Perturbations of self-gravitating, ellipsoidal superfluid-normal fluid mixtures

a r

X

i

v

:a

s t

r o

-

p

h

/

4

3

3

1v

2

2

4

N

o

v

2

Perturbations of self-gravitating,ellipsoidal super?uid-normal ?uid mixtures A.Sedrakian Kernfysisch Versneller Instituut,NL-9747AA Groningen,The Netherlands,?and Institute for Nuclear Theory,University of Washington,Seattle WA 98195-1550I.Wasserman Center for Radiophysics and Space Research,Cornell University,Ithaca,NY 14853We study the perturbation modes of rotating super?uid ellipsoidal ?gures of equilibrium in the framework of the two-?uid super?uid hydrodynamics and Newtonian gravity.Our calculations focus on linear perturbations of background equilibria in which the two ?uids move together,the total density is uniform,and the densities of the two components are proportional to one another,with ratios that are independent of position.The motions of the two ?uids are coupled by mutual friction,as formulated by Khalatnikov.We show that there are two general classes of modes for small perturbations:one class in which the two ?uids move together and the other in which there is relative motion between them.The former are identical to the modes found for a single ?uid,except that the rate of viscous dissipation,when computed in the secular (or “low Reynolds number”)approximation under the assumption of a constant kinematic viscosity,is diminished by a factor f N ,the fraction of the total mass in the normal ?uid.The relative modes are completely new,and are studied in detail for a range of values for the phenomenological mutual friction coe?cients,relative densities of the super?uid and normal components,and,for Roche ellipsoids,binary mass ratios.We ?nd that there are no new secular instabilities connected with the relative motions of the two ?uid components.Moreover,although the new modes are subject to viscous dissipation (a consequence of viscosity of the normal matter),they do not emit gravitational radiation at all.I.INTRODUCTION The problem of the equilibrium and stability of rotating neutron stars is encountered in various astrophysical contexts,ranging from the limiting frequencies of rapidly rotating isolated millisecond pulsars,emission of gravitational waves in neutron star-neutron star or neutron star-black hole binaries,to the generation of γ-rays in the neutron star mergers and X -rays in accreting systems [1].Considerable current interest is attached to the problem of neutron star binary inspiral,which would be the primary source of gravitational wave radiation for detection by future laser interferometers.Such a detection,apart from testing the general theory of relativity,potentially could provide useful information on the equation of state of superdense matter.Also the stability criteria for rapidly rotating neutron stars are essential for placing ?rm upper limits on the frequencies to which millisecond pulsars can be spun up thereby constraining the range of admissible equations of states.High precision,fully relativistic treatments of rapidly rotating isolated neutron stars and binaries comprising two neutron stars have become available in recent years [1].Nevertheless,the development of simpler models that provide a fast and transparent insight into the underlying physics is needed when the basic set of equations is modi?ed to include new e?ects.

A systematic framework for the treatment of the equilibrium and stability of rotating liquid masses bound by self-gravitation in the Newtonian theory is contained in Chandrasekhar’s Ellipsoidal Figures of Equilibrium [2](hereafter EFE).The tensor virial method,developed most extensively by Chandrasekhar and co-workers,transforms the local hydrodynamical equations into global virial equations that contain the full information on the structure and stability of the Newtonian self-gravitating system as a whole.The method describes,in a coherent manner,the properties of solitary ellipsoids with and without intrinsic spin,and ellipsoids in binaries subject to Newtonian tidal ?elds.It is especially useful for studying divergence-free displacements of uniform ellipsoids from equilibrium,in which case the each perturbed virial equation yields (in the absence of viscous dissipation)a di?erent set of normal modes.Recent alternative formulations of the theory of ellipsoids are based mainly on either the energy variation method

[3,4]or the a?ne star model [5]or the (Eulerian)two potential formalism [6].A large class of incompressible and compressible ellipsoidal models has been studied using these methods [5–7].The energy variation method employs the observation that an equilibrium con?guration is possible if the energy of the ellipsoid is an extremum for variations

of the ellipsoidal semiaxis at a constant volume;the ellipsoidal?gure is stable only if the energy is a true local minimum.In the a?ne star method the?gures are described by a time-dependent Lagrangian as a function of a deformation matrix and its derivative.The structure of the star at any particular time is related conformally to the initial unperturbed sphere via a quadratic form constructed from the deformation matrix.

The purpose of this paper is to extend previous studies to a treatment of the oscillation modes of ellipsoidal?gures of equilibrium that contain a mixture of normal?uid and super?uid.Many-body studies of the pair correlations in neutron star matter show that the baryon?uids in their ground state form super?uid condensates in the bulk of the star.The super?uid phases,in the hydrodynamic limit,can be treated as a mixture of super?uid condensate and normal matter.The super?uid rotation is supported by the Feynman-Onsager vortex lattice state,and on the average leads to quasi-rigid body rotation of the super?uid component.The corresponding time-dependent two-?uid hydrodynamic equations are completely speci?ed by two independent velocities for the super?uid and the normal component and corresponding densities of the constituents.The two?uids are coupled to one another by mutual friction forces,which we model phenomenologically according the prescription given by Khalatnikov[17].Because of the additional degrees of freedom in this system,there are twice as many modes as for a single?uid.A natural question is whether the new modes a?ect the stability criteria previously deduced from studying perturbations of a single self-gravitating?uid.

In our treatment of perturbations of a mixture of normal?uid and super?uid,we shall follow most closely Chan-drasekhar’s formulation.However,since the basic equations of motion for the two?uids will include mutual friction between them,the system we study is inherently dissipative.Nevertheless,since the frictional forces only depend lin-early on the relative velocity between the two?uids and vanish in the background,where the two?uids move together, one can still derive relations that resemble Chandrasekhar’s tensor virial equations.Because these equations include dissipation,we shall prefer to regard them as moments of the?uid equations,rather than tensor virial equations.In fact,we shall relegate the derivation of perturbation equations from these moment equations to the Appendix of the paper,and instead derive the necessary equations for the?uid displacements directly by taking moments of linearized equations of motion for the two?uid components.In this paper,we concentrate on two-?uid variants of the classical Maclaurin,Jacobi and Roche ellipsoids.

Although our main aim is to access the oscillation modes and instabilities of neutron stars within the ellipsoidal approximation,the results obtained here may be of signi?cance in other contexts(Ref.[2],the epilogue).One example is the understanding of rapidly rotating nuclei in the spirit of the Bohr-Wheeler model of a charged incompressible liquid droplet[8].In this case,the stability is determined by the competition between the attractive surface tension, the repulsive Coulomb potential,and the centrifugal stretching due to the rotation[9,10].Another example is the stability of rotating super?uid liquid drops of Bose condensed atomic gases,where the stability is determined through an interplay among the pressure of the condensate,the con?ning potential of the magnetic trap and the centrifugal potential[11].

Previous work on the oscillations of super?uid neutron stars concentrated mainly on perturbations of non-rotating or slowly rotating isolated neutron stars[12–15],and used methods entirely di?erent from the one adopted here.The propagation of acoustic waves in neutron star interiors,including those related to the relative motion of neutron-proton super?uids,was studied by Epstein[12],who found the compressional and shear modes related to short-wavelength oscillations of neutron star matter.The small-amplitude pulsation modes of super?uid neutron stars were derived by Lindblom and Mendell[13],who found that the lowest frequency modes were almost indistinguishable from the normal modes of a single?uid star.Their analytical solutions also reveal the existence of a spectrum of modes which are absent in a single?uid star.Subsequent work concentrated on numerical solutions for the radial and non-radial pulsations of the two-?uid stars and identi?ed distinct super?uid modes in the absence of rotation[14].The r modes of slowly rotating two-?uid neutron stars have been derived by Lindblom and Mendell[13],who?nd that they are identical to their ordinary-?uid counterparts to the lowest order in their small-angular-velocity expansion.The linear oscillations of general relativistic stars composed of two non-interacting?uids in a non-rotating static background have been studied by Comer et al.[15].

Our calculations allow arbitrary fast rotation,in the context of(incompressible)Newtonian?uid models.One may anticipate that the e?ects of super?uidity on oscillation modes,if any,should be a?ected by the underlying vortex structure of the rotating super?uid.In our treatment,dissipation arises because of the drag forces experienced by the vortex lines as they move through the normal?uid(and there is no dissipation if the drag force is zero).We ignore the motions related to the isospin degrees of freedom in the core of a neutron star and,hence,the mutual entrainment of the neutron and proton condensates,as well as forces arising from deviations fromβequilibrium.The two-?uid equations used in the remainder of this work can adequately describe the mutual friction of a two-condensate ?uid in the core of a neutron star,since the entrainment e?ect renormalizes the e?ective super?uid densities and the frictional coe?cients,i.e.the phenomenological input in the two-?uid equations.While the ellipsoidal approximation to a super?uid neutron star is restrictive,it allows us to study the e?ects of vorticity on the oscillation modes of a self-gravitating star in a transparent manner,avoiding complications due to the star’s inhomogeneity(multi-layer

composition)[16].

