Using a cosmological CDM simulation, we analyze the differences between the widely-used spin pa- rameters suggested by Peebles and Bullock. The dimensionless spin parameter λ proposed by Peebles is theoretically well-justified but includes an annoying term, the potential energy, which cannot be directly obtained from observations and is computationally expensive to calculate in numerical simulations. The Bullock’s spin parameter λ′ avoids this problem assuming the isothermal density profile of a virialized halo in the Newtonian potential model. However, we find that there exists a substantial discrepancy between λ and λ′ depending on the adopted potential model (Newtonian or Plummer) to calculate the halo total energy and that their redshift evolutions differ to each other significantly. Therefore, we introduce a new spin parameter, λ′′, which is simply designed to roughly recover the value of λ but to use the same halo quantities as used in λ′. If the Plummer potential is adopted, the λ′′ is related to the Bullock’s definition as λ′′ = 0.80 × (1 + z)−1/12λ′. Hence, the new spin parameter λ′′ distribution becomes consistent with a log-normal distribution frequently seen for the λ′ while its mean value is much closer to that of λ. On the other hand, in case of the Newtonian potential model, we obtain the relation of λ′′ = (1 + z)−1/8λ′; there is no significant difference at z = 0 as found by others but λ′ becomes more overestimated than λ or λ′′ at higher redshifts. We also investigate the dependence of halo spin parameters on halo mass and redshift. We clearly show that although the λ′ for small-mass halos with Mh < 2× 1012M⊙ seems redshift independent after z = 1, all the spin parameters explored, on the whole, show a stronger correlation with the increasing halo mass at higher redshifts.

We investigate the dependence of the extended X-ray emission from the halos of optically luminous early-type galaxies on the small-scale (the nearest neighbor distance) and large-scale (the average density inside the 20 nearest galaxies) environments. We cross-match the 3rd Data Release of the Second XMMNewton Serendipitous Source Catalog (2XMMi-DR3) to a volume-limited sample of the Sloan Digital Sky Survey (SDSS) Data Release 7 with Mr < −19.5 and 0.020 < z < 0.085, and find 20 early-type galaxies that have extended X-ray detections. The X-ray luminosity of the galaxies is found to have a tighter correlation with the optical and near infrared luminosities when the galaxy is situated in the low large-scale density region than in the high large-scale density region. Furthermore, the X-ray to optical (r-band) luminosity ratio, LX/Lr, shows a clear correlation with the distance to the nearest neighbor and with large-scale density environment only where the galaxies in pair interact hydrodynamically with seperations of rp < rvir. These findings indicate that the galaxies in the high local density region have other mechanisms that are responsible for their halo X-ray luminosities than the current presence of a close encounter, or alternatively, in the high local density region the cooling time of the heated gas halo is longer than the typical time between the subsequent encounters.