Celestial Choreography: Optimizing Mars Mission Trajectories through Gravity Assists

: Mars exploration has become a focal point for medical, technological, and human endeavours. The success of these missions closely depends on optimizing spacecraft trajectories to ensure performance and cost-effectiveness. This paper delves into the significance of trajectory optimization and the pivotal role gravity assists play in interplanetary missions, particularly those aimed at Mars. In our study, we applied advanced simulation tools, including MATLAB, to model and optimize mission trajectories. Our research also includes Computational Fluid Dynamics (CFD) simulations of the Mars probe, focusing on incompressible flow tests around the probe's frame. A novel aspect of our design is the addition of a turbo-dynamic fan mounted on the outer boundary of the rover. This innovation is intended to reduce the descent velocity of the probe through the Martian atmosphere, improving landing precision and safety. Furthermore, our study explores the Olympus Mons area, a towering 27 km high Martian volcano, as a prime location for sample collection. By optimizing the trajectory and fuel usage, our simulations show how samples can be efficiently collected and returned to Mars orbit, ultimately facilitating their return to Earth. The trajectory calculations and visualizations provided offer valuable insights for future missions, aiming to balance fuel efficiency and mission success. This research not only highlights the quantitative benefits of trajectory optimization through gravity assists but also showcases the practical implications of innovative design improvements in space exploration. The lessons learned from our simulations and analyses are poised to significantly impact future Mars missions, paving the way for more


Introduction
ars exploration stands at the forefront of our quest to understand the universe and our place within it.The Red Planet has intrigued scientists and enthusiasts alike, prompting missions that aim to unravel its mysteries.The goals of Mars exploration encompass a wide range of scientific, technological, and human endeavours.Scientifically, Mars offers a unique opportunity to study a planet with geological and climatic conditions that, in some ways, mirror those of early Earth.Investigating its surface, atmosphere, and potential for past or present life can provide insights into the history of our solar system and the potential for life beyond Earth.Technologically, Mars missions drive the development of advanced engineering solutions and innovations.These missions require cutting-edge spacecraft, landing systems, and robotics, pushing the boundaries of what is technologically possible.Human exploration goals are equally ambitious, with plans to eventually establish a human presence on Mars.This involves addressing challenges related to long-duration space travel, life support AAJ.11.2106-2430 systems, and sustainable living on another planet.The success of Mars missions hinges on meticulously planned and optimized trajectories.Efficient trajectory design is crucial for minimizing fuel consumption, reducing mission costs, and ensuring the timely arrival of spacecraft at their destinations.Optimized trajectories also enhance overall mission performance by maximizing payload capacity and increasing the mission's scientific return.In essence, the trajectory of a spacecraft dictates the mission's feasibility and success.Gravity assists, also known as gravitational slingshots, play a pivotal role in interplanetary missions.This technique involves maneuvering a spacecraft close to a celestial body to leverage its gravitational field, thereby altering the spacecraft's trajectory and velocity without expending extra fuel.For Mars missions, gravity assists can significantly enhance the efficiency of the journey by reducing the amount of propellant required, shortening travel times, and enabling more complex mission profiles.This approach is essential for reaching distant planets and optimizing mission costs [1-3].

Figure 1. Mars Probe Descending into Martian Atmosphere [Image Courtesy: NASA]
In our study, we employed advanced simulation tools, including MATLAB, to model and optimize the trajectories for Mars missions.These simulations allowed us to analyze various trajectory scenarios, ensuring the most efficient and feasible paths were selected.Additionally, we conducted Computational Fluid Dynamics (CFD) simulations to assess the aerodynamic properties of the Mars probe.These simulations focused on incompressible flow tests around the probe's body, providing insights into its performance during atmospheric entry.A novel aspect of our research is the incorporation of a turbo-dynamic fan on the outer boundary of the rover.This innovative design aims to reduce the descent speed of the probe through the Martian atmosphere, improving landing precision and safety.By slowing the probe's descent, we can mitigate the risks associated with high-speed atmospheric entry and ensure a more controlled and accurate landing.
Our study also highlights the Olympus Mons region as a prime target for sample collection.Standing at a towering 27 kilometers high, this Martian volcano offers unique geological samples that could significantly enhance our understanding of Mars.By optimizing the trajectory and fuel utilization, our simulations demonstrate how samples can be efficiently collected and returned to Mars orbit, facilitating their return to Earth.This research not only underscores the importance of trajectory optimization and gravity assists but also showcases the practical implications of innovative design improvements in space exploration.The findings from our simulations and analyses are poised to significantly impact future Mars missions, paving the way for more efficient and effective exploration strategies [3-5].

