Figure 1

Illustration of the shooting problem for determining the optimal value of $(A_2{)}_{*}$. This example corresponds to $\chi =\mu =0$ but it is representative of more general cases.Reuse & Permissions

Figure 2

Evolution of $X_c$ as $A_2$ approaches $(A_2{)}_{*}$. We find that the convergence is logarithmic $X_c\propto \ln |A_2-(A_2{)}_{*}|$ so that a huge precision on the value of $A_2$ is needed to obtain a large value of $X_{\max }$. This example corresponds to $\chi =\mu =0$ but it is representative of more general cases.Reuse & Permissions

Figure 3

Evolution of the parameter $\chi$ as a function of $\mu$.Reuse & Permissions

Figure 4

$M$ as a function of $R_{99}$ for fixed $a>0$. The mass is normalized by $M_a$ and the radius by $R_a$. In the noninteracting limit $M\to 0$, we get $M\sim 9.946/R_{99}$. In the TF limit $M\to +\infty$, we obtain $R_{99}\to 2.998$. The system is always stable. The dotted line corresponds to the approximate analytical mass-radius relation $M=2\sigma R/(\nu R^2-6\pi \zeta )$ (with $R_{99}=2.38167R$) based on the Gaussian ansatz [42]. The radius is given as a function of the mass by $R=(\sigma /\nu M)(1+\sqrt{1+6\pi \zeta \nu M^2/{\sigma }^2})$. In the noninteracting limit $M\to 0$, we get $M\sim 2\sigma /\nu R$, i.e., $M^{\mathrm{Gauss}}\sim 8.955/R_{99}$ and in the TF limit $M\to +\infty$, we get $R\to (6\pi \zeta /\nu {)}^{1/2}$, i.e., $R_{99}^{\mathrm{Gauss}}\to 4.125$. The analytical mass-radius relation has the same qualitative shape as the numerical curve and provides a good quantitative agreement in the noninteracting limit. The agreement is less good in the TF limit where the density profile sensibly differs from a Gaussian.Reuse & Permissions

Figure 5

$M$ as a function of $R_{99}$ in log-log plot for fixed $a>0$. The dashed line corresponds to $M\sim 9.946/R_{99}$.Reuse & Permissions

Figure 6

$M$ as a function of $R_{99}$ for fixed $a<0$. The mass is normalized by $M_a$ and the radius by $R_a$. In the noninteracting limit $M\to 0$ and $R_{99}\to +\infty$, we get $M\sim 9.946/R_{99}$. There exists a maximum mass $M_{\max }\simeq 1.012$ corresponding to a radius $R_{99}^{*}\simeq 5.5$. The configurations with small radius $R<R_{*}$ (i.e., $\mu <{\mu }_{*}$ where ${\mu }_{*}=-1.000$ corresponds to the mass peak) are unstable. The dotted line corresponds to the approximate analytical mass-radius relation $M=2\sigma R/(\nu R^2+6\pi \zeta )$ (with $R_{99}=2.38167R$) based on the Gaussian ansatz [42]. The radius is given as a function of the mass by $R=(\sigma /\nu M)(1\pm \sqrt{1-6\pi \zeta \nu M^2/{\sigma }^2})$. In the noninteracting limit $R_{99}\to +\infty$, we get $M\sim 2\sigma /\nu R$, i.e., $M^{\mathrm{Gauss}}\sim 8.955/R_{99}$ and in the nongravitational limit $R\to 0$ (unstable), we get $M\sim \sigma R/(3\pi \zeta )$, i.e., $M^{\mathrm{Gauss}}\sim 0.5262R_{99}$. On the other hand, $M_{\max }^{\mathrm{Gauss}}=\sigma /\sqrt{6\pi \zeta \nu }\simeq 1.085$ and $R_{*}^{\mathrm{Gauss}}=(6\pi \zeta /\nu {)}^{1/2}$, i.e., $(R_{99}^{*}{)}^{\mathrm{Gauss}}\simeq 4.125$. The analytical mass-radius relation has the same qualitative shape as the numerical curve and provides a good quantitative agreement for any values of the mass.Reuse & Permissions

Figure 7

$M$ as a function of $R_{99}$ in log-log plot for fixed $a<0$. The dashed line corresponds to $M\sim 9.946/R_{99}$.Reuse & Permissions

Figure 8

$E$ as a function of $M$ for fixed $a>0$. The eigenenergy is normalized by $E_a^{\prime }$ and the mass by $M_a$. In the noninteracting limit $M\to 0$, we get $E\sim -0.1628M^2$ and in the TF limit $M\to +\infty$, we obtain $E\sim -0.3183M$. These asymptotes are represented in dashed lines. We have also represented in a dotted line the analytic expression obtained from the Gaussian ansatz. It is given by $E=\sigma /R^2+4\pi \zeta M/R^3-2\nu M/R$ where $R$ is related to $M$ by the equation given in the caption of Fig. 4. In the noninteracting limit $M\to 0$, we get $E\sim -(3{\nu }^2/4\sigma )M^2$ yielding $E^{\mathrm{Gauss}}\sim -0.1592M^2$ and in the TF limit $M\to +\infty$, we get $E\sim -(4{\nu }^{3/2}/(3(6\pi \zeta {)}^{1/2}))M$ yielding $E^{\mathrm{Gauss}}\sim -0.3071M$. The agreement is very good for all values of mass and we hardly see the difference between the two curves.Reuse & Permissions

