Estimation of Pointing Errors of Large Radio Telescopes under Solar Radiation Based on Digital Twin

Abstract

The pointing accuracy of large radio telescopes is affected by antenna structure errors, installation errors, servo errors, gravity, and varying environmental factors. Although an on-line pointing model had been developed in the past for several large radio telescopes to correct these effects, it is also valuable to explore new ways to correct these effects. Therefore, in order to estimate the dynamic pointing error of large radio telescopes under solar radiation, a new way based on digital twin (DT) is used in this paper. Digital models of the Wuqing 70-m radio telescope (WRT70) are created based on the DT operational framework. Finally, the reliability of the WRT70’s DT system is verified, and the dynamic pointing error of WRT70 (staying at the position of the elevation angle of 90° and the azimuth angle of 180°) on two sunny days in different seasons is estimated by the created DT system. The research results show that the pointing error of large radio telescopes under solar radiation fluctuates greatly and the seasonal differences are very striking. The maximum elevation pointing error of WRT70 under solar radiation around the winter solstice is over 40 arcsec, which is equal to 4/11 of the beam width for WRT70 operating at 8 GHz. In this paper, it is feasible to estimate the time-varying pointing errors of large radio telescopes at rest under solar radiation by constructing a DT system, but it is not effective to estimate the dynamic pointing errors of large radio telescopes in operation. It is expected to establish a dynamic pointing error model and calibrate the dynamic pointing errors of large radio telescopes in operation by DT technology in the future.

1. Introduction

Radio telescopes serve as key equipment for receiving and detecting weak sources of radio waves, playing an important role in space communication, deep space exploration, radio astronomy, and more. To observe fainter radio signals from distant space, larger radio telescopes were built, enabling observations with high angular resolution. To date, there are two fully steerable radio telescopes in the world with an aperture of 100 m, the Effelsberg Radio Telescope [1] and the Green Bank Telescope (GBT) [2]. In addition, China is building a 110 m fully steerable radio telescope in Qitai, Xinjiang Province [3], and a 120 m fully steerable radio telescope in Jingdong, Yunnan Province [4].

The Wuqing 70 m radio telescope (WRT70) is a typical wheel-track fully steerable radio telescope, which is shown in Figure 1. Theoretically, the fully steerable radio telescope can point in any direction in horizontal space by driving the antenna to rotate around the vertical azimuth axis and the horizontal elevation axis, separately. In fact, there is always a large deviation between the commanded direction and the actual direction of a large fully steerable radio telescope, due to the influence of structure errors, installation errors, servo errors, gravity, and varying environmental factors [5]. The deviation is defined as the pointing error of radio telescopes, which is usually required not to exceed 1/10 of the beam width (half power beam width, HPBW) [6]. For larger aperture radio telescopes, the beam width is smaller, so the pointing error is critical. Therefore, the excessive pointing error of a large fully steerable radio telescope must be estimated and calibrated effectively.

In order to estimate the dynamic pointing error, the thermal behaviors of radio telescopes under solar radiation have been discussed in the literature around the world. Over the past decades, the thermal behaviors of many radio telescopes around the world have been studied by employing finite-element (FE) calculations and relative experiments, including the thermal deformation of the reflector system of the RT-70 radio telescope [7], the thermal design of the quadripod and backup structure of the IRAM 30-m telescope [8], and the thermal analysis of the alidade of the Medicina 32-m telescope and the Sardinia 64-m telescope [9,10,11]. In addition, an on-line pointing model incorporating the thermal behaviors of the telescopes’ structure has been developed for the GBT to correct its pointing error in real time. Based on the on-line pointing model (without nearby pointing calibrations), the GBT can achieve 9 arcsec root mean squared (RMS) pointing accuracy over the entire sky [12]. In recent years, the thermal behaviors of several radio telescopes also have been researched in China based on FE analysis and experimental measurements, such as the thermal effects of the alidade of the Nanshan 25 m radio telescope and the Miyun 50 m radio telescope [13,14], the nonuniform temperature and the thermal behavior of the Tianma 65 m radio telescope [15,16], the thermal deformations of the alidade, and the reflector system of WRT70 [17,18]. To sum up, extensive studies on the thermal behaviors of radio telescopes have been carried out, but the research on estimating the total pointing error caused by the thermal behaviors of large radio telescopes is poor. Although the pointing model incorporating the thermal model of the telescopes’ structure can correct the pointing error caused by thermal behavior of radio telescopes in real time, the pointing model needs to be updated by regular scanning of radio sources over the entire sky, which will take up part of the observation time [12].