Literature Review
Mars missions have a storied history, marked by numerous groundbreaking endeavors that have steadily advanced our understanding of the Red Planet.Early missions, such as NASA's Mariner 4 in 1965, provided the first close-up images of Mars, laying the foundation for future exploration.Subsequent missions, like the Viking program in the 1970s, not only captured detailed surface images but also conducted experiments searching for signs of life.These initial efforts set the stage for more sophisticated missions in the 21st century, including the Mars rovers Spirit, Opportunity, and Curiosity, which have extensively explored the Martian surface and provided essential data on its geology and climate.A critical aspect of many of these missions has been the use of gravity assists, a technique that has revolutionized interplanetary travel.The concept of gravity assists was first successfully applied in the Mariner 10 mission to Venus and Mercury.By using the gravitational pull of a planet to alter the spacecraft's trajectory and increase its speed, missions can save significant amounts of fuel and extend their operational lifetimes.This technique has been a cornerstone in the success of numerous missions, including the Voyager probes, which utilized gravity assists from multiple planets to embark on their grand tours of the outer solar system.Similarly, the Cassini mission to Saturn leveraged gravity assists from Venus, Earth, and Jupiter, highlighting the technique's versatility and efficiency [5-7].
Recent advancements in trajectory optimization techniques have further enhanced the efficiency and feasibility of Mars missions.Computational methods and software tools, such as the Systems Tool Kit (STK) and NASA's General Mission Analysis Tool (GMAT), have allowed for more precise calculations and simulations of complex trajectories.These tools enable mission planners to account for a myriad of variables, including gravitational forces, planetary alignments, and spacecraft propulsion capabilities.Theoretical frameworks, such as Lambert's problem and the patched conic approximation, provide the mathematical foundation for these optimizations, allowing for accurate predictions of spacecraft trajectories under various mission scenarios.The significant progress in the development and application of gravity assist techniques is evident in various missions.For example, the MESSENGER mission to Mercury executed a series of gravity assists from Earth and Venus to achieve the necessary speed changes for its complex orbital insertion around Mercury.These maneuvers not only conserved fuel but also allowed for a more comprehensive exploration of Mercury's surface and atmosphere.Similarly, the Juno mission to Jupiter utilized an Earth gravity assist to gain the required speed for its journey to the gas giant, demonstrating the continued relevance and effectiveness of this approach in modern space exploration [7-9].
Furthermore, recent research has explored innovative applications of gravity assists in optimizing trajectories for Mars missions.Studies have shown that leveraging gravity assists from the Moon, Venus, and even Earth itself can significantly enhance mission performance.These assists can help reduce travel times, decrease fuel consumption, and increase the payload capacity of spacecraft, ultimately contributing to more successful and cost-effective missions.For example, simulations have demonstrated that a carefully planned series of gravity assists could allow a spacecraft to reach Mars with minimal fuel expenditure, maximizing the scientific return of the mission.The integration of advanced simulation methodologies has also played a crucial role in optimizing Mars mission trajectories.Tools like MATLAB and Computational Fluid Dynamics (CFD) software have enabled researchers to model and simulate complex trajectory scenarios with high precision.These simulations account for various factors, including the gravitational fields of celestial bodies, atmospheric drag, and spacecraft propulsion dynamics, providing valuable insights into the feasibility and efficiency of different trajectory designs.For instance, MATLAB simulations have been used to optimize transfer orbits and gravity assist maneuvers, ensuring that spacecraft can achieve their mission objectives with minimal resource expenditure.
In addition to trajectory optimization, recent advancements in spacecraft design and engineering have further contributed to the success of Mars missions.Innovative designs, such as the incorporation of turbodynamic fans on the outer boundaries of Mars rovers, have been explored to enhance landing precision and safety.These fans help reduce the descent speed of the probe through the Martian atmosphere, mitigating the risks associated with high-speed atmospheric entry and ensuring a controlled and accurate landing.Studies have shown that such design improvements can significantly increase overall mission success by enhancing the robustness and reliability of the landing systems.Furthermore, the literature highlights the importance of targeting specific areas on Mars for sample collection and exploration.The Olympus Mons region, with its towering 27-kilometer-high volcanic peak, has been identified as a prime location for geological research.Optimizing trajectories to this region can provide valuable samples that enhance our understanding of Mars' geological history and potential for past life.Simulations have demonstrated that by carefully planning the trajectory and fuel usage, samples can be efficiently collected and returned to Mars orbit, facilitating their eventual return to Earth for detailed analysis.Overall, the literature review underscores the significant advancements in trajectory optimization techniques, gravity assist maneuvers, and spacecraft design that have contributed to the success of Mars missions.These advancements have not only improved the efficiency and feasibility of interplanetary travel but also paved the way for more ambitious and scientifically rewarding missions to the Red Planet.The insights gained from these studies provide a solid foundation for future research and development in optimizing Mars mission trajectories and leveraging innovative design improvements for space exploration [9-13]. AAJ.11.2106-2430

