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Abstract
Thermal seeing is one of the factors that affect solar telescope observations. A comprehensive analysis method is developed to quantify the thermal seeing effects. A three-dimensional Large Eddy Simulation (LES) turbulence model is used to obtain the transient flow fields around the primary mirror, the secondary mirror and the heat-stop. The thermal seeing is calculated based on the stochastic dynamic influence of turbulence on the light rays. The key parameters of the simulation were calibrated by experiments, and the simulation results were validated by empirical formulas. This method has been applied to evaluate the thermal seeing of the 2m Ring Solar Telescope (2m-RST). Error allocation is performed based on the research results to ensure the Observation effect of 2m-RST.
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1. Introduction
The field of solar physics necessitates meticulous observation of the Sun and its magnetic activities at high resolutions. To advance the scientific objectives in this domain, significant efforts are being dedicated to the development and construction of ground-based solar telescopes with large apertures across the globe [1]. The utilization of larger primary mirrors in these telescopes can enhance the angular resolution and light-gathering capabilities, But the telescope will also have to withstand greater thermal loads. The enormous heat absorbed by the telescope results in optical structure thermal effects and thermal seeing. Optical structure thermal effects refer to the impact on imaging caused by mirror deformation and changes in refractive index within the lens. Thermal seeing is the effect on imaging caused by turbulence in the surrounding air due to the telescope heating it up. An excellent thermal control system is essential to ensure the image quality and scientific performance of a solar telescope [2].
Thermal seeing is a major factor that affects the image quality of ground-based solar telescopes. In order to mitigate the adverse effects of thermal seeing, it is necessary to simulate the thermal seeing in the design process and improve the design scheme based on the simulation results. The solar telescope has a heat-stop that can absorb most of the heat, resulting in a little thermal load on the secondary mirror (M2) located behind the heat-stop. The primary mirror (M1) and heat-stop bear most of the thermal load, so their thermal seeing is significant [3]. The research on mirror seeing is relatively sufficient, Zago [4] proposed an empirical formula between seeing and the temperature rise of the M1. Regarding the heat-stop, a quantitative relationship between the temperature rise of the stop and thermal seeing has not yet been established. Existing studies have only provided temperature control indicators based on the influence range of thermal plumes and engineering experience: 1. The indicator of NST heat-stop is $1^{\circ }\textrm {C}$, but no quantitative basis has been provided [5]. 2. DKIST heat-stop proposes a temperature rise indicator of $6^{\circ }\textrm {C}$ based on engineering experience [6]. 3. EST heat-stop proposes an indicator of $8^{\circ }\textrm {C}$ based on the qualitative criterion of not generating obvious thermal plumes [7]. 4.GREGO heat-stop proposes an indicator of $6^{\circ }\textrm {C}$ based on engineering experience [8]. These studies only consider the impact of heat-stops on the surrounding temperature field, and the corresponding optical effects have not been taken into consideration. Dr. Yangyi Liu performed optical simulations based on Computational Fluid Dynamics(CFD) results with the objective of achieving a wavefront error below $5nm$. The corresponding thermal control index for achieving this target was found to be $6^{\circ }\textrm {C}$ [9]. This research represents the first investigation into the impact of thermal plumes around the heat-stop on imaging. Building upon this study, the present paper has further undertaken the following tasks: 1) By employing simulation methods that are closer to real-world conditions, the accuracy of the simulation has been improved. 2) Quantified the thermal seeing.
