The estimate of sensitivity for large infrared telescopes based on measured sky brightness and atmospheric extinction

In astronomical observations, noise is usually caused by fluctuations of electron numbers produced by a signal and background. The signal may be from a star, and the background generally consists of sky background, instrumental background, dark current, and readout noise. In order to find an accurate signal the background needs to be subtracted. However, in the thermal infrared the background is hard to accurately subtract because it usually fluctuates rapidly. Thus, the variance of the background noise may be added twice into the total noise variance (Léna et al. 2012). Hence, the standard deviation of a fluctuating signal and background, which typically obeys a Poisson distribution, can be expressed as Equation (1):

$$\begin{eqnarray}&&\sigma =\sqrt{Sig\cdot t+2[{B}_{{\rm{sky}}}\cdot t+{B}_{{\rm{ins}}}\cdot t+{n}_{{\rm{pixel}}}\cdot (Dark\cdot t+Ro{n}^{2})]},\end{eqnarray} \tag{ 1 }$$

where Sig is the signal electrons per second, B_sky is the electrons per second from the sky background, B_ins is the electrons per second from the total instrumental background containing the emission from a telescope and relay optics, t is the elementary exposure time in seconds for the detector, Dark is the dark electrons per second at a unit pixel, Ron is the RMS readout noise per pixel, and n_pixel is the number of pixels covered by a star image.

The S/N is shown in Equation (2):

$$\begin{eqnarray}\begin{array}{ll}\frac{S}{N} & =\frac{{n}_{f}\,\cdot\, Sig\,\cdot\, t}{\sqrt{{n}_{f}}\,\cdot\, \sigma }\\ & =\sqrt{Sig\,\cdot\, {n}_{f}\,\cdot\, t}\\ & \sqrt{\frac{Sig\,\cdot\, t}{Sig\,\cdot\, t+2[{B}_{{\rm{sky}}}\,\cdot\, t+{B}_{{\rm{ins}}}\,\cdot\, t+{n}_{{\rm{pixel}}}\,\cdot\, (Dark\,\cdot\, t+Ro{n}^{2})]}}\\ & =\sqrt{{n}_{f}\,\cdot\, t}\sqrt{\frac{{B}_{{\rm{sky}}}}{\beta }}\sqrt{\frac{1}{1+2(\beta +\beta \,\cdot\, {\alpha }_{1}+\beta \,\cdot\, {\alpha }_{2})}},\end{array}\end{eqnarray} \tag{ 2 }$$

where β = B_sky/Sig, α₁ = B_ins/B_sky, α₂ = n_pixel · (Dark · t + Ron²)/(B_sky · t), n_f is the number of superposed frames in multi-frame techniques, and n_f · t can be viewed as the total exposure time.

In Equation (2), the Sig, B_sky and B_ins in a given wave-band can be calculated by

$$\begin{eqnarray}&&Sig\,={\tau }_{{\rm{atm}}}{\tau }_{{\rm{ins}}}\eta {F}_{{\rm{sig}}}\pi {(\frac{D}{2})}^{2}\frac{\lambda }{hc},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{B}_{{\rm{sky}}}={\tau }_{{\rm{ins}}}\eta {L}_{{\rm{sky}}}\pi {(\frac{D}{2})}^{2}{\Omega }_{{\rm{star}}}\frac{\lambda }{hc},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{B}_{{\rm{ins}}}=\eta {L}_{{\rm{ins}}}\pi {(\frac{D}{2})}^{2}{\Omega }_{{\rm{star}}}\frac{\lambda }{hc}.\end{eqnarray} \tag{ 5 }$$

In Equations (3), (4) and (5), τ_atm is the atmospheric transmittance, τ_ins is the total instrumental transmittance, η is the quantum efficiency of a detector, F_sig is the flux of a star, L_sky is the sky brightness, L_ins is the radiance of instruments, Ω_star is the solid angle subtended by the image of a star, D is the telescope diameter, h is the Plank constant, c is the velocity of light, and λ is wavelength.