With the developments of new generation information technologies (New IT), digital twin (DT) has attracted researchers’ attention and has been applied in the field of radio technology [19,20,21]. DT can create a digital model consistent with a physical device to simulate and predict the running state of the actual device [22]. In digital models, the behaviors and rules of a physical device can be represented by multidimensional models and multi-class parameters. Therefore, it will be very promising to use DT to predict the dynamic pointing error of large radio telescopes under solar radiation without taking up the observation time. In this paper, WRT70 is taken as the research object. Based on the DT system, the thermal behaviors of WRT70 under solar radiation can be presented in visual cloud images, and the dynamic pointing error caused by the thermal behaviors can be estimated via the rule model of the pointing error. In Section 2, the operational framework of the DT system of large radio telescopes is introduced. In Section 3, the digital models of WRT70 are constructed, including the geometric model, physical model, behavior model, and rule model. In Section 4, the twin data of WRT70 are verified, and the pointing errors of WRT70 on two specific sunny days are estimated.

2. Operational Framework of the DT System of Large Radio Telescopes

According to the five-dimensional conceptual model of DT proposed by Professor F. Tao of Beihang University, the DT system of large radio telescopes consists of five parts, including physical telescope device, digital telescope model, twin data, service, and connection and interaction between units [23]. The operational framework of the DT system of large radio telescopes is represented in Figure 2.

First, the local state data of large radio telescopes is collected in real-time via a series of sensors installed on the physical telescope device, and stored in the data center of the observation station. Meanwhile, the digital telescope models are created by computer software based on the sensors sensing data. Then, the global state of the physical telescope device is simulated and calculated by the digital telescope models based on the service unit. In addition, the twin data generated by the digital telescope models can be verified by the sensors sensing data, and the digital telescope models can be optimized and updated when necessary. Finally, the behaviors and rules of the physical telescope device can be predicted by analyzing the twin data.

3. Creation of the Digital Models of WRT70

The core task of building a DT system is to create digital models. The creation of the digital models of WRT70 includes geometric modeling, physical modeling, behavior modeling, and rule modeling, which will be described in detail below.

3.1. Geometric Modeling

Geometric modeling refers to the process of constructing a three-dimensional model of one geometric entity inside the computer by CAD software. There are two common CAD modeling methods: the GUI manual modeling and the instruction parametric modeling. In order to facilitate the modification of the geometric model and improve the modeling efficiency, the instruction parametric modeling is adopted here. The antenna’s structure of WRT70 can be divided roughly into five components: alidade structure, elevation cradle, backup structure, main reflector, subreflector, and quadripod. Based on the geometric size and matching feature data of each component, the three-dimensional geometric model of WRT70 is constructed as Figure 3. The alidade serves as the foundation of WRT70, which can move around the vertical azimuth axis on the azimuth rail. The elevation cradle and the above structures are supported by the alidade via two elevation bearings, which can rotate around the horizontal elevation axis as a whole.

The alidade structure is a large spatial A-shaped truss, which is assembled by 52 rectangular beams with different cross-section sizes. The elevation cradle is composed of a diamond-cross truss, a half-big gear, and several diagonal-braced rectangular beams. The backup structure is a complex truss structure with circular symmetry, including an umbrella-support truss, a central main truss, 32 identical radial string beams, 320 ring string beams of different kinds, and hundreds of auxiliary round beams. The main reflector is a paraboloid with a diameter of 70 m, which is composed of 1328 fan-shaped aluminum plates. The subreflector and quadripod include a hyperboloid-type subreflector with a diameter of 6 m and a quadripod truss, where the subreflector is supported by the quadripod.