The quantification of thermal seeing is challenging in current research. In order to simulate thermal seeing more accurately, it is expected to capture finer vortex structures within the thermal turbulence. The slow flow velocities of thermal plumes and the sensitivity of turbulence development to initial conditions render the flow process vulnerable to environmental disturbances. Most existing studies use steady-state simulations based on the RANS model [10], while some adopt LES models to investigate the transient effects of flow fields on seeing [11]. However, there is still a lack of consideration for the influence of environmental disturbances. As a result, the range of thermal plume effects is overestimated, and the vortex structures in the flow field are smoothed out under the assumption of time-averaging. The time-averaging of RANS smoothed out the effects of environmental disturbances. Another challenge is the algorithm for obtaining the seeing from the CFD simulated flow field. The spatial distribution of the refractive index structure constant($Cn^2$) can be calculated from the flow field, and the integral seeing is the integration of $Cn^2$ along the observation direction [12]. This method is based on Kolmogorov spectrum and requires the turbulence structure to be isotropic. The thermal seeing phenomenon is more prominent on the surface of the heat source, but the flow in this region is restricted by the wall and does not satisfy the Kolmogorov spectrum, making this method unsuitable [13]. The third challenge involves performing optical simulations based on flow field data. Current studies evaluate image quality through wavefront error, this does not directly reflect seeing. Evaluating local seeing can be compared with site seeing and consider the impact of each component from a global perspective. Error distribution is conducted to provide a reasonable thermal control index for the optical system.
This article presents a method for evaluating the seeing based on astronomical observations, which involves calculating the seeing based on the statistical analysis of the point spread function (PSF) profile after integration over a certain time period. The algorithm employed Large Eddy Simulation (LES) turbulence model [14] in CFD simulations for three-dimensional unsteady flow, and the ray-tracing algorithm for optical simulations in a gradient refractive index medium. The seeing performance of the M1, the heat-stop, and the M2 of the 2m Ring Solar Telescope (2m-RST) is evaluated under different working conditions. In terms of environmental influences, local wind plays a dominant role. As local wind speed increases, free convection transitions to forced air flow, resulting in a significant improvement in thermal seeing. Eventually, the simulation results have been validated and provide a basis for the thermal control indicators of telescopes.
2. Numerical simulation of thermal seeing
Thermal plumes triggered by concentrated solar irradiation on the telescope structure. The upward thermal plume flow transitions from laminar to turbulent. Turbulences are stochastic and causes random fluctuations in the refractive index of air. Consequently, the path and wavefront of light rays passing through these turbulences are randomly altered, generating thermal seeing. To tackle thermal seeing in the 2m-RST telescope, equipped with a Gregorian optical design, a heat-stop is placed in the optical path behind the M1. This heat-stop filters out the light rays outside the field of view, which carry the majority of the heat load. Hence, the M1 and heat-stop are the two primary components that necessitate assessment to mitigate thermal seeing.
Based on the mechanism of thermal seeing, the algorithm for simulation and evaluation can be divided into the following steps: (1) The flow field distribution around the critical structure is obtained using Computational Fluid Dynamics (CFD) simulation, and the refractive index distribution is subsequently determined. (2) Ray tracing simulation is conducted based on the equation of light propagation in a medium with gradient refractive index. (3) Statistical analysis of light spot diffusion on the imaging plane is conducted for a certain period of time, followed by the fitting of the PSF profile function to estimate the thermal seeing. This integrated analysis method involves multi-physics fields such as fluid dynamics, heat transfer, and optics, which is suitable for simulation using COMSOL Multiphysics software.
2.1 CFD flow field simulation
The thermal plume above the M1 and the heat-stop has a Reynolds number that exceeds the critical value, leading to the transition from laminar to turbulent flow regime. However, accurately solving the physical field in the transition zone and turbulent dissipation range is difficult. The slowly rising thermal plume is highly sensitive to small environmental perturbations, such as fluctuations in the inlet wind speed and heat source temperature. In order to more accurately evaluate the thermal seeing, it is expected to capture the temporal evolution of the turbulent vortex structures generated by the thermal plumes more accurately. We employ the large eddy simulation (LES) model in the COMSOL software rather than other simple models, which has a higher resolution and accuracy for solving natural convection [15].
