Radiative Processes in Astrophysics
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 Field strength (gauss) interstellar medium 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 non- relativistic 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
3 Cosmic ray electrons, # = 10 , $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 location B tcool (gauss) interstellar 10-6 1010yrs medium stellar 1 5days atmosphere Super-massive 104 10-3sec black hole white dwarf 108 10-11sec
neutron star 1012 10-19sec 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 the power per unit freq emitted by each electron is 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 and depend on . However if the limits are sufficiently wide and and the integral is approx. constant, and so 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.