An Innovative External Heat Flow Expansion Formula for Efficient Uncertainty Analysis in Spacecraft Earth Radiation Heat Flow Calculations

: Thermal uncertainty analysis of spacecraft is an important method to avoid overdesign and underdesign problems. In the context of uncertainty analysis, thermal models representing multiple operating conditions must be invoked repeatedly, leading to substantial computational costs. The ray tracing calculation of Earth infrared and albedo radiation heat flux is an important reason for the slow calculation speed. As the rays emitted during external heat flux calculations under different operating conditions are independent and unconnected, the rays produced across various conditions are effectively wasted. In this study, the external heat flow equation is thoroughly expanded and the derived factors are clustered and analyzed to develop a novel formula for calculating external heat flow. When this formula is employed to compute the uncertain external heat flux, only one condition necessitates ray tracing, while the remaining conditions utilize simple matrix operations in place of complex ray tracing. Within the aforementioned procedure, certain matrices demonstrate sparse characteristics. The optimization calculations for these matrices can, therefore, benefit from the application of sparse matrix optimization algorithms. Using a spacecraft as an example, the uncertain external heat flux calculation outcomes of the new and traditional formulas are compared and assessed. The findings reveal that the new formula is highly suitable for estimating uncertain Earth radiation heat flow, with a marked improvement in efficiency. The accuracy is essentially equivalent to that of the traditional formula and the calculation precision can be dynamically adjusted to meet user requirements. The methodology can be further generalized to assess the uncertainties associated with radiative external heat fluxes for other celestial bodies within the solar system. This offers a valuable theoretical framework for addressing the uncertainties in the thermal design of deep space exploration vehicles.


Introduction
As aerospace technology advances and application demands continue to expand, the robustness and reliability of spacecraft thermal control systems are subject to increasingly stringent requirements.During the design process of these control systems, assumptions and simplifications made in the spacecraft thermal analysis model can introduce unknown errors into the design analysis outcomes.Furthermore, uncertainties in the performance of structural materials may arise from material defects or machining precision during the manufacturing process, while the external heat flow absorbed by the structure in operation can also vary.The cumulative impact of these uncertainties can influence the temperature field of the spacecraft structure and, under specific circumstances, may even result in equipment damage or failure.
Spacecraft thermal control systems are typically designed and validated based on extreme conditions.The factors considered under these conditions encompass the corresponding worst-case environmental and spacecraft structural parameters.Subsequent analysis reveals that, to address uncertainties stemming from the remaining parameters, a fixed design margin (also referred to as the uncertainty margin) can be applied to the results (downstream of the model).These margins are derived from statistical analyses of the discrepancies between the predicted temperatures of thermal models and the flight measurement outcomes from select space missions [1,2].
Traditional spacecraft thermal design and verification methods do not rigorously address model uncertainty in the upstream portion of the model.Instead, they handle uncertainty downstream by directly applying fixed design margins to the model output.Research conducted by Ishimoto and Bevans (1968) [3], Howell (1973) [4], and Thunnissen (2004) [5] indicate that the primary goal of performing uncertainty analysis is to provide a more effective method for calculating these design margins.The main concept involves strictly considering model uncertainty upstream and propagating it to the output through the model, subsequently quantifying the uncertainty of the model output.Compared with conventional methods, this approach has the advantage of providing design margins under specific circumstances.
As demonstrated in Figure 1, there is a discernible positive correlation between the radiator size, extending from radiator 5 to radiator 1, and its correlated cooling capacity.Notably, an augmentation in size also results in an escalated mass.The practice of mass reduction is a salient objective in spacecraft design, owing to its substantial potential to curtail both the cost and complexity of the mission.Even marginal mass reduction can translate into remarkable savings in terms of fuel and launch costs.Using the selection of a spacecraft radiator model as an example, conducting thermal uncertainty analysis on the radiator can help avoid overdesign scenarios (radiation radiators 1 and 2) that result from excessively large radiation radiator model choices, effectively reducing development costs.Additionally, this analysis can prevent underdesign situations caused by selecting a radiation radiator model that is too small and has insufficient heat dissipation reliability (radiation radiators 4 and 5).The primary method for spacecraft thermal uncertainty analysis is the Monte Carlo (MC) approach.However, to obtain accurate results, a substantial number of original model simulations are needed, leading to considerable computational costs.This method necessitates numerous spacecraft thermal analyses under various working conditions.If uncertainty analyses were conducted directly based on the spacecraft thermal analysis model, the computational cost would be unacceptably high.Many scholars have conducted research on the balance between the accuracy and efficiency of the thermal uncertainty analysis algorithm.For example, Thunnissen [6] used subset simulation to perform transient thermal uncertainty analysis on the cruise phase of the Mars spacecraft.The tail probability distribution function of the maximum temperature of important components was obtained.The results successfully replicated the MC simulation results and, compared with the MC case, the calculation burden was much lower.Gómez-San-Juan [7] developed a one-dimensional generalized statistical error analysis method.The input uncertainty variables of this method could be arbitrarily distributed and the nonlinear relationship between node temperature and uncertainty variables was incorporated.The calculation time matched that of the statistical error analysis method and the accuracy matched that of the MC approach.Xiong et al. [8] approximated the traditional thermal analysis calculation model of spacecraft based on a radial basis function (RBF) neural network of the improved mind evolutionary algorithm.The trained neural network effectively improved the analysis rate of thermal uncertainty analysis.Kato [9] used a Gaussian process regression simulator and minimum absolute shrinkage selection operator to perform thermal uncertainty analysis on satellites.The simulator reduced the cost of uncertainty quantification.Similar to the above methods, the kriging model [10][11][12][13], RBF method [14], artificial neural network [15,16], support vector regression method [17], and response surface method [18] are all approximate models that replace spacecraft thermal analysis models.Thermal uncertainty analysis can be performed on the approximate model to improve the computational efficiency.However, to obtain an accurate approximation model, the original thermal analysis model of the spacecraft requires repeated calculation.
The existing methods focus on enhancing the computational efficiency of spacecraft thermal uncertainty analysis from a mathematical perspective.In this study, we aim to improve the calculation efficiency of thermal uncertainty analysis by optimizing the original thermal analysis model of spacecraft.During the calculation process of the spacecraft's original thermal analysis model, there are two main steps to obtain the satellite temperature field: radiation model calculation and thermal model calculation.The radiation model calculation comprises three types of external heat flux absorbed by the spacecraft surface elements and the radiative thermal conductance between these elements.These three types of external heat flow include Earth infrared radiation heat flow, Earth albedo radiation heat flow, and solar radiation heat flow.Thermal model calculations encompass the surface element heat capacity, the linear thermal conductivity between the spacecraft surface elements, the power of the heat source within the surface elements, and the solution of the energy balance equation.Reference [7] highlights that the calculation time of the radiation model is significantly longer than that of the thermal model, with the radiation MC ray tracing calculation being the most time-consuming task.The calculation of external heat flow from Earth infrared and albedo radiation is a crucial aspect of radiation model calculation.In thermal uncertainty analysis, the Monte Carlo ray tracing algorithm needs to be employed multiple times to compute the Earth infrared and albedo radiation heat flux, which is one of the main factors contributing to the immense computational cost of thermal uncertainty analysis.
Owing to the uncertainty of spacecraft thermal property parameters, various thermal property parameter samples are generated in thermal uncertainty analysis.When calculating the external heat flux corresponding to each working condition sample, each solution of the external heat flux is independent.Upon completing the external heat flux calculation for one working condition, the subsequent working condition's external heat flux calculation does not utilize the rays emitted by the previous working condition.Instead, the new working condition initiates a fresh set of rays and conducts ray tracing.This process leads to the underutilization of rays from the previous working condition, consequently increasing the calculation time.
To address the issue of slow calculation time resulting from underutilized rays, this study thoroughly expands the formulation of Earth infrared and albedo external heat flow, completely isolating the factors of the formulation to develop a new corresponding expression.This formula is employed for uncertain external heat-flow calculations.Only the first condition emits rays, with ray tracing calculations for Earth radiative external heat flow in this condition generating the external heat-flow expansion (EHFE) matrix.For the remaining thermal uncertainty analysis conditions, new thermal attribute parameter samples are substituted into the EHFE matrix.This new formula is highly suitable for the analysis of the uncertain external heat flux, as it eliminates the need to project rays and perform ray tracing for calculating the external heat flux in each condition.Matrix operations are used instead of the original ray tracing calculation process.This approach to calculating the uncertain external heat flux can effectively reduce the calculation time while maintaining sufficient accuracy.
The structure of this paper is organized as follows: Section 2 introduces the solution model for the spacecraft temperature field, the calculation process for Earth radiation external heat flow, and the ray tracing method for external heat-flow calculation.Section 3 presents the current calculation formula for the Earth infrared and albedo radiation heat flux and provides a detailed description of its expansion to provide a new formula.This section also explains how to analyze an uncertain external heat flux using this new formula.In Section 4, a spacecraft is used as an example, with its composition and calculation parameters being introduced.Section 5 discusses and analyzes the results of the spacecraft's uncertain Earth radiation external heat flow.Finally, conclusions are drawn in Section 6.

