인공위성 운동역학 궤도방정식

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1. Hohmann Transfer

Hohmann 천이는 두 개의 원형궤도 사이를 접하는 타원궤도를 이용하는 것이다. 이 타원궤도의 장반경은 (a_1 + a_2) / 2로 주어진다. 천이 과정에서 속도를 순간적으로 증가시켜 타원궤도로 진입하고, 타원궤도의 apogee에서 다시 속도를 증가시켜 원하는 바깥쪽 원형궤도에 진입한다. 이때 각각의 순간에 필요한 속도 증가량은 계산할 수 있다. Hohmann 천이에 소요되는 시간은 타원궤도의 주기의 반과 같다.
2. Inclination-Change Maneuver

위성의 운동상태 변경 중 가장 에너지가 많이 소모되는 것 중 하나는 궤도면의 변경이다. 두 개의 같은 주기를 가지는 원형궤도 사이의 천이 과정에서 필요한 속도 변화는 두 속도 벡터의 차이로 주어지며, 이는 두 궤도 사이의 경사각 차이에 따라 크게 달라진다. 경사각 변화가 큰 경우 실현이 어려우므로, 지구탈출궤도를 거쳐 다시 원하는 궤도로 진입하는 방법을 사용할 수 있다.
3. Launch to Rendezvous

이미 궤도상에 있는 물체에 접근하기 위해서는 발사 시간을 잘 선택할 필요가 있다. 처음부터 접근 대상과 같은 경사각을 가지는 궤도로 진입하는 것이 효율적이다. 이를 위해서는 접근 대상의 궤도가 발사점의 상공을 지나는 순간에 발사하는 것이 가장 좋다. 이때 발사장의 위도, 접근 대상의 궤도 경사각 등을 고려하여 발사 시간을 계산할 수 있다.
4. Relative Motion and Rendezvous

두 물체 사이의 상대운동을 분석하는 Clohessy-Wiltshire 방정식을 이용하면 랑데부 과정을 쉽게 이해할 수 있다. 이 방정식을 통해 상대 위치와 속도를 시간의 함수로 나타낼 수 있다. 이를 이용하면 초기 조건에서 목표 위치까지 도달하는데 필요한 상대 속도를 계산할 수 있다.
5. Decay Lifetime

저궤도 위성은 공기 저항으로 인해 고도가 점점 낮아져 결국 지구로 떨어지게 된다. 이때 공기 밀도 모델을 이용하여 위성의 수명을 예측할 수 있다. 초기 고도를 높이면 수명이 크게 늘어나지만, 정확한 추락 시간 및 궤도 예측은 어렵다.
6. Earth-Oblateness Effects

지구의 비구형성으로 인해 위성 궤도에 영향을 미치는 효과가 있다. 이로 인해 궤도면이 회전하는 regression of the nodes 현상과 perigee가 회전하는 advance of the perigee 현상이 발생한다. 이를 이용하여 sun-synchronous 궤도와 critical inclination 궤도를 만들 수 있다.
7. Low-Thrust Orbit Transfer

화학 로켓과 달리 추력이 작은 전기 추진 엔진을 사용하여 천이하는 경우, 천이 시간은 길어지지만 연료 소모가 적다. 이때 천이 시간과 속도 변화량을 계산할 수 있다. 추력이 너무 작으면 Van Allen 벨트 통과 시간이 길어져 위험할 수 있으므로 적절한 추력 선택이 필요하다.

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1. Hohmann Transfer

The Hohmann transfer is a fundamental concept in orbital mechanics and spacecraft trajectory design. It describes the most efficient way to transfer between two circular orbits around a central body, such as the Earth or another planet. The Hohmann transfer involves using a two-burn maneuver to transition from the initial orbit to the target orbit, minimizing the required delta-V (change in velocity) and thus the fuel consumption. This makes it a widely used technique in spacecraft operations, satellite deployment, and interplanetary missions. The simplicity and efficiency of the Hohmann transfer make it a crucial tool for mission planners and spacecraft engineers, allowing them to design cost-effective and fuel-optimal trajectories. Understanding and applying the Hohmann transfer is essential for understanding the fundamentals of orbital mechanics and spacecraft dynamics.
2. Inclination-Change Maneuver

The inclination-change maneuver is an important orbital maneuver used to adjust the inclination of a spacecraft's orbit. This is particularly relevant for missions that require the spacecraft to operate in a specific orbital plane, such as Earth observation satellites, communication satellites, or interplanetary probes. The inclination-change maneuver typically involves a series of thrust burns that change the orientation of the spacecraft's orbit relative to the reference plane, such as the equatorial plane or the ecliptic plane. The amount of delta-V required for this maneuver depends on the initial and target inclinations, as well as the spacecraft's orbital parameters. Careful planning and execution of inclination-change maneuvers are crucial for maintaining the desired orbital configuration and ensuring the success of the mission. Understanding the principles and techniques of inclination-change maneuvers is essential for spacecraft mission design and operations.
3. Launch to Rendezvous

