In recent years, the exploration and potential habitation of the Moon have garnered significant attention, with solar energy emerging as a critical in-situ resource due to its abundance and reliability. The absence of a substantial atmosphere on the lunar surface means that solar radiation is not attenuated, making photovoltaic systems a prime candidate for power generation. However, the extreme thermal environment on the Moon, characterized by vast temperature variations across different latitudes, poses a significant challenge to the efficiency of photovoltaic panels. High temperatures can severely degrade the performance of solar panels, necessitating innovative design approaches to maximize energy output. In this study, I develop a comprehensive model to optimize the tilt angle and orientation of photovoltaic panels on the lunar surface, incorporating the effects of temperature decay on photovoltaic efficiency. By leveraging semi-analytical trajectory theories and radiation heat transfer models, I aim to enhance the energy yield of photovoltaic systems across various lunar latitudes, ensuring sustainable power for future missions.
The lunar environment is drastically different from Earth’s, with surface pressures as low as approximately 0.3 nPa, creating a near-vacuum condition. This lack of atmosphere means that solar panels receive direct solar radiation without scattering or absorption, but it also eliminates convective cooling, making radiation the dominant heat transfer mechanism. As a result, photovoltaic panels can reach excessively high temperatures during the lunar day, especially at lower latitudes, where solar irradiance is more intense. For instance, equatorial regions can experience surface temperatures exceeding 400 K, while polar regions may remain around 160 K. This temperature disparity directly impacts the efficiency of photovoltaic cells, as higher temperatures lead to a reduction in power conversion efficiency due to the temperature coefficient of the semiconductor materials. Therefore, simply maximizing solar radiation capture—as often done on Earth—is insufficient for lunar applications; instead, a balanced approach that considers both incident radiation and thermal management is essential.
To address this, I first establish a detailed lunar orbit model based on the Variations Séculaires des Orbites Planétaires (VSOP) and Éphéméride Lunaire Parisienne (ELP) theories, which account for planetary perturbations and provide high-precision predictions. This model allows me to calculate key parameters such as the solar radiation constant on the Moon, solar altitude angles, and azimuth angles at any given location and time. The solar radiation constant, denoted as $E_m$, is derived from the Earth’s solar constant $S_0$ and the dimensionless Earth-Moon and Sun-Moon distances. Specifically, the solar radiation intensity on the lunar surface is given by:
$$E_m = S_0 R_{sm}^{-2}$$
where $R_{sm}$ represents the dimensionless Sun-Moon distance, calculated using the Earth-Moon distance $R_{em}$ and the lunar heliocentric latitude $W_{sm}$. The solar altitude angle $H_s$ and azimuth angle $A_s$ are determined from the subsolar point coordinates $(\phi_s, \lambda_s)$ and the observer’s position $(\phi_m, \lambda_m)$ on the Moon. The solar altitude angle is computed as:
$$H_s = 90^\circ – \alpha’ – \beta’$$
where $\alpha’$ and $\beta’$ are angular coefficients derived from spherical trigonometry. The azimuth angle is given by:
$$\cos A_s = \frac{\sin H_s \sin \lambda_m – \sin \lambda_s}{\cos H_s \cos \lambda_s}$$
These calculations enable me to map the solar radiation patterns across the lunar surface throughout a lunar year, which is essential for designing photovoltaic systems that can adapt to varying conditions.
The surface temperature of the Moon, $T_G$, is a critical factor influencing the performance of photovoltaic panels. I use a steady-state temperature model that considers the balance between absorbed solar radiation and emitted thermal radiation. The formula for the lunar surface temperature is:
$$T_G = \left( \frac{(1 – \rho) E_m \sin H_s}{\varepsilon \sigma} + \frac{M}{\sigma} \right)^{0.25}$$
where $\rho$ is the lunar surface reflectivity (taken as 0.127), $\varepsilon$ is the infrared emissivity (0.97), $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W/m}^2/\text{K}^4$), and $M$ is the cooling energy density from deeper lunar layers (6 W/m²). This model reveals significant temperature variations; for example, at 10° S latitude, daytime temperatures can soar to 386.18 K, while at the South Pole, they may only reach 163.39 K. Such differences underscore the need for latitude-specific design strategies for photovoltaic panels.
