The Stagger-grid: A grid of 3D stellar atmosphere models - VI. Surface appearance of stellar granulation
AAstronomy & Astrophysics manuscript no. sg_gran_v01 c (cid:13)
ESO 2018October 29, 2018
The S tagger -grid: A grid of 3D stellar atmosphere models
VI. Surface appearance of stellar granulation
Z. Magic , and M. Asplund Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germanye-mail: [email protected] Research School of Astronomy & Astrophysics, Cotter Road, Weston ACT 2611, AustraliaReceived ...; Accepted...
ABSTRACT
Context.
In the surface layers of late-type stars, stellar convection is manifested with its typical granulation pattern due to the presenceof convective motions. The resulting photospheric up- and downflows leave imprints in the observed spectral line profiles.
Aims.
We perform a careful statistical analysis of stellar granulation and its properties for di ff erent stellar parameters. Methods.
We employ realistic 3D radiative hydrodynamic (RHD) simulations of surface convection from the S tagger -grid, a com-prehensive grid of atmosphere models that covers a large parameter space in terms of T e ff , log g , and [Fe / H]. Individual granules aredetected from the (bolometric) intensity maps at disk center with an e ffi cient granulation pattern recognition algorithm. From thesewe derive their respective properties: diameter, fractal dimension (area-perimeter relation), geometry, topology, variation of intensity,temperature, density and velocity with granule size. Also, the correlation of the physical properties at the optical surface are studied. Results.
We find in all of our 3D RHD simulations stellar granulation patterns imprinted, which are qualitatively similar to the solarcase, despite the large di ff erences in stellar parameters. The granules exhibit a large range in size, which can be divided into twogroups – smaller and larger granules – by the mean granule size. These are distinct in their properties: smaller granules are regularshaped and dimmer, while the larger ones are increasingly irregular and more complex in their shapes and distribution in intensitycontrast. This is reflected in their fractal dimensions, which is close to unity for the smaller granules, and close to two for largergranules, which is due to the fragmentation of granules. Conclusions.
Stellar surface convection seems to operate scale-invariant over a large range in stellar parameters, which translatesinto a self-similar stellar granulation pattern.
Key words. convection – hydrodynamics – radiative transfer – stars: atmospheres – stars: general– stars: late-type – stars: solar-type
1. Introduction
In the envelopes of cool stars, the energy resulting from thenuclear burning at the center is transported through convectiveenergy transport, which involves the ascension of hot, buoyantplasma towards the surface. At the optical surface, the over-turning convective motions into downdrafts do not come to restimmediately, instead the upflows overshoot well into the visi-ble photosphere due to its inertia, hence leaving an imprint inthe emergent radiation in form of a typical granulation pattern.The visible stellar surface of late-stars is patterned with brightelements (granules) interspersed by the dark intergranular lane.Understanding convection is an important aspect for the energytransport in late-type stars, however, this is a non-trivial task dueto the non-linear and non-local nature of (turbulent) surface con-vection.The Sun shows a distinct granulation pattern on its observ-able (optical) surface, which is the manifestation of convectionthat transports energy to the surface. The solar granulation pat-tern has been subject to manifold observational studies over thelast decades with progressively increasing resolution due to tech-nological advances (e.g. Roudier & Muller 1986; Hirzbergeret al. 1997, 1999a,b; Schrijver et al. 1997; Bovelet & Wiehr2007, 2001; Abramenko et al. 2012). Nowadays high-resolution
Send o ff print requests to : [email protected] solar observations are comparable to the typical numerical reso-lution with a few tens of kilometers. The observational develop-ments were accompanied by improvements in the detection andderivation of statistical properties of solar granulation. More-over, details of the solar small-scale magnetic structures has alsobeen studied (Solanki 1993; Janßen et al. 2003; Carlsson et al.2004; Abramenko & Longcope 2005; Stein & Nordlund 2006;Wiehr & Bovelet 2009).Until the advent of realistic 3D radiative hydrodynamic(RHD) simulations, which involves the (computationally expen-sive) solution of the hydrodynamic equations coupled with a re-alistic radiative transfer (Nordlund 1982; Ste ff en et al. 1989), di-rect comparisons of theoretical predications with the solar gran-ulation properties were absent. In contrast to theoretical 1Dmodels, such 3D models are capable of predicting the typicalstellar granulation pattern imprinted in the (bolometric) intensitymap emerging from the stellar surface. They have revealed thatstellar surface convection is driven by the large-amplitude en-tropy fluctuations in a thin optical surface boundary layer, wherethe energy can escape into space (see Stein & Nordlund 1998;Nordlund et al. 2009). The dark intergranular lanes stem fromthe entropy-deficient plasma that descends into narrow turbu-lent downdrafts, while the granules are warm upflowing plasma.These flows exhibit a distinct asymmetry in its thermodynamicproperties, leading to inhomogeneities and velocities, which has Article number, page 1 of 12 a r X i v : . [ a s t r o - ph . S R ] M a y . . . . gray contour s) over-plotted with the colored-coded contours of the recognized gran-ules with solar ( top panel ) and [Fe / H] = − . bottom ). From left to right : the Sun ( T e ff / log g = / . ff star(6500 K , . , .
