tokens: and model-independent approach. To compare e red and l orb with the data, we follow the procedure of Forero-Romero etal. ( 2013 ), drawing contours in the e red l orb plane to identify those regions of the parameter space that are compatible with current observations. The contours are generated through 10 7 Monte Carlo (MC) iterations, where at each step the values of velocity, mass and distance are drawn from a Gaussian distribution within the intervals shown in Table 7 . The value of v rad was taken from van der Marel etal. ( 2012 ), r from van der Marel Guhathakurta ( 2008 ), while M MW and M M31 are consistent with van der Marel etal. ( 2012 ) and Boylan-Kolchin etal. ( 2013 ). Due to the large discrepancy existing among the and the MC was ran twice using the two different estimates. It turns out that the intervals generated with are extremely large and possess no constraining power, so that we do not consider them in the following analysis. Table 7. Values and intervals used to generate MC contours. M MW , M M31 are expressed in 10 12 h 1 M units, interhalo distance r in h 1 kpc while v rad and v tan in kms 1 . The intervals on r and v rad correspond to the 2 values of van der Marel etal. ( 2012 ), while for v tan they were chosen in agreement with (Sohn etal. 2012 ). In Fig. 2 , we bin the Mod1 objects in the e red l orb plane and compare them to the observational 95percent confidence intervals. The fraction of objects f LG that falls within those boundaries is shown in the last column of Table 8 . This quantity allows us to gauge the ability of a model to account for the observed global dynamics of the system disregarding its transient state. We note that the number for CDM varies between 0.15 and 0.19 which is consistent with Forero-Romero etal. ( 2013 ), who found a 0.08 to 0.12 fraction at the 68percent confidence interval. In the case of modified gravity cosmologies, the results largely confirm the conclusions obtained with the previous analyses of velocities. In fact, models such as FofR04, FofR05, SymmB and to a lesser extent FofR06 produce LG-like pairs within the observational boundaries at substantially lower rates than CDM. In particular, the fraction of Mod1 LGs falling within the 95percent confidence level is reduced by a factor of 3 in SymmB and around a factor of 2 in FofR04 and FofR05, showing from another perspective that the LG dynamics can only be poorly accounted for within these models. These discrepancies are explained by the increased velocity of M31, affecting the values of e red [through the kinetic energy term in equation ( 30 ) and l orb , which tend to shift the distribution farther away from the region allowed by the data]. Figure 2. LG-like pairs versus MC contours showing the 1 and 2 confidence levels. Pairs have been chosen and MC intervals have been generated using the low v tan estimate of Sohn etal. ( 2012 ). Table 8. Energy, angular momentum, spin parameter and fraction of Mod1 pairs falling within
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