Design and Analysis of a Moon-Based Earth-Radiation Measurement System

Abstract

This research project envisions using a lunar observation platform to measure the full-wave (0.2~100 μm) and shortwave (0.2~4.3 μm) radiation of the Earth, achieving an accurate estimation of the overall radiation budget of the Earth. Based on the lunar platform, the system analyzes Earth’s radiation characteristics and geometric attributes, as well as the sampling properties of observation times. Informed by these analyses, an Earth-facing optical radiation measurement system tailored to these specifications is designed. The optical system adopts an off-axis three-mirror configuration with a secondary image plane, incorporating a field stop at the primary image plane to effectively suppress solar stray light, scattered lunar surface light, and background radiation from the instrument itself, ensuring the satisfactory signal-to-noise ratio, detection sensitivity, and observation duration of the instrument. At the same time, stringent requirements are imposed for the surface treatments of instrument components and temperature control accuracy to further ensure accuracy. Simulation analyses confirm that the design satisfies requirements, achieving a measurement accuracy of better than 1% across the entire optical system. This Moon-based Earth-radiation measurement system, with capabilities for Earth-pointing tracking, radiation energy detection, and stray-light suppression, furnishes a more comprehensive dataset, helping to advance our understanding of the mechanisms driving global climate change

1. Introduction

The balance and variability of Earth’s radiation budget are crucial parameters in studying climate change and predicting future climate trends accurately [1]. In the study of the Earth’s climate system, it is essential to observe the Earth as a whole and employ a holistic and global systemic perspective with multi-temporal and spatial-scale analytical methods to investigate the Earth’s overall behavior. Such an approach plays a pivotal role in examining large-scale macroscopic scientific phenomena occurring within the Earth’s system, analyzing their spatial correlations, and understanding the interconnections and mutual influences among different systems [2]. Global climate change study requires comprehensive and accurate measurements of Earth’s energy budget to achieve holistic observations of the planet. This can significantly improve the quality of various regional and global radiation calculations, helping to better quantify Earth’s energy budget and explain macroscopic phenomena such as global dimming or brightening.

In 1975, the Earth Radiation Budget Instrument (ERBI), launched with the American Nimbus series of satellites, initiated pioneering explorations of Earth’s radiation budget measurements in space. In the nearly 50 years since, multiple Earth-radiation-budget measurement instruments have been launched internationally, forming a global polar-orbiting and geostationary Earth-radiation-budget satellite observation system. Figure 1 illustrates the historical development of satellite payloads used for Earth radiation observations.

Currently, Earth-radiation-observing satellites primarily operate in two types of orbits: (1) Low Earth Orbit (LEO), which has a low altitude, limited field of view, and discontinuous temporal and spatial sampling [3]; and (2) Geosynchronous Earth Orbit (GEO), which has a high orbit and good spatiotemporal continuity, but a limited observation angle [4]. Overall, satellite detection platforms suffer from issues such as narrow instantaneous fields of view, orbital drift, and short operational lifetimes. These limitations hinder accurate measurement of Earth’s shortwave reflection and longwave radiation, preventing the determination of the absolute value of Earth’s radiation budget. Consequently, there is an urgent need to develop a new platform capable of conducting long-term, continuous radiation monitoring of the Earth on a global scale.

In comparison to satellite observation platforms, a Moon-based Earth observation platform offers advantages in comprehensiveness, longevity, stability, and continuity. As a stable platform, the Moon can achieve uninterrupted monitoring of almost the entire Earth–Moon space and the Earth’s near-lunar surface on a global scale, with consistent time and continuous coverage relative to GEO or LEO. Unhampered by Earth’s atmospheric interference, this mode of observation facilitates the acquisition of more stable and continuous data [5,6].

With a view to utilizing the Moon as a unique observational platform, we now perform a comprehensive analysis of its characteristics. We will design an optical system specifically adapted to the lunar environment for high-accuracy measurements of Earth’s radiative energy distribution. This system will allow for long-term, large-scale radiation detection of the Earth, providing valuable data on the Earth’s energy balance and the cumulative radiative energy directed toward the Moon. Additionally, we aim to study the instantaneous total radiation budget of the Earth, along with its seasonal and interannual variations. This optical system is designed to precisely measure the total radiation energy of Earth across the 0.2~100 μm spectral range, as well as the reflected shortwave solar radiation energy within the 0.2~4.3 μm band. By utilizing observations from these two channels, it is, further, possible to determine the emitted radiation energy of Earth in the 4.3~100 μm band. The objective is to facilitate long-term, direct, and continuous measurements of Earth’s shortwave reflection and longwave radiation, including totals and variations. By doing so, we will acquire high-accuracy, long-duration, all-weather observational data on the outgoing and reflected energy from the Earth’s hemisphere. These datasets will enable the identification of external factors influencing global climate change (such as the Sun) and changes and interactions within the climate system (including oceans, atmosphere, land, clouds, snow and ice cover, etc.), deepening our understanding of recent global climate change and enhancing our capability to predict future global climate variations [7,8].

