esa-sceva/satcom-chunk-collection huggingface dataset:
## 4 Experimental Verification
### Experimental Result  
#### 4.2.1 Characterization of Different Signals  
According to the PHM dataset's physical signal categories, the time-domain signal in the CWT time-frequency diagram was used to observe and extract features. Different wear  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Cutting Condition** & **Processing Parameter** \\ \hline Spindle speed (r/min) & 10,400 \\ Feed rate (mm/min) & 1555 \\ Axial cutting depth (mm) & 0.2 \\ Radial cutting depth (mm) & 0.125 \\ Feed per tool travel (mm) & 0.001 \\ Cooling condition & Dry cut \\ \hline \hline \end{tabular}
\end{table}
Table 2: Test parameters.  
Figure 5: Schematic of tool wear monitoring device.  
\(10^{2}\) Hz. As wear worsens, the high-energy distribution of the vibration signal extends to the high-frequency region, with a significantly larger high-energy distribution area. During tool cut-in and cut-out, the vibration signals are larger compared to the milling process. However, the CWT time-frequency plots show no significant difference, indicating that this process affects the vibration amplitude without significantly affecting the signal energy distribution. The peak value of the vibration signal follows the same trend as that of the force signal, gradually increasing with the intensification of the wear condition, demonstrating a positive correlation.  
For the acoustic emission signal, the three wear states in the dataset were selected. The same operation as described above was repeated to obtain the CWT time-frequency map and the time-domain image, as shown in Figure 8. As the tool wear condition worsens, cutting force and vibration increase, leading to greater energy release during material deformation and fracture. This results in an enhancement of the high-energy amplitude of the acoustic emission signal and an increase in the energy density at the high-energy distribution in the CWT time-frequency diagram. The acoustic emission signal is larger during tool cut-in and cut-out than during milling, with a significant increase in the energy density of the CWT time-frequency map. This indicates that these processes affect the acoustic emission amplitude and increase the energy share of the low-frequency part. As the wear condition increases, the elastic waves released by the destruction of the tool and workpiece material increase, causing the amplitude of the acoustic emission signal to become larger.  
The analysis above shows that the CWT time-frequency diagrams of force signals, vibration signals, and acoustic emission signals exhibit obvious differences in energy intensity distribution at different wear stages. The time and frequency bands also differ, indicating the feasibility of using two-dimensional CWT time-frequency diagrams instead of the one-dimensional original physical signals. Therefore, CWT time-frequency diagrams of the various types of signals can be effectively used as inputs to the model.  
Figure 7: Analyzing three wear stages of vibration signals.
# Silicon Integrated Nanophotonic Devices for On-Chip Multi-Mode Interconnets
## 3 Silicon Multi-Mode Routing Devices
### Silicon Multi-Mode Splitters
Based on this scheme, the device footprint can be reduced to \(\approx\)120 \(\upmu\)m. The measurement results show low losses (IL \(<\) 0.86 dB) and low crosstalk (XT \(<\) \(-\)15.7 dB). For the aforementioned multi-mode splitters, the splitting ratio is fixed to SR = 50% (i.e., 3-dB power splitting). However, an arbitrary splitting ratio is commonly required for a wide scope of applications, e.g., power distribution [215], power monitoring [210] and box-like filtering [216]. Hence, it is also very important to develop multi-mode splitters with a splitting ratio beyond 50:50. In [214], an ultra-broadband silicon multimode splitter with an arbitrary splitting ratio was proposed for the first time, as shown in Figure 17d. The structure is a direct crossing with an obliquely embedded SWG that partially reflects the incident modes. The SWG is formed by a one-dimensional array of subwavelength nano-holes that can be collectively regarded as an effective medium transllector. The reflectance can be almost the same for all the input modes (i.e., TE\({}_{\rm 0-2}\)) since the effective indices are quite close when w\({}_{\rm 0}>>\lambda\)/n\({}_{\rm s}\). More importantly, any desired splitting ratio, ranging from 0% to 100%, can be achieved by simply changing the SWG duty cycle. Additionally, the splitting ratio spectra can be very uniform over an extremely broad bandwidth (BW \(>\) 415 nm) covering O-, E-, S-, C-, L- and U-bands. The insertion losses and crosstalk are also quite low (IL \(<\) 0.1 dB, XT \(<\) \(-\)20 dB). In [194], a four-mode power splitter was realized by using a uniform trench as a translfactor, whose splitting ratio can also be modified by tuning the trench width. The reported silicon multi-mode splitters have been summarized in Table 7. Among all these schemes, the SWG transflector is the one with the most potential in terms of scalability, flexibility and overall performance. One obstacle might be the adiabatic tapers required by the direct crossing, as discussed in Section 3.2. As a possible way to proceed, extensive efforts should be made to develop a MDM-compatible mode size converter with a shorter conversion length [217; 218], so that a compact and high-performance multi-mode splitter can be realized.
# A Gaussian Process Method with Uncertainty Quantification for Air Quality Monitoring
## Appendix C Approximated Derivatives of SE Kernel  
By specifying a kernel function, we can obtain analytical forms of Equations (28) and (29) immediately. In this paper, we adopt the widely used SE kernel shown in Equation (A1) as an example.  
\[k_{SE}(x,x^{\prime})=s_{f}^{2}\exp\Big{(}-\frac{(x-x^{\prime})^{2}}{2l^{2}} \Big{)}.\] (A1)  
There are two hyperparameters, i.e., the signal variance \(s_{f}\) and length-scale \(l\) are involved. Equations (A2) and (A3) show the expectation (prediction mean) partial derivative (EPD) and covariance partial derivative (CPD) of \(s_{f}\),  
\[\begin{split}&\Big{(}\frac{\partial\mathbf{f}_{s}}{\partial \theta_{s}}\Big{)}_{o}\Big{|}_{\theta_{s}=s_{f}}\\ &=\sum_{i=1}^{n}\sum_{j=1}^{n}\Big{(}k_{oj}\frac{\partial d_{ji}} {\partial s_{f}}+\frac{\partial k_{oj}}{\partial s_{f}}d_{ji}\Big{)}y_{i}\\ &=\sum_{i=1}^{n}\sum_{j=1}^{n}y_{i}\begin{cases}0,\quad j\
eq i \\ 0,\quad j=i\end{cases},\end{split}\] (A2)  
\[\begin{split}&\Big{(}\frac{\partial\text{cov}(\mathbf{f}_{s})}{ \partial\theta_{s}}\Big{)}_{oo}\Big{|}_{\theta_{s}=s_{f}}\\ &=\frac{\partial\mathbf{K}(\mathbf{X}_{s},\mathbf{X}_{s})_{oo}}{ \partial s_{f}}-\sum_{i=1}^{n}\sum_{j=1}^{n}\Big{(}\frac{\partial k_{oj}}{ \partial s_{f}}d_{ji}k_{oi}+k_{oj}\frac{\partial d_{ji}}{\partial s_{f}}k_{oi }-k_{oj}d_{ji}\frac{\partial k_{oi}}{\partial s_{f}}\Big{)}\\ &=2s_{f}-\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\begin{array}{ll}2s_{f} \exp(-\frac{(x_{o}-x_{j})^{2}+(x_{i}-x_{i})^{2}+(x_{o}-x_{i})^{2}}{2l^{2}}), \quad j\
eq i\\ -2s_{f}\exp(-\frac{(x_{o}-x_{j})^{2}+(x_{o}-x_{i})^{2}}{2l^{2}}),\quad\quad \quad j=i\end{array}\right\}.\end{split}\] (A3)  
While the derivatives of \(l\) are given in Equations (A4) and (A5),  
\[\begin{split}&\Big{(}\frac{\partial\mathbf{f}_{s}}{\partial \theta_{s}}\Big{)}_{o}\Big{|}_{\theta_{s}=l}\\ &=\sum_{i=1}^{n}\sum_{j=1}^{n}\Big{(}k_{oj}\frac{\partial d_{ji}} {\partial l}+\frac{\partial k_{oj}}{\partial l}d_{ji}\Big{)}y_{i}\\ &=\sum_{i=1}^{n}\sum_{j=1}^{n}y_{i}\left\{\begin{array}{ll}-\exp(- \frac{(x_{o}-x_{j})^{2}+(x_{j}-x_{i})^{2}}{2l^{2}})\frac{(x_{o}-x_{j})^{2}+(x_ {j}-x_{i})^{2}}{l^{3}},&j\
eq i\\ \exp(-\frac{(x_{o}-x_{j})^{2}}{2l^{2}})\frac{(x_{o}-x_{j})^{2}}{l^{3}},&j=i \end{array}\right\},\\ &\Big{(}\frac{\partial\text{cov}(\mathbf{f}_{s})}{\partial\theta_{s}}\Big{)}_{o 0}\Big{|}_{\theta_{s}=l}\\ &=\frac{\partial\mathbf{K}(\mathbf{X}_{s},\mathbf{X}_{s})_{oo}}{ \partial l}-\sum_{i=1}^{n}\sum_{j=1}^{n}\Big{(}\frac{\partial k_{oj}}{ \partial l}d_{ji}k_{oi}+k_{oj}\frac{\partial d_{ji}}{\partial l}k_{oi}-k_{oj}d _{ji}\frac{\partial k_{oi}}{\partial l}\Big{)}\\ &=\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\begin{array}{ll}\exp(- \frac{(x_{o}-x_{j})^{2}+(x_{j}-x_{i})^{2}+(x_{o}-x_{i})^{2}}{2l^{2}})\quad j \
eq i\\ \ast\frac{(x_{o}-x_{j})^{2}+(x_{j}-x_{i})^{2}-(x_{o}-x_{i})^{2}}{l^{3}}s_{f}^{2 },&j=i\end{array}\right\}.\end{split}\] (A5)
