Coherence analysis of granular intensity

A high resolution spectrogram of the Mg b₂ line from the quiet Sun disc centre is subjected to a coherence analysis. We find that the coherence between intensity fluctuations in the continuum and the wings of the line breaks down at a distance = 0.35 Å from line centre. From this and the r.m.s. intensity contrast as a function of we are led to the following simple model of temperature fluctuation T in the solar photosphere: A lower part (below 50 km, or ₅₀₀₀ > 0.25) with strongly inward increasing T and an upper part (above 50 km) with constant T = 75 K. The two parts are supposed to fluctuate incoherently.Mitteilungen aus dem Kiepenheuer Institut Nr. 166.     

Studies of granular velocities

The two available methods for determining the rms amplitude of the granular convective velocity field, namely the interpretation of line profiles, and direct measurements of velocity fluctuations in highly resolved spectra, give values ( 2 km/sec, and 0.4 km/sec, resp.) which are apparently inconsistent both in magnitude and in their dependence upon optical depth. We give both theoretical and observational evidence for the working hypothesis, that the best resolved spectra mainly show velocity fluctuations due to the oscillation of the solar atmosphere, whereas the contribution of the granular velocity field is greatly reduced because of atmospheric seeing and can be found only as a weak superposition to the oscillatory velocity field. Realistic assumptions for the typical size of the granulation (2.5) and for the seeing parameter (1), together with a simplified model of the granular velocity field, lead to correction factors of 30 to 40 between the true and observed amplitudes of the granular velocities.Mitteilungen aus dem Fraunhofer Institut, Nr. 95.     

Studies of granular velocities

Vertical small scale velocities deduced from highly resolved spectra in the center of the solar disk, obtained with a RCA image intensifier tube, are discussed. As a main result we find that the velocities of small elements (< 2) are remarkably larger in active regions around sunspots than in the quiet Sun. In both cases the velocities decrease with height in the solar atmosphere (Figure 2).Mitteilungen aus dem Fraunhofer-Institut Nr. 145.     

Studies of granular velocities

Spectral observations of solar velocity fields made during a partial solar eclipse are described. Continuum intensity measurements at the lunar limb allow the modulation transfer function to be derived and the true spatial power spectrum of the velocity field to be reconstructed. The oscillatory and granular components are separated by applying spatial filters cutting off at 3.7. The oscillatory component values are in good agreement with those of Canfield (1976) but the granular component has substantially more power and a smaller height gradient. The discrepancy can be resolved by noting the uncertainties of the seeing corrections and of the separation into components in the work of Canfield.However on the basis of this limited material, we would go no further than to claim that the height dependence of the granular velocities remains an open question.Mitteilung aus dem Fraunhofer-Institut Nr. 157.     

Studies of granular velocities

Observations of quiescent solar prominences with the Harvard College Observatory spectrometer abroad Skylab show that prominence material is optically thick in the Lyman alpha line and the Lyman continuum. The color temperature of the Lyman continuum has a mean of 6600 K and an upward gradient toward the top of the prominence. The departure coefficient of the ground state of hydrogen is found to be of the order of unity as expected from theory. The optical depth of the Ciii sheath region is determined directly from the observation of the limb through the prominence and used to infer the mean electron density and the temperature gradient of the sheath. The result implies that the sheath density is about 0.4, and the temperature gradient about 1.4 times the respective value in the Ciii transition zone of the quiet Sun. The Ciii triplet-singlet ratio for the prominence is found to give a density compatible, within the uncertainty of the atomic parameters, with the density obtained from the optical depth. The Oviλ1032 line, which is emitted by both the prominence and the surrounding corona, is used to obtain an estimate of the thickness of the outer transition sheath of the prominence.     