II.PERTURBATION EQUATIONS

The equations of motion for a mixture of two?uids may be summarized simply as

ραDαuα,i=??pα

?x i

+

1

?x i

+2ρα?ilm uα,l?m+Fαβ,i,(1)

where the subscriptα∈{S,N}identi?es the?uid component,and Latin subscripts denote coordinate directions;ρα, pα,and uαare the density,pressure,and velocity of?uidα,φis the gravitational potential,and Fαβis the mutual friction force on?uidαdue to?uidβ.These equations have been written in a frame rotating with angular velocity ?relative to some inertial coordinate reference system.The total time derivative operator

Dα≡??x

j

.(2) The gravitational potentialφis derived from

?2φ=?2(φS+φN)=4πG[ρS(x)+ρN(x)];(3) the individual?uid potentialsφαobey?2φα=4πGρα.The two?uids are coupled to one another via the frictional force Fαβwhich is antisymmetric on interchange ofαandβ.For a normal-super?uid mixture

F SN=?F NS≡ρSωS{β′ν×(u S?u N)+βν×[ν×(u S?u N)]?β′′ν·(u S?u N)},(4) whereβ,β′andβ′′are coupling coe?cients,andωS=νωS≡?×u S;in components we have

F SN,i=?ρSωSβij(u S,j?u N,j),(5) where,from Eq.(4),

βij=βδij+β′?ijmνm+(β′′?β)νiνj.(6) The net rate at which this force does work is

u S·F SN+u N·F NS=?ρSωS{β|ν×(u S?u N)|2?β′′[ν·(u S?u N)]2};(7) there is no dissipation associated with the term proportional toβ′in Fαβ.Throughout this paper,we assume that β,β′andβ′′are independent of position in the?uid mixture.In Eq.(4),we have neglected the e?ects of the vortex tension,and expressed the mutual friction force in terms of the phenomenological coe?cientsβ,β′andβ′′. While these parameters determine the macroscopic behavior of the?uid system,they are not the optimal ones for connecting microscopic parameters of the mixture to its macroscopic motion.Instead,the macroscopic results can be parametrized in terms of frictional coe?cientsηandη′,which connectβandβ′to the drag on individual super?uid vortices via the relations

β=ηρSωS

η2+(ρSωS?η′)2.(8)

The physical meaning ofη’s is apparent from the equation of motion of a single vortex line

ρSωS[(u S?u L)×ν]?η(u L?u N)+η′[(u L?u N)×ν]=0,(9) where u L is the vortex velocity.Equation(9)states that the Magnus force,which represents a lifting force due to the super?ow imposed on the vortex circulation,is balanced by the viscous friction forces along the vortex motion(the term∝η)and perpendicular to the vortex motion(the term∝η′);these latter forces arise from the scattering of the normal quasiparticles o?the vortex line1.The characteristic dynamical relaxation time scale related to the vortex motion can be de?ned as

τD=1

ρSωS

+

ρSωS

dt2=?

?ξα,l

?x i?

??αpα

?x i

+ρα

?

2 ?|?×x|2dt +

?ξα,l

?x l

+ρα

?φ

2

?|?×x|2

?x′

l

1

ραd2ξα,i

?x l

?pα

?x i?x l?

?δpα

?x i

?ραξα,l?2φdt+Fαβ,i.(13)

Then if we de?ne

ξ+=f SξS+f NξNξ?=ξS?ξN,(14) we?nd

ρd2ξ+,i

?x l

?p

?x i?x l?

?δp

?x i

?ρξ+,l?2φdt(15)

whereδp=δp S+δp N,and

ρd2ξ?,i

?x l

?p

?x i?x l?

1

?x i

+

1

?x i

+ρξ?,l(?2δil??i?l)?ρξ?,l?2φdt

?ρωS 1+f S dt.(16)

Equation(15)is identical to what is found for a single?uid,and therefore contains the well-known modes documented by Chandrasekhar:if we de?ne

V i;j= V d3xρξ+,i x j,(17) then we?nd

d2V i;j

dt

+?2V ij??i?k V kj+δijδΠ

?πGρ 2B ij V ij?a2iδij3 l=1A il V ll ,(18)

whereδΠ≡δΠS+δΠN and all other quantities are de?ned exactly as in EFE.All of the new modes of a mixture of normal?uid and super?uid are contained in Eq.(16)for their relative displacements.One noteworthy feature of Eq.(16)is that the Eulerian gravitational potential does not appear.Consequently,the new normal modes of the system only depend on the unperturbed gravitational potential;for perturbations of homogeneous ellipsoids,only the coe?cients A i de?ned by EFE,Chap.3,Eqs.(18)and(40),will appear.

For the most part,we shall be interested in displacements that are linear functions of x i in this paper.For the homogeneous ellipsoids,we can?nd the new modes that result from the di?erential displacements of normal?uid and super?uid by taking the?rst moment of Eq.(16),e.g.by multiplying by x j and integrating over the unperturbed volume.2If we de?ne

U i;j= V d3xρξ?,i x j,(19) then we?nd

d2U i;j

dt

+?2U ij??i?k U kj+δij δΠS f N

?2πGρA i U ij?ωS 1+f S dt,(20) where

δΠα≡ V d3xδpα.(21)

To obtain Eq.(20),various surface terms can be eliminated using the conditions that pαand?αpα=δpα+ξα,l?pα/?x l vanish on the boundary;also,the equation of hydrostatic equilibrium for the unperturbed con?guration,

0=?p

?x i?ρ

?

2 ,(22)

must be used.It is straightforward to compute higher moments of Eq.(16).For example,taking its second moment3 by multiplying by x j x k and integrating over the unperturbed volume gives

d2U i;jk

f S?δΠN,k

f S?

δΠN,j

f N βil

dU l;jk

f S?δΠN

f N βik,(25)

we rewrite the Eq.(26)as

d2U i;j

+?2U ij??i?k U kj+δijδ?Π?2πGρA i U ij?2??βik dU k;j

dt

σ σ2?2(A1+A3)+?2 ±2?(σ2?2A3)=0.(36) The purely rotational modeσ=?decouples only in the spherical symmetric limit where A1=A3.If only axial symmetry is imposed then the characteristic equation is third order:

σ3±2?σ2+ ?2(A1+A3)+?2 σ?4A3?=0.(37) Along the entire sequence parametrized in terms of the eccentricity the three modes are real5and are given by

σ1=2?

3?

13

27+

2?(A1?2A3)??3

9?

2(A1+A3)??2

27

+

2?(A1?2A3)??3

5This is easy to prove directly from Eq.(35).Write the dispersion relation as f(λ2)=0.Then show that(i)f(λ2)→±∞asλ2→±∞,(ii)f(0)>0,and(iii)the two extrema of f(λ2)are both atλ2<0.Thus,the zeros of f(λ2)are all atλ2<0, soσ=iλmust be real.

λ2U3;3=(πρG)?1δ?Π?2A3U33?2??β′′λU3;3,(41)λ2U1;1?2?λU2;1=(πGρ)?1δ?Π+(?2?2A1)U11?2??βλU1;1?2??β′λU2;1,(42)

λ2U2;2+2?λU1;2=(πGρ)?1δ?Π+(?2?2A1)U22?2??βλU2;2+2??β′λU1;2,(43)

λ2U1;2?2?λU2;2=(?2A1+?2)U12?2??βλU1;2?2??β′λU2;2,(44)

λ2U2;1+2?λU1;1=(?2A1+?2)U21?2??βλU2;1+2??β′λU1;1.(45) We add Eqs.(44)and(45)and subtract Eqs.(42)and(43)to?nd the following coupled equations for the toroidal modes(note that A1=A2for Maclaurin spheroids)

λ2+2??βλ+4A1?2?2 (U11?U22)?4?λ(1??β′)U12=0,(46)

λ2+2??βλ+4A1?2?2 U12+?λ(1??β′)(U11?U22)=0.(47) The characteristic equation for the toroidal modes is

λ4+4?βλ3?+λ2(8A1+4?β2?2?8?β′?2+4?β′2?2)

+λ(16A1?β??8?β?3)+16A21?16A1?2+4?4=0.(48) In the frictionless limit this can be written as

(λ2+4A1?2?2)2+4?2λ2=0,(49) which is factorized by writingλ=iσ.The two solutions are then

σ1,2=?±

Equations(51)and(52)can be further combined to a single equation:

λ2+2??βλ?2?2+4A1 (λ2+2??βλ)+4?2λ2(1??β′)2 (U11+U22)

?2 λ2+2??βλ λ2+2?λ?β′′+4A3 U33=0.(53) The solution is found by supplementing these equations by the divergence free condition

U11

a22+

U33

(3?2?2) 1/2.(57) The pulsation modes for a sphere follow in the limit?,?→0:for a sphere A i/(πρG)=2/3,and Eq.(57)reduces to σ2=8/3[σis given in units of(πρG)1/2].This result could be compared with the pulsation modes of an ordinary sphere:σ2=16/15.Thus a super?uid sphere,apart form the ordinary pulsations,shows pulsations at frequencies roughly twice as large as the ordinary ones.

The real and imaginary parts of the dissipative pulsation modes of a super?uid Maclaurin spheroid are shown in the Fig.3.The real parts of the modes are weakly a?ected by mutual friction and closely resemble those of an ordinary Maclaurin spheroid in the frictionless limit.These are located,however,at higher frequencies.The symmetry of the damping rate as a function ofηobserved for the transverse shear and toroidal modes is again observed in Fig. 3. Note that the results in Fig.6were obtained in the caseβ′′=0.The pulsation modes of the super?uid Maclaurin spheroid are stable,as is the case for the ordinary Maclaurin spheroids.

B.Modes of super?uid Jacobi ellipsoid

The sequence of the Jacobi ellipsoids emerges from the Maclaurin sequence at the bifurcation point?=0.813 via a spontaneous breaking of symmetry in the plane perpendicular to the rotation(a1=a2).The super?uid equilibrium?gures are again identical to their ordinary?uid counterparts and the de?ning relations a21a22A12=a23A3 and?2=2B12are unchanged.Ordinary Jacobi ellipsoids are known to be stable against second order harmonic perturbations while they become dynamically unstable against transformation into Poincar`e’s pear shaped?gures through a mode belonging to third order harmonic perturbations.If the sequence of Jacobi ellipsoids is parametrized in terms of the variable cos?1(a3/a1),it is stable between the point of bifurcation from the Maclaurin sequence, cos?1(a3/a1)=54.36,and the point where Poincar`e’s?gures bifurcate,cos?1(a3/a1)=69.82.Here,by an explicit calculation,we verify that super?uid Jacobi ellipsoids do not develop new instabilities via second order harmonic modes of the relative displacements.

