A Dive Into Astrodynamics
- Güney Baver Gürbüz
- 19 Eki 2022
- 5 dakikada okunur
Astrodynamics or orbital mechanics is the study of the motion of artificial bodies moving under the influence of gravity that is caused by large natural bodies. Basically, it is the application that deals with the motion of rockets, satellites, and other spacecraft. There are two terms that should be grasped impeccably to understand the essence of astrodynamics: celestial mechanics and ballistics. Celestial mechanics is the branch of astronomy that deals with the motions of celestial bodies, natural bodies found outside of the Earth's atmosphere. Ballistics is the field of mechanics concerned with the launching, flight behavior, and impact effects of projectiles. To put them all, astrodynamics deals with the ballistics of the artificial bodies and their motions considering the celestial mechanics. However, how engineers utilize astrodynamics is an important point.
Astrodynamics is very fundamental, especially for Aerospace engineers. Engineers dealing with astrodynamics usually generate flight paths that will take spacecraft to get where they need to go. However, its applications vary far and wide as well. Some of the current projects regarding Astrodynamics concentrate on asteroid interception and rendezvous, solar sail and tether system design, in-space propulsion assessment, integrated technology assessment, technologies for the Evolutionary Mission Trajectory Generator, low energy manifold trajectory design, magnetic attitude determination and control, multifunction stochastic optimization, parallel nonlinear optimization, automated trajectory optimization, on-orbit servicing missions, and ground station coverage problems. Some of the prominent government agencies and industry companies such as NASA, The Aerospace Corporation, and DARPA fund and support these projects.
Engineers, while dealing with astrodynamics, benefit mostly from physics. Newton’s and Kepler’s laws are the key components for astrodynamics and help a lot to disseminate this field efficiently.
Kepler was the first successful scientist to model planetary orbits with a high degree of accuracy. After his studies, he published his laws known as the Kepler’s laws. His laws marked an epoch in the history of science and mathematics, steering Newton to find his own way and come up with tremendous and unprecedented ideas. Kepler’s laws of planetary motion are written as follows:
The orbit of every planet is an ellipse with the sun at one of the foci.
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits.
These laws mostly apply today, however, some assumptions regarding non-interference from outside bodies, negligible mass for one of the bodies, and negligible other forces than gravity are necessary to make. On the other hand, Newton’s laws can be used every time and are valid for the calculations of astrodynamics. The laws follow as:
An object at rest remains at rest, and an object in motion remains in motion at a constant speed and in a straight line unless acted on by an unbalanced force.
The acceleration of an object depends on the mass of the object and the amount of force applied.
Whenever one object exerts a force on another object, the second object exerts an equal and opposite on the first.
However, even though these laws can be used in every circumstance, as they are very general, they require some derivations for specific concepts like astrodynamics or space dynamics. Below are the equations derived from Newton’s and Kepler’s laws found:
Escape Velocity:
It is the minimum velocity that a moving body (such as a rocket) must have to escape from the gravitational field of a celestial body (such as the earth) and move outward into space.
Where
G is the universal gravitational constant (6.673 × 10-11 N . m2 / kg2)
M is the mass of the celestial body (kg)
R is the radius of the celestial body (m)
Circular Motion:
Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It is widely used for bodies rotating around planets. The velocity of bodies orbiting around the planets can be found from the following equation.
Where
G is the universal gravitational constant (6.673 × 10-11 N . m2 / kg2)
M is the mass of the celestial body (kg)
R is the radius of the celestial body (m)
As it can be seen, once the circular orbital velocity is known, the escape velocity is easily found by multiplying by √2.
Law of Universal Gravitation:
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The equation of this law can be written as follows:
Where
G is the universal gravitational constant (6.673 × 10-11 N . m2 / kg2)
m1 is the mass of the celestial body (kg)
m2 is the mass of the body orbiting around the celestial body (kg)
r is the distance between two bodies (m)
Elliptical orbits:
Elliptical orbits are much more general and allow for a wider range of initial conditions which existed when a planet/star system forms. The equation below is used to find the velocity of a body following elliptical orbits.
Where
ɥ is the standard gravitational parameter or GM
r is the distance between the orbiting bodies.
a is the length of the semi-major axis.
Orbital period:
Under standard assumptions, the orbital period of a body traveling along an elliptic orbit can be computed as:
Where
ɥ is the standard gravitational parameter or GM
r is the distance between the orbiting bodies.
a is the length of the semi-major axis.
These are some of the equations required to deal with astrodynamics. Engineers use these equations to cover a large range of situations in astrodynamics. However, being an astrodynamic engineer requires much more knowledge and skills. Such as being responsible for the Flight Dynamics and Mission Planning on Planet’s fleet of satellites, dealing with orbital mechanics, state estimation, attitude dynamics, constellation design, maneuver planning, and collision avoidance as well as performing technical analysis and developing systems.
Undoubtedly, with developing technologies and incoming generations, astrodynamics will be an indispensable field that will be utilized to thrive our planet while uncovering the mysteries of the universe.
-Güney Baver Gürbüz
References:
“Astrodynamics and Space Applications Research & Facilities.” School of Aeronautics and Astronautics - Purdue University, engineering.purdue.edu/AAE/research/astrodynamics.
“Astrodynamics.” Astrodynamics - an Overview | ScienceDirect Topics, www.sciencedirect.com/topics/physics-and-astronomy/astrodynamics.
“Astrodynamics: Orbit and Attitude Dynamics and Control of Spacecraft.” Penn State Engineering: Aerospace Engineering | Astrodynamics Research, www.aero.psu.edu/research/research-areas/astrodynamics.aspx.
“Celestial Mechanics.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., www.britannica.com/science/celestial-mechanics-physics.
“Crazy Engineering: Astrodynamics.” NASA, NASA, www.jpl.nasa.gov/videos/crazy-engineering-astrodynamics.
“Newton's Laws of Motion.” NASA, NASA, 25 May 2021, www1.grc.nasa.gov/beginners-guide-to-aeronautics/newtons-laws-of-motion/.
Bate, R. R.; Mueller, D. D.; White, J. E. (1971). Fundamentals of Astrodynamics. Courier Corporation. p. 5. ISBN 978-0-486-60061-1.
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