The parameters for environmental perturbations are determined based on experiments. The temperature rise of both the M1 and the heat-stop is limited to 10$^{\circ }\textrm {C}$. To enhance convergence of the model, we use the Boussinesq and incompressible approximations. The final mathematical model is as Eq. (1). $\rho$, $\mathbf {u}$, $\mathbf {F}$ and $\mathbf {K}$ are density, velocity vector, body force vector and viscous stress tensor [16].
2.2 Optical transmission tracing
The refractive index $n$ is predominantly a function of temperature and pressure, as indicated by the Rueger formula [17] is expressed in Eq. (2).
In Eq. (2), $P$, $T$, $\lambda$, and $e$ denote the local pressure(unit:$Pa$), local temperature(unit:$K$), wavelength unit:$\mu m$), and local vapor pressure(unit:$Pa$), respectively. The formula is applicable within the temperature range of 233-373$K$, pressure range of 80-120$Pa$, and wavelength range of 0.3-1.69$\mu m$. The environmental conditions investigated in this study fall within these ranges, and the refractive index of air is primarily influenced by temperature. Within this range, the error associated with calculations using this formula does not exceed $10^{-7}$.
After obtaining the distribution of the refractive index field, a ray tracing simulation based on the optical equation (Eq. (3)) can be conducted to investigate the impact of thermal turbulence on the transmission trajectory of light in a medium with a gradient refractive index. $ds$ is differential arc length of the ray path. $\vec {r}$ is the coordinate of the point [18].
2.3 Computation of thermal seeing
In most astronomical observations, seeing is quantified using long exposure images, where the exposure time exceeds the time it takes for wavefront phase inhomogeneities to pass through the telescope pupil. This results in an overall blur effect caused by the cumulative image motion effect of seeing. There are several techniques available for quantifying the level of image blurring, with the broadening of PSF by atmospheric turbulence commonly employed as a measure of seeing.
To evaluate thermal seeing, numerical simulation methods are employed based on the measuring principle of seeing. The long-exposure process is discretized into multiple time steps in computational fluid dynamics due to computational constraints, with a time step length of $0.1s$ selected to achieve a $10Hz$ resolution. However, the light propagation process only requires a much shorter time scale of a few tens of $ns$, which is much less than the CFD time step. Therefore, the refractive index field during light propagation can be assumed constant using the Rayleigh frozen hypothesis.
In ray tracing, the M1 is illuminated by collimated rays that are uniformly distributed. At each time step in the computational fluid dynamics simulation, a spot diagram can be generated at the focal plane. The total spread can be statistically analyzed by accumulating spot diagrams over a certain period of time, allowing for the evaluation of thermal seeing. The statistical result can be fit to a Gaussian function. Thermal seeing can be solved based on Eq. (4). $\sigma$ represents the standard deviation, $r_{F W H M}$ represents the half-width and f represents the focal length.
The reliability of the algorithm can be verified by empirical formulas. Lorenzo Zago [4] proposed an empirical equation for mirror seeing, neglecting other factors and primarily considering the impact of temperature field disturbances on the thermal-induced seeing, which has an associated error margin of approximately $\pm 25{\%}$. Equation (5) represent the relationship between mirror seeing and the temperature rise of the M1 under natural convection and forced convection conditions, respectively. $\theta$ represents mirror seeing($^{\prime \prime }$). $D$ is M1 diameter.
3. Numerical simulation of thermal seeing
3.1 2m ring solar telescope
The 2m-RST adopts a Gregorian structure. M1 is ring-shaped, providing the same diffraction limit as a full-aperture circular mirror. This design allows for more space to optimize the F1 heat-stop, and improves the seeing by keeping imaging light away from the heat-stop. Table 1 and Fig. 1 present the relevant parameters and a Structural design of the 2m-RST respectively. The 2m-RST will be installed by the Fuxian Lake in Yunnan Province, where the daytime atmospheric seeing can reach 0.7 under the best conditions. The dome will be designed as fully open, and the lake-land breeze driven by sunlight will improve the thermal seeing.