This paper initially explores a “quasi-point source” observation method for Earth’s top-of-atmosphere radiation energy as observed from the Moon, studying the characteristics of Earth’s atmospheric radiation as observed from the lunar platform. It then proceeds to design an off-axis, three-mirror, Moon-based Earth-radiation observation optical system featuring a secondary image plane, a device tailored specifically for these observations. Using this optical system as a model, simulations are conducted to analyze the effects of solar stray light, lunar surface scattering, and the instrument’s own background radiation. Through these simulations, the system demonstrates effective suppression of stray light and self-radiation, ensuring a measurement accuracy of 1% across the instrument, which satisfies the requirements for Earth-radiation measurement applications.

2. Principle and Composition of the Lunar-Based Earth-Radiation Measurement System

The Moon-based Earth-radiation measurement system, utilizing the Moon as an observation platform, conducts long-term, large-scale radiation monitoring of the Earth, acquiring data on the Earth’s radiation energy balance and the cumulative absolute radiation directed towards the Moon. This information facilitates studies of the Earth’s climate system.

The system components are depicted in Figure 2. The pitch and yaw rotation mechanisms within the system enable tracking of the Earth’s position. The Earth-imaging camera captures images of the Earth and employs image recognition and visual tracking algorithms to identify and track the Earth during the radiation measurement process; the instrument remains pointed at the Earth, and the Earth stays within the field of view of the optical system, meeting the tracking precision requirements for Earth measurements, and thereby ensuring the accuracy of the measurement results. The dual-channel optical system for Earth radiation measurement facilitates the detection of full-spectrum energy radiation from the Earth and the reflection of solar energy, with built-in capabilities to suppress stray light, guaranteeing a measurement accuracy exceeding 1%. This accuracy allows for accurate estimation of the Earth’s overall albedo, as well as its visible and near-infrared reflectance, enabling precise measurement of the Earth’s radiation budget. This is crucial for understanding the global energy balance, climate change, and related ecosystem responses.

3. Lunar Observation Platform: Analysis of Characteristics

As for the lunar observation platform, analyses have been conducted on Earth’s radiation energy, the geometric characteristics of the Earth, and the time-sampling properties of radiation energy observed by optical systems. This research provides crucial foundations for the design of Earth-radiation observation optical systems based on the Moon and the determination of observational parameters.

3.1. Analysis of Earth’s Radiation Energy

The Moon-based Earth-radiation observation optical system comprises two channels: (1) a full-wave channel (0.2~100 μm), designed to measure the total radiation energy emitted by the Earth; and (2) a shortwave channel (0.2~4.3 μm), specifically designed to measure the shortwave solar radiation reflected from the Earth. The reflected solar radiation from both the Earth’s surface and the top of the atmosphere amounts to approximately 99.9 W/m². Meanwhile, the longwave radiation energy emitted from the Earth’s surface to space through the top of the atmosphere is around 239.9 W/m² [9].

In the context of global radiation measurements of the Earth from a lunar observation platform, the Earth can be approximately regarded as a Lambertian source. Utilizing the conversion formula between irradiance and radiance for a Lambertian source, the radiation energy in both channels can be described using the following expressions:

$$L_{earth}={\displaystyle \frac{E_{earth}}{\pi }}$$

(1)

where $L_{earth}$ represents the Earth’s radiance, and $E_{earth}$ denotes the Earth’s irradiance. The Earth, acting as a Lambertian radiator, emits radiation uniformly in all directions.

Table 1. Earth’s radiation energy.

Channel Number	1	2
Waveband Range (μm)	0.2~100	0.2~4.3
Earth Irradiance (W/m2)	339.8	99.9
Earth Radiance (W/m2/sr)	108.2	31.8

illustrates the Earth’s radiative energy for different wavelength ranges.

3.2. Analysis of Geometric Characteristics Relative to the Earth

Data from the Clementine spacecraft in the 1990s suggested the possible presence of water ice beneath the surface layers of permanently shadowed regions at the Moon’s south pole, with the unique geographic and environmental conditions of the lunar south pole rendering it suitable as a site for astronomical observations and a springboard for deep-space exploration [10]. Consequently, a landing site for this Moon-based Earth-radiation measurement system has been designated at the Moon’s south pole, leveraging its position to facilitate radiation-based observations of Earth.

