Dynamics and Structure of Supergranulation

In this paper we investigate the temporal evolution and geometric properties of solar supergranular features. For this purpose we apply an automatic feature-tracking algorithm to a 6-day time series of 18 near-surface flowmaps containing 548 target objects. Lifetimes are calculated by measuring the time elapsing between the birth and death of each target. Using an exponential fit on the lifetime distribution of single supergranules we derived a mean lifetime of 22 hours. Based on the application of segmentation numerical procedures, we estimated characteristic geometric parameters such as area distributions of supergranular cells. We also derive the relationship between measured lifetime and the area of the supergranules.     

On the Fractal Structure of Solar Supergranulation

We employ fractal analysis to study the complexity of supergranulation structure using the Solar and Heliospheric Observatory (SOHO) dopplergrams. Our data consists of 200 visually selected supergranular cells, for which we find a broad, slightly asymmetric dispersion in the size distribution, with the most probable size around 31.9 Mm. From the area–perimeter relation, we deduce a fractal dimension D of about 1.25. This is consistent with that for isobars, and suggests a possible turbulent origin of supergranulation. By relating this to the variances of kinetic energy, temperature and pressure, it is concluded that the supergranular network is close to being isobaric and that it has a possible turbulent origin.     

Structure Properties of Supergranulation and Granulation

We investigate spatial dislocation ordering of the solar structures associated with supergranulation and granulation scales. The supergranular and granular structures are automatically segmented from time-distance divergence maps and from broad-band images, respectively. The spatial dislocation ordering analysis is accomplished by applying the statistical method of Pair Correlation Function, g ₂(r), to segmented features in the solar fields. We compare the computed g ₂(r) functions obtained from both single and persistent, i.e., time-averaged, fields associated with supergranulation and granulation. We conclude that supergranulation and granulation patterns present a different topological order both in single and persistent fields. The analysis carried out on single fields suggests that the granulation behaves as an essentially random distribution of soft plasma features with a very broad distribution in size, while supergranulation behaves as a random distribution of close packed, coherent stiff features with a rather defined mean size.     

Structure and Evolution of Supergranulation from Local Helioseismology

Supergranulation is visible at the solar surface as a cellular pattern of horizontal outflows. Although it does not show a distinct intensity pattern, it manifests itself indirectly in, for example, the chromospheric network. Previous studies have reported significant differences in the inferred basic parameters of the supergranulation phenomenon. Here we study the structure and temporal evolution of a large sample of supergranules, measured by using local helioseismology and SOHO/MDI data from the year 2000 at solar activity minimum. Local helioseismology with f modes provides maps of the horizontal divergence of the flow velocity at a depth of about 1 Mm. From these divergence maps supergranular cells were identified by using Fourier segmentation procedures in two dimensions and in three dimensions (two spatial dimensions plus time). The maps that we analyzed contain more than 10⁵ supergranular cells and more than 10³ lifetime histories, which makes possible a detailed analysis with high statistical significance. We find that the supergranular cells have a mean diameter of 27.1 Mm. The mean lifetime is estimated to be 1.6 days from the measured distribution of lifetimes (three-dimensional segmentation), with a clear tendency for larger cells to live longer than smaller ones. The pair and mark correlation functions do not show pronounced features on scales larger than the typical cell size, which suggests purely random cell positions. The temporal histories of supergranular cells indicate a smooth evolution from their emergence and growth in the first half of their lives to their decay in the second half of their lives (unlike exploding granules, which reach their maximum size just before they fragment).     

The Quest to Understand Supergranulation and Large-Scale Convection in the Sun

Surface granulation of the Sun is primarily a consequence of thermal transport in the outer 1 % of the radius. Its typical scale of about 1?–?2 Mm?is set by the balance between convection, free-streaming radiation, and the strong density stratification in the surface layers. The physics of granulation is well understood, as demonstrated by the close agreement between numerical simulation, theory, and observation. Superimposed on the energetic granular structure comprising high-speed flows, are larger-scale long-lived flow systems (≈?300 m?s^?1) called supergranules. Supergranulation has a typical scale of 24?–?36 Mm. It is not clear if supergranulation results from the interaction of granules or is causally linked to deep convection or a consequence of magneto–convection. Other outstanding questions remain: how deep are supergranules? How do they participate in global dynamics of the Sun? Further challenges are posed by our lack of insight into the dynamics of larger scales in the deep convection region. Recent helioseismic constraints have suggested that convective-velocity amplitudes on large scales may be overestimated by an order of magnitude or more, implying that Reynolds stresses associated with large-scale convection, thought to play a significant role in the sustenance of differential rotation and meridional circulation, might be two orders of magnitude weaker than theory and computation predict. While basic understanding on the nature of convection on global scales and the maintenance of global circulations is incomplete, progress is imminent, given substantial improvements in computation, theory, and helioseismic inferences.     

