The Tianlin Mission: a 6m UV/Opt/IR space telescope to explore the habitable worlds and the universe

In this work, the reflection and transmission spectra of Earth-like planets are generated using the public code petitRADTRANS (Mollière et al. 2019), assuming an Earth-like surface and atmosphere. The temperature-pressure (T-P) profile we adopted is a simplified toy model with a thermal inversion layer included, as shown in Figure 2. The model atmospheres include molecules of oxygen (O₂), ozone(O₃), methane (CH₄), water (H₂O), nitrogen(N₂), and carbon dioxide (CO₂), with their volume mixing ratios (VMR) the same as those of Earth (shown in Table 2). Absorption, emission and scattering dues to H₂O, O₂, O₃, CH₄, CO₂, N₂ are taken into account. Their model-predicted emission and transmission spectral signatures are shown in Figure 1.

The abundance of each molecule in petitRADTRANS is in unit of mass fractions, which is converted to VMR using the following formula:

where $X_{i}$ is the mass fraction of species $i$, $\mu_{i}$ is the mass of the molecule $i$, $\mu$ is 28.7, the atmospheric mean molecular weight of Earth and $n_{i}$ is the VMR of species $i$.

In total, $4\times 2\times 3\times 2=48$ sets of high-resolution transmission and reflection spectra of rocky planets within HZs are generated, considering four types of host stars (G2, G8, K2, K7, cf. Table 3), two types of planets (1R_⊕, 2R_⊕) with the same composition bulk compositions, and three atmospheric H₂O mixing ratio of 0.1, 1 and 10 times of terrestrial value (i.e. dry, standard and wet, respectively), with or without water cloud. We note that the 2R_⊕ planet may have different atmospheric composition then the Earth’s, which will be amended in the future when more knowledge about such planets is obtained. The semi-major axis values tabulated in Table 3 are the conservative inner limits of HZs corresponding to each spectral type following the work by Kopparapu et al. (2014) for G8 to K7 dwarf stars, and are exactly the Earth’s semi-major axis for a G2 dwarf.

For the water cloud, the irregular shapes of water clouds as the condensates in the atmosphere are included, with a mean particle size of 25$\mu$m. For transmission spectra, we also calculate the spectra with a gray cloud deck at 0.1 bar for reference, where is the position of the top of troposphere.

Note that we do not include stars later than K or earlier than G, as they are not our primary targets. Actually, planets around stars with spectral type later than K have much higher signals than those around GK stars, which means that any configuration that can detect spectral signals from Earth-like planets around K star can detect those Earth-like planets around M and less massive stars assuming the planets are outside of the inner working angle (IWAs). The IWA is the smallest angular separation between host and companion sources at which the faint companion is detectable. On the other side, Earth-like planets around F-type stars have too small contrast ratios for solid detection even by a 6m space telescope with the fancy parameters.

For reflection spectra, surface albedo is set to 0.3 and phase angle to 45^∘. For simplicity, we assume all the planets of consideration are beyond the inner working angles (IWAs) of our proposed coronagraph, which is $\sim$25 and 46 mas for the UV and VIS bands respectively. The latter value corresponds to the conservative middle HZ radius of G0 and K9 stars $\sim$40 and 20 pc away, respectively.

Fig. 3 shows as examples the reflection and transmission spectra with or without water cloud of a twin Earth at 1 AU orbit of a G2 star at three different R. For the reflection spectra, those in yellow include the contribution from starlight leakage with a suppression level of 10^-10 for comparisons. In each of the right panel, the yellow transmission spectrum illustrates the case of a twin Earth atmospheres with gray cloud deck presented at 0.1 bar, i.e., the atmosphere below this deck cannot be probed due to either a large absorption opacity or atmospheric diffraction.

An observed planet reflection spectrum $F_{\rm p}^{\rm R,Obs}(\lambda)$ is simply the addition of planet spectrum, stellar spectrum escaped from a stellar light suppression instrument, and noise, while a transmission planet spectrum $F_{\rm p}^{\rm T,Obs}(\lambda)$ consists of model planet spectrum and noise, assuming stellar contribution has been removed clearly by contrasting the in-transit spectrum with out-of-transit spectrum. The leakage stellar signal is estimated by simply multiplying the stellar spectra by the starlight suppression level $C$.

As described in Section 3.1, the model planet spectrum is generated by petitRADTRANS, while the synthetic stellar spectrum $F_{\ast}^{\rm Mod}(\lambda)$ is obtained from PHOENIX library (Husser et al. 2013). Both spectra are then convolved with a Gaussian profile that corresponds to a certain spectral resolution $R$. Then Poisson noise is added to the convolved spectra assuming a Gaussian distribution, assuming a certain SNR at the reference wavelength at 0.88$\mu$m and any other wavelength of interest, (cf. definition of SNR and SNR($\lambda$) in Section 3.4). In the top panels of Fig. 3, the simulated observation spectra with SNR=20 for reflection spectra and SNR=1000 for transmission spectra from 0.8 to 1.05 $\mu$m are overplotted with error bars as examples.

