Editing Radial velocity

{{Short description|Velocity of an object as the rate of distance change between the object and a point}}
{{distinguish-redirect|Radial speed|Circular motion{{!}}radial motion}}
[[File:Radialgeschwindigkeit.gif|thumb|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''' of a target with respect to an observer is the [[temporal rate of change|rate of change]] of the [[vector quantity|vector]] [[displacement (geometry)|displacement]] between the two points. It is formulated as the [[vector projection]] of the target-observer [[relative velocity]] onto the [[relative direction (geometry)|relative direction]] or [[Line of sight|line-of-sight]] (LOS) connecting the two points. 

The '''radial speed''' or '''range rate''' is the [[temporal rate]] of the [[Euclidean distance|distance]] or [[Slant range|range]] between the two points. It is a [[Sign (mathematics)|signed]] [[Scalar (mathematics)|scalar quantity]], formulated as the [[scalar projection]] of the relative velocity vector onto the LOS direction. Equivalently, radial speed equals the [[vector norm|norm]] of the radial velocity, [[Modulo (mathematics)|modulo]] the sign.{{efn|The norm, a nonnegative number, is multiplied by -1 if velocity (red arrow in the figure) and relative position form an [[obtuse angle]] or if relative velocity (green arrow) and relative position are antiparallel.}}

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 <math>\mathbf r \in \mathbb{R}^3</math> defining the instantaneous [[relative position]] of a target with respect to an observer.

Let the instantaneous [[relative velocity]] of the target with respect to the observer be

{{NumBlk|:|<math> \mathbf v = \frac{d\mathbf r}{dt} \in \mathbb{R}^3</math>|{{EquationRef|1}}}}

The magnitude of the position vector <math>\mathbf r</math> is defined as in terms of the [[inner product]]

{{NumBlk|:|<math>r= \|\mathbf r\| = \langle \mathbf r,\mathbf r \rangle^{1/2}</math>|{{EquationRef|2}}}}

The quantity range rate is the [[time derivative]] of the magnitude ([[Norm (mathematics)|norm]]) of <math>\mathbf r</math>, expressed as

{{NumBlk|:|<math>\dot{r}=\frac{d r}{dt}</math>|{{EquationRef|3}}}}

Substituting ({{EquationNote|2}}) into ({{EquationNote|3}})

: <math>\dot{r} = \frac{d \langle \mathbf r,\mathbf r \rangle^{1/2} }{dt}</math>

Evaluating the derivative of the right-hand-side by the [[chain rule]]

: <math>\dot{r} = \frac{1}{2}  \frac{d \langle \mathbf r,\mathbf r \rangle}{dt} \frac{1}{r}</math>

: <math>\dot{r} = \frac{1}{2}  \frac{\langle \frac{d\mathbf r}{dt}, \mathbf r \rangle + \langle \mathbf r,\frac{d\mathbf r}{dt} \rangle}{r}</math>

using ({{EquationNote|1}}) the expression becomes

: <math>\dot{r} = \frac{1}{2}  \frac{\langle \mathbf v,\mathbf r \rangle + \langle \mathbf r,\mathbf v \rangle}{r}</math>

By reciprocity,<ref>{{cite book| last1=Hoffman|first1=Kenneth M.| last2=Kunzel|first2=Ray| year=1971| title=Linear Algebra| edition=Second| publisher=Prentice-Hall Inc.|page=[https://archive.org/details/linearalgebra00hoff_0/page/271 271]| isbn=0135367972|url-access=registration| url=https://archive.org/details/linearalgebra00hoff_0/page/271}}</ref> <math>\langle \mathbf v,\mathbf r \rangle = \langle \mathbf r,\mathbf v \rangle</math>.
Defining the [[unit vector|unit]] relative position vector <math>\hat{r} = \mathbf r/{r} </math> (or LOS direction), 
the range rate is simply expressed as

: <math>\dot{r} = \frac{\langle \mathbf r,\mathbf v \rangle}{r} = \langle \hat{r},\mathbf v \rangle</math>

i.e., the projection of the relative velocity vector onto the LOS direction.

Further defining the velocity direction <math>\hat{v} =\mathbf v/{v} </math>, with 
the [[relative speed]] <math>v =\|\mathbf v\|</math>, we have:
: <math>\dot{r} = \langle \hat{r},v\hat{v} \rangle = v \langle \hat{r},\hat{v} \rangle</math>
where the inner product is either +1 or -1, for parallel and [[antiparallel vector]]s, respectively.


A singularity exists for coincident observer target, i.e., <math>r = 0</math>; in this case, range rate is undefined.

==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.<ref name="IAUInfBull91_c1">''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</ref> However, due to [[Theory of relativity|relativistic]] and [[Cosmology|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.<ref name="Lindegren2003">{{cite journal |last1=Lindegren |first1=Lennart |last2=Dravins |first2=Dainis |date=April 2003 |title=The fundamental definition of "radial velocity" |url=http://www.aanda.org/articles/aa/pdf/2003/15/aah3961.pdf |journal=Astronomy and Astrophysics |volume=401 |issue= 3|pages=1185–1201 |doi=10.1051/0004-6361:20030181 |access-date=4 February 2017 |bibcode=2003A&A...401.1185L|arxiv = astro-ph/0302522 |s2cid=16012160 }}</ref> By contrast, ''astrometric radial velocity'' is determined by [[astrometric]] observations (for example, a [[Secular variation|secular change]] in the annual [[parallax]]).<ref name="Lindegren2003"/><ref>{{cite journal| first1=Dainis|last1=Dravins | first2=Lennart|last2=Lindegren|first3=Søren|last3=Madsen | year=1999|journal=Astron. Astrophys.|volume=348|pages=1040–1051|bibcode=1999A&A...348.1040D |title=Astrometric radial velocities. I. Non-spectroscopic methods for measuring stellar radial velocity|arxiv = astro-ph/9907145 }}</ref><ref name="IAUInfBull91_c2">''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</ref>