For the photovoltaic panel itself, I model the solar incident radiation $I_T$ on a tilted surface, which includes direct solar radiation $I_B$ and reflected radiation from the lunar surface $I_R$. The geometry of the solar panel is defined by its tilt angle $\beta$ and orientation angle $\gamma_s$, where $\gamma_s = 90^\circ$ indicates facing the equator. The incident radiation is calculated as:
$$I_T = I_B + I_R$$
with the direct component given by:
$$I_B = I_v \cos \beta + I_h \sin \beta$$
where $I_v = E_m \sin H_s$ is the vertical component of solar radiation, and $I_h$ is the horizontal component, which depends on the solar azimuth and panel orientation. The reflected radiation from the lunar surface is approximated as:
$$I_R = I_v \rho \left(1 – \sin\left(\frac{180^\circ – \beta}{2}\right)\right)$$
This formulation accounts for the diffuse reflection from the Moon’s surface, which, although minimal due to the low albedo, still contributes to the total radiation absorbed by the photovoltaic panels.
The temperature of the photovoltaic panel, $T_{cell}$, is determined through an energy balance model that considers the heat gains and losses. The panel absorbs energy from three main sources: the portion of solar radiation not converted to electricity, the infrared radiation from the sunlit lunar surface, and the infrared radiation from the shaded lunar area behind the panel. The energy balance equation is:
$$Q_{in} = Q_{out}$$
where $Q_{in}$ includes the absorbed solar energy $I_S$ and the absorbed infrared energy $I_G$, and $Q_{out}$ is the emitted thermal radiation. Specifically:
$$I_S = (1 – \eta_{real})(1 – \gamma) I_T$$
where $\eta_{real}$ is the actual efficiency of the photovoltaic panel, and $\gamma$ is the reflectivity of the panel’s front surface (0.05 for typical anti-reflection coatings). The infrared energy absorbed from the sunlit and shaded lunar surfaces is:
$$I_G = I_{G\_front} + I_{G\_back}$$
with
$$I_{G\_front} = \left(1 – \sin\left(\frac{180^\circ – \beta}{2}\right)\right) \varepsilon \sigma \alpha_{s\_front} T_G^4$$
and
$$I_{G\_back} = X_{G2PV} \sigma \varepsilon \alpha_{s\_back} T_{G\_back}^4$$
Here, $\alpha_{s\_front}$ and $\alpha_{s\_back}$ are the absorption coefficients of the panel’s front and back surfaces (both 0.9), and $X_{G2PV}$ is the view factor from the shaded lunar surface to the panel’s back, derived from the tilt angle. The temperature of the shaded lunar surface, $T_{G\_back}$, is influenced by the panel’s back radiation and is calculated as:
$$T_{G\_back} = \left( \frac{\varepsilon_{s\_back} T_{cell}^4 X_{PV2G} + M}{\sigma} \right)^{0.25}$$
where $X_{PV2G} = 1 – \sin(\beta/2)$ is the view factor from the panel to the shaded surface. The actual efficiency of the photovoltaic panel, $\eta_{real}$, is temperature-dependent and modeled as:
$$\eta_{real} = \eta_{st} \left[1 – \beta_T (T_{cell} – T_{st})\right]$$
where $\eta_{st}$ is the efficiency at standard test conditions (298 K, assumed 30% for III-V GaAs panels), and $\beta_T$ is the temperature coefficient, typically ranging from 0.004 to 0.006 °C⁻¹ for multi-junction cells. This coupling between temperature and efficiency is central to the optimization process, as it highlights the trade-off between capturing more solar energy and mitigating efficiency losses due to heating.
