SpectralModels.md

Spectral models in VTSCat

The following spectral models are used in VTSCat.

Gamma-ray spectral models

Power law

type: pl

$$f_0 \cdot \left(\frac{E}{E_{ref}}\right)^{-\gamma}$$

• f₀: normalisation factor ( = norm in yaml files)
• gamma: spectral index ( = index in yaml files)
• E_ref: normalisation energy ( = e_ref in yaml files)

Power law (with integrated normalisation factor)

type: pl2

$$\frac{dN}{dE} = F_0 \cdot \left(\frac{E}{E_0}\right)^{-\gamma} \cdot \frac{\gamma+1}{E_{max}^{-\gamma+1}-E_{min}^{-\gamma+1}}$$

• F₀: integral flux in the interval E_min to E_max ( = flux in yaml files)

• gamma: spectral index ( = index in yaml files)

• E_min, E_max: energy interval for integral flux calculation

Power law with exponential cutoff

type: ecpl

$$\frac{dN}{dE} = f_0 \cdot \left(\frac{E}{E_{ref}}\right)^{-\gamma}\cdot \exp \left(-\frac{E}{E_{cut}}\right)$$

• f₀: normalisation factor ( = norm in yaml files)
• gamma: spectral index ( = index in yaml files)
• E_ref: normalisation energy ( = e_ref in yaml files)
• E_cut: cutoff energy ( = e_cut in yaml files)

Log-parabola power law

type: lppl

$$\frac{dN}{dE} = f_0 \cdot \left(\frac{E}{E_{ref}}\right)^{-\alpha - \beta \log \frac{E}{E_{ref}}}$$

• f₀: normalisation factor ( = norm in yaml files)
• alpha: spectral index ( = alpha in yaml files)
• beta: pre-factor beta ( = beta in yaml files)
• E_ref: break energy ( = e_ref in yaml files)

Smoothly broken power law

type: brokenpl

$$\frac{dN}{dE} = f_0 \cdot \left(\frac{E}{E_{ref}}\right)^{-\gamma_1} \cdot \frac{1}{1+(E/E_{break}^{\gamma_2-\gamma_1})}$$

Sum of two power laws

type: pl+pl

Sum of power law and power law with exponential cutoff

type pl+ecpl

Cosmic-ray spectral models

Power law

type: plCR

$$\frac{dN}{dE dA dt d\Omega } = f_0 \cdot \left(\frac{E}{E_{ref}}\right)^{-\gamma}$$

• f₀: normalisation factor ( = norm in yaml files)
• gamma: spectral index ( = index in yaml files)
• E_ref: break energy ( = e_ref in yaml files)

Broken power law

type: brokenplCR

$$\frac{dN}{dE dA dt d\Omega } = f_0 \cdot \left(\frac{E}{E_b}\right)^{-\gamma_1} \ \ \mathrm{if} E \leq E_b$$

$$\frac{dN}{dE dA dt d\Omega } = f_0 \cdot \left(\frac{E}{E_b}\right)^{-\gamma_2} \ \ \mathrm{if} E > E_b$$

• f₀: normalisation factor ( = norm in yaml files)
• gamma_1: spectral index below break energy E_b ( = index_1 in yaml files)
• gamma_2: spectral index above break energy E_b ( = index_2 in yaml files)
• E_b: break energy ( = e_break in yaml files)