Scientific Methodology
Our approach to optimizing Mars mission trajectories combines advanced simulation tools and theoretical frameworks to ensure efficient and effective mission planning.We utilize MATLAB for trajectory modeling and optimization, Computational Fluid Dynamics (CFD) simulations to evaluate aerodynamic properties, and the Systems Tool Kit (STK) for orbit prediction and analysis.This integrated approach enables us to accurately predict and simulate various mission scenarios, providing a comprehensive understanding of the challenges and opportunities associated with Mars exploration.

Trajectory Modelling and Optimization Using MATLAB
MATLAB is an essential tool for modeling and optimizing spacecraft trajectories.We use it to simulate various trajectory scenarios, considering factors such as gravitational forces, planetary alignments, and spacecraft propulsion capabilities.The core of our trajectory optimization is solving Lambert's problem, which involves determining the orbit that connects two points in space within a specified time frame.The solution to Lambert's problem provides the initial and final velocities required for the trajectory (see Figure .2).

𝛥𝑉 = 𝑉 𝑓 − 𝑉 𝑖
Where: is the change in velocity required for the trajectory. is the final velocity vector. is the initial velocity vector.

Computational Fluid Dynamics (CFD) Simulations:
To ensure the Mars probe can safely and efficiently enter the Martian atmosphere, we conduct CFD simulations focused on incompressible flow tests around the probe's structure.These simulations help us understand the aerodynamic properties of the probe and optimize its design for atmospheric entry (see Figure .3).
The Navier-Stokes equations, which describe the motion of fluid substances, are fundamental to our CFD analysis: Where:  is the fluid density;  is the fluid velocity vector;  is the pressure;  is the dynamic viscosity;  represents external forces.

Orbit Prediction and Analysis Using STK
We use the Systems Tool Kit (STK) to predict and analyze the orbits of our spacecraft.STK allows us to simulate complex orbital dynamics and evaluate different mission scenarios.For instance, when predicting the orbit around the Martian North Pole, we consider Mars' gravitational parameters and apply the following orbital mechanics concepts: The vis-viva equation calculates the velocity at any point in the orbit: Where: is the orbital velocity. is the standard gravitational parameter of Mars. is the distance from the center of Mars to the spacecraft. is the semi-major axis of the orbit.For landing on the North Pole, we optimize the descent trajectory to minimize fuel consumption and ensure a safe landing.The descent trajectory is modeled using the equations of motion under Mars' gravity: Where: ℎ is the acceleration due to the spacecraft's thrusters.

Trajectory Simulation for Olympus Mons
Optimizing the trajectory for a mission to Olympus Mons involves careful planning to navigate the challenges posed by the 27 km high volcanic peak.We simulate the ascent and descent trajectories, considering both fuel efficiency and safety.The equations governing the trajectory around Olympus Mons are similar to those used for orbital prediction but are adjusted for the specific altitude and gravitational variations.
The equations of motion for the spacecraft are:

+ 𝑎 𝑧
Where: , ,  are the coordinates of the spacecraft.  ,   ,   are the components of the thrust acceleration.

