Velocity of an object as the rate of distance change between the object and a point
"Radial speed" redirects here. Not to be confused with
radial motion.
A plane flying past a radar station: the plane's velocity vector (red) is the sum of the radial velocity (green) and the tangential velocity (blue).
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).
Formulation
Given a differentiable vector $\mathbf {r} \in \mathbb {R} ^{3}$ defining the instantaneous position of a target relative to an observer.
Let
$\mathbf {v} ={\frac {d\mathbf {r} }{dt}}$
(1)
with $\mathbf {v} \in \mathbb {R} ^{3}$, the instantaneous
velocity of the target with respect to the observer.
The magnitude of the position vector $\mathbf {r}$ is defined as
the projection of the observer to target velocity vector onto the ${\hat {\mathbf {r} }}$ unit vector.
A singularity exists for coincident observer target, i.e. $\mathbf {r} ={\begin{bmatrix}0\\0\\0\end{bmatrix}}$. In this case, range rate does not exist as $r=0$.
Applications in astronomy
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.^{
[2]} 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.^{
[3]} By contrast, astrometric radial velocity is determined by
astrometric observations (for example, a
secular change in the annual
parallax).^{
[3]}^{
[4]}^{
[5]}
Spectroscopic radial velocity
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).
The radial velocity of a
star or other luminous distant objects can be measured accurately by taking a high-resolution
spectrum and comparing the measured
wavelengths of known
spectral lines to wavelengths from laboratory measurements. A positive radial velocity indicates the distance between the objects is or was increasing; a negative radial velocity indicates the distance between the source and observer is or was decreasing.
William Huggins ventured in 1868 to estimate the radial velocity of
Sirius with respect to the Sun, based on observed redshift of the star's light.^{
[6]}
Diagram showing how an exoplanet's orbit changes the position and velocity of a star as they orbit a common center of mass
In many
binary stars, the
orbital motion usually causes radial velocity variations of several kilometres per second (km/s). As the spectra of these stars vary due to the Doppler effect, they are called
spectroscopic binaries. Radial velocity can be used to estimate the ratio of the
masses of the stars, and some
orbital elements, such as
eccentricity and
semimajor axis. The same method has also been used to detect
planets around stars, in the way that the movement's measurement determines the planet's orbital period, while the resulting radial-velocity
amplitude allows the calculation of the lower bound on a planet's mass using the
binary mass function. Radial velocity methods alone may only reveal a lower bound, since a large planet orbiting at a very high angle to the
line of sight will perturb its star radially as much as a much smaller planet with an orbital plane on the line of sight. It has been suggested that planets with high eccentricities calculated by this method may in fact be two-planet systems of circular or near-circular resonant orbit.^{
[7]}^{
[8]}
The radial velocity method to detect
exoplanets is based on the detection of variations in the velocity of the central star, due to the changing direction of the gravitational pull from an (unseen) exoplanet as it orbits the star. When the star moves towards us, its spectrum is blueshifted, while it is redshifted when it moves away from us. By regularly looking at the spectrum of a star—and so, measuring its velocity—it can be determined if it moves periodically due to the influence of an exoplanet companion.
Data reduction
From the instrumental perspective, velocities are measured relative to the telescope's motion. So an important first step of the
data reduction is to remove the contributions of
a
monthly rotation of ± 13 m/s of the Earth around the center of gravity of the Earth-Moon system,^{
[9]}
the
daily rotation of the telescope with the Earth crust around the Earth axis, which is up to ±460 m/s at the equator and proportional to the cosine of the telescope's geographic latitude,
small contributions from the Earth
polar motion at the level of mm/s,
contributions of 230 km/s from the motion around the
Galactic center and associated proper motions.^{
[10]}
in the case of spectroscopic measurements corrections of the order of ±20 cm/s with respect to
aberration.^{
[11]}
Sin i degeneracy is the impact caused by not being in the plane of the motion.
See also
Proper motion – Measure of the observed changes in the apparent places of stars in the sky
^Resolution C1 on the Definition of a Spectroscopic "Barycentric Radial-Velocity Measure". Special Issue: Preliminary Program of the XXVth GA in Sydney, July 13–26, 2003 Information Bulletin n° 91. Page 50. IAU Secretariat. July 2002.
https://www.iau.org/static/publications/IB91.pdf
^Resolution C 2 on the Definition of "Astrometric Radial Velocity". Special Issue: Preliminary Program of the XXVth GA in Sydney, July 13–26, 2003 Information Bulletin n° 91. Page 51. IAU Secretariat. July 2002.
https://www.iau.org/static/publications/IB91.pdf