Although Tianlin is proposed to cover a wavelength range of 0.2 to 1.7$\mu$m, here in this work we use only the $0.8-1.0\mu$m part for the simulation. The main purpose of this simulation is to verify whether water feature can be detected in a specific star-planet system and what is the required integration time. Although water has strong features in NIR, but the current available NIR detectors that may be employed for Chinese missions may probably bear large readout noise and dark current, making it more difficult for detecting water features in NIR than in the optical. In the optical, the strongest water features reside in the 0.8-1.0$\mu$m range, thus including other bands will not enhance signals significantly. On the other hand, enlarging wavelength coverage will require much more time for the spectral information retrievals, particularly for high-R cases. Detailed simulation with wider spectral coverage and on other species will be performed in the future.

In summary, the observed reflection and transmission spectra are given respectively by:

and

where $P(\lambda)$ is the added Poisson noise, at a given SNR.

The key of this simulation is to detect H₂O in the simulated observation, thus its mixing ratio is of primary interest. Following the same method of analyzing real observed planet spectra, a spectral retrieval technique is employed to infer from the observed spectra the temperature structure and chemical compositions of the target planet. We define it as a constraint if a posterior shows a localized peak and has a 1$\sigma$ range of less than 1/5 of retrieved posterior values. For H₂O in our simulation, the input of the mixing ratio is $-3.2$ in logarithmic scale, and thus the 1$\sigma$ range is less than 0.6 orders of magnitude. Our criteria of constraint are thus stronger than those defined in Feng et al. (2018) and Smith et al. (2020), in which one order of magnitude 1$\sigma$ range will define a constraint.

The mixing ratio of a certain molecule is thus retrieved using the Nested Sampling Monte Carlo package PyMultiNest (Buchner et al. 2014) of petitRADTRANS from the simulated observed spectra with a certain SNR and spectral resolution $R$. In this way, the lowest SNR values to achieve for a constraint to H₂O VMR are then determined for each spectrum at given $R$.

We use the noise model of Robinson et al. (2016) for the estimation of SNR that could be achieved by the given instrument. The desired SNR values at a certain $\lambda$ are connected to total integration times $t{{}_{\rm int}}$ for a set of technical parameters and a given noise model. Following Equation 6 of Robinson et al. (2016), assuming a background subtraction will be performed to detect the planet signal, the total integration time $t{{}_{\rm int}}$ required for a reflection spectral to reach a given SNR is related to background count rate, planet photon count rate, i.e.,

where $n_{\rm p}$ is the planet photon count rate, $n_{\rm b}=n_{\rm z}+n_{\rm ez}+n_{\rm lk}+n_{\rm D}+n_{\rm R}$ is the total background count rate, $n_{\rm z}$ is local zodiacal light count rate, $n_{\rm ez}$ is exozodiacal light count rate, $n_{lk}$ is the leakage starlight photon count rate from the coronagraph, and $n_{D}$ is dark noise rate. As our observation shall normally stare on one target for long exposures, the detector readout noise rate is significantly smaller than the other noise and thus is not taken into account in this study. The zodiacal and exozodiacal background count rates are given using Equations 8 & 9 and the quoted V band surface brightness in Stark et al. (2019). More detailed calculations of each noise can be found in Robinson et al. (2016). For example, for the case of an Earth-twin around a $V=5$ Sun-like star, when $C=10^{-9}$, $\eta=0.1$, $D=6$ m, we have $n_{\rm p}=2.85\times 10^{-3}$ s^-1, $n_{\rm z}=5.66\times 10^{-2}$ s^-1, $n_{\rm ez}=1.42\times 10^{-1}$, $n_{\rm lk}=7.38\times 10^{-3}$ s^-1, then $n_{\rm b}=0.21$ s^-1. If we require a SNR of 19.8, we need a $t_{\rm int}=5.64\times 10^{3}$ hrs. The corresponding SNR_mol (as described below) for water is 3.

For transmission spectra, as photon noise from host stars vastly dominate against background noises, SNR can be accurately estimated by $n_{\rm p}/n_{\ast}^{0.5}$, where $n_{\ast}=\eta F{{}_{\ast}}t_{\rm int}$ and $n_{\rm p}=\eta F{{}_{\ast}}t_{\rm int}\Delta(R_{\rm p}/R_{\ast})^{2}$. Thus we have

where $\Delta(R_{\rm p}/R_{\ast})^{2}$ stands for the transmission spectral signal of the tracing molecule. Note that currently, we use a wavelength-independent end-to-end throughput $\eta$ for our simulation.

