= Solution
With constant detected rates, $Q=R_QT$, $B=R_BT$. Squaring the previous <signal-to-noise ratio in photon counting> gives
$$
R_Q^2T^2-Z^2(R_Q+2R_B)T-2nZ^2r^2=0.
$$
For $R_Q>0$, nonnegative background and target $Z>0$, choose the positive root:
$$
\boxed{T=\frac{Z^2(R_Q+2R_B)+\sqrt{Z^4(R_Q+2R_B)^2+8nR_Q^2Z^2r^2}}{2R_Q^2}.}
$$
The other root is nonpositive. With no <read noise>, this reduces to $T=Z^2(R_Q+2R_B)/R_Q^2$; in a read-dominated short exposure it approaches $Z\sqrt{2n}\,r/R_Q$. If $R_Q=0$, no positive target stellar signal-to-noise is attainable. A zero target can be assigned zero exposure, with its limiting interpretation when all noise terms also vanish. Real <quantum efficiency> or multiple reads modifies the rates or read-variance term before solving the quadratic.
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