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Article Content
1. Introduction
The pointing model of radio telescopes requires a set of calibrators that have known coordinates, high signal-to-noise ratio (SNR), and uniform distribution across the celestial sphere [1–4]. Current calibrators for radio telescope pointing measurements are predominantly astronomical radio sources that meet these criteria. However, the majority of the astronomical radio sources exhibit relatively weak radiation. Astronomical radio sources can only fulfill the requirements for both high SNR and uniform distribution when observed with large radio telescopes [5, 6]. With smaller radio telescopes, the astronomical radio sources that have enough SNR decrease significantly [7]. It is challenging for smaller radio telescopes to observe sufficient high SNR sources to ensure a uniform distribution. Usually, the uniform distribution of the pointing measurement on a smaller radio telescope relies on the Earth’s rotation [8], which typically takes 12–24 h.
To improve the spatial distribution of pointing measurements, some radio telescopes employed optical pointing telescopes (OPTs) [9, 10]. Optical observation toward stars provides a relatively higher SNR compared with radio sources observation, enabling even small optical telescopes to detect hundreds of calibration sources across the celestial sphere [11]. Moreover, some star trackers are capable of measuring in arbitrary direction [12], thereby fulfilling the requirement for uniform distribution of pointing measurements. However, this OPT method solely determines the direction of the mount and does not account for the antenna radio beam direction. Therefore, observing radio sources is crucial for establishing an optical-radio angle model [9, 13, 14].
With known orbits, satellites can be used for telescope or antenna pointing measurements [15, 16]. Large satellite constellation projects like OneWeb and Starlink have led to dense sky coverage [17, 18]. These satellites may cause potential radio frequency interference (RFI) in radio astronomy observations [19], but they can also serve as pointing calibrators for radio telescope measurements. In comparison to astronomical radio sources, these satellites provide more uniform celestial sphere coverage, and sufficient SNR for both large and smaller radio telescopes. Taking Starlink as an example, the satellite constellation can generate dense trajectories in the sky above a radio telescope within four minutes (as shown in Figure 1(a)). For comparison, Figure 1(b) illustrates the spatial coverage of a 25 m radio telescope’s pointing source catalog.
Figure 1 (a)
Figure 1 (b)
The utilization of satellites as calibrators in pointing measurements still encounters technical challenges, with one major issue being the observation method. The commonly used cross-scan method, incorporating tracking and offset scanning of the target, necessitates the target’s speed to be within the telescope’s tracking capabilities. It may be suitable for observing Geosynchronous Earth Orbit (GEO) and some low-speed Medium Earth Orbit (MEO) satellites. Fast-moving low Earth orbit (LEO) satellites like Starlink are difficult for most radio telescopes to track. Therefore, drift-scan observation becomes a more feasible option, in which the satellites’ own motion across a stationary telescope beam achieves the scanning. This method enables the telescope to conduct pointing observations toward any moving satellite, regardless of its speed, including MEO and LEO satellites. So, the fast-moving LEO satellites, which may constitute a significant fraction of the total satellite population in the sky, can be incorporated into the pointing observation through this drift observation method. However, it yields pointing data in an oblique direction that aligns with the satellite trajectory. This oblique data do not permit the direct derivation of the telescope’s pointing model. To solve this, we developed a technique for fitting the radio telescope’s pointing model using this oblique-scan data. We tested this method on the NSRT-25m telescope and obtained the pointing model.
2. Method
2.1. The Existing Pointing Model and Fitting of Model Parameters
The fundamental pointing model for Altitude–Azimuth (AltAz) mounted radio telescope includes the components related to axis error, encoder error, and gravity deformation factors [20]. This model forms the basis for the pointing models of numerous radio telescopes [9, 21, 22]. Equations (1) and (2) present a pointing model recommended by the Field System (FS) (https://github.com/nvi-inc/fs/blob/main/pdplt/pdplt), which is commonly used by Very Long Baseline Interferometry (VLBI) radio telescopes. In this model, x and y represent the Az and El coordinates of the observation direction; Ex(x, y) and Ey(x, y) represent the corrections predicted by the pointing model in this direction and Pn represent the pointing model parameters.