Table 1. Parameters of the 2m-RST.
At the f1 position, where the heat-stop is located, the power density is extremely high. To minimize heat absorption, the heat-stop of the 2m-RST is designed to be reflective, deflecting light outside the field away from the telescope. The reflectivity should be maximized to reduce heat absorption. As shown in Fig. 2, the heat-stop incorporates a fin array structure in its internal flow-guiding cavity to enhance heat transfer efficiency and prevent excessive temperature rise.
Assuming a reflectivity of $95{\%}$ after coating, the power density and total power of M1, heat-stop, and M2 are presented in Table 2.
Table 2. 2m-RST thermal load analysis.
3.2 Algorithm principle
The geometric model used in CFD simulations needs to be simplified, as shown in Fig. 3(a), with the main consideration being the thermal seeing caused by M1, M2, and heat-stop. To ensure that the flow fields at the inlet and outlet are fully developed when simulating natural wind blowing, the computational domain should have sufficient size, and proper extensions should be applied in the downwind direction. Figures 5(b-c) show the computational domains for natural and forced convection respectively, with small rectangle indicating locally refined mesh regions.
Fig. 3. (a) Simplified geometric model. (b) Area of natural convection. (c) Area of Forced convection.
The simulation combined with optical tracing was only performed in the region shown in Fig. 4(a). The grid density of the thermal-induced turbulence and imaging light overlapping area needs to be refined, and a grid independence study is performed in CFD simulation to obtain accurate simulation results. Y+, as a dimensionless parameter measuring the distance from the wall to the first grid point, is less than 5(Fig. 4(b)), which meets the requirements of the selected turbulence model. The specific parameter settings of CFD simulation are shown in Table 3.
Table 3. Model Selection and Boundary Setting of CFD.
In transient mode, the thermal plume develops, and only fully developed flow conditions are considered. Turbulence sensitivity to initial values is an easily overlooked detail that affects results. The low flow velocity of the plume under natural convection makes its development and dissipation vulnerable to environmental disturbances. The plume of heat-stop is more influenced by the environment than the M1.
Figure 5 depicts the plume development around the heat-stop when the temperature rise in the illuminated area is $4^{\circ }\textrm {C}$, and the cooling liquid is $-1^{\circ }\textrm {C}$. Figures 5(a-b), demonstrates that without environmental disturbance, most of the flow in a large area remains laminar, which is unrealistic. In reality, disturbances always exist, causing laminar flow to transition to turbulence and dissipate into the environment, as shown in Figs. 5(c-d).
Fig. 5. Temperature field. (a) X-Z cross section without environmental disturbance. (b) Y-Z cross section without environmental disturbance. (c) X-Z cross section with environmental disturbance. (d) Y-Z cross section with environmental disturbance.
The setting of environmental disturbance intensity needs to be determined through experiments. In this experiment, a temperature-controlled heating copper block was used in conjunction with a laser rangefinder above it, as show in Fig. 6(a). The dimensions of the copper block is similar to the heat-stop ($10cm$), and the conclusion drawn from this is applicable to the simulation of the heat-stop. The direct measurement of the laser rangefinder is the optical path length, which is subject to random fluctuations due to thermal turbulence. When the temperature of the copper block is raised by $5^{\circ }\textrm {C}$, the signal-to-noise ratio of the optical path disturbance measurement at a distance of $10cm$ above it is about 10, and the instrument accuracy is sufficient [19]. A reference simulation was designed as show in Fig. 6(b), and simulation parameters were adjusted based on the experimental results to make the simulation results close to the experimental results.
