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Space Generative AI with Solar Energy Harvesting

2026 · arxiv_cs

arXiv CS · Papers · 2026

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Space Generative AI with Solar Energy Harvesting

arXiv:2609.01062v1 [cs.AI] 1 Sep 2026

Jierui Zhang, Jianhao Huang, Zhanwei Wang, and Kaibin Huang Abstract—Satellites are emerging as promising platforms for extending generative artificial intelligence (AI) services to remote areas lacking terrestrial infrastructure. However, deploying space generative AI is fundamentally constrained by the limited and time-varying onboard energy supplied by solar energy harvesting (EH). This paper presents a framework for solar-powered space generative AI in which a satellite receives a user prompt, executes a diffusion-based image-generation model, and downlinks the compressed result within a strict time window. To design this framework, we identify fundamental computation–communication (C2 ) trade-offs governed by the shared harvested-energy budgets. Specifically, increasing the number of generation steps improves intrinsic image quality but depletes energy and time available for downlink transmission, whereas prioritizing communication guarantees reliable delivery but sacrifices semantic quality. To balance these trade-offs and maximize end-to-end (E2E) generative performance, we exploit the predictable solar-EH dynamics induced by deterministic orbital motion and develop a joint C2 resource-optimization framework using a tractable two-step approach. First, we characterize the maximum downlink throughput for a fixed generation depth under continuous solar EH. This establishes a separation principle that decouples waiting-time selection from optimal transmit-power control. Next, building on this principle, we formulate a joint C2 utility-maximization problem and derive a closed-form, low-complexity step-selection policy in the dominant constant-power regime. Extensive experiments under realistic orbital dynamics demonstrate that the proposed policy dynamically balances generation quality and transmission reliability. This yields significant E2E performance gains over static computation- and communication-centric baselines across diverse solar EH states. Index Terms—Space computing network, solar energy harvesting, generative AI, space AI.

I. I NTRODUCTION With rapid advances in generative artificial intelligence (AI) such as high-fidelity image generation and complex reasoning, supporting globally accessible generative-AI services is becoming a critical objective for future networks [1]–[4]. However, delivering generative-AI services to remote and underserved areas remains challenging due to the lack of terrestrial AI infrastructure. Although satellites have traditionally served as passive relays connecting remote users to ground networks, this relay-based paradigm may incur significant latency. Enabled by increasing onboard computation capability, satellites are emerging as active AI-service providers for ubiquitous generative-AI service provisioning beyond the coverage of terrestrial networks [5]–[9]. Meanwhile, satellites This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. J. Zhang, J. Huang, Z. Wang, and K. Huang are with the Department of Electrical and Computer Engineering, The University of Hong Kong (HKU), Hong Kong SAR, China. Emails: jrzhang@eee.hku.hk, jianhaoh@hku.hk, zhanweiw@eee.hku.hk, and huangkb@eee.hku.hk (Corresponding author: K. Huang).

can harvest solar energy directly in orbit, avoiding atmospheric attenuation and many ground-level obstructions, thereby supporting onboard tasks through solar energy harvesting (EH). These advantages have motivated two related architectural visions: space data centers, which deploy large-scale computing infrastructure in orbit, and space-ground integrated computing power networks, which coordinate computing resources across satellites, terrestrial edge nodes, and cloud data centers [10]–[12]. A recent example is StarCloud, which launched a prototype orbital computing facility equipped with an H100 GPU for onboard model training and inference, further indicating the feasibility of executing AI workloads in orbit [13]. However, realizing practical space generative AI still requires overcoming a substantial challenge: onboard AI inference and downlink transmission typically rely on timevarying energy supplied by solar EH, while computation and communication are tightly coupled through the shared onboard energy budget. Although sun-synchronous orbits can mitigate solar-energy variation in some cases, such favorable orbits cannot accommodate all space-AI deployments due to limited orbital resources and mission-specific coverage requirements. Hence, space AI systems must also be designed for general orbital conditions, where solar EH can be time-varying. This paper investigates the joint management of computation and communication (C2 ) for space generative AI, with the goal of maximizing end-to-end (E2E) performance under solar-EH constraints. The deployment of generative AI is inherently energyintensive due to its heavy computational workload, posing significant challenges for energy-constrained onboard AI execution [14]. Unlike terrestrial AI systems that typically have access to stable power-grid infrastructure, orbital AI must rely on limited onboard energy resources. To sustain off-grid operations, EH has emerged as a key enabling technology and has attracted extensive research attention, particularly in wireless communications [15]. In satellite systems, solar EH is the primary energy source, but the harvested power is not a static budget. Instead, it varies continuously with orbital motion, solar incidence angle, panel orientation, and sunlight-eclipse transitions [16]–[18]. Although the satelliteground visibility window may be much longer than a single service request, an interactive generative-AI task typically imposes a much shorter execution window to ensure a timely response [19]. This short window limits both the time available for energy accumulation and the duration available for onboard computation and downlink transmission. As a result, the satellite cannot simply wait until sufficient solar energy is harvested, but must complete waiting, computation, and communication within a tight service deadline. During this window, the available onboard energy is strongly determined

by the satellite’s orbital position and solar-EH-induced battery state. Therefore, completing such a task requires careful allocation of limited harvested energy and time between onboard generation and downlink transmission. This allocation problem is particularly challenging for space generative AI because computation and communication jointly determine the final received content quality. Consider a textto-image service where a ground device uploads a lightweight prompt to a satellite, the satellite generates an image onboard using a diffusion model [20], [21], and the generated image is then compressed and downlinked to the ground device. In this pipeline, computation and communication draw from the same harvested-energy buffer. Increasing the number of denoising steps generally improves the intrinsic quality of the generated image, but it consumes more energy and leaves less energy for downlink transmission. Conversely, reserving more energy for transmission increases the delivery capacity and can reduce compression distortion, but it forces the generative process to use fewer denoising steps, which degrades semantic quality. As a result, a computation-centric policy synthesizes a high-quality image that cannot be faithfully delivered, while a communication-centric policy reliably delivers an image whose intrinsic quality is poor. Therefore, maximizing the E2E generation quality requires task-aware allocation of energy between onboard generation and downlink transmission under solar-EH-induced causality constraints. Existing studies provide useful foundations, but they do not fully address the above generate-and-deliver trade-off. A first line of work is EH communications, which studies transmission scheduling under energy-causality constraints. Classical results characterize optimal throughput-maximizing policies through methods such as directional water filling and save-then-transmit strategies [22]–[25]. These works establish important principles for energy-causal transmission, but they typically allocate harvested energy solely to communication, without considering computation that may compete for the same energy buffer. A second related line considers energyaware satellite systems. Several studies have investigated satellite energy management, sunlight-aware task scheduling, and battery-aware edge computing under orbital energy constraints [16]–[18]. These works highlight the importance of orbital energy dynamics, but they mainly focus on routing, task scheduling, or generic computing workloads. They do not model the task-specific quality behavior of generative AI, where the computation depth directly affects the semantic quality of the output. Finally, recent studies have explored joint computation and communication resource allocation, as well as broader sensing–communication–computation integration [26]. Some works minimize download time or information freshness by jointly optimizing onboard processing and transmission [27]–[29], while others focus on energy efficiency, computation offloading, compression, or network throughput [30]–[37]. However, most of these formulations treat computation as a fixed workload or a generic offloading/compression task. They do not capture a key property of diffusion-based generation: the number of denoising

