Glossary

11.5 Orbital maneuvers and transfers

5 min read•august 21, 2024

Orbital maneuvers and transfers are crucial concepts in spacecraft dynamics. They involve applying Newton's laws to plan efficient space missions, predict satellite positions, and design interplanetary trajectories. Understanding these principles allows engineers to optimize fuel usage and mission timelines.

From Hohmann transfers to gravity assists, various techniques offer different trade-offs between fuel efficiency and transfer time. Mastering these methods enables spacecraft engineers to design optimal trajectories for specific mission constraints, balancing factors like requirements and transfer duration.

Fundamentals of orbital mechanics

  • Orbital mechanics forms the foundation of spacecraft dynamics in Engineering Mechanics, applying Newton's laws of motion and gravitation to celestial bodies
  • Understanding orbital mechanics enables engineers to design efficient space missions, predict satellite positions, and plan interplanetary trajectories
  • Key concepts include gravitational forces, orbital energy, and angular momentum conservation

Kepler's laws of planetary motion

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  • First law states planets orbit in ellipses with the Sun at one focus
  • Second law (law of equal areas) describes conservation of angular momentum
  • Third law relates orbital period to semi-major axis, expressed as T2=4π2GMa3T^2 = \frac{4\pi^2}{GM}a^3
  • apply to all orbiting bodies, including artificial satellites
  • These laws provide the basis for predicting orbital motion and designing transfer maneuvers

Orbital elements and parameters

  • Six classical orbital elements uniquely define an orbit (semi-major axis, eccentricity, inclination, right ascension of ascending node, argument of periapsis, true anomaly)
  • represents the closest point to Earth in an orbit, while is the farthest point
  • Orbital period calculated using T=2πa3μT = 2\pi\sqrt{\frac{a^3}{\mu}}, where μ is the gravitational parameter
  • Specific mechanical energy of an orbit given by ε=−μ2a\varepsilon = -\frac{\mu}{2a}
  • Velocity at any point in an orbit determined using the vis-viva equation: v2=μ(2r−1a)v^2 = \mu(\frac{2}{r} - \frac{1}{a})

Types of orbits

  • Circular orbits have constant altitude and velocity (eccentricity = 0)
  • Elliptical orbits vary in altitude and velocity (0 < eccentricity < 1)
  • Parabolic orbits represent escape trajectories (eccentricity = 1)
  • Hyperbolic orbits occur for bodies with excess velocity (eccentricity > 1)
  • Special orbits include geosynchronous, sun-synchronous, and Molniya orbits, each serving specific mission requirements

Hohmann transfer orbits

  • Hohmann transfers represent the most efficient method for transferring between two circular, coplanar orbits in terms of fuel consumption
  • These transfers play a crucial role in orbital mechanics and spacecraft mission design within Engineering Mechanics – Dynamics
  • Understanding Hohmann transfers enables engineers to optimize fuel usage and plan effective orbital maneuvers

Principles of Hohmann transfers

  • Consists of two impulsive burns: one to enter the transfer ellipse, another to circularize at the target orbit
  • Transfer ellipse tangent to both the initial and final orbits at periapsis and apoapsis, respectively
  • Utilizes minimum energy trajectory between two circular orbits
  • Applicable for both orbit raising (expanding transfer ellipse) and orbit lowering (contracting transfer ellipse)
  • Transfer time equals half the orbital period of the transfer ellipse

Delta-v requirements

  • Total delta-v calculated as the sum of two burns: Δvtotal=Δv1+Δv2\Delta v_{total} = \Delta v_1 + \Delta v_2
  • First burn (∆v₁) changes velocity to enter transfer ellipse: Δv1=μr1(2r2r1+r2−1)\Delta v_1 = \sqrt{\frac{\mu}{r_1}}(\sqrt{\frac{2r_2}{r_1+r_2}} - 1)
  • Second burn (∆v₂) circularizes orbit at final radius: Δv2=μr2(1−2r1r1+r2)\Delta v_2 = \sqrt{\frac{\mu}{r_2}}(1 - \sqrt{\frac{2r_1}{r_1+r_2}})
  • Delta-v increases with the ratio of final to initial orbit radii
  • For Earth orbits, typical delta-v values range from hundreds to thousands of meters per second

Transfer time calculations

  • Transfer time equals half the period of the transfer ellipse
  • Calculated using ttransfer=π(r1+r2)38μt_{transfer} = \pi\sqrt{\frac{(r_1+r_2)^3}{8\mu}}
  • Longer transfer times for larger orbit changes
  • Trade-off exists between transfer time and delta-v requirements
  • Hohmann transfers minimize fuel use but may not be optimal for time-critical missions

Non-Hohmann orbital transfers

  • Non-Hohmann transfers provide alternatives to the classic Hohmann maneuver, offering different trade-offs between fuel efficiency and transfer time
  • These techniques expand the toolkit of orbital maneuvers in Engineering Mechanics – Dynamics, allowing for more flexible mission planning
  • Understanding various transfer methods enables engineers to optimize trajectories for specific mission constraints and objectives

