B_d, F_d

calculated eV

Calculation

cosmic rays

observed eV

B₈

1.7 × 10¹⁵

M_Z/α²

the first knee

3 × 10¹⁵

The midpoint between B₈ and F₉

2 × 10¹⁶

Equation (6)

the first ankle

2 × 10¹⁶

F₉ = ${{\nu }^{\prime }}_{\tau }$ (high-mass tau neutrino)

2.35 × 10¹⁷

M_Z/α³

the second knee

3 × 10¹⁷

The midpoint between F₉ and B₉

2.8 × 10¹⁸

Equation (6)

the second ankle

3 × 10¹⁸

B₉

3.2 × 10¹⁹

M_Z/α⁴

the toe

4 × 10¹⁹

F₁₀

4.4 × 10²¹

M_Z/α⁵

beyond the GZK limit (5 × 10¹⁹ eV)

not observed

B₁₁

1.13 × 10²⁸

M_Z/α⁸

Plank mass for gravity

1.22 × 10²⁸