Additional Evidence Supporting a Model of Shallow,High-Speed Supergranulation

Recently, Duvall and Hanasoge (Solar Phys. 287, 71, 2013) found that large-distance separation [Δ] travel-time differences from a center to an annulus [δt _oi] implied a model of the average supergranular cell that has a peak upflow of 240 m?s^?1 at a depth of 2.3 Mm and a corresponding peak outward horizontal flow of 700 m?s^?1 at a depth of 1.6 Mm. In the present work, this effect is further studied by measuring and modeling center-to-quadrant travel-time differences [δt _qu], which roughly agree with this model. Simulations are analyzed that show that such a model flow would lead to the expected travel-time differences. As a check for possible systematic errors, the center-to-annulus travel-time differences [δt _oi] are found not to vary with heliocentric angle. A consistency check finds an increase of δt _oi with the temporal frequency [ν] by a factor of two, which is not predicted by the ray theory.     

Chromospheric rotation

The dependence of solar rotation on the size of the chromospheric tracers is considered. On the basis of an analysis of Ca ii K₃ daily filtergrams taken in the period 8 May–14 August, 1972, chromospheric features can be divided into two classes according to their size. Features with size falling into the range 24 000–110 000 km can be identified with network elements, while those falling into the range 120 000–300 000 km with active regions, or brightness features of comparable size present at high latitudes. The rotation rate is determined separately for the two families of chromospheric features by means of a cross-correlation technique which directly yields the average daily displacement of tracers due to rotation. Before computing the cross-correlation functions, chromospheric brightness data have been filtered with appropriate bandpass and highpass filters for separating spatial periodicities whose wavelengths fall into the two ranges of size, characteristic of the network pattern and of the activity centers. A difference less than 1% of the rotation rate of the two families of chromospheric features has been found. This is an indication for a substantial corotation at chromospheric levels of different short-lived features, both related to solar activity and controlled by the convective supergranular motions.     

Asteroid rotation rates

The dependence of rotational frequency on diameter, taxonomic type, and family membership is analyzed for 217 main-belt asteroids with statistically useful periods extracted from the file published by Harris and Young ((1983). Icarus54, 59–109). It is shown that for asteroids with diameters ? 120 km, mean rotational frequency increases with increasing diameter. This trend is equally present in all subsets of M-, S-, and C-type asteroids, for both family and nonfamily members alike, and cannot be accounted for by observational selection. For asteroids with diameters ? 120 km, mean rotational frequency increases with decreasing diameter; however, within this group there is a subset of asteroids with exceptionally long rotational periods. This marked change in the distribution at diameter ～ 120 km could separate primordial asteroids from their collision products. However, it is probable that the sample is biased in favor of small asteroids with short rotational periods and that the apparent increase of mean rotational frequency with decreasing diameter for small asteroids is at least partly the product of observational selection. An observational program that could test this hypothesis is described. If asteroids of any one diameter are considered, then, on average, M asteroids rotate faster than S asteroids which in turn rotate faster than C asteroids. This shows that asteroids which have been classified by their surface properties alone have different bulk properties. There is also some evidence that for all asteroidal types, of all diameters, family members rotate faster than nonfamily members.     

Physics of rotation

The motion of a rigid body about a fixed point, under the influence of an attractive force, is investigated from a physical basis. The errors in Euler's equations were shown, and appropriate equations were derived. It was also found that by nutation in the direction of decreasing only and oscillatory precession any two bodies, assumed as oblate spheroids, move in the direction of a zero external moment, as their stable equilibrium position. This position corresponding to the minimum kinetic energy of rotation is also reached by any two bodies which were initially acted upon by an external torque.     

Black holes and rotation

In this article, we first consider briefly the basic properties of the non-rotating Schwarzschild black hole and the rotating Kerr black hole Rotational effects are then described in static and stationary spacetimes with arial symmetry by studying inertial forces, gyroscopic precession and gravi-electromagnetism. The results are applied to the black hole spacetimes.     

Lunar-forming collisions with pre-impact rotation

Prior models of lunar-forming impacts assume that both the impactor and the target protoearth were not rotating prior to the Moon-forming event. However, planet formation models suggest that such objects would have been rotating rapidly during the late stages of terrestrial accretion. In this paper I explore the effects of pre-impact rotation on impact outcomes through more than 100 hydrodynamical simulations that consider a range of impactor masses, impact angles and impact speeds. Pre-impact rotation, particularly in the target protoearth, can substantially alter collisional outcomes and leads to a more diverse set of final planet-disk systems than seen previously. However, the subset of these impacts that are also lunar-forming candidates—i.e. that produce a sufficiently massive and iron-depleted protolunar disk—have properties similar to those determined for collisions of non-rotating objects [Canup, R.M., Asphaug, E., 2001. Nature 412, 708-712; Canup, R.M., 2004a. Icarus 168, 433-456]. With or without pre-impact rotation, a lunar-forming impact requires an impact angle near 45 degrees, together with a low impact velocity that is not more than 10% larger than the Earth's escape velocity, and produces a disk containing up to about two lunar masses that is composed predominantly of material originating from the impactor. The most significant differences in the successful cases involving pre-impact spin occur for impacts into a retrograde rotating protoearth, which allow for larger impactors (containing up to 20% of Earth's mass) and provide an improved match with the current Earth-Moon system angular momentum compared to prior results. The most difficult state to reconcile with the Moon is that of a rapidly spinning, low-obliquity protoearth before the giant impact, as these cases produce disks that are not massive enough to yield the Moon.     

