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Concept
Radial velocity
Summary
The radial velocity or line-of-sight velocity, also known as radial speed or range rate, of a target with respect to an observer is the rate of change of the distance or range between the two points. It is equivalent to the vector projection of the target-observer relative velocity onto the relative direction connecting the two points. In astronomy, the point is usually taken to be the observer on Earth, so the radial velocity then denotes the speed with which the object moves away from the Earth (or approaches it, for a negative radial velocity).
Given a differentiable vector defining the instantaneous position of a target relative to an observer.
Let
with , the instantaneous velocity of the target with respect to the observer.
The magnitude of the position vector is defined as
The quantity range rate is the time derivative of the magnitude (norm) of , expressed as
Substituting () into ()
Evaluating the derivative of the right-hand-side
using () the expression becomes
Since
With
The range rate is simply defined as
the projection of the observer to target velocity vector onto the unit vector.
A singularity exists for coincident observer target, i.e. . In this case, range rate does not exist as .
In astronomy, radial velocity is often measured to the first order of approximation by Doppler spectroscopy. The quantity obtained by this method may be called the barycentric radial-velocity measure or spectroscopic radial velocity. However, due to relativistic and cosmological effects over the great distances that light typically travels to reach the observer from an astronomical object, this measure cannot be accurately transformed to a geometric radial velocity without additional assumptions about the object and the space between it and the observer. By contrast, astrometric radial velocity is determined by astrometric observations (for example, a secular change in the annual parallax).
Light from an object with a substantial relative radial velocity at emission will be subject to the Doppler effect, so the frequency of the light decreases for objects that were receding (redshift) and increases for objects that were approaching (blueshift).
Official source
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