3.2. Physical Modeling

Physical modeling refers to the process of loading the boundary load data, measured by sensors from the physical telescope device, into the digital telescope model after equivalent transformation. Our team has installed various sensors on the physical structure of the WRT70 in past work, including 52 thermometers, 2 inclinometers, and 2 encoders. In addition, a weather station was installed around the physical entity of the WRT70 to monitor the environmental data such as atmospheric pressure, air temperature and humidity, solar radiation intensity, etc. The distribution of the above sensors on the WRT70 is shown in Figure 4, where unfortunately the left one of the two-dimensional inclinometers is broken now.

The reflector system (including backup structure, main reflector, subreflector, and quadripod) and the alidade structure of WRT70 are affected significantly by environmental factors. In order to facilitate the setting of boundary conditions and the loading of external loads, FE modeling is considered as the main way of physical modeling. The element number of the whole FE model of WRT70 is huge, and the duration of each FE modeling and simulation is extremely long. Due to the consideration of computer modeling and analysis speed, the reflector system and the alidade of WRT70 are physically modeled separately here. The FE models of the reflector system and the alidade and the marks of the applied boundary loads are shown in Figure 5. The temperature load is applied to the element nodes of the FE models in the form of surface convection heating and equivalent node temperature. The gravitational load is loaded onto the element nodes of the FE models in the form of gravitational acceleration. Furthermore, the corresponding material properties of WRT70 are listed in

Table 1. Material properties of WRT70.

Component	Material	Density	Specific Heat	Thermal Conductivity	Thermal Expansion
		kg ${\mathbf{m}}^{-\mathbf{3}}$	J ${\mathbf{kg}}^{-\mathbf{1}}$ ${\mathbf{K}}^{-\mathbf{1}}$	W ${\mathbf{m}}^{-\mathbf{1}}$ ${\mathbf{K}}^{-\mathbf{1}}$	${\mathbf{10}}^{-\mathbf{6}}$ ${\mathbf{K}}^{-\mathbf{1}}$
Main Reflector	Aluminum (2A12)	2700	960	203	23.21
Backup, Quadripod, Alidade	Steel (16Mn)	7850	465	49.8	11.59

.

3.3. Behavior Modeling

Behavior modeling refers to the process of obtaining the behavior characteristics of a physical entity through digital simulation. Based on geometric modeling and physical modeling, the behavior characteristics of WRT70 can be obtained through FE simulation. At 16:00 Beijing Time on 27 February 2021, the structural shadow distributions, the non-uniform temperature distributions, and the corresponding thermal deformations of WRT70 under solar radiation were visualized by contours, as shown in Figure 6, Figure 7 and Figure 8. The structural surface of WRT70 is coated with white polyurethane paint. According to the relevant research in Reference [24], 0.26 is considered as the solar radiation absorption coefficient of the structural surface of WRT70. The secondary reflections of the structural surface of WRT70 are ignored in this paper.

3.4. Rule Modeling

Rule modeling refers to the process of obtaining the specific rules of the physical device through mathematical modeling and iterative calculation of the behavior characteristic data analyzed by the digital model. The dynamic rule of the pointing error of WRT70 under solar radiation is most desirable to be obtained in this paper. The pointing error of WRT70 is closely related to the tilt of the alidade and the pose variations of the reflector system, which is simply depicted in Figure 9. The tilt of the alidade can be simply calculated by combining the deformation displacement of several special nodes on the alidade, which will not be covered here, while the pose variations of the reflector system can only be solved by mathematical modeling and iterative calculation.

3.4.1. Pose Variation Rule of Reflector System

The pose variations in the main reflector need to be determined by solving the mathematical model of the best-fitting paraboloid of the deformed main reflector. The best-fitting paraboloid is a virtual paraboloid closest to the deformed main reflector, shown in Figure 10.

