NSolve as integration limit

I need the (numerical) solution of equation to be integration limit as below.
How to make it work?

T = 1
u[r_] := 4*(r^-12 - r^-6)

chi[b_, ga_] := [Pi] - 
 2*b*NIntegrate[1/(r^2*Sqrt[1 - b/r^2 - u[r]/(T*ga^2)]), 
  {r, rc /. Last[NSolve[u[rc] ==0.5*ga^2*T*(1 - b^2/rc^2), rc, PositiveReals]],10}]

Sqrt[T/(2*[Pi])]*NIntegrate[Exp[-ga^2]*ga^(2*1 + 3)*
  (1 - Cos[chi[b, ga]]^1)*b, {ga,   0, [Infinity]}, {b, 0, [Infinity]}]

During evaluation of In[6]:= NSolve::ratnz: NSolve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.

During evaluation of In[6]:= NIntegrate::nlim: r = Root[-8.+8. #1^6-1. b^2 ga^2 #1^10+ga^2 #1^12&,6] if condition is not a valid limit of integration.

During evaluation of In[6]:= Throw::nocatch: Uncaught Throw[$Failed] returned to top level.

Out[6]= Hold[Throw[$Failed], Throw[$Failed], Throw[$Failed], Throw[$Failed]]