- #1

- 596

- 49

## Summary:

- Need help normalizing Solar flare proton energy spectrum (Band function) and understanding its units.

I am trying to understand how to normalize a proton energy spectrum from a solar flare.

The spectrum is given by the Band Function, and I cite the paper "Spectral Analysis of the September 2017 Solar Energetic Particle Events" by Bruno in 2019, link to paper. The equation number in the paper is (2):

$$\phi_{\text{Band}}(E)=

\begin{cases}

AE^{-\gamma_{a}}e^{-\frac E {E_0}} & \text{for } E < (\gamma_b - \gamma_a)E_0 \\

AE^{-\gamma_{b}}[(\gamma_{b}-\gamma_{a})E_0]^{(\gamma_{b}-\gamma_{a})}e^{\gamma_{a}-\gamma_{b}} & \text{for } E > (\gamma_b - \gamma_a)E_0

\end{cases} $$

The parameters ##\gamma_{a}##, ##\gamma_{b}## and ##E_0## are empirically fit parameters to his data. The parameter ##A## is a scaling factor.

In the paper, the units of ##\phi_{\text{Band}}## are plotted as ##\frac {1} {\text{MeV} \text{sr} \text{cm}^2}##. I would like to determine the units of the spectrum ##\phi(E)## in simple units of ##\frac {\text{protons}} {\text{cm}^2}## for each energy in MeV at 1 AU from the Sun.

Secondly, I don't understand why the units in Figure 4 of the paper are in "per steradian" when solar flare protons are not isotropic, but travel along the Interplanetary Magnetic Field lines, (if I understand correctly, I'm not a professional solar physicist)? I have read that the flux of protons depends upon whether the point of interest in the solar system is magnetically connected to the source of the flare or not. So why is it plotted in units of ##sr^{-1}## if the flux is not isotropic?

Here is my attempt at normalizing the spectrum:

For simplicity, define ##k=(\gamma_{b}-\gamma_{a})E_0## and ##C=[(\gamma_{b}-\gamma_{a})E_0]^{(\gamma_{b}-\gamma_{a})}e^{\gamma_{a}-\gamma_{b}}##, then

$$1 = \int_{E_{\text{min}}}^{E_{\text{max}}}φ_{Band}(E)dE$$

$$1 = \int_{E_{\text{min}}}^{k}AE^{-\gamma_{a}}e^{-\frac {E} {E_0}}dE+\int_{k}^{E_{\text{max}}}AE^{-\gamma_{b}}CdE$$

$$1 = A \left[ \int_{E_{\text{min}}}^{k}E^{-\gamma_{a}}e^{-\frac {E} {E_0}}dE+C\int_{k}^{E_{\text{max}}}E^{-\gamma_{b}}dE\right]$$

$$A= \frac {1} { \int_{E_{\text{min}}}^{k}\frac{e^{-\frac {E} {E_0}}}{E^{\gamma_{a}}}dE+\frac {C} {1-\gamma_{b}}(E_{\text{max}}^{1-\gamma_{b}}-k_{}^{1-\gamma_{b}})} $$

As can be seen, the second integral in the equation is a simple power law integral, but the first one is not. It has the same form as the generalized Exponential Integral (if you define ##t=E## and ##x=\frac{1}{E_0}## and ##n=\gamma_{a}##)

$$E_{n}(x)=\int_{1}^{∞} \frac{e^{-xt}} {t^n} dt$$

but it does not have the same bounds of integration and ##n## is supposed to be an integer in the exponential integral, but it is not an integer here... This is where I am stuck.

Also, how would the parameter A be determined empirically rather than analytically?

The spectrum is given by the Band Function, and I cite the paper "Spectral Analysis of the September 2017 Solar Energetic Particle Events" by Bruno in 2019, link to paper. The equation number in the paper is (2):

$$\phi_{\text{Band}}(E)=

\begin{cases}

AE^{-\gamma_{a}}e^{-\frac E {E_0}} & \text{for } E < (\gamma_b - \gamma_a)E_0 \\

AE^{-\gamma_{b}}[(\gamma_{b}-\gamma_{a})E_0]^{(\gamma_{b}-\gamma_{a})}e^{\gamma_{a}-\gamma_{b}} & \text{for } E > (\gamma_b - \gamma_a)E_0

\end{cases} $$

The parameters ##\gamma_{a}##, ##\gamma_{b}## and ##E_0## are empirically fit parameters to his data. The parameter ##A## is a scaling factor.

In the paper, the units of ##\phi_{\text{Band}}## are plotted as ##\frac {1} {\text{MeV} \text{sr} \text{cm}^2}##. I would like to determine the units of the spectrum ##\phi(E)## in simple units of ##\frac {\text{protons}} {\text{cm}^2}## for each energy in MeV at 1 AU from the Sun.

Secondly, I don't understand why the units in Figure 4 of the paper are in "per steradian" when solar flare protons are not isotropic, but travel along the Interplanetary Magnetic Field lines, (if I understand correctly, I'm not a professional solar physicist)? I have read that the flux of protons depends upon whether the point of interest in the solar system is magnetically connected to the source of the flare or not. So why is it plotted in units of ##sr^{-1}## if the flux is not isotropic?

Here is my attempt at normalizing the spectrum:

For simplicity, define ##k=(\gamma_{b}-\gamma_{a})E_0## and ##C=[(\gamma_{b}-\gamma_{a})E_0]^{(\gamma_{b}-\gamma_{a})}e^{\gamma_{a}-\gamma_{b}}##, then

$$1 = \int_{E_{\text{min}}}^{E_{\text{max}}}φ_{Band}(E)dE$$

$$1 = \int_{E_{\text{min}}}^{k}AE^{-\gamma_{a}}e^{-\frac {E} {E_0}}dE+\int_{k}^{E_{\text{max}}}AE^{-\gamma_{b}}CdE$$

$$1 = A \left[ \int_{E_{\text{min}}}^{k}E^{-\gamma_{a}}e^{-\frac {E} {E_0}}dE+C\int_{k}^{E_{\text{max}}}E^{-\gamma_{b}}dE\right]$$

$$A= \frac {1} { \int_{E_{\text{min}}}^{k}\frac{e^{-\frac {E} {E_0}}}{E^{\gamma_{a}}}dE+\frac {C} {1-\gamma_{b}}(E_{\text{max}}^{1-\gamma_{b}}-k_{}^{1-\gamma_{b}})} $$

As can be seen, the second integral in the equation is a simple power law integral, but the first one is not. It has the same form as the generalized Exponential Integral (if you define ##t=E## and ##x=\frac{1}{E_0}## and ##n=\gamma_{a}##)

$$E_{n}(x)=\int_{1}^{∞} \frac{e^{-xt}} {t^n} dt$$

but it does not have the same bounds of integration and ##n## is supposed to be an integer in the exponential integral, but it is not an integer here... This is where I am stuck.

Also, how would the parameter A be determined empirically rather than analytically?