共查询到20条相似文献，搜索用时 46 毫秒
1.
Shiro Sakashita 《Astrophysics and Space Science》1974,26(1):183-188
Similarity solution for unsteady accretion flow in a gravitational field of a point mass is obtained. Characteristic features of the flow pattern are discussed. It is shown that the shock waves appeared in the accretion flow propagate outward asr
s
t
2/3. 相似文献
2.
We analytically generalize the well-known solution of steady supersonic spherically symmetric gas accretion onto a star (Bondi 1952) for an iron atmosphere with completely degenerate electrons with an arbitrary degree of relativity. This solution is used for typical physical conditions in the vicinity of protoneutron stars produced by gravitational collapse with masses M 0=(1.4?1.8)M ⊙ and over a wide range of nonzero “iron gas” densities at infinity, ρ∞=(104?5×106)g cm?3. Under these conditions, we determine all accretion parameters, including the accretion rate, whose value is ～(10?50)M ⊙s?1 at M 0=1.8M ⊙ (it is a factor of 1.7 lower for M 0=1.4M ⊙, because the accretion rate is exactly ∝M 0 2 ). We take into account the effect of accreting-gas rotation in a quasi-one-dimensional approximation, which has generally proved to be marginal with respect to the accretion rate. 相似文献
3.
The accretion of hot slowly rotating gas onto a supermassive black hole is considered. The important case where the velocities of turbulent pulsations at the Bondi radius r B are low, compared to the speed of sound c s, is studied. Turbulence is probably responsible for the appearance of random average rotation. Although the angular momentum at r B is low, it gives rise to the centrifugal barrier at a depth r c = l 2 /GM BH ? r B, that hinders supersonic accretion. The numerical solution of the problem of hot gas accretion with finite angular momentum is found taking into account electron thermal conductivity and bremsstrahlung energy losses of two temperature plasma for density and temperature near Bondi radius similar to observed in M87 galaxy. The saturation of the Spitzer thermal conductivity was also taken into account. The parameters of the saturated electron thermal conductivity were chosen similar to the parameters used in the numerical simulations of interaction of the strong laser beam radiation with plasma targets. These parameters are confirmed in the experiments. It is shown that joint action of electron thermal conductivity and free-free radiation leads to the effective cooling of accreting plasma and formation of the subsonic settling of accreting gas above the zone of a centrifugal barrier. A toroidal condensation and a hollow funnel that separates the torus from the black hole emerge near the barrier. The barrier divides the flow into two regions: (1) the settling zone with slow subKeplerian rotation and (2) the zone with rapid supersonic nearly Keplerian rotation. Existence of the centrifugal barrier leads to significant decrease of the accretion rate ? in comparison with the critical Bondi solution for γ = 5/3 for the same values of density and temperature of the hot gas near Bondi radius. Shear instabilities in the torus and related friction cause the gas to spread slowly along spirals in the equatorial plane in two directions.As a result, outer (r > r c) and inner (r < r c) disks are formed. The gas enters the immediate neighborhood of the black hole or the zone of the internal ADAF flow along the accretion disk (r < r c). Since the angular momentum is conserved, the outer disk removes outward an excess of angular momentum along with part of the matter falling into the torus. It is possible, that such outer Keplerian disk was observed by Hubble Space Telescope around the nucleus of the M87 galaxy in the optical emission lines. We discuss shortly the characteristic times during which the accretion of the gas with developed turbulence should lead to the changes in the orientation of the torus, accretion disk and, possibly, of the jet. 相似文献
4.
We investigate the unsteady column accretion of material at a rate \(10^{15} g s^{ - 1} \leqslant \dot M \leqslant 10^{16} g s^{ - 1}\) onto the surface of a magnetized neutron star using a modified first-order Godunov method with splitting. We study the dynamics of the formation and evolution of a shock in an accretion column near the surface of a star with a magnetic field 5×1011≤B≤1013 G. An effective transformation of the accretion flow energy into cyclotron radiation is shown to be possible for unsteady accretion with a collisionless shock whose front executes damped oscillations. The collisionless deceleration of the accreting material admits the conservation of a fraction of the heavy nuclei that have not been destroyed in spallation reactions. The fraction of the CNO nuclei that reach the stellar atmosphere is shown to depend on the magnetic field strength of the star. 相似文献
5.