In Equations (4) and (5), the L_sky and the L_ins can be obtained by measurement or simulation. Based on the black-body radiation theory, for any object another form of radiance can also be expressed by

$$\begin{eqnarray}&&{L}_{{\rm{obj}}}={\epsilon }_{{\rm{obj}}}\cdot {L}_{{\rm{obj}}}^{b}={\epsilon }_{{\rm{obj}}}^{* }\cdot {L}_{{\rm{amb}}}^{b},\end{eqnarray} \tag{ 6 }$$

where ε_obj is the real emissivity, ${\epsilon }_{{\rm{obj}}}^{* }$ is the effective emissivity, L_obj is the radiance of an object, ${L}_{{\rm{obj}}}^{b}$ is the black-body radiance at the object's temperature, and ${L}_{{\rm{amb}}}^{b}$ is black-body radiance at the ambient temperature. By using the effective emissivity a more convenient form of α₁ can be rewritten by

$$\begin{eqnarray}&&{\alpha }_{1}=\frac{{B}_{{\rm{ins}}}}{{B}_{{\rm{sky}}}}=\frac{{\epsilon }_{{\rm{ins}}}^{* }\cdot {L}_{{\rm{amb}}}^{b}}{{\tau }_{{\rm{ins}}}\cdot {\epsilon }_{{\rm{sky}}}^{* }\cdot {L}_{{\rm{amb}}}^{b}}=\frac{{\epsilon }_{{\rm{ins}}}^{* }}{{\tau }_{{\rm{ins}}}\cdot {\epsilon }_{{\rm{sky}}}^{* }},\end{eqnarray} \tag{ 7 }$$

where ${\epsilon }_{{\rm{sky}}}^{* }$ is the effective emissivity of sky, and ${\epsilon }_{{\rm{ins}}}^{* }$ is the effective emissivity of the instruments.

From Equations (3) and (4), F_sig can be expressed by

$$\begin{eqnarray}&&{F}_{{\rm{sig}}}=\frac{{L}_{{\rm{sky}}}\cdot {\Omega }_{{\rm{star}}}}{\beta \cdot {\tau }_{{\rm{atm}}}}.\end{eqnarray} \tag{ 8 }$$

Then the exo-atmospheric magnitude of the stellar flux is obtained by

$$\begin{eqnarray}\begin{array}{lll}{m}_{{\rm{sig}}} & = & -2.5{{\rm{log}}}_{10}(\frac{{F}_{{\rm{sig}}}}{{F}_{0}\times \Delta \lambda })\\ & = & -2.5{{\rm{log}}}_{10}(\frac{{L}_{{\rm{sky}}}\cdot {\Omega }_{{\rm{star}}}}{\beta \cdot {\tau }_{{\rm{atm}}}}\cdot \frac{1}{{F}_{0}\times \Delta \lambda }),\end{array}\end{eqnarray} \tag{ 9 }$$

where F₀ is the spectral flux of zero magnitude in a given wave-band, and Δλ is the bandwidth of filter.

Obviously, if the sky brightness and the atmospheric extinction at astronomical sites are given, the relationship between S/N or m_sig and β can be easily obtained by Equation (2) or (9). Thus, for any S/N, m_sig can be calculated.

In the infrared observations on the ground, the detector noise and the instrument noise make the S/N always lower than its theoretical limit (TL) determined by sky background and stellar intensity. In order to quantitatively describe the impact of noise caused by detectors and instruments on the S/N, it is necessary to introduce a physical quantity which can express the decrease of S/N. From the foregoing discussion we find that when (α₁ + α₂) → 0, S/N reaches its TL. Thus, we obtain Equation (10):

$$\begin{eqnarray}&&\left.\frac{S}{N}\right|_{{\rm{TL}}}=\mathop{\mathrm{lim}}\limits_{({\alpha }_{1}+{\alpha }_{2})\to 0}\frac{S}{N}=\sqrt{{n}_{f}\cdot t}\sqrt{\frac{{B}_{{\rm{sky}}}}{\beta }}\sqrt{\frac{1}{1+2\beta }}.\end{eqnarray} \tag{ 10 }$$

According to Equations (2) and (10) a physical quantity Q_snr called the quality factor of S/N can be defined as follows using Equation (11):

$$\begin{eqnarray}&&{Q}_{{\rm{snr}}}=\frac{\frac{S}{N}}{\frac{S}{N}{|}_{{\rm{TL}}}}=\frac{\sqrt{1+2\beta }}{\sqrt{1+2\beta +2\beta ({\alpha }_{1}+{\alpha }_{2})}}.\end{eqnarray} \tag{ 11 }$$

