cosmic ray large scale anisotropies
cosmic ray small scale anisotropies

Cosmic Ray Propagation in the Interstellar Medium

Ryan Farber, Wheaton College MA

REU 2014



UW Madison

Brief History

The 1936 Nobel Prize in Physics was shared by two discoverers: Carl Anderson for the discovery of the positron and Victor Hess for the discovery of cosmic rays. Victor Hess performed his research in 1912 using hot air balloons to measure changes in radiation levels in the atmosphere. His findings were consistent with previous work by Theodor Wulf in 1909. Wulf invented the electrometer, which he used on the Eiffel Tower and found that higher levels of radiation existed higher in the atmosphere. Since it was thought at that time that radiation came predominantly from decay processes in the Earth, it was not until Hess's measurements that Wulf's findings were taken seriously. Hess ruled out the Sun as the primary source of the radiation by going up in his hot air balloon during a total solar eclipse. The source of this radiation could only be cosmic.

Robert Milikan, who performed the famous Milikan oil drop experiment to measure the charge of an electron, coined the term "cosmic rays" in the 1920s because he thought they were primarily gamma rays. However, in 1927 J. Clay found evidence that cosmic rays are deflected in the geo-magnetic field; cosmic-rays must be charged particles.

Composition of Cosmic Rays

Nearly 100 years later, we have the above chart of the composition of cosmic rays, showing that protons are the dominant constituents of cosmic rays especially at low energies (that is, low for cosmic rays) while Helium and Iron also contribute significantly and even dominate the composition at certain energies.

And so, the scientific spark motivating resarch of cosmic rays has turned from their composition to an as yet unsolved mystery: the cosmic ray anisotropy.


Isotropy is directional-independence; that is, something that is isotropic looks the same from all directions. Its negation, anisotropy, means something has directional-dependece; something that is anisotropic looks different from different directions.

The two images at the header are sky maps revealing the large scale and small scale anisotropies, which have been observed by a host of experiments at an increasing range of energies. These experiments find that the cosmic ray distribution is remarkably uniform, but at the 0.1% level the large scale anisotropies are observed and at a factor of a few smaller in scale the small scale anisotropies are observed. The key motivation, driving eager minds to the scientific enterprise of cosmic ray studies, is that the small scale anisotropies currently have no satisfying solution for why they exist.

Giacinti 2013

One proposed solution for the small scale anisotropies was published in Giacinti 2013, which argues that the small scale anisotropies naturally emerge in the last mean free path of random-walking cosmic rays even in isotropic turbulence so long as there is an initial dipole in this turbulence. Another key point is the assumption that the mean free path of cosmic rays is much smaller than the injection scale of the turbulence; the point of making that assumption is so that the diffusion approximation may be used. The injection scale is the point at which energy is injected into the turbulence which cascades into smaller and smaller eddies.

My REU 2014 project was to determine if the Giacinti 2013 proposed solution would hold up under more physical conditions of propagating a particle by magnetic fields rather than a random walk and in a magnetic field that has anisotropic turbulence.

Since this project focuses so much on turbulence, I think it is best to mention why we know the interstellar medium is turbulent.


Big Power Law in the Sky

The above image is known as the "Big Power Law in the Sky". It was first found in Armstrong et al. 1995 and further work with WHAM (Wisconsin H-Alpha Mapper) data by Chepurnov & Lazarian 2002 confirmed the observation of this power law with an index consistent with the Kolmogorov value (-11/3) which is indicative of turbulence.


The first set of simulations involved the production of the magnetic field to propagate the cosmic rays in. The Cho & Lazarian 2002 MHD code was used by a former student to produce a turbulent magnetic field data cube with a mean magnetic field of 3.0 microGauss solely along the x-direction with a Alfven Mach number of 0.73. The Alfven Mach number compares the strength of the fluctuations of the magnetic field to the mean value of the magnetic field. An Alfven Mach number of 0.73 means the fluctuations are pretty strong but the mean magnetic field is still stronger.

MHD Data Cube

The image above shows particle tracks for particles propagated in a similar MHD data cube by a previous student.

The set of simulations that I performed involved the propagation of the cosmic rays in the magnetic field data cube described above. I used the Xu & Yan 2013 particle propagation code which treats the cosmic rays as test particles, meaning they do not affect the magnetic field but the magnetic field tells the particles how to move.

I performed simulations using the particle propagation code for parameters of injection scale at 100 parsecs (pc) and 10 parsecs (pc), corresponding to cosmic ray (in our case, simply protons) energies of 7.5 PeV and 750 TeV, respectively.


The value of the mean free path was able to be calculated immediately after the particles were propagated. To do this, it is first necessary to find the point at which the particle's motion decoheres. Decoherence occurs when a particle's next step is completely random compared with N steps ago. We used a ninety degree method to find the number of steps to decoherence for a particle, by taking the dot product of its inital momentum with a later momentum and finding the time step at which the angle between those two vectors was 90 degrees.

There are two ways to define the mean free path. One ways is to define the mean free path as the straight line distance connecting the initial point and the point of decoherence and the other way is to define the mean free path as the total distance along the curve that the particle travels to arrive at the point of decoherence.

Our results for our two cases were: mean free path length of 130 pc curve value and 50 pc line value for an injection scale of 100 pc and a mean free path length of 22 pc curve value and 6.5 pc line value for an injection scale of 10 pc.