1.Relative odd modes

The treatment of the oscillations of the Maclaurin spheroid of the previous sections can be readily extended to the Jacobi ellipsoids by lifting the degeneracy in indexes1and2and imposing A1=A2.The equations odd in index3 are

λ2U3;1=?2A3U31?2??β′′λU3;1,(58)

λ2U3;2=?2A3U32?2??β′′λU3;2,(59)λ2U1;3?2?λU2;3= ?2A1+?2 U13?2??βλU1;3?2??β′λU2;3,(60)

λ2U2;3+2?λU1;3= ?2A2+?2 U23?2??βλU2;3+2??β′λU1;3.(61) Combining Eqs.(58)and(60)and,similarly,Eqs.(59)and(61),after some manipulation we?nd

λ2+2??β′′λ λ2+2??βλ +2 λ2+2??βλ A3+ λ2+2??β′′λ (2A1??2) U13

?2?λ(1??β′) λ2+2??β′′λ+2A3 U23=0,(62) λ2+2??β′′λ λ2+2??βλ +2 λ2+2??βλ A3+ λ2+2??β′′λ (2A2??2) U23

+2?λ(1??β′) λ2+2??β′′λ+2A3 U13=0.(63) Equations(62)and(63)are su?cient to determine the symmetric parts of the virials,and any two of Eqs.(58)-(61) can be used to?nd the antisymmetric parts.The sixth order characteristic equation is

λ6+4?(?β+?β′′)λ5+ 4A3+2?(1??β′)2+4?2(?β+?β′′)2?(2A1??2)?(2A2??2) λ4

+ 16A3?β?+8A3?β′′?+8?β′′?2(1??β′)2+16?β2?β′′?3+16?β?β′′2?3

?2?β?(2A1??2)?2?β?(2A2??2)?4?β′′?(2A1??2)?4?β′′?(2A2??2) λ3 + 4A23+8A3?(1??β′)2+16A3?β2?2+32A3?β?β′′?2+8?β′′2?3(1??β′)2+16?β2?β′′2?4

?2A3(2A1??2)?8?β?β′′?2(2A1??2)?4?β′′2?2(2A1??2)?2A3(2A2??2)

?8?β?β′′?2(2A2??2)?4?β′′2?2(2A2??2)+(2A1??2)(2A2??2) λ2 + 16A23?β?+16A3?β′′?2(1??β′)2+32A3?β2?β′′?3?4A3?β?(2A1??2)?4A3?β?(2A2??2)

?4A3?β′′?(2A1??2)?8?β?β′′2?3(2A1??2)?4A3?β′′?(2A2??2)

?8?β?β′′2?3(2A2??2)+4?β′′?(2A1??2)(2A2??2) λ

+ 8A23?(1??β′)2+16A23?β2?2?8A3?β?β′′?2(2A1??2)

?8A3?β?β′′?2(2A2??2)+4?β′′2?2(2A1??2)(2A2??2) =0.(64) The real and imaginary parts of the dissipative odd parity modes are shown in the Fig. 4.The Jacobi sequence is parametrized in terms of cos?1(a3/a1)starting o?from the point of bifurcation of the Jacobi ellipsoid from the Maclaurin sequence.The low frequency mode resembles the rotational mode of the ellipsoid;its frequency decreases with increasing friction.One of the remaining two distinct high frequency modes is almost una?ected by the dissipa-tion,while the other is suppressed close to the bifurcation point in a monotonic manner.The damping rates of the odd modes are maximal atη=1and tend to zero for both large and small friction.The modes are damped along the entire sequence;hence,we conclude that super?uid Jacobi ellipsoids are stable against the odd second harmonic modes of oscillations.

2.Relative even modes

The explicit form of the even parity modes for the Jacobian sequence is

λ2U3;3=(πρG)?1δ?Π?2A3U33?2??β′′λU3;3,(65)λ2U1;1?2?λU2;1=(πGρ)?1δ?Π?2A1U11+?2U11?2??βλU1;1?2??β′λU2;1,(66)

λ2U2;2+2?λU1;2=(πGρ)?1δ?Π?2A2U22+?2U22?2??βλU2;2+2??β′λU1;2,(67)

λ2U1;2?2?λU2;2=(?2?2A1)U12?2??βλU1;2?2??β′λU2;2,(68)

λ2U2;1+2?λU1;1=(?2?2A2)U12?2??βλU2;1+2??β′λU1;1.(69)

These equations can be reduced to a simpler set of equations through manipulations which eliminate the variations of the pressure tensor.Explicitly,in the?rst step we subtract the Eqs.(66)and(67);in the second we sum Eqs.(66) and(67)and subtract twice Eq.(65).The result is

(λ2/2+??βλ??2+2A1)U11?(λ2/2+??βλ??2+2A2)U22

?2?λ(1??β′)U12=0,(70) (λ2/2+??βλ??2+2A1)U11+(λ2/2+??βλ??2+2A2)U22

?(λ2+2??β′′λ+4A3)U33+2?λ(1??β′)(U1;2?U2;1)=0.(71)

Further we add and subtract Eqs.(68)and(69)to?nd

λ2+??βλ?4B12+2(A1+A2) U12+?(1??β′)λ(U11?U22)=0,(72) (λ2+2??βλ)(U1;2?U2;1)??(1??β′)λ(U11+U22)+2(A1?A2)U12=0.(73)

Equations(70)-(73),supplemented by the divergence free condition,Eq.(54),are su?cient to determine the modes. The characteristic equation is of seventh order,excluding the trivial rootλ=0;in the frictionless limit the charac-teristic equation is of third order.The explicit form of these equations is cumbersome and will not be given here. The real and imaginary parts of the dissipative even parity modes are shown in Fig. 5.For each member of the sequence,the eigenvalues of the two high frequency modes are suppressed and that of the low-frequency mode is ampli?ed as the dissipation increases.As in the case of the odd modes the damping rates of the even parity modes are maximal atη=1and tend to zero for both large and small friction.The damping rates are again positive along the entire sequence and we conclude that super?uid Jacobi ellipsoids are stable against the even parity second harmonic modes of oscillations.

C.Modes of super?uid Roche ellipsoid

In this section we extend the previous discussion of isolated ellipsoids to binary star systems,and consider the simplest case–the Roche problem.The classical Roche binary consists of a?nite size ellipsoid(primary of mass M) and a point mass(secondary of mass M′)rotating about their common center of mass with an angular velocity?. The new ingredient in the problem of the equilibrium and stability of the primary is the tidal Newtonian gravitational ?eld of the secondary.Place the center of the coordinate system at the center of mass of the primary with the x1-axis pointing to the center of mass of the secondary and x3-axis along the vector?.The equation of motion for a?uid element of the primary in the frame rotating with angular velocity?is,then(EFE,Chap.8,Sec.55)

ραDαuα,i=??pα

?x i

+

1

?x i

+2ρα?ilm uα,l?m+Fαβ,i,(74)

where the tidal potential of the secondary,up to quadratic terms in x i/R,is

φ′=GM′

R

+

2x21?x22?x23

λ2U i;j?2?il3?3λU l;j=δij(πρG)?1δ?Π?A i U ij?2?λ?βik U k;j

+(?2?φ0)U ij??2δi3U3j+3φ0δi1U1j,(77) where all frequencies are measured in units of(πρG)1/2.Equation(77)is appropriate for?nding the second harmonic modes of oscillations of Roche ellipsoids.

1.Relative odd modes

The equations determining the modes even and odd in index3form separate sets.We start with the modes belonging to l=2and m=?1,1displacements,which are odd in index3;for these,

λ2U3;1=?(2A3+φ0)U31?2??β′′λU3;1,(78)

λ2U3;2=?2(A3+φ0)U32?2??β′′λU3;2,(79)λ2U1;3?2?λU2;3= ?2A1+?2+2φ0 U13?2??βλU1;3?2??β′λU2;3,(80)

λ2U2;3+2?λU1;3= ?2A2+?2?φ0 U23?2??βλU2;3+2??β′λU1;3.(81) On combining Eqs.(78)and(80)and,similarly,Eqs.(79)and(81)we obtain

(λ2+2??β′′λ+2A3+φ0)(λ2+2??βλ)+(λ2+2??β′′λ)(2A1??2?2φ0) U13

?2?(1??β′)λ λ2+2??β′′λ+2A3+φ0 U32=0,(82) (λ2+2??β′′λ+2A3+φ0)(λ2+2??βλ)+(λ2+2??β′′λ)(2A2??2+φ0) U23

+2?(1??β′)λ λ2+2??β′′λ+2A3+φ0 U13=0.(83) The U ij are symmetric under interchange of their indexes,and Eqs.(82)and(83)completely determine the modes [any two of Eqs.[78]-[81]may be used to?nd the antisymmetric parts of U i;j].The real and imaginary parts of the dissipative odd parity modes are shown in Fig.6,for the case of an equal mass binary(P=1).The relative modes of Roche ellipsoids for other values of the mass ratio display behavior similar to the P=1case.The Roche sequence is parametrized in terms of cos?1(a3/a1).The low frequency mode resembles the rotational mode of the ellipsoid;its frequency decreases with increasing friction.The high frequency modes tend towards each other and merge in the limit of slow rotation;in the opposite limit the modes remain una?ected by the dissipation.There are three distinct rates for the damping of oscillations.These are maximal atη=1and tend to zero in both limits of large and small friction,as was the case for the Maclaurin and Jacobi ellipsoids.The damping rates are positive along the entire sequence.We conclude that super?uid Roche ellipsoids do not develop instabilities via the second order harmonic odd modes of relative oscillation.