The “strange star - NDAF” model (NDAF: Neutrino Dominated Accretion Flow) is proposed as an alternative central engine of gamma-ray bursts for unifying the interpretation of the prompt emission and postburst activities of gamma-ray bursts. The structure of NDAF around a strange star is calculated. Different from other central compact objects, the strange star will feed back the phase transition energy of strangization on the accretion flow, with neutrinos as energy carriers. The friction between NDAF and strange star is ignored in this paper. The results indicate: firstly, the structure of NDAF around a strange star is sensitive to accretion rate; secondly, if accretion rate is larger than 0.18 M? s-1, the “strange star - NDAF” model can unify the explanation on the prompt emission and postburst activities of gamma-ray bursts, and the range of allowable accretion rates is wider than that in frictionless “neutron star - NDAF” models; thirdly, the range of annihilation energy of “strange star - NDAF” model is very wide, when the accretion rate is higher than 0.3 M? s-1, the annihilation energy is greater than 1051 erg; finally, if the accretion rate is greater than 0.3 M? s-1, the annihilation energy of “strange star - NDAF” model is larger than what of “black hole - NDAF” model at the same accretion rate by more than one order of magnitude, it is favorable to explaining some extremely energetic gamma-ray bursts. 相似文献
6.
Our goal is to study the regime of disk accretion in which almost all of the angular momentum and energy is carried away by the wind outflowing from the disk in numerical experiments. For this type of accretion the kinetic energy flux in the outflowing wind can exceed considerably the bolometric luminosity of the accretion disk, what is observed in the plasma flow from galactic nuclei in a number of cases. In this paper we consider the nonrelativistic case of an outflow from a cold Keplerian disk. All of the conclusions derived previously for such a system in the self-similar approximation are shown to be correct. The numerical results agree well with the analytical predictions. The inclination angle of the magnetic field lines in the disk is less than 60°, which ensures a free wind outflow from the disk, while the energy flux per wind particle is greater than the particle rotation energy in its Keplerian orbit by several orders of magnitude, provided that the ratio r A/r ? 1, where r A is the Alfvénic radius and r is the radius of the Keplerian orbit. In this case, the particle kinetic energy reaches half the maximum possible energy in the simulation region. The magnetic field collimates the outflowing wind near the rotation axis and decollimates appreciably the wind outflowing from the outer disk periphery. 相似文献
7.
The accretion of hot slowly rotating gas onto a supermassive black hole is considered. The important case where the velocities
of turbulent pulsations at the Bondi radius r
B are low, compared to the speed of sound c
s, is studied. Turbulence is probably responsible for the appearance of random average rotation. Although the angular momentum
at r
B is low, it gives rise to the centrifugal barrier at a depth r
c = l
2
/GM
BH ≪ r
B, that hinders supersonic accretion. The numerical solution of the problem of hot gas accretion with finite angular momentum
is found taking into account electron thermal conductivity and bremsstrahlung energy losses of two temperature plasma for
density and temperature near Bondi radius similar to observed in M87 galaxy. The saturation of the Spitzer thermal conductivity
was also taken into account. The parameters of the saturated electron thermal conductivity were chosen similar to the parameters
used in the numerical simulations of interaction of the strong laser beam radiation with plasma targets. These parameters
are confirmed in the experiments. It is shown that joint action of electron thermal conductivity and free-free radiation leads
to the effective cooling of accreting plasma and formation of the subsonic settling of accreting gas above the zone of a centrifugal
barrier. A toroidal condensation and a hollow funnel that separates the torus from the black hole emerge near the barrier.