Temperature-Field Calculation Model
The order of solving the mathematical thermal model of the spacecraft is as follows.First, radiation calculation is carried out based on a geometric mathematical model.In this process, the MC ray tracing algorithm is used to calculate the radiation heat transfer.QS−i, Qer−i, Qr-I, and Gji are calculated and these outputs are input into the mathematical thermal model.The second step is the solution of the mathematical thermal model to obtain the temperature fields of the spacecraft's surface elements.The expression of the mathematical thermal model is as follows: where the subscripts i and j denote surface element (node) numbers; T is temperature; m is mass; c is specific heat; t is time; Qs−i is the solar heat load on node i; Qer−i is the planetary albedo heat load on node i; Qr-i is the planetary infrared heat load on node i; Q is the internal heat source power; Dji and Gji denote the linear and radiative thermal conductivity between nodes j and i, respectively [19,20].Reference [7] highlights that the calculation time required for the radiation model is significantly longer than that of the mathematical thermal model.The MC radiation calculation is the most time-consuming task.In thermal uncertainty analysis, the computation of the external heat flux utilizing the MC ray tracing method demands considerable computational resources; consequently, there is a need for algorithm optimization.

Earth Radiation Heat Flow Calculation Process
The reverse Monte Carlo (RMC) method is a general approach for computing the Earth infrared and albedo external heat flux absorbed by surface elements.N rays are emitted from the target surface element and ray tracing is conducted on them.The N-ray results are statistically analyzed to determine the Earth infrared and albedo external heat flux for the target surface element.The RMC simulation for a single ray proceeds as follows: (1) Reference coordinate systems, such as body, orbit, and geocentric inertial coordinate systems, are established and orbit parameters are provided.(2) A thermal-physical model of the spacecraft structure is constructed, the thermal analysis surface element grid is partitioned, and the surface equation and radiation characteristics for each thermal analysis surface element are determined.(3) A target surface element of the spacecraft emits a random ray [21].(4) The intersection of the ray with the surface element of the system (spacecraft and radiation source) is calculated and the method for further tracking the ray is determined according to the ray tracing technique.(5) If the ray is absorbed by the radiation source's surface, the external heat flux absorbed by the ray on the target surface element's surface is accounted for using the reverse process.

Ray Tracing Method for External Heat Flux Calculation
At present, there are three ray tracing methods to calculate the external heat flux [22].
1. Collision method: During the ray tracing process, rays are treated as a whole and are not divided into smaller segments.Rays maintain constant energy throughout the tracing process.To calculate the energy absorbed by a surface when a ray strikes it, a random number Rα is selected between 0 and 1 and compared to the surface absorption rate α.If Rα < α, the ray is completely absorbed by the surface and the process ends.If Rα ≥ α, the ray is fully reflected.2. Path length method: In this approach, during ray tracing, a portion of the ray's energy is absorbed by the surface it impacts, while the remaining energy is reflected and continues to be traced.Calculation of the energy absorbed by the surface does not involve random numbers.A small fraction of the ray's energy is absorbed by the surface and the rest is reflected.This method functions by preventing the complete termination of a ray in a single collision, allowing each ray to contribute to the results of multiple subregions.This significantly reduces the statistical uncertainty for a given number of rays.3. When analyzing a closed cavity, the rays emitted may result in an infinite loop based on the path length method of propagation.To avoid this, an energy cutoff fraction k (0 < k< 1) is introduced.When the reflected ray energy q2(1 − α) > kq1, the path length method is used.When the reflected ray energy q2(1 − α) ≤ kq1, ray tracking is halted.q1 is the energy of the initial surface element emitting the ray.Here, q2 is the energy of the ray hitting a surface element.

