If we make the normal isentropic flow assumptions, temperature rise is calculated simply as
temperature rise = k/5 x M^2 x SAT
k is the rise factor for the aircraft system
M is the Mach Number
SAT in degrees K (deg K = deg C + 273)
ie TAT = SAT + temperature rise
The various graphs you find in your OMs will be this equation plotted .. so you can easily reverse a few points to work out the k factor if that interests you.