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1.
A satellite four-body problem is the problem of motion of an artificial satellite of a planet in a region of the space where perturbations due to the gravitational field of the planet are of the same order as perturbations due to influences of two perturbing bodies. In this paper an expansion of the perturbing function into a Fourier series in terms of angular Keplerian elements (
j
,
j
,M
j
:j=0,1,2) (designations are standard) is obtained taking into account a sharp commensurability of the typen/
0=(p+q)/p (n is the mean motion of the artificial satellite and
0 is the angular velocity of rotation of the planet,p andq are integers).The coefficients of the Fourier series are the functions of the positional Keplerian elements (a
j
,e
j
,i
j
;j=0, 1, 2) (designations are standard) and, in particular, are series in terms ofe
j
that, generally speaking, can be written out to an accuracy ofe
j
19
.The expansion obtained can be used for the construction of a semianalytical theory of motion of resonant satellites on the basis of conditionally periodic solutions of the restricted four-body problem. 相似文献
2.
Unrestricted second-order tensor virial equations for linear oscillations of magnetic configurations
M. Goossens 《Astrophysics and Space Science》1979,60(2):401-421
This paper deals with the second-order tensor virial equations for the linear oscillations of a gaseous mass in the presence of a magnetic field. It is shown that the commonly used linearized versions of the tensor virial equations are restricted integral equations that incorporate the linearized equation of motion but not the boundary condition. These restricted equations only allow trial functions that fulfil the boundary condition and are of limited practical value.The unrestricted variational principle for the linear oscillations of a magnetic configuration is used to derive a more general formulation of the second-order tensor virial equations so that the linear trial function i =X
ij
x
j can be used to study the oscillations of a configuration with a magnetic field that extends in the exterior vacuum. The unrestricted virial equations have been applied to Ferraro's model and approximate results for the eigenfrequencies and eigenfunctions have been obtained for nine oscillation modes. 相似文献
3.
Joseph V. Hollweg 《Solar physics》1979,62(2):227-240
We consider a horizontally stratified isothermal model of the solar atmosphere, with vertical and uniform B
0, and v
A
2
v
s
2
. The equations of motion are linearized about a background which is in hydrostatic equilibrium. A homogeneous wave equation results for the motions perpendicular to B
0; this wave equation is similar to the equation for the MHD fast mode. On the other hand, the equation for the parallel motions is inhomogeneous, containing driving terms which arise from the presence of the fast mode; the homogeneous form of this equation is identical to the equation describing vertically-propagating gravity-modified acoustic waves. We demonstrate that a resonance can exist between the (driving) fast wave and the (driven) gravity-modified acoustic wave, in such a way that very large parallel velocities can be driven by small perpendicular velocities. Applications of this resonance to solar spicules, jets, and other phenomena are discussed.The National Center for Atmospheric Research is sponsored by the National Science Foundation. 相似文献
4.
Masaya Saitou 《Astrophysics and Space Science》1989,162(1):47-56
Some peculiarities in the behaviour of a model self-gravitating system described by hydrodynamical equations and isothermal equation of state connected with the presence of thermodynamical fluctuations in real systems were investigated in numerical experiment. The values of density and velocity , , respectively, were computed by numerical code perturbed on each time-step and in each computational cell by random values , for modeling such fluctuations. Perturbed values
i
=
i
+
i
,v
i
=
i
+ v
i
were used to initiate the next step of computations. This procedure is equivalent to an introduction into original hydrodynamical equations of Langevin sources which are random functions. It is shown that these small fluctuations (= v =0,2 =v
2 = 10–8) grow many times in marginally-stable state. 相似文献
5.
Numerical experiments of fictitious small bodies with initial eccentricities e=0.1 have been performed in the overlapping region of the 3/1 mean motion resonance and of thev
6 secular resonance 2.48a2.52AU for different values of the initial inclination 16°i20°. An analysis for thev
6 secular resonance shows that the topology is different from the one found outside the overlapping region: the critical argument for thev
6 resonance in the overlapping region rotates in opposite direction as compared to the purev
6 region. In the 3/1 resonance region the secular resonancev
5 is dominant, and some secondary secular resonances asv
6 –v
16 andv
5 +v
6 are present. 相似文献
6.
