# Self-Gravitating Bose-Einstein Condensates and the Thomas-Fermi Approximation
## 2 Discussion
This is clearly not the case when the TF-approximation is used for a self-gravitating condensate. Therefore, we now aim to find an approximate solution of the GPP system in the spherically symmetric case without resorting to the TF-approximation (see also [7] for another approximation and [8] for a numeric solution in terms of hydrodynamics using the Madelung-representation). Assuming spherical coordinates and a spherically symmetric condensate, the GPE (1) becomes an equation of a single independent variable \(r\). Let us denote \(\partial/\partial r=\partial_{r}\) for brevity, while noting the form of the Laplacian in spherical coordinates, \(\
abla^{2}=\partial_{r}^{2}+(2/r)\partial_{r}\). We can then write the GPE (1) as  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]3  
whereas Poisson's Equation (6) becomes:  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]4  
Let us now consider writing the wavefunction as  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]5  
in which case  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]6 \[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]7  
and the GPE (1) can be rewritten, after dividing through by \(e^{i\phi}\), as  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]8  
Since \(|\Psi|\), \(\phi\), \(V\), \(c\) and \(\mu\) are all real, the real and imaginary parts of this equation can be separated:  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]9 \[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]0  
Equation (29) can be integrated:  
\[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]1  
where \(C\) is an integration constant with the dimensions of \(r|\Psi|^{2}\). It seems that \(C=0\) is not only a valid choice but the only choice that does not result in \(\phi\) becoming singular at the origin (assuming the wavefunction does not vanish at the origin). Therefore, \(\phi=\text{const}\). Under these circumstances, the GPE (1) will read  
\[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]2  
which is readily solvable for \(V\) algebraically:  
\[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]3  
This result can be substituted back into [PERSON]'s equation, yielding a fourth-order ordinary differential equation in \(|\Psi|\):\[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]4 \[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]5  
This equation can be solved numerically. Given that it is a fourth-order homogeneous differential equation in \(|\Psi|\), it has a very large solution space, parameterized by boundary or initial conditions, such as the values of \(|\Psi|\) and its first three derivatives at some value of \(r\). A hint for a suitable solution comes from numerical simulation [20], where we find that apparently stable nonrotating spherically symmetric solutions converge on \(|\Psi|^{2}\propto[\sin(r/r_{0})/(r/r0)]^{2\alpha}\), with \(3\lesssim\alpha\lesssim 4\). This approximate solution has many desirable properties. It is smooth in the interval \(0\geq r/r_{0}\geq\pi\). The corresponding kinetic energy (21) is finite in the same interval. Moreover, it is possible to compute the condensate mass, which is given by  
\[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]6  
and this, too, is finite and well-behaved. Therefore, we find that the following initial approximation for the magnitude of the BEC-Poisson wavefunction:  
\[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]7  
agrees well with a numerical solution (except for very small values of \(r\)), as shown in \[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]8 Furthermore, this choice yields a corresponding solution of Equation (6) for the gravitational potential that is finite, negative, and vanishes at infinity, as expected. The stability of these solutions is confirmed by numerical simulation of a condensate using Equation (35) as an initial approximation.