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why the stars and galaxies are rotating ? what is the origin of the angular momentum ? it is known that the planets in our solar system show a remarkable scaling relation between the angular momentum @xmath1 and the mass @xmath2 for each planet , @xmath3 where the unit mass and the angular momentum are measured respectively by the planck mass @xmath4 g  and the reduced planck constant @xmath5erg@xmath6sec .
only exceptions are the mercury and the venus whose rotations are locked with their orbital rotation around the sun . although the sun by itself deviates from the above scaling , the whole solar system is also on this line . because the relation eq.([jmplanets ] )
holds for the conserved quantities , @xmath1 and @xmath2 , it strongly suggests some fundamental origin of the planet system .    [ cols="^,^ " , ]
we summarize our study and discuss on further extension and possible applications of it .
we started our study from an interesting scaling law but @xmath7 within the cluster series .
the same index is suggested for star series .
see below . ]
@xmath8 which holds for angular momentum @xmath1  and the mass @xmath2  of astronomical objects from the scale of planets toward that of clusters of galaxies . focussing on the cluster series , such as globular clusters , galaxies and clusters of galaxies , we extracted the nature of the fully non - linear stage of dark matter , after the formation of caustic surface , in analogy between sgs and the fluid turbulence .
the essential kolmogorov laws were rederived from the fourier transformed navier - stokes equation and equations for sgs , by demanding the existence of a steady energy flow characterized by a single parameter @xmath9 .
this is an analogue of the case of smoluchowski coagulation equation , which admits scaling solution by demanding the steady mass flow .
then we tried to test our considerations in several cosmological observations , such as the velocity dispersion , mass - luminosity ratio , @xmath10 relation , power spectrum of density fluctuations , and the cosmic magnetic fields .
they all point the concordant value for the constant energy flow per mass : @xmath11@xmath12sec@xmath13 .      1 .
we would like to evaluate the energy flow @xmath9 for dm , which may be associated with the hierarchical coalescence process in which smaller clusters continuously form larger clusters .
this process provides the bottom - up scenario for the formation of large scale structure especially in the cold dm model .
we would like to check whether the value we used @xmath14@xmath15 $ ] is consistent with the hierarchical coalescence evolution of dm .
roughly estimated , this accumulation rate of kinetic energy yields a galaxy within about @xmath16 years .
if the above is the case for dm , then we would like to apply the same analysis to our planet system , which is also considered to have the coalescence evolution as its origin [ 35 ] .
actually in the left half of figure [ fig3 ] , the planet and star systems seem to admit the fitting line with the same slope @xmath17 as the case for dm but with larger parameter value for @xmath9 .
it turns out that the appropriate value for @xmath18  becomes of order @xmath19 , which seems to be too large to be explained from a simple coalescence process as in the case for dm . some violent mechanism , which allow huge energy transfer rate or catastrophic coalescence , is expected in the case for planet formation process .
3 .   in the fluid turbulence
, there holds another relation often called kolmogorov 4/5 law @xmath20which is essential to explain the energy flow takes place from low frequency modes to high frequency modes on average .
then what is the analogous relation for sgs and how is it relevant in the universe ?
although we do not know the answer , it will be important to notice the fact that sgs [ 36 ] shares some common properties with the fluid turbulence [ 37 ] , such as the negative skewness and the exponential distribution function of the velocity - difference , etc .
these common properties will be a good starting point for further discussion .
4 .   it would be interesting if we could actually transform eq.(ftjeans2 ) into the smoluchowski form like eq.([smoluchowski ] ) which directly expresses the coalescence evolution of sgs .
if this is the case , the possible scaling solution , which may asymptotically realize , may correspond to the schechter function [ 3 ] which has a typical form , @xmath21where @xmath22 represents the frequency of the object of luminosity @xmath23 . if the cosmological objects are formed after many coalescence processes , then the smoluchowski equation will be more appropriate than the press - schechter theory [ 3 ] , in which a single collapse determines the population of objects at the corresponding scale
. if this is the case , the index @xmath24 for the kernel @xmath25 ( see just below eq.([smoluchowski ] ) ) should be about @xmath26 .
the exponential factor @xmath27 seems to be a natural tail of the distribution which is gradually evolving without runaway coalescence .
we could not complete the argument on the velocity - luminosity ( or mass ) relation [ 38 ] which holds within a single species of object , from our point of view .
if we simply apply our argument for luminous objects , we would have , from @xmath28 , the luminosity @xmath29as a function of the velocity @xmath30 or the velocity dispersion @xmath31  at that scale , provided appropriate reference ratio @xmath32 is given .
for dm , we have , from @xmath33 , the mass expression @xmath34on the other hand , several tight relations are obtained for each type of galaxies . for spiral galaxies , tally - fisher relation of the form eq.(lv - rel ) holds with the variation of index from 3 for b - band to 4 for k band observations . for elliptical galaxies , a tight relation on the fundamental plane
is established , @xmath35const.@xmath36 where @xmath37 is the average surface brightness .
although we can simply apply our argument to yield @xmath38which is almost consistent with the above , we can not obtain the relation in the form of a plain among three independent parameters ; all quantities are functions of only @xmath39 in our case .
the authors would like to thank osamu hashimoto , hidenori takahashi for relevant discussions on the turbulence , osamu iguchi , yasuhide sota , and tohru tashiro for discussion on the local virial relation , nozomi mori for the cosmic magnetic fields and dynamo theory , masaaki morita , hideaki mouri for turbulence and fate of cosmic vorticity , kiyotaka tanikawa for showing us his paper before publication and providing us relevant references on @xmath10 relation , takayuki tatekawa for higher - order density perturbation theories .
all of their help has been crucial for the completion of this paper .
many scaling relations are observed for self - gravitating systems in the universe . 
 we explore the consistent understanding of them from a simple principle based on the proposal that the collision - less dark matter fluid terns into a turbulent state , i.e. dark turbulence , after crossing the caustic surface in the non - linear stage . 
 the dark turbulence will not eddy dominant reflecting the collision - less property . after deriving kolmogorov scaling laws from navier - stokes equation by the method similar to the one for smoluchowski coagulation equation , 
 we apply this to several observations such as the scale - dependent velocity dispersion , mass - luminosity ratio , magnetic fields , and mass - angular momentum relation , power spectrum of density fluctuations . 
 they all point the concordant value for the constant energy flow per mass : @xmath0 , which may be understood as the speed of the hierarchical coalescence process in the cosmic structure formation .    
 cosmology : theory , gravitation , turbulence , cosmology : dark matter , cosmology : observations , hydrodynamics 98.80.-k 95.35 + d 98.65.-r