Madeleine Pascal 《Celestial Mechanics and Dynamical Astronomy》1985,37(1):81-93
This paper is the continuation of a previous work [6] in which we have obtained the set of all possible equilibria of a gyrostat satellite attracted by n points mass by solving two algebraical equations P1=0 and P2=0. It results that there is a maximum of 24 isolated equilibrium orientations for the satellite. Sufficient conditions of stability for these relative equilibria are given.Here we consider only the elementary case n=1. We show that the coefficients of the two algebrical equations depend on four parameters j1, j3, K and v2. The two first parameters depend only on the direction of the internal angular momentum
of the rotors, the third being only function of the principal moments of inertia of the satellite and the last parameter is a decreasing function of one of the components of
. We show that the two polynomials P1 and P2 are unvariant within two transformations of the parameters j1 and j3. It is then possible to reduce the range of variation of these parameters.For some particular values of the parameters, it is possible to give the minimum number of real roots of equations P1=0 and P2=0. In general cases, a computing program is written to obtain the number of real roots of these equations according to the values of the parameters. We show that among the roots found, few of them corresponds to stable equilibrium orientations. 相似文献
7.
Tsutomu Toichi 《Solar physics》1971,18(1):150-164
In the solar wind, electrostatic ion cyclotron waves can be excited by electrons when the flow velocity becomes supersonic. The waves reduce the proton temperature anisotropy and heat the protons effectively. Temperature equations for T
e
,T
p
and T
p
are solved numerically in the region from 1 AU to the Sun, with the non-thermal proton heating rate included as a parameter. Distributions of T
e
,T
p
, T
p
and the proton heating rate are determined and found to be in good agreement with the proton heating rate expected from the linear growth rate of electrostatic ion cyclotron waves. The electron thermal conductivity is reduced approximately 2–3 times smaller than the usual collisional one due to the plasma wave instabilities. Effective energy exchange rates from proton-proton and electron-proton interactions are 1–10 and 10–100 times larger than the Coulomb collision rates v
ppand v
ep,respectively. 相似文献
8.
The orbital perturbations induced by the librational motion and flexural oscillations are studied for satellites having large flexible appendages. Using a Lagrangian procedure, the equations for coupled motion are derived for a satellite having an arbitrary number of appendages in the nominal orbital plane and two flexible members normal to it. The formulation enables one to study the influence of flexibility on both the orbital and attitude motions. The orbital coordinates are expanded as perturbation series in =(l/a
0)2,l anda
0 being a characteristic length of the satellite and unperturbed semi-major axis of the orbit, respectively. The first order perturbation equations are solved in terms of elastic deformations and librational angles using the WKBJ method in conjunction with the variation of parameter technique. Existence of secular perturbations is noted for certain librational flexural motions. Three specific examples, Alouette II, Radio Astronomy Explorer and Tethered Orbiting Interferometer, are considered subsequently and their possible secular drifts estimated.List of Symbols
A
ij, Bij
coefficients in the eigenfunction expansion ofv
i andw
i respectively, Equation (10)
-
C
k, Dk
constants, Equation (21)
-
EI
i
flexural rigidity of theith appendage
-
E(u0)
2(1+e
0 cosu
0)2
h
0
3
-
F(u0)
perturbation function, Equation (17b)
-
F
,F
,F
functions of librational angles and flexural displacements, Equation (11i)
-
F
,F
,F
F
,F
,F
with change of independent variable fromt tou
0
-
I
xx, Iyy, Izz
principal moments of inertia of the undeformed satellite
- [J
i]
inertia dyadic of the deformedith appendage
- [J
d]
inertia dyadic of the deformed satellite
-
M
mass of the satellite
-
P
R, Pu
functions of librational angles and flexural displacements, Equation (15d) and (15e), respectively
-
R
c
magnitude ofR
c
-
R
c0, R1
unperturbed value and first order perturbation ofR
c, respectively
-
R
c
,R
0
position vectors of the c.m. of the deformed and undeformed satellite, respectively
-
T
kinetic energy of the satellite
-
U
potential energy of the satellite
-
U
e, Ug
elastic and gravitational potential energy, respectively
-
X, Y, Z
orbital co-ordinate axes, located at the c.m. of the deformed satellite
-
Y
1(u0), Y2(u0)
functions ofu
0, Equation (18b) and (18c), respectively
-
a
semi-major axis
-
a
0
unperturbed value ofa
-
e
eccentricity
-
e
0
unperturbed value ofe
-
h
0
unperturbed angular momentum per unit mass of the satellite
-
i
inclination of the orbital plane to the ecliptic
-
i, j, k
unit vectors alongx (or ),y (or ) andz (or ) axes, respectively
-
l
characteristic length of the satellite
-
l
i
length of theith appendage
- [l