2.Relative Even Modes

As is well known,Roche ellipsoids develop a dynamical instability via the second order even parity modes beyond the Roche limit,which is the point of closest approach of the primary to the secondary.If viscous dissipation is allowed for,Roche ellipsoids become secularly unstable at the Roche limit via an even parity mode and before dynamical instability sets in.We have seen that Maclaurin spheroids do not develop any instabilities(i.e.neither dynamical nor secular)via the modes associated with the U ij in the presence of super?uid dissipation.The extension of the theory above to super?uid Roche ellipsoids,as we show now,does not reveal any new instabilities in the presence of super?uid dissipation,again in contrast to the analysis based on the ordinary viscous dissipation.

The explicit form of the even parity modes for the Roche sequence is

λ2U3;3=(πρG)?1δ?Π?(2A3+φ0)U33?2??β′′λU3;3,(84)λ2U1;1?2?λU2;1=(πGρ)?1δ?Π?(2A1??2?2φ0)U11?2??βλU1;1?2??β′λU2;1,(85)λ2U2;2+2?λU1;2=(πGρ)?1δ?Π?(2A2??2+φ0)U22+?2U22?2??βλU2;2+2??β′λU1;2,(86)λ2U1;2?2?λU2;2=(?2+2μ?2A1)U12?2??βλU1;2?2??β′λU2;2,(87)λ2U2;1+2?λU1;1=(?2?φ0?2A2)U12?2??βλU2;1+2??β′λU1;1.(88)

These equations can be reduced to a simpler set of equations through manipulations which eliminate variations of the pressure https://www.sodocs.net/doc/5211374369.html,ing the symmetry properties of the virials we?rst subtract Eqs.(85)and(86),then sum Eqs.

(85)and(86)and subtract twice Eq.(84)to obtain

(λ2/2+??βλ??2?2φ0+2A1)U11?(λ2/2+??βλ??2+φ0+2A2)U22

?2?λ(1??β′)U12=0,(89) (λ2/2+??βλ??2?2φ0+2A1)U11+(λ2/2+??βλ??2+φ0+2A2)U22

?(λ2+2??β′′λ+2φ0+4A3)U33+2?λ(1??β′)(U1;2?U2;1)=0.(90)

Further we add and subtract Eqs.(87)and(88)to?nd

λ2+2??βλ+2(A1+A2)?2?2?φ0 U12+?(1??β′)λ(U11?U22)=0,(91) (λ2+2??βλ)(U1;2?U2;1)??(1??β′)λ(U11+U22)?[3φ0?2(A1?A2)]U12=0.(92)

Equations(91)-(92),supplemented by the divergence free condition,Eq.(54),are su?cient to determine the unknown virials.

The real and imaginary parts of the dissipative even parity modes are shown in Fig.7.For each member of the sequence the eigenvalues of the two high frequency modes are suppressed and that of the low frequency one is ampli?ed with increasing dissipation.In e?ect these modes merge in the slow rotation limit.The modes do not become neutral at any point along the frictionless sequence and,hence,the necessary condition for the onset of dynamical instability is not achieved.Note that our model for the Roche ellipsoids is dynamically unstable as is its classical counterpart, but via the modes governed by the Eq.(18)modi?ed appropriately to include the external tidal potential.We do not repeat the mode analysis for the virials V ij as it is a complete analogue of the analysis in EFE.As in the case of odd modes the damping rates of even parity modes are maximal atη=1and tend to zero in both limits of large and small friction.The damping rates are positive along the entire sequence and we conclude that super?uid Roche ellipsoids are secularly stable against the even parity second harmonic modes of oscillations associated with the relative motions between the super?uid and normal components.

IV.VISCOSITY AND GRA VITATIONAL RADIATION

Above,we neglected viscous dissipation in computing normal modes of a normal?uid-super?uid mixture.However, for a single?uid,viscous dissipation is important for understanding stability,for it is responsible for secular instability. Although viscous terms spoil the calculation of the modes of uniform ellipsoids from moment equations formally,when the dissipative time-scale is long,one can include them perturbatively(e.g.EFE,Chap.5,§37b).

The inclusion of viscosity is slightly more complicated for a mixture of super?uid and normal?uid because viscous dissipation only operates on the normal?uid.The separation of Eq.(11)for the?uid displacements into Eqs.(15) and(16)for the common and di?erential?uid displacements,ξ±,is possible because the only form of dissipation,the mutual friction force,included in Eq.(11)only depends on dξ?/dt.Viscous dissipation depends onξN only,and,in a formal sense,the dynamics no longer separate into the independent dynamics ofξ±.

If we assume that the time-scale associated with viscous dissipation is relatively long,then we can include it perturbatively.The calculation is a bit more subtle than for a single?uid,because we have to deduceξN for the modes.This brings up an issue that we glossed over earlier,in setting up the calculation of modes in§II:even when computing the modes that arise from the equation forξ?alone,the equation forξ+must be satis?ed,and viceversa. The simplest way for this to work is forξ+to vanish whenξ?is nonzero and vice versa.In fact it is easy to show that this is a reasonable solution provided that the Eulerian pressure perturbations areδpα=?ξα,l?pα/?x l,a situation that arises naturally for adiabatic perturbations,where the Lagrangian pressure perturbations are?αpα=?Γα?ξα,l/?x l, and the perturbations are solenoidal,so that?ξα,l/?x l=0,as is true for all modes considered in this paper.6To see how this works,consider a mode of Eq.(16),and examine under what conditions Eq.(15)will be satis?ed.Then, using

δpα=?ξα,l ?pα

?x l

(93)

and[substituting the de?nition ofξ+,andρα=fαρinto Eq.(12)]

δφ≡?G V d3x′ρ(x′)ξ+,l(x′)?|x?x′| (94)

it is easy to see that Eq.(15)only depends onξ+.But since the normal modes of Eq.(16)have di?erent frequencies than normal modes of Eq.(15),we must haveξ+=0whenξ?=0.Since,in general,

ξS=ξ++f Nξ?ξN=ξ+?f Sξ?,(95) we conclude that,for modes of Eq.(16),ξS=f Nξ?andξN=?f Sξ?,when Eulerian pressure perturbations are given by Eq.(93).Similarly,since Eq.(16)only depends onξ?,for modes withξ+=0,we must haveξ?=0 and,therefore,ξS=ξN=ξ+,assuming Eq.(93).In particular,if the kinematic viscosityνis held constant[see EFE,Chap.5,Sec.36,Eq.(111)],for modes withξ+=0,the viscous dissipation rate is smaller by a factor of f N than it is for a single?uid with the same background and displacementξ+,as might have been expected qualitatively (i.e.for small normal?uid density,the viscous dissipation must be diminished).For perturbations withξ?=0and displacements that are linear functions of the coordinates,we must add

?5f Sν 1dt+1dt (96)

to the right hand side of Eq.(20)to include the e?ects of viscous dissipation.

Perturbations withξ?=0emit no gravitational radiation becauseξ+=0for them,and therefore there are no perturbations of the quadrupole moment or any other net mass currents associated with them.Gravitational radiation is emitted by the modes in which the two?uids move together at the same rate as for a single?uid(e.g.ref.[18]). Thus,none of the new modes of a super?uid-normal?uid mixture found here is a?ected by gravitational radiation at all.

V.CONCLUSIONS

Despite a number of simplifying assumptions,the study of the oscillation modes of uniform ellipsoids is useful for understanding the equilibrium and stability of real neutron stars,at least qualitatively.Moreover,the theory is simple enough that it can be extended readily to include modi?cations to the underlying physics;here,we have considered new features that arise because a neutron star contains a mixture of normal?uid and super?uid coupled by mutual friction.The theory of uniform ellipsoids is interesting from the viewpoint of mathematical physics because it is solvable exactly,and it may also have applications to other physical systems,such as the physics of trapped,rotating Bose-Einstein condensates.

In this paper we have extended previous treatments of the oscillation modes of ellipsoidal?gures of equilibrium to the case of a mixture of normal?uid and super?uid.The basic equations of motion for the two?uid hydrodynamics include mutual friction exactly,since the frictional forces depend linearly on the relative velocity between the two ?uids,and vanish in the background where the two?uids move together.In addition the?uids are coupled via mutual gravitational attraction,which is also treated without further approximations.As a result our relations closely resemble Chandrasekhar’s tensor virial equations,even though they are intrinsically dissipative due to the mutual friction.While we have developed these moment equations for general underlying equilibria,they are most useful for perturbations around uniform backgrounds,in which case the moment equations of various orders decouple and yield exact solutions for the normal modes.

Quite generally,there are two classes of modes for small perturbations,one class in which the two?uids move together and the other in which there is relative motion between them.The former are identical to the modes found for a single?uid.As a result our models of super?uid Maclaurin and Roche ellipsoids undergo dynamical instabilities with respect to these modes which are indistinguishable from what is found for their classical counterparts.When ordinary viscous dissipation is included they are also subject to secular instabilities related to the modes in which two ?uids move together.If the kinematic viscosity is held constant,then the rate of viscous dissipation,when computed in the“low Reynolds number”approximation,is diminished by a factor f N,the fraction of the total mass in the normal?uid(see however below).

The modes involving the relative motion between the?uids are completely new and are shown to be stable along

the entire sequences of the incompressible Maclaurin,Jacobi and Roche ellipsoids independent of the magnitude of the phenomenological mutual friction.These modes also do not become neutral at selected points along any sequence

and the necessary condition for the point of bifurcation is not achieved.Our main conclusion is that mutual friction does not drive secular instabilities in incompressible and irrotational ellipsoids.In addition we?nd that even though

the new modes are subject to viscous dissipation(a consequence of viscosity of the normal matter),they do not emit gravitational radiation,and are therefore immune to any instabilities associated with gravitational radiation,

irrespective of their modal frequencies.

