Mathematica
ClearAll[a, e, f]
delT = 5.215*60*60;
r1 = 200 + 6378;
r2 = 130000;
mu = 3.986*10^5;
eq1 = r1 - a*(e - 1) == 0;
eq2 = delT - Sqrt[a^3/mu]*
(e*Sinh[f] - f) == 0;
eq3 = r2 - a*(e*Cosh[f]-1)==0;
sol = NSolve[{eq1, eq2, eq3},
{a, e, f}, Reals]
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{{a->12029.39633,
e->1.546827108,
f->2.721303232}}
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