i]
matrix of direction cosines ofx
i, vi andw
i
-
l
,l
,l
direction cosines ofR
c
-
m
0, mi
mass of the main body andith appendage, respectively
-
p
i
2
-
q
m, Qm
generalized co-ordinate and force, respectively
-
r
1
R
1/Rc0
-
r
position vector of an element of the body referred toxyz axes
-
r
u
position vector of an element after deformation, referred to axes
-
r
c
x
c
i+y
c
j+z
c
k, position vector of the c.m. of the deformed body referred toxyz axes
-
s
x
i/li
-
t
time
-
u
true anomaly
-
u
0, u1
unperturbed value and the first order perturbation ofu, respectively
-
u
elastic displacement vector
-
u
c
u–r
c
-
velocity of an element relative to axes
-
v
i, wi
flexural deformations
-
x, y, z
body co-ordinate axes with origin at the c.m. of the undeformed satellite
-
x
i
distance of an element of theith appendage from the root
-
j
jth eigenfunction (normalized) of a cantilever
-
angle between the line of nodes and vernal equinox
-
,
,
components of nondimensionalized angular velocity of the satellite
- , ,
pitch (spin), yaw and roll, respectively
- i
nominal inclination of theith appendage in the orbital plane
-
-
small parameter, (l/a
0)2
- j
jth eigenvalue of a cantilever
-
gravitational constant
-
jk
constant, Equation (11j)
- , ,
body co-ordinate axes with origin at the c.m. of the deformed satellite
-
(
i +
j +
k), angular velocity of the satellite 相似文献
9.
In this paper we are concerned with incompressible MHD turbulence in a rotating system and have derived an equation for the rate of change of vorticity covariance of MHD turbulent flow. The result derived shows that the defining scalars (r,t), (r,t), and (r,t) for the rate of change of vorticity covariance solely depend on the defining scalars of the tensorsW
ij, Pik,j, Fkj,i, Tik,j, andR
kj,ialready defined in the text. 相似文献
10.
BUSS observations of the profiles of two well observed spectral lines in the ultraviolet spectrum of CMi (Procyon; F5 IV–V) are analysed with a Fourier transform method in order to determine values of various parameters of the velocity field of the upper photosphere. We find a microturbulent line-of-sight velocity componentL
= 0.9 ± 0.4 km s–1, a macroturbulent velocity componentL
M = 5.3 ± 0.2 km s–1, and a rotational velocity componentv
R sini=10.0±1.2 km s–1. In these calculations a single-moded sinusoidal isotropic macroturbulent velocity function was assumed. The result appears to be sensitive to the assumed shape of the macroturbulence function: for an assumed Gaussian shape the observations can be described withv
R sini=4 km s–1 andL
M = 11.6 ± 2.7 km s–1. A comparison is made with other results and theoretical predictions. 相似文献
11.
We have analyzed data on sunspot groups compiled during 1874–1981 and investigated the following: (i) dependence of the `initial' meridional motion (v
ini()) of sunspot groups on the life span () of the groups in the range 2–12 days, (ii) dependence of the meridional motion (v(t)) of sunspot groups of life spans 10–12 days on the age (t) of the spot groups, and (iii) variations in the mean meridional motion of spot groups of life span 2–12 days during the solar cycle. In each of the latitude intervals 0°–10°, 10°–20° and 20°–30°, the values of both v
ini() and v(t) often differ significantly from zero. In the latitude interval 20°–30°, the forms of v
ini() and v(t) are largely systematic and mutually similar in both the north and south hemispheres. The form of v(t) suggests existence of periodic variation in the solar meridional motion with period of 4 days and amplitude 10–20 m s–1. Using the anchoring depths of magnetic structures for spot groups of different and testimated earlier, (Javaraiah and Gokhale, 1997), we suggest that the forms of v
ini() and v(t) may represent radial variation of meridional flow in the Sun's convection zone, rather than temporal variation of the flow. The meridional flows (v
e(t)) determined from the data during the last few days (i.e., age t: 10–12 days) of spot groups of life spans of 10–12 days are found to have magnitudes (10–20 m s–1) and directions (poleward) similar to the those of the surface meridional plasma flows determined from the Dopplergrams and magnetograms. The mean meridional velocity of sunspot groups living 2–12 days seems to vary during the solar cycle. The velocity is not significantly different from zero during the rising phase of the cycle and there is a suggestion of equatorward motion (a few m s–1at lower latitudes and 10 m s–1at higher latitudes) during the declining phase (last few years) of the cycle. The variation during the odd numbered cycles seems to anticorrelate with the variation during the even numbered cycles, suggesting existence of 22-year periodicity in the solar meridional flow. The amplitude of the anticorrelation seems to be depending on latitude and the cycle phase. In the latitude interval 20°–30° the `surface plasma meridional motion', v
e(t), is found to be poleward during maximum years (v
e(t) 20 m s–1at 4th year) and equatorward during ending years of the cycle (v
e(t) –17 m s–1at 10th year). 相似文献
12.