()
()The functions Ex(x, y) and Ey(x, y) depend on variables x and y, with the function form determined by parameters Pn. In pointing measurement, measuring points are evenly distributed in the x − y coordinate space. At each measuring point, the values of [x, y, ex, ey] are observed, where ex and ey represent the measured pointing errors. By defining the cost function as Ex(x, y) − ex and Ey(x, y) − ey and the optimizing parameters as P1 − P22, the pointing model can be determined through minimizing this cost function. Given that the pointing model equations (1) and (2) exhibit linearity with respect to parameter Pn, it is possible to determine the value of parameter Pn through linear regression [23, 24]. The specific steps are as follows.
The cross-scan observation obtains values [x, y, ex, ey] at each measuring point. With a total of n points measured, the values of ex and ey are then compiled into a vector exy with dimensions (2n, 1) as follows.
()The terms in the pointing model equations (1) and (2) are expressed as equations (4) and (5).
()
()If the parameter Pi is absent in equation (1), VXPi(x, y) is assigned a value of zero; similarly, if the parameter Pi does not appear in equation (2), VYPi(x, y) is set to zero. The specific forms of the 22 terms are presented in the Table A1.
Construct the matrix Mxy of dimensions (2n, m) based on VXPi(x, y) and VYPi(x, y), where m represents the total number of Pn parameters, which is 22 in this case.
()Formulate a vector P of dimension (m, 1) with the Pn parameters.
()According to the pointing model (equations (1) and (2)), a relationship exists.
()The Mxy matrix is generated from x and y data in the pointing measurement, while the exy vector can be derived from the measured ex and ey data. Subsequently, the P vector can be solved using multiple linear regression based on the observed data in pointing measurement.
2.2. Observation Principle of Oblique Drift-Scans
In drift-scan observation, an observation point (x, y) is selected on the satellite trajectory. The antenna moves to this coordinate and remains stationary. A local relative coordinate system is then established with this observation point as the origin, as illustrated in Figure 2. Under ideal conditions, without any pointing error, the contour of the antenna beam should be a circle centered at the origin. However, in the cases with pointing errors, this circle may shift to any nearby location.
Figure 2
During measurement, as the satellite moves along the vector 
from the origin of the coordinate system and intersects with the beam, the radio telescope continuously records the received power over time. By combining theoretical orbit of the satellite with this power-time curve, the spatial position A(a, b) corresponding to maximum received power can be determined. Thus, for each drift-scan observation point, a set of [x, y, a, b] data can be obtained.
Assuming that the satellite trajectory is straight in the vicinity and the antenna beam contours are perfectly circular, the actual antenna pointing position should lie on a straight line through point A and perpendicular to line OA (the “perpendicular” hereafter). Evidently, it is not possible to derive the exact value of ex or ey from a set of [x, y, a, b] data; only the linear relationship between the variables can be determined. Therefore, the pointing model fitting method described in Section 2.1 cannot be directly applied.
We propose the following method to determine the pointing model parameters based on [x, y, a, b] data. Assuming that point P[Ex(x, y), Ey(x, y)] is the position predicted by the pointing model, we define the variable residual as the distance between the point P and the “perpendicular” of drift-scan measurement. If the pointing model parameters are accurate and measurement error is ignored, the residual value should be zero. Therefore, by minimizing the residual, we can obtain the correct pointing parameters and establish the antenna pointing model. Equation (9) illustrates the manner in which the variable residual is derived from the measurement data.
()In the equation, a and b are obtained in the measured dataset; Ex(x, y) and Ey(x, y) are functions of x and y, which are also included in the measurement data set. Given a set of measurement data [x, y, a, b], the residual value only depends on pointing model parameters. Therefore, this method is able to determine the parameters based on the [x, y, a, b] data.
This method can be considered as a generalization of the conventional cross-scan pointing measurement approach. In a particular case where b = 0, indicating that the satellite trajectory is parallel to the ex direction, equation (9) simplifies to residual = Ex − a. This facilitates a direct determination of ex through the measured variable a. Such a scenario exhibits similarities to the conventional cross-scan pointing measurement approach.
2.3. Pointing Model Fitting Method Using Oblique Drift-Scan Data
In the pointing model fitting algorithm described by equation (8), the ex and ey data are required to construct the vector exy. However, in oblique drift-scan (as described in Section 2.2), ex and ey are not directly observable. Instead, the directly observed variables are a and b. Consequently, it is necessary to establish the relationship between observed [a, b] and the pointing model.