The results summarized above hold within a combined framework of two-?uid super?uid hydrodynamics,Newtonian

gravity,and the ellipsoidal approximation,as formulated in EFE.Each of these elements of our approach contains a number of simplifying approximations which need to be relaxed in realistic applications to neutron stars.For example,

to treat the mixture of neutron and proton super?uids in the neutron star cores,the one-constituent two-?uid super?uid hydrodynamics must be replaced by the hydrodynamics of the multi-constituent super?uid mixtures,in which case

the mutual entrainment of the super?uids and deviations fromβ-equilibrium must be accounted for(see Ref.[13]for a discussion of these e?ects and their impact on neutron star oscillations within the real energy functional method).

We anticipate that these e?ects can be incorporated in the tensor virial approach in a perturbative manner and the results of previous sections will hold in leading order of the perturbation expansion.On the other hand,relaxing the

incompressible approximation and,hence,including the partial pressures of the super?uid and the normal?uid,will lead to non-perturbative e?ects as the pressure terms signi?cantly alter the balance between gravitational attraction

and centrifugal stretching.As is well known,the points of the onset of the dynamical and secular instabilities of

compressible ellipsoids depend on the adiabatic index of the underlying polytropic equation of state.As noted in Sec.3,the conclusions reached with respect to the stability of the incompressible ellipsoids should be veri?ed for

the compressible models anew.The di?erences in the pressures(or equations of states)of the normal and super?uid phases,however,are typically small in neutron stars,since the condensation energy is negligible compared to the

degeneracy pressures of interacting Fermi-liquids.The coupling between the partial pressures of the noraml?uid and super?uid,therefore,can be treated perturbatively.

Another e?ect that needs to be included is the density dependence of the mutual friction coe?cients and the kine-matic viscosity.For example,in neutron stars the kinematic viscosity is density dependent in general,explicitly as a

result of to the density dependence of the phase space of normal quasiparticles undergoing collisions,implicitly because of the density dependence of the in-medium scattering amplitudes(see for further details Ref.[19]).The rami?cation

for comparison of the secular instabilities of the normal?uid and super?uid ellipsoids is that the modi?cations of the rate of the viscous dissipation will depend in a non-trivial manner on the fraction of the normal?uid in the system.

The entrainment,β-nonequilibrium,compressibility,e.t.c.will couple the relative and center-of-mass modes in general.Such a“mixing”,as discussed in Sec.5,impliesξ+=0for the relative modes and,similarly,ξ?=0for the center-of-mass modes.Therefore,the mutual friction might tend to drive the center-of-mass modes secularly unstable; if they emit gravitational radiation,the mutual friction will suppress the gravitational radiation induced instabilities.

One of the important issues to be addressed by the future work is the magnitude of the“mixing”of the modes for general equilibria and the corresponding times scales.

ACKNOWLEDGMENTS

A.S.acknowledges the Nederlandse Organisatie voor Wetenschappelijk Onderzoek for its support at KVI Groningen via the Stichting voor Fundamenteel Onderzoek der Materie,the Institute for Nuclear Theory at the University of Washington for its hospitality and the Department of Energy for partial support.I.W.acknowledges partial support for this project from NASA.

APPENDIX A:“VIRIAL”EQUATIONS AND PERTURBATIONS

The two?uids need not occupy the same volume,and we shall suppose that?uidαoccupies a volume Vα.Taking the zeroth moment of Eq.(1)amounts to integrating over Vα;doing so,we obtain7the“?rst order‘virial’equation”8

d

+ Vαd3xFαβ,i.(A1)

?x i

Apart from inertial forces,which do not couple the two?uids,there are two forces that do couple them:gravity and friction.The net,mutual gravitational force between the?uids only vanishes if they(i)occupy the same volume and (ii)have densities that are proportional to one another(i.e.ρS∝ρN).The mutual friction force is nonzero as long as the?uids move relative to one another.Thus we see that,for a two?uid mixture,the zeroth moment of Eq.(1)is not trivial,as it would be for a single?uid(as in EFE).Note,though,that the center of mass motion of the combined system is trivial;i.e.,if the center of mass of the combined system starts out at x=0with zero velocity,it does not move.9

Taking the?rst moment of Eq.(1)results in the second order“virial”equation

d

2 Vαd3xραuα,i uα,j

Mα,ij≡?G|x?x′|3

Mαβ,ij≡?G Vαd3x Vβd3x′ρα(x)ρβ(x′)x j(x i?x′i)

=?δG Vαd3x Vβd3x′ρα(x)ρβ(x′)(x i?x′i)

?x i

=?G Vαd3xρα(x)ξα,l(x)?|x?x′|3

?x i

+G Vβd3xρβ(x)ξβ,l(x)?|x?x′|3(A5)

which is manifestly antisymmetric on α?β.Assuming V α=V β=V and ρα=f αρ(x )in the background equilibrium,we can simplify this to

?δ V α

d 3xρα?φβ?x l V d 3x ′ρ(x ′)(x i ?x ′i )dt

2 f α V d 3xρξα,i =2?ilm ?m d ?x l V

d 3x ′ρ(x ′)(x i ?x ′i )|x ?x ′|3=?φ(x )

?x l ?x i

.(A9)Second,recall that the potential at any interior point of a homogeneous ellipsoid is (Theorem 3in Chap.3of EFE)φ(x )=?πGρ I ?3 k =1A k x 2k ,

(A10)

where I is a constant;consequently

?2φ

dt f S V d 3xρ(x )ωS βij (ξS,j ?ξN,j ) ,

(A15)

where S αβ=0if α=β,1if α=S and β=N ,and ?1if α=N and β=S ,and [see Eq.

(4)]βij =βδij +β′?ijm νm +(β′′?β)νi νj .It is clear that the centers of mass of the two ?uids remain stationary if the ?uid

displacements are identical.However,there may be other conditions under which they remain stationary.For example, the integrated mutual friction force will be zero as long as

f S V d3xρ(x)ωSβij(ξS,j?ξN,j)=0;(A16)

for a background with uniform density,vorticity and frictional coupling coe?cients,this is guaranteed if

V d3xξS= V d3xξN,(A17)

which is less restrictive than the requirement of identical displacements.We found the same condition for the vanishing of the integrated,mutual gravitational force for perturbations of uniformly dense ellipsoids.Equation(A17)may be true,in fact,for all of the perturbations considered in our paper,since both sides may vanish identically.

Next,consider variations of the second order virial equation.Most of the terms are varied exactly as for single ?uids;one exception is

δMαβ,ij=?Gfαfβ V d3xρ(x)ξα(x)?|x?x′|3

+ V d3xρ(x)[ξα,l(x)?ξβ,l(x)]?|x?x′|3 ,(A18)

where we have specialized to backgrounds with proportional densities and identical bounding volumes.The?rst term in the brackets can be combined withδMα,ij and we?nd

δMα,ij+(1?δαβ)δMαβ,ij=?Gfα V d3xρ(x)ξα(x)?|x?x′|3

?Gfαfβ V d3xρ(x)[ξα,l(x)?ξβ,l(x)]?|x?x′|3.(A19) The last equation can be written more compactly in terms of the functions

B ij≡G V d3x′ρ(x′)(x i?x′i)(x j?x′j)?x i≡?G V d3x′ρ(x′)x′j(x i?x′i)

+fαfβ V d3xρ(ξα,l?ξβ,l)?2D j

?x l

=B ij?x j?φ

?x i

?x l

+fαfβ V d3xρ(ξα,l?ξβ,l) ?B ij?x i?x j?2φ

=a2j x j A j?3 k=1A jk x2k B ij

πGρ

(πGρ)?1?2D j

?x l =2B ij (δil x j +δjl x i )?2a 2i δij A il x l .(A25)

Using these results and the de?nitions [EFE,Chap.2,Eqs.(122),(124),and (125)]

V α,i ;j ≡ V d 3xρξα,i x j V α,ij =V α,i ;j +V α,j ;i ,(A26)

we ?nd

δM α,ij +(1?δαβ)δM αβ,ij

dt ?dξN,k

dt V d 3xρx j (ξS,k ?ξN,k ) ? V d 3xρu j (ξS,k ?ξN,k ) .(A30)

For backgrounds in which there are no ?uid motions,the last term is absent and

δ V αd 3xx j F αβ,i =?S αβf S ρωS βik dV α,k ;j dt ,(A31)

using the de?nition in Eq.

(A26).

When there are no ?uid motions of the unperturbed star in the rotating frame,the second order virial equations are

f S d 2V S,i ;j

dt +?2f S V S,ij ??i ?k f S V S,kj +δij δΠS

?f S πGρ 2B ij V S,ij

?a 2i δij

3 l =1

A il V S,ll ?a 2j f S f N πGρ 2A ij (V S,ij ?V N,ij )+δij

3 l =1A il (V S,ll ?V N,ll ) ?f S ωS βik

dV S,k ;j dt

,f N d 2V N,i ;j dt +?2f N V N,ij ??i ?k f N V N,kj +δij δΠN

如何写先进个人事迹

如何写先进个人事迹 篇一：如何写先进事迹材料 如何写先进事迹材料 一般有两种情况：一是先进个人，如先进工作者、优秀党员、劳动模范等；一是先进集体或先进单位，如先进党支部、先进车间或科室，抗洪抢险先进集体等。无论是先进个人还是先进集体，他们的先进事迹，内容各不相同，因此要整理材料，不可能固定一个模式。一般来说，可大体从以下方面进行整理。 (1)要拟定恰当的标题。先进事迹材料的标题，有两部分内容必不可少，一是要写明先进个人姓名和先进集体的名称，使人一眼便看出是哪个人或哪个集体、哪个单位的先进事迹。二是要概括标明先进事迹的主要内容或材料的用途。例如《王鬃同志端正党风的先进事迹》、《关于评选张鬃同志为全国新长征突击手的材料》、《关于评选鬃处党支部为省直机关先进党支部的材料》等。 (2)正文。正文的开头，要写明先进个人的简要情况，包括：姓名、性别、年龄、工作单位、职务、是否党团员等。此外，还要写明有关单位准备授予他(她)什么荣誉称号，或给予哪种形式的奖励。对先进集体、先进单位，要根据其先进事迹的主要内容，寥寥数语即应写明，不须用更多的文字。 然后，要写先进人物或先进集体的主要事迹。这部分内容是全篇材料