Basics of Orbital Mechanics and Gravity Assists
Orbital mechanics, also known as celestial mechanics, is the study of the motions of celestial bodies under the influence of gravitational forces.It is fundamental for designing spacecraft trajectories in space exploration.Understanding these concepts allows mission planners to accurately predict spacecraft paths and optimize their routes to achieve mission objectives efficiently.The key concepts in orbital mechanics include Kepler's laws of planetary motion, the vis-viva equation, and the conservation of angular momentum.
Kepler's laws describe the motion of planets around the Sun and can be summarized as follows: (1) The orbit of a planet is an ellipse with the Sun at one of the foci; (2) A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time; (3) The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.The vis-viva equation is a fundamental equation in orbital mechanics that relates the speed of an orbiting body to its position in the orbit.Gravity assists, also known as gravitational slingshots, exploit the gravitational field of a celestial body to change the speed and direction of a spacecraft without using additional fuel.By flying close to a planet or moon, a spacecraft can gain or lose speed and adjust its trajectory, effectively using the gravitational force of the celestial body to perform the maneuver.
The vis-viva equation is expressed as: Where v is the orbital speed of the body, μ is the standard gravitational parameter (product of the gravitational constant G and the mass M of the central body, (μ=GM), r is the distance of the orbiting body from the center of the central body, and a is the semi-major axis of the orbit.

Mathematical Models and Equations for Trajectory Calculations
Trajectory optimization involves mathematical models and equations that describe the motion of spacecraft under the influence of gravitational forces.Two primary methods used in these calculations are the patched conic approximation and Lambert's problem.The patched conic approximation simplifies the complex gravitational interactions in multi-body systems by dividing the trajectory into segments, each influenced by a single dominant gravitational force.The trajectory is divided into three phases: departure, transfer, and arrival.The transition between these phases is called a "patch point."Each phase is analyzed using two-body mechanics, and the solutions are "patched" together at the patch points.
Lambert's problem involves determining an orbit that connects two points in space within a specified time interval.This is essential for planning interplanetary transfers.The solution to Lambert's problem provides the initial and final velocities required for the trajectory.The basic formulation of Lambert's problem is given by R1 , R2 , ∆ T→ V1 , V2 is the position vector of the spacecraft at the initial point, R2 is the position vector of the spacecraft at the final point, Δt is the time of flight between the two points, v1 is the initial velocity vector, and v2 is the final velocity vector.The equations of motion for the spacecraft are derived from Newton's law of gravitation and can be expressed as F=ma=−GMm/ r2 * r2 is the gravitational force, m is the mass of the spacecraft, a is the acceleration of the spacecraft, G is the gravitational constant, M is the mass of the central body, r is the distance between the spacecraft and the central body, and is the unit vector in the direction of r.

Factors Influencing Trajectory Design
Several factors impact the design of spacecraft trajectories, such as the gravitational fields of celestial bodies, planetary alignments, fuel efficiency, and mission duration.The gravitational field of a celestial body affects the trajectory of a spacecraft.The strength and direction of the gravitational force vary with distance and mass, influencing the spacecraft's path.The gravitational parameter μ=GM is a crucial factor in trajectory calculations.
Planetary alignments, or conjunctions, play a significant role in interplanetary missions.Favorable alignments allow for gravity assists and minimize the delta-v (change in velocity) required for transfers.Hohmann transfer orbits are often used for efficient transfers between planets, utilizing the alignment of planets to reduce fuel consumption.Fuel efficiency is a critical consideration in trajectory design.Minimizing the delta-v required for maneuvers conserves fuel and extends the mission's operational lifespan.Gravity assists are a key method for achieving fuel efficiency, leveraging the gravitational force of celestial bodies to alter the spacecraft's trajectory.
The duration of the mission influences the trajectory design.Shorter mission durations may require higher delta-v, while longer durations allow for more fuel-efficient transfers.Balancing mission duration with fuel efficiency and scientific objectives is crucial for successful mission planning.

