and this is the solution for T_e:
T_e:
14.54253001
17.80607304
15.99105488
17.19057572
15.42830198

My question is know, how exactly did you come to this values of T_e?
If i calculated it step by step I come to different solutions for T_e.
Maybe someone can show me, what of the data I have to put in the different equations to come to the solutions of the for loop.
Thank you very much.

Ok thanks. But as it is no for loop, do you still know how the values of T_e are caluculated? How did the program come to the solutions with the code and the data I put in? When I calculate each value of T_e, I come to different outputs.

Even tough you don't use the i, the code in your loop still runs 9999 times.

At the end of the loop, you assign your result (T_enew) to T_e. Since T_e is used in the first line of the loop to calculate v_e, each new iteration of the loop uses an updated value of T_e to compute v_eand thus T_enew.
Here a simple example to ilustrate this principle:

a <- 12
for(i in 1:5){
b <- a/2 # Do some operations based on a
a <- b # assign the results of the previous line back to a
print(a) # Just to ilustrate what happens to a in each iteration
}
#> [1] 6
#> [1] 3
#> [1] 1.5
#> [1] 0.75
#> [1] 0.375
a
#> [1] 0.375

I think that might explain why you get a different value of T_e if you simply paste the individual values in your formula.