Bd, Fd

calculated eV

Calculation

cosmic rays

observed eV

B8

1.7 × 1015

MZ2

the first knee

3 × 1015

The midpoint between B8 and F9

2 × 1016

Equation (6)

the first ankle

2 × 1016

F9 = ν τ (high-mass tau neutrino)

2.35 × 1017

MZ3

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

MZ4

the toe

4 × 1019

F10

4.4 × 1021

MZ5

beyond the GZK limit (5 × 1019 eV)

not observed

B11

1.13 × 1028

MZ8

Plank mass for gravity

1.22 × 1028