According to the positional relationship depicted in Figure 3, the half-field of view angle of the Earth relative to the Moon-based Earth-radiation-observing optical system is represented by Equation (2).

$$\theta =\arcsin \left({\displaystyle \frac{R_E}{\sqrt{{R_M}^2+{D_{E-M}}^2}}}\right)$$

(2)

The Moon-based Earth-radiation observation optical system is positioned at point S at the lunar south pole. Here, $R_E$ represents the Earth’s radius, which is 6371 km, $R_M$ denotes the Moon’s radius, measuring 1738 km, and $D_{E-M}$ signifies the Earth–Moon distance. Due to the elliptical orbit of the Moon around the Earth, this distance $D_{E-M}$ fluctuates between its nearest point (perigee) at roughly 363,300 km and its farthest point (apogee) at about 405,500 km. Based on Equation (2), the resultant range for the half-angle of the field of view for Earth observations made from the lunar south pole is determined to be within 0.9° to 1.0°.

3.3. Analysis of Time Sampling Characteristics for Moon-Based Observations

Due to the large Earth–Moon distance, when observing the Earth from the Moon, the entire Earth occupies a relatively small field of view, which can be approximated as a point source. With its geometric characteristic of fully covering the Earth-facing hemisphere near the lunar surface, the Moon-based platform can achieve measurements by treating the Earth as a “quasi-point” radiation source. The term “quasi-point” refers to the small field of view of the Earth under the Moon-based observation platform. However, this innovative observation approach raises a novel question: how does the interval between observations influence the sampling of radiation energy at the top of the atmosphere [11]? To address this, an analysis is conducted on the impact of temporal variations on the results of radiation measurements.

3.3.1. The Influence of Earth–Sun Angle Variation on Measurement Results

Spatially, as the angle vector among the Earth, Sun, and Moon changes over time, this leads to variations in the illuminated region of the Earth during radiation measurements. Consequently, the reflected solar radiation from the Earth is affected, as illustrated in the following, Figure 4:

The Moon-based Earth-radiation measurement system is equipped with a ground-pointing tracking mechanism, ensuring that the optical system consistently targets the Earth for radiation measurements. The scenario depicted in Figure 5 primarily results from variations in the Earth–Sun angle. With the Moon’s south pole chosen as the landing site, orbital analyses were conducted using simulation software, revealing the temporal changes in the Earth–Sun angle.

Based on the orbital calculation results, the Earth–Sun angle varies within the range of 0° to 180°, with a rate of change of 0.008° per minute. Starting from an initial observation condition in which the Earth–Sun angle is 90°, and considering the pattern of Earth–Sun angle variation, we deduced the relative changes in Earth’s reflected radiative energy over time during the process of Earth’s radiation measurement by the optical system. This is shown in

Table 2. The impact of Earth–Sun angle variation on radiation measurement.

Time Variation (min)	Variation in the Earth–Sun Angle (°)	Relative Variation of Measurement Results (%)
1	0.008	0.014
2	0.016	0.028
5	0.040	0.070
10	0.080	0.140
20	0.160	0.280

:

3.3.2. The Influence of Earth’s Rotation on Measurement Results

In addition to the temporal variation in the Sun–Earth–Moon angular vector, Earth’s rotation induces temporal variations in the observed terrain. Distinct terrestrial features possess differing albedos, which fluctuate based on factors such as terrain type, surface conditions, humidity, and seasonal changes. These disparities in albedo directly influence the measurement outcomes of instruments detecting Earth’s reflected radiant energy.

Table 3. Typical albedo of terrestrial features.

Terrestrial Features	Albedo (%)
Land	30
Ocean	10
Snow-Covered Areas	85
Deserts	25

presents the albedo of terrestrial features.