steps is a controllable quality knob that should be jointly adapted with downlink transmission. Moreover, solar EH is rarely considered in generative-service-oriented satellite designs. Consequently, existing EH communication and satellite C2 frameworks are insufficient for optimizing E2E generative performance under solar EH. The key challenges of optimizing such a framework arise from two factors: (1) the time-varying profiles of both the wireless channel and solar EH, and (2) the coupling between onboard generation and subsequent downlink transmission under the solar-EH profile. For the first challenge, deterministic orbital motion and geometric relationships make these profiles predictable over the task execution window and provide tractable mathematical models, thereby enabling proactive resource allocation. For the second challenge, onboard generation and downlink transmission jointly determine the E2E performance, requiring the system to manage generation quality, waiting time, and communication throughput under a shared solar-EH-powered energy buffer, rather than merely scheduling transmit power as in conventional EH communication. To study this joint C2 optimization for space generative AI, we decompose the E2E performance into two coupled components: onboard generation quality and downlink throughput. We first fix the generation step and optimize the waiting time and transmit power to characterize the maximum achievable downlink throughput under continuous energy causality. Building on this characterization, we then formulate a joint C2 utility that balances onboard generation quality and downlink throughput, and derive a closed-form, low-complexity step-selection rule in the dominant constantpower regime. Specifically, the key contributions and findings are summarized as follows: Communication Throughput Maximization under Solar EH: We fix the generation step and investigate a communication-throughput maximization problem. This problem not only serves as the inner subproblem of the subsequent joint C2 optimization, but also becomes critical when the generated content has a large downlink data volume. Unlike classical EH communication, which provides general structural solutions for arbitrary EH profiles, our problem involves an orbital solar-EH profile and a waiting-time decision introduced by onboard task execution. To solve the problem, we instantiate a tractable, representative solar-EH model based on orbital geometry, prove a separation principle between waitingtime selection and transmit-power control, and derive the optimal energy-feasible transmit-power policy using the lower convex envelope (LCE) principle. 2 • Joint C Optimization under Solar EH: Beyond the fixed-step communication-throughput optimization, we further consider the general case where onboard generation quality and downlink throughput must be jointly balanced. Specifically, we approximate the relationship between onboard generation quality (measured by CLIP score [38], [39]) and generation steps, and then formulate •

a joint C2 utility maximization problem over the generation step, waiting time, and transmit-power policy. This problem is challenging because these control variables are tightly coupled through the shared solar-EH-powered energy buffer. By decomposing the problem with respect to the generation step, the inner subproblem is connected to the throughput-maximization problem derived above. In the dominant constant-power regime, this decomposition yields a reduced single-variable problem, for which we derive a closed-form, low-complexity step-selection rule using the Lambert W function. The resulting rule enables lightweight onboard adaptation and reveals how the optimal generation step changes with the solar-EH state and system parameters. • Experimental Results: We conduct comprehensive experiments to evaluate the proposed space generative AI framework under realistic low-Earth-orbit (LEO) parameters. The results show that the proposed joint optimization scheme adapts between computation-centric and communication-centric schemes according to the solarEH state. It approaches the more suitable extreme policy in energy-scarce or energy-abundant regimes, while outperforming both extremes in moderate-energy regimes through balanced onboard generation and downlink transmission. Overall, it achieves more robust E2E generation performance than fixed extreme allocation schemes. The remainder of this paper is organized as follows. Section II introduces the models and metrics. Section III solves the communication throughput maximization problem. Section IV investigates a joint C2 optimization problem and derives the closed-form solution. Experimental results are provided in Section V, followed by concluding remarks in Section VI. II. S YSTEM M ODEL Consider a space generative AI framework powered by solar EH, as shown in Fig. 1. In this framework, a static ground device offloads prompt-based generation tasks to a satellite. Then, the solar-EH-powered satellite completes a sequence of onboard operations within a short task execution window of duration T , including waiting, content generation, and result transmission. The solar EH model, onboard computation model, space-ground communication model, and performance metrics are detailed in the following subsections. A. Solar EH Model In this subsection, we model satellite solar EH based on orbital geometry. Two Cartesian coordinate systems, the orbital reference frame (Fo ) and the satellite body frame (Fb ) [40], are as shown in Fig. 1(a). • Orbital Reference Frame (Fo ): The origin is located at the center of the Earth. The Yo -axis is orthogonal to the orbital plane. The Xo -axis points towards the projection of the sun vector onto the orbital plane, and the Zo -axis completes the right-handed system. The sun vector is So = [cos β, sin β, 0]T ,

(1)

where β denotes the solar elevation angle. Satellite Body Frame (Fb ): The origin is the satellite’s center of mass. We adopt a standard nadir-pointing attitude [41], where zb points towards the Earth (nadir), yb is aligned with the negative orbit normal (cross-track) and xb is aligned with the velocity vector (along-track). The angular velocity ω of theq satellite is determined by GM Kepler’s third law, i.e., ω = (RE +H)3 , where G is the gravitational constant, M is the mass of the Earth, RE is the Earth radius, and H is the orbital altitude [42]. Consider a solar panel rigidly mounted on the satellite body, with unit normal vector n = [nx , ny , nz ]T ∈ R3 expressed in Fb . Assume that t = 0 corresponds to the orbital noon and that n = [0, 0, −1]T (i.e., the anti-nadir direction); then the solar-EH profile (the instantaneous harvested power) is1  π  , P0 cos(ωt), 0 ≤ t ≤   2ω   π 3π ≤t≤ , (2) P (t) = 0, 2ω 2ω     P0 cos(ωt), 3π ≤ t ≤ 2π . 2ω ω Here, P0 = η · γ · A · cos β, where η is the power conversion efficiency, γ is the solar irradiance per unit area, and A is the area of the solar panel [16]. Integrating (2) from the orbitalnoon reference yields the following representative cumulativeenergy profile: P π 0  sin(ωt) + E0 , 0≤t≤ ,    2ω ω P0 π 3π E(t) = (3) + E0 , ≤t≤ ,  ω 2ω 2ω     P0 2 + sin(ωt) + E , 3π ≤ t ≤ 2π , 0 ω 2ω ω where E0 ≥ 0 denotes the initial energy at task arrival. •

B. Onboard Generative AI Model Although generative AI supports various modalities, textto-image generation naturally fits the considered scenario: a power-limited ground user uploads only a lightweight prompt, while the solar-EH-powered satellite generates and downlinks the larger image result. Hence, we consider a typical text-toimage generation task, as described below. 1) Generative AI Model: We adopt a representative latent diffusion model (LDM) [20] as the generative backbone for text-to-image generation. The inference pipeline consists of three distinct phases as follows. First, a pre-trained text encoder τθ processes the input prompt to produce conditional embeddings c. Second, initialized with random Gaussian noise zn in the latent space, the diffusion backbone iteratively refines the latent representation over n steps using the denoising diffusion implicit model (DDIM) scheduler conditioned on c [21]. Finally, the VAE decoder D projects the clean latent z0 back into the pixel space to generate the final image x. 1 The orbital period of LEO satellite (tens of minutes) is negligible relative to that of Earth’s revolution around the Sun. Hence, we assume that the sunlight direction and the orbital plane remain constant within a single orbital period.