According to equation (9), the following relationship can be derived:
()Construct a vector L of dimension (n, 1) and a matrix Mab of dimension (n, 2n) as follows:
()
()Based on equations (8) and (10), there exists the subsequent relationship.
()The term Mab·Mxy is generated from the variables a, b, x, and y, while the vector L is generated from the variables a and b. So far, we have successfully established a linear regression model for pointing parameters based on the oblique drift-scan measurement data [x, y, a, b].
In certain scenarios, this linear regression model can also transform into the same form as the cross-scan illustrated by equation (8). For instance, when b = 0, indicating that the scanning direction is parallel to the ex direction and the maximum power point corresponds to a = ex. Under such circumstances, the (n, n) elements on the right half of the matrix Mab constitute an identity matrix, with the remaining elements being zero. At the same time, the vector L’s elements become ex. This is consistent with the linear regression model of cross-scan described in Section 2.1.
2.4. Experiments With Simulated Data
A set of virtual oblique drift-scan data [a, b] was generated by simulated oblique scans based on cross-scan observation data. The oblique-scan directions were randomly assigned in the simulation. We employed the method described in Section 2.3 to derive the pointing parameters from the simulated oblique scan data. Moreover, we compared these parameters with those derived from the cross-scan data. The results obtained from both oblique drift-scan and cross-scan methods are presented in Figure 3.
Figure 3 (a)
Figure 3 (b)
Figure 3 (c)
Figure 3 (d)
Figure 3(b) displays the “perpendiculars” of the oblique-scan data. It can be observed that the original data exhibit a rather disordered distribution. After correction using the fitted pointing model, the “perpendiculars” converge near the zero point. This suggests that the fitted pointing model successfully minimized the average distance from each “perpendicular” to the zero point. Subsequently, the pointing models derived by cross-scan and oblique scan were employed to correct ex and ey data, respectively. The residuals are shown in Figures 3(c) and 3(d). The residual distributions obtained by the two pointing models exhibit close similarity. The root mean square (RMS) values on ex and ey are as follows: cross-scan [5.85, 7.12] arcseconds and oblique scan [6.07, 7.49] arcseconds. Compared with the cross-scan results, the oblique-scan RMS values show an increase of merely 3.9% in the ex direction and 5.3% in the ey direction. This indicates that the method proposed in this study achieved nearly the same accuracy as the conventional cross-scan method.
To validate the degradation of the oblique scan method to the cross-scan method in scenarios where a = 0 or b = 0, we generated oblique scan data [a, b] using two specific directions (0 and 90 degrees). The result of oblique-scan fitting method is depicted in Figure 4. The residuals of ex and ey show an exact match with those obtained from the cross-scan method, and the statistical RMS values are identical. This demonstrates the equivalence of the two methods in this special scenario.
Figure 4 (a)
Figure 4 (b)
Figure 4 (c)
Figure 4 (d)
3. Observation
To experimentally validate the proposed oblique drift-scan pointing measurement method, an observation was conducted utilizing the NSRT-25m telescope in conjunction with the Global Navigation Satellite System (GNSS) constellation.
3.1. The NSRT-25m Radio Telescope
The NSRT-25m is a meridian mount telescope with a 25 m-diameter main reflector. An L-band receiver is installed at its prime focus, enabling it to observe frequencies between 1.0 and 1.4 GHz. Figure 5 displays a photograph of the NSRT-25m telescope. The telescope’s azimuth angle is fixed at either 180 degrees or 0 degrees, and the elevation angle can move within a zenith angle range of ±60 degrees. Because the oblique drift-scan method does not require tracking, the meridian mount of the NSRT-25m telescope fulfills the observational requirements. The GNSS satellites, which are evenly distributed across the sky, transmit signals at frequencies of 1.2 and 1.5 GHz. The NSRT-25m telescope can observe the 1.2 GHz signal.
Figure 5
The pointing model of the telescope is based on equations (1) and (2), with some modifications. First, because of the meridian mounting configuration, the azimuth-related terms can be replaced by constant values. Second, considering that the telescope beam’s full width at half maximum (FWHM) is approximately 0.7 degrees, it is feasible to neglect terms that have both high spatial frequencies and small amplitudes. Equation (14) presents the final pointing model.