The occurrence at a heliocentric distance of 1 AU of alpha particle streaming velocities larger than proton streaming velocities,v
/v
p
>1 (Ogilvie, 1975) is investigated on the basis of the theory suggesting the existence in the solar wind of an accelerating force acting preferentially on the alpha particles.Accurate solution of the three-fluid model equations for the quiet solar wind indicates that anecessary andsufficient condition for (v
/v
p
)1 AU>1 is the presence of a relativelyweak accelerating forceacting in a limited region in the vicinity of 1 AU. If the force is effectiveonly at small heliocentric distances, the alpha particle streaming velocity excess vanishes at distances less than 1 AU, because of the (equalization) action of the dynamical friction force. 相似文献
13.
If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations
and
where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, , and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example,
where is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of
: the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral
indicative of a transport of angular momentum towards the equator.With the help of a formalism developed earlier we evaluate, for solid body rotation, the velocity correlations
and
for several values of an arbitrary parameter, D, left unspecified by the theory. The most striking result of these calculations is the increase of
with D. Next we calculate the turbulent viscosity coefficients defined by
whereC
ro
0
and C
o
0
are the velocity correlations for solid body rotation. In these calculations it was assumed that 2 was a linear function of r. The arbitrary parameter D was chosen so that the meridional flow vanishes at the surface for the rotation laws specified below. The coefficients v
ro
i
and v
0o
i
that allow for the calculation of C
ro
and C
0o
for any specified rotation law (with the proviso that 2 be linear) are the turbulent viscosity coefficients. These coefficients comply well with intuitive expectations: v
ro
1
and –v
0o
3
are the largest in each group, and v
0o
3
is negative.The equations for the meridional flow were first solved with
0 and
2 two linear functions of r (
0
1
= – 2 × 10 –12 cm –1) and (
2
1
= – 6 × 10 12 cm –1). The corresponding angular velocity increases slightly inwards at the poles and decreases at the equator in broad agreement with heliosismic observations. The computed meridional motions are far too large ( 150m s–1). Reasonable values for the meridional motions can only be obtained if
o (and in consequence ), increase sharply with depth below the surface. The calculated meridional motion at the surface consists of a weak equatorward flow for gq < 29° and of a stronger poleward flow for > 29°.In the Sun, the Taylor-Proudman balance (the Coriolis force is balanced by the pressure gradient), must be altered to include the buoyancy force. The consequences of this modification are far reaching: is not required, now, to be constant along cylinders. Instead, the latitudinal dependence of the superadiabatic gradient is determined by the rotation law. For the above rotation laws, the corresponding latitudinal variations of the convective flux are of the order of 7% in the lower SCZ. 相似文献
14.
Satoshi Hinata 《Solar physics》1982,80(1):173-183
Current dissipation models of coronal loop heating are studied. Turbulent current dissipation is shown to lead to a time dependent process because of an enormous mass motion induced in the current layer. A stationary heating process involves only ohmic heating, which requires a large current layer. To insure MHD stability, the loop must be composed of many elements with the oppositely directed currents. A stationary current dissipation process induces the plasma motion across the magnetic field into the loop and down the loop with the speeds v
104 cm s–1 and v
104 cm s–1, respectively. The pressure of the loop is also estimated to be proportional to the current density: p/J=6.3 × 10-8dyn/statamp. 相似文献
15.
Mohamed Adel Sharaf 《Astrophysics and Space Science》1987,129(1):19-38
In this paper of the series, the third step of the author's regularization approach will be started by establishing the expansions of the functionX
n
(r)
(, ,u) in terms of the sectorial variables
j
(i)
introduced in Paper IV (Sharaf, 1982) to regularize the highly-oscillating perturbation force of some orbital systems. The literal analytical expressions for the Fourier expansion of the function will be explored in terms of
j
(i)
for anyn positive integer,r any real number whatever the types and the number of sectors forming the divisions situation of the elliptic orbits may be. The basic computational materials of the theory will also be given and for which the method of solution, the recurrence formulae, and the general computational sequence for the coefficients are considered. 相似文献
16.