Studies of granular velocities

Continuum brightness and Doppler velocity fluctuations in the lines 6301.5 and 6302.5 Å of Fei, measured in two selected spectrograms, are analysed by standard statistical (power- and coherence spectrum) methods. It is shown qualitatively that the oscillatory component of the velocity fluctuations (at spatial wavelengths > 4) decreases, while the supposedly granular component (at spatial wavelengths < 4) as well as the coherence between brightness and velocity fluctuations increases with optical depth.The spatial resolution of the spectrograms is estimated by comparing the observed power spectrum of brightness fluctuations with spectra found in the literature, assuming the combined instrumental and seeing spread function to be Gaussian. The resolution thus determined is = 1.24 ± 0.07. If the measured values are corrected accordingly, we obtain a true brightness rms of 10 to 14%, depending upon the shape of the power spectrum chosen for comparison, and a velocity rms at continuum optical depth of 1.3 km/sec. It is shown, however, that using the same correction function for the velocity power spectrum as for the brightness possibly gives rise to misestimating the velocity rms.Mitteilungen aus dem Fraunhofer Institut Nr. 100.     

Studies of granular velocities

Observations with the image intensifier tube at the Coudé refractor in Capri seem to confirm that the velocity field near sunspots contains more small scale fluctuations than in non-active regions, and that these fluctuations decrease more rapidly with height.Mitteilung aus dem Fraunhofer Institut Nr. 130.     

Studies of granular velocities

The process of measuring granular velocity fields with an instrument having finite spectral and spatial resolution is investigated for the case that (1) a weak Fraunhofer line is used, (2) the velocity is constant with height in the solar atmosphere, (3) the original Doppler shifts are of the same order of magnitude as the intrinsic width of the line (width observed with infinitely high spectral and spatial resolution), (4) continuum brightness and line strength fluctuations are superimposed onto the velocity field.It is shown that using a spectral instrumental profile which is large compared to both the intrinsic line width and the rms Doppler shifts (as in the case of filtergrammes), the shift-induced brightness signal is always a linear function of the shift and corrections for finite spatial resolution can be applied to the measured shifts in the usual straightforward way.If the spectral instrumental profile is not large (as in the case of slit-spectrogrammes), the observed line profile is shown to depend upon the spatial resolution as well. It is altered (broadened, made asymmetric) by (1) spatially unresolved Doppler shifts and higher moments of the Doppler shift amplitude distribution, (2) by local correlation between continuum brightness, line strength, and velocity fluctuation. A value of the Doppler shift which is unaffected by nonlinearities, can be measured at a certain position in the line wing. Knowledge of the intrinsic line width is necessary, however, to determine this position, as well as the order of magnitude of the nonlinearity effects producing asymmetries in the observed line profile. Finally, the conditions are discussed under which a complete deconvolution of a spectrum could be accomplished.On leave from Fraunhofer Institut, Freiburg.     

A solar wind model with the power spectrum of Alfvénic fluctuations

A new solar wind model has been developed by including in the model the Alfvénic fluctuation power spectrum equation proposed by Tu et al. (1984). The basic assumptions of the model are as follows: (1) for heliocentric distances r > 10 R _⊙, the radial variation of the power spectrum of Alfvénic fluctuations is controlled by the spectrum equation (1), (2) for heliocentric distances r < 10 R _⊙, the radial variation of the fluctuation amplitude is determined by the Alfvén wave WKB solution, (3) no energy cascades from the low-frequency boundary of the Alfvénic fluctuation power spectrum into the fluctuation frequency range, and the energy which cascades from the high-energy boundary of the spectrum into the higher frequency range is transported to heat of the solar wind flow. Some solutions of this model which, on one hand, describe the major properties of the Alfvénic fluctuations and the high-speed flow observed by Helios in the space range between 0.3–1 AU and, on the other hand, are consistent with the observational constraints at the coronal base have been obtained under the following conditions: (1) the spectrum index of the fluctuations is near to -1 for almost the whole frequency range at 10 R _⊙, (2) the particle flux density at 1 AU is not greater than 3 × 10⁸ cm^?2 s^?1, (3) the solution is for spherically-symmetric flow geometry or the solution passes through the outermost of the three critical points of the rapidly diverging flow geometry with f _max = 7. Some solutions passing through the innermost critical point of the rapidly diverging flow geometry with f _max = 7 have been found, however, with too low pressure at the coronal base to compare with the observational constraints. Heat addition or other kind of momentum addition for r < 10 R _⊙ is required to modify this model to yield better agreement with observations. A cascade energy flux function which leads to Kolmogorov power law in the high-frequency range of Alfvénic fluctuations is presented in Appendix A. More detailed discussions about the characteristics, the boundary conditions and the solution of the spectrum equation (1) are given in Appendix B.     

