  We give an explicit formula for the arithmetic intersection number of CM
cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating
the intersection number on the infinite level. Our CM cycles are constructed by
choosing two separable quadratic extensions $K_1,K_2/F$ of non-Archimedean
local fields $F$. Our formula works for all cases, $K_1$ and $K_2$ can be
either the same or different, ramify or unramified. As applications, this
formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a
comparison of integrals. This formula can also be used to recover Gross and
Keating's result on lifting endomorphism of formal modules.
