Orbital constraints and efficiency modeling for calibration satellites in long-range radar applications

Orbit design is a critical aspect of the calibration satellite system, with its effective configuration being integral for successful satellite calibration. This paper addresses the challenge of designing satellite orbits while considering the constraints imposed by satellite-ground equipment performance and launch capabilities. Key factors constraining calibration satellite orbits are analyzed, and orbital constraint equations are developed, focusing on radar performance, calibration time, and regional coverage. The study also introduces a model to assess the efficiency of satellite orbits for calibration purposes, which is rigorously validated through simulation. This study uniquely addresses the integration of radar performance with satellite orbit design, an area not extensively explored in current literature.


Introduction
The process of satellite calibration employs artificial satellites orbiting close to the Earth.Fundamentally, this process involves tracking and measuring space-specific satellite targets with radar systems to gather data.Concurrently, precise orbital data of these satellites corresponding to radar measurement arcs are obtained.By comparing radar measurement data with satellite orbital data, an optimization approach is employed to resolve radar error coefficients, thereby calibrating the equipment.In addition, satellites designated for calibration purposes can perform precise calibration tasks, such as range finding, velocity measurement, and radar cross-section (RCS) analysis.This calibration technique is particularly applicable to remote sensing radar systems covering distances in the order of thousands of kilometers and has seen extensive application in countries like the United States and Russia.
A satellite's orbit plays a pivotal role in its calibration capability.Primarily, an orbit dictates the satellite's positional distribution, which in turn influences the calibration's quality and accuracy due to its relationship with ground-based radar.This relationship affects vital calibration aspects, such as distance.Furthermore, an orbit determines the satellite's motion, which, coupled with the Earth's rotation, influences the timing and distribution of the satellite's coverage over ground-based systems, thereby affecting calibration timing.Optimal orbital design enhances the calibration satellite's coverage and calibration time for ground-based radars, thus improving calibration efficiency.
In designing satellite calibration orbits, the typical approach involves selecting appropriate orbital parameters.This selection is based on analyses of orbital perturbations, payload capacities, and operational lifespans in accordance with the satellite's mission and payload.Hence, the satellite fulfills user requirements post-orbit deployment.In satellite orbit theory, research predominantly concentrates on orbital dynamics.For instance, Noullez et al. [1] proposed a calculation model of high-precision satellite operation period, which can quickly complete the design of low-orbit satellite orbit parameters IOP Publishing doi:10.1088/1742-6596/2764/1/012082 2 under high-order and non-axisymmetric gravity models.Ortore et al. [2] constructed a satellite orbit parameter design model that generates appropriate orbit distance and revisit frequency in a specified area.Based on the circular repeat orbit equation, Lee [3] designed a satellite orbit to shorten the revisit time of a specific area by iterative numerical analysis and matching with the target point.
Yuan et al. [4] focused on specific challenges in aerospace measurement and control.This study establishes a dynamic system error model for radar, identifying and applying optimization methods to resolve error coefficients.The validity and practicality of this approach for satellite calibration are substantiated through detailed data analysis and calculations.Furthermore, Jin et al. [5] introduced a method and model for pulse measurement in radar satellite calibration.This study critically examines related engineering aspects and validates the effectiveness of the calibration process through simulations.
In addition, in [6] and [7], there is a detailed exploration of the practical aspects of radar satellite calibration.These studies analyze the radar's systematic error model, the selection process for calibration satellites, and the precision of ephemeris data.They also provide specific methodologies for satellite calibration.
Although research on satellite orbit design and radar calibration technology is well-established internationally, there is a notable lack of studies specifically focusing on the distinct mission requirements of calibration satellites.This gap highlights the need for more targeted research in this specialized area.
Therefore, this research aims to refine satellite orbit design for enhanced calibration efficiency and to develop a comprehensive efficiency model, addressing a gap in current methodologies.

Foundational assumptions for orbit analysis
To effectively analyze satellite orbits, it's imperative to first establish constraints and adopt certain simplifications.These constraints are intended to keep the analysis within practical boundaries while focusing on the principal factors influencing satellite orbit design.Key assumptions include:  Single satellite For a designated ground area, the presence of either a singular or multiple satellites in an orbit significantly impacts the coverage time and efficiency.According to the actual needs of the project, this paper only studies the radar calibration problem based on a single satellite.
 Calibration equipment performance on satellite The capabilities of on-board calibration equipment substantially affect the satellite's orbital altitude.This paper restricts the power and sensitivity of the satellite's calibration equipment, acknowledging that unlimited enhancement of these parameters would lead to significant variations in orbital altitude.
 Radar configuration The number, distribution, and capabilities of radar systems are crucial in selecting an appropriate calibration orbit.This analysis assumes a predetermined number of radar systems, utilizing both stationary and mobile units, and focuses primarily on a predefined operational area.
 Satellite launch dynamics To minimize costs, this study assumes that the launch of calibration satellites is carried out in an economical manner.The satellites are equipped with limited fuel, primarily employing direct orbit insertion and minimizing or completely avoiding in-orbit maneuvers.
 Power supply constraints of the satellite Given the small size, lightweight, and limited battery capacity of calibration satellites, this paper assumes a straightforward power management approach without the complexities of larger, more powerintensive satellites.

