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Radial velocity

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From Wikipedia, the free encyclopedia

Velocity of an object as the rate of distance change between the object and a point

"Radial speed" redirects here; not to be confused withradial 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).

Theradial velocity orline-of-sight velocity of a target with respect to an observer is therate of change of thevector displacement between the two points. It is formulated as thevector projection of the target-observerrelative velocity onto therelative direction orline-of-sight (LOS) connecting the two points.

Theradial speed orrange rate is thetemporal rate of thedistance orrange between the two points. It is asigned scalar quantity, formulated as thescalar projection of the relative velocity vector onto the LOS direction. Equivalently, radial speed equals thenorm of the radial velocity,modulo the sign.^[a]

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

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Given a differentiable vector $\mathbf {r} \in \mathbb {R} ^{3}$ defining the instantaneousrelative position of a target with respect to an observer.

Let the instantaneousrelative velocity of the target with respect to the observer be

\mathbf {v} ={\frac {d\mathbf {r} }{dt}}\in \mathbb {R} ^{3}

The magnitude of the position vector $\mathbf {r}$ is defined as in terms of theinner product

r=\|\mathbf {r} \|=\langle \mathbf {r} ,\mathbf {r} \rangle ^{1/2}

The quantity range rate is thetime derivative of the magnitude (norm) of $\mathbf {r}$ , expressed as

{\dot {r}}={\frac {dr}{dt}}

Substituting (2) into (3)

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

Evaluating the derivative of the right-hand-side by thechain rule

{\dot {r}}={\frac {1}{2}}{\frac {d\langle \mathbf {r} ,\mathbf {r} \rangle }{dt}}{\frac {1}{r}}

{\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}}

using (1) the expression becomes

{\dot {r}}={\frac {1}{2}}{\frac {\langle \mathbf {v} ,\mathbf {r} \rangle +\langle \mathbf {r} ,\mathbf {v} \rangle }{r}}

By reciprocity,^[1] $\langle \mathbf {v} ,\mathbf {r} \rangle =\langle \mathbf {r} ,\mathbf {v} \rangle$ .Defining theunit relative position vector ${\hat {r}}=\mathbf {r} /{r}$ (or LOS direction), the range rate is simply expressed as

{\dot {r}}={\frac {\langle \mathbf {r} ,\mathbf {v} \rangle }{r}}=\langle {\hat {r}},\mathbf {v} \rangle

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

Further defining the velocity direction ${\hat {v}}=\mathbf {v} /{v}$ , with therelative speed $v=\|\mathbf {v} \|$ , we have:

{\dot {r}}=\langle {\hat {r}},v{\hat {v}}\rangle =v\langle {\hat {r}},{\hat {v}}\rangle

where the inner product is either +1 or -1, for parallel andantiparallel vectors, respectively.

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

Applications in astronomy

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In astronomy, radial velocity is often measured to the first order of approximation byDoppler spectroscopy. The quantity obtained by this method may be called thebarycentric radial-velocity measure or spectroscopic radial velocity.^[2] However, due torelativistic andcosmological 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 byastrometric observations (for example, asecular change in the annualparallax).^[3]^[4]^[5]

Spectroscopic radial velocity

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Light from an object with a substantial relative radial velocity at emission will be subject to theDoppler 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 astar or other luminous distant objects can be measured accurately by taking a high-resolutionspectrum and comparing the measuredwavelengths of knownspectral 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 ofSirius 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 manybinary stars, theorbital 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 calledspectroscopic binaries. Radial velocity can be used to estimate the ratio of themasses of the stars, and someorbital elements, such aseccentricity andsemimajor axis. The same method has also been used to detectplanets around stars, in the way that the movement's measurement determines the planet's orbital period, while the resulting radial-velocityamplitude allows the calculation of the lower bound on a planet's mass using thebinary mass function. Radial velocity methods alone may only reveal a lower bound, since a large planet orbiting at a very high angle to theline 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]

Detection of exoplanets

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Main article:Doppler spectroscopy