Ground device

Satellite

1 Prompt uplink

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4 Image downlink

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(a) Space generative AI framework.

(b) Operations and protocol.

Fig. 1: System model of space generative AI.

2) Computation Energy Consumption: We assume that, for each task, the computation power Pcomp is fixed. Let Ω1 and Ω2 denote the number of floating point operations (FLOPs) of each DDIM step and the FLOPs of remaining operations for a single inference (e.g., decoder of the VAE). Let ν denote the computation speed. Then, the computation time is t2 = c1 n + c2 ,

(4)

where c1 = Ων1 and c2 = Ων2 . Since the satellite may not possess sufficient energy to execute the task immediately upon reception, a waiting phase t1 may be required. Hence, the cumulative computation energy consumption is given by  treq ≤ t < treq +t1 , 0, Ecomp (t) = Pcomp (t − treq − t1 ), treq +t1 ≤ t < a, (5)  Pcomp t2 , a ≤ t ≤ b, where a ≜ treq + t1 + t2 and b ≜ treq + T . C. Space-ground Communication Model We focus on downlink transmission because the promptuplink traffic is negligible. Consider a static ground device located on the Earth’s surface. We define the delivery capacity as the achievable number of bits delivered over the communication window [a, b]. In this work, the satellite can exploit non-causal information because both the orbital trajectory and the space-ground channel evolution are predictable. Hence, the delivery capacity and the corresponding transmit power policy can be determined in advance (detailed in Section III). Then, the generated image x is compressed into x′ such that its data size does not exceed the delivery capacity. The compressed image is then transmitted according to the optimized power policy. The instantaneous channel gain g(t) is g(t) = Gtx Grx LFSPL LAL ,

(6)

where Gtx and Grx denote the transmit and receive 2 antenna ℓ gains, respectively [6], [43]. LFSPL = 4πd(t) represents the large-scale free-space path-loss factor, where ℓ = fcc is the downlink carrier wavelength, c is the speed of light, fc is the carrier frequency, and d(t) is the time-varying distance between the satellite and the ground device. Note that d(t) is

predictable based on the satellite’s orbital dynamics, thereby leading to a predictable channel. LAL accounts for additional environmental attenuation, such as rain fading. Small-scale fading is omitted, due to the nature of the considered remote area [6]. According to Shannon’s theorem [44], the instantaneous communication rate is given by Pcomm (t)g(t) , (7) r(t) = B log2 1 + N0 B where Pcomm (t) is the transmit power, N0 is the noise power spectral density, and B is the bandwidth. Then, the cumulative communication energy consumption is given by 0, treq ≤ t < a, Ecomm (t) = R t (8) P (τ ) dτ, a ≤ t ≤ b. comm a D. Performance Metrics

We aim to maximize the E2E generation quality, measured by the E2E CLIP score, i.e., the semantic similarity between the text prompt and the image received at the ground device [38], [39]. However, directly characterizing this E2E metric in closed form is challenging, as it depends jointly on the generative model, compression process, and downlink transmission. Therefore, we instead consider two tractable factors, corresponding to computation and communication, respectively, that jointly determine the E2E metric: • Onboard CLIP score: This metric measures the semantic similarity between the text prompt and the onboardgenerated image, reflecting its intrinsic quality. Empirical results show that the CLIP score S increases with the number of DDIM steps n. • Communication throughput: This metric is defined as the total number of bits delivered over the communication window [22], given by Z b R= r(t) dt. (9) a

A higher communication throughput allows a lower compression ratio and thus better preserves the quality of the generated image. It is related to waiting time, computation time, and transmit power policy.

Remark 1. Under solar-EH constraints, these two factors exhibit an inherent C2 trade-off: allocating more energy to onboard generation improves intrinsic image quality, but reduces the energy available for downlink transmission, and vice versa. This motivates the analytical design in the following two sections: Section III studies communication-throughput maximization when fixing computation, while Section IV jointly optimizes C2 for E2E performance maximization. III. C OMMUNICATION T HROUGHPUT M AXIMIZATION UNDER S OLAR EH In this section, we maximize communication throughput under solar EH, assuming a fixed computation time. This formulation constitutes a critical subproblem of the joint optimization framework presented in Section IV. A. Problem Formulation The communication throughput maximization problem is formulated in (10), where we fix n and thus t2 is also fixed. Z b (P1)

max

t1 , Pcomm (·)

r(τ ) dτ

(10a)

s.t. Pcomp (t−treq −t1 ) ≤ E(t), ∀t ∈ [treq +t1 ,a], (10b) Z t Pcomm(τ )dτ ≤ E(t)−Pcomp t2 , ∀t ∈ [a,b], a

(10c)

where r(t) is given in (7). Since the waiting time t1 is continuous and the transmit power Pcomm (·) is a continuoustime function, directly solving problem (P1) is highly complex. A possible approach is to discretize the feasible range of t1 and, for each discretized value, apply the continuousEH transmission result in [45]. However, this discretization generally yields only an approximate solution and incurs high computational complexity when a fine time resolution is required. Therefore, we first investigate the structural properties of the optimal policy in the following subsection. B. Properties of Optimal Solution To derive the exact solution of (P1), we first analyze the properties of the optimal solution. The property of the optimal waiting time is characterized in the following proposition. Proposition 1 (Optimal Waiting Time). To achieve the optimal solution, the waiting time t1 should be minimized, provided that the energy causality constraint for computation (10b) is satisfied. Specifically, the optimal value t∗1 is given by n t∗1 = inf t1 ∈ [0, T − t2 ] o (11) P comp (t − treq − t1 ) ≤ E(t), ∀ t ∈ [treq + t1 , a] . Proof. Let t∗1 denote the minimal feasible waiting time defined in (11). We show that any t̂1 > t∗1 cannot yield a higher throughput than t∗1 . Consider a feasible policy (t̂1 , P̂comm (·)) achieving a throughput R̂. Since t∗1 < t̂1 , the computation phase completes earlier, expanding the feasible time window

for communication. Let a∗ = treq + t∗1 + t2 and â = treq + t̂1 + t2 . By extending P̂comm (t) as zero over [a∗ , â), the resulting policy remains feasible for (P1) and achieves the same throughput R̂. Since reducing t1 does not shrink the feasible set of communication policies, t∗1 is optimal. Remark 2 (Separation Principle). Proposition 1 reveals that the optimization of t1 and Pcomm (·) can be decoupled without compromising optimality. This separation significantly reduces the problem complexity, as it allows the optimal waiting time to be determined prior to solving for the power control. Having determined the optimal waiting time t∗1 , we proceed to optimize the communication policy. Let [a, b] denote the communication interval corresponding to t∗1 . We define the feasible set for the transmit power Pcomm (·) as n B ≜ Pcomm : [a, b] → [0, ∞) Z t o (12) Pcomm (τ ) dτ ≤ Ē(t), ∀ t ∈ [a, b] , a

where Ē(t) ≜ E(t)−Pcomp t2 represents the residual harvested energy available for communication. Since the task execution window T is relatively small, the channel gain over the window can be assumed to be unchanged, i.e., g(t) ≡ g. For simplicity, we let α ≜ Ng0 B , ϕ(P ) ≜ B log2 (1 + αP ). Note that a is a linear function of t1 . Consequently, (P1) reduces to the following problem: Z b (P2) max ϕ Pcomm (τ ) dτ. (13) Pcomm (·)∈B