The height variation of granular and oscillatory velocities

Previous observations of spatially-resolved vertical velocity variations in ten lines of Fe i spanning the height range 0 h 1000 km are re-analyzed using velocity weighting functions. The amplitudes and scale heights of granular and oscillatory velocities are determined, as well as those of the remaining unresolved velocities. I find that the optimal representation of the amplitude of the outward-decreasing granular velocities is an exponentially decreasing function of height, with a scale height of 150 km and a velocity at zero height of 1.27 km s^–1. The optimal representation of the same quantities for oscillatory velocities is an exponential increase with height, with a scale height of 1100 km and a velocity at zero height of 0.35 km s^–1. The remaining unresolved velocities decrease with height, with a scale height of 380 km and a velocity at zero height of 2.3 km s^–1.     

On the detection of subphotospheric convective velocities and temperature fluctuations

A procedure is outlined for estimating the influence of large-scale convective eddies on the wave patterns of five-minute oscillations of high degree. The method is applied to adiabatic oscillations, with frequency ω and wave number k, of a plane-parallel polytropic layer upon which is imposed a low-amplitude convective flow. The distortion to the k - ω relation has two constituents: one depends on the horizontal component of the convective velocity and has a sign which depends on the sign of ω/k; the other depends on temperature fluctuations and is independent of the sign of ω/k. The magnitude of the distortion is just at the limit of present observational sensitivity. Thus there is reasonable hope that it will be possible to reveal some aspects of the large-scale flow in the solar convection zone.     

Measuring the power spectrum of density fluctuations at intermediate redshift with X-ray background observations

The precision of intensity measurements of the extragalactic X-ray background (XRB) on an angular scale of about a degree is dominated by spatial fluctuations caused by source confusion noise. X-ray source counts at the flux level responsible for these fluctuations, ∼10⁻¹² erg cm⁻² s⁻¹, will soon be accurately measured by new missions, and it will then be possible to detect the weaker fluctuations caused by the clustering of the fainter, more distant sources which produce the bulk of the XRB. We show here that measurements of these excess fluctuations at the level of (Δ I/I )∼2×10⁻³ are within reach, improving by an order of magnitude on present upper limits. Since it is likely that most (if not all) of the XRB will be resolved into sources by AXAF , subsequent optical identification of these sources will reveal the X-ray volume emissivity in the Universe as a function of redshift. With these ingredients, all-sky observations of the XRB can be used to measure the power spectrum (PS) of the density fluctuations in the Universe at comoving wavevectors k _c∼0.01–0.1 Mpc⁻¹ at redshifts where most of the XRB is likely to originate ( z ∼1–2) with a sensitivity similar to, or better than, the predictions from large-scale structure theories. A relatively simple X-ray experiment, carried out by a large-area proportional counter with a 0.5–2 deg² collimated field of view scanning the whole sky a few times, would be able to determine the PS of the density fluctuations near its expected peak in wavevector with an accuracy better than 10 per cent.     

Short-period intensity fluctuations of integral sunlight

An attempt has been made to detect short-period solar luminosity fluctuations in the vicinity of 5 min, analogous to the observed velocity oscillation. Using silicon photodiodes to monitor integral sunlight, an upper limit for the amplitude of the intensity fluctuations of 3 x 10^-5 rms was found.Proceedings of the 14th ESLAB Symposium on Physics of Solar Variations, 16–19 September 1980, Scheveningen, The Netherlands.     