Initial Map at Source

In the image above, you can see the initial t = 0 sky map, which has a completely random angular distribution. Over time, as the particles travel to a certain radius from the center, it begins to be influenced by the mean magnetic field along the x-direction, as can be seen in the image below at a radius of 100 pc.

Forward Tracking Map at 100 pc

At the scale of our simulation box, the Earth would be like a speck in a single grid; in other words, our resolution is simply too coarse to gain the information on the propagation of cosmic rays in turbulence at the appropriate injection scales and at the same time resolve the angular distribution of cosmic rays at Earth.

We solve this problem by backtracking the cosmic rays, which is a valid method when Liouville's theorem is applicable. Liouville's theorem says that the phase space distribution is constant along the trajectories of a system. In other words, the cosmic rays' angular distribution at a certain radius from the source will maintain that same angular distribution when the radius is shrunken down. Therefore, a question of critical importance is when Liouville's theorm can be applied.

Liouville's theorem can be applied when the first adiabatic invariant of plasma physics, the particle's magnetic moment, is conserved.

Histogram of Standard Deviation

As can be seen from the histogram plot above, the magnetic moment is conserved and therefore it is possible to backtrack the cosmic rays to find their angular distribution as observed from Earth.

Sky Maps

Map Backtracked to Earth with Dipole

To produce the map above, cosmic rays were forward tracked until they reached a radial displacement from the center of 200 pc. Then, they were fit with a dipole at that radius and backtracked "to Earth".

There are several possible reasons the cosmic rays would have a dipole distribution. One of the simpler explanations is that there is simply some small excess of supernova in one direction and therefore a corresponding deficit of supernova in another direction. Another possible reason is the dipole results from convection; at lower energies, the Compton-Getting effect could explain the dipole but our energies are too great for this to be very plausible in our case.

Map Backtracked to Earth with Dipole

The map above reveals the small scale anisotropies by subtracting out the dipole, leaving the higher multipole moments to dominate.

Angular Power Spectrum

Angular Power Spectrum

Information complementary to the sky maps may be obtained by looking at the angular power spectrum. The graph above displays the angular power spectrum for particles forward tracked various distances and then backtracked to Earth.

You can see in the plot above that the dipole is a hundred to a thousand times stronger than the higher multipole moments. An additional observation is the curious fact, whose cause or explanation is under current investigation, that the power decreases with increasing displacement.

Angular Power Spectrum with Ahlers plot

Above, the angular power spectrum is overlayed on Ice-Cube data at 20 TeV and Ahlers 2014 theoretical predicted curve. We had to scale our angular power spectra to the dipole for comparison between our spectra and the Ice-Cube and Ahlers angular power spectra. Our simulation's angular power spectrum matches nicely with the experimental data and with Alhers' theoretical curve.


2014 is an exciting year in cosmic ray research with interesting results on the cosmic ray anistropies published recently and with research on this topic ongoing.

I investigated this summer the feasibility of the proposed solution to the cosmic ray small scale anisotropies of Giacinti 2013. I have found that the assumption that the mean free path is much less than the injection scale to be incorrect in more realistic conditions of particle propagation in the interstellar medium.

However, an evaluation of the main point of Giacinti 2013, that the small scale anisotropies emerge in the last mean free path, requires further analysis of the sky maps and angular power spectrum.


[1] Amenomori et al. ICRC, 2011.

[2] Abbasi et al., ApJ, 746, 33, 2012.

[3] Abdo et al., PRL, 101, 221101, 2008.

[4] Abbasi et al., ApJ, 740, 16, 2011.

[5] Giacinit, G. & Sigl, G., AIP Proceedings, 1505, 458, 2012.

[6] Armstrong et al., ApJ, 443, 209, 1995.

[7] Chepurnov, A. & Lazarian A., 710, 853, 2010.

[8] Cho, J., & Lazarian, A., Physical Review Letters, 88, 24 2002.

[9] Xu S. & Yan H., ApJ, 779, 140, 2013.

[10] Ahlers M., Physical Review Letters, 112, 022101, 2014.


First, I want to thank my Mom and Dad for 21 years of love and support, and I also appreciate constant support from my two sisters Christina and Meahgan, the rest of my family, and my friends.

Great thanks is due to the National Science Foundation (NSF) and the UW Madison Astrophysics REU program for supporting me and providing me the opportunity to participate in this exciting research at the breaking edge of contemporary science.

Special thanks is awarded to Alex Lazarian, my mentor, for choosing me and making my summer doing this research a reality.

And I give my deepest thanks and appreciation to Vanessa Lopez-Barquero and Paolo Desiati for helping me so very much with this project.

About Me

I would currently call myself Grade 15-16 Summer; in other words, I will be a senior at Wheaton College in Norton, MA this fall.

I shall return to be commanding officer of the Physics Club ("Supreme Overlord" is the official title for that position in our club), which I helped to resurrect in my Sophomore year.

I should be graduating in the spring of 2015 with a major in Physics, a second major in Mathematics, and a minor in Astronomy and I shall hopefully be going to graduate school in the fall of 2015 to study theoretical physics or theoretical astrophysics.

When not doing physics, I enjoy studying metaphysics and ancient history and trying to develop my own video games.

I greatly appreciate the reading of physical, preferably hard-covered, books.


University of Wisconsin - Madison

Madison, WI 53706

Email me: ryan at a-bcommunications dot com

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