Since Ecomm (t) and Pcomm (t) have a one-to-one mapping, we use them interchangeably in the analysis. The property of optimal transmit power is given by the following theorem. Theorem 1 (Optimal Policy via Lower Convex Envelope [45]). ∗ (t) for t ∈ The optimal cumulative energy consumption Ecomm [a, b] is given by the lower convex envelope (LCE) of Ē(t), anchored at the initial point (a, 0) and the final point (b, Ē(b)). C. Optimal Waiting Time and Transmit Power Based on the preceding results, this subsection introduces the optimal solutions for (P1) under the representative solarEH profile in (3). According to Proposition 1 and Theorem 1, a minimal possible waiting time and an LCE of the solar-EH profile are required. Leveraging the closed-form expression of the EH curve, we can determine a closed-form solution efficiently. For analytical clarity, we assume that Pcomp t2 < Pω0 , π , and E0 = 0. Other cases are either infeasible or t2 < T < 2ω can be handled using the same procedures. To provide a unified expression for transmit power when 0 ≤ treq ≤ 2π ω − T , we first investigate two specific cases for treq as follows. 1) Case 1: Task Starts from Orbital Noon: We consider the scenario when treq = 0 and derive the exact solution to (P1). π We start with finding t∗1 . Since E(t) is concave over [0, 2ω ], tangency between E(t) and the computation-related segment is impossible. Therefore, the minimal feasible t1 occurs in either case as follows: (1) Immediate start. It is feasible if and only

E(t) Ec (t)

Energy (J)

1000

Ecomp (t) Ecomm (t)

Wait Comp.

at (a, 0) as

Comm.

s ≜

750 500 250 0

Time (minutes)

(a) treq = 0. 2000

Energy (J)

1600

E(t) Ec (t)

Ecomp (t) Ecomm (t)

Tangency Wait

Comp. Comm.

1200 800 400 0 65

Time (minutes)

3π . (b) treq = 2ω

Fig. 2: Energy evolution and time allocation under the optimal policy for two representative task-request times. Here, Ec (t) = Ecomp (t) + Ecomm (t) denotes the total cumulative energy consumption.

if Pω0 sin(ωt2 ) ≥ Pcomp t2 , and renders t∗1 = 0. (2) Terminal touch at t1 + t2 . It is feasible if Pω0 sin(ωt2 ) < Pcomp t2 . In this case, a positive wait is necessary, and t∗1 is characterized ∗ by tightness at the end of the computation ωPcompwindow: E(t1 + ∗ t2 ) = Pcomp t2 ⇐⇒ sin ω(t1 +t2 ) = P0 t2 , which yields ωP 1 comp t∗1 = arcsin t2 − t 2 , (14) ω P0 π well-defined when Pcomp t2 < Pω0 and t∗1 ∈ 0, 2ω − t2 . In summary, the optimal t∗1 is given by ( P0 0, ∗ ω sin(ωt2 ) ≥ Pcomp t2 , (15) t1 = 1 ωPcomp P0 ω arcsin( P0 t2 )−t2 , ω sin(ωt2 ) < Pcomp t2 . ∗ For Pcomm (t), since Ē(t) is concave on [a, b], its LCE is the affine function interpolating (a, 0) and b, Ē(b) . Hence,

t−a t−a ∗ Ē(b) = Ecomm (t) =

E(b) − Pcomp t2 . (16)

b−a b−a Taking the derivative yields the constant optimal power: ∗ Pcomm (t) ≡

P0 sin(ωb)−Pcomp t2 E(b)−Pcomp t2 = ω . (17) b−a b−a

For some parameters, the optimal policy is shown in Fig. 2(a).2 2) Case 2: Task Starts from Orbital Dawn: We consider the 3π scenario when treq = 2ω . Under the assumption Pcomp t2 < P0 , the energy-causality constraint is satisfied even with t1 = ω ∗ (t), we first investigate a tangent 0. Hence t∗1 = 0. For Pcomm construction. Define the slope of the supporting line anchored

2 Since the actual time window is too narrow to clearly observe power variations, we employ an extended time window to better visualize the solution. This scaling is intended purely for qualitative illustration rather than rigorous quantitative analysis. This also applies to Fig. 2(b).

inf

t∈(a, b]

Ē(t) − 0 E(t) − Pcomp t2 = inf . (18) t−a t−a t∈(a, b]

Since Ē is convex on (a, b], the infimum is achieved at a unique point τ ∈ (a, b]. If no interior tangency exists, the minimizer occurs at the boundary, and we set τ ≜ b. When an interior tangency exists, τ satisfies the equal-slope condition Ē(τ )−0 = Ē ′ (τ ). With explicit form of Ē(t), it becomes τ −a P0 ω 2 + sin(ωτ ) − Pcomp t2 = P0 cos(ωτ ). (19) τ −a After transforming it to P0 2+sin(ωτ ) −ωPcomp t2 −P0 ω(τ − a) cos(ωτ ) = 0, it follows that the left-hand side is monotone; the equation can therefore be solved efficiently by bisection. ∗ ∗ Therefore, Ecomm (t) and the corresponding Pcomm (t) are   t − a Ē(τ ), t ∈ [a, τ ], ∗ (20) Ecomm (t) = τ −a Ē(t), t ∈ [τ, b],  E(τ ) − Pcomp t2  Ē(τ ) = , t ∈ [a, τ ], ∗ (21) Pcomm (t) = τ − a τ −a Ē ′ (t) = P cos(ωt), t ∈ (τ, b]. 0

∗ Note that when t ∈ (τ, b], Pcomm (t) follows a cosine-power ∗ (t) control. If no interior tangency exists (i.e., τ = b), Pcomm Ē(b) ∗ reduces to a constant-power policy Pcomm (t) ≡ b−a . For some parameters, the optimal policy is shown in Fig. 2(b). 3) Optimal Solutions: Based on the previous discussions of the two special cases, we have identified the basic structure of the solutions, which naturally extends to the general case. We first examine t∗1 w.r.t. different treq . From (14), define ωP 1 comp τ0 ≜ arcsin t2 − t 2 . (22) ω P0

The causality constraint requires treq + t1 ≥ τ0 . If τ0 > 0, it implies t1 ≥ τ0 −treq . Otherwise, if τ0 ≤ 0, then treq +t1 ≥ τ0 always holds. Therefore, we have t∗1 = max{0, τ0 − treq }.

