November 1
Monday, 7: Synchrotron Radiation for Beginners
An accelerated electron emits electromagnetic radiation. The most effective way to accelerate an electron is to use electromagnetic forces. Since electrons have mass, they can also be accelerated gravitationally. However, electrons are seldom in freefall for long. In addition, gravitational accelerations are frequently small. An acceleration g = 980 cm s-2 makes an electron radiate with a power 2e2 g 2 = 5.5 10-45 erg s-1 . (1) P = 3c3 This amount of power would be produced if the electron passed a proton at a distance 1/2 e2 b= = 510 cm . (2) me g Thus, if a free electron is falling near the Earth's surface, if there's a proton within five meters, the electrostatic acceleration will be greater than the gravitational acceleration.1 An electron experiences an electromagnetic force if it passes a positively charged ion; the light it emits in this case is bremsstrahlung. In the limit that the relative velocity v of the electron and ion is small (v c), the electron experiences a pure electric force directed toward the ion. In the case of relativistic bremsstrahlung, the electron experiences a mixture of electric and magnetic fields in its instantaneous rest frame as the ion zips past. There exist circumstances in which the electron experiences a purely magnetic force. Suppose, in some inertial frame of reference, there exists a uniform magnetic flux density B, with a negligibly small electric field strength E. In the universe, it is not difficult to find magnetic fields, ranging in strength from the microgauss fields of intergalactic space to the teragauss fields near the most highly magnetized neutron stars. They aren't usually uniform in strength, but the constant B approximation is a useful place to start. If the velocity of the electron is small, then (as we've seen in Problem Set 2) the electron has a circular orbit in a plane perpendicular to B, with
1
This is just another way of stating that gravity is a pathetically feeble force.
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angular frequency cyc = B Be = 1.8 107 s-1 me c 1 gauss . (3)
The radiation produced in this non-relativistic case is called cyclotron radiation. In the limit v c, the cyclotron radiation is monochromatic, with frequency = cyc /(2). Thus, the Earth's magnetic field, with B 0.5 gauss, would produce radio waves with 1 MHz. Producing visible light by the cyclotron process would require a much higher magnetic flux density: B 2 108 gauss. The power produced by cyclotron radiation is Pcyc
4 2e2 cyc r2 2 2 vcyc = r0 c = 3 3c 3 c 2
B2
2
(4) , (5)
= 1.6 10
-15
erg s
-1
2 cyc
B 1 gauss
where vcyc = cyc c = cyc r is the orbital velocity of the electron in the magnetic field B. For a relativistic electron moving through a magnetic field, the emitted radiation becomes more complicated, and thus more interesting. As we noted just a week ago, in an inertial frame where a charged particle is at highly relativistic speeds, with a Lorentz factor = (1 - v 2 /c2 )-1/2 1, (6)
the radiation from the charged particle is strongly beamed in the particle's direction of motion. The angular width of the beamed radiation is 1/. Not only is the direction of radiation altered in the relativistic limit, but so is the total power radiated and the spectrum of radiation. To begin, consider a charged particle of mass m and charge q in a uniform magnetic field B. There is no electric field: E = 0. The velocity of the particle, v = c, is not necessarily small compared to the speed of light. The equation of motion, in its correct relativistic form, is d v q (mv) = q(E + B) = v B . dt c c The rate at which work is done on the charged particle is d v (mc2 ) = qv (E + B) = 0 . dt c 2 (8) (7)
The constancy of the charged particle's energy implies that = constant. Thus, the velocity vector v is constant in length, although changing in direction, and the relativistic equation of motion can be written in the form q dv = vB . dt mc (9)
Since the acceleration of the charged particle is proportional to the cross product of v and B, it's convenient to separate v into a component parallel to B and a component perpendicular to B. If we look at v , the velocity component parallel to B, we find, from equation (9), dv =0, dt (10)
2 and thus v is constant. Since v = (v 2 + v )1/2 is also constant, it follows that v must also be constant. The equation of motion for v ,
q dv = v B , dt mc
(11)
then represents motion in a circle with a constant acceleration and a constant angular frequency qB . (12) B = mc Note that the frequency is lower by a factor 1/ than it would be in the non-relativistic limit. The combination of constant velocity parallel to B and circular motion perpendicular to B means that the net motion of the charged particle is helical (Figure 1), with a pitch angle given by tan = v /v . As the charged particle winds along its helix, it emits radiation at a rate P = 2q 2 4 2 a , 3c3 (13)
where I am using the relativistically correct equation, taking into account that the acceleration is entirely perpendicular to the charged particle's motion. The magnitude of the acceleration is a =
2 v qB = v B = ( c) r mc
.
