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Orbit determination predicts how objects in space orbit each other using methods such as initial orbit determination, least squares, and sequential processing. Applications range from GPS to predicting collisions with Earth. The least squares method is most commonly used and takes into account errors formed due to unknown forces and interactions during an orbit. The sequential processing method is the most accurate but requires extensive data and complex mathematics. State estimation reference and linearization are used when data is too small for nonlinear methods.
In astronomy, orbit determination means predicting how objects in space orbit each other. There are several methods for making these predictions. The initial orbit determination method is the simplest method and requires two measurements to find the direction and velocity of an orbiting body. The least squares method is more accurate but requires many estimates of the same orbit to produce a prediction of the orbit’s direction, speed, and error. The sequential processing method is the most accurate and requires many estimates of the orbital error of previous models. This method produces new orbital models that take into account various factors causing orbit error, such as small collisions with space dust.
Applications of orbit determination range from global positioning satellites (GPS) to binary star orbits. Orbit error can cause serious problems with the GPS system and must be monitored constantly. Objects scheduled to collide with Earth are expected to be predicted by pre-impact orbital determination methods.
Determining the initial orbit has been used throughout history and developed independently by many astronomers. It was used by Johannes Kepler to derive his three laws of planetary motion. The first accurate orbit model for the planet Mars was also developed using initial orbit determination.
Since it was first developed by Carl Friedrich Gauss in 1801, the least squares method has replaced the use of initial orbit determination. An orbital period is one complete cycle of one orbit. The method of least squares shows that between complete orbital periods there are always errors that are formed due to unknown forces and interactions of the orbiting body during the journey. The determination of the initial orbit does not take into account previous data. It is only the first step in modern orbit determination because the least squares method calculates the orbit error.
The sequential processing method is most preferred due to computer modeling. With this method and Sherman’s theorem, astronomers develop orbital models using computers to find future position, speed, direction, and orbital error with very limited data. Sherman’s theorem requires another mathematical step for the sequential processing method, called linearization.
The complex mathematics and extensive data required for using the sequential processing method are often not available, so astronomers produce estimates for the sequential processing method. This reduces the difficulty of determining the orbit but slightly increases the orbit error. This process is called referencing the state estimate. Astronomers use state estimation reference and linearization only when the orbital data they are studying is too small to use nonlinear methods of sequential processing.
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