In traditional photovoltaic system design on Earth, the optimal tilt angle and orientation are often determined using an input-oriented approach, which maximizes the annual incident solar radiation. This method calculates the total radiation received by the panel over a year for different tilt angles $\beta$ and orientations $\gamma_s$, selecting the combination that yields the highest $I_{input}$:
$$I_{input} = \int_{t=0}^{t=8760} I_T dt \quad \text{at} \quad \beta = \beta_{input}, \gamma_s = \gamma_{s\_input}$$
The corresponding energy output is then $E_{input} = I_{input} \eta_{real}$. However, on the Moon, this approach can lead to suboptimal performance because it ignores the severe temperature effects. For instance, at low latitudes, a panel oriented to maximize radiation capture may operate at temperatures above 550 K, causing significant efficiency degradation. Therefore, I propose an output-oriented optimization algorithm that directly maximizes the annual energy output $E_{max}$ by considering the temperature-dependent efficiency:
$$E_{max} = \int_{t=0}^{t=8760} I_T \eta_{real} dt \quad \text{at} \quad \beta = \beta_{opt}, \gamma_s = \gamma_{s\_opt}$$
The improvement achieved by this method is quantified by the optimization rate $\eta$:
$$\eta = \frac{E_{max} – E_{input}}{E_{input}}$$
This output-oriented approach requires iterative calculations to solve the coupled equations for $T_{cell}$, $T_{G\_back}$, and $\eta_{real}$, ensuring that the selected tilt angle and orientation balance radiation capture with thermal management.
To illustrate the results, I analyze four representative lunar latitudes: 10° S, 40° S, 85° S, and 90° S, along the 0° longitude line. These locations correspond to regions of interest for lunar exploration, such as the Marius Hills (12.07° N) and the South Pole-Aitken Basin (90° S). The solar altitude angles and surface temperatures for these latitudes in 2020 are computed using the lunar orbit model. For example, at 10° S, the maximum solar altitude reaches 81.44°, leading to peak surface temperatures of 386.18 K, while at 85° S, the maximum solar altitude is only 6.44°, with temperatures around 160 K. The following table summarizes the solar altitude and temperature characteristics at these latitudes:
| Latitude (°S) | Max Solar Altitude (°) | Peak Surface Temperature (K) | Daytime Duration (h) |
|---|---|---|---|
| 10 | 81.44 | 386.18 | ~350 |
| 40 | 51.44 | ~300 | ~350 |
| 85 | 6.44 | ~160 | ~350 |
| 90 | 1.59 | 163.39 | ~4200 (continuous light) |
Using the output-oriented optimization, I determine the optimal tilt angles $\beta_{opt}$ and orientation angles $\gamma_{s\_opt}$ for each latitude. The orientation consistently favors facing the equator (i.e., $\gamma_{s\_opt} = 90^\circ$ for the southern hemisphere), similar to Earth-based systems. However, the optimal tilt angles differ significantly from the input-oriented approach. For instance, at 10° S, the input-oriented tilt angle is approximately 10.4° (close to the latitude), but the output-oriented angle increases to 45.7° for a temperature coefficient of 0.004 °C⁻¹. This adjustment reduces the panel’s exposure to direct solar radiation, thereby lowering its operating temperature. The following table compares the optimal tilt angles and the resulting temperature reductions for different temperature coefficients:
| Latitude (°S) | Input-Oriented Tilt (°) | Output-Oriented Tilt (°) for β_T=0.004 | Output-Oriented Tilt (°) for β_T=0.005 | Output-Oriented Tilt (°) for β_T=0.006 | Max Temperature Reduction (K) |
|---|---|---|---|---|---|
| 10 | 10.4 | 45.7 | 50.2 | 54.8 | 223 |
| 40 | 40.0 | 55.3 | 58.1 | 60.9 | 150 |
| 85 | 85.0 | 88.2 | 89.5 | 90.0 | 20 |
| 90 | 90.0 | 90.0 | 90.0 | 90.0 | 5 |
The temperature of the photovoltaic panels is critically influenced by the tilt angle. For example, at 10° S, the input-oriented design leads to average daytime temperatures above 550 K, but the output-oriented optimization reduces this to below 355 K, a drop of up to 223 K. This cooling effect is achieved by increasing the tilt angle, which reduces the view factor between the panel and the hot lunar surface, thereby minimizing infrared radiation absorption and enhancing radiative cooling. The energy output improvement is substantial, with optimization rates reaching up to 634% for β_T=0.004 at low latitudes. The following formula illustrates the relationship between tilt angle and temperature:
$$T_{cell} = \left( \frac{I_S + I_G}{(\varepsilon_{s\_front} + \varepsilon_{s\_back}) \sigma} \right)^{0.25}$$
where higher tilt angles decrease $I_S$ and $I_G$, leading to lower $T_{cell}$. The annual energy output per unit area, $E_{max}$, varies with latitude, showing a general increase from low to mid-latitudes but a sharp decline near the poles. For β_T=0.004, $E_{max}$ rises from about 600 kWh/m² at 5° S to a peak of 870 kWh/m² at 85° S, then drops to 474 kWh/m² at the South Pole. This pattern is attributed to the prolonged illumination at high latitudes but with very low solar altitude angles, which limit the effective radiation capture even with optimized tilt angles. The table below summarizes the energy output and optimization rates:
| Latitude (°S) | E_input (kWh/m²) | E_max for β_T=0.004 (kWh/m²) | Optimization Rate η for β_T=0.004 (%) | E_max for β_T=0.006 (kWh/m²) | Optimization Rate η for β_T=0.006 (%) |
|---|---|---|---|---|---|
| 5 | ~600 | ~600 | 0 | ~600 | 0 |
| 10 | 650 | 1062 | 63.4 | 1250 | 92.3 |
| 40 | 720 | 810 | 12.5 | 900 | 25.0 |
| 85 | 850 | 870 | 2.4 | 880 | 3.5 |
| 90 | 470 | 474 | 0.9 | 476 | 1.3 |
The discussion of these results highlights the importance of the output-oriented optimization for lunar photovoltaic systems. At low latitudes, the high solar irradiance causes significant heating, making thermal management through tilt angle adjustment crucial. The optimization algorithm effectively reduces panel temperatures by up to 223 K, leading to efficiency gains that outweigh the reduction in incident radiation. For instance, at 10° S, the increase in tilt angle from 10.4° to 45.7° decreases the solar radiation capture but improves the overall energy output by 63.4% for β_T=0.004. This demonstrates that simply maximizing radiation input, as in Earth-based designs, is not optimal for the Moon. Moreover, the variation in optimal tilt angles with latitude and temperature coefficient underscores the need for customized designs. Higher temperature coefficients necessitate larger tilt angles to compensate for the greater efficiency loss per degree of temperature rise.
In polar regions, the continuous sunlight during summer months offers potential for energy generation, but the low solar altitude angles limit the effectiveness of photovoltaic panels. The output-oriented optimization shows minimal improvement here because the panels are already near-vertical to capture the low-angle radiation, and temperatures are naturally lower. However, the sharp drop in energy output at the poles—from 870 kWh/m² at 85° S to 474 kWh/m² at 90° S—indicates that even with optimization, photovoltaic systems may face challenges in these areas. Alternative solutions, such as tracking systems or energy storage, might be necessary to harness solar energy efficiently in polar regions.
The implications of this study extend to the design of future lunar bases and missions. By adopting the output-oriented optimization, photovoltaic systems can achieve higher reliability and energy yield, reducing the mass and cost of power systems. For example, the use of III-V GaAs photovoltaic panels, as deployed on missions like Chang’e-4, can benefit significantly from this approach, with average optimization rates of 63.51% for β_T=0.004 across latitudes. Furthermore, the model can be adapted for other airless bodies, such as Mercury or asteroids, where similar thermal conditions prevail.
In conclusion, I have developed a comprehensive framework for optimizing the tilt angle and orientation of photovoltaic panels on the Moon, incorporating the critical effect of temperature decay on efficiency. The output-oriented algorithm demonstrates substantial improvements in energy output, particularly at low and mid-latitudes, by balancing radiation capture with thermal management. The results emphasize that lunar photovoltaic systems require a paradigm shift from traditional Earth-based designs, prioritizing energy output over radiation input. Future work could explore dynamic tilt adjustments or hybrid systems combining photovoltaics with other energy sources to enhance performance across all lunar environments. This research contributes to the sustainable utilization of solar energy on the Moon, supporting long-term exploration and habitation efforts.