Doppler spectroscopy

Doppler spectroscopy(also known as theradial-velocity method,or colloquially, thewobble method) is anindirect methodfor findingextrasolar planetsandbrown dwarfsfromradial-velocitymeasurements via observation ofDoppler shiftsin thespectrumof theplanet's parent star. As of November 2022, about 19.5% of known extrasolar planets (1,018 of the total) have been discovered using Doppler spectroscopy.

Diagram showing how a smaller object (such as anextrasolar planet) orbiting a larger object (such as astar) could produce changes in position and velocity of the latter as they orbit their commoncenter of mass(red cross).

Doppler spectroscopy detects periodic shifts in radial velocity by recording variations in the color of light from the host star. When a star moves towards the Earth, its spectrum is blueshifted, while it is redshifted when it moves away from us. By analyzing these spectral shifts, astronomers can deduce the gravitational influence of extrasolar planets.

History

Exoplanets discovered by year (as of February 2014). Those discovered using radial velocity are shown in black, whilst all other methods are in light grey.

Otto Struveproposed in 1952 the use of powerfulspectrographsto detect distant planets. He described how a very large planet, as large asJupiter,for example, would cause its parent star to wobble slightly as the two objects orbit around their center of mass.He predicted that the small Doppler shifts to the light emitted by the star, caused by its continuously varying radial velocity, would be detectable by the most sensitive spectrographs as tinyredshiftsandblueshiftsin the star's emission. However, the technology of the time produced radial-velocity measurements with errors of 1,000m/sor more, making them useless for the detection of orbiting planets.The expected changes in radial velocity are very small – Jupiter causes theSunto change velocity by about 12.4 m/s over a period of 12 years, and the Earth's effect is only 0.1 m/s over a period of 1 year – so long-term observations by instruments with a very highresolutionare required.

Advances in spectrometer technology and observational techniques in the 1980s and 1990s produced instruments capable of detecting the first of many new extrasolar planets. TheELODIE spectrograph,installed at theHaute-Provence Observatoryin Southern France in 1993, could measure radial-velocity shifts as low as 7 m/s, low enough for an extraterrestrial observer to detect Jupiter's influence on the Sun.Using this instrument, astronomersMichel MayorandDidier Quelozidentified51 Pegasi b,a "Hot Jupiter"in the constellation Pegasus.Although planets had previously been detected orbitingpulsars,51 Pegasi b was the first planet ever confirmed to be orbiting amain-sequencestar, and the first detected using Doppler spectroscopy.

In November 1995, the scientists published their findings in the journalNature;the paper has since been cited over 1,000 times. Since that date, over 1,000 exoplanet candidates have been identified, many of which have been detected by Doppler search programs based at theKeck,Lick,andAnglo-AustralianObservatories (respectively, the California, Carnegie and Anglo-Australian planet searches), and teams based at theGeneva Extrasolar Planet Search.

Beginning in the early 2000s, a second generation of planet-hunting spectrographs permitted far more precise measurements. TheHARPSspectrograph, installed at theLa Silla Observatoryin Chile in 2003, can identify radial-velocity shifts as small as 0.3 m/s, enough to locate many possibly rocky, Earth-like planets.A third generation of spectrographs is expected to come online in 2017.With measurement errors estimated below 0.1 m/s, these new instruments would allow an extraterrestrial observer to detect even Earth.

Procedure

Properties (mass and semimajor axis) of planets discovered through 2013 using radial velocity, compared (light gray) with planets discovered using other methods.

A series of observations is made of the spectrum of light emitted by a star. Periodic variations in the star's spectrum may be detected, with thewavelengthof characteristicspectral linesin the spectrum increasing and decreasing regularly over a period of time. Statistical filters are then applied to the data set to cancel out spectrum effects from other sources. Using mathematicalbest-fittechniques, astronomers can isolate the tell-tale periodicsine wavethat indicates a planet in orbit.

If an extrasolar planet is detected, aminimum massfor the planet can be determined from the changes in the star's radial velocity. To find a more precise measure of the mass requires knowledge of the inclination of the planet's orbit. A graph of measured radial velocity versus time will give a characteristic curve (sine curvein the case of a circular orbit), and the amplitude of the curve will allow the minimum mass of the planet to be calculated using thebinary mass function.

The Bayesian Kepler periodogram is a mathematicalalgorithm,used to detect single or multiple extrasolar planets from successiveradial-velocitymeasurements of the star they are orbiting. It involves aBayesian statistical analysisof the radial-velocity data, using apriorprobability distributionover the space determined by one or more sets of Keplerian orbital parameters. This analysis may be implemented using theMarkov chain Monte Carlo(MCMC) method.

The method has been applied to theHD 208487system, resulting in an apparent detection of a second planet with a period of approximately 1000 days. However, this may be an artifact of stellar activity.The method is also applied to theHD 11964system, where it found an apparent planet with a period of approximately 1 year. However, this planet was not found in re-reduced data,suggesting that this detection was an artifact of the Earth's orbital motion around the Sun.

Although radial-velocity of the star only gives a planet's minimum mass, if the planet'sspectral linescan be distinguished from the star's spectral lines then the radial-velocity of the planet itself can be found and this gives the inclination of the planet's orbit and therefore the planet's actual mass can be determined. The first non-transiting planet to have its mass found this way wasTau Boötis bin 2012 whencarbon monoxidewas detected in the infrared part of the spectrum.

Example

The graph to the right illustrates thesine curveusing Doppler spectroscopy to observe the radial velocity of an imaginary star which is being orbited by a planet in a circular orbit. Observations of a real star would produce a similar graph, althougheccentricityin the orbit will distort the curve and complicate the calculations below.