Analysis of Ray Tracing Methods: Uncertainty Issues
The present state of algorithms for uncertainty thermal analysis in spacecraft appears to exhibit a challenge in terms of computational efficiency.Owing to the uncertainty in the solar absorbance and infrared emissivity of spacecraft coatings, conducting thermal uncertainty analysis necessitates invoking the thermal model multiple times and iteratively calculating the external heat flow.It generates huge computational costs.Employing traditional methods for external heat flow calculation requires re-emitting rays each time, resulting in the issue of wasted rays.This study presents a detailed analysis of ray tracing, in which every emitted ray is fully utilized during the thermal uncertainty analysis.When the surface properties of the spacecraft coating are altered, rays emitted by the previous model are employed, thus eliminating the need to re-emit rays for calculating the Earth radiated external heat flow under the given condition.
In this investigation, the Earth radiative external heat flow calculation equation is fully expanded.By substituting the intricate process of ray tracing with rudimentary matrix operations, there is a substantive reduction in computational expenditures.This allows for accurate and expedited uncertainty thermal analysis.This methodology aids in the development of more proficient and dependable thermal control systems; it also enhances the performance of onboard instrumentation and systems.Moreover, it facilitates extended mission durations, bolsters safety, and promotes cost efficiency.

Earth Infrared Radiation External Heat Flow: Collision Method
This paper is based on the energy conservation relationship of the Earth infrared radiation and the basic principle of the RMC method emission of rays from the spacecraft target surface element and ray tracing using the collision method.Nei beams of rays emitted from the surface element Ei are tracked, with Nai beams reaching the Earth surface and being absorbed.Then, the density of the external heat flow from the Earth infrared radiation absorbed by the surface element Ei can be expressed as: where ρ is the Earth average solar albedo with a value of 0.35; S is the solar constant with a value of 1353 W/m 2 ; E i r  is the infrared emissivity of the surface element Ei coating [23- 28].

Earth Infrared Radiation External Heat Flow: Pathlength Method and Cutoff Factor
Rays are emitted from the spacecraft target surface element and ray tracing is performed using the latter two methods.Nr beams of rays emitted from the spacecraft's surface element Ei are tracked, with A r N rays reaching the Earth surface directly.Then, the density of the external heat flow from the Earth infrared direct radiation absorbed by the surface element Ei can be expressed as: where r,A i q denotes the density of the Earth infrared direct radiation heat flow to the surface element Ei; r E i  , the infrared absorbance of the spacecraft surface element Ei, is equal to the coating infrared emissivity [23][24][25][26][27][28].
Nr beams of rays emitted from the spacecraft's surface element Ei are tracked, with B r N beams reaching the Earth surface after multiple reflections.Then, the external heat flow from the Earth infrared indirect radiation absorbed by the surface element Ei can be expressed as: where r rE 1 (1 ) , with k being the number of reflections [23][24][25][26][27][28].
In summary, the total external heat flow density of the Earth infrared radiation absorbed by the surface element Ei is r r,A r,B i i i q q q =+ (5)

Earth Albedo Radiation External Heat Flow: Collision Method
This paper is based on the energy conservation relationship of Earth albedo radiation and the basic principles of the RMC method.By emitting rays from the spacecraft target surface element and using the collision method for ray tracing, Nei beams of rays emanating from surface element Ei are tracked, with Nai rays reaching the Earth surface and being absorbed.The angle θse between the outer normal vector of the ray's intersection with the Earth and the direction vector of the sun rays is greater than 90°.Then, the density of the external heat flow from the Earth albedo radiation absorbed by surface element Ei can be expressed as: where  is the solar absorptance of the surface element Ei coating [23][24][25][26][27][28].

Earth Albedo Radiation External Heat Flow: Path Length Method and Cutoff Factor
Rays are emitted from the target surface element of the spacecraft and ray tracing is performed using the latter two methods.Ner beams of rays from the spacecraft's surface element Ei are tracked, with A er N beams reaching the Earth's surface directly.The angle θse between the outer normal vector of the intersection of the ray and the Earth and the direction vector of the sun's rays is greater than 90°.Then, the external heat flow from the Earth albedo direct radiation absorbed by the surface element Ei can be expressed as: where er,A i q denotes the density of direct Earth albedo radiation heat flow to the surface element Ei; er E i  is the solar absorptance of spacecraft surface element Ei; θse is the angle between the direction vector of the sun's rays and the outer normal vector of rays emitted by surface element Ei hitting the Earth surface intersection [23][24][25][26][27][28].
Ner beams of rays emitted from surface element Ei are tracked and B er N beams are reflected several times to reach the Earth surface.The angle θse between the outer normal vector of the intersection of the ray and the Earth and the direction vector of the sun's rays is greater than 90°.Then, the external heat flow from the Earth albedo indirect radiation absorbed by the surface element Ei can be expressed as: where er er E 1 ( 1) , with k being the number of reflections [23][24][25][26][27][28].
In summary, the total external heat flow density of the Earth albedo radiation absorbed by surface element Ei is:

Comparative Analysis of Calculation Formulas for Various Methods
The collision method is employed to compute the Earth radiative external heat flow.Random numbers must be generated and compared with the absorption rate corresponding to the impacted surface element to determine whether the rays are absorbed or reflected.In the external heat flow calculation equations (Equations ( 2) and ( 6)), only the absorption rate of the target surface element is available; information on the coating of the surface element struck during ray propagation is absent.For this type of ray tracing, separating absorbance and reflectance from the equation is unfeasible.If the surface properties of the spacecraft coating change, rays emitted by the previous working model cannot be utilized.
In contrast, the second and third ray tracing methods provide information on the coatings impacting surface elements during ray propagation, as found in Equations ( 4) and ( 8) of the Earth radiative external heat flow calculation.These two methods enable the separation of absorptivity and reflectance from the equation.There is no need to reemit rays when the surface properties of the spacecraft coating change, as the separated expansion formula can be used for direct calculations.