(14)
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Figure 1: Helical motion of a charged particle in a magnetic field. This leads to a radiated power of P = 2q 2 4 2 2 q 2 B 2 2q 4 2 2 2 c 2 2 2 = B . 3c3 mc 3m2 c3 (15)
If the charged particle is an electron (and it usually is), we can write 2 2 2 P = r0 c 2 B 2 , 3 (16)
larger by a factor of 2 than the non-relativistic result. The electrons in a magnetic field usually have a wide range of pitch angles , so the power radiated by an individual electron, 2 2 P = r0 c 2 2 sin2 B 2 , 3 (17)
ranges from zero for electrons with = 0 (or v = 0) to a maximum for electrons with = /2 (or v = 0). If we assume that v has a probability distribution that is uniform over solid angle, then sin2 = 2/3, and 4 2 P = r0 c 2 2 B 2 . 9 (18)
The observed spectrum of synchrotron radiation is strongly affected by the fact that the radiation emitted by a relativistic electron is strongly beamed. Thus, an observer will only be able to see light from a highly relativistic electron during the part of its orbit when it's moving almost straight 4
toward the observer. Thus, once for every turn of the helix, which occurs with period tB 2/B , the observer will detect a brief burst of radiation, with duration t tB . Since the power observed as a function of time is nearly a delta function, doing a Fourier transform tells us that the power observed as a function of angular frequency should be nearly uniform, up to a critical frequency c 1/t. The more detailed argument of Rybicki and Lightman tells us that the duration of a pulse seen by a distant observer should be t 1 1 , 3 sin B (19)
where is the pitch angle.2 Thus, we expect radiation from a synchrotron electron to have a more-or-less uniform spectrum up to a cutoff frequency c eB 1 3 B sin 2 sin . t me c (20)
There is a notable difference in the spectra of (highly non-relativistic) cyclotron radiation and (highly relativistic) synchrotron radiation. The cyclotron radiation is all emitted at the cyclotron frequency cyc = eB/me c. The synchrotron radiation is emitted over a broad range of frequencies up to a limiting frequency c 2 cyc cyc . Thus, synchrotron radiation is a mechanism for extracting photons of high energy from a magnetic field of fixed B, and thus fixed cyclotron frequency cyc .
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Wednesday, November 9: Advanced Synchrotron Radiation
Suppose that an electron (charge -e, = mass = me ) with velocity v is in a region of constant magnetic flux density B. The velocity of the electron can also be expressed as the dimensionless velocity v/c, as the Lorentz factor (1 - 2 )-1/2 , or as the electron energy = me c2 . The electron
You might not have guessed -3 without doing the detailed analysis. A factor of comes from geometrical considerations; the beam is only pointed toward you for a fraction 1/ of the complete orbital period. A factor of -2 comes from the fact that the electron is moving toward the observer at highly relativistic speeds during the pulse, so the duration of the pulse is strongly blueshifted, by a factor 1 - v/c 2/ 2 .
-1 2
5
will move on a helix whose axis lies along the B vector (see Figure 1). As the electron is accelerated, it emits electromagnetic radiation with power 2 2 P = B 2 r0 c 2 2 sin2 = 2UB T c 2 2 sin2 , 3 (21)
2 where T = (8/3)r0 is the Thomson cross-section of the electron, and UB = B 2 /(8) is the energy density of the magnetic field. In the limit of an extremely relativistic electron, 1, and
P 2UB T c 2 sin2 , which becomes
(22)
4 P UB T c 2 (23) 3 when averaged over all angles, since sin2 = 2/3. From the discussion of the previous lecture, we expect the spectral distribution of the power to be roughly constant at a value dP P , d c (24)
up to a critical frequency c 2 cyc sin , and drop off rapidly at higher frequencies. I was prowling through various textbooks trying to find a brief yet clear derivation of the actual shape of dP/d. I was discouraged when Frank Shu stated (in The Physics of Astrophysics, Volume I ), "The formal manipulations required for synchrotron theory can get formidable."3 The full equations are laid out in Rybicki and Lightman. I will merely cite the results, then stride forward to apply them to something astrophysically interesting. Physicists, by convention, have defined 3 3 eB sin = 2 cyc sin . c 2 2 me c 2 (25)
In terms of this critical frequency c , the power spectrum of light from a highly relativistic synchrotron electron is 3 dP 3 e B sin = F (/c ) . (26) d 2 me c2
Malcolm Longair (High Energy Astrophysics, Volume II ), chimes in with "It is a very major undertaking to work out properly all the properties of synchrotron emission...".
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6
The synchrotron function F (x) is a dimensionless function which describes the shape of the synchrotron spectrum. The calculations in section 6.4 of Rybicki and Lightman reveal that F (x) = x
x
K5/3 (x )dx ,
(27)
where K5/3 is a modified Bessel function.4 If you are a true Bessel function fanatic, you may want to prove that F (x) x1/3 when x 1 (that is, when c ), and that F (x) x1/2 e-x when x 1, giving the desired cutoff when c . A plot of F (x) is shown in Figure 2. The synchrotron function
Figure 2: The synchrotron function F (x), shown as log F versus log(/c ). is skewed. The modal (most probable) value of x is xmod 0.29; the mean value of x is x 1.32. If you had a population of electrons with the identical pitch angle and Lorentz factor , you would expect the spectrum of light they emit to look like Figure 2, with dP dP = ne 1/3 (28) dV d d in the low frequency limit where c . The mean frequency of the emitted light would be = 1.32c = 2.0 2 cyc sin . (29) However, when we look at the spectrum of actual synchrotron emitting sources, such as the Crab Nebula (Figure 3), we do not see a 1/3 power
4
The factor of
3/(2) in equation (26) is then required for proper normalization.
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Figure 3: Spectrum of the Crab Nebula. (Note: what is plotted is F .) law for I . Frequently, the spectrum is reasonably well fit by a power law, I -s , over a wide range of frequencies, but we usually find s = -1/3. The fact that s = -1/3 can be easily explained: not all electrons have the same energy . In general, high energy electrons (large ) are less common than low energy electrons (smaller ). If you look, for instance, at the energy distribution for charged particles in cosmic rays, you find that a power law is a good fit: n( )d -p d , (30)
where n( )d is the number density of electrons with energies in the range + d . The fit is usual good over a wide range of energies: 1 < < 2 , 5 2 with 2 1 . Since = me c , a power-law distribution of energies implies a power-law distribution of Lorentz factors: n()d -p d , (31)
for 1 < 2 . To find the power radiated per unit volume per unit frequency, we need to compute 2 dP dP = (32) n() d . dV d d 1 Taking ...