THE THIRD SIGNATURE OF GRANULATION IN BRIGHT-GIANT AND SUPERGIANT STARS

Solar granulation can be seen directly and has been extensively observed and modeled. Stellar granulation cannot be seen directly, making its study more challenging and far less complete. But whether spatially resolved or otherwise, granulation is a measurable physical link to envelope convection, that powerful dynamic process that mixes chemicals, transports energy, and generates and shapes magnetic fields. Stellar granulation sends us two types of signals. Because of its variable granular structure, there is a small statistical variation in a star's brightness. This effect is now beginning to produce results (Guenther et al. 2008; Kallinger & Matthews 2010). Second, the motions of the granulation introduce Doppler shifts that broaden, shape, and position spectral lines, and it is here that our investigation fits in. Based on observational technique and historical development, we have three spectroscopic signatures or characteristics that have been used.

The first signature of granulation is the macroturbulence component of spectral line broadening. The dispersion of an assumed velocity distribution is commonly used as a measure of its strength, in this case, a measure of the dispersion of granulation velocities. Variations across the H-R diagram have been documented, with the general result that macroturbulence velocities are larger for stars of higher effective temperature and higher luminosity (e.g., Gray 2005a).

The second signature is the bisector of a line profile, which is characterized by its shape and velocity span. Velocity spans generally vary in concert with the macroturbulence strength, as do shape changes, including the position of the blue-most part of the "C" shaped bisectors (Gray 2005b). Bisectors having a reversed curvature are seen for hotter stars (Gray & Nagel 1989; Gray 1989, 2005a, 2010c; Landstreet 1998). An explanation of the origin of bisector shapes is given in previous publications (Gray 2010a, 2010b).

Recently, differential shifts of spectral lines, the third signature, have been exploited to get additional information on granulation as well as radial velocities corrected for convective blueshifts (Hamilton & Lester 1999; Allende Prieto et al. 2002; Gray 2009, hereafter Paper I; Ramírez et al. 2009, 2010). The basic third-signature plot consists of line bisectors portrayed on an absolute velocity scale. The physical interpretation of the differential shifts is predicated on weaker lines being formed deeper in the photosphere than stronger lines. Therefore, weaker lines are displaced by the characteristic velocities of granulation in deeper layers, while stronger lines respond to velocities higher in the photosphere. Granules dominate the light because they are typically brighter and cover more surface area than the intergranular lanes. Since the granules, at least in solar-type stars, are rising but decelerating, a net blueshift is introduced and that shift declines with height in the photosphere. In other words, weaker spectral lines are more blueshifted than stronger lines.

Earlier work showed that the line-core points of third-signature plots delineate a common shape, i.e., the plots differ from one star to the next by a scale factor (Paper I). In effect, the scale factor is a measure of the vigor of the granulation. As one might expect from the first and second signatures, the scale factor was found to be larger in hotter and in more luminous stars (dwarfs through giants). Further, a comparison of individual bisector shapes to the relation shown by the core points for the same star yields a "flux deficit" parameter that is a measure of the flux contrast between rising granules and intergranular lanes (Gray 2010a). An extension of this analysis to stars on the hot side of the granulation boundary (Gray 2010b) suggest that the granulation boundary is not a statement of where convection does or does not affect the photospheric velocity fields, but instead a statement of the relative velocity dispersion compared to the velocity gradient through the photosphere.

The study of the F8 supergiant, γ Cyg, which shows reversed bisectors, revealed that not all third-signature plots (core points) have the same shape, i.e., are scalable (Gray 2010c). Instead of the slope declining with height, as seen for solar-type stars, the opposite was found for γ Cyg. In other words, the granulation velocities fall away more rapidly with height than for the cooler, fainter stars. The important question then arises: how does the granulation gradient behave in evolved stars more generally? To address this question, we took spectroscopic observations of 18 F8-M1.5 stars in the upper part of the H-R diagram and constructed their third-signature plots, as presented below.

Spectroscopic observations were procured using the coude spectrograph at the Elginfield Observatory, University of Western Ontario. Aspects of this facility are discussed in numerous previous publications (e.g., Gray 1986, 2005b) and need not be reiterated here. Suffice it to say that the resolving power is ∼100,000, the observations spanned the wavelength range from ∼6217 Å to ∼6272 Å in the ninth order of the diffraction grating. An absolute wavelength scale was established using water-vapor lines inside the coude spectrograph (see Gray & Brown 2006; Paper I). Signal-to-noise ratios (S/Ns) varied widely on individual exposures according to the apparent magnitude of the star and sky conditions. Table 1 gives information for the program stars, including the number of exposures and the total S/N when all exposures are combined. Absolute magnitudes were computed from the V magnitude and revised Hipparcos parallaxes as listed in the SIMBAD internet database. In some of the plots below, in order to give a more complete picture of variations across the H-R diagram, we include results for lower-luminosity stars from Paper I. Figure 1 shows the positions of the program stars in the H-R diagram.