Earth Radiative EHFE Algorithm
The Monte Carlo method is employed to generate NS working conditions for the thermal uncertainty analysis of the spacecraft.Radiative model calculations are significantly more time-consuming than thermal model calculations, primarily due to multiple instances of ray tracing and the utilization of the intersection evaluation algorithm.Uncertainty analysis necessitates multiple applications of the external heat flow calculation model, as well as numerous ray tracing and intersection evaluations.In each of these NS conditions, emitted rays are independent, with no interconnection between them.Consequently, rays from the previous working condition are not utilized in the subsequent working condition.This arises from the incomplete separation of individual variables within the external heat flow solution equation.For the Earth radiative external heat flow solution, the uncertainty variables predominantly involve the infrared emissivity and solar absorbance of the spacecraft surface coating.
As analyzed in Section 3.1, employing the latter two ray tracing methods allows the aforementioned uncertainty variables to be separated.Therefore, this paper adopts the path length method with the introduction of a cutoff factor, where the cutoff factor is set to 0.1.The external heat flow equation is fully expanded to isolate the uncertainty variables within the equation, constituting a new formula for calculating the Earth radiative external heat flow.This approach facilitates an uncertainty analysis of the external heat flow absorbed by the spacecraft.

Earth Infrared Radiation EHFE Equation
A ray originating from an initial surface element that directly reaches Earth is classified as a component of Earth infrared direct radiation external heat flow.Such rays are depicted in Figure 2a.Conversely, if a ray undergoes multiple reflections on the spacecraft's surface prior to ultimately arriving at Earth, it is regarded as a constituent of Earth infrared radiation indirect external heat flow.Corresponding rays are demonstrated in Figure 2b-d.To fully separate the coating's infrared emissivity and the corresponding infrared spectrum's reflectance from the equation, the Earth infrared radiated external heat flow is further expanded, as demonstrated in Equation (10).
The second term indicates that the rays are reflected once on the spacecraft's surface before reaching Earth, representing a single reflection of Earth infrared radiation external heat flow, as shown in Figure 2b.The third term signifies that the rays are reflected twice on the spacecraft's surface before reaching Earth, corresponding to a secondary reflection of Earth infrared radiation external heat flow, as depicted in Figure 2c.The nth term implies that the rays are reflected n − 1 time on the spacecraft's surface before reaching Earth, representing the n − 1th reflection of Earth infrared radiation external heat flow.
The Earth infrared radiation EHFE in Equation ( 10) is split into a matrix multiplicative form.The applicable formula is r-old r r r (1,1) The element composition of the Ar matrix is shown in Equation ( 14).
The Cr matrix corresponds to each common factor of the EHFE formula.One of the uncertainty parameters is the infrared emissivity where The Ar matrix consists of the infrared spectral reflectance of the surface elements.As described below, the first According to the relationship α + ρ = 1, the elements of the Ar matrix are derived from the Hr matrix, where the elements of the Hr matrix represent the corresponding surface element infrared emissivities.
Prior to conducting the uncertainty analysis, the Earth infrared radiation external heat flow absorbed by the primary target surface element must be determined using the RMC method.The number of intersecting face elements and the corresponding infrared emissivities for each ray are recorded and summarized in the intersecting face element number matrix Ir and matrix Hr, respectively.Subsequently, uncertainty analysis of Earth infrared radiation external heat flow is performed for each operating condition without employing ray tracing, as in the traditional method.Instead, a sample of coated infrared emissivities is generated using random numbers.The Cr matrix is then updated to create the Cr 1 matrix and the Hr matrix is updated to form the Hr 1 matrix based on the Ir matrix.The Ar 1 matrix is calculated using the Hr 1 matrix, which is subsequently utilized to compute the Dr 1 matrix.Finally, by substituting the resulting matrices into Equation (17), the Earth infrared radiation external heat flow, accounting for uncertainties, can be obtained.
In this paper, it is crucial to note that the elements constituting the rows of matrices Ir, Hr, Ar, and Dr can be permuted in any desired sequence.The current study organizes the elements within each row of the matrices in a specific manner to facilitate concise and clear communication.During the actual computational process, the elements in each row of Ir, Hr, Ar, and Dr are generated directly based on the sequence of rays emitted by the surface elements.
The equation for the Earth infrared radiation external heat flow taking into account uncertainties is as follows: The comparison of two processes for calculating the Earth infrared radiation external heat flow considering parameter uncertainties is in Figure 3.
The main steps of the conventional method for solving the Earth infrared radiative external heat flow considering uncertainties are as follows: (1) Samples of all surface element infrared emissivity and corresponding infrared spectral reflectance of the spacecraft are generated using random numbers.(2) The RMC method is used to perform ray tracing to solve for the Earth infrared radiation external heat flow.
(3) The first two steps are repeated to generate multiple r i q samples for statistical analysis.
The main steps in solving the Earth infrared radiation external heat flow considering uncertainties using the EHFE formula are as follows: (1) The RMC method incorporating ray tracing is used to solve the Earth infrared radiation external heat flow and to obtain the Ir matrix and Hr matrix.(4) Equation ( 17) allows solution for r-new i q .
(5) Steps ( 2)-( 4) are repeated to generate multiple r-new i q samples to be used for statistical analysis.C D N =   and matrix operations are performed.Utilizing matrix operations instead of the original ray tracing approach can effectively reduce the computational cost associated with thermal uncertainty analysis.

