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directed and elliptic flow are defined by the and coefficients in the fourier expansion of the azimuthal distribution of particles relative to the reaction plane : 1 + 2 v_1 ( -_r ) + 2 v_2 ( 2(-_r ) ) , [ fourier ] where @xmath5 is the azimuthal angle of a particle and @xmath6 is the azimuthal angle of the reaction plane .
the coefficients are usually evaluated differentially as a function of @xmath7 , @xmath8 , and centrality .
three methods of analysis will be discussed . in the conventional method
@xcite of correlating particles with an event plane the flow coefficients are given by : v_n^obs = ( n(_i-_n ) ) , [ conventional ] where @xmath9 is the observed event plane of order @xmath10 .
since the observed event plane is not the true reaction plane , the observed coefficients have to be corrected by dividing by the resolution of the event plane , which is estimated by measuring the correlation of the event planes of subevents .
the flow coefficients can also be obtained by the pair - wise correlation  @xcite of all the particles without refering to an event plane .
this two - particle correlation method produces the squares of the coefficients , so that one has to take the square root of the correlation effect : v_n^2 = ( n(_i-_j ) ) _ i j .
[ pairwise ]    recently , a multiparticle correlation method has been proposed  @xcite using cumulants . for four particles ,
one calculates the four - particle correlation minus twice the square of the two - particle correlations : v_n^4 = & - ( n(_1+_2-_3-_4 ) ) + & + ( n(_1-_3 ) )  ( n(_2-_4 ) ) + & + ( n(_1-_4 ) )  ( n(_2-_3 ) ) .
[ cumulant ] here one must take the fourth root of the result , and naturally the statistical errors are larger than for the two - particle analysis
. however , this method has the advantage of eliminating two - particle non - flow effects which are not correlated with the reaction plane , such as hbt and resonance decay .
na49 directed and elliptic flow for pions from a minimum bias trigger as a function of rapidity .
the open points are reflected .
the data are still preliminary . ]    from na49 , initial flow results from part of the data set were published  @xcite a few years ago .
the full set of data has now been analyzed  @xcite .
the and values for pions as a function of rapidity are shown in fig .
[ na49mby ] .
the values have been integrated over @xmath8 by taking mean values weighted with the cross section .
the values have been corrected for conservation of momentum  @xcite . before this correction , crossed mid - rapidity at + 0.6% , while after the correction , with no adjustable parameters , it can be seen that the curve now crosses mid - rapidity at zero .
other non - flow effects  @xcite have been verified to be small using the following tests .
short - range correlations , like hbt , were determined to be not large by using striped sub - events where only physically separated particles were correlated .
coulomb effects were determined to be not large by looking at @xmath11 and @xmath12 particles separately .
neutral resonances , like @xmath13 , were shown to be not important by correlating @xmath12 with an event plane determined only by other @xmath12 particles .
the pion flow values integrated over both @xmath8 and @xmath7 are show in fig .
[ na49vcen ] as a function of centrality bin .
na49 directed and elliptic flow for pions as a function of centrality bin .
the centrality increases to the left .
the data are still preliminary . ]
wa98 has measured  @xcite the directed flow of pions near target rapidity in the plastic ball .
these data are plotted in fig .
[ wa98 ] reflected into the forward hemisphere , together with the na49 data .
it can be seen that at projectile rapidities the directed flow is huge .
wa98 also has shown that for protons is just as large , but in the positive direction .
this indicates that looking at projectile rapidity fragments might be a good way to determine the reaction plane .
however , even though is large , the resolution of the event plane also depends on the square root of the multiplicity , which in this rapidity region is low .
na49 ( circles ) and wa98 ( squares ) pion data are compared .
plotted is the directed flow versus rapidity .
the vertical lines are at mid - rapidity and beam rapidity .
the data are preliminary . ]
na49 ( circles ) and na45 ( squares ) charged particle data are compared .
plotted is the elliptic flow at mid - rapidity versus the centrality .
the centrality increases to the left because the abscissa is the cumulative most central percentage of all events that are in the bin .
the data are preliminary . ]    using silicon drift chambers na45 has measured  @xcite the flow coefficients for charged particles . in fig .