的主体，要下功夫写好，关键是要写得既具体，又不繁琐；既概括，又不抽象；既生动形象，又很实在。总之，就是要写得很有说服力，让人一看便可得出够得上先进的结论。比如，写一位端正党风先进人物的事迹材料，就应当着重写这位同志在发扬党的优良传统和作风方面都有哪些突出的先进事迹，在同不正之风作斗争中有哪些突出的表现。又如，写一位搞改革的先进人物的事迹材料，就应当着力写这位同志是从哪些方面进行改革的，已经取得了哪些突出的成果，特别是改革前后的．经济效益或社会效益都有了哪些明显的变化。在写这些先进事迹时，无论是先进个人还是先进集体的，都应选取那些具有代表性的具体事实来说明。必要时还可运用一些数字，以增强先进事迹材料的说服力。 为了使先进事迹的内容眉目清晰、更加条理化，在文字表述上还可分成若干自然段来写，特别是对那些涉及较多方面的先进事迹材料，采取这种写法尤为必要。如果将各方面内容材料都混在一起，是不易写明的。在分段写时，最好在每段之前根据内容标出小标题，或以明确的观点加以概括，使标题或观点与内容浑然一体。 最后，是先进事迹材料的署名。一般说，整理先进个人和先进集体的材料，都是以本级组织或上级组织的名义；是代表组织意见的。因此，材料整理完后，应经有关领导同志审定，以相应一级组织正式署名上报。这类材料不宜以个人名义署名。 写作典型经验材料-般包括以下几部分： (1)标题。有多种写法，通常是把典型经验高度集中地概括出来，一

关于时间管理的英语作文 manage time

How to manage time Time treats everyone fairly that we all have 24 hours per day. Some of us are capable to make good use of time while some find it hard to do so. Knowing how to manage them is essential in our life. Take myself as an example. When I was still a senior high student, I was fully occupied with my studies. Therefore, I hardly had spare time to have fun or develop my hobbies. But things were changed after I entered university. I got more free time than ever before. But ironically, I found it difficult to adjust this kind of brand-new school life and there was no such thing called time management on my mind. It was not until the second year that I realized I had wasted my whole year doing nothing. I could have taken up a Spanish course. I could have read ten books about the stories of successful people. I could have applied for a part-time job to earn some working experiences. B ut I didn’t spend my time on any of them. I felt guilty whenever I looked back to the moments that I just sat around doing nothing. It’s said that better late than never. At least I had the consciousness that I should stop wasting my time. Making up my mind is the first step for me to learn to manage my time. Next, I wrote a timetable, setting some targets that I had to finish each day. For instance, on Monday, I must read two pieces of news and review all the lessons that I have learnt on that day. By the way, the daily plan that I made was flexible. If there’s something unexpected that I had to finish first, I would reduce the time for resting or delay my target to the next day. Also, I would try to achieve those targets ahead of time that I planed so that I could reserve some more time to relax or do something out of my plan. At the beginning, it’s kind of difficult to s tick to the plan. But as time went by, having a plan for time in advance became a part of my life. At the same time, I gradually became a well-organized person. Now I’ve grasped the time management skill and I’m able to use my time efficiently.

英语演讲稿：未来的工作

英语演讲稿：未来的工作 这篇《英语演讲稿范文：未来的工作》，是特地，希望对大家有所帮助！ 热门演讲推荐：竞聘演讲稿 | 国旗下演讲稿 | 英语演讲稿 | 师德师风演讲稿 | 年会主持词 | 领导致辞 everybody good afternoon:. first of all thank the teacher gave me a story in my own future ideal job. everyone has a dream job. my dream is to bee a boss, own a pany. in order to achieve my dreams, i need to find a good job, to accumulate some experience and wealth, it is the necessary things of course, in the school good achievement and rich knowledge is also very important. good achievement and rich experience can let me work to make the right choice, have more opportunities and achievements. at the same time, munication is very important, because it determines whether my pany has a good future development. so i need to exercise their municative ability. i need to use all of the free time to learn

最新小学生个人读书事迹简介怎么写800字

小学生个人读书事迹简介怎么写800字 书,是人类进步的阶梯,苏联作家高尔基的一句话道出了书的重要。书可谓是众多名人的“宠儿”。历来,名人说出关于书的名言数不胜数。今天小编在这给大家整理了小学生个人读书事迹,接下来随着小编一起来看看吧! 小学生个人读书事迹1 “万般皆下品,惟有读书高”、“书中自有颜如玉,书中自有黄金屋”,古往今来,读书的好处为人们所重视,有人“学而优则仕”,有人“满腹经纶”走上“传道授业解惑也”的道路……但是,从长远的角度看,笔者认为读书的好处在于增加了我们做事的成功率,改善了生活的质量。 三国时期的大将吕蒙,行伍出身,不重视文化的学习,行文时,常常要他人捉刀。经过主君孙权的劝导,吕蒙懂得了读书的重要性,从此手不释卷,成为了一代儒将,连东吴的智囊鲁肃都对他“刮目相待”。后来的事实证明,荆州之战的胜利,擒获“武圣”关羽,离不开吕蒙的“运筹帷幄,决胜千里”,而他的韬略离不开平时的读书。由此可见,一个人行事的成功率高低,与他的对读书,对知识的重视程度是密切相关的。 的物理学家牛顿曾近说过,“如果我比别人看得更远,那是因为我站在巨人的肩上”,鲜花和掌声面前,一代伟人没有迷失方向,自始至终对读书保持着热枕。牛顿的话语告诉我们,渊博的知识能让我们站在更高、更理性的角度来看问题,从而少犯错误,少走弯路。

读书的好处是显而易见的,但是,在社会发展日新月异的今天,依然不乏对读书,对知识缺乏认知的人,《今日说法》中我们反复看到农民工没有和用人单位签订劳动合同,最终讨薪无果;屠户不知道往牛肉里掺“巴西疯牛肉”是犯法的;某父母坚持“棍棒底下出孝子”,结果伤害了孩子的身心,也将自己送进了班房……对书本,对知识的零解读让他们付出了惨痛的代价,当他们奔波在讨薪的路上,当他们面对高墙电网时,幸福,从何谈起?高质量的生活,从何谈起? 读书,让我们体会到“锄禾日当午,汗滴禾下土”的艰辛;读书,让我们感知到“四海无闲田,农夫犹饿死”的无奈;读书,让我们感悟到“为报倾城随太守,西北望射天狼”的豪情壮志。 读书的好处在于提高了生活的质量,它填补了我们人生中的空白,让我们不至于在大好的年华里无所事事,从书本中,我们学会提炼出有用的信息,汲取成长所需的营养。所以,我们要认真读书,充分认识到读书对改善生活的重要意义,只有这样,才是一种负责任的生活态度。 小学生个人读书事迹2 所谓读一本好书就是交一个良师益友,但我认为读一本好书就是一次大冒险,大探究。一次体会书的过程,真的很有意思,咯咯的笑声,总是从书香里散发;沉思的目光也总是从书本里透露。是书给了我启示,是书填补了我无聊的夜空,也是书带我遨游整个古今中外。所以人活着就不能没有书,只要爱书你就是一个爱生活的人,只要爱书你就是一个大写的人,只要爱书你就是一个懂得珍惜与否的人。可真所谓

关于坚持的英语演讲稿

关于坚持的英语演讲稿 Results are not important, but they can persist for many years as a commemoration of. Many years ago, as a result of habits and overeating formed one of obesity, as well as indicators of overall physical disorders, so that affects my work and life. In friends to encourage and supervise, the participated in the team Now considered to have been more than three years, neither the fine rain, regardless of winter heat, a day out with 5:00 time. The beginning, have been discouraged, suffering, and disappointment, but in the end of the urging of friends, to re-get up, stand on the playground. 成绩并不重要，但可以作为坚持多年晨跑的一个纪念。多年前，由于庸懒习惯和暴饮暴食，形成了一身的肥胖，以及体检指标的全盘失常，以致于影响到了我的工作和生活。在好友的鼓励和督促下，参加了晨跑队伍。现在算来，已经三年多了，无论天晴下雨，不管寒冬酷暑，每天五点准时起来出门晨跑。开始时，也曾气馁过、痛苦过、失望过，但最后都在好友们的催促下，重新爬起来，站到了操场上。 In fact, I did not build big, nor strong muscles, not a sport-born people. Over the past few years to adhere to it, because I have a team behind, the strength of a strongteam here, very grateful to our team, for a long time, we encourage each other, and with sweat, enjoying common health happy. For example, Friends of the several run in order to maintain order and unable to attend the 10，000 meters race, and they are always concerned about the brothers and promptly inform the place and time, gives us confidence and courage. At the same time, also came on their own inner desire and pursuit for a good health, who wrote many of their own log in order to refuel for their own, and inspiring. 其实我没有高大身材，也没健壮肌肉，天生不属于运动型的人。几年来能够坚持下来，因为我的背后有一个团队，有着强大团队的力量，在这里，非常感谢我们的晨跑队，长期以来，我们相互鼓励着，一起流汗，共同享受着健康带来的快