The experimental and simulation results are shown in the Fig. 7. In the experiment, the strength of environmental disturbances under natural convection is measured by using a copper block and Fig. 7(a) shows the experimental results. In Fig. 7(a), the optical path length is affected by optical turbulence, resulting in high-frequency random small vibrations. In addition to these small vibrations, there are also random large-amplitude mutations (marked with red circles), caused by turbulence intermittent phenomenon. Figure 7(b) shows the experiment results removing the data affected by the intermittent turbulence, as they would interfere with turbulence statistics [20] and affect measurement results. Figure 7(c) shows the simulation results. It could be made out that the experimental and simulation results could match well with similar amplitude of the optical path vibration and similar rising trend. However, due to limitations in experimental equipment, the optical path length between the experiment and simulation is slightly different. Comparative studies were conducted at different temperature increases, wind speeds, and heights. A $5{\%}$ random fluctuation in the surface temperature rise of the heat source and a $2{\%}$ random fluctuation in the inlet wind speed have been determined. Other simulation parameters were adjusted based on these settings to better simulate the effects of environmental disturbances. It is worth noting that although intermittency was ignored in the data processing, it is important to consider intermittency when analyzing actual engineering problems.
Fig. 7. Optical path length. (a) Experimental results. (b) Experiment results removing the data affected by the intermittent turbulence marked red in (a). (c) Simulation results.
According to Eq. (4), the refractive index is dependent on both air pressure and temperature. As shown in Fig. 8, the temperature field exhibits a strong correlation with the refractive index field, indicating that the refractive index is primarily influenced by temperature.
Fig. 8. Natural convection nephogram. (a) Pressure nephogram. (b) temperature nephogram. (c) refractive index nephogram.
Based on the refractive index field(Fig. 8(c)), ray tracing was performed in a medium with a graded refractive index using the Comsol geometric optical module. This method is based on geometric optics and ignores the diffraction effect of light waves. Point sources were placed uniformly on the M1(Fig. 9(a)), and parallel light rays were emitted from them in various directions to simulate different fields of view. The Fig. 9(b) illustrates the area of ray tracing and the direction of ray propagation. The color legend represents the length of the light propagation path.
Figure 10 displays the point diagram of Comsol optical simulation results for fields of view of $0^\prime$, $0.83^\prime$, $1.6^\prime$, and $2.5^\prime$, respectively corresponding to the four images labeled 1, 2, 3, and 4 in the picture. The color axis on the right side of Fig. 10(a) represents the distance between the initial release position of the light ray and the optical axis. Transverse and longitudinal coordinates represent the scale of the focal plane. The rms value above the point diagram is the rms statistical value of the distance between each point and the centroid position, which measures the dispersion degree of the point diagram. The circle in the lower left corner as show in Fig. 10(a) represents the Airy spot after considering diffraction effects. The scatter plots for all four fields of view are smaller than the Airy spot, so the influence of optical aberrations can be ignored when evaluating thermal seeing. The optical model was designed using Zemax software, and the point diagrams of Comsol and Zemax are similar as show in Fig. 10, indicating high accuracy of Comsol optical simulation.
Figure 11(a) shows the spot diagrams of four fields obtained by ray tracing based on the refractive index field. Figure 11(b) shows the superposition of 20 frames integrated over 2 seconds at a frame rate of 10 frames per second. The corresponding thermal seeing($\theta$) is shown above the spot diagram in Fig. 11(b). After being affected by thermal turbulence disturbance, the diffraction spots of the four fields expand significantly and become similar. The elongation of the PSF in the long-exposure image is slightly wider than that in a single-frame image. To simplify matters, only the $0^\prime$ field of view was focused on in the follow-up study.
We performed a statistical analysis on the spot diagrams for the $0^\prime$ field of view using a sample size of 3,600 points. The frequency distribution of all sample points is presented in Fig. 12(a). Fitting the distribution yielded a standard deviation of $\sigma =21.5\mu m$, indicating consistency with the Gaussian function approximation, as show in Fig. 12(b). Solving Eq. (4) yielded $\theta =0.799^{\prime \prime }$.
Fig. 12. Gaussian function fitting scatter plot. (a) Frequency statistics. (b) Gaussian function fitting.
