## 3 Optical Indices
### Telescope Aperture and Overall Throughput
#### 3.3.2 Total Noise  
Noise is expressed here as the standard deviation of noise in spectral bins. Although the noise of atmospheric signals cannot be obtained in actual measurements, it hasa proportional relationship with OOT noise according to Equation (4), so it is necessary to estimate atmospheric signal noise and evaluate the performance of the instrument by measuring OOT noise. The noise sources to be included in the calculation of \(\sigma_{\text{OOT}}(\lambda)\) are [27]: (1) photon noise from the target source, (2) zodiacal background noise, (3) thermal-radiation noise from the instrument, (4) dark current noise, and (5) readout noise. Photon noise and background noise belong to uncorrelated astrophysical noise sources that produce shot noise obeying Poisson distribution; standard deviation is the square root of the signal. Therefore, the noise component (noise/signal) decreases with stronger signals and longer integration time. Zodiacal background radiation is related to the ecliptic latitude location of the target star. On the basis of the photometric model [28], zodiacal spectral radiance \(L_{zodi}(\lambda)\) in \(\text{W}/\text{m}^{2}/\upmu\text{m}/\text{sr}\) was established as  
\[L_{zodi}(\lambda)=3.5\times 10^{-14}B_{\lambda}(5500K)+3.58\times 10^{-8}B_{ \lambda}(270K), \tag{15}\]  
where \(B_{\lambda}(270K)\) is the Planck function for the thermal-emission component of the interplanetary dust cloud, and \(B_{\lambda}(5500K)\) for the scattered-light component. Zodiacal light is transmitted through the instrument and converges on the focal plane, producing photoelectrons in each spectral bin per exposure as  
\[S_{zodi}(\lambda)=L_{zodi}(\lambda)\tau(\lambda)tQE(\lambda)A_{pix}\Omega_{ pix}\delta\lambda\frac{\lambda}{hc}*C(\lambda), \tag{16}\]  
where \(A_{pix}\) is the area of a pixel, and \(\Omega_{pix}\) is the solid angle at the pixel. \(*\) denotes a convolutional operation, and \(C(\lambda)\) is the boxcar function related to a spectrometer slit image. Infrared thermal-radiation noise from the instrument is the sum of the thermal emissions of every optical element subsequent transmitted, of which the energy is calculated from each emissivity element \(\epsilon(\lambda)\) multiplied by Planck function \(B_{\lambda}(T)\). The spectral radiance of the instrument's thermal-radiation component reaching the detector is  
\[L_{emm}(\lambda)=\sum_{i=1}^{N-1}\left(\epsilon_{i}(\lambda)B_{\lambda}(T_{i}) \prod_{k=i+1}^{N}\tau_{k}(\lambda)\right)+\epsilon_{N}(\lambda)B_{\lambda}(T_{ N}), \tag{17}\]  
where \(N\) is the number of optical elements, and \(\epsilon_{i}(\lambda)\) and \(T_{i}\) are the emissivity and temperature of the \(i\)-th element, respectively. Similar to the zodiacal background, thermal-radiation noise from the instrument is measured as  
\[S_{emm}(\lambda)=L_{emm}(\lambda)\tau(\lambda)tQE(\lambda)A_{pix}\Omega_{pix} \delta\lambda\frac{\lambda}{hc}*C(\lambda). \tag{18}\]  
It was assumed that there was no correlation noise in the simplified model, and the effects of limb darkening and jitter noise were negligible [29]. The theoretical variance of the total noise composed of the above-mentioned sources per exposure is  
\[\sigma^{2}_{total}(\lambda)= \sigma^{2}_{star}(\lambda)+\sigma^{2}_{zodi}(\lambda)+\sigma^{2 }_{emm}(\lambda)+\sigma^{2}_{dc}(\lambda)+\sigma^{2}_{rn}(\lambda) \tag{19}\] \[= S_{\text{OOT}}(\lambda)+S_{zodi}(\lambda)+S_{emm}(\lambda)+N_{ pix}(\lambda)I_{dc}t+2N_{pix}(\lambda)\sigma^{2}_{rpix},\]  
where \(\sigma^{2}_{star}(\lambda)\) is photon noise from the target source, \(\sigma^{2}_{zodi}(\lambda)\) is zodiacal-background noise, \(\sigma^{2}_{emm}(\lambda)\) is thermal-radiation noise from the instrument, \(\sigma^{2}_{dc}(\lambda)\) is dark-current noise, \(\sigma^{2}_{rn}(\lambda)\) is readout noise of the detector with correlated double sampling, \(N_{pix}(\lambda)\) is the number of pixels corresponding to each spectral resolution element, \(I_{dc}\) is the single-pixel dark current, and \(\sigma^{2}_{rpix}\) is the single-pixel readout-noise variance.