The barrier divides the flow into two regions: (1) the settling zone with slow subKeplerian rotation and (2) the zone with
rapid supersonic nearly Keplerian rotation. Existence of the centrifugal barrier leads to significant decrease of the accretion
rate Ṁ in comparison with the critical Bondi solution for γ = 5/3 for the same values of density and temperature of the hot gas near Bondi radius. Shear instabilities in the torus and
related friction cause the gas to spread slowly along spirals in the equatorial plane in two directions.As a result, outer
(r > r
c) and inner (r < r
c) disks are formed. The gas enters the immediate neighborhood of the black hole or the zone of the internal ADAF flow along
the accretion disk (r < r
c). Since the angular momentum is conserved, the outer disk removes outward an excess of angular momentum along with part of
the matter falling into the torus. It is possible, that such outer Keplerian disk was observed by Hubble Space Telescope around
the nucleus of the M87 galaxy in the optical emission lines. We discuss shortly the characteristic times during which the
accretion of the gas with developed turbulence should lead to the changes in the orientation of the torus, accretion disk
and, possibly, of the jet. 相似文献
8.
A two-stage growth of the giant planets, Jupiter and Saturn, is considered, which is different from the model of contraction of large gaseous protoplanets. In the first stage, within a time of ～3 × 107 years in Jupiter's zone and ～2 × 108 years in Saturn's zone, a nucleus forms from condensed (solid) material having the mass, ～1028 g, necessary for the beginning of acceleration. The second stage may gravitating body, and a relatively slow accretion begins until the mass of the planet reaches ～10 m⊕. Then a rapid accretion begins with the critical radius less than the radius of the Hill lobe, so that the classical formulae for the rate of accretion may be applied. At a mass m > m1 ≈ 50 m⊕ accretion proceeds slower than it would according to these formulae. When the planet sweeps out all the gas from its nearest zone of feeding (m = m2 ≈ 130 m⊕), the width of the exhausted zone being of the whole zone of the planet) growth is provided the slow diffusion of gas from the rest of the zone (time scale increases to 105?106 years and more). The process is terminated by the dissipation of the remnants of gas. In Saturn's zone m1 > m2 ≈ 30 m⊕. The initial mass of the gas in Jupiter's zone is estimated. Before the beginning of the rapid accretion about 90% of the gas should have been lost from the solar system, and in the planet's zone less than two Jupiter masses remain. The highest temperature of Jupiter's surface, ≈5000°K, is reached at the stage of rapid accretion, m < 100 m⊕, when the luminosity of the planet reaches 3 × 10?3 L⊙. This favors an effective heating of the inner parts of the accretionary disk and the dissipation of gas from the disk. The accretion of Saturn produced a temperature rise up to 2000?2400° K (at m ≈ 20?25 m⊕) and a luminosity up to 10?4 L⊙. 相似文献
9.
P. Stogianidis D. P. Georgiou G. A. Georgantopoulos 《Astrophysics and Space Science》1985,109(2):309-326
the influence of radiation absorption on the flow-field of an unsteady laminar boundary layer due to free convection is considered. The flow is that of an incompressible viscous dissipative and electrically conducting fluid past an infinite vertical porous plate, when the flow is subjected to the action of a transverse magnetic field and the mainstream is oscillating around a mean value and an oscillating suction. The radiation is absorbed by a second material in a small concentration within the fluid and the absorption rate is proportional to the local concentration. The solution of the problem is obtained in the form of power series of Eckert numberE, which is very small for incompressible fluids, analytical expression for the velocity, temperature and the induced magnetic field are given for both the steady and the unsteady flows. The solution is also given in a number of figures that present the combined influence of the absorption withE,G r , and the frequency of oscillation. In addition the influence of the absorption on the skin friction the heat flux is given. The results indicate the importance of the absorption which can be significant under certain conditions. 相似文献
10.
Ru-liang Liu 《Chinese Astronomy and Astrophysics》1983,7(3):195-200
The nuclear mass M, bolometric luminosity Lbol, and the Eddington luminosity LEdd are determined for fifty Type-I Seyfert galaxies from data on the luminosity and line-width of nuclear Balmer line Hβ, assuming a mass accretion flow model. It is emphasized that the full width at zero intensity (FWZI), rather than the full width at half maximum (FWHM) should be used to give the velocity in the central broad line region. The histograms of M and of Lbol/LEdd are shown in Figs. 1 and 2, for ready comparison with Type-II Seyferts and QSOs.Adding the data on the X-ray luminosities LX for 38 Seyferts on our list, we have obtained the following correlations: Lx(2–10KcV)α(FWZI)2, with γ = 0.50, and Lx(2–10KcV)α(M), with γ = 0.70 See Figs. 3 and 4. The implications of our statistical results are briefly discussed in terms of the spherically symmetric flow model of mass accretion. 相似文献
11.