This theoretical star's velocity shows a periodic variance of ±1 m/s, suggesting an orbiting mass that is creating a gravitational pull on this star. UsingKepler'sthird law of planetary motion,the observed period of the planet's orbit around the star (equal to the period of the observed variations in the star's spectrum) can be used to determine the planet's distance from the star ($r$) using the following equation:

$r^{3}={\frac {GM_{\mathrm {star} }}{4\pi ^{2}}}P_{\mathrm {star} }^{2}\,$

where:

• ris the distance of the planet from the star
• Gis thegravitational constant
• M_staris the mass of the star
• P_staris the observed period of the star

Having determined$r$,the velocity of the planet around the star can be calculated usingNewton'slaw of gravitation,and theorbit equation:

$V_{\mathrm {PL} }={\sqrt {GM_{\mathrm {star} }/r}}\,$

where$V_{\mathrm {PL} }$is the velocity of planet.

The mass of the planet can then be found from the calculated velocity of the planet:

$M_{\mathrm {PL} }={\frac {M_{\mathrm {star} }V_{\mathrm {star} }}{V_{\mathrm {PL} }}}\,$

where$V_{\mathrm {star} }$is the velocity of parent star. The observed Doppler velocity,$K=V_{\mathrm {star} }\sin(i)$,whereiis theinclinationof the planet's orbit to the line perpendicular to theline-of-sight.

Thus, assuming a value for the inclination of the planet's orbit and for the mass of the star, the observed changes in the radial velocity of the star can be used to calculate the mass of the extrasolar planet.

Radial-velocity comparison tables

PlanetMass	Distance AU	Star's Radial Velocity Due to the Planet (vradial)	Notice
Jupiter	5	12.7 m/s
Neptune	0.1	4.8 m/s
Neptune	1	1.5 m/s
Super-Earth(5 M🜨)	0.1	1.4 m/s
L 98-59 b(0.4M🜨)	0.02	0.46 m/s
Super-Earth(5 M🜨)	1	0.45 m/s
Earth	0.09	0.30 m/s
Earth	1	0.09 m/s

Ref:

Planets

Planet	Planet Type	Semimajor Axis (AU)	Orbital Period	Star's Radial Velocity Due to the Planet (m/s)	Detectable by:
51 Pegasi b	Hot Jupiter	0.05	4.23 days	55.9	First-generation spectrograph
55 Cancri d	Gas giant	5.77	14.29 years	45.2	First-generation spectrograph
Jupiter	Gas giant	5.20	11.86 years	12.4	First-generation spectrograph
Gliese 581c	Super-Earth	0.07	12.92 days	3.18	Second-generation spectrograph
Saturn	Gas giant	9.58	29.46 years	2.75	Second-generation spectrograph
L 98-59 b	Terrestrial planet	0.02	2.25 days	0.46	Third-generation spectrograph
Neptune	Ice giant	30.10	164.79 years	0.281	Third-generation spectrograph
Earth	Habitable planet	1.00	365.26 days	0.089	Third-generation spectrograph (likely)
Pluto	Dwarf planet	39.26	246.04 years	0.00003	Not detectable

For MK-type stars with planets in the habitable zone

Stellar mass (M☉)	Planetary mass (ME)	Lum. (L0)	Type	RHAB (AU)	RV (cm/s)	Period (days)
0.10	1.0	8×10−4	M8	0.028	168	6
0.21	1.0	7.9×10−3	M5	0.089	65	21
0.47	1.0	6.3×10−2	M0	0.25	26	67
0.65	1.0	1.6×10−1	K5	0.40	18	115
0.78	2.0	4.0×10−1	K0	0.63	25	209

Limitations

The major limitation with Doppler spectroscopy is that it can only measure movement along the line-of-sight, and so depends on a measurement (or estimate) of the inclination of the planet's orbit to determine the planet's mass. If the orbital plane of the planet happens to line up with the line-of-sight of the observer, then the measured variation in the star's radial velocity is the true value. However, if the orbital plane is tilted away from the line-of-sight, then the true effect of the planet on the motion of the star will be greater than the measured variation in the star's radial velocity, which is only the component along the line-of-sight. As a result, the planet'strue masswill be greater than measured.

To correct for this effect, and so determine the true mass of an extrasolar planet, radial-velocity measurements can be combined withastrometricobservations, which track the movement of the star across the plane of the sky, perpendicular to the line-of-sight. Astrometric measurements allows researchers to check whether objects that appear to be high mass planets are more likely to bebrown dwarfs.

A further disadvantage is that the gas envelope around certain types of stars can expand and contract, and some stars arevariable.This method is unsuitable for finding planets around these types of stars, as changes in the stellar emission spectrum caused by the intrinsic variability of the star can swamp the small effect caused by a planet.

The method is best at detecting very massive objects close to the parent star – so-called "hot Jupiters"– which have the greatest gravitational effect on the parent star, and so cause the largest changes in its radial velocity. Hot Jupiters have the greatest gravitational effect on their host stars because they have relatively small orbits and large masses. Observation of many separate spectral lines and many orbital periods allows thesignal-to-noise ratioof observations to be increased, increasing the chance of observing smaller and more distant planets, but planets like the Earth remain undetectable with current instruments.

Left:A representation of a star orbited by a planet. All the movement of the star is along the viewer's line-of-sight; Doppler spectroscopy will give a true value of the planet's mass.
Right:In this case none of the star's movement is along the viewer's line-of-sight and the Doppler spectroscopy method will not detect the planet at all.

See also

• Methods of detecting exoplanets
• Systemic (amateur extrasolar planet search project)

External links

• California and Carnegie Extrasolar Planet Search
• The Radial Velocity Equation in the Search for Exoplanets ( The Doppler Spectroscopy or Wobble Method )