The launch to rendezvous process is a critical aspect of spacecraft operations, particularly in missions involving the docking or berthing of spacecraft, such as in the case of crewed spacecraft visiting the International Space Station or cargo resupply missions. This process involves the precise timing and coordination of the launch of the spacecraft, its trajectory, and the rendezvous with the target spacecraft or station. It requires a deep understanding of orbital mechanics, including the calculation of transfer orbits, phasing, and the timing of maneuvers to ensure a successful and safe rendezvous. The launch to rendezvous process also involves complex guidance, navigation, and control systems to precisely monitor and adjust the spacecraft's trajectory throughout the mission. Mastering the launch to rendezvous process is essential for the success of a wide range of space missions, from human spaceflight to robotic exploration, and it continues to be an area of active research and development in the field of spacecraft engineering.
4. Relative Motion and Rendezvous

Relative motion and rendezvous are fundamental concepts in spacecraft dynamics and mission design. Understanding the relative motion between two spacecraft, or between a spacecraft and a celestial body, is crucial for a wide range of space operations, such as docking, proximity operations, and formation flying. The study of relative motion involves the analysis of the relative position, velocity, and acceleration between the two objects, taking into account factors like orbital mechanics, perturbations, and control systems. Rendezvous, on the other hand, is the process of bringing two spacecraft together, often with the goal of docking or berthing. This requires precise trajectory planning, guidance, and control to ensure a safe and successful rendezvous. The ability to accurately model and predict relative motion, as well as to execute rendezvous maneuvers, is essential for the success of many space missions, from human spaceflight to robotic exploration and on-orbit servicing. Continued research and development in this area are crucial for advancing the capabilities of space systems and enabling new and more complex space operations.
5. Decay Lifetime

The decay lifetime of an object in orbit is an important consideration in spacecraft and mission design. Decay lifetime refers to the amount of time it takes for an object, such as a satellite or spacecraft, to re-enter the Earth's atmosphere and burn up or land on the surface due to atmospheric drag and other perturbations. Understanding and accurately predicting the decay lifetime of an orbiting object is crucial for a variety of reasons. It helps mission planners ensure that spacecraft and satellites remain in their intended orbits for the required duration of the mission, and it also informs decisions about end-of-life disposal and deorbiting procedures. Accurate decay lifetime models also contribute to the management of the space environment, as they help track and predict the re-entry of space debris, which is an increasing concern for the safety of space operations. Continued research and development in the field of orbital decay and lifetime prediction is essential for the sustainable and responsible use of space.
6. Earth-Oblateness Effects

The Earth's oblateness, or deviation from a perfect sphere, is an important factor in the study of orbital mechanics and spacecraft dynamics. The Earth's oblate shape, caused by its rotation, has a significant impact on the motion of objects in Earth orbit. These Earth-oblateness effects can lead to perturbations in the orbits of satellites and spacecraft, causing changes in their inclination, eccentricity, and other orbital parameters over time. Understanding and accurately modeling these effects is crucial for mission planning, satellite operations, and the prediction of long-term orbital behavior. Accounting for Earth-oblateness effects is particularly important for missions that require precise orbital control, such as Earth observation, navigation, and communication satellites. Continued research and development in the modeling and simulation of Earth-oblateness effects, as well as the incorporation of these effects into spacecraft design and operations, are essential for ensuring the success and longevity of space missions.
7. Low-Thrust Orbit Transfer

Low-thrust orbit transfer is an important concept in spacecraft propulsion and mission design. It involves the use of low-thrust propulsion systems, such as electric propulsion or solar electric propulsion, to gradually change a spacecraft's orbit over an extended period of time. This approach is in contrast to the more traditional high-thrust chemical propulsion systems, which provide large, discrete changes in velocity. Low-thrust orbit transfer offers several advantages, including improved fuel efficiency, the ability to perform complex maneuvers, and the potential for longer-duration missions. However, it also presents challenges in terms of trajectory planning, guidance and control, and the management of the spacecraft's power and thermal systems. Understanding the principles and techniques of low-thrust orbit transfer is crucial for the design and operation of a wide range of space missions, from station-keeping and orbit-raising of satellites to interplanetary and deep-space exploration. Continued research and development in this area will help unlock new capabilities and enable more ambitious and cost-effective space missions in the future.

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