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All exposures were normalized by a flat-field lamp to remove individual CCD pixel differences and the continuum was fixed by placing a cubic spline through high points or obvious portions of continuum. A sample of the spectroscopic data is given in Figure 2. Note in particular the large increase in line broadening from dwarfs to supergiants and the concomitant increase in line blending. Line blending also increases markedly with declining effective temperature.

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A number of the program stars are known variables in brightness and/or radial velocity, and some are in binary systems. In most binary cases, the separation is sufficient to prevent a composite spectrum and in those few cases where it is not, the secondary is usually substantially fainter than the program star itself. A previously unrecognized (early-type) companion apparently contributes substantial light to our exposures of μ Gem. The two bright giants from Paper I, β Dra and ι Aur, were redone, non-Fe i lines were added (see next section), and the results slightly revised.

Since we seek absolute line shifts as a function of line strength, we need to know the absolute wavelengths of the spectral lines we measure. The absolute wavelengths of Fe i lines in this spectral region were taken from Nave et al. (1994). Individual line positions are uncertain to ∼10 m s⁻¹, according to Nave et al. Fourteen Fe i lines prove to be useful for our program stars and they are marked in Figure 2 with arrows. Not all of these lines remain unblended in all stars, so the actual number used varies from star to star. These are essentially the same lines used in Paper I. We supplemented the Fe i lines with 21 others in the field (V i, Ni i, Si i, Ti i, Sc ii, and Fe ii), marked by line segments in Figure 2, whose absolute wavelengths we determined using third-signature plots of ε Eri (K2 V), η Dra (G8 III), and α Ari (K2 III). Basically, we altered the wavelengths until the core points of the bisectors of the non-Fe i lines lay on the third-signature plot for the star as defined by the Fe i lines and the mean curve in Figure 14 of Paper I. These lines and the wavelengths we determined are listed in Table 2. The uncertainty is ∼0.0006 Å or ∼30 m s⁻¹.

Bisectors of the spectral lines were then derived in the usual manner, with difference in wavelength from the absolute wavelength of the line being converted to velocity units. Naturally the Doppler shift of the space motion of the star gives the same zero offset to each bisector for a given star and so the relative shift of the bisectors with line strength, which we study here, is not compromised. In the cases of α Sco (M1.5 Iab) and μ Gem (M3 III), the lines are so shallow and blended we could not compute meaningful bisectors and instead measured only the positions of the line cores by fitting vertically oriented parabolae to them. The results from all exposures of each star were averaged using weights proportional to their continuum photon count.

Aside from line blending, errors on bisector points arise mainly from the photometric errors, which in turn is mainly photon noise (see Table 1). The velocity component of the bisector error is basically the photometric error divided by the local slope of the profile (Gray 1983) and therefore is larger in the line core and the far wing portions. In addition, because the lines are broader in higher-luminosity stars, the slopes are smaller and the errors larger for a given photometric S/N in the continuum. Line blending is also larger in higher-luminosity stars causing even larger errors and reducing the number of useful lines. These errors are manifested as larger scatter in the third-signature plots of higher-luminosity stars.

The third-signature plots contain the essential results of our investigation. To reiterate from an earlier publication, the core points of the bisectors have less chance of being distorted by blends than other parts of the bisector, so we concentrate on core points in the analysis. A selection of core-point third-signature plots is given in Figures 3–6. Data for stars not listed in Table 1 (Classes III and V) are from Paper I, but with the non-Fe i lines added (◊ symbols). Although the atomic excitation potentials range from 0.0 to 5.6 eV, there is no obvious correlation between excitation potential and velocity shift.

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The slope of the relations in these plots is typically less steep for higher-luminosity stars, i.e., the span of velocities through the photosphere is larger. But a rather striking change occurs toward the highest luminosities, namely, the slope reverses and the velocity spanned is much larger, as illustrated in the plots for 22 Vul, 47 Cyg, and α Sco. Interestingly, the G8 supergiant ε Gem does not show such a reversal. In addition, some of the stars show transition behavior, β Aqr (Figure 3), ι Cnc (Figure 4), perhaps γ¹ And (Figure 5), and μ Gem (Figure 6), for example, where the weak line portions show an upturn or reversal.

