the rateat which they are radiating away their energy. The energy E = #mc2 so

the half-life of a synchrotron emitting electron

Synchrotron Loss Time (cooling):

typical cooling times

10-19sec

10-11sec

10-3sec

5days

1010yrs

tcool

1012

108

104

1

10-6

TypicalB

(gauss)location

interstellarmediumstellar

atmosphereSuper-massive

black hole

white dwarf

neutron star

Power Law Energy DistributionIn a wide range of astrophysical applications, the energy spectrum of relativisticelectrons is a power-law as might be produced by a stochastic accelerationmechanism.A good example is the Fermi mechanism which operates in supernovae remnants:electrons scatter off turbulent magnetic ``bubbles’’ and are pushed towardsequipartition but before they can achieve statistical equilibrium they escape theremnant around energies of 1015 eV. The resulting energy distribution:

Where p is the spectral index (~2.5 for cosmic rays). To compute the emissivityor the emission coefficient we assume (1) uniform magnetic field (2) power lawenergy distribution (3) isotropic velocities

Given the frequency spectrum for electrons of a given energy:

Tricky integralGet a good approximation by assuming that all the electrons radiate at theircritical 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 anumerical 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 thespectrum no matter how high the upper limit E2 is

we can derive a lot about a spectrum simply using the fact that the electric fieldis a function of & only through #& (beaming & ~ 1/#)

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 onany physical parameters (except t) - but we can still derive the generaldependence of the spectrum on $.

the fourier transform of the E-field is:

where & is the polar angle about the direction of motion (beam)

the spectrum

definition - energy/unit freq/ unit solid angle

using definition (from part I)

can show that:

integrating over solid angle anddividing by orbital period (bothindependent 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 theintegral over $ to the previous result for P

we do not know what is until we specify F(x), but we canassume it is a non-dimensional arbitrary value, and still determine C.

thus, for high relativistic case the power per unit freq emittedby each electron is

from and

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

the total power radiated by per unit volume per unit freq by such adistribution is given by the integral over times the singleparticle radiation formula over all energies

often the number density of particles with energies between E and E+dE(or and ) can be expressed:

change variablesand note

the limits on the integral correspond to the limits and dependon . However if the limits are sufficiently wide andand the integral is approx. constant, and so

this means that the spectral index s is related to the particle distributionindex p, by

when electrons are moving at velocities close to the speedof light two effects alter the nature of the radiation

1. the radiation is beamed

an electron moving with Lorentz factor towards anobserver emits radiation into a cone, of opening angle which means an observer will only seeradiation from a small portion of the orbit when thecone is pointed towards us - a pulse of radiation whichbecomes shorter for more energetic electrons.

2. the pulse is foreshortenedfor an electron moving at v~c a photon emitted at theend of the pulse almost catches up with the photon fromthe start.