The radial velocity method to detectexoplanets 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

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From the instrumental perspective, velocities are measured relative to the telescope's motion. So an important first step of thedata reduction is to remove the contributions of

theEarth's elliptic motion around the Sun at approximately ± 30 km/s,
amonthly rotation of ± 13 m/s of the Earth around the center of gravity of the Earth-Moon system,^[9]
thedaily 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 Earthpolar motion at the level of mm/s,
contributions of 230 km/s from the motion around theGalactic Center and associated proper motions.^[10]
in the case of spectroscopic measurements corrections of the order of ±20 cm/s with respect toaberration.^[11]
Sin i degeneracy is the impact caused by not being in the plane of the motion.

Notes

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^The norm, a nonnegative number, is multiplied by -1 if velocity (red arrow in the figure) and relative position form anobtuse angle or if relative velocity (green arrow) and relative position are antiparallel.

References

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^Hoffman, Kenneth M.; Kunzel, Ray (1971).Linear Algebra (Second ed.). Prentice-Hall Inc. p. 271.ISBN 0135367972.
^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 Archived 2022-10-09 at Ghost Archive
^^a ^bLindegren, Lennart; Dravins, Dainis (April 2003)."The fundamental definition of "radial velocity""(PDF).Astronomy and Astrophysics.401 (3):1185–1201.arXiv:astro-ph/0302522.Bibcode:2003A&A...401.1185L.doi:10.1051/0004-6361:20030181.S2CID 16012160. Retrieved4 February 2017.
^Dravins, Dainis; Lindegren, Lennart; Madsen, Søren (1999). "Astrometric radial velocities. I. Non-spectroscopic methods for measuring stellar radial velocity".Astron. Astrophys.348:1040–1051.arXiv:astro-ph/9907145.Bibcode:1999A&A...348.1040D.doi:10.1088/0004-637X/709/1/168.
^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 Archived 2022-10-09 at Ghost Archive
^Huggins, W. (1868). "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 II".Philosophical Transactions of the Royal Society of London.158:529–564.Bibcode:1868RSPT..158..529H.doi:10.1098/rstl.1868.0022.
^Anglada-Escude, Guillem; Lopez-Morales, Mercedes; Chambers, John E. (2010). "How eccentric orbital solutions can hide planetary systems in 2:1 resonant orbits".The Astrophysical Journal Letters.709 (1):168–78.arXiv:0809.1275.Bibcode:2010ApJ...709..168A.doi:10.1088/0004-637X/709/1/168.S2CID 2756148.
^Kürster, Martin; Trifonov, Trifon; Reffert, Sabine; Kostogryz, Nadiia M.; Roder, Florian (2015). "Disentangling 2:1 resonant radial velocity oribts from eccentric ones and a case study for HD 27894".Astron. Astrophys.577: A103.arXiv:1503.07769.Bibcode:2015A&A...577A.103K.doi:10.1051/0004-6361/201525872.S2CID 73533931.
^Ferraz-Mello, S.; Michtchenko, T. A. (2005). "Extrasolar Planetary Systems".Chaos and Stability in Planetary Systems. Lecture Notes in Physics. Vol. 683. pp. 219–271.Bibcode:2005LNP...683..219F.doi:10.1007/10978337_4.ISBN 978-3-540-28208-2.
^Reid, M. J.; Dame, T. M. (2016)."On the rotation speed of the Milky Way determined from HI emission".The Astrophysical Journal.832 (2): 159.arXiv:1608.03886.Bibcode:2016ApJ...832..159R.doi:10.3847/0004-637X/832/2/159.S2CID 119219962.
^Stumpff, P. (1985). "Rigorous treatment of the heliocentric motion of stars".Astron. Astrophys.144 (1): 232.Bibcode:1985A&A...144..232S.

External links

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The Radial Velocity Equation in the Search for Exoplanets ( The Doppler Spectroscopy or Wobble Method )Archived 2021-12-02 at theWayback Machine

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