On the spectrum of granular and intergranular regions

A very high quality wiggly-line spectrogram was analyzed by making high-resolution spectral scans of numerous small solar features. An attempt from the line profiles to detect a magnetic field difference between the granular and intergranular regions, resulted in a field increase of 20 ± 15 G in the darker regions of the granular field. Line width increases apparently due to small-scale turbulent velocities are seen in the darker regions. It is postulated that in general darker regions show increased turbulent velocities. Conspicuous asymmetries in line profiles are seen in dark intergranular regions. It is suggested that these are the result of velocity gradients in the downward flow of material. An ionized Cr line showed a conspicuous increase in equivalent width in the darker regions of the granular field, thus indicating a decrease in electron pressure in these areas.     

Non-Gaussian cosmic microwave background temperature fluctuations from peculiar velocities of clusters

We use numerical simulations of a (480 Mpc  h ⁻¹)³ volume to show that the distribution of peak heights in maps of the temperature fluctuations from the kinematic and thermal Sunyaev–Zeldovich (SZ) effects will be highly non-Gaussian, and very different from the peak-height distribution of a Gaussian random field. We then show that it is a good approximation to assume that each peak in either SZ effect is associated with one and only one dark matter halo. This allows us to use our knowledge of the properties of haloes to estimate the peak-height distributions. At fixed optical depth, the distribution of peak heights resulting from the kinematic effect is Gaussian, with a width that is approximately proportional to the optical depth; the non-Gaussianity comes from summing over a range of optical depths. The optical depth is an increasing function of halo mass and the distribution of halo speeds is Gaussian, with a dispersion that is approximately independent of halo mass. This means that observations of the kinematic effect can be used to put constraints on how the abundance of massive clusters evolves, and on the evolution of cluster velocities. The non-Gaussianity of the thermal effect, on the other hand, comes primarily from the fact that, on average, the effect is larger in more massive haloes, and the distribution of halo masses is highly non-Gaussian. We also show that because haloes of the same mass may have a range of density and velocity dispersion profiles, the relation between halo mass and the amplitude of the thermal effect is not deterministic, but has some scatter.     

A power spectrum description of Galactic neutral hydrogen

Angular power spectra for neutral hydrogen towardsl = 140°,b = 0° on scales from ≈ 1° to 1/10° are presented. The spectra are generally well fitted by a power-law dependence on radius in theuv-plane, with an index between ≈ -2.2 and ≈ -3.0. The more distant hydrogen, which corresponds to larger physical scales, tends to have a more negative index than does the nearer material. There is no preferred angular scale apparent at any velocity. For the distant hydrogen there is no obvious difference between the slope of power spectra for emission aligned perpendicular compared with that parallel to the Galactic plane.     

A study of interstellar medium of dwarf galaxies using H i power spectrum analysis

We estimate the power spectrum of H  i intensity fluctuations for a sample of eight galaxies (seven dwarf and one spiral). The power spectrum can be fitted to a power-law     for six of these galaxies, indicating turbulence is operational. The estimated best-fitting value for the slope ranges from  ∼−1.5  (AND IV, NGC 628, UGC 4459 and GR 8) to  ∼−2.6  (DDO 210 and NGC 3741). We interpret this bi-modality as being due to having effectively 2D turbulence on length-scales much larger than the scale-height of the galaxy disc and 3D otherwise. This allows us to use the estimated slope to set bounds on the scale-heights of the face-on galaxies in our sample. We also find that the power-law slope remains constant as we increase the channel thickness for all these galaxies, suggesting that the fluctuations in H  i intensity are due to density fluctuations and not velocity fluctuations, or that the slope of the velocity structure function is ∼0. Finally, for the four galaxies with '2D turbulence' we find that the slope α correlates with the star formation rate (SFR) per unit area, with larger SFRs leading to steeper power laws. Given our small sample size, this result needs to be confirmed with a larger sample.