Most of the Earth’s surface consists of oceans and land. Given this typical scenario, we analyze the transformation of the terrain in the observation area from land to sea. Likewise, we initiate our measurements with the angle between the Earth and the Sun set at 90°. Under these conditions, changes in the Earth’s rotation angle lead to alterations in the relative characteristics of the observed region:

$$\Delta S=\left\{\begin{array}{l}{\displaystyle \frac{\Delta \theta }{{90}^{°}}}·100\%(0^{°}\le \Delta \theta \le {90}^{°})\\ 100\%({90}^{°}<\Delta \theta <{270}^{°})\\ {\displaystyle \frac{{360}^{°}-\Delta \theta }{{90}^{°}}}·100\%({270}^{°}\le \Delta \theta \le {360}^{°})\end{array}\right.$$

(3)

Here, $\Delta \theta$ represents the Earth’s rotation angle. Based on Equation (3) and the variation in reflectance due to the change in terrain from land to sea, the Earth’s rotation leads to the following relative changes in measurement outcomes:

$$\Delta R=\Delta S·\Delta \rho$$

(4)

$\Delta \rho$ denotes the change in reflectivity caused by the variation in terrain features. The Earth completes one full rotation in a period of one day, with a rotational speed of 0.25° per minute. Figure 6 provides a schematic of how the Earth’s rotation affects the terrain scenes.

Table 4. The rotation of the Earth, leading to variations in its radiation.

Time Variation (min)	Earth’s Rotational Angle (°)	Relative Variation of Measurement Results (%)
1	0.25	0.06
2	0.50	0.12
5	1.25	0.30
10	2.50	0.60
20	5.00	1.20

shows the impact of Earth’s rotation, over time, on measurement results.

The analysis above reveals that changes in terrain features due to Earth’s rotation can directly impact the measurement accuracy of shortwave channels. To facilitate rapid measurement of Earth’s radiation from a Moon-based Earth observation radiometer, with the aim of capturing the evolution of climatic events within a short timeframe and achieving high measurement accuracy (better than 1%), and thereby revealing hidden phenomena and patterns of climate change, a requirement of a time resolution better than 5 min is imposed on the instrument.

4. Results

This Moon-based Earth-radiation observation optical system, utilizing the Moon as an observation platform, conducts long-term, macroscopic radiation monitoring of the Earth, acquiring data on the Earth’s radiation energy imbalance and the absolute cumulative radiation energy of the Earth as a whole in the lunar direction. This system is designed to investigate the instantaneous total of Earth’s radiation budget as well as its seasonal and interannual variations. Equipped with capabilities for Earth radiation energy detection, suppression of stray light, signal acquisition, and data reading, the Moon-based Earth-radiation observation optical system serves as a pivotal tool in understanding Earth’s radiant-energy dynamics.

4.1. Detector Selection

The spectral range requirements of the observation channels require a device covering 0.2–100 μm and specifically emphasize the 0.2–4.3 μm band; only thermoelectric detectors possess such characteristics, and they are divided into two types: thermopiles and thermistors. Climate research requires accurate radiation measurements; therefore, absolute radiation detectors are needed. These detectors incorporate a heating wire structure into the thermal detector, allowing the incident light power to be substituted for by electrical heating power, and thereby enabling the calculation of the incident light power through the measurement of electrical power. The detector has an effective photosensitive diameter of 3.5 mm, a time constant better than 5 min, and a radiation power detection precision better than 0.1 μW.

The absolute radiation detector used in the optical system employs a structure consisting of a platinum resistance thermometer combined with a heating wire. The platinum resistance thermometer is used to sense the radiation energy absorbed by the detector, while the heating wire provides controllable electrical heating power. This detector operates on the principle of electrical substitution. When the temperature change in the cavity caused by the incident radiation power is replicated by an equivalent electrical power, the unknown optical power (P_E) can be calibrated by precisely measuring the equivalent electrical power (P₀), ensuring that the temperature increase due to the incident radiation equals the temperature increase caused by the electrical heating [12]. The measurement principle of the detector is shown in Figure 7.

$$P_0={\displaystyle \frac{P_E}{\alpha }}$$

(5)

where $\alpha$ denotes the absorptance of the detector. Absolute radiometric detectors feature self-calibration capabilities, enabling them to undergo electrical calibration by applying a known amount of heating power to the heating filament. This process serves to calibrate the detector’s response to incident light power.

4.2. Analysis and Selection of Optical Structural Configurations

The selection of an optical system structure is decisive in optical design, with different configurations impacting performance parameters and dictating the overall dimensions, mass, and cost of the instrument [13]. Choosing the appropriate optical structure tailored to specific application requirements is crucial for optimizing performance, controlling costs, and fulfilling environmental constraints.

For Moon-based Earth-radiation observation systems, which must cope with a broad spectral range (0.2~100 μm), weak target signals, strong stray solar and lunar-surface light, and high internal background radiation, a comprehensive assessment leads to the off-axis three-mirror configuration as the most ideal choice. This structure enables wide-spectrum coverage, an absence of central obstruction for high energy utilization, and, through the design of a secondary image plane, effectively suppresses off-axis stray light (from the Sun and Moon’s surfaces) as well as the instrument’s own background radiation. This enhances the system’s signal-to-noise ratio and detection sensitivity, thereby facilitating continuous and prolonged radiation measurements of the Earth with improved accuracy and sensitivity.

