Light-time correction is a slight angular shift in the apparent position of a celestial object from its geometric position on the celestial sphere caused by the object's motion during the time it takes its light to reach Earth.
Thus light-time correction moves a celestial object back from where it is at a particular instant of Earth time to where it was when its light left it, so that its calculated position does indeed predict its observed position (at least due to this effect). It depends only on the motion of the celestial object during its light-time--it does not depend on the motion of Earth. It is calculated iteratively. An approximate light-time is calculated by dividing the object's geometric distance from Earth by the speed of light. Then the object's velocity is multiplied by this approximate light-time to determine its approximate displacement through space during that time. Its previous position is used to calculate a more precise light-time. This process is repeated until the desired precision is reached.
A celestial object's geometric position is calculated via Sir Isaac Newton's theory of universal gravitation, which assumes that an object's gravity affects all other objects simultaneously, no matter how far away they are.
A false light-time correction of the Sun is often mentioned as a form of the aberration of light. There is an apparent 20.5" (arc-second) shift of the Sun's position along the ecliptic due to Earth's movement along its orbit during the 8.3 minutes that it takes the Sun's light to reach Earth. This approximates the aberration of light from other stars, which obviously have vastly greater light-times, thus the phenomena are totaly different. Furthermore, the geometric position of the Sun is always calculated at an instant of Earth time, thus Earth does not get a chance to move. Actually, during its light-time the Sun does move slightly around the barycenter (center of mass) of the solar system as a counter-balance to the massive jovian planets (Jupiter, Saturn, Uranus, and Neptune). Thus the true light-time correction of the Sun is extremely small, usually much smaller than 0.03".
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