0) and K-dwarf (4500 K , . cfist -grid employing the co bold -code, studied the im-pact of granulation (see Ludwig et al. 2009; Freytag et al. 2012;Kuˇcinskas et al. 2013; Tremblay et al. 2013). Trampedach et al.(2013) established a grid of 3D RHD models with solar metal-licity. Also, e ff orts are being made with the muram -code (seeVögler & Schüssler 2003; Beeck et al. 2013a,b). We have com-puted the S tagger -grid, a grid of 3D RHD model atmospheres(Magic et al. 2013a,b, hereafter Paper I and II), and used our 3Dmodels for several applications (Magic et al. 2014b,a, hereafterPaper III and IV).In the present work, we perform an extensive analysis ofthe stellar granulation properties based on the S tagger -grid. Wewant to address the following key question: How do the stellargranulation properties change for di ff erent stellar parameters?First, we explain the granule recognition method that we haveused to detect the individual granules from the (bolometric) in-tensity maps of the 3D simulations (Sect. 2). Then, we discusssuccessively the various properties of individual granules, suchas their diameter (Sect. 3), the fractal dimension (Sect. 4), thegeometry (Sects. 5), the variation of the intensity, temperature,density, velocity with granule size (Sect. 6), and finally the prop-erties at the optical surface (Sect. 7).
2. The granule recognition method
Several methods for detecting granules in observed solar imageshave been developed over the years. Classically, a single-levelclip of an intensity image is used for the granule recognition,where the small and large features are filtered out by spatial pass-band Fourier filtering. These "Fourier-based recognition" tech-niques have been the most commonly applied ones in the past,and are fast, but also inaccurate (see Roudier & Muller 1986;Hirzberger et al. 1997). Another possible approach is to tracegranules with a single fixed relative intensity-level, e.g., between0 .
97 and 1 .
03, as proposed by Abramenko et al. (2012). How-ever, in this work we prefer the more robust "multiple level track-ing" algorithm that was developed by Bovelet & Wiehr (2001).It is a simple, yet very powerful tool to extract the granules fromthe (bolometric) intensity map alone. The basic idea behind thismethod is to find (granular) shapes repeatedly for decreasing in-tensity level clips, thereby increasing their filling factors, until apredefined threshold filling factor is matched. One obtains un-ambiguously the granules with a single input parameter being thefilling factor for the upflows, f up . Since the latter is basically thesame for all stellar parameters with f up ≈ / ff star, a K-giant,and a K-dwarf as well as their metal-poor ([Fe / H] = −
3) analogsin Fig. 1.Following the multiple level tracking algorithm, we tracedthe granules in our simulations and computed the respective fill-
Article number, page 2 of 12. Magic and M. Asplund: The Stagger-grid – VI. Surface appearance of stellar granulation
Fig. 2: Comparison of the solar granules detected from di ff erent variables. Left figure : the visible (bolometric) intensity map (orangecontour); middle figure : the averaged vertical velocity (Eq. 1; up / down: blue / red); and right figure : the integrated temperature excess(Eq. 2; gray contour).ing factor, f i , of the considered intensity-level. We started at therelative intensity ¯ I = . .
97 in steps of 0 .
01 until the threshold with f up = .
60 wasreached for all stellar parameters. We chose the threshold valuebeing slightly lower than the average filling factor ( f up ≈ / I smoothed by a boxcar aver-age with window size of 5 pixels, and computed the root-mean-square (RMS) for I rms = I − ˆ I . Then, the bright points were de-tected at the 2 σ -threshold relative to the other granules. We per-formed the granule-recognition for each stored snapshot of thetime-series, thereby leading to large sample of granules for anindividual simulation (e.g., Sun ∼ Besides the intensity map, we also detected granules from thevertical velocity and the temperature excess with the multiple-level-tracking algorithm. The emergent (bolometric) intensity iscomputed for disk-center, therefore, it is a 2D representation ofthe 3D granulation structure present in the superadiabatic regionof the convection zone. To account for the depth-dependenceof the coherent granulation structures, we averaged the verticalvelocity and integrated the temperature excess as well.We averaged the vertical velocity on layers of constantRosseland optical depth from the optical surface to the peak(maximum) of the superadiabatic gradient (the granulation canbe found in these layers best) over ten equidistant layers in stepsof ∆ log τ Ross = .