Obviously, 0 < Q_snr < 1. The larger Q_snr is, the better is the sensitivity of the telescope. For the given β and Q_snr in Equation (11), α₁ + α₂ can be calculated with Equation (12):

$$\begin{eqnarray}&&{\alpha }_{1}+{\alpha }_{2}=(\frac{1}{{Q}_{{\rm{snr}}}^{2}}-1)(1+\frac{1}{2\beta }).\end{eqnarray} \tag{ 12 }$$

From Equations (10), (11) and (12), one can obtain the following facts for a given sky background:

• For 2β ⪡ 1, $S/N{|}_{{\rm{TL}}}\approx \sqrt{{n}_{f}\cdot t\cdot {B}_{{\rm{sky}}}/\beta }\propto {\beta }^{-1/2}$. The intensity of the stellar flux is far greater than the sky background, and the sensitivity is limited by the signal shot noise.
• For 2β ⪢ 1, which is called the background-limited domain, ${\alpha }_{1}+{\alpha }_{2}\approx ({Q}_{{\rm{snr}}}^{-2}-1)$ and $S/N{|}_{{\rm{TL}}}\approx \sqrt{{n}_{f}\cdot t\cdot {B}_{{\rm{sky}}}/2}/\beta \propto {\beta }^{-1}$. We take the partial derivative of S/N|_TL with respect to β, and then obtain ∂(S/N|_TL)/(S/N|_TL) = ∂β/β. Likewise, taking the partial derivative of the m_sig in Equation (9) with respect to β, we can obtain ∂(m_sig) = 2.5 × log₁₀(e)× ∂β/β. Thus, in this domain, the slight reduction relative to the magnitude limited by sky background can be approximately obtained by

  $$\begin{eqnarray}&&\Delta {m}_{{\rm{sig}}}\approx 2.5\times {{\rm{log}}}_{10}(e)\times \frac{\Delta (\frac{S}{N}{|}_{{\rm{TL}}})}{\frac{S}{N}{|}_{{\rm{TL}}}}=1.086\times (1-{Q}_{{\rm{snr}}}).\end{eqnarray} \tag{ 13 }$$

  Based on the desired Δm_sig, Equation (13) can be used to roughly estimate Q_snr. However, this only applies to the circumstances where S/N approaches TL. By combining Equations (12) and (13), we can use Q_snr to obtain reasonable requirements for detector noise and instrument noise. Thus, according to the Q_snr, the background-limited condition of astronomical sites (BLCAS) can be defined mathematically as

  $$\begin{eqnarray}&&\left\{\begin{array}{l}2\beta \gg 1\\ {Q}_{{\rm{snr}}}\to 1\end{array}.\right.\end{eqnarray} \tag{ 14 }$$

  As illustrated in Equation (13), if we can accept that the maximum magnitude reduction with respect to the magnitude limited by sky background is about 0.05^m, the minimum of Q_snr should be about 0.95. Hence, in the background limited domain, the upper limit of α₁ + α₂ is 0.11, which determines the maximum noise allowed from instruments and detectors.

The sealing windows of the AMIRM are close to the air and far from the cooler which is at −40° C, so the temperature of the windows is significantly affected by air-temperature. Figure 3 shows that the simulating temperatures of the outer window are affected by the airtemperature of −10° C.

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Figure 4 illustrates that the simulating average temperatures of the windows vary with air temperature and are far higher than −40° C under the usual air-temperature conditions. Hence, calibration is necessary to eliminate the fluctuating thermal radiation of the windows. In the M'-band, the multivariate equation (Zhao 2017) obtained by calibrating in a temperature environmental chamber is shown in Equation (15):

$$\begin{eqnarray}\begin{array}{lll}{I}_{{\rm{adu}}} & = & 4.875\times {10}^{5}\cdot t\cdot {L}_{{\rm{sky}}}^{{M}^{^{\prime} }}+8.489\times {10}^{3}\cdot t\cdot {L}_{{\rm{amb}}}^{{M}^{^{\prime} }}\\ & & +2.549\cdot t+707,\end{array}\end{eqnarray} \tag{ 15 }$$

where I_adu is the reading (the unit is in ADU) of the AMIRM, t is exposure time, ${L}_{{\rm{sky}}}^{{M}^{^{\prime} }}$ is radiance of sky, ${L}_{{\rm{amb}}}^{{M}^{^{\prime} }}$ is black-body radiance at the air-temperature. The rms error of Equation (15) is 4.25 ADU.

