model {
   X ~ dpois(lambda)
   lambda <- ls + lB
   ls <- r * T 
   r ~ dgamma(1, 0.00001)  # gamma, but indeed  dexp(0.00001)
   lB <- rB * T

   # experiment with background only
   lB0 <- rB * TB
   XB ~ dpois(lB0)
   rB ~ dgamma(1, 0.00001) # vague priors also on the background
}