Orbit modeling
Orbit modeling is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity.
Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects.
Newton, at the time he formulated his laws of motion and of gravitation, applied them to the first analysis of perturbations,[1] recognizing the complex difficulties of their calculation.
[1] Many of the great mathematicians since then have given attention to the various problems involved; throughout the 18th and 19th centuries there was demand for accurate tables of the position of the Moon and planets for purposes of navigation at sea.
The differences between the Keplerian orbit and the actual motion of the body are caused by perturbations.
These perturbations are caused by forces other than the gravitational effect between the primary and secondary body and must be modeled to create an accurate orbit simulation.
Analytical solutions (mathematical expressions to predict the positions and motions at any future time) for simple two-body and three-body problems exist; none have been found for the n-body problem except for certain special cases.
[2] Due to the difficulty in finding analytic solutions to most problems of interest, computer modeling and simulation is typically used to analyze orbital motion.
This form of the equation is particularly useful when dealing with parabolic trajectories, for which the semi-major axis is infinite.
An alternate approach uses Isaac Newton's law of universal gravitation as defined below: where: Making an additional assumption that the mass of the primary body is much greater than the mass of the secondary body and substituting in Newton's second law of motion, results in the following differential equation Solving this differential equation results in Keplerian motion for an orbit.
Orbit models are typically propagated in time and space using special perturbation methods.
[2] Special perturbation methods are the basis of the most accurate machine-generated planetary ephemerides.
and these are integrated numerically to form the new velocity and position vectors as the simulation moves forward in time.
[6] Its advantages are that perturbations are generally small in magnitude, so the integration can proceed in larger steps (with resulting lesser errors), and the method is much less affected by extreme perturbations than Cowell's method.
Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as rectification.
, is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extra significant digits.
[8][9] In 1991 Victor R. Bond and Michael F. Fraietta created an efficient and highly accurate method for solving the two-body perturbed problem.
In 1989 Bond and Gottlieb embedded the Jacobian integral, which is a constant when the potential function is explicitly dependent upon time as well as position in the Newtonian equations.
The Jacobian constant was used as an element to replace the total energy in a reformulation of the differential equations of motion.
In 1991, Bond and Fraietta made further revisions by replacing the Laplace vector with another vector integral as well as another scalar integral which removed small secular terms which appeared in the differential equations for some of the elements.
To account for variations in gravitational potential around the surface of the Earth, the gravitational field of the Earth is modeled with spherical harmonics[12] which are expressed through the equation: where where: When modeling perturbations of an orbit around a primary body only the sum of the
The magnitude of acceleration it imparts to a spacecraft in Earth orbit is modeled using the equation below:[13] where: For orbits around the Earth, solar radiation pressure becomes a stronger force than drag above 800 km (500 mi) altitude.
The force of a rocket engine is modeled by the equation:[15] Another possible method is a solar sail.
Solar sails use radiation pressure in a way to achieve a desired propulsive force.
The primary non-gravitational force acting on satellites in low Earth orbit is atmospheric drag.
[13] Drag will act in opposition to the direction of velocity and remove energy from an orbit.
The force due to drag is modeled by the following equation: where Orbits with an altitude below 120 km (75 mi) generally have such high drag that the orbits decay too rapidly to give a satellite a sufficient lifetime to accomplish any practical mission.
[13] Density of air can vary significantly in the thermosphere where most low Earth orbiting satellites reside.
[13] Magnetic fields can play a significant role as a source of orbit perturbation as was seen in the Long Duration Exposure Facility.
[12] Like gravity, the magnetic field of the Earth can be expressed through spherical harmonics as shown below:[12] where where:
