Comparison of the SRP spherical model between LEO and GEO satellites

Solar Radiation Pressure (SRP) is a phenomenon caused by the pressure exerted by solar photons on a satellite’s surface when it is exposed to sunlight. It is a form of radiation force and can significantly impact the motion and behaviour of satellites in space. SRP influences a satellite’s orbit by causing changes in its semimajor axis, eccentricity, inclination, argument of perigee, right ascension of the ascending node, and mean anomaly. SRP models are used to simulate the effects of solar radiation pressure on satellites. These models are essential for accurately predicting satellite trajectories and orbital behaviour. There are several types of SRP models, such as spherical model, flat model, box-wing model, faceted model, and analytical SRP models. This research focuses on Telkom 1 and LAPAN A1 satellites, both belonging to Indonesia and positioned in Geostationary Earth Orbit (GEO) and Low Earth Orbit (LEO) orbits, respectively. The study aims to find a comparison of the effects of SRP spherical model on LEO and GEO satellites. Our modelling shows that the semimajor axis and eccentricity are sensitive to SRP, while the inclination and right ascension of the ascending node are not significantly affected. Comparing the effects of SRP on LEO and GEO satellites, we concluded that both LEO and GEO orbit experience the most significant fluctuations in January (perihelion), likely due to the influence of solar radiation pressure.


Introduction
Solar Radiation Pressure (SRP) is the force exerted by sunlight on satellites and other objects in space due to the transfer of momentum from solar photons [1].When photons from the sun hit a satellite's surface, they create a tiny but continuous pressure, influencing the satellite's motion and orbit.SRP occurs when sunlight collides with a satellite, transferring momentum to the satellite's surface [2].It remains a constant force acting on the satellite as long as it is exposed to sunlight.The strength of SRP depends on the size and reflective properties of the satellite's surface; larger and more reflective surfaces experience a stronger SRP.One of the primary effects of SRP is its influence on the orbits of satellites.Solar photons striking the satellite's surface create pressure, leading to changes in the satellite's orbital parameters over time.SRP affects a satellite's orbit by causing changes in its semimajor axis, eccentricity, inclination, argument of perigee, right ascension of the ascending node, and mean anomaly [3].Consequently, satellites in specific orbits, such as Geostationary Earth Orbit (GEO), require regular adjustments known as station-keeping maneuvers.These maneuvers are essential to counteract the effects of SRP and maintain the satellite's precise position and functionality.

Spherical model for GEO satellites
GEO satellites, positioned approximately 35,786 kilometers above the equator, have an orbital period matching Earth's rotational period, making them appear stationary relative to the Earth's surface.This stationary nature is ideal for applications requiring continuous coverage.However, maintaining their specific orbital slots poses a challenge.The continuous pressure from SRP can cause gradual changes in orbital parameters, necessitating station-keeping maneuvers using onboard thrusters to counteract these effects.These maneuvers are crucial to prevent drifting and ensure satellites remain within their designated positions.Moreover, GEO satellites experience significant thermal variations due to prolonged exposure to sunlight.As the satellite moves through Earth's shadow, rapid temperature changes can affect the satellite's structure and materials, influencing how SRP is absorbed and reflected, leading to variations in the applied forces.Hence, accurate thermal modeling is essential for understanding these effects [6].
In the context of GEO satellites, the spherical model offers a fundamental approximation of SRP effects.This model assumes the satellite's surface uniformly absorbs or reflects all incoming solar radiation, simplifying the calculations related to radiation pressure.It estimates basic orbital parameter changes due to SRP, including variations in the satellite's semimajor axis, eccentricity, inclination, argument of perigee, right ascension of the ascending node, and mean anomaly.While the spherical model does not capture all the intricacies of a real GEO satellite's behavior, it provides a foundational understanding of how solar radiation affects these parameters [7].
During the initial stages of mission planning, engineers often use the spherical model to estimate the perturbations caused by SRP.By understanding the basic effects of solar radiation, mission planners can make initial estimations about the satellite's orbital behavior.Additionally, approximating SRP effects using the spherical model enables engineers and mission planners to anticipate potential collisions.This knowledge facilitates the development of proactive collision avoidance strategies, ensuring GEO satellites maintain their precise positions and avoid space debris.However, real-world GEO satellites require more detailed simulations.Engineers conduct complex analyses, considering factors such as satellite shape, material properties, and orientation to create precise simulations.These simulations enable engineers to make accurate adjustments to satellite trajectories, compensating for SRP-induced changes and ensuring GEO satellites remain within their designated orbital parameters [8].
Despite the advantage of a fixed position relative to the Earth's surface, GEO satellites face unique challenges due to their location and the influence of gravitational and solar perturbations.The most significant challenge is the need for station-keeping, which involves periodic adjustments to the satellite's position and velocity.GEO satellites must maintain specific position in space to remain synchronized with the Earth's rotation.Gravitational perturbations from the Moon and the Sun, coupled with solar radiation pressure, can gradually shift the satellite from its designated orbital slot.To counteract these effects, satellites need to perform station-keeping maneuvers, adjusting thrusters to correct their position and ensuring they stay within acceptable orbital parameters.