The expression of the best fitting paraboloid is given as

$$z=\frac{x^2+y^2}{4f_{\mathrm{m}}}-\frac{x\Delta x_{\mathrm{m}}}{2f_{\mathrm{m}}}-\frac{y\Delta y_{\mathrm{m}}}{2f_{\mathrm{m}}}+\Delta z_{\mathrm{m}}+y\Delta {\varphi }_{\mathrm{mx}}-x\Delta {\varphi }_{\mathrm{my}}-\frac{x^2+y^2}{4f_{\mathrm{m}}}\frac{\Delta f_{\mathrm{m}}}{f_{\mathrm{m}}},$$

(1)

where $\Delta x_{\mathrm{m}}$, $\Delta y_{\mathrm{m}}$, $\Delta z_{\mathrm{m}}$ are the offsets of the best fitting paraboloid relative to the designed paraboloid along the X-axis, Y-axis, and Z axis, respectively. $\Delta {\varphi }_{\mathrm{mx}}$ and $\Delta {\varphi }_{\mathrm{my}}$ are the rotations of the best fitting paraboloid relative to the designed paraboloid at the apex around the X axis and Y axis, respectively. $\Delta f_{\mathrm{m}}$ is the offset of the focal length of the best-fitting paraboloid relative to the focal length of the designed paraboloid. The relationship between the above six key parameters and the displacement of each node on the main reflector could be expressed as [25]

$$A_{4\times 6}·\chi =B_{4\times 1},$$

(2)

where

$$A_{4\times 6}=\left[\begin{array}{cccccc}\sum _{i=1}^N\frac{x_i^2}{2f_{\mathrm{m}}}& \sum _{i=1}^N\frac{x_i{y}_i}{2f_{\mathrm{m}}}& -\sum _{i=1}^N{x}_i& -\sum _{i=1}^N{x}_i{y}_i& \sum _{i=1}^N{x}_i^2& \sum _{i=1}^N\frac{x_i{z}_i}{f_{\mathrm{m}}}\\ \sum _{i=1}^N\frac{x_i{y}_i}{2f_{\mathrm{m}}}& \sum _{i=1}^N\frac{y_i^2}{2f_{\mathrm{m}}}& -\sum _{i=1}^N{y}_i& -\sum _{i=1}^N{y}_i^2& \sum _{i=1}^N{x}_i{y}_i& \sum _{i=1}^N\frac{y_i{z}_i}{f_{\mathrm{m}}}\\ \sum _{i=1}^N\frac{x_i{z}_i}{2f_{\mathrm{m}}}& \sum _{i=1}^N\frac{y_i{z}_i}{2f_{\mathrm{m}}}& -\sum _{i=1}^N{z}_i& -\sum _{i=1}^N{y}_i{z}_i& \sum _{i=1}^N{x}_i{z}_i& \sum _{i=1}^N\frac{{z_i}^2}{f_{\mathrm{m}}}\\ \sum _{i=1}^N\frac{x_i}{2f_{\mathrm{m}}}& \sum _{i=1}^N\frac{y_i}{2f_{\mathrm{m}}}& -N& -\sum _{i=1}^N{y}_i& \sum _{i=1}^N{x}_i& \sum _{i=1}^N\frac{z_i}{f_{\mathrm{m}}}\end{array}\right],$$

(3)

$$\chi ={\left[\Delta x_{\mathrm{m}}\Delta y_{\mathrm{m}}\Delta z_{\mathrm{m}}\Delta {\varphi }_{\mathrm{mx}}\Delta {\varphi }_{\mathrm{my}}\Delta f_{\mathrm{m}}\right]}^T,$$

(4)

$$B_{4\times 1}={\left[\sum _{i=1}^N{x}_i\left(z_i-z_i^{\prime }\right)\sum _{i=1}^N{y}_i\left(z_i-z_i^{\prime }\right)\sum _{i=1}^N{z}_i\left(z_i-z_i^{\prime }\right)\sum _{i=1}^N\left(z_i-z_i^{\prime }\right)\right]}^T.$$

(5)

According to Equation (2), the pose variations in the main reflector, including $\Delta x_{\mathrm{m}}$, $\Delta y_{\mathrm{m}}$, $\Delta z_{\mathrm{m}}$, $\Delta {\varphi }_{\mathrm{mx}}$, and $\Delta {\varphi }_{\mathrm{my}}$, could be determined.

