{\rm mag}_{{\rm p},i} < \Delta {\rm mag_{obs}}$, where $\Delta {\rm mag_{obs}}$ expresses the faintest planet that can be detected at the desired S/N in a certain exposure time.

Exposure time calculator
------------------------

Assuming negligible read noise and dark counts, we approximate the exposure time required to image a planet with $\Delta {\rm mag_p} = \Delta {\rm mag_{obs}}$ as $$\label{tau_equation}
    \tau = \left({\rm S/N}\right)^2 \left(\frac{{\rm CR_p} + 2\, {\rm CR_b} }{{\rm CR_p}^2}\right),$$ where $\rm{S/N}$ is the signal to noise ratio desired for the planet, ${\rm CR_p}$ is the photon count rate for the planet, ${\rm CR_b}$ is the photon count rate for the background, and $\Delta {\rm mag_{obs}} <  \Delta {\rm mag_{floor}}$, where the systematic noise floor $\Delta {\rm mag_{floor}}$ is the dimmest planet detectable at the given S/N [@2005ApJ...624.1010B]. The factor of two in front of the background count rate is due to the necessity of background subtraction. The planet count rate is given by

$$\label{CRp_equation}
    {\rm CR_p} = F_0\, 10^{-0.4\left({\rm m}_V + \Delta {\rm mag_{obs}}\right)}\, A\, \Upsilon\, T\, \Delta \lambda,$$

where $F_0$ is the zero-magnitude flux at $V$ band, $m_V$ is the stellar apparent $V$-band magnitude, $A = \pi D^2 / 4$ is the effective collecting area of the 