rOrb[t_] =
  {-1769.42 - 5230.58 Cos[0.0001 t] - 536.31 Sin[0.0001 t],
   1233.56 - 6233.56 Cos[0.0001 t] + 4499.51 Sin[0.0001 t],
   4187.62 + 5812.38 Cos[0.0001 t]};

T[t_] := Simplify[ rOrb'[t] / Norm[ rOrb'[t] ], t ϵ Reals];

vOrb[t_] := D[rOrb[t], t];

vdust[x_, y_, z_] :=
  1.7 {E^(-4 10^(-8) x^2), E^(-4 10^(-8) y^2), e^(-1.6  10^(-9) z^2)};

ρdust = 2 10^-6;

NIntegrate[((vOrb[t] - vdust[rOrb[t]]).T[t]) ρdust rOrb[t], {t,
  0, 20000 π}]

vdust[{x_, y_, z_}] :=  1.7 {E^(-4 10^(-8) x^2), E^(-4 10^(-8) y^2),
                             E^(-1.6  10^(-9) z^2)};