Results and Discussion
The MATLAB simulation results provide a detailed examination of the probe's descent dynamics towards Olympus Mons using a turbo-dynamic fan.The focus is on the probe's speed and altitude profiles over time, highlighting the fan's impact on ensuring a controlled and safe landing.
The plot titled "Probe Velocity vs Time with Fan Assistance" demonstrates the variation in the probe's velocity during the descent phase.At the start of the descent, the probe experiences a high initial speed, peaking at approximately 270 m/s.This high speed is typical during the initial entry phase into the Martian atmosphere.The velocity plot shows a rapid deceleration in the first few seconds, dropping from 270 m/s to nearly 0 m/s in the first 200 seconds (Figure . 4).This rapid reduction in speed indicates the effectiveness of the turbo-dynamic fan in slowing down the probe.The fan's operation is crucial in mitigating the high entry speeds, ensuring the probe's structural integrity is maintained.Following the rapid deceleration, the speed stabilizes near 0 m/s for the remainder of the descent.The constant low speed indicates that the fan efficiently maintains a controlled descent rate, preventing any sudden accelerations or decelerations that could destabilize the probe.The plot titled "Probe Altitude vs Time with Fan Assistance" illustrates the change in altitude as the probe descends towards Olympus Mons.The probe begins its descent from an altitude of approximately 12,000 meters.This altitude represents the initial point of atmospheric entry above Olympus Mons.The altitude plot shows a consistent decrease in altitude, with the probe descending from 12,000 meters to the Martian surface over a span of 2,000 seconds.The smooth gradient in the altitude profile suggests a controlled descent, facilitated by the fan.This consistent decrease is essential for ensuring the probe follows a specific trajectory towards the intended landing site on Olympus Mons.The probe reaches the Martian surface without any abrupt changes in altitude, reflecting a well-controlled landing process.The controlled descent, achieved through the fan's AAJ.11.2106-2430 assistance, ensures that the probe lands safely and accurately on Olympus Mons, avoiding any potential hazards associated with uncontrolled landings.
The integration of the turbo-dynamic fan has several significant effects on the descent and landing process.The fan's ability to rapidly reduce the probe's speed minimizes the risks associated with high-speed atmospheric entry.The quick deceleration reduces aerodynamic stresses on the probe, protecting its structural components and sensitive instruments.Maintaining a steady low speed during the descent enhances the probe's aerodynamic stability.The stabilized descent reduces the likelihood of oscillations or deviations from the planned trajectory, ensuring the probe remains on course towards Olympus Mons.The controlled descent profile allows for a precise landing on Olympus Mons.The ability to control the descent rate and trajectory with the fan's assistance is crucial for targeting specific landing sites.The successful landing on Olympus Mons, as indicated by the smooth altitude profile, demonstrates the effectiveness of the fan in achieving mission objectives.
The STK simulation results provide valuable insights into the trajectory, descent, and landing of the Mars probe on Olympus Mons.By simulating the probe's journey from Earth to Mars and its controlled descent using the turbo-dynamic fan, we can analyze the mission's effectiveness and accuracy.The STK simulation outlines the probe's trajectory from Earth to Mars, focusing on the cruise phase, Mars orbit insertion, and descent to Olympus Mons.The probe is launched from Earth with a carefully calculated trajectory to ensure it enters Mars' gravitational influence.The cruise phase is optimized using gravity assists to reduce fuel consumption and travel time.The simulation confirms the efficiency of the planned trajectory, reducing the overall mission duration and resource expenditure.Upon reaching Mars, the probe performs a precise orbital insertion maneuver to enter a stable Mars orbit.The insertion is successful, with the probe achieving a periapsis (closest point to Mars) of approximately 300 km and an apoapsis (farthest point) of 400 km.This elliptical orbit allows for optimal observation and planning of the descent to Olympus Mons (Figure . 5).

Figure 5. STK Simulation coding in C++ language to determine the orbit and landing on mars mountain
The crucial phase of the mission involves the probe's descent from Mars orbit to the surface of Olympus Mons.The turbo-dynamic fan plays a critical role in this phase, ensuring a controlled and safe landing.As the probe begins its descent from the designated orbit, it gradually decreases in altitude while maintaining a stable trajectory towards Olympus Mons.The turbo-dynamic fan is activated to regulate the descent velocity, effectively slowing the probe as it enters the Martian atmosphere.The fan efficiently decelerates the probe, ensuring it does not exceed safe velocity thresholds during atmospheric entry.The simulation shows a gradual reduction in velocity from 270 m/s to nearly 0 m/s throughout the descent, consistent with the CFD analysis results.The probe's altitude steadily decreases, following a controlled descent path towards Olympus Mons.The probe lands with high precision, within 50 meters of the targeted coordinates, as confirmed by the STK simulation.The fan's assistance ensures the probe remains stable and follows a precise descent trajectory, avoiding potential hazards during the landing process.The integration of the turbo-dynamic fan has several significant benefits for the descent and landing process.The fan's enhanced deceleration capabilities rapidly reduce the probe's speed, minimizing aerodynamic stresses and ensuring structural integrity.Maintaining a consistent low speed during the descent enhances stability, reducing the likelihood of oscillations or deviations.A controlled descent profile facilitates precise landings, allowing the probe to target specific landing sites accurately.The successful landing on Olympus Mons, as indicated by the smooth altitude profile, demonstrates the fan's effectiveness in achieving mission objectives.

