# Extended Field Interactions in [PERSON]'s Equation Revision
## 3 Potential Applications
### Intergalactic Gravitational Potential  
The behavior of gravitational potential in a system where the radial component of the vorticity varies inversely with distance from the center is explored in this Section. The assumption that the radial component is proportional to \(1/r\) leads to a solution of the potential in terms of three terms: a term proportional to \(1/r\), a term proportional to \(\ln(r)/r\), and a term proportional to \(r^{2}\). This result sheds light on the behavior of the gravitational potential in idealized and real fluids and is consistent with theoretical predictions and astronomical observations. The obtained intergalactic gravitational potential is consistent with the one caused by dark matter and presents the Navarro-Frenk-White (NFW) profile equation for describing the density distribution of dark matter halos in cosmological simulations. If the source has a gravitational origin, Equation (8) must be written in the form  
\[\partial_{rr}^{2}\Phi+\bigg{(}\frac{2}{r}+\widetilde{\omega}_{r}\bigg{)} \partial_{r}\Phi+\bigg{[}\frac{1}{r^{2}}\partial_{r}(r^{2}\widetilde{\omega} _{r})\bigg{]}\Phi=-4\pi G\rho. \tag{9}\]  
Figure 1: Family of potential curves as a function of distance \(r\). Each curve represents a different case under consideration of basic radial dependencies, such as \(\widetilde{\omega}=\widetilde{\omega}(r)\), illustrating how variations in \(\widetilde{\omega}_{r}\) influence the potential. Arbitrary units. Assuming \(\widetilde{\omega}_{r}=1/r\), the solution for the potential then follows as  
\[\Phi(r)=\frac{c_{1}}{r}+\frac{c_{2}\ln(r)}{r}+\frac{r^{2}}{9} \tag{10}\]  
and its representation in Figure 2.  
Based on this outcome, it can be concluded that the gravitational potential \(\Phi(r)\) is expressible as a composite of three terms: the classical term proportional to \(1/r\), a term proportional to \(\ln(r)/r\), and a term proportional to \(r^{2}\). The coefficients of these terms are given by \(c_{1}\), \(c_{2}\), and \(1/9\), respectively. The assumption that \(\widetilde{\omega}_{r}=1/r\) allowed us to solve the potential \(\Phi(r)\) using the equations governing the behavior of gravitational fields. This result provides insight into the behavior of the gravitational potential in a system where the radial component of the vorticity is assumed to vary inversely with the distance from the center of the system. A similar result was obtained in the work of [PERSON] (2015) in the context of dark matter models, where the gravitational potential's characteristics were explored within the framework of modified gravity theories. [PERSON]'s research, akin to this study, underscores the intricate relationship between field oscillations and gravitational effects, highlighting the adaptability and comprehensiveness of these theoretical frameworks within astrophysical settings. This is consistent with the theoretical predictions and astronomical observations ([PERSON], 2015). In an idealized inviscid fluid, the circulation of a vortex is conserved and does not decay with distance. In a real fluid with viscosity, the circulation can decay with distance due to viscosity effects. In 2D viscous flows, the decay rate of circulation with distance can be approximated as \(1/r\). In 3D flows, the decay rate can be more complex and can depend on the specific flow conditions and properties. A modified version of the gravitational potential equation that accounts for the spread of dark matter in the cosmos may be used to determine the intergalactic gravitational potential caused by dark matter. A useful equation for describing the density distribution of dark matter halos in cosmological simulations is the Navarro-Frenk-White (NFW) profile. The source of the NFW profile is as follows:  
\[\rho(r)=\frac{\rho_{0}}{(r/r_{s})(1+r/r_{s})^{2}}, \tag{11}\]  
Figure 2: Intergalactic gravitational potential \(\Phi(r)\) as a function of the radial distance \(r\) (in arbitrary units).