Bi-elliptic transfers

  • Involves three burns: initial burn to highly elliptical , intermediate burn at apoapsis, final burn to circularize
  • Can be more efficient than Hohmann transfers for large orbit changes (typically when final orbit radius > 11.94 times initial radius)
  • Total delta-v calculated as sum of three burns: Δvtotal=Δv1+Δv2+Δv3\Delta v_{total} = \Delta v_1 + \Delta v_2 + \Delta v_3
  • Intermediate apoapsis can extend beyond the final orbit radius
  • Longer transfer time compared to Hohmann, but potentially lower total delta-v for certain orbit ratios

Low-thrust spirals

  • Utilizes continuous low-thrust propulsion (electric propulsion) instead of impulsive burns
  • Gradually raises or lowers orbit through many revolutions
  • Total delta-v higher than but enables use of more efficient propulsion systems
  • Transfer time significantly longer than impulsive transfers
  • Orbit evolves along a spiral trajectory, continuously changing semi-major axis and eccentricity
  • Useful for missions with high specific engines and less stringent time constraints

Gravity assist maneuvers

  • Utilizes gravitational field of a planet or moon to change spacecraft's velocity and trajectory
  • Can provide significant delta-v without propellant expenditure
  • Hyperbolic flyby trajectory around assisting body
  • Velocity change in heliocentric frame depends on planet's orbital velocity and flyby geometry
  • Multiple gravity assists can be chained together for complex interplanetary trajectories (Grand Tour missions)
  • Requires precise timing and trajectory planning to align with planetary positions

Orbital plane changes

  • Orbital plane changes represent a critical aspect of spacecraft maneuvering in Engineering Mechanics – Dynamics
  • These maneuvers allow satellites to modify their orbital inclination, essential for various mission requirements and orbit corrections
  • Understanding plane changes helps engineers design efficient mission profiles and optimize fuel usage for inclination adjustments

Simple plane change maneuvers

  • Involves changing the inclination of an orbit without altering its size or shape
  • Performed by applying thrust perpendicular to the orbital plane
  • Delta-v requirement calculated using Δv=2vsin⁡(Δi2)\Delta v = 2v \sin(\frac{\Delta i}{2}), where v is orbital velocity and Δi is the change in inclination
  • Most efficient when performed at the nodes (intersection of initial and desired orbital planes)
  • Plane changes require significant delta-v, especially for large inclination changes
  • Often combined with other maneuvers to reduce overall fuel consumption

Combined plane change and altitude adjustment

  • Integrates plane change with orbit raising or lowering to reduce total delta-v
  • Three main types: single-impulse, two-impulse, and three-impulse combined maneuvers
  • Single-impulse combines plane change and altitude change in one burn, optimal for small inclination changes
  • Two-impulse maneuver splits delta-v between two burns, one at each node
  • Three-impulse (bi-elliptic) maneuver can be more efficient for large plane changes and altitude adjustments
  • Optimization involves finding the best distribution of plane change between burns
  • Can significantly reduce fuel requirements compared to separate plane change and altitude adjustment maneuvers

Key Terms to Review (18)