Earth Albedo Radiation EHFE Equation
A ray originating from the initial surface element that directly strikes the Earth is considered.The angle θse between the outer normal vector of its intersection with the Earth and the direction vector of the sun's rays is greater than 90°.In this case, the ray constitutes a component of the Earth albedo direct radiation external heat flow.This type of ray is illustrated in Figure 2a.If the ray is reflected multiple times on the spacecraft's surface before reaching the Earth and the angle θse between the outer normal vector of its intersection with the Earth and the direction vector of the sun's rays remains greater than 90°, then the ray is considered part of the Earth albedo indirect radiation external heat flow.Such rays are depicted in Figure 2b-d.To accurately isolate the coating solar absorptance, the Earth albedo radiation external heat flow is further expanded as in Equation (18).The second term of this equation signifies that the ray impacts the Earth's surface after a single reflection on the spacecraft's surface, corresponding to the one-time reflection of the Earth albedo radiation external heat flow, with such rays displayed in Figure 2b.The third term indicates that the ray reaches the Earth's surface after two reflections on the spacecraft's surface, representing a two-time reflection of the Earth albedo radiation external heat flow, with rays of this type shown in Figure 2c.The nth term suggests that the ray collides with the Earth's surface after n − 1 reflections on the spacecraft's surface, which corresponds to the n−1th reflection of the Earth albedo radiation external heat flow.
The Aer matrix consists of the solar spectral reflectance of the surface elements.As described below, the first In accordance with α + ρ = 1, the elements of the Aer matrix are derived from the Her matrix, where the Her matrix elements represent the corresponding surface element solar absorbance.
Prior to conducting the uncertainty analysis, it is essential to solve the Earth albedo radiative external heat flow using the RMC method.The number of faces intersected by each ray and their respective solar absorbance values are recorded and consolidated into the intersecting face number matrix Ier and matrix Her, respectively.Additionally, the angle θse between the outer normal vector at the final intersection of each ray with the Earth and the direction vector of the sun's rays is recorded, resulting in the generation of matrix Cer.In contrast to the traditional method, which requires multiple ray tracing for each operating condition, the uncertainty analysis of the Earth albedo external heat flow employs random numbers to generate samples of coated solar absorbance.The Cer matrix is then updated to the Cer 1 matrix and the Her matrix is updated to the Her 1 matrix based on the Ier matrix.Subsequently, the Aer 1 matrix is calculated using the Her 1 matrix and the Der 1 matrix is obtained from the Aer 1 matrix.Finally, by substituting the resulting matrix into Equation (25), the Earth albedo external heat flow, accounting for uncertainties, can be determined.
It is imperative to note that the elements within each row of matrices Ier, Her, Aer, and Der, as well as those within each column of matrix Cer, can be reordered in any arbitrary arrangement.This paper presents the elements of the matrices in a specific configuration to facilitate clarity and conciseness in the presentation.During the actual computational analysis, the elements in each row of matrices Ier, Her, Aer, and Der, and those in each column of matrix Cer, are generated directly based on the sequence of rays emitted by the spacecraft surface elements.(1,1) The uncertain Earth albedo external heat flow is calculated as follows: 11 er-new er er er The comparison of two processes for calculating the Earth albedo external heat flow taking into account parameter uncertainties is in Figure 4.
The main steps in solving the Earth albedo radiative external heat flow considering uncertainties using traditional methods are as follows: (1) Random numbers are used to generate samples of the solar absorbance and reflectance of the corresponding solar spectrum for all surface elements of the spacecraft.
(2) Ray tracing using the RMC method is used to solve the Earth albedo radiation external heat flow.
(3) Steps ( 1) and ( 2) are repeated to generate multiple er i q samples for statistical analysis.
The main steps for solving the Earth albedo radiation external heat flow considering uncertainties using the EHFE formula are as follows: (1) Ray tracing using RMC is used to solve the Earth albedo radiation external heat flow and to obtain the Ier, Her, and Cer matrices.(4) Equation ( 25) is used to solve er-new i q .

Experimental Model
To validate the accuracy and computational efficiency of the EHFE formulation method, this paper employs the spacecraft thermal analysis model depicted in Figure 5 as a representative example for uncertain orbital external heat flow calculations.Thermal desktop (TD) software, currently a prevalent tool for spacecraft thermal analysis, is utilized to construct the model, which comprises the satellite body, solar panels, antennae, and truss.The numbers assigned to the spacecraft's surface elements correspond to the experimental analyses delineated below.As the orbital space external heat flow exclusively impacts the spacecraft's outer surface, its internal structure is not taken into consideration for this study.The spacecraft orbit and attitude parameters are presented in Table 1.As shown in Figure 5, the spacecraft is discretized into many surface elements.The surface of the spacecraft encompasses both a front face element and a reverse face element.Upon assuming a negligible temperature difference between the front and reverse face elements, they are assigned the same number.Since the external heat flux inside the satellite body is zero, the satellite body's reverse surface element is not considered in this study.The uncertainty input parameters selected for this study [29] and their average values are presented in Table 2.The input parameters are normally distributed and their standard deviation is 0.05.One surface element from each structure of the spacecraft is chosen as the subject of investigation.The surface element numbers and their corresponding components are detailed in Table 3.The initial positioning of the spacecraft, as depicted in Figure 6, situates it within the sunlit sector.At this location, uncertain Earth infrared and albedo radiation external heat flow analyses are conducted using both the TD conventional method and the EHFE formulation algorithm.The obtained results are compared and analyzed to validate the accuracy and efficiency of the Earth radiative EHFE formula.Initial Earth infrared and albedo radiation external heat flow ray tracing conditions, the values of infrared emissivity, and the solar absorptance of each spacecraft component coating for this condition are the mean values of the uncertainty variables corresponding to Table 2.In the computation of the Earth infrared radiation external heat flow, the satellite body surface elements emit 1 × 10 6 rays, whilst other structural surface elements discharge 2 × 10 6 rays.A similar procedure is adopted for calculating Earth albedo radiation external heat flow.
The matrices, namely, Ir, Cr, Dr, Ier, Cer, and Der related to the Earth radiative EHFE equation, are derived from the outcomes of the external heat flow ray tracing for these initial working conditions.
The matrices Cr 1 , Dr 1 , Cer 1 , and Der 1 , corresponding to the remaining uncertainty conditions, are generated from matrices Ir, Cr, Dr, Ier, Cer, and Der and samples of infrared emissivity and solar absorbance matching the new conditions.Notably, this process obviates the necessity for a comprehensive ray tracing operation.
To obtain uncertainty margins and output response confidence intervals for accurately calculating the Earth infrared and albedo external heat flow, an accurate external heat flow response corresponding to P at both ends of its probability density function is needed.The literature [7] states that for the obtained P0.95 to effectively lie between the actual P0.94 and P0.96 with 95% confidence, the number of model runs should be at least 1900.The solution formula is as follows: where Nmin is the minimum number of runs the model should run, P is the percentile to be calculated, ΔP is the allowable deviation from that value, and Nσ is the confidence level that the expected calculated percentile P lies at the actual P − ΔP and P + ΔP.Consequently, both the original TD model and the EHFE formula are executed 2000 times in this study to acquire the requisite statistical data pertaining to Earth radiative external heat flow.The computer employed for the analysis was equipped with a 3.50 GHz W-2265 CPU and 64 GB of RAM.