1, i.e.˜ v z ( x , y ) = (cid:88) i = v z ( x , y , z i ) . (1) The granules are determined on the zero velocity contour of˜ v z ( x , y ). The averaged vertical velocity is independent of anyinput parameter, however, it relies on the assumption that the co-herent granular structures are given in the superadiabatic region.One could also use the vertical velocity present at the opticalsurface, however, then one would neglect the depth-dependence,while our approach by averaging over several layers does.We computed the temperature fluctuations for each lay-ers by normalizing it with the horizontal average, i.e. δ T z =∆ T z / (cid:104) T (cid:105) z . Then, we determined the temperature excess, δ T = δ T > max( (cid:104) δ T (cid:105) z ) /
5, which are the (positive) temperature fluc-tuations above the threshold of one-fifth of the maximum meanhorizontal temperature fluctuations. The relative threshold al-lows the method to work for di ff erent stellar parameters. We in-tegrated the temperature fluctuations from just above the opticalsurface to the bottom of the simulation box with Θ ( x , y ) = (cid:90) bot (cid:104) τ (cid:105) = . δ T ( z ) dz . (2)The granules are retrieved with the clip-level of the unity (av-erage) contour of Θ ( x , z ). In contrast to the intensity map, thismethod is independent of any assumption on the filling factor.We remark that the temperature excess, δ T , is a convenient quan-tity to illustrate the topology of the superadiabatic convectivecells (under-dense regions with heat-excess).In Fig. 2, we compare the detected granules from a solarsimulation snapshot with the three methods. Since the under-lying variables exhibit distinct features, the recognized granulesdi ff er slightly, in particular, the boundaries between close gran-ules is interpreted di ff erently. Nonetheless, one can clearly seethat the granules correlate with bright upflowing regions of sig-nificant T -excess.The above mentioned granule detection methods assume thatbright regions in the intensity maps associate with the hotter andlighter stellar plasma leading to the upflowing granules, which isnormally fulfilled, as we demonstrate in Sect. 7.2. The remain-ing dark regions in the intensity maps inherently consist of cool,dense gas, which one usually refers as the (negatively buoyant)downdrafts. Article number, page 3 of 12a) (b)
Fig. 3:
Left figure : The linear and logarithmic histogram of the granule area, A , ( top panel ; blue and black line, respectively) andthe area contribution, f ac , ( bottom ) derived from our solar simulation. The bin sizes for the histograms are 0 .
186 Mm and 0 .
041 dex.We indicated the location of the mean granule area averaged over all granules ( dashed ) and the maximum of f ac ( solid line ). Rightfigure : The logarithmic histograms of the granule size, d gran , smoothed with a moving-average over 10 elements. Furthermore, weindicated the mean granule diameters ( dotted lines , see Fig. 4) and the dominant scales, d ac ( filled square s). Note the di ff erence inabscissa between the top and bottom panel.
3. Diameter of granules
From the area of the granules we determined the equivalent di-ameter with d gran = (cid:112) A gran /π, which is the diameter of a circlethat has the same area A gran . With the granule size we refer to d gran or A gran in the following. Furthermore, we determined theunique barycenter, x bc = (cid:80) x i A i / A gran , where the summation runsover all pixels enclosed by the contour of the granule, and x i isthe vector pointing to the cell i , and A i is the pixel area. Thegranule size is the first property we want to address, therefore,we show the histogram of the granule areas of the Sun in Fig. 3a.The range in granule sizes is very large (typically spanning fourorders of magnitude), therefore, a histogram considering a linearequidistant granule size for the histogram bins would overesti-mate the smallest values by employing very large steps (see Fig.3a). This would result in a bottom-heavy distribution and themisleading conclusion of a dominant a large number of smallgranules (see Roudier & Muller 1986; Hirzberger et al. 1997;Abramenko et al. 2012). Therefore, we advise against a linearbinning of the histograms for starkly varying quantities like thegranule size, and instead consider logarithmic granule area forthe histograms. In Fig. 3a the histogram exhibits a maximumclose to the mean granule size, which we refer to as the modeof granule size, i.e. d h = max (cid:2) p ( A ) (cid:3) . The distribution aroundthe mode of the granule size is very asymmetric with a long tailtowards smaller size. These two regimes (separated by d h ) rep-resent on the one hand the oversized fragmenting granules andthe other hand the resulting fragments. The (fragmented) small-scale granules were found in high-resolution solar observationsby Abramenko et al. (2012). The fragmentation of granules is a continuous process, therefore, the distribution of granule sizes isalso continuous, and it covers a fairly large range.Another possibility to quantify the granule size distribu-tion is the area contribution function, which is given by f ac = n i A i / A tot , with n i being the number of elements within the area-bin A i , and A tot = (cid:80) n i A i being the total area of all granules. Thearea contribution function is in principle a histogram of granulesize, which is weighted with the contribution of area to the to-tal area (Roudier & Muller 1986). We noted above that a lineargranule size is overestimating the histogram for smaller gran-ules. The contribution function has the intrinsic advantage thata large number of small granules contributes only little to f ac ,since their area is small. It depicts the dominant granule size,which contributes most to the radiation (radiative losses occurmostly from granules, which are hotter and have a larger area),independently of the linear or logarithmic bin sizes. In Fig. 3awe show the area contribution function resulting from the so-lar simulation. We also label the dominant granule size withthe maximum, i.e. d ac = max (cid:2) f ac (cid:3) , which leads to the furtherdefinition A ac = π ( d ac / . Similar to the above finding withthe mode d h , the dominant granule size divides the distributioninto two regimes at a very similar value, which further supportsthe location of the "typical" granule size. The decline towardslarger granule sizes is similar, but the lower part is noticeablysmaller than the histogram (both are not expected to coincideentirely due to their di ff erent definitions). We remark that the de-clining tail of f ac towards smaller granules sizes illustrates thatemploying a logarithmic scale for the histograms of the gran-ule size is essential to yield correct conclusions on the declin-ing distribution of the smallest granules. We find for the so- Article number, page 4 of 12. Magic and M. Asplund: The Stagger-grid – VI. Surface appearance of stellar granulation
Fig. 4: The mean granule size vs. e ff ective temperature for dif-ferent stellar parameters.lar simulation a dominant scale of A ac = .
