Harvard Astronomy 201b

ARTICLE: The Acceleration of Cosmic Rays in Shock Fronts (Bell 1978)

In Journal Club 2013 on April 9, 2013 at 5:20 am
Cartoon picture of Bell's diffusive shock acceleration, courtesy of Dr. Mark Pulupa's space physics illustration: http://sprg.ssl.berkeley.edu/~pulupa/illustrations/

Cartoon picture of Bell’s diffusive shock acceleration, courtesy of Dr. Mark Pulupa’s space physics illustration: http://sprg.ssl.berkeley.edu/~pulupa/illustrations/

Summary By Pierre Christian

Disclaimer

WordPress does not do equations very well!!! For ease of viewing, I attach here the PDF file for this paper review. It is identical in content with the one on WordPress, but since it is generated via Latex, the equations are much easier to read. On the other hand, the WordPress version has prettier pictures – take your pick… Here’s the Latex-ed version: Latex-ed Version

Here is the handout for said Journal Club: Handout

Bell’s paper can be accessed here: The Acceleration of Cosmic Rays in Shock Fronts

Introduction: Cosmic Rays in Context

In order to appreciate Bell’s acceleration mechanism, it is important for us to first learn some background information about the cosmic rays themselves. These are highly energetic particles, mostly protons (with a dash of heavier elements) that are so energetic that at the highest energies one proton has the energy equivalent to the fastest tennis serves. What can produce such energetic particles?

Our first clue comes from the composition of these cosmic rays. First, there is no electrically neutral cosmic ray particle. Electromagnetic forces must therefore be important in their production. Second, there is an overabundance of iron and other heavy elements in the cosmic ray population compared to the typical ISM composition (Drury, 2012).

The favored cosmic ray production sites are supernova remnants (Drury, 2012). Why not extra-galactic sources? Well, it has been shown that cosmic ray intensity is higher in the inner region of the Milky Way and goes down as we move radially to the outer disk. In addition, cosmic ray intensity is also shown to be less in the Magellanic clouds compared to the Milky Way. Additionally, strong shocks amplify magnetic fields to large proportions, and due to stellar nucleosynthesis, there is an overabundance of iron and heavy elements. I should also mention that a recent Fermi result conclusively proves that supernova remnants produce cosmic rays.

The observed cosmic ray spectrum is given in the figure below. We can see that the bulk of cosmic rays follow a broken power law. In particular, there are two breaks in this power law, one at 3 \times 10^{15} eV, called the ‘knee’ and one at 3 \times 10^{18} eV, called the ‘ankle’. Bell’s contribution is the derivation of a cosmic ray power law spectrum via acceleration in the shocks of supernova remnants. Bell’s model did not explain why the ‘ankle’ and the ‘knee’ exist, and to my knowledge the reason for their presence is still an open question. One explanation is that galactic accelerators cannot efficiently produce cosmic rays to arbitrarily high energies. The knee marks the point where galactic accelerators reach their energetic limits. The ankle marks the point where the galactic cosmic ray intensity falls below the intensity of cosmic rays from extragalactic sources, the so called ultra high-energy (UHE) cosmic rays (Swordy, 2001).

Observed cosmic ray spectrum from many experiments. Originally published by Swordy (2001), and modified by Dr. William Hanlon of the University of Utah (http://www.physics.utah.edu/~whanlon/spectrum.html). This image shows the three powerlaw regimes and the corresponding two breaks: the knee at 3 x 10^{15} eV and the ankle at 3 x 10^{18} eV.

Observed cosmic ray spectrum from many experiments. Originally published by Swordy (2001), and modified by Dr. William Hanlon of the University of Utah (http://www.physics.utah.edu/~whanlon/spectrum.html). This image shows the three power law regimes and the corresponding two breaks: the knee at 3 x 10^{15} eV and the ankle at 3 x 10^{18} eV.

Bell’s Big Ideas

Bell saw the potential of using the large bulk kinetic energy produced in objects such as a supernova remnant to power the acceleration of particles to cosmic ray energies. In order to harness the energy in bulk fluid motions, Bell needed a mechanism to transfer this energy to individual particles. Cited in Bell’s paper, Jokipii and Fisk had in the late sixties and early seventies deduced a mechanism using shocks as a method for accelerating particles. Bell modified and perfected Jokipii’s and Fisk’s mechanisms into something that could be applicable in the acceleration of cosmic ray particles in strong magnetized shocks.

