Doppler imaging

Last updated November 16, 2019

Inhomogeneous structures on stellar surfaces, i.e. temperature differences, chemical composition or magnetic fields, create characteristic distortions in the spectral lines due to the Doppler effect. These distortions will move across spectral line profiles due to the stellar rotation. The technique to reconstruct these structures on the stellar surface is called Doppler-imaging, often based on the Maximum Entropy image reconstruction to find the stellar image. This technique gives the smoothest and simplest image that is consistent with observations.

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

The Doppler effect 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 described the phenomenon in 1842.

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 identify atoms and molecules. These "fingerprints" can be compared to the previously collected "fingerprints" of atoms and molecules, and are thus used to identify the atomic and molecular components of stars and planets, which would otherwise be impossible.

To understand the magnetic field and activity of stars, studies of the Sun are not sufficient. Therefore, studies of other stars are necessary. Periodic changes in brightness have long been observed in stars which indicate cooler or brighter starspots on the surface. These spots are larger than the ones on the Sun, covering up to 20% of the star. Spots with similar size as the ones on the Sun would hardly give rise to changes in intensity. In order to understand the magnetic field structure of a star, it is not enough to know that spots exist because their location and extent are also important.

The Sun, or Sol, is the star at the center of the Solar System. It is a nearly perfect sphere of hot plasma, with internal convective motion that generates a magnetic field via a dynamo process. It is by far the most important source of energy for life on Earth. Its diameter is about 1.39 million kilometers, or 109 times that of Earth, and its mass is about 330,000 times that of Earth. It accounts for about 99.86% of the total mass of the Solar System. Roughly three quarters of the Sun's mass consists of hydrogen (~73%); the rest is mostly helium (~25%), with much smaller quantities of heavier elements, including oxygen, carbon, neon, and iron.

Starspots are stellar phenomena, so-named by analogy with sunspots. Spots actually at the size of sunspots would be very hard to detect on other stars because they are too small to cause detectable fluctuations in brightness. The commonly observed starspots are in general much larger than those on the Sun: up to about 30% of the stellar surface may be covered, corresponding to starspots 100 times larger than those on the Sun.

History

Doppler imaging was first used to map chemical peculiarities on the surface of Ap stars. For mapping starspots it was first used by Steven Vogt and Donald Penrod in 1983, when they demonstrated that signatures of starspots were observable in the line profiles of the active binary star HR 1099 (V711 Tau); from this they could derive an image of the stellar surface.

A binary star is a star system consisting of two stars orbiting around their common barycenter. Systems of two or more stars are called multiple star systems. These systems, especially when more distant, often appear to the unaided eye as a single point of light, and are then revealed as multiple by other means.

Criteria for Doppler Imaging

In order to be able to use the Doppler imaging technique the star needs to fulfill some specific criteria.

The stellar rotation needs to be the dominating effect broadening spectral lines, ${\textstyle V\sin i=10-100~{\mbox{km}}\,{\mbox{s}}^{-1}}$ .

Stellar rotation angular motion of a star about its axis

Stellar rotation is the angular motion of a star about its axis. The rate of rotation can be measured from the spectrum of the star, or by timing the movements of active features on the surface.

The projected equatorial rotational velocity should be at least ,

V\sin i<10~{\mbox{km}}\,{\mbox{s}}^{-1}

. If the velocity in lower, spatial resolution is degraded, but variations in the line profile can still give information of areas with higher velocities. For very high velocities ,

V\sin i>100~{\mbox{km}}\,{\mbox{s}}^{-1}

., lines become too shallow for recognizing spots.

The inclination angle, i, should preferably be between 20˚-70˚.

When i =0˚ the star is seen from the pole and therefore there is no line-of-sight component of the rotational velocity, i.e. no Doppler effect. When seen equator-on, i =90˚ the Doppler image will get a mirror-image symmetry, since it is impossible to distinguish if a spot is on the northern or southern hemisphere. This problem will always occur when i ≥70˚; Doppler images are still possible to get but harder to interpret.

Theoretical Basis

In the simplest case, dark starspots decrease the amount of light coming from one specific region; this causes a dip or notch in the spectral line. As the star rotates the notch will first appear on the short wavelength side when it becomes visible towards the observer. Then it will move across the line profile and increase in angular size since the spot is seen more face-on, the maximum is when the spot passes the star's meridian. The opposite happens when the spot moves over to the other side of the star. The spot has its maximum Doppler shift for;

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, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

In astronomy, the meridian is the great circle passing through the celestial poles, as well as the zenith and nadir of an observer's location. Consequently, it contains also the north and south points on the horizon, and it is perpendicular to the celestial equator and horizon. A celestial meridian is coplanar with the analogous terrestrial meridian projected onto the celestial sphere. Hence, the number of celestial meridians is also infinite.

\Delta \lambda =v\sin i\cos l\sin L~{\mbox{km}}\,{\mbox{s}}^{-1}

Where l is the latitude and L is the longitude. Thus signatures from spots at higher latitudes will be restricted to spectral line centers, which will also occurring when the rotation axis is not perpendicular to the line of sight. If the spot is located at high latitude it is possible that it will always be seen, in which case the distortion in the line profile will move back and forth and only the amount of distortion will change.