Analysis of orbit constraints
In the context of a satellite calibration system designed to cater to multiple users, several constraints must be considered to ensure the system's effectiveness and efficiency: IOP Publishing doi:10.1088/1742-6596/2764/1/0120823  Inclination angle for comprehensive coverage The satellite's orbit must be capable of providing comprehensive coverage.This necessitates an orbital inclination angle exceeding 55 degrees to ensure all areas fall within the satellite's calibration scope.
 Tracking duration for calibration accuracy Effective calibration requires sustained tracking of the satellite.As noted in [8], calibration data acquired over periods exceeding 120 seconds yield more stable and precise results.Consequently, the calibration orbit must allow for continuous radar tracking of the satellite for at least 120 seconds.
 Orbit calibration period To facilitate ongoing correction of radar system errors, the interval between successive satellite orbit calibrations must be reasonably short.In addition, selecting a regressive orbit, where feasible based on orbit height and inclination, aids in the comparative analysis and comprehensive utilization of calibration data.

 Mitigation of atmospheric resistance impact
The orbital selection should aim to reduce the effects of high-level atmospheric resistance.Minimizing this impact is crucial for enhancing the satellite's operational lifespan.
 Additional influencing factors Other factors influencing the choice of orbit include the location of the launch site, the prevailing space environment, the measurement and control systems, and conditions related to the satellite's launch and deployment.
By carefully considering these constraints, the satellite calibration system can be optimized to provide effective and efficient service to a wide range of users.

Orbital constraint formulation for satellite calibration
Given the utilization of on-board launch mode for the calibration satellite and its straightforward energy management approach, the satellite needs to maintain stability in relation to the sun.Therefore, this study proposes the adoption of a near-Earth circular orbit for the calibration satellite.For the sake of simplification in our calculations, the Earth is assumed to be a homogeneous sphere.

The equation for radar performance constraint.
The orbit of the satellite faces constraints primarily related to the operating range and duration of individual radar systems.Given the finite power and sensitivity of the satellite's transponder, the minimum distance between the satellite and the radar during calibration transit must remain within the operational range of the ground-based radar.
The constraint can be mathematically expressed as follows:    (1) Where  represents the semi-major axis of the orbit. denotes the Earth's radius, approximately 6378.14 kilometers. is the operational range of the radar.Further, by considering the radar's elevation angle (λ) and the required calibration time (T), the maximum allowable height (h) of the satellite's orbit (h=a−R e ) can be determined, as shown in Table 1


The equation for ascending node movement The trajectory of the satellite's subsatellite point, especially the ascending node, shifts westward due to Earth's rotation.This movement is most pronounced at the equatorial ascending node.The westward movement angle per orbit is given by:     (2) Where T Ω represents the time for the satellite to pass through the ascending node twice in succession [9], with ω e being a constant.The interval T Ω is calculated as: In this equation,  Is a constant geopotential coefficient, R e is the Earth's radius, a is the semi-major axis of the orbit, and i is the orbital inclination angle.
Furthermore, the rate of change of the right ascension of the ascending node ( ) for circular orbits is: )  is the average speed of the orbit.In polar orbits (inclination = 90°),  equals zero, indicating movement solely due to the Earth's rotation.In prograde orbits (inclination < 90°),  is negative, leading to westward node regression.Conversely, in retrograde orbits (inclination > 90°),  is positive, resulting in eastward node movement.
For different satellite orbit heights and inclination angles,  can be calculated, as shown in Table 2. Calibration cycle analysis The concept of a calibration cycle plays a vital role in the functioning of a satellite-based radar calibration system.The analysis commences by designating the ascending node at the initial time as S 0 , and the ascending node at the first intersection period of the satellite's motion as S 1 .The interval of continuous adjacent trajectories is defined by Δλ.
The phase shift angle α for the trajectory of the satellite's second orbit relative to its initial trajectory is given by:   2 (5) Where I represents the number of orbital cycles approximating a day.To manage the calibration cycle effectively, the interval Δλ of each contiguous adjacent trajectory is divided into N segments, corresponding to the number of days within the maximum calibration period.The minimum width γ of each segment is then calculated as: IOP Publishing doi:10.1088/1742-6596/2764/1/0120825 When the daily phase shift angle of the trajectory equals the width angle, meaning the trajectories through Δλ are sequentially arranged over N days, the time interval T Ω for this arrangement can be expressed as: Where D N is referred to as the "node day" [10] and is determined by: In scenarios where the daily trajectory phase shift angle equals an integer multiple C of the width angle, leading to a continuous phase shift over N days, T Ω is recalculated as: This detailed calibration cycle analysis aids in optimizing the satellite's orbit to ensure effective radar calibration over predetermined intervals.

