# Self-Gravitating Bose-Einstein Condensates and the Thomas-Fermi Approximation
## 2 Discussion
This can be achieved by setting  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]4  
This clarifies the role of the chemical potential in the case of the GPP system in the TF-approximation: its presence ensures that the gravitational potential takes on the standard form outside the condensate and vanishes at infinity. The energy density of the time-independent GPE in the Thomas-Fermi limit is given by  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]5  
or, after substituting the solution for \(V\) from the GPE (1),  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]6  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]7To find the total energy, we integrate over the condensate volume:  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]8 \[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]9  
or, after restoring units,  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]0  
The positive sign of the total energy implies that the solution for a self-gravitating BEC using the TF-approximation is inherently unstable. This result is based on the assumption that the kinetic energy can be neglected. Now that we have an explicit solution for \(|\Psi|^{2}\), this assumption can be verified by direct substitution into the energy functional (2). When we do so we find that, using the solution given by Equation (13), the condensate kinetic energy,  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]1 \[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]2  
is divergent for any \(r_{0}>0\). The implication of this divergence is that the assumption behind the TF-approximation, namely that the kinetic term in the GPE (1) can be neglected, is maximally violated. The divergence of the kinetic energy density (represented by the integrand in Equation (21)) in the Thomas-Fermi approximation, near the condensate surface, and the logarithmic divergence of the total kinetic energy are well known (the author wishes to thank the anonymous referees and the Academic Editor of _Galaxies_ for stressing this point) [7; 8; 17; 18]. However, in the case of self-gravitating systems, as we have seen, the problem gets worse: the total energy of a self-gravitating condensate is positive, hence the condensate is gravitationally unstable. This has especially important implications for numerical simulations of self-gravitating Bose-Einstein condensates that use this approximation to model the initial state (see, e.g., [16; 20]). Not only is the magnitude of the total kinetic energy ill-defined (which makes it dependent on nonphysical simulation parameters, such as the numerical integration step size or even small rounding errors), the positive total energy is especially troublesome, as the stability of a self-gravitating system is dependent on \(E<0\). This finding agrees with the author's experience using numerical simulation code [20] that was designed to model the (time-dependent) GPP system, using the TF-approximation to model the initial condensate density. It was perplexing that versions of the code ran differently in different programming environments (e.g., single vs. double precision, FORTRAN vs. C), processors (Intel x86 vs. GPGPU) and operating systems (Linux vs. Windows). This is clearly not permissible: the results, apart from accuracy and rounding issues, should not be dependent on such factors. Indeed, the present study arose as a result of systematically analyzing the failure of these algorithms to produce consistent results. Numerically stable simulations require an initial state that is not dependent on nonphysical parameters and has a well-defined total energy that is consistent with stability conditions.