4.3. Optical System Design and Optimization

Based on the analysis of the characteristics of the lunar observation platform and designed to ensure a measurement accuracy exceeding a specification of better than 1%, the design parameters for the Moon-based Earth-radiation observation optical system are presented in

Table 5. Optical system design specification parameters.

Performance Specifications	Parameter Values
Wavelength Range	0.2~100 μm @Longwave channel 0.2~4.3 μm @Shortwave channel
Field of View	±1.1°
Entrance Pupil Diameter	35 mm
Focal Length	60 mm
Time Constant	Better then 5 min
Detector Dimensions	$\mathrm{\Phi }$ 3.5 mm

.

Using the specified parameters of the optical system, Zemax software is employed to conduct optical modeling and simulation, facilitating an evaluation and iterative optimization of the system’s performance in order to attain a rational optical layout. The optical system adopts an off-axis three-mirror anastigmatic configuration featuring a secondary image plane. An intermediate image plane is formed between the second and third mirrors, and at this plane, a field stop (filter) is introduced to effectively suppress off-axis stray light and mitigate the intrusion of instrumental background radiation into the detector, thereby enhancing measurement accuracy [14]. Figure 8 shows a schematic of the optical system structure for both the full-wave and shortwave channels, where different colors of the light rays represent different field angles.

The primary and secondary mirrors of the optical system adopt a hyperbolic surface profile, while the third mirror employs an ellipsoidal shape. Two spectral channels share the same optical system, with a 1 mm thick fused silica filter placed at the primary image plane for the shortwave channel, effectively cutting off radiation in the 4.3~100 μm wavelength band. The effective diameter of the detector’s sensitive area is 3.5 mm, which is capable of fully receiving the Earth’s radiative energy. The detailed parameters of the mirror and the filter are shown in

Table 6. Parameters of the optical system mirror and filter.

Optical Components	Diameter (mm)	Curvature Radius (mm)	Conic Constant
Primary Mirror	44	166.15	−1.63
Secondary Mirror	22	232.77	−2.00
Third Mirror	40	75.69	−0.19
Filter	6	Infinity	/

.

It is important to note that the Moon-based Earth-radiation observation optical system is not an imaging system; its post-design performance assessment primarily relies on the footprint map of the image plane. This map illustrates whether the final image height meets the operational requirements. As depicted in Figure 9, under full-field-of-view conditions, the imaging beams of the full-wave channel concentrate within a circular region with a radius of 1.66 mm, whereas those of the shortwave channel fall within a circle of 1.68 mm radius. Given that the detector receiving surface diameter measures 3.5 mm, the designed optical system demonstrates its capability to adequately capture the target radiation energy, thus fulfilling the specifications for Earth-radiation measurement purposes.

Moreover, optical aberrations and diffraction effects significantly influence the performance of the MTF; the MTF not only reveals the system’s capability in reproducing details and image sharpness, but also serves as a pivotal criterion for optical design and performance evaluation. As illustrated in Figure 10, which depicts the MTF curves of an optical system’s dual spectral channels, it is evident that at the system’s Nyquist frequency of 0.14 line pairs per millimeter (with a pixel size of 3.5 μm), both the full-wave and shortwave channels exhibit MTF values above 0.95, indicating effective correction of optical aberrations within the system.

4.4. Tolerance Analysis

For a designed optical system, the tolerance situation is a crucial factor in evaluating its feasibility. Excessive tolerances can escalate manufacturing and alignment demands, leading to increased costs and potentially unstable image quality. Conversely, overly loose tolerances may severely degrade image quality. In summary, unreasonable tolerance allocation can result in performance deficiencies in the system. Therefore, rational tolerance distribution is of great importance for an optical system [15]. In the case of these Moon-based Earth-radiation measurement optics, to meet the requirements for terrestrial radiation measurement, the tolerance analysis dictates that the image plane dimensions should be less than 3.5 mm to ensure satisfactory performance.

Table 7. Tolerance allocation for optical component fabrication.

Optical Components	Curvature Radius	Conic Constant	$\mathbf{RMS} \mathbf{Surface} \mathbf{Figure} (\mathit{\lambda }=633 \mathbf{n}\mathbf{m}$)	Thickness (mm)
Primary Mirror	±0.1	±0.05	1/10	±0.1
Secondary Mirror	±0.1	±0.1	1/10	±0.1
Third Mirror	±0.05	±0.01	1/10	±0.1
Filter	/	/	1/5	±0.1

and

Table 8. Alignment tolerance allocation for optical systems.