06 Mm , which is d ac = .
62 Mm. Observational findings have similar values with d ac ≈ ff erence arises probably from atmospherical and in-strumental e ff ects (Stein & Nordlund 1998).In the following, we discuss the resulting granule sizes fordi ff erent stellar parameters. We show in Fig. 3b the histogramsof the granule diameters, d gran . And we show the mean gran-ule size for di ff erent stellar parameters in Fig. 4. For higher T e ff and [Fe / H] the mean granule sizes are slightly larger, whilethese are significantly larger for giants (lower log g ), since thepressure scale height scales with the surface gravity (see Pa-per I). In general, the shapes of the histogram are similar to thesolar one and also exhibit a distinct maximum in their granulesizes. In the case of dwarfs, the peak of d h is less pronounced to-wards lower T e ff and we find increasingly a bimodal distributionin the histograms of the granule diameter with a distinct secondpeak from the small-scale granules, in particular, for cooler mod-els (Fig. 3b). The second peak at smaller granule sizes varieswith stellar parameter, so that in some cases the two peaks arereversed. In these models the granules fragment into smallerpieces more e ffi ciently (see Sect. 4). However, most of radiationstill emerges from the larger peak, since the area contributionfunction exhibits a single peak, which is located at larger gran-ule size (see Fig. 3b, where we have marked d ac ). Furthermore,the decline towards larger fragmenting granules is steeper withhigher T e ff , which means that these granules are prone to disin-tegrate within a smaller range of granule sizes. The lower halfof the histograms are similar despite a shift and the second peak.In Paper I, we estimated the typical granule size from thelocation of the maximum of the 2D spatial power-spectrum ofthe intensity, d Int . Furthermore, we found d Int to correlate wellwith pressure scale height just below the optical surface. In Fig.5, we compare the mean granule size, d gran , with the estimatedgranule size from the power-spectrum of the intensity d Int . Thetwo correlate very well for all of our atmosphere models. Fig. 5: The length-scale at the maximum of the temporally av-eraged 2D spatial power-spectrum of the intensity, d Int , vs. themean granule size, d gran , for di ff erent stellar parameters.Our findings carry some uncertainty that might be rooted inthe granule detection method or in the simulation boundaries.However, we have confirmed that our results are robust, sincethese are qualitatively similar to those by Beeck et al. (2013b).They used also the multiple level tracking algorithm, and foundalso an asymmetric distribution exhibiting a dominant granulesize with an extended tail for the small-scale granules. More-over, the mean granule diameters and the filling factors theyfound are similar to our results.
4. Fractal dimension
The fractal dimension is a suitable, measurable value to quantifythe complexity of a geometrical shape (Mandelbrot 1977). In thecase of granules, this is given by the area-perimeter relation P = kA D / (3)with k being a shape factor and D the fractal dimension (Roudier& Muller 1986). In planar geometry, ideal objects have an inte-ger fractal dimension, e.g., circles or squares have D = ff erent shape factors k = √ π and 4,respectively. However, real objects are of fractal nature. It isan important measure for the regularity of granules; more regu-lar ones will have a lower D , while more irregular granules willhave higher area-perimeter ratios.We show the 2D histogram of the area and perimeter deter-mined from the granules of the solar simulation in Fig. 6a with atight correlation. At the dominant granule size, we find a distinctchange in the slope of the correlation, indicating a multi-fractalnature of granulation. Therefore, we determined two fractal di-mensions with two separate linear least-square fits. The first oneis performed for the small-scale granules ( A < ¯ A ), and the re-sulting fractal dimension is very close to unity with D = . Article number, page 5 of 12a) (b)
Fig. 6:
Left figure : We show the (smoothed) histogram of the area-perimeter relation determined from the solar simulation. Weindicated the histogram of the perimeter ( gray line ), the location of the mean granule area ( black dashed ) and the maximum of f ac ( black solid line ). Also the linear fit for the small-scale ( red solid ) and the large granules ( red dashed lines ) are also included. Rightfigure : The fractal dimensions, D and D , for di ff erent stellar parameter. Furthermore, we included linear fits for D and D , andthe slopes are ∆ ≈ .