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In Figure 5 we illustrate a one-hour sequence of the M'-band at the darkest sky brightness of the zenith which had been measured from 21:35 (6:00) to 22:35 (7:00) Beijing time at Ali (Daocheng). The low frequency fluctuations of sky brightness are the sky noise usually subtracted by a chopping technique in the infrared astronomical observations.

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The histogram for the above time sequence is shown in Figure 6, and the mean, standard deviation, minimum, and maximum are shown in Table 2. The M'-band sky brightness at Mauna Kea is also listed as a reference in Table 2, which is calculated by us based on the simulated data of spectral (between 4.605 μm and 4.755 μm) emission of the sky on Gemini Observatory's website ¹ (Lord 1992).

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The spectral flux of the M'-band zero magnitude is 2.2 × 10⁻¹⁵ W · cm⁻² · μm⁻¹ (Léna et al. 2012). Thus, the magnitude per square arc-second of the mean sky brightness in Table 2 is 2.867 mag · arcsec⁻² at Ali, 2.868 mag · arcsec⁻² at Daocheng, and 2.064 mag · arcsec⁻² at Mauna Kea.

The apertures of equipment used to measure the thermal infrared atmospheric radiation are usually very small. Therefore, it is generally impossible to measure atmospheric extinction using infrared standard stars in astronomical site surveys. For this reason we present a convenient method for the thermal infrared-band to measure extinction based on the atmospheric radiation transfer equation, as shown in Equation (16) (Zhao et al. 2018; Wang et al. 2020):

$$\begin{eqnarray}&&{I}_{{\rm{sky}}}=a\cdot (1-{e}^{-{\overline{O}}_{d}\cdot AM}),\end{eqnarray} \tag{ 16 }$$

where I_sky is ADU readings of M'-band sky calibrated by Equation (15), AM is air mass at any zenith angles, and ${\overline{O}}_{d}$ is the average optical depth in M'-band at the zenith.

Figure 7 shows that the measured sky brightness at different air masses is fit by Equation (16), and the ${\overline{O}}_{d}$ obtained by fitting is 0.18 at Ali, and 0.23 at Daocheng. From ${e}^{-{\overline{O}}_{d}\cdot AM}$, the transmittance at the zenith is 0.84 at Ali and 0.80 at Daocheng. As a reference, by using LBLRTM (Line-By-Line Radiative Transfer Model), we calculate the spectral transmittance at the zenith with radiosondes data of Naqu2, and the transmittance results are shown in Figure 8. By integrating the spectral transmittance in M'-band, the mean transmittance is 0.82 at Naqu.

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The atmospheric extinction can be calculated by

$$\begin{eqnarray}\begin{array}{lll}\Delta {m}_{{\rm{ext}}} & = & {m}_{{\rm{atm}}}-{m}_{{\rm{exoatm}}}\\ & = & -2.5{{\rm{log}}}_{10}({e}^{-{\overline{O}}_{d}\cdot AM})\approx 1.086\times {\overline{O}}_{d}\cdot AM,\end{array}\end{eqnarray} \tag{ 17 }$$

where m_atm is the magnitude through the atmosphere, and m_exoatm is the magnitude beyond the atmosphere. Atmospheric extinctions at a unit air mass calculated by Equation (17) are given in Table 3.

The results of atmospheric extinction at Ali and Daocheng are slightly different from MKO because the wave-band of our filter is narrower than MKO's.

Under the circumstances that an infrared telescope is used at its diffraction limit, we have that Ω_star = π(1.22λ/D)². According to Equation (4), the electrons per second at a unit pixel from the sky background at Ali, Daocheng, and Mauna Kea are about 2.567 × 10⁶, 2.565 × 10⁶, and 5.376 × 10⁶ respectively. If an airy disk is resolved at the Nyquist frequency, n_pixel ≈ 4 and ${\alpha }_{2}=4\times (Dark\cdot t+Ro{n}^{2})/({\tau }_{{\rm{ins}}}\eta {L}_{{\rm{sky}}}{\pi }^{2}(0.61\lambda)t\frac{\lambda }{hc})$. The limiting magnitude at S/N = 3 is denoted by m_3σ . Under the condition that the instrumental emission is ignored (α₁ = 0), we calculate the m_3σ by Equations (2) and (9), which is shown in Table 5. The curves that relate S/N and magnitudes at Ali are shown in Figure 9, where the two five-pointed stars mean S/N = 3.