Shock wave and thermodynamic data for rock-forming and volatile-bearing minerals are used to determine minimum impact velocities (vcr) and minimum impact pressures (pcr) required to form a primary H2O atmosphere during planetary accretion from chondritelike planetesimals. The escape of initially released water from an accreting planet is controlled by the dehydration efficiency. Since different planetary surface porosities will result from formation of a regolith, vcr and pcr can vary from 1.5 to 5.8 km/sec and from 90 to 600 kbar, respectively, for target porosities between 0 and ～45%. On the basis of experimental data, hydration rates for forsterite and enstatite are derived. For a global regolith layer on the Earth's surface, the maximum hydration rate equals 6 × 1010 g H2O sec?1 during accretion of the Earth. Attenuation of impact-induced shock pressure is modeled to the extent that the amount of released water as a function of projectile radius, impact velocity, weight fraction of water in the target, target porosity, and dehydration efficiency can be estimated. The two primary processes considered are the impact release of water bound in hydrous minerals (e.g., serpentine) and the subsequent reincorporation of free water by hydration of forsterite and enstatite. These processes are described in terms of model calculations for the accretion of the Earth. Parameters which lead to a primary atmosphere/hydrosphere are: an accretion time of ? 1.6 × 108years, the use of an accretion model defined by Weidenschilling (1974, 1976), a mean planetesimal radius of 0.5 km, a hydration rate of 6 × 1010 g H2O sec?1 inferred from a mean porosity of ～ 10% for the upper 1 km of the accreting Earth, and values for the dehydration efficiency, DE, of 0.55 and 0.07 for the maximum and minimum pressure decay model, respectively. Conditions which prohibit the formation of a primary atmosphere include an accretion time much longer than 1.6 × 108 years, a hydration rate for forsterite and enstatite well in excess of 6 × 1010 g H2O sec?1, and a dehydration efficiency DE < 0.07. We conclude that the concept of dehydration efficiency is of dominant importance in determining the degree to which an accreting planet acquires an atmosphere during its formation. 相似文献
12.
We elaborate the model of accretion disks of young stars with the fossil large-scale magnetic field in the frame of Shakura and Sunyaev approximation. Equations of the MHD model include Shakura and Sunyaev equations, induction equation and equations of ionization balance. Magnetic field is determined taking into account ohmic diffusion, magnetic ambipolar diffusion and buoyancy. Ionization fraction is calculated considering ionization by cosmic rays and X-rays, thermal ionization, radiative recombinations and recombinations on the dust grains. Analytical solution and numerical investigations show that the magnetic field is coupled to the gas in the case of radiative recombinations. Magnetic field is quasi-azimuthal close to accretion disk inner boundary and quasi-radial in the outer regions. Magnetic field is quasi-poloidal in the dusty “dead” zones with low ionization degree, where ohmic diffusion is efficient. Magnetic ambipolar diffusion reduces vertical magnetic field in 10 times comparing to the frozen-in field in this region. Magnetic field is quasi-azimuthal close to the outer boundary of accretion disks for standard ionization rates and dust grain size a d=0.1 μm. In the case of large dust grains (a d>0.1 μm) or enhanced ionization rates, the magnetic field is quasi-radial in the outer regions. It is shown that the inner boundary of dusty “dead” zone is placed at r=(0.1–0.6) AU for accretion disks of stars with M=(0.5–2)?M ⊙. Outer boundary of “dead” zone is placed at r=(3–21) AU and it is determined by magnetic ambipolar diffusion. Mass of solid material in the “dead” zone is more than 3?M ⊕ for stars with M≥1?M ⊙. 相似文献
13.