06 and ∆ ≈ . A > ¯ A ), and the fractal dimension is distinctivelylarger with D = .
86 (see Fig. 3a). This means that large gran-ules feature increasingly larger perimeters.The fractal dimension has been determined from solar ob-servations. In agreement with our result, Roudier & Muller(1986) found two distinct fractal dimensions with D = .
25 and D = .
15 with a Fourier-based recognition (FBR) method. Ourfractal dimensions also coincide with Hirzberger et al. (1997),who determined for smaller granules D ≈ . D ≈ . D = . D = .
96. On the other hand, Bovelet & Wiehr(2001) determined with their multiple layer tracking method dis-tinctively lower values for the fractal dimensions with D = . D = .
28. Their smaller fractal dimension is similar to ours,however, the second one for the larger granules is much lower.They find that the FBR method is recognizing smaller granulesas a larger single one compared to the multiple layer trackingmethod. However, since we do not use a FBR method, our re-sults should be similar to their findings. We also performed a sin-gle linear fit, which resulted in D = .
10, but the latter is clearlyinsu ffi cient to depict the larger granules (not shown). In manyof these cases, the fractal dimensions are slightly larger thanour values, which probably originates from the di ff erent granulerecognition method (FBR), but also from the reduced resolutionof their observations that include atmospheric e ff ects.Figure 6b shows the variation of the area-perimeter ratiobased on D and D for di ff erent stellar parameters. The branch- ing at the dominant granule size scale is always given with aslope close to unity for the smaller granules and a steeper slopefor larger granules, in particular for lower T e ff and higher [Fe / H].The fractal dimension for the smaller granules is close to unitywith the average value being D = . ± .
02, and thereforebasically universal for all simulations. This means that gran-ules smaller than the dominant granule size are mostly regularlyshaped. Furthermore, a value close to unity implies that theperimeter increases to the square root with the area, i.e. P ∝ A / ,for the smaller granules (see Eq. 3). The second dimensionsare clearly larger, being on average D ≈ . ± . T e ff and [Fe / H], and lowerlog g . D never exceeds 2 in every of our simulations. Thelarger granules above the dominant granule size are irregularlyshaped, and D ∼ P ∝ A . This is in principle the manifestation of the frag-mentation of oversized, unstable granules. When we considera hotter / cooler dwarf ( T e ff = / g = . D are ∼ . ∼ .
9. If we comparetwo granules with the same (larger) area, A , from both dwarfs,then the granules of the cooler dwarf will exhibit much largerperimeters, P hot ( A ) < P cool ( A ), i.e. its granules will be in gen-eral more fragmented. This might be due to the higher densitiesand the lower vertical velocities, thereby shifting the balance ofthe characteristic length scales (see Paper III).The granulation pattern in our simulations exhibit a strik-ing self-similarity despite the large variations in the horizon-tal length scales and convective flow properties (see Fig. 1).This observation is backed by the linear correlation of the area-perimeter relations, and the similar fractal dimensions between Article number, page 6 of 12. Magic and M. Asplund: The Stagger-grid – VI. Surface appearance of stellar granulation
Fig. 7: The smoothed distribution of the geometrical shapefactors for roundness, circularity, elongation and ellipticity vs.granule area derived from the solar simulation. We outlinedthe mean ( solid ), the standard deviation around mean ( dashed )and the extrema ( dotted lines ). Furthermore, we included also a(smoothed) histogram of the shape factor ( blue lines ), in orderto render its distribution. The maximum of the latter is indicated( horizontal blue line ). The vertical lines indicate mean and dom-inant granule area ( red vertical dashed and solid line ).the di ff erent stellar parameters. Surface convection appears tooperate scale-invariant over large ranges. This is true, in partic-ular, for the small-scale granules. Furthermore, the branchingof the two fractal dimensions is taking place at the dominantgranule size for all stellar parameters, since above the latter thegranules cannot be supported by the pressure excess and start tofragment, thereby increasing the granule perimeter and becom-ing more irregular. Therefore, the branching area between D and D can be regarded as the maximal granule size, with gran-ules of larger sizes being unstable.