Optical Components	TILT X/(°)	TILT Y/(°)	DEC X/mm	DEC Y/mm	Distance/mm
Primary Mirror	±0.1	±0.1	±0.1	±0.1	±0.5
Secondary Mirror	±0.3	±0.3	±0.05	±0.05	±0.5
Third Mirror	±0.1	±0.1	±0.1	±0.1	±0.3
Filter	±0.1	±0.1	±0.05	±0.05	±0.5

show the tolerances allowed for optical system manufacturing and assembly.

5. Discussion

For a Moon-based Earth-radiation measurement optical system, stray light is a critical parameter that can impact the detection of weak target signals and the duration of continuous Earth observations. Thus, the analysis, measurement, and correction of stray-light factors are of paramount importance [16]. In the process of stray-light analysis, it is first necessary to establish the material properties and surface optical characteristics of the optical and mechanical components. In the optomechanical structure as described in

Table 9. Surface Properties of Components in the Moon-Based Earth-radiation observation Optical System.

Object Number	Component Name	Surface Treatment	Surface Properties
1	Light Baffle	Sprayed with Black Paint	0.2~100 μm: Absorption 95%, Lambertian Reflection 5%
2	Primary–Secondary Mirror Chamber	Sprayed with Black Paint	0.2~100 μm: Absorption 95%, Lambertian Reflection 5%
3	Primary Mirror	Coated with Metal Film	0.2~100 μm: Specular Reflection 90%, Absorption 7%, Lambertian Reflection 3%
4	Secondary Mirror	Coated with Metal Film	0.2~100 μm: Specular Reflection 90%, Absorption 7%, Lambertian Reflection 3%
5	Field Stop	Polished	0.2~100 μm: Specular Reflection 80%, Absorption 10%, Lambertian Reflection 10%
6	Third Mirror	Coated with Metal Film	0.2~100 μm: Specular Reflection 90%, Absorption 7%, Lambertian Reflection 3%
7	Third Mirror Chamber	Polished	0.2~100 μm: Specular Reflection 80%, Absorption 10%, Lambertian Reflection 10%
8	Detector Front Baffle	Polished	0.2~100 μm: Specular Reflection 80%, Absorption 10%, Lambertian Reflection 10%
9	Detector	Coated with Black Gold	0.2~100 μm: Absorption 98%, Lambertian Reflection 2%

, the setup of the light source is illustrated, and ray tracing can be performed on the optomechanical system. Figure 11 shows the structural model for the stray-light simulation and analysis of the optical system.

The light baffles and primary–secondary mirror chamber are treated with black paint, leveraging their high absorption properties to suppress off-field stray light from the solar and lunar surfaces. Meanwhile, the polishing of the third mirror chamber and the detector front baffle tube aims to minimize the instrument’s own background radiation. Moreover, the design of the field stop and the detector front baffle effectively constrains the width of the light beam entering the optical system and ultimately contributes to the imaging process. This dramatically reduces the energy of non-imaging rays incident upon the detector, playing a pivotal role in suppressing stray light as to the instrument.

5.1. Solar Stray-Light Analysis

As the primary source of out-of-field stray light for the optical system, the Sun’s radiation energy is approximately 5.45 × 10⁴ times that of Earth’s radiation (with solar irradiance at 1362 W/m² and the minimum Earth irradiance received in the shortwave channel being 0.025 W/m²). The analysis method for solar stray light initially determines the optical system’s stray-light suppression capability at varying off-axis angles. Subsequently, by relating the intensity of the stray-light source to that of the target source, the final stray-light coefficient is derived. The ultimate results from the stray-light simulation analysis serve to ascertain the operating conditions for the instrument. Figure 12 shows a schematic of the simulation and analysis of the optical system without stray-light and at a 10-degree off-axis angle.

The solar stray-light suppression capability of the Moon-based Earth-radiation measurement optical system is represented by Equation (6):

$$PS{T}_{solar}={\displaystyle \frac{{\mathrm{\Phi }}_{{\theta }_{solar}}}{{\mathrm{\Phi }}_0}}$$

(6)

where, ${\mathrm{\Phi }}_{{\theta }_{solar}}$ is the detector count value obtained from the simulation analysis when the Sun is at an off-axis angle of ${\theta }_{solar}$, and ${\mathrm{\Phi }}_0$ is the detector count value from the effective field of view simulation analysis.