The subreflector of WRT70 is fixed to the top of the quadripod. The pose variations in the subreflector can be almost equivalent to the ones of the top of the quadripod. There are four key nodes on the top structure of the quadripod, shown in Figure 11. Based on the displacement of the key nodes 1 to 4, the pose variations in the subreflector could be equivalently converted. The offsets $\Delta x_{\mathrm{s}}$, $\Delta y_{\mathrm{s}}$ and $\Delta z_{\mathrm{s}}$ of the actual subreflector relative to the designed subreflector along X axis, Y axis, and Z axis could be expressed as Equations (6)–(8), respectively [18]. The rotations $\Delta {\varphi }_{\mathrm{sx}}$ and $\Delta {\varphi }_{\mathrm{sx}}$ of the actual subreflector relative to the designed subreflector at the focus around X axis and Y axis could be expressed as Equations (9) and (10), respectively, [18]. ($\Delta x_1$, $\Delta y_1$, $\Delta z_1$), ($\Delta x_2$, $\Delta y_2$, $\Delta z_2$), ($\Delta x_3$, $\Delta y_3$, $\Delta z_3$), and ($\Delta x_4$, $\Delta y_4$, $\Delta z_4$) are the displacement of the key nodes 1 to 4, respectively.

$$\Delta x_{\mathrm{s}}=(\Delta x_1+\Delta x_2+\Delta x_3+\Delta x_4/4,$$

(6)

$$\Delta y_{\mathrm{s}}=(\Delta y_1+\Delta y_2+\Delta y_3+\Delta y_4/4,$$

(7)

$$\Delta z_{\mathrm{s}}=(\Delta z_1+\Delta z_2+\Delta z_3+\Delta z_4+)/4,$$

(8)

$$\Delta {\varphi }_{\mathrm{sx}}=\arctan \left[\left(\Delta z_2-\Delta z_4\right)/y_{24}\right],$$

(9)

$$\Delta {\varphi }_{\mathrm{sy}}=arctan\left[\left(\Delta z_3-\Delta z_1\right)/x_{13}\right],$$

(10)

3.4.2. Pointing Error Rule of WRT70

The pointing error of radio telescopes can be subdivided into the elevation pointing error, horizontal cross-elevation pointing error, and vertical cross-elevation pointing error. According to the geometric relationship of the alidade shown in Figure 9, the elevation pointing error $\Delta {\xi }_{1\mathrm{r}}$ caused by the rotation of the elevation cradle can be expressed as [26]

$$\Delta {\xi }_{1\mathrm{r}}=\left[\left(y_{\mathrm{N}}+y_{\mathrm{M}}\right)/2-y_{\mathrm{P}}\right]/h_1,$$

(11)

the elevation pointing error $\Delta {\xi }_{1\mathrm{t}}$ caused by the tilt deformation of the alidade can be expressed as [26]

$$\Delta {\xi }_{1\mathrm{e}}=\left[\left(y_{\mathrm{M}}-y_{\mathrm{U}}\right)+\left(y_{\mathrm{N}}-y_{\mathrm{V}}\right)-\right]/2h_2,$$

(12)

the horizontal cross-elevation pointing error $\Delta {\xi }_{1\mathrm{h}}$ caused by the alidade deformation can be expressed as [26]

$$\Delta {\xi }_{1\mathrm{h}}=\left(y_{\mathrm{M}}-y_{\mathrm{N}}\right)/l_{{\mathrm{M}}^{\prime }{\mathrm{N}}^{\prime }}+\left(y_{\mathrm{F}}-y_{\mathrm{E}}\right)/l_{{\mathrm{F}}^{\prime }{\mathrm{E}}^{\prime }},$$

(13)

and the vertical cross-elevation pointing error $\Delta {\xi }_{1\mathrm{v}}$ caused by the alidade deformation can be expressed as [26]

$$\Delta {\xi }_{1\mathrm{v}}=\left(z_{\mathrm{N}}-z_{\mathrm{M}}\right)/l_{{\mathrm{M}}^{\prime }{\mathrm{N}}^{\prime }},$$

