Abstract
We extended the modified Lemaitre-Tolman model taking into account the effect of angular momentum and dynamical friction. The inclusion of these quantities in the equation of motion modifies the evolution of a perturbation, initially moving with the Hubble flow. Solving the equation of motions we got the relationships between mass, $M$, and the turn-around radius, $R_0$. Knowing $R_0$, the quoted relation allows the determination of the mass of the object studied. The relationships for the case in which also the angular momentum is taken into account gives a mass $\simeq 90$ \% larger than the standard Lemaitre-Tolman model, and two times the value of the standard Lemaitre-Tolman model, in the case also dynamical friction is taken into account. As a second step, we found relationships between the velocity, $v$, and radius, $R$, and fitted them to data of the Local Group, M81, NGC 253, IC342, CenA/M83, and to the Virgo clusters obtained by Ref.[New Astronomy 11(4):325, A&A 488(3):845]. This allowed us to find optimized values of the mass and Hubble constant of the objects studied. The fit gives values of the masses smaller with respect to the $M-R_0$ relationship method, but in any case 30-40\% larger than the $v-R$ relationship obtained from the standard Lemaitre-Tolman model. Differently from mass, the Hubble parameter becomes smaller with respect to the standard Lemaitre-Tolman model, when angular momentum, and dynamical friction are introduced. This is in agreement with Ref.[New Astronomy 11(4):325, A&A 488(3):845], who improved the standard Lemaitre-Tolman model taking into account the cosmological constant. Finally, we used the mass, $M$, and $R_0$ of the studied objects to put constraints to the dark energy equation of state parameter, $w$. Comparison with previous studies show different constraints on $w$.
Highlights
While the mass-to-light (M/L) ratios of group of galaxies was in the past estimated through the virial theorem to be typically of the order of ≃170M⊙/LB,M⊙ [1], new measurements based on high quality data, and estimating methods different from the Virial theorem [2] give much smaller results in the range10–30M⊙/LB,M⊙
The relationships for the case in which the angular momentum is taken into account gives a mass ≃ 90% larger than the standard Lemaitre– Tolman model, and two times the value of the standard Lemaitre–Tolman model, in the case dynamical friction is taken into account
By means of methods used by observers, [3] showed that ≃20% of the studied groups were not gravitationally bound. [4,5] proposed an alternative approach to the virial theorem based on the Lemaitre–Tolman (LT) model [6,7] giving a good description of a central core gravitationally bound located inside an homogeneous region whose density decreases till reaching the background value
Summary
While the mass-to-light (M/L) ratios of group of galaxies was in the past estimated through the virial theorem to be typically of the order of ≃170M⊙/LB,M⊙ [1], new measurements based on high quality data, and estimating methods different from the Virial theorem [2] give much smaller results in the range. We will further extend the modified Lemaitre–Tolman (MLT model) by taking into account the effect of angular momentum (JLT model) and dynamical friction (JηLT model) The effect of these two quantities on the spherical collapse model (SCM) and its effect on the clusters of galaxies structure and evolution, the turn-around, the threshold of collapse, their mass function, their mass–temperature relation, have been studied in [13,14,15,16,17,18,19,20,21]. The model describes the evolution of a spherical symmetric over density which initially expands with the Hubble flow, detaches from it, when the density overcomes a critical value, reaches a maximum radius, dubbed turn-around radius, and collapse and virialize.
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