Differential rotation of coronal holes

Using KPNO helium 10830 Å synoptic charts of Carrington rotations 1716 through 1739, and by assembling a time sequence representing single latitude zone, rotational properties of coronal holes for five zones of latitudes (±10°, ±20° – ±40°, and ±40° – ±60°) have been examined. It seems that the rotation period of coronal holes is a function of latitude, thus reflecting differential rotation of coronal holes.     

The equatorial photospheric rotation rate

The equatorial photospheric rotation rate has been observed on 14 days in 1978–1980. The resulting rotation rate, = 14.14±0.04°/day, is 2% slower than the rate as observed for long-lived sunspots.Stationed at Kitt Peak National Observatory.Operated by the Association of Universities for Research in Astronomy, Inc. under contract with the National Science Foundation.     

Differential rotation in F stars

Differential rotation can be detected in single line profiles of stars rotating more rapidly than about v sin i = 10km s-1 with the Fourier transform technique. This allows to search for differential rotation in large samples to look for correlations between differential rotation and other stellar parameters. I analyze the fraction of differentially rotating stars as a function of color, rotation, and activity in a large sample of F-type stars. Color and rotation exhibit a correlation with differential rotation in the sense that more stars are rotating differentially in the cooler, less rapidly rotating stars. Effects of rotation and color, however, cannot be disentangled in the underlying sample. No trend with activity is found. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectroscopic determinations of solar rotation

Spectral line shift data obtained from full-disk magnetograms recorded at Mt. Wilson are analyzed for differential rotation. The method of analysis is discussed and the results from the data for 1966 through 1968 are presented. The average equatorial velocity over this period is found to be 1.93 km/sec or 13.76 deg/day (sidereal). This corresponds to a sidereal period of 26.16 days. The average results are = 2.78 × 10^-6 - 3.51 × 10^-7 sin² B - 4.43 × 10^-7 sin⁴ B rad/sec, whereB is the solar latitude. This indicates a smaller decrease of angular velocity with latitude than found by earlier investigators. Variations from day to day are caused by large-scale short-lived velocity fields on the solar surface. There also appear to be secular variations.Currently at the Kitt Peak National Observatory, Tucson, Arizona.     

The Faraday rotation of sunspots

In this paper the numerical solutions of the Unno-Beckers's equations for the magneto-sensitive line Fei 5250.216 are used to demonstrate the importance and role of Faraday rotation in sunspot magnetic fields and to study the influence of this effect on the measurements of the azimuth of the transverse field. We propose a method to determine the intrinsic direction of the transverse field with the observed azimuthal angle of the plane of linear polarization.     

The rotation of small asteroids

The addition of the unbiased sample of R. P. Binzel and J. D. Mulholland (Icarus56, 519–533) nearly triples the sample size of photoelectrically determined rotational parameters for main belt asteroids with estimated diameters (D) ≤30 km. Nonparametric stattistical tests which require no assumptions about the distributions or variances of the samples are used to examine rotational parameters for all D ≤ 30 km asteroids. A comparison of photoelectric and photographic results shows that the techniques have a highly significant difference in the range of detected frequencies. This difference does not allow photographic and photoelectric observations to be combined for meaningful statistical tests since a photographic bias toward smaller sample variances can induce statistical results that appear overly significant. Photographic observations also show a highly significant bias toward detecting asteroids with larger lightcurve amplitudes. The fit of a Maxwellian to the observed rotational frequency distribution can be rejected at a highly significant confidence level but the observed distribution can be acceptably fit by two Maxwellian distributions, which is consistent with the hypothesis that there are separate populations of slow and fast rotating asteroids. The frequency distributions of <15 km main belt asteroids and Earth and Mars crossers are not found to differ significantly. However, the larger mean lightcurve amplitude of the Earth and Mars crossing asteroids is found to be statistically significant. This latter result is interesting in view of the lack of any strong inverse amplitude versus diameter relation for small asteroids. No significant diameter dependence on rotational frequency is seen among only D ≤ 30 km asteroids. However, the inverse frequency versus diameter relation for D ≤ 120 km asteroids found by S. F. Dermott, A. W. Harris, and C. D. Murray (Icarus, in press) is found to be statistically significant using a linear least-squares analysis of photoelectric data only. No significant diameter dependence on rotational lightcurve amplitude is seen using linear least-squares analysis of photoelectric data for D≤30 and D≤90 km asteroids. However, a significant inverse amplitude versus diameter relation is found when this analysis is extended to D≤120 km asteroids. This finding may be consistent with the hypothesis of Dermott et al. that near 120 km there is a transition between primordial asteroids and their collisional fragments.