It is obvious that the SNR is $\lambda$-dependent. We set the reference point of SNR at $\lambda_{0}$=0.88 $\mu$m, just outside of the water band. Considering the integration time is constant in a single bandpass, SNR($\lambda$) can be derived using the following two equations for reflection and transmission spectra

and

Note that the total integration time required to achieve a given SNR is different from the SNR_mol, which is the minimum SNR required to detect a certain molecule as defined for example by Brandt & Spiegel (2014). The main difference between SNR and SNR_mol is the definition of the signal. The signal involved in SNR_mol is the molecular spectral feature imprinted on the continuum, i.e., the difference between the continuum level and the valley value (or the peak value) for the case of emission (or transmission) spectrum. For a given planet spectrum, the two SNRs are numerically connected. Taking the R=150 clear emission spectrum as an example (the blue curve in the top-left of Fig. 3), the signal of SNR is $F_{\rm p}/F_{\star}$ and equals to $\sim 4\times 10^{-10}$ at 0.88$\mu$m , while the signal of SNR_mol for H₂O is $F_{\rm p}/F_{\star}(0.88\mu{\rm m})-F_{\rm p}/F_{\star}(0.94\mu{\rm m})$ $\approx 4\times 10^{-10}-2\times 10^{-10}=2\times 10^{-10}$. So, for this case, the SNR at 0.88$\mu$m is about twice of SNR${}_{\rm H_{2}O}$. Based on this noise-free spectrum, we generate a series of noisy spectra with different SNR${}_{\rm H_{2}O}$, which are used to determine the abundance of H₂O via the retrieval analysis. The smallest SNR${}_{\rm H_{2}O}$ that allows a “constraint” of H₂O abundance is the SNR_mol, and the SNR(0.88$\mu$m) can be calculated from the corresponding noisy spectra. The last step is to calculate $T_{\rm int}$ using Eq. 4.

Our simulation suggests that under the optimized instrumental performance, Tianlin is expected to yield emergent spectra with SNR${}_{\rm mol}=5$ for H₂O with its coronagraph after integrating $10^{2}-10^{3}$ hours for the vast majority of the standard and wet Earth-like planets in the HZs of G8-K7 stars, assuming the phase angle of 90^∘. Increasing $D$ to 6m will make Tianlin capable of measuring the water vapor of a nearby twin Earth in less than $10^{3}$ hrs.

The main results of our simulation on reflection spectra are presented in Fig. 4, which shows the dependence of $t_{\rm int}$ on $D$, $C$ and $\eta$ for a twin Earth with the Earth-like water vapor content. The dependence on dark current rate $d$ is very weak until $d$ exceeds 2e^-/h/pixel, and even lower dark current is already accomplished by modern detector developing companies such as Teledyne e2v²²2https://www.teledyne-e2v.com/. This is reasonable because in this regime, noise is dominated by local zodiacal and exozodiacal light backgrounds.

It is clear from Fig. 4 that $t_{\rm int}$ is very sensitive to $C$, the contrast of a coronagraph or an external starshade, especially when $C>10^{-9.5}$. It must be better than $10^{-9}$ to constrain H₂O of a twin Earth in a maximum affordable integration time, i.e., less than 10⁴ hours, or about 1 year. Therefore, the starlight suppression performance, particularly the contrast $C$ is critical for the achievement of Tianlin’s objective. Such limits can be achieved in a laboratory experiment with a raw contrast $10^{-9}$ or better at visible wavelengths (Trauger & Traub 2007; Dou & Ren 2016; Mejia Prada et al. 2019; Llop-Sayson et al. 2020; Harness et al. 2021), and the improved post-processing analysis could further reduce residual starlight speckles. It will be certainly more challenging to realize such a high contrast on board a spacecraft, mainly due to fast line-of-sight jitter and slow wavefront drift. As shown in Gaudi et al. (2020), the self-induced pointing jitter for HabEx-like telescope at the L2 point with tiny thermal and gravitational gradient disturbances should be nonexistent, while the reaction wheels or microthrusters may induce pointing jitter in the order of 10${}^{-4}{{}^{\prime\prime}}$ rms. The simulation done for HabEx shows that such jitter won’t affect the raw contrast at IWA to the level of $3^{-10}$. On-board testing in space for $>10^{8}$ starlight suppression will be carried out on the near-future Nancy Grace Roman Space Telescope (Akeson et al. 2019), and $\sim 10^{8}$ contrast ratio on the Chinese Space Station Telescope (cf. CSST whitebook, private communication).