Based on the optical system’s capability to suppress solar stray light, the final solar stray-light coefficient is deduced as follows:

$$V_{solar}=PS{T}_{solar}·{\displaystyle \frac{\cos {\theta }_{solar}·E_{solar}}{E_{earth-short}}}$$

(7)

where $E_{solar}$ is the solar irradiance, taken as 1362 W/m², and $E_{earth-short}$ is the Earth’s irradiance in the shortwave channel, taken as 0.025 W/m². Based on the simulation analysis outcomes, the final status of the stray light is determined. Based on the analysis results from the stray-light simulation software LightTools, the final stray-light conditions are shown in

Table 10. Results of solar stray-light simulation analysis.

Off-Axis Angle (°)	Stray-Light Suppression Capability	Stray-Light Coefficient (%)
3	1.34 × 10−6	7.27
5	8.35 × 10−7	4.53
10	5.01 × 10−8	0.27
15	3.34 × 10−8	0.18
20	1.67 × 10−8	0.09
30	1.33 × 10−8	0.06
45	6.68 × 10−9	0.03
60	5.01 × 10−9	0.01
80	4.18 × 10−9	0.01

.

To ensure radiation measurement accuracy of the Moon-based Earth-radiation measurement optical system exceeding 1%, a stray-light coefficient of no more than 0.3% is required. According to the simulation analysis results, with the Sun’s angle relative to the optical axis exceeding 10°, the stray-light coefficient is ≤0.27%, meeting the measurement requirements. Figure 13 shows a schematic of the conditions under which the instrument can observe the Earth in the presence of solar stray-light.

5.2. Lunar Surface Stray-Light Analysis

The Moon-based Earth-radiation measurement optical system situated at the lunar south pole would encounter stray light from the Moon’s surface. These stray-light angles deviate minimally from the optical axis, significantly increasing the system’s demand for stray-light suppression. This is a primary reason why the optical system adopts an off-axis three-mirror anastigmatic configuration with a secondary image plane. By incorporating a field stop at the primary image plane, the majority of lunar surface stray light is suppressed. Additionally, the inclusion of an extinction tube before the detector further enhances the suppression of such stray light.

Orbital calculations reveal that at the lunar south pole, the range of solar elevation angles varies between ±6.6°, while the Earth elevation angle ranges from 0 to 11.5°. Figure 14 and Figure 15 show the variations in the solar and Earth pitch angles.

The lunar irradiance can be expressed as

$$E_{lunar}=E_{solar}·\sin {\theta }_{height}$$

(8)

where $E_{solar}$ represents the solar irradiance of 1362 W/m², and ${\theta }_{height}$ is the solar elevation angle, with a range derived from orbital calculations. This yields a lunar irradiance range of 0 to 156.5 W/m²; here, the maximum lunar irradiance received, 156.5 W/m², is used for analysis to ensure coverage across the entire range of solar elevation angles during measurements. The lunar surface is not a good reflector, and the stray light it generates primarily falls within the 4.3~100 μm wavelength range. Consequently, lunar surface stray light primarily affects measurements in the full-wave channel. The process for analyzing this stray light is similar to that described above for solar stray light, and the ultimate stray-light coefficient is

$$V_{lunar}=PS{T}_{lunar}·{\displaystyle \frac{E_{lunar}·\cos {\theta }_{lunar}}{E_{earth-total}}}$$

(9)

where ${\theta }_{lunar}$ denotes the angle at which lunar surface light deviates from the optical axis, $PS{T}_{lunar}$ represents the lunar surface stray-light suppression capability, $E_{lunar}$ is the lunar irradiance, taken as 156.5 W/m², and $E_{earth-total}$ is the minimum Earth irradiance in the full-wave channel, taken as 0.084 W/m².

Table 11. Results of lunar surface stray-light simulation analysis.

Off-Axis Angle (°)	Stray-Light Suppression Capability	Stray-Light Coefficient (%)
2	1.34 × 10−6	2.74
3	1.34 × 10−6	0.25
5	8.35 × 10−7	0.15
10	5.01 × 10−8	0.01
11.5	4.68 × 10−8	0.01

shows the stray-light conditions at different angular deviations between the lunar surface and the optical system’s optical axis.

To ensure that the radiation measurement accuracy of the Moon-based Earth-radiation measurement optical system exceeds 1%, a lunar surface stray-light coefficient of no more than 0.3% is required. According to the simulation analysis results, with the lunar surface angle relative to the optical axis exceeding 3°, the stray-light coefficient is ≤0.25%, meeting the measurement requirements. Figure 16 shows a schematic of the conditions under which the instrument can observe the Earth in the presence of lunar surface stray-light.