5. Geometrical properties
To quantify the geometrical properties of the complex granuleshapes, we followed Hirzberger (2002) and determined f r = π A / P , (4) f c = d gran / d MF , (5) f l = w MF / d MF , (6) f e = b / a . (7)The roundness factor (Eq. 4) is the area-perimeter relation thatmeasures the deviation from a perfect circle, and is also knownas the isoperimetric quotient. The isoperimetric inequality, f s ≤
1, holds for any arbitrary shape, and yields equality only for thecircle. The circularity factor (Eq. 5) is the ratio between granulesize, d gran , and the maximal Feret-diameter, d MF , which is thediameter of the principal axis, i.e. the maximum diameter at thebarycenter for all degrees of rotation. It quantifies the evenness along the boundary, where only an even shape will lead to a valueclose to 1. The elongation factor (Eq. 6) is determined with w MF being the width perpendicular to d MF , i.e. it is the aspectratio of the principal axis. Finally, the ellipticity factor (Eq. 7)is obtained by a = ξ + ( ξ − A /π ) / and b = A / ( π a ) with ξ = [( A /π ) / + P /π ] /
3, and compares the shape with an ellipse (seeHirzberger 2002).The geometrical properties are shown for the Sun in Fig. 7.For granules smaller than the dominant granule size, the geo-metrical properties are in general very similar. Above d ac onefinds a transition, in particular, f r and f e are dropping towardszero above the dominant granule size, since the granules start tofragment and split, and the perimeter is increasing much fasterthan the area ( f r ∝ P − ). This is in agreement with the second,larger fractal dimension, D , discussed in Sect. 4. The shape fac-tors f c and f l are independent of the perimeter. The histogramsof the shape factors are symmetrically distributed around a well-defined maximum with di ff erent widths. However, the roundnessfactor is an exception, it exhibits a skewed distribution that cov-ers almost the whole range between zero and unity, and the max-imum is located at f r = .
77. As given in Fig. 7, the contributionsarise from di ff erent granules sizes. Smaller granules tend be ingeneral more regular at their boundaries (larger f r ) with smoothedges, while the fragmenting granules that are larger than d ac have increasingly irregular, complex boundaries (small f r ) thatare fringed and convoluted (see Fig. 1). The granule shapesare in overall regular, circular shapes ( f c = .
68 and f l = . d gran . Furthermore, the granules are quiteelongated with f e = .
33. When we compare our four shape fac-tors with those by Hirzberger (2002), then these are qualitativelyvery similar, only the maximum of the roundness factor is atmuch lower values with f r = .
1, which might due to di ff erencesin the recognition methods. Therefore, we remark that our so-lar simulation harbors a realistic granulation pattern. Since theshape factors are very similar for di ff erent stellar parameters, werestrict ourself to the discussion of the solar values only.
6. Properties with granule size
In Fig. 8a, we show the mean (bolometric) intensities of gran-ules (seen at the disk-center), (cid:68) I g (cid:69) , that are normalized by thetemporal average of the entire simulation, (cid:104) I (cid:105) , against their gran-ule sizes. Smaller granules are darker (5 − d ac , onefinds the brightest granules (5 −
15 %). This means that the mostabundant granules with sizes similar to d ac cover most of thestellar surface and are the brightest, i.e. these dominate the bolo-metric intensity not only due to their size and abundance, butalso brightness. Therefore, most of the radiative energy is lost inthese granules. The mean intensities of granules larger than d ac are lower than the maximal, since these large fragmenting (ex-ploding) granules develop dark spots due to pressure excess andmass flux reversal (see Stein & Nordlund 1998), which is thenreducing the mean intensity. Hirzberger et al. (1997) finds alsoa similar granule size dependence for the mean intensity in theobserved solar granules.For higher T e ff , smaller granules are dimmer and the largerones are brighter, while for di ff erent log g the changes are onlysubtle. In the case of metal-poor simulations, the same small-scale granules are darker for hotter T e ff and brighter for cooler T e ff compared to the solar case, which correlates with the en-hancement of the intensity contrast at lower metallicity (see Pa- Article number, page 7 of 12a) (b)
Fig. 8:
Left figure : Mean normalized intensity vs. granule size. Furthermore, we indicated the mean granule diameters ( dottedlines ) and the dominant scales, d ac ( filled square s). Note the di ff erence in abscissa between the top and bottom panel. Right figure :Mean intensity contrast vs. granule size. Furthermore, we indicated the dominant scales, d ac ( filled square s). Note the di ff erence inabscissa between the top and bottom panel.per I). Due to the lack of metals at lower metallicity, the impor-tance of neutral hydrogen as primary electron-donors increasesfor higher T e ff , and since the electron density is controlling theformation of negative hydrogen – the dominant opacity source –the opacity is therefore more sensitive to an increase in temper-ature.The intensity contrast of the granules vs. their size is shownin Fig. 8b. The trends are similar to the intensity with stel-lar parameters. The intensity contrast is lower for small gran-ules, typically reach a maximum at the mean granule size, anddecreasing above it. Higher intensity contrast correlates withmore complex substructures in the granules with dark spots andbright edges. These arise due to di ff erences in the temperatureexcess of the granules originating from the granular dynamics(e.g., Hirzberger et al. 1997; Stein & Nordlund 1998). We averaged the temperatures and densities of the recognizedgranules (Sect. 2) on layers of constant Rosseland optical depthat the optical surface ( τ Ross = ff erent stellar parameters.Since the densities are essentially the same as the temperatures,we refrain from showing them. To improve the comparison thedisplayed mean values of the granules, (cid:68) T g (cid:69) and (cid:68) ρ g (cid:69) , are nor-malized to the temporal and horizontal averages, (cid:104) T (cid:105) and (cid:104) ρ (cid:105) , atthe surface ( τ Ross =
1) of the whole simulation. In general, largergranules feature higher mean temperatures and lower densities.An inverse correlation between the temperature and density is tobe expected (from ideal gas law follows T ∼ p /ρ ). The temper-ature excess peaks around the mean granule diameters (1 − dottedlines ) and the dominant scales, d ac ( filled square s). Note thedi ff erence in abscissa between the top and bottom panel.(1 −
15 %). The T -peak and ρ -minimum are increasing for higher T e ff and lower log g . The smallest granules exhibit lower-than- Article number, page 8 of 12. Magic and M. Asplund: The Stagger-grid – VI. Surface appearance of stellar granulation
Fig. 10: Mean rms vertical velocity of granules vs. granule size,which is obtained on layers of constant Rosseland optical depth.Furthermore, we indicated the mean granule diameters ( dottedlines ) and the dominant scales, d ac ( filled square s). Note thedi ff erence in abscissa between the top and bottom panel.average temperatures and higher-than-average densities, sincethese are small granule fragments located in the downdrafts. Fur-thermore, we find a tight correlation between the mean gran-ule temperature and intensity with typical values around ∼
97 %with very small variation of di ff erent stellar parameters, whilethe density is anti-correlated with the intensity by values around ∼ − (cid:104) p th (cid:105) , exhibits very similar dependence with the granulesize as the density, while the mean entropy resembles the tem-perature, but on a smaller scale (not shown). In Fig. 10, we show the mean rms of the vertical velocity de-rived for the individual granules on layers of constant Rosselandoptical depth at the optical surface. The rms velocity are in-creasing for higher T e ff , lower log g and higher [Fe / H]. Theseare in general flat for lower T e ff , while for higher T e ff , one canfind a distinct peak close to the mean granule diameter. We haveseen above (Sect. 6.2) that these granules with mean diametershave lower densities due to higher temperatures, therefore, theselighter granules will experience a larger buoyancy acceleration.We remark that the characteristic variations of the rms velocitiesarise mainly from the upflowing material. We find that the lowermean densities around the mean granule diameters are not de-creasing the mean upwards directed vertical mass flux, since thehigher velocities are raising the upwards mass transport. Fig. 11: Overview of the rms deviation of the geometrical depthfor optical depth unity that is normalized with the pressure scaleheight for di ff erent stellar parameters.
7. The optical surface
The optical surface is defined as the layer with optical depthunity ( τ Ross = ff erent stellar pa-rameters, encompassing the Sun, a turno ff star, a K-giant and aK-dwarf. Furthermore, we also illustrate the vertical velocity atthe optical surfaces, showing that the downflows are located inthe intergranular lanes, while the granules are flowing upwardsat the surface.The level of corrugation di ff ers for the di ff erent stellar pa-rameters. The level of corrugation can be quantified with thetemporal averaged rms deviation of the geometrical depth forthe layers of constant optical depth unity, i.e. (cid:104) z rms ( τ Ross = (cid:105) .The solar simulation is slightly corrugated with ∼
33 km, whichis close to the value found by Stein & Nordlund (1998) with ∼
30 km. Compared to the solar radius this is a very small rel-ative variation: ∼ × − %. In comparison the Earth surfacehas a tolerance of 0 .
17 % from a spheroid, which is ∼
30 timeslarger than the (quiet) Sun. The turno ff and giant simulation ex-hibit much larger corrugated optical surfaces compared to theSun with 300 and 23 000 km respectively, while the dwarf modelhas a very smooth optical surface with 3 km. One can estimatethat the turno ff star has approximately twice the solar radius,while the K-giant is twenty times higher, making the relativevariations 0 .
02 and 0 .
17 %, respectively. The K-dwarf would
Article number, page 9 of 12 e ff = K , log g = . T e ff = K , log g = . T e ff = K , log g = . T e ff = K , log g = . µ = .
