## 3 Optical Indices
### Telescope Aperture and Overall Throughput
#### 3.3.3 Selection of Telescope Aperture and Overall Throughput  
According to the \(S_{OOT}(\lambda)\) and \(\sigma_{OOT}(\lambda)\) deduced above, \(SNR_{A_{gt}}(\lambda)\), \(SNR_{A_{se}}(\lambda)\) and \(N_{transit}\) can be estimated as in Equations (5), (7), and (9). The observation feasibility of the selected astronomical targets is judged by comparing \(N_{transit}\) with the maximal number \(N_{limit}\) of possible transits during the mission, \(N_{limit}=L_{tel}/P\), where \(P\) is the orbital period of the planet and \(L_{tel}\) is the operation lifetime of the mission. The ExoSim simulator [30] was used as the test tool to calculate the required transit number achieving the \(SNR_{goal}\) under different throughputs \(\tau\) and telescope apertures \(D_{tel}\), where \(D_{tel}=2\sqrt{A_{tel}/\pi}\). Planetary system GJ 1214 b [31] was assumed as an example of an observation target, of which the parameters are shown in Table 1. GJ 1214 b is considered to have abundant atmospheric molecular species, which makes it an excellent example for detecting the atmospheric characteristics by transit spectroscopy. It is also representative of a warm super-Earth orbiting an M dwarf star, imposing challenges to transit observation. When GJ 1214 b proved to be a feasible sample for the optical system, a large number of planetary targets were detectable. Planetary atmospheric contribution \(A_{pt}\), as shown in Equation (3), adopts further approximation assuming that the full-band starlight is completely absorbed by the atmosphere when \(n_{H}(\lambda)=5\)[23]. The energy distribution of stellar radiation adopts the Phoenix model [32], and the linear limb darkening coefficient was set to 0.2674 [33]. The emissivity of the optical element was set to 0.03, and operating temperature was 70 K. Detector properties refer to the indices of Teledyne H1 RG detector [34] with a single exposure time of 20 s. Systematics could be considered and tested by ExoSim. Uncertainty from systematic noise sources is quantified in ExoSim by capturing complex time-domain effects and performing Monte Carlo simulations to obtain the spectral distribution of transit depths. The test results are shown in Table 2 with a fixed telescope diameter. \(SNR_{1}\), adopting the mean value of \(SNR_{A_{gt}}\) and \(SNR_{A_{se}}\), was calculated as the average of each channel over the wave band in a single transit observation, which varies with the overall throughput of optical system. Under the condition of \(SNR_{goal}=7\), numbers of transits \(N_{transit}\) were also calculated and are listed in the table, as well as the average \(N_{transit}\) in two channels. The change in \(D_{tel}\) made \(SNR_{1}\) and \(N_{transit}\) different. Using the same method, the relationship between the optical-instrument parameters and the average \(N_{transit}\) of two channels is shown in Figure 3. A larger telescope aperture means a larger collection area for the optical system, which leads to an increase in \(SNR_{1}\) and decrease in \(N_{transit}\). In addition, \begin{table}
\begin{tabular}{c c c c c} \hline Exoplanet system & \multicolumn{3}{c}{Super-Earth GJ 1214 b} \\ Star features & M4.5 dwarf, 0.2213 R\({}_{\odot}\), 0.176 M\({}_{\odot}\), magK = 8.782 \\ Star Phoenix model & T = 3000 K, log g = 5.0, [Fe/H] = 0 \\ Planetary parameters & P = 1.58 d, 6.55 M\({}_{\odot}\), 2.84 R\({}_{\odot}\), T\({}_{p}\) = 604 K, \(\mu\) = 18 g/mol, T\({}_{14}\) = 3161 s \\ \(\mathbf{A_{pt}}\) & & 282.78 ppm \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of GJ 1214 b exoplanet system.