(14)

where $y_{\mathrm{M}}$ and $z_{\mathrm{M}}$ are the displacements of ${\mathrm{A}}_1$-tower top point M along the Y-axis and the Z-axis, respectively. $y_{\mathrm{N}}$ and $z_{\mathrm{N}}$ are the displacements of ${\mathrm{A}}_2$-tower top point N along the Y-axis and the Z-axis, respectively. $y_{\mathrm{P}}$ is the Y-axis displacement of point P. $h_1$ is the distance between the horizontal elevation axis and the elevation driver. $h_2$ is the height of the horizontal elevation axis from the azimuth rail. Furthermore, l is the width of the alidade. In particular, the elevation pointing error $\Delta {\xi }_{1\mathrm{r}}$ caused by the rotation of the elevation cradle and the elevation pointing error $\Delta {\xi }_{1\mathrm{t}}$ caused by the tilt deformation of alidade can be merged into the total elevation pointing error $\Delta {\xi }_{1\mathrm{e}}$ of alidade due to the consistent pointing direction of both.

In order to estimate the pointing error caused by the pose variations in the reflector system of radio telescopes, the concept of beam deviation factor (BDF) was first proposed by Ruze in 1965. BDF is defined as the ratio of the beam deviation of radio telescopes to the focal-line deviation of the parabolic reflector. The expression of BDF could be given as [27]

$$\mathrm{BDF}=\frac{48}{1+2\tau }\frac{f^2}{D^2}\left[1-\frac{32f^2}{D^2}(\tau -1)+\tau \right]+\frac{32f^2}{D^2}\left\{-1+\frac{16f^2}{D^2}(\tau -1)\left[-ln\frac{f^2}{D^2}+ln\left(\frac{1}{16}+\frac{f^2}{D^2}\right)\right]\right\},$$

(15)

where D is the diameter of the main reflector or subreflector, f is the focal length of the primary focal or Cassegrain focal, and $\tau$ is the taper parameter.

Based on BDF and geometric optical path of reflector system, the elevation pointing error $\Delta {\xi }_{2\mathrm{e}}$ caused by the pose variations ($\Delta y_{\mathrm{m}},\Delta y_{\mathrm{s}},{\phi }_{\mathrm{mx}},{\phi }_{\mathrm{sx}}$) of the reflector system can be expressed as [27]

$$\Delta {\xi }_{2\mathrm{e}}=-{\mathrm{BDF}}_{\mathrm{m}}\frac{\Delta y_{\mathrm{m}}}{f_{\mathrm{m}}}+\left(1+\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{m}}\right)\Delta {\phi }_{\mathrm{mx}}+\left(\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{m}}-\frac{\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{s}}}{M}\right)\frac{\Delta y_{\mathrm{s}}}{f_{\mathrm{m}}}-\frac{\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{s}}}{M}\frac{2c}{f_{\mathrm{m}}}{\phi }_{\mathrm{sx}},$$

(16)

the horizontal cross-elevation pointing error $\Delta {\xi }_{2\mathrm{h}}$ caused by the pose variations ($\Delta x_{\mathrm{m}},\Delta x_{\mathrm{s}},$${\phi }_{\mathrm{my}},{\phi }_{\mathrm{sy}}$) of the reflector system can be expressed as [27]

$$\Delta {\xi }_{2\mathrm{h}}=\left[-\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{m}}\frac{\Delta x_{\mathrm{m}}}{f_{\mathrm{m}}}+\left(1+\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{m}}\right)\Delta {\phi }_{\mathrm{my}}+\left(\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{m}}-\frac{\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{s}}}{M}\right)\frac{\Delta x_{\mathrm{s}}}{f_{\mathrm{m}}}-\frac{\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{s}}}{M}\frac{2c}{f_{\mathrm{m}}}{\phi }_{\mathrm{sy}}\right]cosEL,$$