(23)

3π ∗ − T , E(t) is Next, we examine Pcomm (t). When treq ≤ 2ω ∗ concave over [a, b], constant power applies: Pcomm (t) ≡ Ē(b) b−a . 3π When 2ω − T ≤ treq ≤ 2π − T , E(t) is convex over [a, b], ω

and the tangent construction (case 2) applies, yielding (24). D. Discussion

1) The Meaning of τ0 : The quantity τ0 is a virtual threshold time defined by the endpoint-tightness condition: E(τ0 +t2 ) = Pcomp t2 . It indicates the earliest computation-start threshold implied by the harvested-energy curve. If τ0 > 0, starting before τ0 violates the computation energy-causality constraint, and the satellite must wait until τ0 . If τ0 ≤ 0, this threshold lies before the considered task horizon, so immediate computation is already feasible. Thus, (23) holds. 2) Constant-Power Control Versus Cosine-Power Control: The optimal power structure is determined by the convexity

 E(b) − Pcomp t2   , t ∈ [a, b],   b−a ∗  E(τ ) − Pcomp t2 Pcomm (t) = , t ∈ [a, τ ],    τ −a  P (t), t ∈ (τ, b),

CLIP Score

0.30 0.28 0.26

0.22

Data Fitted curve 0

DDIM Steps n

S(n) ≈ c3 n + c4 ,

Fig. 3: CLIP score versus DDIM steps. 3π of E(t) over [a, b]. If E(t) is concave, i.e., treq ≤ 2ω − T , the LCE is the chord from (a, 0) to (b, Ē(b)), yielding constant3π −T ≤ treq ≤ 2π power control. If E(t) is convex, i.e., 2ω ω −T , the policy is constant on [a, τ ] and then follows cosine∗ (t) = P (t) = P0 cos(ωt) on power control, i.e., Pcomm (τ, b). This cosine-power control is a distinctive feature of the considered solar-EH-driven space AI system, where the optimal transmit power can track the orbital solar-EH profile. Nevertheless, constant-power control applies in most cases, covering approximately three quarters of the task-start range. 3) Extension to a General Solar-EH Profile: Although the previous subsection evaluates a representative profile for demonstration, the properties established in Sec. III-B hold for any solar-EH profile. For example, for an arbitrary solar panel normal vector n = [nx , ny , nz ]T , the solar-EH profile is

P (t) =η · γ · A · max(−nx cos β sin(ωt)

− ny sin β − nz cos β cos(ωt), 0).

(25)

See Appendix A for the derivation. For a sanity check, substituting n = [0, 0, −1]T into (25) recovers (2). The corresponding E(t) can be obtained by integrating P (t). The solution derivations under such profiles are omitted for brevity. IV. J OINT C2 U TILITY O PTIMIZATION UNDER S OLAR EH In this section, we aim to optimize the energy and time resources between computation and communication (C2 ) under solar EH. A. Problem Formulation 1) Metric Design: We consider two critical yet inherently conflicting objectives for space generative AI (see Section II-D): (onboard) CLIP score and communication throughput. To evaluate the overall performance, we define the joint C2 utility as a composite metric of these two terms: U = S + λR, where λ is a balancing coefficient.

3π − T ≤ treq ≤ 2π if 2ω ω − T.

(24)

2) Approximation of CLIP Score: As shown in Fig. 3, the CLIP score increases rapidly at early DDIM steps, while additional steps provide only marginal gains. Motivated by this saturation behavior, we restrict the DDIM step n to the effective set N ≜ {nmin , nmin + 1, . . . , nmax }, and approximate the onboard CLIP score within this set as

0.32

0.24

3π if treq ≤ 2ω − T,

(26)

(27)

where c3 > 0 and c4 are fitting coefficients. 3) Problem Description: We aim to maximize the joint C2 utility by optimizing the DDIM steps, waiting time, and power control under the given solar-EH profile. A critical constraint is energy causality, ensuring that total energy consumption never exceeds the cumulative harvested energy. Let a(n) ≜ treq +t1 +t2 (n) denote the start of the communication phase. Accordingly, we formulate the joint optimization over the DDIM step n ∈ N , waiting time t1 , and transmit-power policy Pcomm (·) as follows: Z b (P3) max c3 n + c 4 + λ ϕ Pcomm (t) dt (28a) n,t1 ,Pcomm (·)

a(n)

s.t. Pcomp (t − treq − t1 ) ≤ E(t), Z t

∀t ∈ [treq + t1 , a(n)],

(28b) (28c)

Pcomm(τ )dτ ≤ E(t)−Pcomp t2(n), (28d) a(n) ∀t ∈ [a(n), b].

(28e)

B. Problem Decomposition The main difficulty of (P3) lies in the coupling among computation, communication, and solar EH. Computation and communication draw from the same solar-EH-powered energy buffer, creating a time-causal resource conflict between the two stages. To this end, we decompose (P3) with respect to the DDIM step and characterize the resulting optimality structure in the following proposition. Proposition 2 (Decomposition of (P3)). For any fixed n ∈ N , let R∗ (n) denote the maximum throughput obtained by solving the communication-throughput maximization problem (P1) with t2 = t2 (n). Then, the optimal DDIM step of (P3) is given by n∗ = arg max {c3 n + c4 + λR∗ (n)} . n∈N

(29)

Proof. For each fixed n, (P3) reduces to (P1); maximizing c3 n + c4 + λR∗ (n) over n ∈ N yields the result. This decomposition reduces the original mixed discretecontinuous problem to an outer one-dimensional search over

1.0 0.8 0.6

Uf(n)

n ∈ N and an inner communication-throughput maximization problem. The resulting search over N , referred to as exhaustive search, yields an exact solution to (P3). More importantly, the decomposition separates the role of computation from the communication-control subproblem, thereby providing the basis for the closed-form analysis in the sequel.

0.4 0.2 0.0

C. Closed-form Analysis Although exhaustive search can solve (P3) exactly over the discrete set N , such a search-based solution provides limited insight into how the optimal DDIM step n depends on system parameters. It also requires repeatedly solving the inner communication problem for all feasible n. To reveal the structure of the C 2 trade-off and enable lightweight onboard adaptation, we analyze the dominant constant-power regime, in which the reduced problem admits a closed-form characterization. Specifically, we consider the following problem: (P4)

max n∈N

c3 n + c4 + λR(n),

(30)

where R(n) = b − a(n) ϕ Pcomm (n) . Here, Pcomm (n) = Eb −Pcomp t2 (n) with Eb ≜ E(b). For analytical tractability, we b−a(n) relax n from the discrete set N to the interval [nmin , nmax ], and later recover the integer solution by evaluating the nearest feasible DDIM steps. We set t1 = 0 and impose the feasibility conditions Ecomp (a(nmax )) < E(a(nmax )) and a(n) < b, which ensure sufficient computation energy and a non-empty communication window, respectively. We first propose the following proposition to describe the properties of R(n).