6, including the vertical velocity to illustrate the up- anddownflows ( blue and red ; each with a range of 8 km / s) for a selection of stars: Sun (5777 K / . ff (6500 K / . ∼ × − %.To illustrate the systematic variation of the corrugation, weoverview the standard deviation of (cid:104) z ( τ Ross = (cid:105) , which is nor-malized by the (total) pressure scale height in Fig. 11. Thecorrugation increases primarily with lower surface gravity andhigher e ff ective temperatures, since from hydrostatic equilib-rium ( d p / dz = ρ g ) follows dz ∝ / g and the dominant negativehydrogen opacity source is very temperature sensitive ( κ H − ∝ T ; Stein & Nordlund 1998). Furthermore, at higher metallicitythe corrugations are also larger due to the lower densities. In order to study the surface properties more closely, we deter-mined the temporally averaged (2D) histograms for the temper-ature, T , density, ρ , and intensity fluctuations, δ I , as a func-tion of vertical velocity at the optical surface on layers of geo-metrical depth ( (cid:104) τ Ross (cid:105) =
1) or constant Rosseland optical depth( τ Ross = ff erent properties between the up- and downflows (see SN98). All the thermodynamic propertiesexhibit a bimodal distribution due to the inherent asymmetricnature of the convective energy transport. On the one hand, thestellar plasma in the upflows has hotter temperatures and lowerdensities with brighter intensities located in the granules. On theother hand, the downflows are composed by cooler temperaturesand higher densities with darker intensities found in the inter-granular lane. Furthermore, the (slower) upflows correlate withhigher entropy and ionization, while the (faster) downflows as-sociate with lower entropy and ionization. In Fig. 13 we showthe mean values of the histograms for T , ρ and δ I , which ex-hibits typically a s-shape. Moreover, we display the contoursof one-fifth of the maximal probability for the T and ρ . Thenone can obtain a temperature jump from the histograms derivedon layers of constant geometrical depth (left panel in Fig. 13),where we determined the height of the optical surface on thetemporal average, i.e. (cid:104) τ Ross (cid:105) =
1. At a higher e ff ective tem-perature the density decreases, while the velocity rises, therebyleading to an enhancement in the overshooting of the convec-tive upflows, which is also known as "naked granulation" (Nord-lund & Dravins 1990). In agreement with the latter, we find thatthe T -jump becomes more distinct for hotter T e ff , and the bi- Article number, page 10 of 12. Magic and M. Asplund: The Stagger-grid – VI. Surface appearance of stellar granulation
Fig. 13: Correlation of the relative intensity fluctuations, temperatures, and densities with the vertical velocities at the optical surfacefor di ff erent stellar parameters (top, middle and bottom panel respectively) shown by the distribution of their histogram (normalizedhistogram at 0.2 is shown with thick contour lines). We indicated the mean value (solid line), and for the up- and downflowseparately their mean (horizontal dashed line), range (vertical dotted line) and standard deviation (vertical solid line). Furthermore,we show the geometrical averages taken at the height with (cid:104) τ Ross (cid:105) = τ Ross = τ Ross = T -sensitive due to thenegative hydrogen opacity, and layers with similar temperaturesare mapped during transformation to the optical depth (see PaperII). The fluctuations of the upflows are broader in temperatureand narrower for the density (see Fig. 13), while the downflowsfeature a broad distribution in ρ and smaller ranges in T .
8. Conclusions
We derived extensive details of stellar granulation by applyingthe multiple layer tracking algorithm for the detection of gran-ules imprinted in the emergent (bolometric) intensity map, whichwas originally developed for solar observations. This methodworks very reliable for di ff erent stellar parameters. Then, wedetermined for the individual detected granules properties: di-ameter, intensity, temperature, density, velocity and geometry.The granule diameters span a large range, therefore, we advisethe use of a logarithmic equidistant histogram, since otherwise,the smaller scales are under-resolved, which leads to the misin-terpretation of a large population of small granules. A distin- guished dominant granule size can always be determined withthe maximum of the area contribution function, which is of-ten very close the maximum of the diameter distribution. Fur-thermore, we find two distinct fractal dimensions (slopes of thearea-perimeter relation) that are divided at the dominant granulesize. For smaller granules the fractal dimension is always veryclose to unity, which points out that these are evenly shaped.The larger granules have distinctively larger fractal dimensionsclose to 2, primarily depending on the e ff ective temperature. Forlower T e ff we find fractal dimension being larger. In the caseof the solar simulation, the dual fractal dimensions we find isin contradiction to the finding by Bovelet & Wiehr (2001), whofinds only a single fractal dimension with the same method fromsolar observations, and the discrepancy might root in observa-tional constraints. The bifurcation of the fractal dimension abovethe dominant granule size arises simply due to the fragmentationof granules, which will inevitably entail that the perimeter in-creases. We studied also the properties prevailing at the opticalsurface in our stellar atmosphere simulations. We find that thecorrugation of the optical surface increases for higher T e ff andlower log g . Also, we revealed the systematic correlation of theintensity, temperature and density with the vertical velocity as anatural consequence of the convective energy transport. Article number, page 11 of 12 eferences
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