(17)

the vertical cross-elevation pointing error $\Delta {\xi }_{2\mathrm{v}}$ caused by the pose variations ($\Delta x_{\mathrm{m}},\Delta x_{\mathrm{s}},$${\phi }_{\mathrm{my}},{\phi }_{\mathrm{sy}}$) of the reflector system can be expressed as [27]

$$\Delta {\xi }_{2\mathrm{v}}=\left[-\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{m}}\frac{\Delta x_{\mathrm{m}}}{f_{\mathrm{m}}}+\left(1+\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{m}}\right)\Delta {\phi }_{\mathrm{my}}+\left(\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{m}}-\frac{\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{s}}}{M}\right)\frac{\Delta x_{\mathrm{s}}}{f_{\mathrm{m}}}-\frac{\mathrm{B}\mathrm{D}{\mathrm{F}}_{\mathrm{s}}}{M}\frac{2c}{f_{\mathrm{m}}}{\phi }_{\mathrm{sy}}\right]sinEL,$$

(18)

where M is equivalent to $(c+a)/(c-a)$, a is the half of the real axis length of the hyperbolic subreflector, c is the half of the focal length of the hyperbolic subreflector, $f_{\mathrm{m}}$ is the focal length of the parabolic reflector, and $EL$ is the elevation angle of WRT70.

Affected by varying environmental factors, the deformation of the alidade and reflector system WRT70 is the most significant. Ignoring the influence of azimuth rail and elevation cradle, the sum of the pointing errors caused by the deformation of the alidade and reflector system can be taken as the total pointing error of WRT70. Combining Equations (11)–(14) and Equations (16)–(18), the total elevation pointing error $\Delta {\xi }_{\mathrm{e}}$ of WRT70 can be expressed as

$$\Delta {\xi }_{\mathrm{e}}=\Delta {\xi }_1\mathrm{e}+\Delta {\xi }_{2\mathrm{e}},$$

(19)

the total horizontal cross-elevation pointing error $\Delta {\xi }_{\mathrm{h}}$ of WRT70 can be expressed as

$$\Delta {\xi }_{\mathrm{h}}=\Delta {\xi }_1\mathrm{h}+\Delta {\xi }_{2\mathrm{h}},$$

(20)

and the total vertical cross-elevation pointing error $\Delta {\xi }_{\mathrm{v}}$ of WRT70 can be expressed as

$$\Delta {\xi }_{\mathrm{v}}=\Delta {\xi }_1\mathrm{v}+\Delta {\xi }_{2\mathrm{v}}.$$

(21)

4. Verification of Structural Temperature Data and Estimation of Pointing Error for WRT70 under Solar Radiation

Based on the DT system created above, the pointing error of WRT70 (keeping at the position of the elevation angle of 90° and the azimuth angle of 180°) on two specific sunny days (27 February 2021, and 5 June 2021) is estimated in this paper. To verify the reliability of the above DT system, the structural temperature data and the pointing error data caused by the alidade deformation of the WRT70 are verifed below.

4.1. Verification of Structural Temperature Data

The distribution of the 52 temperature sensors on WRT70 is shown in Figure 4, with 20 temperature sensors (T01 to T20) on alidade, 24 temperature sensors (T21 to T44) on backup structure, and 8 temperature sensors (T45 to T52) on quadripod. The simulated temperature data and the measured temperature data for several representative locations (T01, T03, T05 on alidade, and T49, T51, T52 on backup structure) on 27 February 2021 and 5 June 2021 are compared, respectively, in Figure 12 and Figure 13.

The comparison results in Figure 12 and Figure 13 show that the simulated temperature data and the measured temperature data at several representative locations of the alidade and backup structure are in good agreement. The temperature difference at different locations of the WRT70 on 27 February 2021 is larger than that on 5 June 2021.