()Even though this pointing model has been considerably simplified compared with the standard pointing model, the basic principle of the oblique drift-scan method can still be verified.
3.2. The Selection of Satellites
Due to the frequency range of the NSRT-25m telescope, it is unable to observe signals from Starlink satellites (Ku/Ka/V band). Therefore, alternative satellites within the telescope’s frequency range were chosen for the preliminary verification observations in this study. Future research studies may employ a radio telescope that covers the frequency of Starlink satellites to finalize the verification observations.
The GNSS satellites transmit signals at specific frequencies of 1.2 and 1.5 GHz, with the 1.2 GHz signal observable by the NSRT-25m telescope. The frequency of 1.2 GHz is utilized by the GPS, GLONASS, Galileo, and Beidou constellations. Consequently, by setting the receiver frequency to 1.2 GHz during observations, a relatively large number of satellites are expected to be captured.
The proposed method aims to address the challenge of observing fast-moving LEO objects. However, the method’s assumptions are oblique drift-scan and known trajectories, without restricting the orbit type to LEO. Although most GNSS satellites are MEO objects, they can still satisfy the assumptions of our method. As a result, given their substantial presence within the telescope’s frequency range and their oblique motion behavior, GNSS satellites serve as a favorable alternative in this verification observation.
3.3. Observation Method
The positions of GNSS satellites are calculated using the two-line elements (TLEs) orbit parameters. These TLE data are obtained from the CelesTrak website (https://celestrak.com/NORAD/elements/gnss.txt) and updated before each observation to keep the data up-to-date. The pyephem (https://rhodesmill.org/pyephem/) package is used for reading TLE data and computing satellite trajectories in the AltAz coordinate system. The coordinates [x, y] of the observation point are determined by solving the intersection between the telescope’s motion path and the satellite’s trajectory. The telescope is driven to the observation point and remains stationary until the satellite crosses the radio beam. During the satellite moving through the radio beam, the signal power received by the telescope is recorded at a sampling rate of 2 Hz.
This observation consists of two stages. The observations of stage one were carried out on 12-JAN-2022 and 28-JAN-2022, without employing a pointing model. Thereafter, based on the data from the stage one, a pointing model was established. With the implementation of this pointing model, the stage two observations were carried out on 30-MAR-2022 to evaluate the pointing model.
3.4. Data Processing and Result
The trajectories of the satellites during the stage one observation are shown in Figure 6. Because the telescope is meridian mounted, the Az angle of pointing is approximately 0 or 180 degrees. An illustrative power curve for satellite drift-scan is presented in Figure 7. By fitting a Gaussian profile to the power-time curve, the time corresponding to the maximum power point can be determined. Subsequently, by combining this information with TLE orbit data, the satellite’s position corresponding to the maximum power ([a, b] in equation (9)) can be calculated. The Az angle of the telescope remains fixed at either 0 or 180 degree, the El angle corresponds to the telescope motion command angle. The values can be combined into pointing data [x, y, a, b], which are the input data for the oblique drift-scan pointing model fitting. Table A2 displays the data of stage one observation.
Figure 6
Figure 7
We utilized the oblique drift-scan model fitting method in conjunction with the pointing model equation (14) to fit the data of stage one. The Random Sample Consensus (RANSAC) algorithm was used for outlier removal. The pointing model parameters P0 − P4 were determined to be [1.451, 2.796, 2.752, 1.705, −0.021]. Figure 8(a) illustrates the “perpendicular” of the stage one data. Similar to the simulation experiment in Section 2.4, it can be observed that the “perpendiculars’” directions show a more concentrated distribution toward zero point after applying the pointing model correction. Table A3 shows the pointing data of stage two, which was observed with the pointing model established through the stage one data. Figure 8(b) shows the pointing offset and the “perpendicular” of stage two, alongside that of stage one for comparison. The pointing offset of stage two is observed to be predominantly clustered around the zero point, exhibiting an RMS value of 0.08 degrees, which is approximately 11% of the beam FWHM.
