Stochasticity of cosmic rays from supernova remnants and the ionization rates in molecular clouds

Cosmic rays are the only agent able to penetrate into the interior of dense molecular clouds. Depositing (part of) their energy through ionization, cosmic rays play an essential role in determining the physical and chemical evolution of star-forming regions. To a first approximation their effect can be quantified by the cosmic-ray induced ionization rate. Interestingly, theoretical estimates of the ionization rate assuming the cosmic-ray spectra observed in the local interstellar medium result in an ionization rate that is one to two orders of magnitude below the values inferred from observations. However, due to the discrete nature of sources, the local spectra of MeV cosmic rays are in general not representative for the spectra elsewhere in the Galaxy. Such stochasticity effects have the potential of reconciling modeled ionization rates with measured ones. Here, we model the distribution of low-energy cosmic-ray spectra expected from a statistical population of supernova remnants in the Milky Way. The corresponding distribution for the ionization rate is derived and confronted with data. We find that the stochastic uncertainty helps with explaining the surprisingly high ionization rates observed in many molecular clouds.

Figure 1

Left: The surface density of observed SNRs $g({\rho }_G,{\varphi }_G)$ in the Galaxy. The cross and filled circle marks are respectively the position of the solar system and the observer at ${\rho }_G=6.5\mathrm{kpc}$ in the spiral arms (referred to as the LOC and SPA point respectively, see text for more details). Right: The probability density with respect to $r_s$ which is the distance between the observer and an SNR projected onto the midplane of the Galactic disk.

Figure 2

Relative intensities of CR protons ($j/j_{\odot }$) at $E=10\mathrm{MeV}$ (left panel) and $E=30\mathrm{GeV}$ (right panel) in a particular realization of SNRs.

Figure 3

Intensities and corresponding stochastic fluctuations of CR protons (upper panels) and electrons (lower panels) in comparison with data from Voyager 1 [25] (blue) and AMS-02 [82, 83] (orange). Results are presented for an observer in the local ISM (the LOC position, left column) and in a spiral arm (the SPA position, right column). The dotted and solid-black curves are respectively the expectation values and the median of the intensities. The shaded red or green regions are the 95% uncertainty ranges of the intensities.

Figure 4

Schematic view for the one-dimensional transport models describe in Sec. 4. The shaded area represents the molecular cloud made up by molecular hydrogen. The ordered magnetic field threading the cloud is denoted as ${\mathbf{B}}_0$. The $x$-direction is chosen to be along ${\mathbf{B}}_0$. We denote the CR intensity at the cloud border and in the ISM respectively as $j(x=0,E)=j(x=L_{cl},E)=j_b(E)$ and $j(x\to \pm \infty ,E)=j_{\mathrm{ISM}}(E)$ (see text for more details).

Figure 5

Stochastic fluctuations of the ionization rate for the local ISM (left) and for the chosen point in a spiral arm (right). The dashed and solid-black lines correspond to the median intensities predicted from the diffusive and ballistic model. Data for the ionization rate are from [10] (filled-blue circles), [94] (green triangle), [95] (red triangles are upper limits), [96] (asterisk), and [11] (black squares are data points while inverted yellow triangles are upper limits).

Figure 6

Ionization rate in 50 simulated realizations of sources around the SPA point, represented by solid lines, from CR protons (left column) and CR electrons (right column). Note that solid lines with the same shade of color mean that the result have been derived with the same realization of SNRs. The dashed black lines are the stochastic median of the ionization rate values. Data for the ionization rate are from [10] (filled blue circles), [94] (green triangle), [95] (red triangles are upper limits), [96] (asterisk), and [11] (black squares are data points while inverted yellow triangles are upper limits).