General Idea
The general idea of Bell’s mechanism is that a particle with gyroradius, much larger than the shock’s width can move between the upstream (region the shock is moving into) and downstream regions (region that the shock had already passed in). Every crossing increases the particle’s energy slightly, and after many crossings, the particle is accelerated up to cosmic ray energies. This requires the particle to be confined within a certain region around the shock; there must be a mechanism that keeps the particle bouncing around the shock.

Confining
As a shock wave wades through the ISM, it produces turbulence in its wake. In the frame of the shock, the bulk kinetic energy of the upstream plasma is processed into smaller scales (turbulence and thermal) in the downstream region. This turbulence cascades down to smaller scales (via a magnetic Kolmogorov spectrum) before being dissipated far downstream. This magnetic turbulence can scatter fast moving charged particles and deflect them. In particular, a particle trying to escape downstream can be deflected by this turbulence back upstream.

If a charged particle saw some magnetic inhomogeneity (a random fluctuation) in the plasma, it is possible for the particle to scatter off these inhomogeneities. It is also known that much like how a particle moving in air produces sound waves, fast moving charged particles in an MHD fluid can produce Alfven waves. In the upstream region, energetic particles will excite Alfven waves. Now, a particle moving much faster than the Alfven speed will essentially see these waves as magnetostatic inhomogeneities. Furthermore, there is a resonant reaction that can scatter a particle with Larmor radius comparable to the wavelength of the inhomogeneities. In the upstream region of the shock, energetic particles will then scatter off Alfven waves that they themselves generate, deflecting them back into the downstream region.

At this point I would like to point out that the Alfven wave scatterings are elastic. Bell used confusing wording that made it seem that Alfven waves decrease the energy of the upstream particles enough so that they get overtaken by the shock. This is not true; magnetostatic interaction necessarily conserve energy because static magnetic fields can do no work (force is always perpendicular to velocity). What happens is that, in the frame of the shock, particles scatter with Alfven waves through a multitude of small angle scatterings, so that the angle between the particle motion and the background magnetic field undergoes a random walk. After several steps in this random walk, some particles will be scattered backwards towards the downstream direction.

Energy Source
Bell emphasizes that there are two different scattering centers in the process. The downstream scattering centers are the turbulence excited by the shock while the upstream scattering centers are the Alfven waves produced by the energetic particles. What is important is that these two waves are moving at different speeds in the frame of the shock. The energy powering the particle acceleration is exactly harnessed from the speed differential between the upstream and downstream scattering centers. Look at the ‘Intuitive Derivation of Equation (4)’ section for more details on this process.

The Power Law Spectra of Cosmic Rays

Bell found the differential energy spectrum of cosmic rays to be:
N(E) dE = \frac{\mu - 1} {E_0} \left( \frac{E} {E_0} \right) ^{-\mu} dE \; ,
where E_0 is the energy of the particle as it is injected in the acceleration process and \mu is defined to be:
\mu = \frac{2 u_2 + u_1} {u_1 - u_2} + O\left(\frac{u_1 - u_2} {c} \right) \; .
Where O\left(\frac{u_1 - u_2} {c} \right) denotes terms of order \left(\frac{u_1 - u_2} {c} \right) and higher.
Now, u_1 and u_2 are respectively the velocity of the scattering centers. Each are given by:
u_1 = v_s - v_A \; ,
u_2 = \frac{v_s} {\chi} + v_W \; ,
where v_A is the Alfven speed, v_W the mean velocity of scattering centers downstream, v_s the shock velocity, and \chi the factor by which the gas is compressed at the shock (\chi = 4 for high Mach number shocks obeying Rankine-Hugoniot jump conditions). Combining these equations we get:
\mu = \frac{ (2 + \chi) + \chi(2 v_W / v_s - 1/M_A)} {(\chi - 1) - \chi (v_W/v_s + 1/M_A)} \; ,
where M_A is the Alfven Mach number of the shock. In particular, for typical shock conditions this gives us a slope of \sim -2 to \sim -2.5, in agreement with the observed power law of the cosmic ray spectrum.