Figure 8 (a)
Figure 8 (b)
4. Discussion
The method proposed in this study has achieved preliminary success in constructing a radio telescope pointing model utilizing oblique-scan data. Nevertheless, achieving precise pointing measurement of the radio telescope via satellite observations requires further investigation. Several challenges require attention, including high-precision satellite orbits, receiver saturation, and observation frequencies, as all these factors can influence the measurement accuracy. These unresolved issues might account for why the RMS of the NSRT-25m experiment slightly exceeds 10% of the beam FWHM.
The TLE orbit parameters are employed in this study to calculate the satellite positions within the AltAz coordinate system. However, it is important to note that when calculating the Low Earth Orbit (LEO) satellites, atmospheric and gravity disturbances may introduce greater errors compared with other types of satellites. To improve the precision of satellite position calculations, data from multiple sources such as space debris telescopes, satellite antennas, Doppler measuring antennas, radars, and other relevant devices can be considered. For GNSS satellites, higher precision orbit data can be obtained through broadcast ephemeris. However, since other constellations do not provide these data, this method is not a primary focus.
The oblique drift-scan principle in this paper assumes that the satellite trajectory is a straight line and the beam cross-section is a perfect circle. However, in reality, the antenna beam may exhibit elliptical or triangular components due to the aberrations and minor lobes. Furthermore, the satellite trajectory may exhibit significant curvature in certain regions. These factors deviate from the assumptions in this paper and need to be addressed by establishing more complex models. In the future study, more complex measurement models can be introduced, and potentially, nonlinear solving methods are needed. In the nonlinear solving process, the results obtained by the linear regression in this paper can be used as initial values. Utilizing of these initial values is expected to improve the speed and convergence probability of the nonlinear solving process.
The high spatial density of satellites also benefits the offset pointing technique. Taking the satellite trajectories within a 4-min window as an example, there is a 50% probability of satellite presence within a 1.5-degree vicinity for any direction and a 90% probability within a 4.3-degree vicinity. Given the angular speed of the satellites, each measurement consumes approximately 100 ms. The spatial density and measurement speed are comparable to the method with OPT.
The Starlink constellation’s transmitting frequency is approximately 11 GHz. Considering the high SNR of the measurement, the uncertainty of an individual measurement is expected to satisfy the pointing requirement of the antenna at higher frequency. Future iterations of Starlink satellites could potentially incorporate Ka or V band signals, thereby providing better accuracy for pointing measurements. As satellite signals can easily saturate astronomical receivers, installing a dedicated satellite-observation receiver may be a practical solution. Future research could involve equipping the antenna with a Ku/Ka/V band receiver and exploring Starlink signals for pointing measurement.
The satellite drift-scan method offers an appealing approach for the pointing measurement of meridian radio telescopes. First, it enables the constraining of pointing directions in two-dimensional space without necessitating a servo system. Previously, the meridian radio telescopes relied on wobbling scanning for elevation pointing measurement, which required a servo system to perform the scanning. Second, the high SNR of satellite signals can effectively reduce errors caused by receiving system fluctuations. The measured results on the NSRT-25m meridian radio telescope show that the pointing model generated by the satellite drift-scan does provide a good pointing correction.
5. Conclusion
To take advantage of the high SNR and dense spatial distribution of satellites in radio telescope pointing measurement, a pointing measurement method using oblique drift-scan of satellites was proposed. The measurement principle was described, and the algorithm for fitting the pointing model parameters was developed to handle the oblique-scan data. This method was validated using both simulated data and experimental observations conducted on the NSRT-25m meridian radio telescope. The pointing model obtained from the NSRT-25m experiment provided effective pointing corrections for subsequent observations.
Conflicts of Interest
The authors declare no conflicts of interest.
Funding
This research was supported by the National Key R&D Program of China no. 2021YFC2203501 and the Scientific Instrument Developing Project of the Chinese Academy of Sciences, Grant no. PTYQ2022YZZD01. The NSRT-25m Telescope was partly supported by the Fund for Astronomical Telescopes and Facility Instruments and administrated by the Chinese Academy of Sciences (CAS).
Acknowledgments
This research was supported by the National Key R&D Program of China no. 2021YFC2203501 and the Scientific Instrument Developing Project of the Chinese Academy of Sciences, Grant no. PTYQ2022YZZD01. The NSRT-25m Telescope was partly supported by the Fund for Astronomical Telescopes and Facility Instruments and administrated by the Chinese Academy of Sciences (CAS).