## 5 Results and Analysis
### SURAA: Parametric Configuration  
The matrix, _spal_004_, is a rectangular matrix with fewer rows and a large number of dense columns. It also has a fairly high _npr variance_. Due to the high density, it is highly suitable for dynamic parallelism. In the results presented in this paper, the parameter _max_child_threads_ is set to 8192 for _spal_004_. During the course of conducting our experiments, we found that configuring this parameter to 2048, 4096, and 8192 varies the performance of the SURAA tool. The typical trend was that the highest value provided the best performance for _spal_004_ due to the increase in dynamic parallelism. To illustrate, _spal_004_ achieved 19.5 GFLOP/s in our experiments with 8192 kernels, while it provided 13.60 GFLOP/s when the parameter was set to 2048. This indicates that tweaking of the _max_child_threads_ affects the performance of the SpMV computation. However, larger values for _max_child_threads_ cause larger overheads due to the spawning of the dynamic kernels, and, therefore, setting it to higher values should be done on a necessity basis. In our benchmark dataset, for 23 of the 26 matrices, there are no more than 2048 rows, in total, whose _npr_ value is greater than 32. The three matrices are _spal_004_, _rail4284_ and _12 month1_, with _spal_004_ the biggest exception containing 8192 rows which have more than 32 _npr_. Therefore, we have set _max_child_threads_ to 2048 for all the matrices except for the _spal_004_ matrix. An automatic configuration of the tool for this parameter can be done during the set up phase by calculating the number of rows with _npr_\(\geq\) 32 and setting  
Figure 13: Throughout with respect to _npr_ variance and _nnz_ for (**a**) BSR; (**b**) SELL-P; (**c**) WPK1; (**d**) WPK2; (**e**) CSR; (**f**) HYB; (**g**) CSR5; and (**h**) SURAA.  
the _max_child_threads_ parameter to an appropriate value. In future work, we will further investigate this behavior.
## 3 Results and Findings from the Existing Literature
### Techno-Economic Analysis (TEA)
In essence, the study prompts a critical evaluation of the total costs, emissions, and environmental risks associated with e-fuels and DACCS to make informed decisions regarding sustainable and environmentally responsible aviation practices. [PERSON] and [PERSON] [66] focused on assessing the production costs and life cycle emissions of e-fuels produced through different electrolysis technologies and Fischer-Tropsch synthesis. The study investigated both current alkaline and future solid oxide electrolysis methods. The research aimed to provide insights into the economic feasibility and environmental impact of e-fuels as an alternative to traditional vehicle fuels. The findings suggested that e-fuel production costs, under optimistic assumptions for various parameters, were estimated to be approximately 1.17 V/leq diesel or 3.12 V ct/MJ in 2015. The studies collectively stress the necessity for process optimisation, regulatory support, and innovative technological solutions to make SAF a viable, sustainable, and economically competitive alternative to conventional aviation fuels. Table 1 summarises and compares the processes and their associated costs of production of SAF.