2.3.3.
The equation for regional coverage constraints.The concept of regional coverage in satellite calibration pertains to the satellite's ability to periodically calibrate all radars within a specified area.This coverage is constrained by the radar's operational range and its elevation angle.The coverage angle d of the satellite's orbit, accounting for these factors, is given by:  sin sin  (10) where R m represents the maximum operational range of the radar, a is the semi-major axis of the orbit, and λ is the radar's elevation angle.
In addition, the satellite's longitudinal coverage of the Earth is closely linked to the required coverage period.For a desired N-day coverage, the width angle λ and the orbital coverage angle d must satisfy the following relationship:  2 (11)

Model for calibration efficiency of satellite orbits
Calibration satellites are typically launched alongside other satellites, subject to various orbital constraints.Calibration efficiency measures how well-suited a satellite's orbit is for calibrating specific radars or areas.This efficiency is determined by how the orbit adheres to the required calibration parameters, including coverage angle, orbital period, and the ability to maintain consistent calibration intervals.

Efficiency model of radar calibration.
The calibration efficiency of a satellite orbit with respect to a single radar system is quantified by two primary factors: the average calibration time per orbit and the average number of calibrations per day.This efficiency can be mathematically represented as:       (12) Where c i indicates the calibration efficiency for the radar identified as i; k i is the radar's calibration coefficient; t i specifies the average calibration time; N i represents the average number of calibrations per day; η it and η iN are the respective weights for the average calibration time and the number of calibrations.
There are m radars, of which n radars must be calibrated.The calibration efficiency model is as follows: ∑   (13)  denotes the weight of the radar i.The expression of  is as follows: If there is a radar  0 k = 0 in m important radar, then  0.
2.4.2.Model for regional calibration efficiency.The efficiency of a satellite orbit for calibrating a specific region is evaluated based on two criteria: the extent of regional coverage and the calibration IOP Publishing doi:10.1088/1742-6596/2764/1/0120826 period.Greater area coverage and shorter calibration periods contribute to higher regional calibration efficiency.The model for regional calibration efficiency is expressed as:     (15) In this equation, A is the regional calibration efficiency; l stands for the regional calibration coefficient; D is the covered area; T denotes the calibration cycle; δ D and δ T are the weights attributed to the coverage area and calibration cycle, respectively.

Setting the conditions for simulation
The simulation exercise is conducted under the following stipulations:  The calibration duration for a single satellite orbit to radar must be at least "120 seconds" [ 5].


The calibration interval should not exceed ten days.


The calibration targets include three radar stations: P1, P2, and P3.The performance parameters for these radars are detailed in Table 3.

Radar station
Operating distance Angle of elevation For the simulation, three distinct satellite orbits, labeled A, B, and C, are considered.The specifics of these orbits are outlined in Table 4.

Calibration weight assignment
Assuming that the P3 radar must be calibrated, according to the importance of the task, the assignment of weight coefficients to each radar station is shown in Table 5. is assumed that δ D is set to 1 and δ T to 10 for the purposes of this simulation."δ D =1" represents the coverage of China's region."δ T =10" means that it takes 10 days to cover all radars.

Simulation analysis
The purpose of simulation is to verify the correctness of the model.Figure 1 illustrates how satellite orbit A covers the radar stations P1, P2, and P3.  6 shows the average calibration time and times of satellite orbit A to radar.2. depicts the coverage offered by satellite orbit B to the radar stations P1, P2, and P3.   3 shows the extent to which satellite orbit C can provide coverage to the radar stations P1, P2, and P3.