Proposition 3 (Properties of R(n)). R(n) is concave and decreases monotonically w.r.t. n. Proof. See Appendix B. The monotonicity of R(n) reflects the fundamental resource trade-off: increasing n consumes more energy, thereby reducing the power available for transmission. Furthermore, the concavity of R(n) implies that the marginal degradation in throughput accelerates as energy becomes scarcer. Corollary 1 (Concavity of U (n)). The utility function U (n) is concave w.r.t. n. Proof. Since S ′′ (n) = 0 and λ > 0, U ′′ (n) = λR′′ (n) ≤ 0.

Hence, U (n) is concave.

Fig. 4 illustrates the result of Corollary 1 and the fundamental C 2 trade-off. As the DDIM step n increases, S(n) improves due to greater computational investment. However, this consumes more energy and reduces the budget available for transmission, thereby degrading the throughput R(n). As a result, U (n) exhibits unimodal behavior, increasing initially before declining. The maximizer therefore represents the optimal balance between computation and communication, and

λ = 3.0 × 10 −7 λ = 4.0 × 10 −7 λ = 4.2 × 10 −7 λ = 4.5 × 10 −7

Fig. 4: The C 2 trade-off. The normalized utility, Ũ (n), is plotted.

shifts as the weighting parameter λ varies. Corollary 1 enables the following closed-form characterization. Theorem 2 (Closed-form Step Selection). In the dominant constant-power regime, under the feasibility conditions stated above and assuming that a real stationary point exists, equation U ′ (n) = 0 admits the following solution:  −C  α∆E W0 −De  , if ∆E > 0, a1 + a2 D W0 −De−C +1 (31) ñ = W−1 −De−C  E  a1 + a2 α∆ , if ∆ < 0, E D −C W−1 −De

T −t1 −c2 , a2 c1

2 c3 where a1 ≜ ≜ ≜ 1 + aλB ln 2, D ≜ 1 + αPcomp , and ∆E ≜ Eb − Pcomp (T − t1 ). Wk (·) denotes the Lambert W function with branch k [46]. The solution n∗ to problem is given by: n∗ = arg maxn∈S U (n), (P4) n nmax ⌉ is the candidate set. Here, where S ≜ ⌊[ñ]nmin ⌋, ⌈[ñ]nmax min b [x]a ≜ min(max(x, a), b) denotes the clipping operator. ⌊·⌋ and ⌈·⌉ denote the floor and ceiling functions, respectively. The degenerate case ∆E = 0, for which U (n) becomes affine, is omitted for brevity.

1 c1 , C

Proof. See Appendix C. The condition ∆E ≷ 0 creates a physical boundary equivalent to Pcomm ≷ Pcomp , which can be verified by ∆E ≷ 0 ⇐⇒ Eb ≷ Pcomp (T − t1 ) ⇐⇒ Pcomm ≷ Pcomp .

This distinction is pivotal because the Lambert W function branches, W0 and W−1 , exhibit opposing monotonicity. Consequently, n∗ responds differently to system parameters in the communication- and computation-dominant power regimes. Remark 3 (Complexity Analysis). The closed-form rule in Theorem 2 reduces online decision-making to evaluating at most two candidate integers around ñ. Therefore, its online complexity is O(1). In contrast, the exact exhaustive-search solution requires evaluating all feasible DDIM steps in N , leading to a complexity of O(|N |). V. E XPERIMENTAL R ESULTS

A. Experimental Settings 1) System and communication settings: We investigate a space generative AI framework, and the relevant system parameters are summarized in Table I. The communication and energy-harvesting settings largely follow the specifications in [6] and [16], respectively.

TABLE I: System Parameters

Orbital angular velocity Computation power Channel bandwidth Noise power spectral density Additional attenuation factor Transmit/receive antenna gain Carrier frequency Task execution window Energy conversion efficiency Solar irradiance constant Solar panel area Weighting parameter Minimum DDIM step Maximum DDIM step Initial energy at task arrival

ω Pcomp B N0 LAL Gtx and Grx fc T η γ A λ nmin nmax E0

Value 2π/5400 rad/s 300 W 5 × 103 Hz −174 dBm/Hz −5 dB 53 dB and 0 dB 12 GHz 3s 0.19 1353 W/m2 2.0 m2 4.3869 × 10−7 4 10 400 J

2) Metrics: To evaluate the proposed framework, we adopt the following two metrics: (1) Communication Throughput: It is defined as the total number of bits delivered over the transmission window [a,b], and is used to quantify communication performance. (2) E2E CLIP Score: We measure the semantic similarity between the original text prompt and the image received at the ground device [38], to indicate the overall E2E generation quality. The reported metrics are averaged across diverse channel realizations and multiple queries. 3) Benchmarking schemes: We consider two categories of benchmarks. The first category is used to validate the proposed solver, while the second category evaluates the E2E performance gains of our joint C 2 control. • Solver benchmark: We compare the proposed closed-form solution with exhaustive search over the feasible set N = {nmin , . . . , nmax }. It verifies that the closed-form solution guarantees optimality while avoiding the O(|N |) search complexity. • Computation-centric scheme: It enforces a fixed, high DDIM step n = nmax to prioritize image fidelity, potentially at the expense of transmission reliability in energy-scarce regimes. • Communication-centric scheme: It prioritizes energy allocation for the downlink transmission to ensure high throughput, while only guaranteeing the minimal required DDIM step n = nmin . B. Validation of Throughput Maximization (P1) To isolate the gains derived from the optimal waiting time and transmit power, in this subsection, we fix the DDIM step to n = 5 and investigate the communication throughput. The following three baselines are considered: (1) Fixed waiting time and optimal transmit power. t1 = 0.4 s, with the optimal power obtained from Theorem 1. (2) Optimal waiting time and fixed transmit power. t∗1 is obtained from Proposition 1, with Pcomm = 10 W. (3) Fixed waiting time and fixed transmit power. Both the waiting time and transmit power are fixed: t1 = 0.4 s and Pcomm = 10 W.

Ours (opt. waiting + opt. comm.) Fixed waiting + opt. comm.

Opt. waiting + fixed comm. Fixed waiting + fixed comm.

Average Throughput (KiB)

Symbol

30 28 26 24 22 20 18

0.0

0.5

1.0

1.5

2.0

A (m2)

2.5

3.0

3.5

4.0

(a) Average communication throughput versus A. 30

Average Throughput (KiB)

Parameter

28 26 24 22 20 18

β (◦)

(b) Average communication throughput versus β. Fig. 5: Communication throughput across different schemes.