Comparative Analysis
The SRP spherical model is used for both LEO and GEO satellites to assess its impact on orbital parameters such as semimajor axis, eccentricity, inclination, argument of perigee, right ascension of the ascending node, and mean anomaly.Before comparing the effects of SRP on LEO and GEO satellites, we will present a comparison of orbital parameters affected by SRP and those without SRP.These comparative plots will illustrate which orbital parameters are sensitive to SRP and which ones are not.In Figure 1, it is evident that the semimajor axis due to SRP has a narrower range compared to the semimajor axis without SRP.However, they exhibit a similar trend and periodicity.Moving on to the next orbital parameter, eccentricity, Figure 2 shows that the eccentricity due to SRP has larger values compared to the eccentricity without SRP.Both curves display a similar pattern of gradual changes, but the eccentricity due to SRP increases and then decreases to its starting value with small fluctuations.These changes occur periodically.Meanwhile, the eccentricity without SRP changes over time, displaying constant small fluctuations.In Figure 3, it can be observed that the inclination due to SRP has similar values to the inclination without SRP.Figure 4 also demonstrates that there is no difference between the right ascension of the ascending node due to SRP and that without SRP.Based on the fourth set of plots comparing orbital parameters affected by SRP and those unaffected, it can be concluded that semimajor axis and eccentricity are sensitive to SRP, while the inclination and right ascension of the ascending node are not significantly impacted.
The following plots will compare the effects of SRP on LEO and GEO satellites.These figures display multiple plots together in the same graph, called STL (Seasonal and Trend decomposition using Loess).This statistical technique decomposes time series data into three components: seasonality, trend, and residual.The trend component offers a broad direction for the overall data, while seasonality reveals regular and predictable patterns recurring at fixed intervals.The residual component represents random fluctuations or unpredictable changes.These figures are illustrated as follows:   displays the plot of the GEO satellite's semimajor axis, featuring the actual data, data trend, seasonality, and residual.The presence of two peaks in the season, occurring in January during perihelion, suggests a potential influence from the Sun or perturbations originating from the Sun.One likely explanation could be the effect of solar radiation pressure.A similar observation can be made for LEO satellites, as demonstrated in Figure 6.By decomposing the eccentricity data into these components, an STL plot offers valuable insights into the underlying patterns and trends in the satellite's orbital eccentricity.Researchers and scientists use this information to analyze the impact of external factors, such as solar radiation pressure or gravitational perturbations, on the satellite's orbital behavior and to make predictions about its future movements.These patterns and trends are illustrated in Figure 7 and Figure 8.In Figure 9 and Figure 10, the seasonal component represents the regular and predictable patterns that occur at fixed intervals of time.These patterns can be influenced by various factors, such as gravitational interactions with the Moon and the Sun or Earth's axial tilt.Detecting trends in inclination is vital for recognizing long-term changes in the satellite's orbital plane, which could be caused by external forces or perturbations.Residuals often caused by various factors, including short-term perturbations from other celestial bodies or irregularities in the satellite's orbit.By examining these components, researchers can gain insights into how external factors, such as solar radiation pressure or gravitational influences, impact the satellite's orbital inclination.

Conclusion
LEO satellites can have varying semimajor axis, inclinations, and eccentricities.The spherical model assists in understanding the impact of SRP-induced changes in these parameters, affecting the satellite's orbital plane and shape of the orbit.Meanwhile, GEO satellites are typically placed in equatorial orbits with low inclinations and nearly circular orbits.The spherical model helps in assessing minor eccentricity variations, ensuring the satellite remains within the desired orbital parameters.In summary, the differences in altitude, orbital period, inclination, and eccentricity between LEO and GEO satellites influence the frequency and precision of predictions made using the spherical model.While LEO applications focus on rapid, short-term changes, GEO applications emphasize long-term stability and periodic adjustments.
Based on our modelling, semimajor axis and eccentricity are sensitive to SRP, while the inclination and right ascension of the ascending node are not significantly impacted by SRP.When we compared the effects of SRP on LEO and GEO satellites, it was concluded that the semimajor axis, eccentricity, and inclination of both LEO and GEO have the biggest fluctuation in January (perihelion).This can be associated with the effect of solar radiation pressure.

Figure 1 .
Figure 1.Comparison of semimajor axis due to SRP (blue) and semimajor axis without SRP (red) over the course of a year.

Figure 2 .
Figure 2. Comparison of eccentricity due to SRP (blue) and eccentricity without SRP (red) over the course of a year.

Figure 3 .
Figure 3.Comparison of inclination due to SRP (blue) and inclination without SRP (red) over the course of a year.

Figure 4 .
Figure 4. Comparison of right ascension of the ascending node due to SRP (blue) and right ascension of the ascending node without SRP (red) over the course of a year.

Figure 5 .
Figure 5.The STL plot illustrates the GEO satellite's semimajor axis throughout the year (January-December 2004).

Figure 6 .
Figure 6.The STL plot illustrates the LEO satellite's semimajor axis throughout the year (January-December 2004).

10th 7 Figure 5
Figure5displays the plot of the GEO satellite's semimajor axis, featuring the actual data, data trend, seasonality, and residual.The presence of two peaks in the season, occurring in January during perihelion, suggests a potential influence from the Sun or perturbations originating from the Sun.One likely explanation could be the effect of solar radiation pressure.A similar observation can be made for LEO satellites, as demonstrated in Figure6.

Figure 7 .
Figure 7.The STL plot illustrates the GEO satellite's eccentricity throughout the year (January-December 2004).

Figure 8 .
Figure 8.The STL plot illustrates the LEO satellite's eccentricity throughout the year (January-December 2004).

Figure 9 .
Figure 9.The STL plot illustrates the GEO satellite's inclination throughout the year (January-December 2004).

Figure 10 .
Figure 10.The STL plot illustrates the LEO satellite's inclination throughout the year (January-December 2004).