关于管理的英语演讲

1.How to build a business that lasts100years 0:11Imagine that you are a product designer.And you've designed a product,a new type of product,called the human immune system.You're pitching this product to a skeptical,strictly no-nonsense manager.Let's call him Bob.I think we all know at least one Bob,right?How would that go? 0:34Bob,I've got this incredible idea for a completely new type of personal health product.It's called the human immune system.I can see from your face that you're having some problems with this.Don't worry.I know it's very complicated.I don't want to take you through the gory details,I just want to tell you about some of the amazing features of this product.First of all,it cleverly uses redundancy by having millions of copies of each component--leukocytes,white blood cells--before they're actually needed,to create a massive buffer against the unexpected.And it cleverly leverages diversity by having not just leukocytes but B cells,T cells,natural killer cells,antibodies.The components don't really matter.The point is that together,this diversity of different approaches can cope with more or less anything that evolution has been able to throw up.And the design is completely modular.You have the surface barrier of the human skin,you have the very rapidly reacting innate immune system and then you have the highly targeted adaptive immune system.The point is,that if one system fails,another can take over,creating a virtually foolproof system. 1:54I can see I'm losing you,Bob,but stay with me,because here is the really killer feature.The product is completely adaptive.It's able to actually develop targeted antibodies to threats that it's never even met before.It actually also does this with incredible prudence,detecting and reacting to every tiny threat,and furthermore, remembering every previous threat,in case they are ever encountered again.What I'm pitching you today is actually not a stand-alone product.The product is embedded in the larger system of the human body,and it works in complete harmony with that system,to create this unprecedented level of biological protection.So Bob,just tell me honestly,what do you think of my product? 2:47And Bob may say something like,I sincerely appreciate the effort and passion that have gone into your presentation,blah blah blah-- 2:56(Laughter) 2:58But honestly,it's total nonsense.You seem to be saying that the key selling points of your product are that it is inefficient and complex.Didn't they teach you 80-20?And furthermore,you're saying that this product is siloed.It overreacts, makes things up as it goes along and is actually designed for somebody else's benefit. I'm sorry to break it to you,but I don't think this one is a winner.

关于工作的优秀英语演讲稿

关于工作的优秀英语演讲稿 Different people have various ambitions. Some want to be engineers or doctors in the future. Some want to be scientists or businessmen. Still some wish to be teachers or lawers when they grow up in the days to come. Unlike other people, I prefer to be a farmer. However, it is not easy to be a farmer for Iwill be looked upon by others. Anyway,what I am trying to do is to make great contributions to agriculture. It is well known that farming is the basic of the country. Above all, farming is not only a challenge but also a good opportunity for the young. We can also make a big profit by growing vegetables and food in a scientific way. Besides we can apply what we have learned in school to farming. Thus our countryside will become more and more properous. I believe that any man with knowledge can do whatever they can so long as this job can meet his or her interest. All the working position can provide him with a good chance to become a talent. 1 ————来源网络整理，仅供供参考

立春节气的特点是什么_二十四节气立春特点

立春节气的特点是什么_二十四节气立春特点 二十四节气，每一个节气都代表着季节气候的转变。立春是春天的头一个节气，那么大家知道立春节气会有哪些特征变化吗?下面是小编给大家整理的有关立春节气的特点，欢迎大家来参阅。 立春的节气特点 每年2月4日前后，太阳到达黄经315度时交立春节气。“立春”是预示季节转换的节气。这个节气，早在战国末期(约公元前239年)就确定了。 “立春”是二十四节气中的第一个节气。在古时，古人认为，春季开始叫“立春”，夏季开始叫“立夏”，秋季开始叫“立秋”，冬季开始叫“立冬”。其农事意义是“春种、夏长、秋收、冬藏”。 春夏秋冬，以春为首。春季六节气，“立春”居先，标志着春天的到来、四季的开始，这是古时的说法。 立春物候 立春第一候是“东风解冻”，东风送暖，化解了大地冰封已久的寒冻。第二候是“蛰虫始振”，“蛰”是隐藏的意思，指潜伏在地下的小虫都自冬眠中苏醒过来。第三候是“鱼陟负冰”，陟是上升的意思，鱼儿囚为水温渐暖，所以竞相浮游到水面，但水中仍有未溶解的碎冰。在岸上观看，就如同鱼儿背负着冰块在水中游动。 立春气候

从现代气候学标准看，以“立春”作为春季的开始，不能和当时全国各地的自然物 候现象(即春季应有的景象)吻合。如2月初“立春”后，我国华南已经花红柳绿了，然而 华北大地仍会大雪纷飞。现在比较科学的划分，是把候(5天为一候)平均气温在10摄 氏度以上的始日，作为春季开始。 相连的广州起步北上，开始旅程，2月中旬先是到了昆明、下旬则越过云贵高原 进人四川盆地，3月中旬到达武汉三镇、下旬越过郑州和济南，4月上旬抵达京津地区、中旬跨过山海关来到塞外沈阳城、下旬经由长春到达哈尔滨，5月20日前后，才能光 临我国最北的地方—漠河。另外，由于我国幅员辽阔，地势复杂，不但春临大地的时 间有早有迟，春姑娘在各地停留的时间也有长短之别，如西北地区土壤干燥，春季升 温迅速，整个春季仅40多天，可谓来去匆匆;而云南昆明等地则四季如春，被誉为“春城”。 虽然“立春”节气后，不能说春天已到，但确实预示着春天就要来了，农业生产大 忙季节即将来临。这不，一些农谚即是很好的佐证。如:“立春一日，水热三分”、“春到 三分暖”……以安徽省为例，“立春”节气以后，虽处于冬季之尾，但有时气温仍较低， 还会发生低温冻害。如安徽省极端最低气温的极值，就出现在1969年2月6日的固 镇县，为一24.3‘C。因此，仍应加强冬作物，尤其是温室大棚的保温防寒工作。不过，有时冬季偏暖年份，南方湿热空气势力强于北方冷空气，在“立春” 立春时节 立春一到，人们明显地感觉到白昼长了，太阳暖了，气温增高了，日照延长了， 降雨也开始了。农谚提醒人们“立春雨水到，早起晚睡觉”。立春时竹北方备耕也开始了。 “一年之计在干春”，在中国是流传最为广泛的格言式谚语之一。在《增广贤文》中，它一共是四句:“一年之计在于春，一日之计在于晨，一家之计在于和，一生之计 在于勤。”其中前两句，强调的都是时间的紧迫性和重要性。 阳春之季，万象更新，对于农事耕作，这是个关键时刻。对于一个人来说，青年 时期也是人生的春天。青春的创造力是无穷的，爱惜青春，珍视青春，就能创造出知识，创造出财富;反之，浪费青春，虚度年华，除了惭愧之外，将一无所得。“一年之 计在于春”，强调的是要在一年开始时多做并做好工作，为全年的工作打下坚实的基础。 节气特征现象

个人先进事迹简介

个人先进事迹简介 01 在思想政治方面,xxxx同学积极向上,热爱祖国、热爱中国共产党,拥护中国共产党的领导.利用课余时间和党课机会认真学习政治理论,积极向党组织靠拢. 在学习上,xxxx同学认为只有把学习成绩确实提高才能为将来的实践打下扎实的基础,成为社会有用人才.学习努力、成绩优良. 在生活中,善于与人沟通,乐观向上,乐于助人.有健全的人格意识和良好的心理素质和从容、坦诚、乐观、快乐的生活态度,乐于帮助身边的同学,受到师生的好评. 02 xxx同学认真学习政治理论,积极上进,在校期间获得原院级三好生,和校级三好生,优秀团员称号,并获得三等奖学金. 在学习上遇到不理解的地方也常常向老师请教,还勇于向老师提出质疑.在完成自己学业的同时,能主动帮助其他同学解决学习上的难题,和其他同学共同探讨,共同进步. 在社会实践方面,xxxx同学参与了中国儿童文学精品“悦”读书系,插画绘制工作,xxxx同学在班中担任宣传委员,工作积极主动,认真负责,有较强的组织能力.能够在老师、班主任的指导下独立完成学院、班级布置的各项工作. 03 xxx同学在政治思想方面积极进取,严格要求自己.在学习方面刻苦努力,不断钻研,学习成绩优异,连续两年荣获国家励志奖学金；作

为一名学生干部,她总是充满激情的迎接并完成各项工作,荣获优秀团干部称号.在社会实践和志愿者活动中起到模范带头作用. 04 xxxx同学在思想方面,积极要求进步,为人诚实,尊敬师长.严格 要求自己.在大一期间就积极参加了党课初、高级班的学习,拥护中国共产党的领导,并积极向党组织靠拢. 在工作上,作为班中的学习委员,对待工作兢兢业业、尽职尽责 的完成班集体的各项工作任务.并在班级和系里能够起骨干带头作用.热心为同学服务,工作责任心强. 在学习上,学习目的明确、态度端正、刻苦努力,连续两学年在 班级的综合测评排名中获得第1.并荣获院级二等奖学金、三好生、优秀班干部、优秀团员等奖项. 在社会实践方面,积极参加学校和班级组织的各项政治活动,并 在志愿者活动中起到模范带头作用.积极锻炼身体.能够处理好学习与工作的关系,乐于助人,团结班中每一位同学,谦虚好学,受到师生的好评. 05 在思想方面,xxxx同学积极向上,热爱祖国、热爱中国共产党,拥护中国共产党的领导.作为一名共产党员时刻起到积极的带头作用,利用课余时间和党课机会认真学习政治理论. 在工作上,作为班中的团支部书记,xxxx同学积极策划组织各类 团活动,具有良好的组织能力. 在学习上,xxxx同学学习努力、成绩优良、并热心帮助在学习上有困难的同学,连续两年获得二等奖学金. 在生活中,善于与人沟通,乐观向上,乐于助人.有健全的人格意 识和良好的心理素质.