As illustrated in Fig. 5, our proposed strategy consistently achieves the highest communication throughput across the entire range of panel areas A and solar elevation angles β. In Fig. 5(a), the throughput grows logarithmically with A, ∗ (t) demonstrating that our joint optimization of t∗1 and Pcomm effectively converts increased harvested energy into higher data volume. In Fig. 5(b), we observe that throughput remains stable at low elevation angles but degrades as β approaches 90◦ , reflecting the reduction in the effective solar harvesting area as the sun vector becomes orthogonal to the orbital plane. The persistent performance gap between our scheme and the baselines demonstrates the necessity of minimizing the waiting phase and adopting the LCE principle. C. Evaluation of the Closed-form Solution for (P4) As shown in Fig. 6, the pre-rounding closed-form solution closely tracks the exhaustive-search optimum across the considered settings. This demonstrates that the closed-form rule maintains solution accuracy while reducing complexity. Analyzing the communication-dominant scenario reveals key resource trade-offs. First, n∗ decreases as the throughput weight λ increases (Fig. 6(a)), shifting resources toward transmission. A similar trend is observed for the bandwidth B (Fig. 6(c)). Conversely, higher initial energy E0 allows for more computation, increasing n∗ (Fig. 6(b)). Interestingly, n∗ exhibits a non-monotonic relationship with the deadline T (Fig. 6(d)). Under tight deadlines (T < 1.5 s), transmission is severely bottlenecked, so the system prioritizes computation to preserve generation quality. As T relaxes further, n∗ steadily decreases to fully exploit the extended window for throughput.

Exhaustive (n ∗ ) Closed-form (ñ)

6 4

67 ×10

0.2

8 6 4

Exhaustive (n ∗ ) Closed-form (ñ) 2

0.6

E0 (J)

0.8

1.03 ×10

(b) n∗ versus E0 .

DDIM Steps

(a) n∗ versus λ.

0.4

Average Throughput (KiB)

Exhaustive (n ∗ ) Closed-form (ñ)

Ours Comm-centric Comp-centric

Exhaustive (n ) Closed-form (ñ) ∗

100

150

T (s)

(d) n∗ versus T .

Fig. 6: Comparison of the optimal DDIM steps obtained by exhaustive search and the pre-rounding closed-form solution under different system parameters.

250

300

350

Ours Comm-centric Comp-centric

0.32

×10

200

θ (orbital angle, deg)

(a) Average communication throughput.

6 4

B (Hz)

Average E2E CLIP Score

DDIM Steps

10 8

0.31 0.30 0.29 0.28 0.27 0.26 0.25

100

150

200

θ (orbital angle, deg)

250

300

350

Average E2E CLIP Score

(b) Average E2E CLIP score. 0.32

Fig. 8: System performance at different orbital angles (θ ∈ [0◦ , 360◦ ]).

0.31

Ours (w/ Solar EH) Comm-centric (w/ Solar EH) Comp-centric (w/ Solar EH)

0.30 0.29 0.28 0.27 0.26

0.0

0.2

0.4

0.6

0.8

1.0

P0 (W)

1.2

1.4

×103

Average E2E CLIP Score

(a) Average E2E CLIP score versus P0 . 0.32 0.31

Ours (w/ Solar EH) Comm-centric (w/ Solar EH) Comp-centric (w/ Solar EH)

0.30 0.29 0.28 0.27 0.26

E0 (J)

8 ×102

(b) Average E2E CLIP score versus E0 . Fig. 7: Performance comparison of various schemes under solar EH.

D. Performance Evaluation In this subsection, we evaluate the holistic performance of our joint C2 optimization scheme. 1) Comparison with Other Schemes: As shown in Fig. 7, the proposed joint C2 policy achieves the highest E2E CLIP score across the considered energy range. In the lowenergy regime, it reduces the DDIM step and approaches the communication-centric baseline, preserving sufficient energy for transmission. In the high-energy regime, it allocates more resources to generation and approaches the computationcentric baseline. The largest gain appears in the moderateenergy regime, where neither extreme allocation is optimal. The proposed policy effectively balances onboard generation quality and downlink throughput. The sharp transition of each

scheme occurs when the available energy crosses its schemedependent feasibility threshold for completing both generation and transmission. Below the corresponding threshold, insufficient energy remains for downlink transmission after generation, resulting in transmission failure. 2) Impact of Orbital Position: Fig. 8 illustrates the communication throughput and E2E CLIP score for independent task executions initiated at different orbital angles, each with initial energy E0 and the corresponding solar-EH profile. Although the communication-centric baseline achieves the highest throughput by minimizing the computation load, its E2E CLIP score remains lower than the other schemes in sunlit regions because the onboard generation quality is limited by the minimum DDIM step. In contrast, the computation-centric baseline achieves a high E2E CLIP score in sunlit regions, but suffers a sharp performance drop during the eclipse. This is because the fixed high DDIM step consumes excessive computation energy when no solar energy is harvested, leaving insufficient energy for downlink transmission. As a result, transmission fails and the E2E CLIP score drops to zero. The proposed joint C 2 policy avoids both drawbacks by adapting the DDIM step to the orbital energy state: it allocates more resources to generation when sunlight is available and scales back computation during the eclipse to preserve downlink transmission. Consequently, it maintains the highest E2E CLIP score over the full orbit. 3) Impact of Solar EH: Fig. 9 shows the impact of solar EH under different E0 . In the resource-constrained regime, solar EH substantially improves the E2E CLIP score because the harvested energy helps satisfy the minimum generationand-transmission requirement and can further support a larger DDIM step or higher downlink throughput. The visible steplike transitions are caused by discrete DDIM step selection:

Average E2E CLIP Score

Ours (w/ Solar EH) Ours (w/o Solar EH)

0.32

Ground Truth

Comp-centric

Comm-centric

Ours

A photo of magpie. CLIP score: 0.3462

CLIP score: 0.3193

CLIP score: 0.2942

CLIP score: 0.3169

A photo of junco, snowbird. CLIP score: 0.3464

CLIP score: 0.3198

CLIP score: 0.2189

CLIP score: 0.3247

0.31 0.30 0.29 0.28 0.27 0.26

E0 (J)

8 ×102

Fig. 9: Impact of solar EH on the average E2E CLIP score.

Average E2E CLIP Score

(a) Visual performance with solar EH. w/ Solar EH

0.32

w/o Solar EH

Ground Truth

Comp-centric

Comm-centric

Ours

CLIP score: 0.2942

CLIP score: 0.3096

CLIP score: 0.2189

CLIP score: 0.3081

0.3101

0.31

0.3052

Transmission failed

0.3005

(Energy Depleted)

0.30 0.29 0.2807

0.2807 0.0000

0.28 0.27

Ours

Comm-centric

Scheme

A photo of magpie. CLIP score: 0.3462

Comp-centric

Fig. 10: Comparison of the average E2E CLIP score of different schemes in a specific scenario, evaluated with and without solar EH.