Results and Discussion
Uncertainty analysis focuses on the mean, standard deviation, probability density distribution, and confidence interval (CI) of the output response.To ascertain whether the EHFE formula can supplant the original ray tracing method, 2000 statistical comparisons of the aforementioned model output responses must be analyzed.

Earth Infrared and Albedo External Heat Flow Uncertainty Analysis
The TD model and the EHFE formula are executed for 2000 cycles.In Figure 7, the split-edge violin plots illustrate the outcomes of two distinct methodologies employed in determining the Earth radiative external heat flux in the context of spacecraft thermal analysis.Denoted by blue dashed lines, the quartiles demarcate the statistical findings of each approach.The median is represented by the red dashed line, whereas the confidence interval, ranging from the 1st to the 99th percentile, is indicated by the pink dashed line.Furthermore, the contour effect within the violin plots signifies the probability density magnitude corresponding to a specific data point.The larger the contour, the higher the probability density associated with the given point's statistical data.
Upon careful examination of the eight split-edge violin plots, it becomes evident that the quartiles, medians, and confidence intervals (ranging from 1% to 99%) of the statistical data derived from both TD and EHFE methodologies exhibit a remarkable resemblance.Furthermore, the contours within the violin plots, which correspond to these two approaches, seem to be nearly indistinguishable.Drawing on the qualitative comparison of the split-edge violin plots, the observed concordance between the outcomes substantiates the notion that the fundamental mechanisms and performance metrics of TD and EHFE exhibit analogous characteristics.A detailed quantitative assessment of the statistical findings is provided below.
The resulting mean values, standard deviations, and errors of the external heat flows absorbed by the spacecraft components due to Earth infrared and albedo radiation are presented in Tables 4 and 5.For Earth infrared radiation, the maximum relative error between the two models is 4.37%, corresponding to a surface element of 15 standard deviations, with an absolute error of 0.157 W/m 2 .Regarding Earth albedo radiation, the maximum relative error between the two models is 5.09%, corresponding to a surface element of 15 standard deviations and an absolute error of 0.402 W/m 2 .Although the relative errors are high, the absolute errors are all less than 1 W/m 2 , which constitutes an acceptable margin of error.The relative errors in the external heat flow mean values obtained for each surface element are less than 1%.The Earth radiative EHFE equation demonstrates good agreement with the uncertainty-related numerical simulation results of the TD ray tracing model.6  and 7.For Earth infrared radiation, the maximum relative error between the two models is 3.41%, corresponding to surface element 1: lower limit of the 99.7% confidence interval, with an absolute error of 3.06 W/m 2 .For Earth albedo radiation, the maximum relative error between the two models is 4.45%, corresponding to surface element 21, the lower limit of the 99.7% confidence interval, with an absolute error of 2.47 W/m 2 .Comparison of the relative errors of the two confidence results reveals that applying the EHFE formula for uncertainty analysis yields a better confidence interval result of 95.4% than the 99.7% confidence interval result.Figure 9 shows probability density plots of the Earth infrared and albedo radiation external heat flow absorbed by the four surface elements of the spacecraft.The probability density functions derived from the two models generally exhibit good agreement and smooth curves, but there are minor discrepancies in the head and tail sections.For Earth infrared radiation, surface elements 1, 15, and 40 exhibit small deviations at their peak values between the two model results.For Earth albedo radiation, all surface elements display minor discrepancies at their peak values between the two model outcomes.The probability density function obtained from the EHFE formulation model successfully approximates the probability density function of the TD ray tracing external heat flow calculation model.

Discussion
When conducting uncertainty analyses for spacecraft thermal properties, variations in the parameters of the spacecraft surface coating's thermal properties necessitate a new ray tracing calculation for Earth infrared and albedo radiation external heat flux.This iterative process is time-consuming.In this paper, the equations for calculating the Earth infrared and albedo radiation external heat flow absorbed by the surface element are fully expanded.The parameters of the formula are clearly evident in the expanded formula matrix.For uncertainty analysis in thermal properties, the samples generated by the new operating conditions can simply be substituted into the corresponding parameter terms of the matrix, eliminating the need for additional ray tracing.The new method proposed in this paper requires only one ray tracing step to obtain the expanded matrix.Employing matrix operations instead of additional ray tracing for the remaining uncertain operating conditions is a key factor in the increased computational speed offered by the new method.
The Hr, Her matrix, and intersecting facet index Ir, Ier matrix have more than half of their elements equal to zero.When obtaining the Hr, Her, Ir, and Ier matrices during the initial operating condition external heat flux calculations using ray tracing, as well as when acquiring the updated Hr 1 , Her 1 matrix for external heat flux uncertainty calculations, employing sparse matrix storage and computation techniques can save a significant amount of computer memory and further enhance the computational speed for external heat flux uncertainty calculations.
This study applies the EHFE formula for uncertainty calculations of the Earth radiative external heat flux and identifies the reasons for discrepancies with the TD ray tracing calculation results.In the uncertain external heat flux calculation, the expansion formula algorithm avoids emitting rays for new conditions, but its calculation matrix is derived from rays emitted during the ray tracing associated with the initial conditions of the external heat flux calculation.When new operating conditions are considered, the thermal property parameters of the spacecraft surface coating change.In the ray propagation process, the path remains unchanged based on the initial operating conditions, although the energy propagated along the path should change.
The ray tracing method employed in this study is the path length method with the introduction of a cutoff factor.When the reflected ray energy (q2(1-α)) is less than or equal to kq1, ray tracking should be stopped.For new operating conditions, the TD ray tracing external heat flux calculation model takes into account the cutoff factor, ensuring that the ray is terminated at the appropriate location.However, the calculation using the EHFE formula still relies on the ray propagation path of the original operating conditions.Consequently, the reflected energy differs from the original operating conditions, resulting in rays traveling along longer (Figure 10) and shorter paths (Figure 11), i.e., the rays are cut off at incorrect locations.For Figure 10a, the energy at the ray at the path extension is less than or equal to kq1.At this juncture, the ray should terminate tracking.The legitimate trajectory of the ray is exhibited in Figure 10b.For Figure 11a, the energy at the ray where the path is shortened is greater than kq1.Consequently, the ray's tracing should be sustained.The accurate propagation path of the ray is elucidated in Figure 11b.This discrepancy accounts for the differences between the results of the EHFE formula and the TD ray tracing external heat flux calculation model.
The smaller the value of the cutoff factor (k), the less energy is used to terminate ray propagation, and the smaller this error becomes.Such errors can be avoided by using the path length method of ray tracing for external heat flux calculations.However, due to the disappearance of truncated energy, infinite tracing of rays in a dead loop may occur.
Furthermore, when performing uncertainty analysis calculations for external heat flux, the coating solar absorptivity, infrared emissivity, ray emission points, and ray emission directions are generated as samples based on random number seeds.The differences in these samples between the two models also contribute to the occurrence of random errors.The above analysis focuses on the accuracy and efficiency of the EHFE formula for calculating the uncertainty in the Earth radiative external heat flux.When using the TD method for thermal uncertainty analysis calculations, it is necessary to perform ray tracing for each of the N operating conditions, which is highly time-consuming.In contrast, the method proposed in this paper requires ray tracing for only one operating condition, generating the corresponding matrices.The remaining operating conditions involve updating the matrices to obtain the corresponding Earth infrared and albedo radiation external heat flux, resulting in significantly higher computational efficiency.The error generated in comparison with the TD model can be dynamically adjusted for calculation accuracy based on the size of the cutoff factor.