自我管理演讲稿英语翻译

尊敬的领导，老师，亲爱的同学们， 大家好！我是5班的梁浩东。今天早上我坐车来学校的路上，我仔细观察了路上形形色色的人，有开着小车衣着精致的叔叔阿姨，有市场带着倦容的卖各种早点的阿姨，还有偶尔穿梭于人群中衣衫褴褛的乞丐。于是我问自己，十几年后我会成为怎样的自己，想成为社会成功人士还是碌碌无为的人呢，答案肯定是前者。那么十几年后我怎样才能如愿以偿呢，成为一个受人尊重，有价值的人呢？正如我今天演讲的题目是：自主管理。 大家都知道爱玩是我们孩子的天性，学习也是我们的责任和义务。要怎样处理好这些矛盾，提高自主管理呢？ 首先，我们要有小主人翁思想，自己做自己的主人，要认识到我们学习，生活这一切都是我们自己走自己的人生路，并不是为了报答父母，更不是为了敷衍老师。 我认为自主管理又可以理解为自我管理，在学习和生活中无处不在，比如通过老师，小组长来管理约束行为和同学们对自身行为的管理都属于自我管理。比如我们到一个旅游景点，看到一块大石头，有的同学特别兴奋，会想在上面刻上：某某某到此一游话。这时你就需要自我管理，你需要提醒自己，这样做会破坏景点，而且是一种素质低下的表现。你设想一下，如果别人家小孩去你家墙上乱涂乱画，你是何种感受。同样我们把自主管理放到学习上，在我们想偷懒，想逃避，想放弃的时候，我们可以通过自主管理来避免这些，通过他人或者自己的力量来完成。例如我会制定作息时间计划表，里面包括学习，运动，玩耍等内容的完成时间。那些学校学习尖子，他们学习好是智商高于我们吗，其实不然，在我所了解的哪些优秀的学霸传授经验里，就提到要能够自我管理，规范好学习时间的分分秒秒，只有辛勤的付出，才能取得优异成绩。 在现实生活中，无数成功人士告诉我们自主管理的重要性。十几年后我想成为一位优秀的，为国家多做贡献的人。亲爱的同学们，你们们？让我们从现在开始重视和执行自主管理，十几年后成为那个你想成为的人。 谢谢大家！

关于工作的英语演讲稿

关于工作的英语演讲稿 【篇一：关于工作的英语演讲稿】 关于工作的英语演讲稿 different people have various ambitions. some want to be engineers or doctors in the future. some want to be scientists or businessmen. still some wish to be teachers or lawers when they grow up in the days to come. unlike other people, i prefer to be a farmer. however, it is not easy to be a farmer for iwill be looked upon by others. anyway,what i am trying to do is to make great contributions to agriculture. it is well known that farming is the basic of the country. above all, farming is not only a challenge but also a good opportunity for the young. we can also make a big profit by growing vegetables and food in a scientific way. besides we can apply what we have learned in school to farming. thus our countryside will become more and more properous. i believe that any man with knowledge can do whatever they can so long as this job can meet his or her interest. all the working position can provide him with a good chance to become a talent. 【篇二：关于责任感的英语演讲稿】 im grateful that ive been given this opportunity to stand here as a spokesman. facing all of you on the stage, i have the exciting feeling of participating in this speech competition. the topic today is what we cannot afford to lose. if you ask me this question, i must tell you that i think the answer is a word---- responsibility. in my elementary years, there was a little girl in the class who worked very hard, however she could never do satisfactorily in her lessons. the teacher asked me to help her, and it was obvious that she expected a lot from me. but as a young boy, i was so restless and thoughtless, i always tried to get more time to play and enjoy myself. so she was always slighted over by me. one day before the final exam, she came up to me and said, could you please explain this to me? i can not understand it. i

关于时间管理的英语演讲

Dear teacher and colleagues: my topic is on “spare time”. It is a huge blessing that we can work 996. Jack Ma said at an Ali's internal communication activity, That means we should work at 9am to 9pm, 6 days a week .I question the entire premise of this piece. but I'm always interested in hearing what successful and especially rich people come up with time .So I finally found out Jack Ma also had said :”i f you don’t put out more time and energy than others ,how can you achieve the success you want? If you do not do 996 when you are young ,when will you ?”I quite agree with the idea that young people should fight for success .But there are a lot of survival activities to do in a day ,I want to focus on how much time they take from us and what can we do with the rest of the time. As all we known ,There are 168 hours in a week .We sleep roughly seven-and-a-half and eight hours a day .so around 56 hours a week . maybe it is slightly different for someone . We do our personal things like eating and bathing and maybe looking after kids -about three hours a day .so around 21 hours a week .And if you are working a full time job ,so 40 hours a week , Oh! Maybe it is impossible for us at

关于人英语演讲稿(精选多篇)

关于人英语演讲稿(精选多篇) 关于人的优美句子 1、“黑皮小子”是我对在公交车上偶遇两次的一个男孩的称呼代号。一听这个外号，你也定会知道他极黑了。他的脸总是黑黑的；裸露在短袖外的胳膊也是黑黑的；就连两只有厚厚耳垂的耳朵也那么黑黑的，时不时像黑色的猎犬竖起来倾听着什么；黑黑的扁鼻子时不时地深呼吸着，像是在警觉地嗅着什么异样的味道。 2、我不知道，如何诠释我的母亲，因为母亲淡淡的生活中却常常跳动着不一样的间最无私、最伟大、最崇高的爱，莫过于母爱。无私，因为她的爱只有付出，无需回报；伟大，因为她的爱寓于

普通、平凡和简单之中；崇高，是因为她的爱是用生命化作乳汁，哺育着我，使我的生命得以延续，得以蓬勃，得以灿烂。 3、我的左撇子伙伴是用左手写字的，就像我们的右手一样挥洒自如。在日常生活中，曾见过用左手拿筷子的，也有像超级林丹用左手打羽毛球的，但很少碰见用左手写字的。中国汉字笔画笔顺是左起右收，适合用右手写字。但我的左撇子伙伴写字是右起左收的，像鸡爪一样迈出田字格，左看右看，上看下看，每一个字都很难看。平时考试时间终了，他总是做不完试卷。于是老师就跟家长商量，决定让他左改右写。经过老师引导，家长配合，他自己刻苦练字，考试能够提前完成了。现在他的字像他本人一样阳光、帅气。 4、老师，他们是辛勤的园丁，帮助着那些幼苗茁壮成长。他们不怕辛苦地给我们改厚厚一叠的作业，给我们上课。一步一步一点一点地给我们知识。虽然

他们有时也会批评一些人，但是他们的批评是对我们有帮助的，我们也要理解他们。那些学习差的同学，老师会逐一地耐心教导，使他们的学习突飞猛进。使他们的耐心教导培养出了一批批优秀的人才。他们不怕辛苦、不怕劳累地教育着我们这些幼苗，难道不是美吗？ 5、我有一个表妹，还不到十岁，她那圆圆的小脸蛋儿，粉白中透着粉红，她的头发很浓密，而且好像马鬓毛一样的粗硬，但还是保留着孩子一样的蓬乱的美，卷曲的环绕着她那小小的耳朵。说起她，她可是一个古灵精怪的小女孩。 6、黑皮小子是一个善良的人，他要跟所有见过的人成为最好的朋友！这样人人都是他的好朋友，那么人人都是好友一样坦诚、关爱相交，这样人与人自然会和谐起来，少了许多争执了。 7、有人说，老师是土壤，把知识化作养分，传授给祖国的花朵，让他们茁壮成长。亦有人说，老师是一座知识的桥梁，把我们带进奇妙的科学世界，让

二十四节气解读

二十四节气解读 立春： 立春，二十四节气[1]之一，“立”是“开始”的意思，立春就是春季的开始，每年2月4日或5日太阳到达黄经315度时为立春。《月令七十二候集解》：“正月节，立，建始也……立夏秋冬同。”古代“四立”，指春、夏、秋、冬四季开始，其农业意义为“春种、夏长、秋收、冬藏”，概括了黄河中下游农业生产与气候关系的全过程。 雨水： 雨水，表示降水开始，雨量逐步增多。每年2月18日前后，太阳到达黄经330度，为“雨水”节气。雨水，表示两层意思，一是天气回暖，降水量逐渐增多了，二是在降水形式上，雪渐少了，雨渐多了。《月令七十二候集解》中说：“正月中，天一生水。春始属木，然生木者必水也，故立春后继之雨水。且东风既解冻，则散而为雨矣。惊蛰： 惊蛰——春雷乍动，惊醒了蛰伏在土壤中冬眠的动物。这时气温回升较快，渐有春雷萌动。每年公历的3月6日左右为惊蛰。二十四节气之一。蛰是藏的意思。“惊蛰”是指钻到泥土里越冬的小动物被雷震苏醒出来活动。“惊蛰”节气日，地球已经达到太阳黄经345度，一般在每年3月4日～7日。 春分： 春分，古时又称为“日中”、“日夜分”、“仲春之月”，在每年的3月21日前后(20日～22日)交节，农历日期不固定，这时太阳到达黄经0°。据《月令七十二候集解》：“二月中，分者半也，此当九十日之半，故谓之分。”另《春秋繁露·阴阳出入上下篇》说：“春分者，阴阳相半也，故昼夜均而寒暑平。”有《明史·历一》说：“分者，黄赤相交之点，太阳行至此，乃昼夜平分。”所以，春分的意义，一是指一天时间白天黑夜平分，各为12小时；二是古时以立春至立夏为春季，春分正当春季三个月之中，平分了春季。 清明： 清明的意思是清淡明智。“清明”是夏历二十四节气之一，中国广大地区有在清明之日进行祭祖、扫墓、踏青的习俗，逐渐演变为华人以扫墓、祭拜等形式纪念祖先的一个中国传统节日。另外还有很多以“清明”为题的诗歌，其中最著名的是唐代诗人杜牧的七绝《清明》。 谷雨： 谷雨是二十四节气之一。谷雨指雨水增多，大大有利于谷类农作物的生长。每年4月19日～21日视太阳到达黄经30°时为谷雨。