# Self-Gravitating Bose-Einstein Condensates and the Thomas-Fermi Approximation
## 2 Discussion  
A self-gravitating Bose-Einstein condensate is described by a combination of the Gross-Pitaevskii equation and [PERSON]'s equation for gravity. In units such that the BEC particle mass is \(m=1\) and also \(\hbar=1\), the time-independent Gross-Pitaevskii equation can be written in an attractively simple form:  
\[-\frac{1}{2}\
abla^{2}\Psi+(V+c|\Psi|^{2}-\mu)\Psi=0 \tag{1}\]  
where \(\Psi\) is the BEC wavefunction, \(V\) is the gravitational potential, \(c\) is the BEC coupling coefficient and \(\mu\) is the chemical potential, the presence of which guarantees the conservation of energy. We normalize the wavefunction such that the number of particles is \(\int_{V}|\Psi|^{2}=N\). The energy functional from which with the GPE (1) can be derived using the variational principle (cf. [18; 19]; note the additional factor of \(1/2\) in front of \(V\), required to avoid double counting the gravitational potential energy between two regions of the condensate as \(V\) is itself a function of \(|\Psi|^{2}\)) is given by  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]  
In the Thomas-Fermi (TF) approximation [6; 7; 8], kinetic energy is neglected. Therefore, the time-independent GPE takes on the following simplified form:  
\[(V+c|\Psi|^{2}-\mu)\Psi\simeq 0 \tag{3}\]  
If \(V\) is not dependent on \(\Psi\), this equation can be solved directly for \(|\Psi|^{2}\):  
\[|\Psi|^{2}\simeq\frac{\mu-V}{c} \tag{4}\]  
Moreover, if we require the wavefunction to vanish at infinity, we must have  
\[\mu-V\to 0 \tag{5}\]  
at infinity. If \(V\) is dependent on \(\Psi\), the situation becomes somewhat more complicated. In particular, in the GPP system, the relationship between \(V\) and \(\Psi\) is given by [PERSON]'s equation:  
\[\
abla^{2}V=4\pi G|\Psi|^{2} \tag{6}\]  
where \(G\) is the gravitational constant. Solving the GPE (1) for \(V\) in the TF-approximation,  
\[V\simeq\mu-c|\Psi|^{2} \tag{7}\]  
and substituting this solution back into [PERSON]'s Equation (6), we get  
\[\
abla^{2}(\mu-c|\Psi|^{2})\simeq 4\pi G|\Psi|^{2} \tag{8}\]  
If \(\mu=\text{const. }\), we are left with  
\[\left[\
abla^{2}+\frac{4\pi G}{c}\right]|\Psi|^{2}\simeq 0 \tag{9}\]which is an homogeneous Helmholtz-type equation for \(|\Psi|^{2}\), spherically symmetric solutions of which are  
\[|\Psi|^{2}\simeq C_{1}\frac{\sin kr}{r}+C_{2}\frac{\cos kr}{r} \tag{10}\]  
where \(k^{2}=4\pi G/c\), while \(C_{1}\) and \(C_{2}\) are integration constants. To avoid solutions that are singular at the origin \(r=0\), we must set \(C_{2}=0\). On the other hand, \(\sin kr/r\) (and thus, \(|\Psi|^{2}\)) vanishes at \(r=\pi/k\). Therefore, we set \(r_{0}=\pi/k\) as the radius of the condensate. This determines \(C_{1}\) since we require that  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]0  
This integral can be readily evaluated:  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]1  
hence \(C_{1}=N/4r_{0}^{2}\). Therefore, the TF-approximation for the GPP is given by the Lane-Emden type solution  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]2  
for \(0\leq r\leq r_{0}=\sqrt{\pi c/4G}\). Given \(|\Psi|^{2}\), we can solve the GPE (1) for \(V\):  
\[\mathcal{E}=\frac{1}{2}|\
abla\Psi|^{2}+\left(\frac{1}{2}V-\mu\right)|\Psi|^{ 2}+\frac{1}{2}c|\Psi|^{4} \tag{2}\]3  
again for \(0\leq r\leq r_{0}\). At \(r_{0}\) and beyond, the condensate vanishes, and the gravitational potential becomes that of a point mass \(M\) (where \(M=Nm\) is the total mass of the condensate), i.e., \(V=-GN/r\) (\(r_{0}\leq r\)). At the boundary, these two forms must agree.