The hottest of our program stars, 41 Cyg (F5 II, not shown in the figures), has its weaker lines more blueshifted, but the relation has the convex curvature exhibited by γ Cyg (F8 Iab; Gray 2010c). Both of these stars are on the hot side of the granulation boundary.

For many of the third-signature plots, it is possible to match the standard curve from Paper I, shifting and scaling the velocity coordinate as explained in Paper I. Some of these are shown as the solid lines in Figures 3–5. The shift yields the radial velocity of the star with the convective blueshifts effectively removed. (The differential gravitational redshift has not been removed.) The scale factor is relative to the solar third-signature plot and therefore represents the velocity range exhibited by granulation through the depth of the star's photosphere normalized to the solar range. However, scale factors may have dubious meaning for transition-type plots, where only the stronger-line portion is used to fix the scale. Naturally, the stars showing reversed slope are not scalable. Table 3 summarizes relevant results. A "t" is appended to the scale factors in Table 3 to indicate the additional uncertainty stemming from the uncertain meaning of the transition-type behavior. Errors are estimated by scaling to the extreme limits of possible agreement with the standard plot. There is also a possible systematic error arising from the shallowing of spectral lines caused by rotation and macroturbulence (macroturbulence increases systematically with luminosity), or light from a companion, as appears to be the case for μ Gem. When there is greater line broadening, the core points of the third-signature plot are raised slightly. Although this results in a small slope change, the main effect is to misinterpret the vertical shift as a velocity shift. The effect is largest in the supergiants and will lower the derived velocity ∼70 m s⁻¹, which is at or below the measurement noise. On the other hand, a zero-level change from an early-type companion can cause a serious change in the apparent slope and deduced scale factor. Numerical experiments show that a 10% variation in zero level can alter the scale factor ∼25% of its value. The only star on our list that has significant zero-level contamination, as far as we can tell, is μ Gem (Figure 7), and we have not attempted to find a scale factor for it because it does not show the solar shape.

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The last column in Table 3 serves as an indication of intrinsic velocity variability, as seen in our data, but since the times of observation are relatively few, larger variation cannot be ruled out.

Scale factors are shown as a function of spectral type in Figure 7. Values for dwarfs and giants are from Paper I. Throughout the G-star span, evolved stars have scale factors ∼1.5 times larger than dwarfs. But toward the right of the diagram, in the K-star range, they seem to converge with the dwarfs, showing little or no difference. Supergiant scale factors are not generally larger than those of bright giants. (Perhaps supergiants do have larger values in the K-star range, but with only one K supergiant in our sample, this remains uncertain.)

Third-signature plots showing reversed slope occur, at least in our sample, for the highest luminosity stars, namely, 22 Vul, 47 Cyg, and α Sco (but see below), while stars of lesser luminosity, β Aqr, ι Cnc, and μ Gem, show what appears to be the beginning of a reversal for the weaker lines. The velocities spanned in the reversed plots are several times larger than in the solar-type plots.

While the scale factor is a measure of the range in granulation flow velocities through the photosphere, the macroturbulence dispersion is a measure of the spread or dispersion in granulation velocities. Values of macroturbulence dispersion were taken from previous publications (Gray 1986; Gray & Toner 1987) and are plotted against the scale factors in Figure 8. There appears to be a multiplicity of relationships, one for each luminosity class. Along each dashed line shown in the figure, there is a strong correlation with effective temperature (hotter to the right). There are too few supergiants to map out their relationship, if it exists. In any case, generally speaking, the dispersion of granulation velocities increases with luminosity, effective temperature, and scale factor.

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The third-signature plot for α Sco changes over time. We grouped and averaged the results for α Sco from 2008 April (seven exposures) and 2009 April (five exposures) to give the result plotted in Figure 9. There is considerable scatter because (1) the spectral lines are broad, so their positions are not sharply defined and (2) there is enormous line blending that distorts the line positions. Nevertheless, we can see that the velocities mainly became more negative and the spectral lines grew slightly shallower (ΔF/F_c ∼ −0.03) over the one-year interval. Weaker lines show much larger velocity excursions than stronger lines, and there is even a hint of velocity shift of the opposite sign for the strongest lines.

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Variation cannot be ruled out for the other supergiants. Unfortunately, we have only three exposures of 22 Vul within a few weeks time span, insufficient to make a definitive statement. The same situation applies to 47 Cyg.

From third-signature plots, we deduce some of the physical characteristics of stellar granulation. Since the strongest lines we measure have cores formed high in the photosphere, that is, high in the overshoot region of convection, the velocities are expected to be small, approaching zero. The weakest lines are formed near the bottom of the photosphere, and so the velocity difference observed in the third-signature plot tells us the size of the granulation rise velocities in the deep layers of the photosphere. The weakness of this simple approach is that the strongest lines we happen to have on the plots might not have cores deep enough to be formed at the top of the photosphere. The scale factor tells us similar information, avoiding the necessity of having line strengths spanning the full photospheric range of depths, but requires that we know the granulation velocities of the solar photosphere.