In the following chapters, we will further investigate the thermal seeing of the M1, heat-stop, and M2 separately. The analysis will be conducted under different pointing angles and wind speeds.
3.3 Mirror seeing
The M1 of the solar telescope is located in front of the heat-stop and has a high thermal power load. It heats up a large volume of air, which greatly affects the thermal seeing. When thermal equilibrium is reached, there is a stable temperature field distribution inside the mirror body. To obtain the temperature field distribution in the M1, a fluid-thermal coupling simulation was performed, using the steady solve as the initial condition for the subsequent transient simulation. The environmental disturbance parameters in the simulation were based on experimental findings. Figure 13(a) shows the hot plume distributions on the M1 under different temperature rises. Figure 13(b) shows the relationship between thermal seeing and temperature rise, where the cross points are the simulation results and the red and green lines represent the upper and lower bounds given by Eq. (5), which provides the error range of the empirical equation. Simulation results are consistent with empirical formulas, except for temperatures below $1^{\circ }\textrm {C}$ where the results are larger. The effectiveness of the algorithm was validated within a certain level of accuracy.
Figure 14(a) presents the thermal plumes around the M1 of the telescope under natural convection conditions, while Fig. 14(b) shows the corresponding thermal seeing results. As the pointing angle of the telescope increases, there is less overlap between the thermal plumes and the imaging light, leading to an improvement in thermal seeing.
Under natural convection conditions, thermal plumes slowly rise. Natural convection can be transformed into forced convection by wind. This transformation will have a significant impact on thermal seeing. Figure 15(a) displays thermal plumes around the M1 at different wind speeds. With increasing wind speed, thermal plumes shift from vertical to horizontal direction along the wind direction, reducing their overlap area with imaging light. Under superposed forced convection conditions, heat transfer efficiency increases, resulting in a decrease(Fig. 15(b)) in the surface temperature rise of the M1 and a marked improvement in thermal seeing, as show in Fig. 15(c). In the given data, lower wind speed data points fall within the range predicted by Eq. (5). As wind speed increases, the data points deviate more from the predicted values, which can be attributed to the fact that the empirical formula does not account for pressure perturbations. Moreover, the thermal seeing does not monotonically decrease with increasing wind speed. Enhanced pressure perturbations at high wind speeds can lead to a deterioration of thermal seeing. However, the influence of pressure perturbations on thermal seeing is approximately one order of magnitude smaller than that of temperature fluctuations. Aside from thermal seeing factors, the effects of strong wind-induced frame vibrations and thin mirror surface deformation on image quality should be considered.
Fig. 15. (a) Temperature field of M1. (b) Wind speed vs temperature rise. (c) Wind speed vs thermal seeing.
To mitigate the thermal seeing effect of the M1, it is necessary to reduce the influence of the thermal plumes above it on the imaging light. There are two main technical solutions: 1) blowing air on the back of the M1 to cool it down. Due to the time required for the M1 to reach the set temperature and the error of the temperature control system, it is preferable to have a surface temperature of the M1 slightly lower than the ambient temperature, with a temperature difference of −1$^{\circ }\textrm {C}$, as shown in Fig. 16(a), in which case the cold plumes develop in a direction opposite to that of the light path(black line), and the overlap area is smaller. The resulting thermal seeing value is $0.15^{\prime \prime }$, which is smaller than the thermal seeing value of $0.52^{\prime \prime }$ corresponding to a temperature difference of +1$^{\circ }\textrm {C}$. 2) Using the air knife technique, which involves setting air outlets along the inner edge of the annular M1 to blow air radially, causing the airflow to blow against the M1 surface. As show in Fig. 16(b), the streamline chart represents the velocity field, while the cross section represents the temperature field. The air knife technique forces natural convection to transition to forced convection, which can more quickly control the surface temperature of the M1 and reduce the range of influence of the thermal plumes. The thermal seeing can be controlled to $0.0591^{\prime \prime }$ with the air knife technique, which is significantly better than the value of $0.52^{\prime \prime }$ without it. Therefore, the following suggestions are proposed for M1 temperature control: 1) The M1 temperature should be slightly lower than the ambient temperature. 2) To optimize the thermal seeing effect, the air knife technique is more effective, and both techniques can be used simultaneously.