4.2. Verification of Pointing Error Data Caused by Alidade Deformation

The pointing error caused by alidade deformation can be subdivided into four subitems ($\Delta {\xi }_{1\mathrm{r}}$, $\Delta {\xi }_{1\mathrm{t}}$, $\Delta {\xi }_{1\mathrm{h}}$, and $\Delta {\xi }_{1\mathrm{v}}$) to calculate according to Equations (11)–(14). Meanwhile, the elevation pointing error $\Delta {\xi }_{1\mathrm{r}}$ caused by the rotation of elevation cradle and the horizontal cross-elevation pointing error $\Delta {\xi }_{1\mathrm{h}}$ caused by alidade deformation can be measured by elevation and azimuth encoders, and the elevation pointing error $\Delta {\xi }_{1\mathrm{t}}$ caused by the tilt deformation of alidade and the vertical cross-elevation pointing error $\Delta {\xi }_{1\mathrm{v}}$ caused by alidade deformation can be measured by two-dimensional inclinometers. The calculated data and the measured data for the four subitems of the pointing error caused by alidade deformation of the WRT70 on 27 February 2021 and 5 June 2021 are compared in Figure 14.

The comparison results in Figure 14 show that the calculated data and the measured data for the four subitems ($\Delta {\xi }_{1\mathrm{r}}$, $\Delta {\xi }_{1\mathrm{t}}$, $\Delta {\xi }_{1\mathrm{h}}$, and $\Delta {\xi }_{1\mathrm{v}}$) of the pointing error caused by alidade deformation of the WRT70 are all in agreement to some degree. The fluctuation range in the elevation pointing error $\Delta {\xi }_{1\mathrm{r}}$ caused by the rotation of the elevation cradle on 27 February 2021 is over 20 arcsec according to the calculated data and the measured data of the elevation encoder. The fluctuation range in the elevation pointing error $\Delta {\xi }_{1\mathrm{t}}$ caused by the tilt deformation of the alidade on 27 February 2021 is close to 15 arcsec according to the calculated data and the measured data around the X axis of the two-dimensional inclinometer. The fluctuation range of the horizontal cross-elevation pointing error $\Delta {\xi }_{1\mathrm{h}}$ caused by alidade deformation on 5 June 2021 is over 20 arcsec according to the calculated data and the measured data of the azimuth encoder. The fluctuation range in the vertical cross-elevation pointing error $\Delta {\xi }_{1\mathrm{v}}$ caused by alidade deformation on 5 June 2021 is over 10 arcsec according to the calculated data and the measured data around the Y axis of the two-dimensional inclinometer.

4.3. Estimation of Total Pointing Error of WRT70

According to the rule model of the DT system established above, the total pointing errors of WRT70 under solar radiation can be calculated. Figure 15 depicts the estimated results of the total pointing error of WRT70 under solar radiation on 27 February 2021 and 5 June 2021.

The above results show that the pointing error of WRT70 under solar radiation varies greatly and fluctuates sinusoidally. The elevation pointing error of WRT70 around the winter solstice is over 40 arcsec, which is about seven times larger than the estimated average pointing error of 6 arcsec in elevation direction for WRT70 [28]. The horizontal cross-elevation pointing error of WRT70 around the winter solstice fluctuates little and much less than that around the summer solstice.

5. Conclusions

In this paper, the DT system of a large radio telescope is constructed and the total pointing error of WRT70 under solar radiation is estimated. According to the research results, the following conclusions can be drawn.

(1) The DT system of WRT70 constructed in this paper is reliable to a great degree. The simulated temperature data and the measured temperature data at several representative locations of WRT70 are proved to be in good agreement. The calculated data and the measured data for the four subitems of the pointing error caused by alidade deformation of the WRT70 are all in agreement to some degree.

(2) The structural deformation of large radio telescopes estimated by the DT system can be used to predict the total pointing error. However, it will need to be validated by tracking a source in a subsequent study.

(3) The pointing error of large radio telescopes under solar radiation fluctuates greatly, and the seasonal differences are very striking, which should never be ignored. The maximum elevation pointing error of WRT70 around the winter solstice is over 40 arcsec, exceeding the acceptable pointing error of 11 arcsec for WRT70 operating at 8 GHz.

This study provides a method to estimate the time-varying pointing errors of large radio telescopes at rest under solar radiation, which is different from the on-line pointing model used in the GBT. It will be promising for establishing a dynamic pointing error model and calibrating the dynamic pointing errors of large radio telescopes in operation by DT technology in the future.