Conclusions
In this paper, we fully expand the factors in the traditional Earth infrared and albedo radiation external heat flux calculation formulas and perform cluster analysis on them.The factors are decomposed into the Earth infrared and albedo direct radiation external heat flux, first-order reflected Earth infrared and albedo radiation external heat flux, second-order reflected Earth infrared and albedo radiation external heat flux, ..., and (n − 1)order reflected Earth infrared and albedo radiation external heat flux.These components are then recombined to form a new external heat flux calculation formula.
In the calculation of N different operating conditions for the uncertainty external heat flux, the EHFE formula requires ray tracing calculation for only one operating condition to obtain the external heat flux and generate the corresponding matrix during this process.For the remaining operating conditions, there is no need for further ray tracing; simply substituting the new uncertainty samples into the corresponding positions in the matrix yields the uncertainty results for the Earth radiative external heat flux.The EHFE formula replaces the original complex ray tracing calculation with simple matrix operations and the calculation matrix in the EHFE formula contains sparse matrices.By combining sparse matrix storage and calculation techniques for external heat flux uncertainty calculations, a high computational efficiency is achieved.
Taking a specific spacecraft as an example, the external heat flux uncertainty calculation is performed.Comparison of the mean, standard deviation, probability distribution function, confidence interval, probability density function, and calculation time indicates that the calculation accuracy of the EHFE formula is comparable to that of the traditional external heat flux calculation formula, while the calculation time is significantly reduced.
The matrix required for the EHFE formula is calculated using the path length method of ray tracing with the introduction of a cutoff factor.Due to the presence of this factor, the ray propagation path may be elongated or shortened, resulting in certain differences between the traditional Earth radiation external heat flux ray tracing calculation results.Users can adjust the cutoff factor to dynamically control the calculation accuracy.This EHFE formula can provide a reference for improving the computational efficiency of

Figure 1 .
Figure 1.Optimization of radiation radiator design based on uncertainty analysis.
red spectrum reflectance of the τth intersection of the jn−1th ray in the nth term of the corresponding EHFE for the surface element coating.

Figure 2 .NN
Figure 2. Detailed classification of statistical rays for the calculation of the Earth infrared and albedo radiation external heat flow (n = 4).(a) Statistical rays of the Earth infrared and albedo direct radiation external heat flow; (b) one reflection of statistical rays of the Earth infrared and albedo radiation external heat flow; (c) secondary reflection of statistical rays of the Earth infrared and albedo radiation external heat flow; (d) three-time reflection of statistical rays of the Earth infrared and albedo radiation external heat flow.
Assuming n = 3 in the equation, the Earth infrared radiation external heat flow consists of the Earth infrared direct radiation external heat flow, the primary reflective Earth infrared radiation external heat flow, and the secondary reflective Earth infrared radiation external heat flow.For the EHFE equation, the Dr matrix is the infrared spectral reflectance combination matrix.

NKK
rows of the Ar matrix correspond to the expansion coefficients of the first term of the EHFE equation for the Earth infrared direct radiation external heat flow.As the rays strike the Earth directly, all elements are 1of the second term of the EHFE equa- tion.The elements in the first column represent the reflectance of the infrared spectrum for the surface element where the rays intersect for the first time; the second and third columns are of the third term of the EHFE equation.The elements in the first column represent the reflectance of the infrared spectrum for the surface element where the rays intersect for the first time.The elements in the second column represent the reflectances of the infrared spectrum of the surface element where the rays intersect for the second time.The elements in the third column are 1

( 2 )
Samples of all surface element infrared emissivity and corresponding infrared spectral reflectance of the spacecraft are generated using random numbers.(3)Based on the new infrared emissivity sample and the intersecting surface element numbering matrix Ir, the corresponding elements of matrices Ar, Hr, and Cr are updated.Matrices Ar 1 , Hr 1 , and Cr 1 that take into account the uncertainty parameters are generated.Matrix Ar 1 is used to obtain matrix Dr 1 .

Figure 3 .
Figure 3.Comparison of two processes for calculating the Earth infrared radiation external heat flow considering parameter uncertainties: (left) conventional; (right) based on the EHFE equation.In summary, the equation for solving the original Earth infrared radiative external heat flow is expanded.The clustering analysis of each item of the equation constructs a new solution equation r-old solar spectrum for the face element of the τth intersection of the jn−1th ray in the nth term of the corresponding expansion.According to Equation (18), the surface element Ei emits a total of Ner rays,

N
rows of the Aer matrix correspond to the expansion coefficients of the first term of the EHFE equation.As the rays all strike the Earth, all elements are 1term of the EHFE equation.The elements in the first column represent the reflectance of the solar spectrum for the surface element where the rays intersect for the first time, and the second and third columns are 1term of the EHFE equation.The elements in the first column represent the reflectance of the solar spectrum for the surface element where the rays intersect for the first time.The elements in the second column represent the reflectance of the solar spectrum for the surface element where the rays intersect for the second time; the elements in the third column are 1.