Based on the aforementioned analyses of solar and lunar surface-based stray light, the operational conditions for the Moon-based Earth-radiation measurement optical system used to observe the Earth are as follows: (1) the Sun must be deviated by more than 10° from the optical axis; and (2) the Earth elevation angle should exceed 3°. Employing an off-axis three-mirror anastigmatic design with a secondary image plane effectively suppresses stray light from small off-axis angles, ensuring extended observation periods for the instrument. This approach enables the acquisition of continuous and prolonged radiation energy data from the Earth, which is instrumental in studying global climate change, revealing hidden phenomena and patterns of climate variation, and fostering the development of new models and theories of the Earth’s climate system.

5.3. Instrument-Based Background Radiation Analysis

The full-wave channel of the Moon-based Earth-radiation measurement optical system covers a spectral range of 0.2~100 μm. When measuring radiation from the Earth, the instrument’s own background radiation can impact its signal-to-noise ratio and detection sensitivity. Mitigating the effects of instrument-based background radiation primarily involves two strategies: (1) minimizing the radiation energy produced by the instrument itself that enters the detector via reflection or diffraction; and (2) controlling variations in background radiation energy during the measurement process of the optical system [17].

The optical system employs a field stop at the primary image plane, which effectively intercepts a substantial amount of background radiation generated by components such as the light baffle, primary–secondary mirror chamber, main mirror, and secondary mirror located in front of the field stop. Furthermore, the third mirror chamber and the detector front baffle tube, serving as the system’s sensitive areas, undergo polished surface treatments to decrease their emissivity, thereby minimizing the emission of radiation energy.

Below, through a simulation analysis of the instrument’s self-generated background radiation, temperature control requirements for each component are outlined to ensure that variations in background radiation during the measurement process have no impact on measurement accuracy. The Moon-based Earth-radiation measurement optical system, located at the lunar south pole, maintains its operational temperature within a suitable range through active thermal control mechanisms, with the working temperature set near an ambient temperature of around 300 K. Therefore, the initial temperature of each component is set to 300 K for the background radiation simulation analysis. The temperature variations for each element $\Delta T$ are then determined, and based on these temperature changes and the resultant alteration in the power received by the detector ${\Delta \mathrm{\Phi }}_T$, the influence of background radiation fluctuations on measurement accuracy is assessed. This is shown in

Table 12. Simulation analysis of instrument-generated background radiation.

Object Number	Component Name	Temperature Variation ΔT (K)	Impact of Background Radiation (%)
1	Light Baffle	10	0
2	Primary–Secondary Mirror Chamber	0.1	0.047
3	Primary Mirror	0.1	0.046
4	Secondary Mirror	0.1	0.049
5	Field Stop	0.5	0.055
6	Third Mirror	0.05	0.043
7	Third Mirror Chamber	0.02	0.033
8	Detector Front Baffle	0.01	0.064
9	Detector	0.01	0.028

.

By positioning the field stop at the primary image plane of the optical system and applying specialized surface treatments to sensitive areas, along with appropriate temperature control measures, a substantial suppression of the system’s own radiation has been achieved, thereby ensuring high measurement accuracy throughout the optical system.

6. Conclusions

This paper proposes using the Moon as an observation platform to view the Earth as a “quasi-point” radiation source, thereby achieving a spatially consistent and continuous angular measurement of the outward radiation energy emitted by the entire hemispherical Earth. Leveraging the unique vantage point of the Moon, detailed analyses were conducted of Earth’s radiation energy and of geometric characteristics of the observation point relative to the Earth, as well as the temporal sampling attributes of the optical system. Based on these analyses, parameters for the Moon-based Earth-radiation measurement optical system were confirmed, leading to an in-depth system design and simulation analysis. The optical system adopted an off-axis three-mirror anastigmat configuration with a secondary image plane, complemented by appropriate surface treatments for all components, effectively suppressing extraneous solar and lunar surface-based stray light, as well as the instrument’s own background radiation. The confirmed operational conditions ensure prolonged Earth-observation periods, ensuring prolonged Earth-observation durations, enabling high detection sensitivity and a measurement accuracy of 1% across the entire optical system. The paper concluded with a brief overview of the entire Moon-based Earth-radiation measurement system, elucidating its operational principles and composition; it is designed to provide comprehensive, stable, and long-term observational data on Earth’s radiation budget and climate change. This research furnishes crucial foundations for the Moon-based Earth-radiation measurement system and the determination of observation parameters.