[ na45 ] their elliptic flow values at mid - rapidity are plotted versus centrality , together with the same quantity for pions from na49 .
the agreement is good .
na45 also has a rich detector which identifies high @xmath8 pions .
when correlated with an event plane determined from their tpc , the @xmath8 dependence falls nicely on a line extrapolated from the lower @xmath8 results of na49 .
however , when they correlate pairs of high @xmath8 pions from the rich using eq .  [ pairwise ] ,
their results are somewhat higher  @xcite .
na50 has analyzed the neutral transverse energy from their calorimeter in the rapidity region from 1.1 to 2.3 .
na50 has shown elliptic flow as a function of centrality  @xcite , but it is hard to compare their results with the other experiments because of the different method of analysis .
star results for elliptic flow for charged particles  @xcite and identified particles  @xcite have been published .
the flow is larger than at the sps and comes close to the hydrodynamic predictions  @xcite for the mid - central and central data .
the identified particle flow shows a variation with mass as predicted by hydrodynamics .
however , the high @xmath8 behavior  @xcite exhibits a saturation effect well below where hydro would saturate . it has been suggested that at high @xmath8 elliptic flow could be due to parton energy loss ( jet quenching )  @xcite .
the doubly integrated elliptic flow values as a function of centrality are shown in fig .
[ star ] , using the conventional event plane method ( eq .  [ conventional ] ) .
the asymmetric error bars indicate the systematic uncertainty due to possible non - flow effects . a four - particle correlation analysis  @xcite using eq .
[ cumulant ] is also show in the figure .
indeed , when two - particle non - flow effects are eliminated , the points do come down to about the limits of the systematic errors .
it should be noted that part of the decrease could be due to fluctuation effects  @xcite which also would go in the same direction .
however , the agreement with hydro for mid - central and central collisions is not affected .
star elliptic flow for charged particles versus centrality .
the results using the conventional event plane method ( circles ) and the four - particle cumulant method ( squares ) are shown . the later are preliminary . ]
the phenix results  @xcite for charged particle elliptic flow as a function @xmath8 for a minimum bias trigger is shown together with the star results in fig .
[ phenix ] .
phenix uses the pair - wise method of eq .
[ pairwise ] .
the agreement is excellent .
however , at the moment , the doubly - integrated elliptic flow values as a function of centrality are about 50% higher for phenix than for star .
only about half of this discrepancy can be accounted for by the higher @xmath8 threshold of phenix .
star ( circles ) and phenix ( squares ) charged particle data for a minimum bias trigger are compared .
plotted is the elliptic flow versus the transverse momentum .
the phenix data are preliminary . ]
the phobos elliptic flow values  @xcite as a function of centrality are shown in fig .
[ phoboscen ] , together with the star values .
the agreement is excellent .
[ phoboseta ] shows phobos elliptic flow for a minimum bias trigger measured over a wide range of pseudorapidity . where there are star measurements , in a narrow range near mid - rapidity , the agreement is good with the flat star distribution .
a 3d hydro calculation  @xcite gets rather flat distributions over a wide range of pseudorapidity for all reasonable initial conditions .
thus the fast fall off away from midrapidity in fig .
[ phoboseta ] is not understood at present in a model with thermalization .
star ( circles ) and phobos ( squares ) charged particle data are compared .
plotted is the elliptic flow versus centrality .
the phobos data are preliminary and the error bars are only statistical . ]
phobos charged particle elliptic flow for a minimum bias trigger plotted versus pseudorapidity .
the data are preliminary and the error bars are only statistical . ]
good agreement between the various experiments has been achieved .
there might be some indication that the pair - wise method for high @xmath8 particles indicates non - flow effects .
the four - particle method at rhic indicates that there are non - flow effects for peripheral collisions .
the elliptic flow increases from the sps to rhic , and at rhic is close to the hydro limit .
this is taken to be a signature for early time thermalization at rhic .
i wish to thank alexander wetzler , thomas peitzmann , kirill filimonov , roy lacey , steve manly , inkyu park , and peter steinberg for supplying some of the data and graphs .
comments by hans - georg ritter , nu xu , sergei voloshin , jean - yves ollitrault , and peter seyboth are also appreciated .
this work was supported by the division of nuclear physics of the office of science of the u.s.department of energy .
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the results on directed and elliptic flow for @xmath0 at the full energy of the sps ( @xmath1  gev / a ) and from the first year of @xmath2 at rhic ( @xmath3  gev ) are reviewed . 
 the different experiments agree well and a consistent picture has emerged indicating early time thermalization at rhic .    
 = 0.5 in    elliptic flow from non - central collisions has become a prime hadronic signature for pressure at early times in relativistic heavy ion collisions  @xcite . 
 parton cascade model calculations  @xcite show that the elliptic flow builds up in the first few @xmath4 and then remains constant . in this model 
 the amount of elliptic flow produced is proportional to the assumed parton - parton scattering cross section . 
 thus it is thought that rescattering converts the initial space anisotropy of the overlap region of non - central collision to the momentum anisotropy of elliptic flow . 
 as the initial lens - shaped overlap region expands it becomes more spherical , quenching the driving force that produces the elliptic flow . 
 thus elliptic flow is sensitive to the number of interactions and is considered to be a measure of the degree of thermalization at early time .