CLIP score: None

Transmission failed (Energy Depleted)

A photo of junco, snowbird. CLIP score: 0.3464

CLIP score: None

(b) Visual performance without solar EH.

once the available energy exceeds a threshold that supports an additional DDIM step while preserving reliable transmission, n increases, resulting in a sudden improvement in the E2E CLIP score. As E0 becomes sufficiently large, the curves with and without solar EH converge because the initial energy alone is enough to support both generation and transmission, making additional harvested energy less influential. Fig. 10 further compares different schemes with and without solar EH in a representative scenario. With solar EH, the proposed scheme achieves the highest E2E CLIP score by jointly adapting computation and communication. Without solar EH, the computation-centric baseline becomes transmission-infeasible because its fixed high DDIM step depletes the energy budget before downlink transmission, causing transmission failure. The communication-centric baseline remains stable but quality-limited because it always prioritizes transmission and uses the minimum DDIM step. In contrast, the proposed scheme scales back computation when solar EH is unavailable, preserving transmission reliability while maintaining a competitive E2E CLIP score. The qualitative examples in Fig. 11 are consistent with these results: solar EH improves visual quality, and without solar EH, the proposed scheme still delivers semantically aligned images while the computationcentric baseline fails to transmit. VI. C ONCLUDING R EMARKS This work establishes a framework for space generative AI and reveals the inherent coupled C 2 bottleneck under solar EH. Specifically, it characterizes how a satellite should jointly manage waiting, diffusion-based generation, and downlink transmission when C 2 stages draw from the same time-causal harvested-energy buffer. First, computation should start at the earliest energy-feasible time. Second, constant transmit power is optimal in most orbital states, while dawn may require

Fig. 11: Qualitative comparison of generated images.

solar-tracking control. Third, adaptive generation depth is most beneficial under moderate energy, where neither computationnor communication-centric extremes are optimal. These results advance satellite edge intelligence from generic computationoffloading design towards task-oriented generative service provisioning under physical orbital constraints. Future work should extend this framework to multi-user and multi-task satellite systems, where heterogeneous generative requests compete for shared harvested energy. It is also important to incorporate practical hardware constraints such as finite battery capacity, peak transmit power, and accelerator-dependent inference energy. Finally, more refined E2E quality models are needed to jointly capture generation quality, compression distortion, channel effects, and humanperceived semantic fidelity. A PPENDIX A. Derivation for Eq. (25) As the satellite orbits the Earth with angular velocity ω, the Fb rotates relative to the Fo around the Yo . The rotation matrix from Fo to Fb at time t is:   − sin(ωt) 0 − cos(ωt) . 0 −1 0 R(t) =  (32) − cos(ωt) 0 sin(ωt) The sun vector in Fb is derived as Sb (t) = R(t)So :   − cos β sin(ωt) . − sin β Sb (t) =  − cos β cos(ωt)

(33)

The instantaneous harvested power is proportional to the effective projection area: P (t) = η · γ · A · max (n · Sb (t), 0) .

(34)

Û (u) = −a2 c3 u + a1 c3 + c4 + λf (u),

B. Proof of Proposition 3 For R(n), with u ≜ T − t1 − c1 n − c2 and ∆E ≜ Eb − Pcomp (T − t1 ), it becomes α∆E f (u) = B u log2 1 + αPcomp + , u > 0. (35) u Calculating its first derivative yields "

′

The second derivative is ( d ′′ E log2 1 + αPcomp + α∆ f (u) = B u ) du α∆E 1 − ln 2" u 1 + αPcomp + α∆E /u B − α∆E /u2 = ln 2 1 + αPcomp + α∆E /u !# d α∆E − . du u 1 + αPcomp + α∆E /u

(36)

(40)

where f (u) is from (35). Consequently, it suffices to solve Û ′ (u) = 0 for the stationary point ũ, from which ñ is recovered. Using the expression for f ′ (u) from (36), the derivative Û ′ (u) is explicitly given by " α∆E ′ Û (u) = − a2 c3 + λB log2 1 + αPcomp + u # (41) 1 α∆E − . ln 2 u(1 + αPcomp + α∆E /u) Setting Û ′ (u) = 0 gives E − ln 1+αPcomp + α∆ u

a c α∆E = 2 3 ln 2. λB u 1 + αPcomp + α∆E /u (42) 2 c3 Let D ≜ 1 + α Pcomp , K ≜ aλB ln 2, and C ≜ 1 + K. Then α∆E E = K. ln D + α∆ − u u D + α∆E /u α∆E α∆E E Let x ≜ D + α∆ u . Thus u = x − D and u = x−D , x ̸= D. This further simplifies the equation as

ln x − (37)

α∆E = K. ux

x−D E Plugging α∆ into it, we have ux = x

ln x +

Compute the remaining derivative. Let ψ(u) ≜ 1 + αPcomp + α∆E α∆E /u and q(u) ≜ uψ(u) . Then ψ ′ (u) = −α∆E /u2 , and d −1 u ψ(u)−1 q ′ (u) = α∆E · du = α∆E −u−2 ψ −1 + u−1 · −1 ψ −2 ψ ′ (u) α∆E α∆E α∆E /u2 =− 2 + · u ψ(u) u ψ(u)2 2 2 α∆E α ∆ =− 2 + 3 E2 . u ψ(u) u ψ(u)

C. Proof of Theorem 2 Let u ≜ T − t1 − c1 n − c2 , and conversely, n = a1 − a2 u where a1 ≜ T −tc11−c2 , a2 ≜ c11 . This transforms U (n) to

Taking the inner product yields the results.

E f (u) = B log2 1 + αPcomp + α∆ u # u d α∆E + ln 1 + αPcomp + u ln 2 |du {z } −α∆E /u2 = 1+αPcomp +α∆E /u " E = B log2 1 + αPcomp + α∆ u # α∆E 1 − . ln 2 u(1 + αPcomp ) + α∆E

Therefore, f ′ (u) ≥ f ′ (+∞) > 0. Hence, we have R′′ (n) ≤ 0 and R′ (n) < 0, which completes the proof.

D = 1 + K = C. x

Transform it and we have x = e C−D/x = eC e−D/x . Then, D D y C x eD/x = eC . Let y ≜ D x , x = y , then y e = e and thus (−y) e−y = −D e−C . Hence −y = Wk −De−C , and D −C x = y = −D/Wk −De , with branch index k chosen to satisfy domain constraints. For brevity, we denote ζk ≡ ζk E E Wk −De−C , and then u = −α∆ = − α∆ D D ζk +1 . We −D ζk

(38)

Plugging back, we have α∆ B α∆E α 2 ∆2 E f ′′ (u) = − 2 − − 2 + 3 E2 ln 2 u ψ(u) u ψ(u) u ψ(u) B α2 ∆2E = − 3 (39) ln 2 u ψ(u)2 Bα2 ∆2E 1 =− ≤ 0. ln 2 u3 1 + αPcomp + α∆E /u 2

1 further investigate the term y = D x = 1+ α∆E . All constituent Du parameters are positive, with the exception of ∆E . If ∆E > 0, it follows that y < 1 (implying −y > −1); consequently, the solution corresponds to the principal branch W0 . If ∆E < 0, the branch W−1 is selected. Therefore,  −C  α∆E W0 −De  − , if ∆E > 0,  D W0 −De−C +1 ũ = (43) W−1 −De−C   E − α∆ , if ∆ < 0. E D −C W−1 −De

Substituting (43) into ñ = a1 − a2 ũ yields the results in (31). Due to the concavity of the objective function, the optimal integer solution n∗ is guaranteed to lie in the candidate set S defined in Theorem 2. This completes the proof.

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