Damping of Alfven Waves

Bell notices that Alfven waves are damped by two important processes: collision of charged particles with neutral particles and the sound cascade. The first effect can be explained by noting that the magnetic field is flux frozen only to the charged particles in the fluid. If there is a significant amount of neutral particles, the charged particles that are ‘waving’ along with the Alfven waves can collide and scatter with these neutral particles (because the neutral particles do not ‘wave’ along with the magnetic field). This transfers energy from the hydromagnetic Alfven waves to heating the fluid, effectively damping the Alfven waves. Bell notes that the spectral index he derived is only good up to an energy cutoff, above which the spectrum becomes more steep. This energy cutoff is:
\frac{E_{crit}}{\rm{GeV}} = 0.07 \left( \frac{100 u_1} {c} \right)^{4/3} \left(\frac{n_e}{cm^{-3}} \right)^{-1/3}\left(\frac{n_H}{cm^{-3}} \right)^{-2/3} \left( \frac{f(0, p) - f_0(p)} {f_{gal}(p)} \right)^{2/3} \; .
Note that f_0(p) , the background particle distribution, is exactly f_{gal}(p) (the Galactic particle distribution) if the shock is moving through undisturbed interstellar gas. This condition is false for objects that generates multiple shock fronts. Also note that due to compression in the shockwave, f(0,p) can be many times the galactic value.

The second damping mechanism is due to the sound cascade. Alfven waves can interact with and lose energy to magnetosonic waves of lower wavelengths. This requires the sound speed to be less than the Alfven speed:
T < 3100 \left( \frac{B} {3 \mu G} \right)^2 \left( \frac{n_e} {cm^{-3}} \right) ^{-1} K \; .
However, this damping mechanism does not completely remove the Alfven waves, but merely limits the wave intensity. If this process is important, it will allow particles upstream to travel further from the shock before crossing back downstream. Bell does not believe that this will hamper the acceleration process in most physical cases.

Injection Problem

Throughout this paper, Bell assumed that the accelerated particles are sufficiently energetic to be able to pass through the shock. In order for this assumption to be valid, the gyroradius of the particle needs to be larger than the shock width. The typical thermal energy of particles in the fluid is much too low to satisfy this condition. Therefore, there is a need for a pre-acceleration phase where thermal particles become energized enough for them to participate in Bell’s acceleration (Bell, 1978).

A lot of work has been put into answering this question and to my knowledge it is still an open problem. One method due to Malkov and Volk is to note that even when the gyroradius of the particle is smaller than the shock width, there is still a leaking of mildly suprathermal particles from downstream to upstream. These particles will then excite Alfven waves and accelerate in much the same way as Bell’s diffusive shock acceleration (Malkov & Voelk, 1995).

What About Electrons?

Why do we observe a deficit of electrons in the cosmic ray composition? There are two primary reasons. The first is that the injection problem is much more severe for an electron. Recall that the gyroradius is:
r_{gyro} = \frac{mv_{\perp}} {|q|B} \; .
As a consequence of electrons being much less massive than protons (protons are heavier by a factor of about 2000), they have a much smaller gyroradius. In particular, this means that an electron needs to be much more energetic than a proton for it to be able to pass through a shock.

The second reason for the electron deficit is efficient radiative cooling processes. Due to its tiny mass, both inverse Compton and synchrotron cooling are extremely efficient is taking kinetic energy away from an electron. In particular, close to a shock where magnetic fields can be amplified to many times their background value, synchrotron radiation becomes extremely effective. As such, not only is it more difficult for electrons to join in the acceleration process, electrons also lose kinetic energy much more efficiently than protons or other heavier nuclei.

However, it should be noted that there are methods for electrons to participate in the acceleration. One of the cutest ones involves electrons hitching a ride on ions. It turns out that the acceleration timescale of ions is comparable to their electron stripping timescale in typical shock conditions. So, ions that still harbor electrons can be accelerated via Bell’s diffusive shock acceleration before the electrons, now also possessing a large kinetic energy due to the acceleration, get stripped from the ions and join in the acceleration party.