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Table 5. Limiting Magnitudes of 10 m Telescopes without Instrumental Emission

Site	Scientific detector			Commercial detector			${m}_{3\sigma }^{{\rm{BLCAS}}}$
	α2	Qsnr	m3σ	α2	Qsnr	m3σ
Ali	1.5026 × 10−4	0.9999	13.0122m	0.6633	0.7755	12.7361m	13.0123m
Daocheng	1.5026 × 10−4	0.9999	12.9596m	0.6634	0.7755	12.6835m	12.9597m
Mauna Kea	1.5018 × 10−4	0.9999	12.5716m	0.6305	0.7832	12.3063m	12.5716m

In Table 5, ${m}_{3\sigma }^{{\rm{BLCAS}}}$ is 3σ magnitude limited by the atmosphere. The scientific detector has little impact on the limiting magnitude, which is less than 0.0001^m at all three sites. The commercial detector has a significant impact on the limiting magnitude, which is about 0.3^m.

As discussed in Section 2.2, for Q_snr ≥ 0.95, the calculated 3σ magnitudes are greater than 12.956^m, 12.904^m, and 12.516^m at Ali, Daocheng, and Mauna Kea respectively, which decrease at most by 0.056^m relative to ${m}_{3\sigma }^{{\rm{BLCAS}}}$. Obviously, the Q_snr can clearly reflect whether the sky background limited sensitivity is approached. Hence, based on the requirements of observations, we can easily select appropriate detectors using Q_snr. Figure 10 shows the α₂ varying with different detector noise and the curve of α₂ = 0.11 determined by Q_snr = 0.95.

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From Figure 10 we should select the detector such that the readout noise is less than 250 e⁻ for Dark = 10⁻¹ e⁻ · s⁻¹ · pixel⁻¹, and 200 e⁻ for Dark = 10⁵ e⁻ · s⁻¹ · pixel⁻¹. The dark current just provides a small contribution to α₂ because it is far less than the sky background electrons (about 2.6 × 10⁶ e⁻ · s⁻¹ · pixel⁻¹) in M'-band for the above two detectors. The dark current significantly decreases with decreasing the temperature of detectors. Therefore, for deep cryogenic detectors, we should pay special attention to readout noise.

For thermal infrared observations, in addition to detector noise, we should also strictly control instrumental thermal emission, which is determined by the real emissivity and temperature of the instruments. By using the above analysis, we can select a deep cryogenic detector such that the dark current can be ignored and the readout noise is 200 e⁻. According to the measured sky brightness in Table 2 and Equation (6), the effective emissivity of sky is 0.1554 of −5.4° C at Ali, and 0.1898 of −10° C at Daocheng. In order to meet Q_snr ≥ 0.95, based on Equation (7) and the readout noise of 200 e⁻, the effective emissivity of instruments should be less than 0.003 at Ali, and less than 0.004 at Daocheng. Obviously, if the real emissivity is equal to the effective emissivity, it is not necessary to refrigerate the instruments. For three instrumentswith given real emissivity, the required cooling temperatures are also calculated by Equation (6), which are shown in Table 6.

In Table 6, all elements have the same cooling temperatures for ease of calculation. However, the openair elements of a telescope are hard to cool down to the same temperature as the relay optics. Therefore, the relay instrument should be cooled down to a lower temperature than in Table 6, and the real emissivity of the telescope should be kept as low as possible.

By taking the case of Ali site we calculate more possible combinations of detector noise and instrument noise, which are shown in Figure 11. The required cooling temperatures of instruments that vary with the instrumental real emissivity are calculated by Equation (6), which are shown in Figure 12.

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The black five-pointed star in Figure 11 presents the noise of the scientific detector on Keck II. The redbrown five-pointed star in Figure 12 demonstrates that the instrumental thermal noise can be ignored for the instruments cooled down to −135° C. From the above two figures, when the thermal emission of instruments is strictly controlled, we are allowed to choose a less-than-ideal detector whose maximum readout noise is about 250 e⁻. In other words an excellent thermal-control design of instruments makes it easier to select a required detector.