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Given this function, which is essentially J = constant*E^d[(E^k+const)/(const)]^(const)

This was done for 3 species, including helium, electrons, and protons (so the function is similar but different for all 3).

note: d, k are also constants. These were figured out in Matlab using least squares regression, but my question isn't related to this, so I didn't feel it necessary to include the entire equation typed out. If needed, I can include this.

Next, I found the energy density (aka kinetic energy density) by integrating this intensity over a solid angle. So in essence, integrate the above function J, and that'll be times the constant 4pi which comes out of the solid angle.I assumed the arrival of cosmic rays to be isotropic.

Great. So now I need to find the ionization rate. And I have no idea how to do this. So given some cosmic ray spectrum that has J proportional to some value of E, and/or given the energy density (by integrating the spectrum), how on earth do I go about finding the ionization rate?

Thanks!