Apogee: Apogee is the point in an orbit where an object is farthest from the Earth or the celestial body it is orbiting. This term is crucial in understanding orbital maneuvers and transfers, as it directly influences the speed and trajectory of the spacecraft. When a satellite reaches apogee, its gravitational potential energy is at a maximum, and its kinetic energy is at a minimum, which affects how it can be maneuvered into different orbits.
Bi-impulsive transfer: A bi-impulsive transfer refers to a maneuver in orbital mechanics where two impulses are applied at different points in time to change an object's orbit. This method is often used in space missions to efficiently transition between orbits by utilizing the gravitational influence of celestial bodies. The dual impulse approach allows for precision in achieving desired orbital parameters while minimizing fuel consumption.
Delta-v: Delta-v is a measure of the change in velocity needed for an object to perform a specific maneuver in space, typically represented in meters per second (m/s). It is crucial for understanding how much energy or thrust is required to execute maneuvers such as changing orbits, rendezvousing with other spacecraft, or landing on celestial bodies. Delta-v plays a significant role in mission planning and trajectory analysis as it directly influences the efficiency and feasibility of various orbital operations.
Geostationary orbit: A geostationary orbit is a circular orbit around the Earth where a satellite has an orbital period that matches the Earth's rotation period, allowing it to appear stationary relative to a fixed point on the Earth's surface. This unique alignment is crucial for communication and weather satellites, as it enables continuous monitoring and transmission from a specific location.
Gravity assist: Gravity assist is a spaceflight technique that uses the gravitational pull of a celestial body to change the speed and direction of a spacecraft. By flying close to a planet or moon, a spacecraft can gain energy and velocity without using additional fuel, allowing for more efficient travel through the solar system. This maneuver not only helps in reaching distant destinations but also enables spacecraft to carry out complex mission profiles with reduced energy costs.
Hohmann transfer: A Hohmann transfer is an efficient orbital maneuver used to transfer a spacecraft between two circular orbits with different altitudes using the least amount of fuel. This maneuver utilizes two engine burns to move the spacecraft from its initial orbit to a higher or lower orbit, optimizing the energy needed for the transition. It is crucial for understanding how to plan and execute orbital maneuvers, particularly when dealing with various orbital elements.
Impulse: Impulse is defined as the product of the average force acting on an object and the time duration over which that force acts. It connects directly to momentum, as it causes a change in momentum and is integral in analyzing how forces influence the motion of objects. Understanding impulse is essential for solving problems related to collisions, motion of rigid bodies, and maneuvers in different environments, including those involving orbital dynamics.
Kepler's Laws: Kepler's Laws are three fundamental principles that describe the motion of planets in their orbits around the sun. These laws provide crucial insights into the nature of planetary motion, specifically how objects move in elliptical paths, the relation between the distance from the sun and orbital period, and the dynamics of orbital transfers and maneuvers. They form a foundational understanding of celestial mechanics and are essential for predicting the trajectories of satellites and other celestial bodies.
Launch window: A launch window is a specific time period during which a spacecraft must be launched to achieve its intended mission objectives, often aligning with particular orbital elements and transfer trajectories. This timing is crucial because it ensures that the spacecraft can rendezvous with another celestial body or enter a desired orbit efficiently, considering factors such as the alignment of planets, gravitational assists, and energy requirements for maneuvers.
Low Earth Orbit: Low Earth Orbit (LEO) is a region of space typically defined as an altitude between 160 kilometers (about 99 miles) and 2,000 kilometers (approximately 1,200 miles) above the Earth's surface. Satellites in this orbit have shorter orbital periods, allowing for quick data transmission and frequent access to the Earth's surface, which makes LEO ideal for various applications like telecommunications and Earth observation.
Newton's Law of Universal Gravitation: Newton's Law of Universal Gravitation states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This principle is fundamental in understanding how celestial bodies interact, influencing concepts like orbital elements, the characteristics of elliptical orbits, and the mechanics behind orbital maneuvers and transfers.
Orbital energy equation: The orbital energy equation describes the total mechanical energy of an object in orbit, which is the sum of its kinetic and potential energy. This equation is crucial for understanding how celestial bodies move in elliptical orbits and how spacecraft can perform maneuvers to change their trajectories or transfer between different orbits.
Perigee: Perigee is the point in an orbit where an object is closest to the Earth. This term is particularly relevant when discussing orbital maneuvers and transfers, as understanding perigee helps in planning efficient trajectories for satellites and space missions. When an object reaches its perigee, it experiences the highest gravitational pull from the Earth, which can be advantageous for various operations such as propulsion boosts or satellite deployment.
Reaction Wheels: Reaction wheels are devices used in spacecraft for attitude control by changing the spacecraft's orientation without using thrusters. They operate on the principle of conservation of angular momentum, allowing for precise adjustments in a spacecraft's position during orbital maneuvers and transfers. These wheels provide a non-propellant method to stabilize and control spacecraft orientation, which is crucial for tasks like aligning instruments and adjusting orbits.
Thrusters: Thrusters are propulsion devices used in spacecraft and other vehicles to create thrust, enabling them to maneuver in space. These devices play a critical role in orbital maneuvers and transfers, allowing spacecraft to change their velocity and direction without relying solely on the main engines. By generating controlled forces, thrusters help maintain the desired trajectory and orientation during missions.
Transfer orbit: A transfer orbit is a trajectory that spacecraft use to move from one orbit to another, typically designed to efficiently transition between orbits with minimal energy expenditure. It usually involves an elliptical path that connects two circular orbits, allowing a spacecraft to change its altitude and velocity. Understanding transfer orbits is essential for planning orbital maneuvers and optimizing mission profiles in space exploration.
Transfer Window: A transfer window is a specific period during which spacecraft can change orbits or transfer from one trajectory to another, typically involving the least amount of energy. This concept is crucial for mission planning as it defines the optimal timing for launches and maneuvers to ensure that the spacecraft can reach its intended destination efficiently. The timing of these windows can significantly affect fuel consumption and mission success.
Tsiolkovsky Rocket Equation: The Tsiolkovsky rocket equation is a fundamental formula in astronautics that relates the velocity of a rocket to its mass and the effective exhaust velocity of its propulsion system. This equation demonstrates how the change in momentum of the rocket is influenced by the mass of the rocket and the speed at which it expels fuel, making it essential for understanding orbital maneuvers and transfers. By using this equation, engineers can calculate the necessary fuel requirements for achieving specific trajectories and velocities, which is critical when planning space missions.
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