In addition, Fig. 4 suggests that the second key technical parameter $D$ should not be smaller than 6m. It requires more than 1550 and 670 hours for $D=4$m or 6m, assuming $C=10^{-10}$ and $\eta=0.1$. These two numbers increase dramatically to 13,000 and 5600 hours for $C=10^{-9}$. Therefore, a primary mirror with $D=4$m seems not large enough to secure robust detection of atmospheric biosignatures from Earth-like planets, especially in the cases that the planet is smaller than the Earth, or the star is larger and brighter than the Sun, or the planet system is more than 10 pc from us. Enlarging $D$ to 6 m will result in a gain in photons by a factor of $\sim$2, which will consequently increase largely the number of planets that can be observed and characterized. It is also shown that the higher $\eta$ is, the shorter $t_{\rm int}$ is required. Increasing $\eta$ seems to have a similar effect as increasing $D$, because both parameters are directly related to the number of photons received.

On the other hand, our results show that using classic low resolution ($R\sim 150$) transit spectroscopy, much longer $T_{\rm int}$ is required to achieve the same SNR_mol. For example, as shown by the starting points of the lines in Fig. 5, a 6m telescope needs to integrate $\sim 50$ and $>180$ hours to constrain H₂O VMR at 0.94$\mu$m for an Earth-sized planet orbiting a K7 and K2-G2 dwarf star $\sim$10 pc away, respectively. Unfortunately, for transit spectroscopy, the total usable integration time is strictly limited by the number of transits that could be covered in the mission’s lifetime and the duration of each transit, with the latter about 13 hrs for the Sun-Earth system. For Tianlin, the conservative mission lifetime is 5 years, with possible 5 years extension. Therefore, the total integration time is approximately 65 and 130 hr, for a habitable planet around a G2 and K7 star, respectively. Increasing $D$ to 8m will significantly shorten the required $T_{\rm int}$, and thus make it possible to detect H₂O with SNR$\sim$5 on an Earth-like planet in the HZ of K type star in mission lifetime, however for the same case around G types stars, one may need $D\geq$8m to detect H₂O.

Lastly, we have studied the advantages of using high-$R$ technique for transit spectroscopy. Our simulations show that increasing $R$ will significantly shorten the required $T_{\rm int}$ time, at least for nearby bright stars. As shown in Fig. 5, for a constraint of water content on an Earth-like planet around a G2 star at 10 pc, Tianlin with $D$ of 6m demands $T_{\rm int}\sim$380, 600 and 210 hours with R=150, 1500 and 15,000, respectively. Increasing R will lead to two contrary changes of SNR and thus $t_{\rm int}$. On one hand, the molecular signal per pixel decreases linearly with increasing $R$, while the detector noise does not change, and thus the SNR will decrease proportionally to the square root of $R$ approximately. On the other hand, when $R$ increases, the total integrated signal of many individual lines of a molecule will first increase quickly because the mixed individual lines at low $R$ will gradually separate from each other, and the peak intensity of each line will increase as well. Nevertheless, the increase will turn slow and eventually stop after all individual lines are separated and the instrumental lines’ width becomes smaller than the intrinsic widths of planet lines. The combination of these two effects results in a decrease and then an increase of $t_{\rm int}$ when $R$ changes from $\sim$100 to $\sim$1000, followed by a steady decrease to $R\sim$15000 for G2 and G8 host star, and to $R\sim$20000 for K2 host stars. This trend is reasonable given that the median separation of every two adjacent water lines in the wavelength range from 0.8 to 1.05$\mu$m is found to be 0.6 AA, corresponding to a resolution of $\sim 13000-16000$. Specifically, when $R=150$, only 0.07% lines could be separated well from their neighbours and thus resolved. This fraction only increases to 1.9% for $R=1000$, and is 53% and 61% for $R=15000$ and 20000, respectively.

We point out one important bonus advantage for the case of $R>15000$ which has not been taken into account in the above analysis. The molecular signals from individual lines can be combined by cross-correlating the observed spectra with a model template spectrum. In this way, the significance of a molecular species is subsequently enhanced by the square root of $N_{\rm lines}$, which is a multiplication factor that takes into account the number and strength of the individual planet lines involved (Snellen et al. 2015). For the 0.94$\mu$m H₂O band, we find that $N_{\rm lines}\geq 100$. Therefore, the required $T_{\rm int}$ in theory for a twin-Earth should be less than 100 hours using the cross-correlating technique. Simulation will be performed in the future to confirm this number.

The high-resolution spectroscopy can be also used in conjunction with the high contrast imaging technique as well to achieve a contrast ratio of $10^{-10}$ in ideal cases (Snellen et al. 2015; Wang et al. 2018). More realistic simulation will be further performed in order to identify the $R$ value which is the best for detecting molecular signals, and in the meanwhile can be realized in a space telescope at reasonable expenses.