m = 0.35;

J0 := BesselJ[0, m];

J1 := BesselJ[1, m];

α = 0.85 2  π 15.235*^6/2.9979*^8;

P0dark = ((J0^2) + 2  (J1^2)) (1 - c)/2 + 
        2 (J1 Sin[α])^2 c

P0bright = ((J0^2) + 2  (J1^2)) (1 - c)/
            2 + ((J0^2) + 2  (J1^2) Cos[α]^2) c

contrast = ((P0bright - P0dark)/(P0bright + P0dark)) // 
            Expand // Together // FullSimplify

FindRoot[contrast == 0.984, {c, 0.9, 0.999}]

{J0^2 (1 - c)/2, 2 (J1 Sin[α])^2}  /. %