Recent observations constrained the tangential velocity of M31 with respect to the Milky Way ( MW ) to be v _ { M 31 ,tan } < 34.4 km s ^ { -1 } and the radial velocity to be in the range v _ { M 31 ,rad } = -109 \pm 4.4 km s ^ { -1 } ( 29 ) .
In this study we use a large volume high resolution N-body cosmological simulation ( Bolshoi ) together with three constrained simulations to statistically study this kinematics in the context of the \Lambda CDM .
The comparison of the ensembles of simulated pairs with the observed LG at the 1- \sigma level in the uncertainties has been done with respect to the radial and tangential velocities , the reduced orbital energy ( e _ { tot } ) , angular momentum ( l _ { orb } ) and the dimensionless spin parameter , \lambda .
Our main results are : ( i ) the preferred radial and tangential velocities for pairs in \Lambda CDM are v _ { r } = -80 \pm 20 km s ^ { -1 } , v _ { t } = 50 \pm 10 km s ^ { -1 } , ( ii ) pairs around that region are 3 to 13 times more common than pairs within the observational values , ( iii ) 15 \% to 24 \% of LG-like pairs in \Lambda CDM have energy and angular momentum consistent with observations while ( iv ) 9 \% to 13 \% of pairs in the same sample show similar values in the inferred dimensionless spin parameter .
It follows that within current observational uncertainties the quasi-conserved quantities that characterize the orbit of the LG , i.e . e _ { tot } , l _ { orb } and \lambda , do not challenge the standard \Lambda CDM model , but the model is in tension with regard to the actual values of the radial and tangential velocities .
This might hint to a problem of the \Lambda CDM model to reproduce the observed LG .