The full story is somewhat more complex. To start with, we are dealing with an average over the visible hemisphere of the star, which implies true velocities a few percent higher than given directly from the observations. The details here will depend on the limb darkening (see Chiavassa et al. 2009; Neilson 2012) and center-to-limb variations of the effects of granulation on the spectrum, including possibly corrugation-type effects (Dravins & Nordlund 1990). The "depth of formation," even for the cores of lines, spans a considerable range of depth and therefore velocities. In regions where the velocity gradients are large, we are forced to deal with averages. Should we wish to actually scale from the solar velocities, we have numerous choices, e.g., Jin et al. (2009), Nordlund et al. (2009), Kostik et al. (2009), de la Cruz Rodriguez (2011), which leads to some ambiguity. On the other hand, if one is computing models, internal consistency then dictates that the model for the star has the observed scale factor compared to the model for the Sun. Nevertheless and details aside, this general interpretation of third-signature plots is likely to be sound for stars showing the same generic behavior as the Sun's.

Differences in metallicity might be suspected as a factor of significance. While it is true that lower metallicity makes for a more transparent photosphere because there are fewer electron donors, the actual geometrical span of the photosphere changes by only a small amount because the decline in absorption coefficient is offset by an increase in density. In effect, the optical-depth span of the photosphere is shifted slightly deeper relative to the geometrical depth. This would have two consequences. The stronger would result in an overcorrection for convective blueshifts, making the deduced absolute radial velocity too high by ∼100 m s⁻¹ dex⁻¹ in [Fe/H]. Second, the slope of the third signature would be altered if the convective velocity gradient changes significantly, i.e., this is a second-order effect. In practice, the program stars are near solar metallicity, showing nowhere near a one dex deviation from solar values, and so the effects of metallicity on the results given here are expected to be minimal. However, these ideas should be verified with three-dimensional hydrodynamical models before being considered definitive.

What is to be made of the supergiants' reversed-slope cases? Our views of stellar granulation are naturally shaped by the solar case, and much of what has been said above is predicated on the solar-type behavior. In particular, the hotter material in rising granules is assumed to dominate the integrated light and produce the blueshifts. But this rather straightforward interpretation could be misleading if the basic structure of granulation differs between dwarfs and supergiants. In the thinner photospheres of supergiants, radiative exchange or other processes might possibly smooth out the (horizontal) temperature differences, or differences in size and characteristics of convection cells might alter the asymmetry balance, giving more weight to the falling material compared to dwarf-star case. As a result, both rising and falling material in supergiants might influence the core position of the spectral lines, reducing or even reversing the apparent velocities in the third-signature plots. Perhaps the explanation of the reversed slopes in supergiant third-signature plots lies here. Evidence for significant changes in granulation with position in the H-R diagram has been found in photometric variations (Mathur et al. 2011) and is also seen in models (e.g., Robinson et al. 2003; Collet et al. 2007; Freytag & Höfner 2008; Stothers 2010), but as far as we know, no one has computed third-signature plots for bright giants or supergiants.

Time variation of the third signature, as in Figure 9, opens a new page in the dynamic behavior of stellar photospheres. Two common explanations for time variations of cool supergiants are (1) pulsation and (2) giant convection cells accompanied by resonance. Curiously, the weaker lines in Figure 9 show the greater velocity variation. This implies that the deeper layers are moving more than the higher layers. Such behavior seems implausible for pulsation given the outward decline in density. Large convection cells, on the other hand, offer a simple explanation: the cells penetrate the lower photosphere where the weak lines are formed, but do not rise as high as the upper photosphere thus leaving the cores of strong lines relatively untouched. If the cell is large enough to dominate the light from the visible hemisphere, the star will appear to show expansion as the cell rises, followed by apparent contraction as the cell falls. This is similar to the explanation proposed for the variation of Betelgeuse (Gray 2008). Time variations of third-signature plots offer interesting and powerful constrains on models of granulation in evolved stars.

The case of α Sco (Figure 9) might also point to a more general explanation of third-signature plots with reversed slope. Namely that all reversed-slope cases are time variable and our observations have caught only the reversed-slope phases. We expect to bring more details to light in future publications.

We are grateful to M. Debruyne for technical assistance at the Elginfield Observatory and to the Natural Sciences and Engineering Research Council of Canada for financial support.