Regional calibration simulation.
The simulation examines the regional calibration capabilities of three different satellite orbits labeled A, B, and C.
The inclination angle i of Satellite A is set at 98.5°, well above the minimum requirement of 55°.Its regression period is established at three days.Given the height of Satellite A's orbit, the operational IOP Publishing doi:10.1088/1742-6596/2764/1/0120829 range of the radar, and the specified calibration time, the constraint equation yields a coverage angle d A of 7.72°.Consequently, Satellite A's orbit is capable of calibrating radars located at latitudes up to 89.67°.In contrast, Satellite B's orbit can service radars up to a latitude of 85°, while Satellite C's orbit does not provide effective coverage.
Figure 4 shows the coverage dimensions achieved by the different satellite orbits, illustrating their respective capabilities in calibrating radar systems across various latitudes.The analysis reveals that Orbits A and B can provide effective coverage over the region of China, whereas Orbit C falls short in this regard.These findings align with the results derived from the applied constraint equations.
In summary, when considering the factors of radar operational range, the required elevation angle for accurate performance, calibration time, and calibration period, Orbit A emerges as the most suitable option for calibrating the P1, P2, and P3 radar stations and covering the specified land area.
It should be noted that for the same satellite orbit, different mission requirements set different parameters, which may lead to different evaluation results.For example, for some precision tracking radars, the calibration time is long, and it may be more advantageous to choose a satellite orbit with a relatively high altitude; for some radars with long service time, the calibration period is short, and it may be more advantageous to choose a satellite orbit with relatively low altitude.

Conclusion
Radar calibration plays a crucial role in the range testing, daily operation, and maintenance of radar systems.The use of dedicated satellites for error calibration in radar systems is a practice that has gained widespread adoption in technologically advanced nations like the United States and Russia.This paper addresses the challenges in designing orbits for calibration satellites, examining a range of constraints that affect the selection of satellite orbits.It delves into the impact of radar performance, calibration intervals, and regional coverage on orbit determination, providing equations that reflect constraints such as operational distance of radar, elevation angle limitations, calibration timing, and coverage requirements.The analysis of the calibration orbit encompasses both qualitative and quantitative aspects, introducing two models for evaluating calibration efficiency, one for radar and another for regional applications.Through simulation, the validity of the proposed satellite orbit constraint equations and the calibration efficiency models has been tentatively confirmed.

1 .
Radar calibration simulation.The simulation explores the extent of coverage provided by different satellite orbits to the specified radar stations P1, P2, and P3.

Figure 1 .
Figure 1.Coverage of orbit A for radar stations P1, P2, and P3.Table6shows the average calibration time and times of satellite orbit A to radar.

Figure 4 .
Figure 4. Coverage dimension simulation.The analysis reveals that Orbits A and B can provide effective coverage over the region of China, whereas Orbit C falls short in this regard.These findings align with the results derived from the applied constraint equations.In summary, when considering the factors of radar operational range, the required elevation angle for accurate performance, calibration time, and calibration period, Orbit A emerges as the most suitable option for calibrating the P1, P2, and P3 radar stations and covering the specified land area.It should be noted that for the same satellite orbit, different mission requirements set different parameters, which may lead to different evaluation results.For example, for some precision tracking radars, the calibration time is long, and it may be more advantageous to choose a satellite orbit with a relatively high altitude; for some radars with long service time, the calibration period is short, and it may be more advantageous to choose a satellite orbit with relatively low altitude.

Table 1 .
. Satellite orbit altitude under different operating distances and calibration time constraints.The equation for assessing radar performance in satellite calibration.The calibration period is an essential aspect of satellite-based radar calibration, denoting the maximum time between two consecutive calibrations for a radar station.As latitude increases, the satellite's ground coverage overlaps more, necessitating a consistent recalibration schedule across different latitudes.A radar station at the equator is utilized as a baseline for analyzing the calibration period's influence on the orbit.
Note: When R = 1000 km, the maximum T is 256 s ( < 300 s ), at this time, h is about 165 km.

Table 2 .
The  values corresponding to different satellite orbit heights and inclination angles.

Table 4 .
Parameters of satellite orbits.

Table 5 .
Weight coefficient of radar.

Table 6 .
The average calibration time and times of orbit A to radar.

Table 7
shows the average calibration time and times of satellite orbit B to radar.

Table 7 .
The average calibration time and times of orbit B to radar.

Table 8
shows the average calibration time and times of satellite orbit C to radar.

Table 8 .
The average calibration time and times of orbit C to radar.Based on the simulation outcomes and using the radar calibration efficiency model, the calibration efficiency values for each of the three satellite orbits are shown in Table9.

Table 9 .
Calibration efficiency of satellite orbit to radar.