# Field and Numerical Investigation of Taihu Resort Cut Slope Failure in Suzhou, China
## 3 Study Landslide Area
### Relationship between Slope Displacement and Precipitation Amount  
In order to investigate the relationship between slope displacement and rainfall, slope displacement monitoring data were collected by the Suzhou Civil Architecture Design Institute (SCADI), and precipitation data were obtained from the China Meteorological Data Sharing Service System. The cumulative displacements in depth are illustrated in Figure 6a-c, and the horizontal displacement and meteorological records are illustrated in Figure 7, whereas the displacements were monitored during the period from August 2012 to July 2013. The evolution of the ground displacement values could be divided into three periods as follows. In the first period, from August to November, the monitoring point of Section S5 recorded very small displacements as a straight line during August and September, except for the Section S6 and S7 displacements, which started increasing irregularly from August to the middle of November. Heavy rainfall occurred at the beginning of November, with 110 mm of precipitation. The displacement of the three sections started increasing significantly in November with an intensity of rainfall of 10 mm/day \(\leq\) rainfall \(\leq\) 80 mm/day, with total accumulated displacement values of 10 mm, 12 mm, and 11 mm in Sections S5, S6, and S7, respectively. In the next period, from  
Figure 5: Taihu Resort landslide Photos of (**a**) Second phase of the overall slope collapse in July 2013; (**b**) Dimensions of the slope collapse; (**c**) Cross-section of the Taihu Resort landslide at the second landslide.  
December to February, the slope remained stable, and a small decrease in the cumulated displacement was recorded from December to February. In the third period, from the end of February to July, based on the meteorological data, the monitoring area entered the monsoon rainfall in March. Therefore, the slope was affected by heavy rainfall, as can be seen from the monitoring points at Sections S6 and S7, which developed at a rate of 0.31 mm/day (Figure 7). Section S6 recorded an increase of 0.23 mm/day with an intensity of rainfall 50 mm/day \(\leq\) rainfall \(\leq\) 120 mm/day. Until July 2013, under the influence of continuous rainfall, the maximum cumulative displacements at Sections S5-S7 reached 14 mm and 17 mm, respectively. Accordingly, the slope stability at Section 6 (S6) and CX1-1 reached a critical state in July. Hence, the process went into global failure almost eight months after the completion of remedial work. Through a detailed field investigation, it was determined that rainfall was one of the main factors that induced failure on the Taihu slope.  
Figure 6: Horizontal cumulative displacement of Taihu slope from August 2012 to July 2013. (**a**) Monitoring point CX1–1, (**b**) Monitoring point CX2–1, (**c**) Monitoring point CX3–1.
# [PERSON] on Korean Block Ciphers
## 2 Related Works
### Target Block Ciphers
#### 2.2.3. LEA  
In WISA'13, a lightweight block cipher LEA was presented [14]. LEA is a 128-bit block cipher supporting three key lengths, i.e., 128-bit, 192-bit, and 256-bit. LEA-128, LEA-192, and LEA-256 require 24, 28, and 32 rounds, respectively. In each round, 192-bit round keys are required. The word size is 32-bit and primitive operations are 32-bit wise addition, rotation, and exclusive-or. The parameters for the LEA block cipher are shown in Table 4.  
The key schedule of the LEA uses constant value \(\delta\) representing the ASCII codes of 'L,' 'E,' and 'A', and the key schedule is slightly different depending on the parameters. The key schedule of LEA-128 for \(0\leq i<24\) is as follows:  
\[\begin{array}{ll}\delta[0]=0x3 efe9 db,&\delta[1]=0x44626b02\\ \delta[2]=0x79e27c8a,&\delta[3]=0x78 df30 ec\\ \delta[4]=0x715 ea49e,&\delta[5]=0x785 da0a\\ \delta[6]=0x04 ef22a,&\delta[7]=0x5c40957\end{array} \tag{12}\]  
\[\begin{array}{ll}K[0]=\text{ROL}_{1}(K[0]\boxplus\text{ROL}_{i}(\delta[i \text{ mod }4]))\\ K[1]=\text{ROL}_{3}(K[1]\boxplus\text{ROL}_{i+1}(\delta[i\text{ mod }4]))\\ K[2]=\text{ROL}_{6}(K[2]\boxplus\text{ROL}_{i+2}(\delta[i\text{ mod }4]))\\ K[3]=\text{ROL}_{11}(K[3]\boxplus\text{ROL}_{i+3}(\delta[i\text{ mod }4]))\\ RK_{i}=(K[0],K[1],K[2],K[1],K[3],K[1])\end{array} \tag{13}\]  