Oblique Shocks

Figure 2 of Bell's paper, oblique shock geometry.

Figure 2 of Bell’s paper, showing oblique shock geometry.

What if the shock normal is not parallel to the magnetic field? In general, since the plasma can move both parallel and perpendicular to the magnetic field lines, we have to include the electric field in the description. However, Bell claimed that we can Lorentz transform to a frame where the shock front is stationary and all bulk fluid motion is parallel to the magnetic field as long as the angle between the shock’s normal and the magnetic field is less than cos^{-1} (v_s / c) (better argued in Drury, 1983). In this frame, the electric field would be non-existent and we can use much of our previous calculations. If velocity downstream and upstream are \vec{w_1} and \vec{w_2} respectively (look at Figure 2 of Bell’s paper, reproduced preceding this paragraph), the energy increase due to one crossing is:

E_{k+1} = E_{k} \left( \frac{1 + \vec{v}_{k1} \cdot (\vec{w}_1 - \vec{w}_2)} {1 + \vec{v}_{k2} \cdot (\vec{w}_1 - \vec{w}_2)} \right ) \; ,
Since \vec{w}_1 and \vec{w}_2 are not parallel, the increase per crossing is smaller than that of the parallel shock. However, the particle’s gyration around magnetic field lines would allow the particle to cross the shock multiple times when it drifts close to the shock (remember, the particle’s gyration radius must be large compared to the shock’s width for it to participate in the acceleration process!). In total, Bell claims that these two effects cancel each other out, resulting in the same power law spectrum.

Conclusion

In this paper, Bell described a novel method to use the bulk kinetic energy in shocks around supernova remnants to accelerate particles to cosmic ray energies. The analytically calculated particle energy power law spectrum agrees with the observed spectrum. However, the study of cosmic ray acceleration is still far from over. An issue left open by this paper is the ‘seed’ of the accelerating particles. Because in standard SNR condition thermal particles do not possess enough energy to participate in the acceleration mechanism, Bell’s process requires a pre-acceleration mechanism. This injection mechanism is still unknown, and we would like to direct interested readers to Malkov & Voelk, 1995 for a hypothesis. The subject of ultra high energy cosmic rays (UHECR) is also left undiscussed. In particular, although most observed cosmic rays are accelerated by Milky Way’s SNR’s, these UHECRs are thought to originate from extragalactic sources. Both the site of their acceleration and the mechanism for said acceleration is still a mystery.

Derivation of Equation (4)

Equation (4) in Bell’s paper is a cornerstone equation which Bell’s argument rests upon. For some reason he did not show the derivation of this equation. The easiest way to derive it is to follow Bell’s advice and perform Lorentz transforms of the energy in the rest frame of the scattering center of the new region. This Lorentz boost will be in the direction parallel to the shock’s normal. For a single upstream to downstream crossing, the particle energy is increased by:
E = \frac{E' + vp'} {\sqrt{1 - v^2/c^2}} = \frac{E' + v \gamma m v_{k1}} {\sqrt{1 - v^2/c^2}} = E' \frac{1 + v v_{k1}/c^2} {\sqrt{1 - v^2/c^2}}
where E' is the energy in the original region (upstream) and E' is the energy in the downstream region; v_{k1} is the velocity at which the particle is crossing from upstream to downstream; and v is the difference of the velocity between scattering centers upstream and downstream parallel to the shock’s normal (direction of Lorentz boost) given by: v = (u_1 - u_2) \cos \theta_{k1} , where \theta_{k1} is the angle the motion is making with the shock’s normal. Putting this together gives:
E_{downstream} = E_{upstream} \left( \frac{1 + v_{k1}(u_1 - u_2) \cos \theta_{k1}/c^2} {\sqrt{1 - ((u_1 - u_2) \cos \theta_{k1})^2/c^2}} \right) \; .
Now, equation (4) in Bell’s paper is the total energy change of the particle if it crosses from upstream to downstream and to upstream again. This means we have to once again perform a Lorentz boost, this time from downstream to upstream. This is the exact same procedure, but now we solve for E' instead of E (inverse Lorentz transformation) since we are measuring everything in the frame of the upstream scattering centers. This gives:
E_{final} = E_{initial} \left( \frac{1 + v_{k1}(u_1 - u_2) \cos \theta_{k1}/c^2} {\sqrt{1 - ((u_1 - u_2) \cos \theta_{k1})^2/c^2}} \right) \left(\frac{\sqrt{1 - ((u_1 - u_2) \cos \theta_{k1})^2/c^2}} {1 + v_{k2}(u_1 - u_2) \cos \theta_{k2}/c^2} \right )
= E_{initial} \left( \frac{1 + v_{k1}(u_1 - u_2) \cos \theta_{k1}/c^2} {1 + v_{k2}(u_1 - u_2) \cos \theta_{k2}/c^2} \right ) \; ,
which is exactly equation (4)!

