Impact of distance from galactic centre on the value of energy in the cosmic ray spectrum where knee is observed?

Question

This question is based on the recommendation and great explanation by @Kyle_Kanos. Is it known what causes the "knee" in the observed Cosmic Ray spectrum?

Accepting the reason for the occurrence of knee around a few PeV energies of Ultra High Energy Cosmic Rays (UHECR) to be the shift in majority source from intergalactic to extra-galactic due to Larmor radius becoming comparable to the sscale height perpendicular to the galactic plane. Does that mean that observers farther from the centre would observe the value of knee energy to be lower due them getting more vulnerable to lower energy extra-galactic sources than the knee value for observer closer to the centre?

Answer 1

No, the location of the knee is not going to change based on an observer's location in the galaxy. It will always be at ~PeV energy.

First, note that the knee is observed in a plot of the flux, which is the particle counts per area per steradian per energy per second: $$ \left[J(E)\right]=\frac{\#}{\text{eV}\,\text{m}^2\,\text{sr}\,\text{sec}}.$$ It is not a pure count of events, which means that even if sources were obscured to some degree, the measured flux would not change. What would change with the pure counts of events is the error bars: less counts means larger error bars while more counts means smaller error bars.

Second, we have a problem of sources. From the Hillas criterion, we know that the sources of >PeV CRs must be either highly compact with massive magnetic fields or enormously extended objects. In addition to this, we also have to have worry about efficiency of the accelerator--for instance energy losses (e.g., synchrotron) and confinement of the particles. Within our galaxy, there are no potential sources of >PeV accelerators, which points to their being extra-galactic in origin.

As there are two different sources of CRs with different acceleration mechanisms (diffusive shock acceleration vs linear accelerator), then we should expect that there is a difference in the slopes of the flux plots with a break at the point where sub-PeV accelerators cease and supra-PeV accelerators arise. But this transition occurs due to energetics of the sources, not your location!

For instance, the image below, adapted from arXiv:0204357, shows how the spectrum of the differing population generate the knee. The red curve shows the galactic portion of CR iron ions while the black line shows the extra-galactic portion of CR protons combining together to form the whole spectrum.

Hence, the only thing that should impact the location of the knee is the energy spectrum of the particles themselves, which is going to depend on the source accelerating them.

For a good resource, see Strong, Moskalenko & Ptuskin (2007), particularly section 2.3

Answer 2

The position of the knee should shift with your location in the galaxy, yes, but not to first order. Since the galaxy is disk-like, the appropriate Lamor radius is the thickness of the disk, which is very roughly independent of distance from the center. Or at least to a sufficient approximation, given that this anyway is just a back-of-the-envelope estimate.

Answer 3

For a highly relativistic proton $v\approx c$, moving with momentum $p\approx\gamma mc$, the Larmor radius is given by - \begin{equation} r_L = \frac{p}{eB} \approx \frac{\gamma m c}{eB} \approx \frac{E}{eBc} \end{equation} So, higher the energy, lesser will be the deflection due to linearly increasing Larmor radius with energy for a relativistic particle. Due to this, for energies larger than the value for which the Larmor radius becomes comparable to the scale height of the galaxy in the direction perpendicular to the disk plane, the extragalactic cosmic rays cease to deflect away significantly, hence start to dominate the cosmic ray spectrum after that value. So, this 'knee' of sudden change could be caused by shift in majority source from intergalactic to extragalactic.\ So, the location of this knee should depend majorly on the galactic scale height at that location and average strength of magnetic field within the galaxy. The scale height(for a spiral galaxy similar to milky way) is further dependent on size of the galaxy and distance from the centre. I will use a few approximations to estimate the proportionality expression for scale height in terms of these quantities.\ Denoting D as the size of galaxy, d as the distance from the centre and t as the scale height there. I will assume that the shape of galaxies of these categories are about the same, therefore for a galaxy of twice the size 2D, the scale height at twice the distance 2d, from the centre should be twice 2t. Also, scale height is assumed to be decreasing linearly with distance from centre, which gives- \begin{equation} \frac{t2}{t1}=\frac{D2}{D1}\Big(\frac{D_1(d-D_2)}{D_2d_1-D1D_2}\Big)=\frac{D_2-d}{D_1-d_1} \end{equation} The Larmor radius, which is a measure of deflection at knee energy value, should be of order of this scale height. So, \begin{equation} \frac{E_1}{eB_1c}\approx t1 \end{equation} Therefore, \begin{equation} \frac{E_1}{E_2}\frac{B_2}{B_1} = \frac{t1}{t2}= \frac{D1-d1}{D_2-d} \end{equation}