Unlike the key schedule, the round function is performed as follows for \(0\leq i<r\) regardless of the parameter for the plaintext \(P=\{X_{0}[0],X_{0}[1],X_{0}[2],X_{0}[3]\}\):  
\[\begin{array}{ll}X_{i+1}[0]=\text{ROL}_{9}((X_{i}[0]\oplus RK_{i}[0])\boxplus (X_{i}[1]\oplus RK_{i}[1]))\\ X_{i+1}[1]=\text{ROL}_{5}((X_{i}[1]\oplus RK_{i}[2])\boxplus(X_{i}[2]\oplus RK _{i}[3]))\\ X_{i+1}[2]=\text{ROL}_{3}((X_{i}[2]\oplus RK_{i}[4])\boxplus(X_{i}[3]\oplus RK _{i}[5]))\\ X_{i+1}[3]=X_{i}[0]\end{array} \tag{14}\]  
Key schedule and round function structure of LEA are shown in Figures 4 and 5.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cipher** & **Block Size (bits)** & **Key Size (bits)** & **Word Size (bits)** & **Keywords** & **Rounds** \\ \hline LEA-128 & 128 & 128 & 32 & 4 & 24 \\ \hline LEA-192 & 128 & 192 & 32 & 6 & 28 \\ \hline LEA-256 & 128 & 256 & 32 & 8 & 32 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters of LEA.  
Figure 3: Round function of CHAM.
## 4 Proposed Approach
### Data Balancing and Data Splitting  
To address class imbalances in the dataset, the Synthetic Minority Oversampling Technique (SMOTE) [26] is utilized to create synthetic samples for underrepresented classes. This method works by interpolating between an existing minority class instance \(x_{i}\) and one of its \(k\) nearest neighbors \(x_{nm}\). The synthetic instance \(x_{new}\) is generated as follows:  
\[x_{new}=x_{i}+\lambda\cdot(x_{nm}-x_{i}) \tag{1}\]  
where \(\lambda\) represents a randomly generated number, uniformly drawn from the interval [0, 1]. This ensures that the synthetic data points are strategically placed along the line segment between \(x_{i}\) and \(x_{nm}\).  
Despite the variability in synthetic data points due to random selection introduced by \(\lambda\), SMOTE's ability to interpolate between data points allows for meaningful sample generation, which aids in preserving the relationships within the feature space. This allows the model to better capture the complexities of the minority class, ensuring that the synthetic samples remain relevant and contribute to the model's improved performance in classification tasks, especially for imbalanced datasets [49].  
For the next step, the problem is reframed as a binary classification task to distinguish between malicious and benign network traffic, and then the dataset is divided into training and testing sets using an 80-20 split. This stratified approach allows for a robust assessment of the model's performance and generalizability, contributing to the development of effective security solutions.
## 2 Hybrid Charging Scheme  
### Charging by Mobile Charger with Omnidirectional Antenna  
Figure 1a depicts the charging infrastructure sensor nodes by a mobile charger. The wireless charging model in [20] for wireless sensor networks operating at 1 GHz is given as  
\[p_{r}=\frac{G_{t}G_{r}\eta}{L_{p}}\left(\frac{\lambda}{4\pi(d+\beta)}\right)^{ 2}p_{0} \tag{1}\]  
where \(p_{r}\), \(G_{t}\), \(G_{r}\), \(\eta\), \(\lambda\), \(d\), \(\beta\), \(L_{p}\), and \(p_{0}\) are the received power, antenna gain of the transmitter, antenna gain of the receiver, rectifier efficiency, wavelength, distance between the transmitter and receiver, parameter characterizing charging model, polarization loss, and transmit power, respectively. Because Equation (1) is used for the case between directional antennas, it is simplified in [10; 11] by merging the parameters into a single parameter \(\alpha\) to give an omnidirectional power equation, as follows  
\[P_{r}(\overset{\rightarrow}{r}_{j},\overset{\rightarrow}{r}_{charger}(t))= \frac{\alpha}{(d_{j}(t)+\beta)^{2}}\qquad j=1,...,N_{T}\] (2a) where \[\overset{\rightarrow}{r}_{charger}(t)=\left(x_{charger}(t)\,y_{charger}(t) \right)\quad\overset{\rightarrow}{r}_{j}=\left(x_{j}\,y_{j}\right) \tag{2b}\] \[d_{j}(t)=\sqrt{\left(x_{j}-x_{charger}(t)\right)^{2}+\left(y_{j}-y_{charger}(t )\right)^{2}} \tag{2c}\]  
The term \(P_{r}(\overset{\rightarrow}{r}_{j},\overset{\rightarrow}{r}_{charger}(t))\) in Equation (2a) represents the received power, \(\overset{\rightarrow}{r}_{charger}(t)\) in Equation (2b) represents the two-dimensional location of the mobile charger at time \(t\), and \(d_{j}(t)\) is the distance between the \(j\)-th node located at \(\overset{\rightarrow}{r}_{j}\) and the mobile charger.  