A More Intuitive Derivation of Equation (4)

Although the previous derivation is valid, I think it is not very physically illuminating. Here is a more intuitive derivation of the amount of energy increase a particle receives due to crossing from upstream to downstream. Note that we will not be using Lorentz transforms for this derivation, so it is only accurate to the lowest order in (u_1- u_2)/c .

What is the amount of momentum increase a particle receive due to crossing from upstream to downstream? Suppose p' is the momentum of the particle after it crosses and p is the momentum of the particle before it crosses:
p' = p + \Delta p
= p + \gamma m \Delta v
= p + \gamma m (u_1 - u_2) \cos \theta_{k1} \; .
Now, what is the change of particle energy due to this increase in momentum?
\Delta E = \int \frac{dp}{dt} dl = \int dp v_{k1} \sim v_{k1} \Delta p \;.
The change of energy therefore is:
E' = (E + v_{k1} \Delta p)
= (E + v_{k1} \gamma m (u_1 - u_2) \cos \theta_{k1})
= E ( 1 + v_{k1} m (u_1 - u_2) \cos \theta_{k1}/c^2) \; ,
where we have used E = \gamma m c^2 on the last line. This is the amount of energy the particle possess after it crosses from the upstream to the downstream. The increase in energy is:
\Delta E = E (v_{k1} m (u_1 - u_2) \cos \theta_{k1}/c^2) \; .
Why is this derivation more illuminating than the Lorentz boosts? For one, it is easy to see where the extra energy comes from. It is obvious from our \Delta E equation that it comes from the difference in velocity of the scattering centers upstream and downstream (u_1 - u_2) . Therefore, cosmic ray acceleration is literally a process where energetic particles steal energy from the bulk fluid motion!!!

Another illumination comes from the fact that this equation is exactly the same if we apply it backwards, from downstream to upstream, as long as we change v_{k1} to the particle velocity going back upstream and \theta_{k1} to the angle the motion makes with the shock’s normal (\theta now defined in the downstream region). Now, u_1-u_2 will have to be changed to u_2-u_1 , giving a minus sign. However, the angle must be inverted, \cos\theta \rightarrow - \cos\theta , netting another minus sign. The energy change is always positive no matter if the particle crosses from upstream to downstream or downstream to upstream. Particles always gain and never lose energy when the cross a shock!!! In particular, the total energy gain by a particle after many crossings is just the product of a bunch of these (v_{k} m (u_1 - u_2) \cos \theta_{k}/c^2) terms!

To get to equation (4), we have to perform two crossings, one upstream to downstream and another downstream to upstream.
Some coordinate change trickery is also required. In particular, \theta_{k1} and \theta_{k2} are both measured in the upstream frame. Therefore, the extra minus sign from \cos\theta \rightarrow - \cos\theta is not present. Therefore:
E_{final} = E_{initial} (1 + v_{k1} m (u_1 - u_2) \cos \theta_{k1}/c^2) (1 - v_{k2} m (u_1 - u_2) \cos \theta_{k2}/c^2) \; ,
where the minus sign comes from the u_2 - u_1 term of the second crossing. It is now obvious to see that this is exactly equation (4) if we Taylor expand the denominator to linear order in (u_1 - u_2)/c . Since equation (7) in Bell’s paper only goes to linear order in (u_1 - u_2)/c , this approximation will give us the same differential energy spectrum as equation (9) of Bell’s paper.

