Hot and dense matter is produced in heavy-ion collisions at relativistic energies. Anisotropic expansion of this matter results in azimuthal asymmetry of particle production relative to the reaction plane, the so-called azimuthal anisotropic flow. Azimuthal anisotropic flow as result of non-central nuclear collisions is a very informative observable closely linked with the properties of the created matter.
To quantify the anisotropic flow a standard Fourier decomposition of the distribution of the azimuthal angle of particles with respect to the n-th harmonic reaction planes is used:
(1) where is the n-th harmonic flow coefficient and is the n-th harmonic symmetry plane determined by the initial geometry of the system. The second harmonic in this decomposition is called the elliptic flow.
2. The NA49 experiment
The scientific goal of the NA49 experiment , which recorded data in 1994-2002, was to create an extended "quark-gluon-plasma" state of strongly interacting matter and search for indications of a critical point. NA49 was one of the fixed target experiments at the CERN SPS accelerator.
Four large volume time projection chambers (TPC) are the main components of the experimental setup . They were used for measurement and identification of charged particle tracks. The ring calorimeter (RCAL) with pseudorapidity acceptance is a cylinder shaped calorimeter, subdivided into 240 cells, configured in 10 radial rings and 24 azimuthal sectors. It was used for transverse energy and elliptic flow measurements.
3. Elliptic flow measurement
The current analysis used Pb+Pb collision events at 40 GeV energy with statistics of 360K events before selection cuts. 120K events passed the interaction vertex selection. Track selection is based on information from the TPCs. Transverse momenta of all charged particles are restricted to GeV/c and pseudorapidity in the range . Only tracks with number of charge clusters in the TPCs of more than 55% of the maximal possible number are used to avoid "track splitting". A cut on the so-called track impact parameter, the distance between the reconstructed main vertex and the back extrapolated track in the target plane and , is applied to reduce the contribution of secondary particles. The number of possible measured points in one of the Vertex TPCs was required 20 and in the MTPC 30. To select tracks with good fit quality a cut is applied. Event classes are based on the TPC multiplicity distribution parameterized with the Modified Wounded Nucleon model also known as MC-Glauber .
Event plane corrections
Evaluation of the symmetry plane angle is based on flow vectors :
(3) where the weights are the transverse momentum of tracks from the TPCs or the energy deposited in the modules of the RCAL and is the azimuthal angle of positively charged particles or the calorimeter cells.
Then, the event plane angle is estimated according to the formula:
Next, the recentering procedure  was applied for each 5% centrality window in order to correct non-uniform detector acceptance:
Finally, the flattening procedure was applied to remove higher order fluctuations :
Event plane resolution correction
Two and three subevent methods were used to determine the event plane resolution correction for elliptic flow measurement from the TPCs and RCAL using the following formulae :
(4) where subevent A and B are random subevents from the TPCs and C a subevent from RCAL. Results for are plotted in Fig. 1.
Elliptic flow coefficients of negatively charged pions measured with the event plane method  are shown in Fig . 2 and Fig . 3 for different event classes and detectors. The values of were calculated as:
(4) where is the azimuthal angle of negatively charged pions, is the event plane angle determined with Eq.(4) with applied corrections from Eqs.(5),(6). Results from TPC and RCAL were further corrected by the resolution factor Eqs.(8),(9) appropriate for the respective detector.
Measured elliptic flow of negatively charged pions from the NA49 experiment is compared with results from the STAR collaboration in Fig. 4. Results on measured by STAR at RHIC  and reanalyzed from NA49 data using the TPC event plane are seen to be consistent. The values of are systematically lower than . Further investigations are required.
Elliptic flow is measured using the event plane method with the event plane estimated from the TPC and RCAL detectors. Detector non-uniformity is corrected using recentering and flattening procedures. Results are compared with published data from the STAR experiment. In future, these results will be used as reference for flow measurements from the lead ion energy scan program of the NA61/SHINE experiment at the CERN SPS .
We thank P. Seyboth for providing information about the RCAL. Also this work was partially supported by the Ministry of Science and Education of the Russian Federation, grant N 3.3380.2017/4.6, and by the National Research Nuclear University MEPhI in the framework of the Russian Academic Excellence Project (contract No. 02.a03.21.0005, 27.08.2013).