# LTE: Lightweight Transformer Encoder for Orbit Prediction
## 2 Materials and Methods
### Data Preprocessing
\begin{table}
\begin{tabular}{c l l l} \hline \hline \multicolumn{1}{c}{**Element’s Name**} & \multicolumn{1}{c}{**Description**} & \multicolumn{1}{c}{**Dataset**} & \multicolumn{1}{c}{**Range**} \\ \hline \multirow{3}{*}{Semi-major axis (S)} & A value that indicates the size of the orbit, expressed in kilometers, and signifies how far the satellite is from & K3 & 7052.951\(\sim\)7073.345 \\  & in kilometers, and signifies how far the satellite is from & K3 & 6892.097\(\sim\)6915.121 \\  & the Earth. & K5 & 6928.669\(\sim\)6937.687 \\ \hline \multirow{3}{*}{Eccentricity (\(E\))} & A value that characterizes the orbit as an elliptical shape, ranging between 0 and 1. & K3 & 0.000001\(\sim\)0.003911 \\  & ranging between 0 and 1. & K3A & 0.000002\(\sim\)0.004223 \\  & & K5 & 0.000410\(\sim\)0.003133 \\ \hline \multirow{3}{*}{Inclination (\(I\))} & The angle by which the orbital plane is tilted concerning & K3 & 89.1185\(\sim\)98.19598 \\  & the equatorial plane. & K3A & 97.45451\(\sim\)97.59076 \\  & & K5 & 97.59100\(\sim\)97.63480 \\ \hline \multirow{3}{*}{RAAN (\(R\))} & The angle between the vernal equinox and the ascending & K3 & 0.00034\(\sim\)359.99997 \\  & node of the satellite’s orbit. & K3 & 0.00005\(\sim\)359.99992 \\  & & K5 & 0.00004\(\sim\)359.99992 \\ \hline \multirow{3}{*}{Argument of perigee (\(A\))} & The angle formed between the point where the satellite & K3 & 0.00016\(\sim\)359.99936 \\  & is nearest to the Earth and the satellite’s ascending node & K3A & 0.00028\(\sim\)359.99987 \\  & in its orbital path. & K5 & 0.00014\(\sim\)359.99990 \\ \hline \multirow{3}{*}{Mean anomaly (\(M\))} & The angle at a specific point in the orbit, measured from & K3 & 0.00079\(\sim\)359.999989 \\  & the perigee. & K3A & 0.00107\(\sim\)359.99975 \\ \cline{1-1}  & & K5 & 61.20162\(\sim\)298.27813 \\ \hline \end{tabular}
\end{table} 
Table 1: Description of the six orbital elements used in our study from the TLE set, which consists of twelve elements. The table includes the name and symbol of each element, a brief description, and the range of each element, as obtained from the data of each satellite. These six orbital elements collected from the TLE set of each satellite exhibit values that have different ranges. The ranges of these six orbital elements in each dataset are provided in Table 1. As shown in this table, Eccentricity ranges from 0.000016 to 0.004229, while the Semi-major axis ranges from 6892.097 to 6915.121. As a result, the influence of Eccentricity is weaker compared to the Semi-major axis during model training. To address this, we applied normalization to the data of the six orbital elements collected from the TLE sets of the K3, K3A, and K5 satellites. Each orbital element was normalized by subtracting the minimum value from each data point and then dividing by the range, which is computed as the difference between the maximum and minimum values. The equation for normalization is shown in Equation (2). Here, \(\mathcal{Z}\) represents the normalized value, \(x\) denotes the individual data point, \(\min(x)\) refers to the minimum value within the column containing the data point to be normalized, and \(\max(x)\) refers to the maximum value within the same column. \[\mathcal{Z}=\frac{(x-\min(x))}{(\max(x)-\min(x))} \tag{2}\]  
We note that, since 2016 is a leap year, which occurs once every four years and includes an extra day, the 29 February, data measured on this day exist. This single day's data are excluded from our research to develop a more consistent model, as including them would require the manual adjustment of inputs for that day every leap year. By excluding these data, we ensure uniformity across all years and enhance the model's generalization capability.