Derivation of Equations (1) & (13)

In this section we shall derive Equation (1) of the paper:
\frac{\partial n} {\partial t} + u_2 \frac{\partial n} {\partial x} = \frac{\partial} {\partial x} \left( D(x) \frac{\partial n} {\partial x} \right) \; .
This equation describes the evolution of the particle density, n(x, t) in the downstream region. The two important mechanisms are advection due to the fluid flow (with advection velocity u_2 ) and diffusion.

Start with a flux continuity equation of the particle density n with its flux, n u_2 .
\frac{\partial n} {\partial t} + \nabla \cdot (n \vec{u_2}) = \rm{Source} - \rm{Sink} \; .
Perform a product rule:
\frac{\partial n} {\partial t} + n \nabla \cdot \vec{u_2} + \vec{u_2} \cdot \nabla n = \rm{Source} - \rm{Sink} \; .
Now, n \nabla \cdot \vec{u_2} = 0 since the advection velocity downstream is u_2 regardless of position, so:
\frac{\partial n} {\partial t} + \vec{u_2} \cdot \nabla n = \rm{Source} - \rm{Sink} \; .
This takes care of advection. What about diffusion? If we know how diffusion affects \partial n / \partial t , we can put diffusion into the Source – Sink term.

Let us consider again a flux continuity equation, but only considering diffusion processes (no advection or other source/sink terms) this time:
\frac{\partial n} {\partial t} + \nabla \cdot \vec{J} = 0 \; .
By the phenomenological Fick’s first law:
\vec {J} = -D(\vec{x}) \nabla n \; ,
where D(\vec{x}) is called the \textbf{diffusion coefficient} and can vary with respect to position in the shock. Plugging this to the previous equation gives:
\frac{\partial n} {\partial t} = \nabla \cdot (D(\vec{x}) \nabla n) \; .
We can interpret this equation as an extra contribution to \partial n / \partial t due to diffusion. In particular, this is the Sink-Source due to diffusion. The complete continuity equation with both advection and diffusion is therefore:
\frac{\partial n} {\partial t} + \nabla \cdot (n \vec{u_2}) = \nabla \cdot (D(\vec{x}) \nabla n) \; .
Now, since things do not vary wildly perpendicular to the shock’s normal, the only important terms in these derivatives are the ones parallel to the shock’s normal (the x component in Figure (1) of the paper). Similarly, \vec{u_2} = u_2 \hat{x} and D(\vec{x}) = D(x) , since we assume that the advection velocity and diffusion coefficient do not vary perpendicular to the shock’s normal.
Therefore, the continuity equation becomes:
\frac{\partial n} {\partial t} + u_2 \frac{\partial n} {\partial x} = \frac{\partial} {\partial x} \left( D(x) \frac{\partial n} {\partial x} \right) \; ,
equation (1) of Bell’s paper. Note that I only need to change the advection velocity from u_2 to u_1 if I want to find a similar equation in the upstream region, giving equation (13) of the paper.

References

Bell, A. R. 1978, MNRAS, 182, 443

Drury, L. O. . 2012, Astroparticle Physics, 39, 52

Drury, L. O. 1983, Reports on Progress in Physics, 46, 973

Malkov, M. A., & Voelk, H. J. 1995, A&A, 300, 605

Swordy, S. P. 2001, Space Sci. Rev., 99, 85

Further Readings

Blandford, R., & Eichler, D. 1987, Phys, Rep., 154, 1

Schure, K. M., Bell, A. R., OC Drury, L., & Bykov, A. M. 2012, Space Sci. Rev., 173, 491

Skilling, J., 1975a. Mon. Not. R. astr. Soc., 172, 557.

Skilling, J., 1975b. Mon. Not. R. astr. Soc., 173, 245.

Skilling, J., 1975c. Mon. Not. R. astr. Soc., 173, 255.

Wentzel, D. G., 1974. A. Rev. Astr. Astrophys., 12, 71.

Cas A, a supernova remnant where shocks are accelerating cosmic rays, courtesy of wikipedia.

Cas A, a supernova remnant where shocks are accelerating cosmic rays, courtesy of wikipedia.

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