( 2 )
Random numbers are used to generate samples of the solar absorbance and reflectance of the corresponding solar spectrum for all surface elements of the spacecraft.(3) Based on the new solar absorbance samples and the intersecting surface element numbering matrix Ier, the corresponding elements of matrices Aer, Her, and Cer are updated.Matrices Aer 1 , Her 1 , and Cer 1 that take into account the uncertainty parameters are generated.Matrix Aer 1 is used to obtain matrix Der 1 .

Figure 4 .
Figure 4. Comparison of two processes for calculating the Earth albedo external heat flow taking into account parameter uncertainties: (left) conventional; (right) based on the EHFE equation.In summary, the primitive Earth albedo radiation external heat flow solution equation is expanded.The clustering analysis of each item of the equation leads to the construction of a new solution formula er-old

Figure 6 .
Figure 6.Spacecraft position of TD at the starting moment.

Figure 7 .
Figure 7. Split-edge violin plots of surface elements absorbing Earth infrared and albedo radiation external heat flow.(a,b) Earth infrared radiation; (c,d) Earth albedo radiation.

Figure 8
Figure 8 displays the probability distribution of the Earth infrared and albedo radiation external heat flow absorbed by the four surface elements of the spacecraft.The head and tail of the probability distribution curves obtained from the uncertainty calculations exhibit slight differences between the TD model and the EHFE formula model, while the rest of the curves largely overlap.The confidence interval results at 95.4% and 99.7% confidence levels can be derived from the curves in the figure.The chosen levels correspond to the 2σ and 3σ confidence levels of the normal distribution and are typical values employed for risk assessment in uncertainty analysis.The results are presented in Tables6 and 7.For Earth infrared radiation, the maximum relative error between the two models is 3.41%, corresponding to surface element 1: lower limit of the 99.7% confidence interval, with an absolute error of 3.06 W/m 2 .For Earth albedo radiation, the maximum relative error between the two models is 4.45%, corresponding to surface element 21, the lower limit of the 99.7% confidence interval, with an absolute error of 2.47 W/m 2 .Comparison of the relative errors of the two confidence results reveals that applying the EHFE formula for uncertainty analysis yields a better confidence interval result of 95.4% than the 99.7% confidence interval result.

Figure 8 .
Figure 8. Probability distribution of surface elements absorbing Earth infrared and albedo radiation external heat flow.(a-d) Earth infrared radiation; (e-h) Earth albedo radiation.

Figure 9 .
Figure 9. Probability density plots of surface elements absorbing Earth infrared and albedo radiation external heat flow.(a-d) Earth infrared radiation; (e-h) Earth albedo radiation.Table8presents the time required to conduct 2000 uncertainty analyses for each method.Compared with the TD ray tracing method, the EHFE formula for uncertain Earth infrared radiation heat flow calculations can be accelerated by a factor of approximately 22. Uncertain Earth albedo external heat flow calculations can be expedited by a factor of nearly 14.The computational efficiency is significantly improved.In the Earth radiation external heat flow's uncertainty analysis, TD-based ray tracing uses 100% of CPU capacity.EHFE-based calculations of external heat flow use about 80% of CPU capacity.EHFE achieves good acceleration results, while not calling up all of the computer's CPU.

Figure 10 .
Figure 10.EHFE ray propagation path extension under actual working conditions.(a) EHFE ray propagation path; (b) TD baseline ray propagation path.

Nomenclature
QS-ithe solar heat load on node i (W)Irthe intersecting face element number matrix Qer-i the planetary albedo heat load on node i (W) Hr infrared emissivity matrix of intersecting surface elements from Ir Qr-i the planetary infrared heat load on node i (W) Ar infrared reflectance matrix of intersecting surface elements from Ir Q the internal heat source power (W) Cr each common factor of the EHFE formula Dji the linear thermal conductivity between nodes j and i (W/K) Der a combination matrix of the solar spectral reflectance of the coating Gji the radiative thermal conductivity between nodes j and i (W/K 4 ) Ier the intersecting face number matrix T temperature (K) Her solar absorbance matrix of intersecting surface elements from Ier m mass (kg) Aer solar spectral reflectance matrix of intersecting surface elements from Ier c specific heat (Jkg −1 K −1 ) Cer the multiplication of each common factor of the EHFE formula by cosθse N rays' number emitted from the target surface element Nmin the minimum number of model runs Rα a random number between 0 and 1 Nσ number of standard deviations q1 the energy of the initial surface element emitting the ray (W/m 2 ) P percentile q2 the energy of the ray hitting a surface element (W/m 2 ) ΔP deviation from percentile k radiation absorbed by the surface element Ei (W/m2) ρ the Earth average solar albedoS the solar constant (W/m 2 ) θse the angle between the outer normal vector of the ray's intersection with the Earth and the direction vector of the sun rays (deg) E  i r the infrared emissivity of the surface element Ei coating Acronyms r,A i q the density of the Earth infrared direct radiation heat flow to the surface element Ei (W/m 2 ) MC Monte Carlo r,B i q the external heat flow from the Earth infrared indirect radiation absorbed by the surface element Ei (W/m 2 ) RBF radial basis function er i q the density of the external heat flow from the Earth albedo radiation absorbed by surface element Ei (W/m 2 ) EHFE external heat flow expansion er,A i q the density of direct Earth albedo radiation heat flow to the surface element Ei (W/m 2 flow from the Earth albedo indirect radiation absorbed by the surface element Ei (W/m 2 ) TD thermal desktop Dr the infrared spectral reflectance combination matrix CI confidence interval

Table 1 .
Orbit and attitude parameters.

Table 2 .
Average value of uncertainty variables.

Table 3 .
Spacecraft experimental analysis surface element numbers and their components.

Table 4 .
Mean and standard deviation of the Earth infrared radiation external heat flow absorbed by the surface elements of the two models.

Table 5 .
Mean and standard deviation of the Earth albedo radiation external heat flow absorbed by the surface elements of the two models.

Table 6 .
Confidence intervals for Earth infrared radiation external heat flow absorbed by the two model surface elements.

Table 7 .
Confidence intervals for Earth albedo radiation external heat flow absorbed by the two model surface elements.

Table 8 .
Time required for the calculation of the uncertainty in the Earth radiation external heat flow for each method.