The effect of the onset of post-sunset conditions on thermal proton flow is examined for mid-latitudes by numerical solution of the equations of continuity, momentum and energy balance for H+ and O+. Results are calculated for a dipole magnetic field tube situated at L = 4 and acceleration terms are included in the momentum equations. Proton flow into the ionosphere results from decay of the F2-layer. Changes in temperatures and temperature gradients following sunset may not enhance the H+ flow. Under extreme conditions the H+ flow remains subsonic. It seems unlikely that an interhemispheric flux of protons can directly maintain the nighttime F2-layer. 相似文献
14.
The exact nonstationary solution for the variation of the magnetic field in the Schwarzschild metric with a given spherically symmetric flow is obtained. Initially a homogeneous magnetic field increases with time, changing into a quasi-radial field. On the assumption of equipartition between the magnetic and kinetic energies of a falling gas, in the relativistic case, estimates of the stationary field and the intensity of synchrotron radiation are presented. A considerable part of the radiation is formed in the relativistic regionr?(2.5 to 7.7)r g (r g is the gravitational radius of a black hole). Estimates are made for radiation from the relativistic region in the case of disc type accretion. 相似文献
15.
The results of longitudinal magnetic field measurements B z in the hot accretion spot in three classical T Tauri stars (CTTS) are reported. In all three stars the magnetic field is detected at a level above 2σ in the formation region of the narrow component of the He I 5876 Å emission line. In the case of DS Tau the longitudinal field B z in the hot spot was also measured from the narrow emission components of the Na I D lines, implying +0.8 ± 0.3 kG, which is equal to the B z field component measured from the He I 5876 Å line. Our results suggest that the 6-m telescope of the Special Astrophysical Observatory can be used to study magnetic fields in the hot spots of CTTS with magnitudes down to 13m, making it possible to double the number of stars of this type with measured B z values in the accretion zone. 相似文献
16.
We investigate the relation between black hole mass, M bh, and jet power, Q jet, for a sample of BL Lacs and radio quasars. We find that BL Lacs are separated from radio quasars by the FR I/II dividing line in M bh–Q jet plane, which strongly supports the unification scheme of FR I/BL Lac and FR II/radio quasar. The Eddington ratio distribution of BL Lacs and radio quasars exhibits a bimodal nature with a rough division at L bol/L Edd~0.01, which imply that they may have different accretion modes. We calculate the jet power extracted from advection-dominated accretion flow (ADAF), and find that it requires dimensionless angular momentum of black hole j???0.9???0.99 to reproduce the dividing line between FR I/II or BL Lac/radio quasar if dimensionless accretion rate $\dot{m}=0.01$ is adopted, which is required by the above bimodal distribution of Eddington ratios. Our results suggest that black holes in radio galaxies are rapidly spinning. 相似文献
17.
We model the subnebulae of Jupiter and Saturn wherein satellite accretion took place. We expect each giant planet subnebula to be composed of an optically thick (given gaseous opacity) inner region inside of the planet’s centrifugal radius (where the specific angular momentum of the collapsing giant planet gaseous envelope achieves centrifugal balance, located at rCJ ∼ 15RJ for Jupiter and rCS ∼ 22RS for Saturn) and an optically thin, extended outer disk out to a fraction of the planet’s Roche-lobe (RH), which we choose to be ∼RH/5 (located at ∼150 RJ near the inner irregular satellites for Jupiter, and ∼200RS near Phoebe for Saturn). This places Titan and Ganymede in the inner disk, Callisto and Iapetus in the outer disk, and Hyperion in the transition region. The inner disk is the leftover of the gas accreted by the protoplanet. The outer disk may result from the nebula gas flowing into the protoplanet during the time of giant planet gap-opening (or cessation of gas accretion). For the sake of specificity, we use a solar composition “minimum mass” model to constrain the gas densities of the inner and outer disks of Jupiter and Saturn (and also Uranus). Our model has Ganymede at a subnebula temperature of ∼250 K and Titan at ∼100 K. The outer disks of Jupiter and Saturn have constant temperatures of 130 and 90 K, respectively.Our model has Callisto forming in a time scale ∼106 years, Iapetus in 106-107 years, Ganymede in 103-104 