Radial velocities
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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 Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
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 \in \mathbb^3 defining the instantaneous position of a target relative to an observer. Let with \mathbf \in \mathbb^3, the instantaneous
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of the target with respect to the observer. The magnitude of the position vector \mathbf is defined as The quantity range rate is the time
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the magnitude (
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
) of \mathbf, expressed as Substituting () into () : \frac = \frac Evaluating the derivative of the right-hand-side : \frac = \frac \frac \frac : \frac = \frac \frac using () the expression becomes : \frac = \frac \frac Since : \langle \mathbf,\mathbf \rangle = \langle \mathbf,\mathbf \rangle With : \hat =\frac The range rate is simply defined as : \frac = \frac = \langle \hat,\mathbf \rangle the projection of the observer to target velocity vector onto the \hat unit vector. A singularity exists for coincident observer target, i.e. \mathbf = \begin0 \\ 0 \\ 0\end. 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.''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 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 Astrometry is a branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. It provides the kinematics and physical origin of the Solar System and this galaxy, the Milky Way. Histor ...
observations (for example, a secular change in the annual
parallax Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects ...
).''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


Spectroscopic radial velocity

Light from an object with a substantial relative radial velocity at emission will be subject to the
Doppler effect The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...
, so the frequency of the light decreases for objects that were receding (
redshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
) and increases for objects that were approaching (
blueshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in fr ...
). The radial velocity of a
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
or other luminous distant objects can be measured accurately by taking a high-resolution
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
and comparing the measured
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
s of known
spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
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 Sirius is the list of brightest stars, brightest star in the night sky. Its name is derived from the Ancient Greek language, Greek word , or , meaning 'glowing' or 'scorching'. The star is designated α Canis Majoris, Latinisation ...
with respect to the Sun, based on observed redshift of the star's light. In many
binary star A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in wh ...
s, the
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
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 Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
es of the stars, and some
orbital element Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same ...
s, such as eccentricity and
semimajor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...
. The same method has also been used to detect
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s around stars, in the way that the movement's measurement determines the planet's orbital period, while the resulting radial-velocity
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
allows the calculation of the lower bound on a planet's mass using the
binary mass function In astronomy, the binary mass function or simply mass function is a function that constrains the mass of the unseen component (typically a star or exoplanet) in a single-lined spectroscopic binary star or in a planetary system. It can be calculated ...
. 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.


Detection of exoplanets

The radial velocity method to detect
exoplanet An exoplanet or extrasolar planet is a planet outside the Solar System. The first possible evidence of an exoplanet was noted in 1917 but was not recognized as such. The first confirmation of detection occurred in 1992. A different planet, init ...
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 Data reduction is the transformation of numerical or alphabetical digital information derived empirically or experimental data, experimentally into a corrected, ordered, and simplified form. The purpose of data reduction can be two-fold: reduce the ...
is to remove the contributions of *the Earth's elliptic motion around the sun at approximately ± 30 km/s, *a monthly rotation of ± 13 m/s of the Earth around the center of gravity of the Earth-Moon system, *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. *in the case of spectroscopic measurements corrections of the order of ±20 cm/s with respect to aberration. *
Sin i degeneracy In astronomy, minimum mass is the lower-bound calculated mass of observed objects such as planets, stars and binary systems, nebulae, and black holes. Minimum mass is a widely cited statistic for extrasolar planets detected by the radial veloci ...
is the impact caused by not being in the plane of the motion.


See also

* * * * *
Bistatic range rate Bistatic radar is a radar system comprising a transmitter and receiver that are separated by a distance comparable to the expected target distance. Conversely, a conventional radar in which the transmitter and receiver are co-located is called ...
*
Doppler effect The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...
*
Inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
* Orbit determination *
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although ...


References


The Radial Velocity Equation in the Search for Exoplanets ( The Doppler Spectroscopy or Wobble Method )


Further reading

* * Renze, John; Stover, Christopher; and Weisstein, Eric W. "Inner Product." From MathWorld—A Wolfram Web Resource.http://mathworld.wolfram.com/InnerProduct.html {{DEFAULTSORT:Radial Velocity Astrometry Concepts in astronomy Orbits