Celestial mechanics play an important role in planetary and satellite orbits. The two-body problem is a crucial part of celestial mechanics. This article discusses the reduction of the two-body problem into a one-body problem, Keplerian motion, and Keplerian satellite orbits.
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The Reduction Two-Body Problem
The two-body is the problem of predicting the motion of two massive objects that are viewed abstractly as point particles. The problem assumes that two objects only interact with one another and the only force that affects each object arises from the other object, ignoring all other objects.
In celestial mechanics, the two-body problem is reduced to a one-body problem to solve problems involving the motion of two objects in a simplified manner. Two freely moving point masses exerting force on each other in an isolated dynamical system represent the two-body problem.
The first object and the second object are of mass m1 and m2 and located at position vectors r1 and r2, respectively. The first object initially exerts a force of f21 on the second object, and the second object exerts an opposite and equal force of f12=−f21 on the first object by Newton’s third law.
All other forces are ignored and the equations of motion for the two objects are obtained as m1d2r1/dt2= −f21 for the first object and m2d2r2/dt2= f21 for the second object. The center of mass of the system is located at rcm= (m1r1 + m2r2)/m1 + m2, which can also be written as r1 = rcm – (m2r/m1 + m2) and r2 = rcm + (m1r/m1+ m2), where r = r2 – r1.
Substituting the above two equations in m1d2r1/dt2= −f21 and m2d2r2/dt2= f21, respectively, and considering that the center of mass of an isolated system does not accelerate, both equations yield µd2r/dt2 = f, where µ = (m1m2)/m1 + m2, which is known as the reduced mass.
Thus, the original two-body problem was effectively converted into an equivalent one-body problem, where the force f is similar to the force acting on each object in the original problem and the mass µ is less than either m2 or m1. The one-body problem is similar to that of a single particle with position vector r and mass m∗ subject to an external force F.
Keplerian Motion
Kepler's laws of planetary motion consist of three laws that describe the motions of the planets in the solar system. However, these laws only apply to the two-body problem, where the number of objects interacting in the space is limited to two.
The orbits described by both Newton’s and Kepler’s laws are referred to as Keplerian orbits as both laws agree in the two-body theory, while the Newtonian theory is used to describe multiple-body interactions without applying relativistic effects.
In several cases, such as while approximating the positions of planets relative to the Sun, the moon relative to Earth, and spacecraft in planetary and solar orbits, the effects of multiple bodies can be considered as slight perturbations superimposed on the two-body theory.
Two bodies are assumed whose motions have to be characterized, with the bigger body among the two being designated as the primary body while the other body is designated as the smaller body.
If r1 is the position vector of the primary body with respect to an arbitrary inertial origin, r2 is the position vector of the secondary body, r = r2 – r1 is the vector from primary to secondary body, and r =| r | is the magnitude, then the equations of force by Newton’s laws at both bodies are m1d2r1/dt2 = Gm1m2r/r3 and m2d2r2/dt2 = –Gm1m2r/r3, where G is the Newton gravitational constant.
The body mass on the left-hand side of each equation divides out with the mass on the right-hand side. These two equations are then subtracted to obtain d2r/dt2= – G (m1 + m2)r/r3. Thus, Newton's law for a single body moving around a primary body is d2r/dt2= – r/r3, with G(m1 + m2) remaining constant. All three of Kepler’s law of planetary motion follows from this expression.
Similarly, adding the two above equations demonstrates that the net force acting on the pair is zero as viewed in an inertial frame, m1d2r1/dt2 + m2d2r2/dt2 = 0. The net momentum of the pair, which is constant, can be obtained through the integration of this equation. A second integration yields the motion of the center of mass/barycenter, which is located at c = m1r1 + m2r2/m1 + m2.
The barycenter is non-accelerating and acts as an inertial frame in which the velocity and position of the center of mass and bodies can be determined at all times from initial velocities and positions.
Keplerian Satellite Orbits
A Keplerian orbit is primarily an orbit for which all orbital elements except the mean anomaly (M) are constant. Thus, the ellipse maintains a constant shape, orientation, and size with respect to the stars.
Although these orbits are simple when viewed from space, Keplerian orbits become complicated when viewed from Earth as the Earth rotates beneath the fixed orbit, which leads to two daily passes of the satellite, including one as the orbit descends and one as the orbit ascends, above any point on Earth with the time of ascend and descend changing throughout the year.
The Keplerian orbits are perturbed by several factors, including the non-spherical gravitational field of Earth, third-body interactions/the gravitational attraction of other bodies, radiation pressure from the Sun, drag and lift forces from the atmosphere, particle flux from the solar wind, and electromagnetic forces.
All these factors, except the non-spherical gravitational field of Earth, lead to random orbit perturbations, which can be corrected by periodic orbital element observations and orbit adjustment using on-board thrusters. The Earth’s non-spherical gravitational field leads to secular changes/changes linear with time in the orbital elements.
However, Keplerian orbits are not suitable as the satellite passes over the same location throughout the year at different times of the day, which results in several issues, including difficulties in solar panel orientation and less useful dusk and dawn images. Additionally, the data also do not fit into operational schedules.
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References and Further Readings
Fitzpatrick, R. (2012). An introduction to celestial mechanics. Cambridge University Press. https://books.google.co.in/books/about/An_Introduction_to_Celestial_Mechanics.html?id=K5wdRGI3w8kC&redir_esc=y#:~:text=This%20accessible%20text%20on%20classical,features%20of%20solar%20system%20dynamics.
Fundamentals of Orbital Mechanics [Online] Available at https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/MPG%20Book/Release/Chapter7-OrbitalMechanics.pdf (Accessed on 09 October 2023)
Satellite Orbits [Online] Available at https://www.atmosp.physics.utoronto.ca/people/strong/phy499/section2_05.pdf (Accessed on 09 October 2023)
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