By use of the dispersion equation given by Song, Wu, and Dryer (1987) for a cylinder plasma with mass motion and gravity included, we investigate the linear current instabilities developed in loop prominences. The results indicate that the mode of linear instability depends mainly on whetherv
s
2
> or not, wherev
s
is the sonic velocity at heightz, =GM/(R +z) is the gravity potential,G the gravitational constant,M andR the mass and the radius of the Sun respectively. Ifv
s
2
> , then the sausage instability will be dominant. Otherwise, the kink instability will be more important. A possible explanation of knot structure, which appears sometimes in solar loop prominences has been given. 相似文献
17.
Eugene Rabe 《Celestial Mechanics and Dynamical Astronomy》1973,8(2):177-179
As previously shown (Rabe, 1970), two classes of small periodic librations exist, in the plane, elliptic restricted problem, for an infinite sequence of easily specified oscillation frequenciesZ
j
. The present paper considers the dependence ofZ on the eccentricitye of the primary motion, in addition to its dependence on the mass parameter , and determines the resulting relations between ande, for any given periodic frequencyZ
j
. These relationships are obtained from the conditionD (Z
j
,, e)=0, where the basic determinantD has been expanded up to terms of orderZ
20, 5, ande
4.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972. 相似文献
18.
The motion of spin particles in the Schwarzschild field is examined in this paper. It is shown that Mathisson-Papapetrou equations under additional conditions
, where
is the Lie derivative of the Killing vector
j
, permit only radial motion, motion in the equatorial plane and in the equilibrium points. All these types of motion are considered more in detail. 相似文献
19.
The recent discovery of localised intense magnetic fields in the solar photosphere is one of the major surprises of the past few years. Here we consider the theoretical nature of small amplitude motions in such an intense magnetic flux tube, within which the field strength may reach 2 kG. We give a systematic derivation of the governing expansion equations for a vertical, slender tube, taking into account the dependence upon height of the buoyancy, compressibility and magnetic forces. Several special cases (e.g., the isothermal atmosphere) are considered as well as a more realistic, non-isothermal, solar atmosphere. The expansion procedure is shown to give good results in the special case of a uniform basic-state (in which gravity is negligible) and for which a more exact treatment is possible.The form of both pressure and velocity perturbations within the tube is discussed. The nature of pressure perturbations depends upon a critical transition frequency,
p
, which in turn is dependent upon depth, field strength, pressure and density in the basic (unperturbed) state of the tube. At a given depth in the tube pressure oscillations are possible only for frequencies greater than
p
for frequencies below
p
exponentially decaying (evanescent) pressure modes occur. In a similar fashion the nature of motions within the flux tube depends upon a transition frequency,
v
. At a given depth within the tube vertically propagating waves are possible only for frequencies greater than
v
; for frequencies below
v
exponentially decaying (evanscent) motions occur.The dependence of both
v
and
p
on depth is determined for each of the special cases, and for a realistic solar atmosphere. It is found that the use of an isothermal atmosphere, instead of a more realistic temperature profile, may well give misleading results.For the solar atmosphere it is found that
v
is zero at about 12 km above optical depth 5000= 1, thereafter rising to a maximum of 0.04 s–1 at some 600 km above 5000 = 1. Below 5000 = 1, in the convection zone,
v
has a maximum of 0.013 s–1. The transition frequency,
p
, for the pressure perturbations, is peaked at 0.1 s–1 just below 5000 = 1, falling to a minimum of 0.02 s–1 at about one scale-height deeper in the tube 相似文献
20.
We have investigated how the latitude dependence of the solar wind velocity (SWV) influenced the cosmic-ray (CR) modulation and distribution in the heliosphere. The dependence proposed by Fry and Akasofu (1987) is used:v
SW=v
O+v
1(1-cos
n
m
, where the SWV,v
SW is a function of the heliomagnetic latitude
m
andv
0 andv
1 are constants. An estimation of the diffusion and drift terms in the transport equation is made, which shows that towards the poles the effects of the drift transfer decrease, while the diffusion terms in the equation increase due to the change of the interplanetary magnetic field (IMF) geometry. The numerical solutions of the two-dimensional (2-D) transport equation show that when the SWV changes with latitude: (1) The CR intensities away from the neutral sheet are larger for both IMF polarity periods in comparison with the case when the SWV does not change with the latitude. (2) The latitude gradients are negative during negative magnetic polarity periods. (3) The Voyager 1 and Voyager 2 long-time observations showing greater galactic CR intensities nearer the Sun at greater distances, could be explained by the proposed model. 相似文献