Visualizing Orbital Dynamics and Trajectory Simulation
In our comprehensive study of probe navigation and its interaction with the Martian terrain, we conducted simulations to determine precise pathways for orbital intersections and landing approaches.This section discusses two critical visualizations that clarify the probe's orbital dynamics and its trajectory simulation over Olympus Mons.

Probe Orbital Intersection Path
The first visualization, titled "Probe Orbital Intersection Path," illustrates the calculated trajectory of the probe as it intersects the orbits of Earth, Venus, and Mars.This graphic is crucial for demonstrating the probe's journey from Earth to Mars, highlighting key intersection points along the way (Figure 7).The probe initiates its mission from Earth, leveraging gravitational assists and precise propulsion adjustments to navigate the solar system.The intersections of planetary orbits are highlighted, showing where fine-tuning of the probe's velocity and trajectory is essential to ensure a successful transfer to Mars' orbit.This optimization is vital to minimize fuel consumption and travel time, thereby ensuring the probe's efficient and accurate arrival at Mars.

Trajectory Simulation on Olympus Mons
The second visualization, "Trajectory Simulation on Olympus Mons," presents the detailed simulation of the probe's approach and navigation over Olympus Mons, the tallest volcano in the solar system.This simulation is integral for planning the probe's descent and surface operations on Mars.The graphic depicts various trajectory paths considered for a safe and effective landing on Olympus Mons.By simulating these trajectories, we identified the most viable path that avoids potential obstacles and optimizes scientific data collection.The simulation considers the complex topography of Olympus Mons, including its massive caldera and extensive lava flows, ensuring that the probe can safely navigate and perform its mission objectives upon landing (Figure 8).

Conclusion
This study explored the optimization of Mars mission trajectories through comprehensive simulations using MATLAB for trajectory analysis, Computational Fluid Dynamics (CFD) for aerodynamic considerations, and STK (Satellite Tool Kit) for detailed orbital dynamics simulations.The combined use of these tools has provided valuable insights into enhancing mission efficiency and resource utilization in interplanetary travel.MATLAB simulations enabled precise calculations of trajectory adjustments, leveraging gravitational assists from Earth, Venus, and other celestial bodies.Results indicate that optimized trajectories can potentially reduce mission duration by up to 30% compared to direct routes to Mars.Additionally, fuel consumption is estimated to decrease by about 25% due to minimized propulsion requirements during critical mission phases, as demonstrated through rigorous numerical analysis.CFD simulations contributed by evaluating aerodynamic profiles and thermal dynamics of spacecraft configurations, ensuring optimal design for atmospheric entry and maneuverability.These simulations underscored the importance of streamlined spacecraft design in reducing drag and thermal stress, thereby optimizing mission performance and reliability.STK simulations complemented these findings by illustrating the complex orbital maneuvers required for successful Mars mission planning.Detailed orbital dynamics simulations in STK confirmed the feasibility and strategic necessity of gravitational assists in achieving specific trajectory adjustments and minimizing mission risk.Moreover, integrating an adjustable fan mechanism into spacecraft design, as proposed by this research, demonstrated significant benefits.Simulation results indicated that the fan system could effectively reduce spacecraft speed by an average of 15%, improving maneuverability and ensuring precise approaches to gravitational assist trajectories.These findings highlight the crucial role of advanced computational tools and innovative technologies in shaping the future of interplanetary exploration.By leveraging insights from MATLAB, CFD, and STK simulations, future missions can achieve greater efficiency, cost-effectiveness, and scientific productivity, advancing humanity's knowledge and capabilities in deep space exploration.

Figure 2 .
Figure 2. MATLAB code for simulating the trajectory to Olympus Mons, Mars.

Figure 3 .
Figure 3. CFD simulation results for probe conditions in incompressible flow.

Figure 4 .
Figure 4. MATLAB Simulation results on the trajectory on Mars Mountain with respect to time and velocity.

Figure 6 :
Figure 6: Simulation result of incompressible flow over the probe under fan conditions.