In [10], linear programming is used to determine the optimal stopping spots of the charger with omnidirectional antenna as follows  
\[\min T_{cm}\,=\min\sum_{k=1}^{\infty}\Delta t_{k} \tag{3}\]  
subject to  
\[\sum_{k=1}^{\infty}\Delta t_{k}P_{r}(\overrightarrow{r}_{j},\overrightarrow{r }_{charger}(t=t_{acc}))\geq ET_{S}\quad j=1,...,N_{T}\]  
where \(T_{cm}\) is the time for charging by the mobile charger, \(\Delta t_{k}\) is the charging duration at the \(k\)-th stop location, \(t_{acc}\) is the accumulated time until the charger is located at a particular location, and \(ET_{S}\) represents the target energy level of the sensor nodes. Further, \(N_{T}\) is the total number of sensor nodes. In Equation (3), the transition of the mobile charger from one stopping spot to another is not specified. Moreover, the charging energy is evaluated for all the sensor nodes.
## 3 Mission Application and Trajectory Simulations
### Case of Unconstrained Clock Angle  
Consider the general case of an unconstrained sail clock angle, in which the optimal control law is given by Equation (19). The proposed optimization procedure was used to parametrically analyze the Earth-Venus transfer as a function of the characteristic acceleration value \(a_{c}\in[0.1,\,0.2]\,\mathrm{mm}/\mathrm{s}^{2}\). The results are summarized in Figure 8, which shows that the flight time \(\Delta t\) depends strongly on \(a_{c}\). Note, in fact, that changing from a value of \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\) to a value of \(a_{c}=0.2\,\mathrm{mm}/\mathrm{s}^{2}\) results in a reduction in the flight time of more than 60 days.  
Figure 8: Minimum flight time as a function of the characteristic acceleration \(a_{c}\in[0.1,\,0.2]\,\mathrm{mm}/\mathrm{s}^{2}\) in an Earth–Venus orbit-to-orbit transfer. The black dot refers to the special case of \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\) discussed in the first part of the section.  
Figure 7: Ecliptic projection and isometric view of the rapid transfer trajectory in an Earth–asteroid 433 Eros mission scenario, when \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\) and unconstrained sail clock angle \(\delta\). The legend is reported in Figure 5.  
The second mission scenario analyzed an Earth-Mars interplanetary transfer. In that case, we again assumed the same characteristic acceleration reported in Ref. [1], that is, \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\). The minimum flight time for an orbit-to-orbit transfer was about 752 days, and the SFGIS-propelled spacecraft completed one revolution around the Sun before reaching the target planet orbit, as shown in Figure 9. The optimal control law \(\delta=\delta(t)\) is shown in Figure 10, where the clock angle remains close to the value \(\delta=0\) (or \(\delta=360^{\circ}\), which is equivalent in terms of the direction of \(\hat{m}\) in the RTN reference frame) during the entire flight.  
The last scenario analyzed in this subsection was an Earth-Mercury transfer with \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\). The low-thrust transfer to Mercury is a challenging problem, typically requiring a long flight time and numerous revolutions around the Sun to complete [32; 33; 34; 35]. In this case, an SFGIS-propelled spacecraft required about 780 days to reach Mercury's heliocentric orbit, with four complete revolutions around the Sun. The  
Figure 10: Time variation of the (unconstrained) sail clock angle \(\delta\) in a minimum-time Earth–Mars mission scenario when \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\). The legend is reported in Figure 4.  
Figure 9: Ecliptic projection and isometric view of the rapid transfer trajectory in an Earth–Mars mission scenario, when \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\) and the sail clock angle \(\delta\) is unconstrained. The legend is reported in Figure 5.  
spacecraft transfer trajectory and the time variation in the sail clock angle are shown in Figures 11 and 12, respectively.