years, and Titan in 104-105 years. Callisto takes much longer to form than Ganymede because it draws materials from the extended, low density portion of the disk; its accretion time scale is set by the inward drift times of satellitesimals with sizes 300-500 km from distances ∼100RJ. This accretion history may be consistent with a partially differentiated Callisto with a ∼300-km clean ice outer shell overlying a mixed ice and rock-metal interior as suggested by Anderson et al. (2001), which may explain the Ganymede-Callisto dichotomy without resorting to fine-tuning poorly known model parameters. It is also possible that particulate matter coupled to the high specific angular momentum gas flowing through the gap after giant planet gap-opening, capture of heliocentric planetesimals by the extended gas disk, or ablation of planetesimals passing through the disk contributes to the solid content of the disk and lengthens the time scale for Callisto’s formation. Furthermore, this model has Hyperion forming just outside Saturn’s centrifugal radius, captured into resonance by proto-Titan in the presence of a strong gas density gradient as proposed by Lee and Peale (2000). While Titan may have taken significantly longer to form than Ganymede, it still formed fast enough that we would expect it to be fully differentiated. In this sense, it is more like Ganymede than like Callisto (Saturn’s analog of Callisto, we expect, is Iapetus). An alternative starved disk model whose satellite accretion time scale for all the regular satellites is set by the feeding of planetesimals or gas from the planet’s Roche-lobe after gap-opening is likely to imply a long accretion time scale for Titan with small quantities of NH3 present, leading to a partially differentiated (Callisto-like) Titan. The Cassini mission may resolve this issue conclusively. We briefly discuss the retention of elements more volatile than H2O as well as other issues that may help to test our model. 相似文献
18.
Aurora Sicilia-Aguilar Lee Hartmann César Briceño María Eugenia Contreras James Muzerolle Nuria Calvet 《Astrophysics and Space Science》2004,292(1-4):457-463
We are studying the young clusters Tr37 and NGC7160 in the Cep OB2 region as part of a program to understand the evolution of accretion disks at the ages of disk dissipation and planet formation. Here, we present the first identifications of low mass (spectral types K-M) members of the clusters and study the presence and characteristics of their accretion disks, finding evidences of disk evolution. Using optical photometry and spectroscopy, we have identified ～70 members in Tr37 and ～20 in NGC7160, confirming age estimates of 3 and 10 Myr respectively. Accretion rates are ～10?8 M ⊙yr?1 in Tr37. We have not found any accreting members in NGC7160, suggesting that disk accretion generally ends before the age of 10 Myr, which is consistent with the results from other populations. 相似文献
19.
In a previous paper (hereafter referred to as Paper I) we have tried to show that superdense cores in the nuclei of disk galaxies can be formed by accretion of gas ejected by the evolved stars which populate the central bulge of these galaxies. Solving the equations for radial flow of a magnetized gas, we found that the accretion of an explodable mass at the core can be achieved over a time-scale ranging from a few times 107 and a few times 108 yr. It was shown, however, that the accretion process is seriously inhibited if the gas possesses sufficient rotational velocity but lacks any dissipative, mechanism within the system. Since rotational velocity is an observed parameter of the stars which shed the gas to be accreted, one must consider the existence of some dissipative force in it in order that the accretion process may be efficient. In the present paper, therefore, we have solved the problem of the flow of a rotating, viscous (variable), magnetized gas. With plausible assumptions regarding some of the parameters involved, the time-scale for the accretion of an explodable mass (109
M
) at the core again turns out to be ranging between a few times 107 and a few times 108yr. Such time-scale has been proposed by several authors as that for repeated explosions in nuclei of these galaxies. It has also been proposed by many authors that the spiral arms are generated and destroyed in disk galaxies over the same time-scale. Our solution also yields a nearly linear rotational velocity law which is usually observed in the central regions of these galaxies. 相似文献
20.
Chuan-Yi Tu 《Solar physics》1987,109(1):149-186
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. 相似文献