3.4 Thermal seeing of the remaining portion
The heat-stop induces thermal seeing that is analogous to the M1. The thermal seeing performance of the heat-stop under various conditions of temperature rise, pointing angle and wind speed is presented in Fig. 17. The heat of the heat-stop is mainly taken away by the internal cooling liquid, and the surface temperature rise does not decrease significantly with the increase of wind speed. The results in Fig. 17(c) did not consider the influence of temperature rise on wind speed, and the results were simulated when the temperature rise was $5^{\circ }\textrm {C}$.
Fig. 17. (a) Heat-stop thermal seeing vs temperature rise. (b) Heat-stop thermal seeing vs zenith angle. (c) Heat-stop thermal seeing vs wind speed.
The M2 mirror is situated downstream of the heat stop and has a much smaller thermal load than M1. Under natural convection conditions, the temperature rise of M2 is only $0.35^{\circ }\textrm {C}$. The imaging light area is scarcely influenced by the thermal plume. The M2 seeing is only 0.083$^{\prime \prime }$, compared to the M1 seeing of 0.52$^{\prime \prime }$.
The integrated seeing of the three components is shown in the Table 4. The seeing of the M1 with a $1^{\circ }\textrm {C}$ temperature increment is comparable to that of the heat-stop with a $10^{\circ }\textrm {C}$ increment under natural convection conditions, while the seeing of the M2 is negligible. The overall effect of these three components is 0.8$^{\prime \prime }$, which slightly deteriorates from the optimal condition at the station. In most cases, there is a lake-land breeze blowing along the lakeshore, which significantly improves the thermal seeing.
Table 4. Thermal seeing of the 2m-RST.
4. Conclusion
This study employed a comprehensive analysis method that combines multiple physical fields to quantify the thermal seeing of the solar telescope. The analysis method is generic, but the conclusion only applies to the 2m-RST. The key parameters of the simulation were calibrated by experiments, and the simulation results were validated by empirical formulas. The simulation process is closer to the actual situation, and the method for evaluating seeing is similar to traditional observation methods. Using seeing as an indicator can better reflect the impact of thermal issues in solar telescopes on observations, which is in line with the traditional understanding of peers. The quantification of the seeing simulation has been accomplished, and the accuracy is adequate to offer guidance for engineering design.
According to the 2m-RST telescope, as the pointing angles and environmental wind speeds increase, the thermal seeing is improved even further. The thermal seeing of each component and the combined effect under natural convection conditions are presented in the Table 4. As a result, the thermal control indices for a $1^{\circ }\textrm {C}$ temperature rise of the M1 and a $10^{\circ }\textrm {C}$ temperature rise of the heat-stop are provided. This study has laid the foundation for the design of the 2m-RST and future large ground-based solar telescopes.
Funding
National Key Research and Development Program of China; Tsinghua Initiative Scientific Research Program; Tsinghua University Education Foundation; Tsinghua-Jiangyin Innovation Special Fund; Key Technology Research on the heat stop of Large Solar Telescope (U1931124).
Acknowledgments
A special acknowledgment should be shown to Prof. Zhenyu Jin and Prof. Zhong Liu from the Yunnan Observatories, Chinese Academy of Sciences, for his guidance we benefited greatly. Meanwhile, we are very grateful to the reviewers for their valuable advice. We also would like to express our gratitude to Dr. Xu Tan, Dr. Xiaojuan Zhang, Dr. Xiqun Wang, Dr. Bin Wang, Dr. Gang Hu, for their help.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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