Doppler imaging can also be made for changing chemical abundances across the stellar surface; these may not give rise to notches in the line profile since they can be brighter than the rest of the surface, instead producing a dip in the line profile.

Zeeman-Doppler imaging

The Zeeman-Doppler imaging is a variant of the Doppler imaging technique, by using circular and linear polarization information to see the small shifts in wavelength and profile shapes that occur when a magnetic field is present.

Binary stars

Another way to determine and see the extent of starspots is to study stars that are binaries. Then the problem with i =90° is reduced and the mapping of the stellar surface can be improved. When one of the stars passes in front of the other there will be an eclipse, and starspots on the eclipsed hemisphere will cause a distortion in the eclipse curve, revealing the location and size of the spots. This technique can be used for finding both dark (cool) and bright (hot) spots.

Related Research Articles

Sunspots are temporary phenomena on the Sun's photosphere that appear as spots darker than the surrounding areas. They are regions of reduced surface temperature caused by concentrations of magnetic field flux that inhibit convection. Sunspots usually appear in pairs of opposite magnetic polarity. Their number varies according to the approximately 11-year solar cycle.

Differential rotation is seen when different parts of a rotating object move with different angular velocities at different latitudes and/or depths of the body and/or in time. This indicates that the object is not solid. In fluid objects, such as accretion disks, this leads to shearing. Galaxies and protostars usually show differential rotation; examples in the Solar System include the Sun, Jupiter and Saturn.

An RS Canum Venaticorum variable is a type of variable star. The variable type consists of close binary stars having active chromospheres which can cause large stellar spots. These spots are believed to cause variations in their observed luminosity. Systems can exhibit variations on timescales of years due to variation in the spot surface coverage fraction, as well as periodic variations which are, in general, close to the orbital period of the binary system. Some systems exhibit variations in luminosity due to their being eclipsing binaries. Typical brightness fluctuation is around 0.2 magnitudes. They take their name from the star RS Canum Venaticorum.

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A stellar magnetic field is a magnetic field generated by the motion of conductive plasma inside a star. This motion is created through convection, which is a form of energy transport involving the physical movement of material. A localized magnetic field exerts a force on the plasma, effectively increasing the pressure without a comparable gain in density. As a result, the magnetized region rises relative to the remainder of the plasma, until it reaches the star's photosphere. This creates starspots on the surface, and the related phenomenon of coronal loops.

In astrophysics, Zeeman–Doppler imaging is a tomographic technique dedicated to the cartography of stellar magnetic fields, as well as surface brightness and temperature distributions.

Ap and Bp stars are chemically peculiar stars of types A and B which show overabundances of some metals, such as strontium, chromium and europium. In addition, larger overabundances are often seen in praseodymium and neodymium. These stars have a much slower rotation than normal for A and B-type stars, although some exhibit rotation velocities up to about 100 kilometers per second.

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Phi Phoenicis, Latinized from φ Phoenicis, is a binary star system in the southern constellation of Phoenix. It is faintly visible to the naked eye with an apparent visual magnitude of 5.1. Based upon an annual parallax shift of 10.48 mas as seen from Earth, it is located around 310 light years from the Sun. It is moving away from the Sun with a radial velocity of 10.4 km/s.

Xi Phoenicis, Latinized from ξ Phoenicis, is a visual binary star system in the southern constellation of Phoenix. It is faintly visible to the naked eye, having an apparent visual magnitude of 5.70. Based upon an annual parallax shift of 14.61 mas as measured from Earth, it is located around 223 light years from the Sun. The system is moving away from the Sun with a radial velocity of about +10 km/s.

References

Vogt et al. (1987),"Doppler images of rotating stars using maximum entropy image reconstruction ", ApJ, 321, 496V
Vogt, Steven S., & G. Donald Penrod, "Doppler Imaging of spotted stars - Application to the RS Canum Venaticorum star HR 1099" in Astronomical Society of the Pacific, Symposium on the Renaissance in High-Resolution Spectroscopy - New Techniques, New Frontiers, Kona, HI, June 13–17, 1983 Publications of the Astronomical Society of the Pacific, vol. 95, Sept. 1983, p. 565–576.
Strassmeier,( 2002 ),"Doppler images of starspots", AN, 323, 309S
Korhonen et al. (2001), "The first close-up of the ``flip-flop phenomenon in a single star", A&A, 379L, 30K
S.V.Berdyugina (2005), "Starspots: A Key to the Stellar Dynamo", Living Reviews in Solar Physics, vol. 2, no. 8
K.G.Strassmeier (1997), "Aktive sterne. Laboratorien der solaren Astrophysik", Springer, ISBN 3-211-83005-7
Gray, "The Observation and Analysis of Stellar Photospheres", 2005, Cambridge University Press, ISBN 0521851866
Collier Cameron et al., "Mapping starspots and magnetic fields on cool stars"

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