===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 [[Electromagnetic spectrum|spectrum]] and comparing the measured [[wavelength]]s of known [[spectral line]]s 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.<ref>{{cite journal | last=Huggins | first=W. | title=Further observations on the spectra of some of the stars and nebulae, with an attempt to determine therefrom whether these bodies are moving towards or from the Earth, also observations on the spectra of the Sun and of Comet&nbsp;II | journal=[[Philosophical Transactions of the Royal Society of London]] | date=1868 | volume=158 | pages=529–564 | doi=10.1098/rstl.1868.0022| bibcode=1868RSPT..158..529H}}</ref>

[[File:Planet reflex 200.gif|thumb|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 star]]s, the [[orbit]]al 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 [[mass]]es of the stars, and some [[orbital element]]s, such as [[eccentricity (orbit)|eccentricity]] and [[semimajor axis]]. The same method has also been used to detect [[planet]]s 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 [[Sightline|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.<ref name="Anglada-Escude">{{cite journal | first1=Guillem |last1 = Anglada-Escude |first2=Mercedes |last2=Lopez-Morales|first3= John E.|last3=Chambers | title = How eccentric orbital solutions can hide planetary systems in 2:1 resonant orbits | journal = The Astrophysical Journal Letters | arxiv = 0809.1275 | doi = 10.1088/0004-637X/709/1/168 | volume=709 | issue=1 | pages=168–78 | bibcode = 2010ApJ...709..168A|year = 2010 |s2cid = 2756148 }}</ref><ref name="KursterAA2015">{{cite journal|first1=Martin |last1=Kürster| first2=Trifon |last2=Trifonov |first3=Sabine |last3=Reffert| first4=Nadiia M. | last4=Kostogryz |first5=Florian |last5=Roder | journal=Astron. Astrophys. |year=2015|pages=A103 |doi=10.1051/0004-6361/201525872 | volume=577 |title=Disentangling 2:1 resonant radial velocity oribts from eccentric ones and a case study for HD 27894 | arxiv=1503.07769 | bibcode=2015A&A...577A.103K|s2cid=73533931}}</ref>

===Detection of exoplanets===
{{Main|Doppler spectroscopy}}
[[File:The radial velocity method (artist’s impression).jpg|thumb|The radial velocity method to detect exoplanets]]
The radial velocity method to detect [[exoplanet]]s 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
*the [[Earth's orbit|Earth's elliptic motion]] around the Sun at approximately ± 30&nbsp;km/s,
*a [[Orbit of the Moon|monthly rotation]] of ± 13&nbsp;m/s of the Earth around the center of gravity of the Earth-Moon system,<ref name="FMelloLNP683">{{cite book|first1=S. |last1=Ferraz-Mello |first2=T. A. | last2=Michtchenko |title=Chaos and Stability in Planetary Systems |chapter=Extrasolar Planetary Systems |pages=219–271|series=Lecture Notes in Physics | volume=683| year=2005| doi=10.1007/10978337_4| bibcode=2005LNP...683..219F|isbn=978-3-540-28208-2 }}</ref>
*the [[Earth's rotation|daily rotation]] of the telescope with the Earth crust around the Earth axis, which is up to ±460&nbsp;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&nbsp;km/s from the motion around the [[Galactic Center]] and associated proper motions.<ref name="Reiadarxiv1608">{{cite journal|arxiv=1608.03886|first1= M. J. |last1=Reid|first2=T. M. |last2=Dame|title=On the rotation speed of the Milky Way determined from HI emission|year=2016|doi=10.3847/0004-637X/832/2/159|volume=832|issue= 2 |journal=The Astrophysical Journal|page=159|bibcode = 2016ApJ...832..159R |s2cid= 119219962 |doi-access= free }}</ref>
*in the case of spectroscopic measurements corrections of the order of ±20&nbsp;cm/s with respect to [[Relativistic aberration|aberration]].<ref name="StumpffAA">{{cite journal|first1=P. |last1=Stumpff|year=1985|journal=Astron. Astrophys. |bibcode=1985A&A...144..232S |title=Rigorous treatment of the heliocentric motion of stars |volume=144 |issue=1| pages=232}}</ref>
*[[Sin i degeneracy]] is the impact caused by not being in the plane of the motion.

==See also==
* {{annotated link|Proper motion}}
* {{annotated link|Peculiar velocity}}
* {{annotated link|Relative velocity}}
* {{annotated link|Space velocity (astronomy)}}
* [[Bistatic range rate]]
* [[Doppler effect]]
* [[Inner product]]
* [[Orbit determination]]
* [[Lp space]]

==Notes==
{{Notelist}}

==References==
{{Reflist}}

==Further reading==
* {{citation|last1=Hoffman|first1=Kenneth M.|last2=Kunzel|first2=Ray|title=Linear Algebra|edition=Second|publisher=Prentice-Hall Inc.|year=1971|isbn=0135367972|url-access=registration|url=https://archive.org/details/linearalgebra00hoff_0}}
* Renze, John; Stover, Christopher; and Weisstein, Eric W. "Inner Product." From MathWorld—A Wolfram Web Resource.http://mathworld.wolfram.com/InnerProduct.html

==External links==
*[https://www.relativitycalculator.com/radial_velocity_equation.shtml The Radial Velocity Equation in the Search for Exoplanets ( The Doppler Spectroscopy or Wobble Method )]

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