We developed a mathematical model to derive time scales and the presence of BS stars. The model is based on the variation of mass through a circle into the cluster defined by a radius, and at a time; this mass cross is translated into a differential equation that it can be integrated for a given radius (r) and a determined time (t). From this equation we can derive the different time scales that allows us to reach conclusions like: clusters not containing blue strugglers (BS) stars disappear younger than those clusters containing BS. In clusters containing BS stars, the volume which takes up half of the cluster mass is bigger than the one corresponding to clusters without BS stars but the time to catch it up is shorter. We also studied, by means of this equation, the core collapse of stars of the cluster and the region where this concentration is stopped/retained; this region is identified by means of the relation $c/ch$, being $c=\log(rt/rc)$ and $ch=\log(rc/rh)$. Where rt and rc are the tidal and the core radius respectively, and rh is the radius where half of the cluster mass is concentrated. The model also drove us to the conclusion that the number of the blue straggler stars in a cluster follows a distribution function whose components are the ratio between relaxation time and the age, labelled as $\it f$, and a factor, named $\varpi$, which is an indicator of the origin of the BS; $\varpi$ increases as the number of BS increase but it is limited to$\sim$5.0. The mentioned distribution function is expressed as $\it NBS$ $\sim$ $\it f^3$($\frac{1}{e^{\frac{f}{\varpi}}-1}$). The validity of this function was carried out by means of matching the number of observed blue straggler (BS) stars to the number of predicted ones in the available sample of OC.

The James Webb Space Telescope (JWST) has made startling discoveries regarding the early universe. It has revealed galaxies as soon as 300 million years after the Big Bang, challenging current galaxy formation models. Additionally, it has identified massive, bright galaxies in the young universe, contradicting the standard {\Lambda}CDM model's age estimate of 13.8 Gyr. This prompts a reevaluation of galaxy formation and cosmological models. There is a strong tension between JWST high-redshift galaxy observations and Planck Cosmic Microwave Background (CMB) satellite measurements. Even alternative cosmological models, including those incorporating dark matter baryon interaction, f(R) gravity, and dynamical dark have failed to resolve this tension. One possible solution is that the Universe's age exceeds predictions by the {\Lambda}CDM model. The study challenges this by introducing a method based on blue straggler stars (BSs) within GCs, comparing ages with other models. The ages obtained are compared with those of different models to certify that they are equally valid. These values are comparable within the error ranges except for the clusters: NGC104, NGC 5634, IC 4499, NGC 6273 and NGC 4833, finding their respective ages to be between 14.7 and 21.6 Gyr, surpassing the commonly accepted age of the Universe. These results inferred an age for the Universe of around 26 Gyr, close to 26.7 Gyr. This value aligns with that suggested by the cosmological model named Covarying Coupling Constants + TL (CCC+TL). Such a value is consistent with early universe observations from the James Webb Space Telescope (JWST). The results of the present paper reinforce the advocating for a critical review of models encompassing dark mass, dark energy, and the dynamics of the Universe, particularly in explaining the presence of primitive massive galaxies, very old GCs, and very old and poor metallic stars.

This work aims to investigate the behaviour of the lithium abundance in stars with and without detected planets. Our study is based on a sample of 1332 FGK main-sequence stars with measured lithium abundances, for 257 of which planets were detected. Our method reviews the sample statistics and is addressed specifically to the influence of tides and orbital decay, with special attention to planets on close orbits, whose stellar rotational velocity is higher than the orbital period of the planet. In this case, tidal effects are much more pronounced. The analysis also covers the orbital decay on a short timescale, with planets spiralling into their parent star. Furthermore, the sample allows us to study the relation between the presence of planets and the physical properties of their host stars, such as the chromospheric activity, metallicity, and lithium abundance. In the case of a strong tidal influence, we cannot infer from any of the studies described that the behaviour of Li differs between stars that host planets and those that do not. Our sample includes stars with super-solar metallicity ([Fe/H]>0.15 dex) and a low lithium abundance (A(Li) <1.0 dex). This enabled us to analyse scenarios of the origin and existence of these stars. Considering the possible explanation of the F dip, we show that it is not a plausible scenario. Our analysis is based on a kinematic study and concludes that the possible time that elapsed in the travel from their birth places in the central regions of the Galaxy to their current positions in the solar neighbourhood is not enough to explain the high lithium depletion. It is remarkable that those of our high-metallicity low-lithium stars with the greatest eccentricity (e>0.2) are closest to the Galactic centre. A dedicated study of a set of high-metallicity low-Li stars is needed to test the migration-depletion scenario.