Figure 9

$E$ as a function of $M$ for fixed $a<0$. The eigenenergy is normalized by $E_a^{\prime }$ and the mass by $M_a$. In the noninteracting limit $M\to 0$, we get $E\sim -0.1628M^2$. The eigenenergy corresponding to the maximum mass $M_{\max }\simeq 1.012$ is $E_{*}\simeq -0.36$. We have also represented in a dotted line the analytic expression obtained from the Gaussian ansatz. It is given by $E=\sigma /R^2-4\pi \zeta M/R^3-2\nu M/R$ where $R$ is related to $M$ by the equation given in the caption of Fig. 6. In the noninteracting limit $M\to 0$ with $E\to 0$, we get $E\sim -(3{\nu }^2/4\sigma )M^2$ yielding $E^{\mathrm{Gauss}}\sim -0.1592M^2$ and in the nongravitational limit $M\to 0$ with $E\to -\infty$, we get $E\sim -({\sigma }^3/(3(3\pi \zeta {)}^2))M^{-2}$ yielding $E^{\mathrm{Gauss}}\sim -0.3927/M^2$. The eigenenergy corresponding to the maximum mass $M_{\max }^{\mathrm{Gauss}}\simeq 1.085$ is $E_{*}^{\mathrm{Gauss}}=-5\sigma \nu /(18\pi \zeta )\simeq -0.4166$.Reuse & Permissions

Figure 10

$E_{\mathrm{tot}}$ (obtained with the Gaussian ansatz) as a function of $M$ for fixed $a>0$. The energy is normalized by $E_a$ and the mass by $M_a$. This yields $E_{\mathrm{tot}}=\sigma M/R^2+2\pi \zeta M^2/R^3-\nu M^2/R$ where $R$ is related to $M$ by the equation given in the caption of Fig. 4. In the noninteracting limit $M\to 0$, we get $E_{\mathrm{tot}}\sim -({\nu }^2/4\sigma )M^3$ yielding $E_{\mathrm{tot}}^{\mathrm{Gauss}}\sim -0.05307M^3$ and in the TF limit $M\to +\infty$, we get $E_{\mathrm{tot}}\sim -(2{\nu }^{3/2}/(3(6\pi \zeta {)}^{1/2}))M^2$ yielding $E_{\mathrm{tot}}^{\mathrm{Gauss}}\sim -0.1536M^2$.Reuse & Permissions

Figure 11

$E_{\mathrm{tot}}$ (obtained with the Gaussian ansatz) as a function of $M$ for fixed $a<0$. The energy is normalized by $E_a$ and the mass by $M_a$. This yields $E_{\mathrm{tot}}=\sigma M/R^2-2\pi \zeta M^2/R^3-\nu M^2/R$ where $R$ is related to $M$ by the equation given in the caption of Fig. 6. In the noninteracting limit $M\to 0$ with $E_{\mathrm{tot}}\to 0$, we get $E_{\mathrm{tot}}\sim -({\nu }^2/4\sigma )M^3$ yielding $E_{\mathrm{tot}}^{\mathrm{Gauss}}\sim -0.05307M^3$ and in the nongravitational limit $M\to 0$ with $E_{\mathrm{tot}}\to +\infty$, we get $E_{\mathrm{tot}}\sim ({\sigma }^3/(3(3\pi \zeta {)}^2))M^{-1}$ yielding $E_{\mathrm{tot}}^{\mathrm{Gauss}}\sim 0.3927M^{-1}$. The energy corresponding to the maximum mass $M_{\max }^{\mathrm{Gauss}}\simeq 1.085$ is $E_{\mathrm{tot}}^{\mathrm{Gauss}}=-{\sigma }^2{\nu }^{1/2}/(3(6\pi \zeta {)}^{3/2})\simeq -0.09045$. The curve makes a spike at $M=M_{\max }$ corresponding to $\mu ={\mu }_{*}$. For $M<M_{\max }$, there are two solutions for the same mass but the stable state ($\mu >{\mu }_{*}$) corresponds to the state of lowest total energy $E_{\mathrm{tot}}$, as expected (note that it corresponds to the state of highest eigenenergy $E$, i.e., highest chemical potential $\alpha$, in Fig. 9).Reuse & Permissions

Figure 12

${\rho }_0$ as a function of $M$ for fixed $a>0$. The central density is normalized by ${\rho }_a$ and the mass by $M_a$. In the noninteracting limit $M\to 0$, we get ${\rho }_0\sim 4.400\times {10}^{-3}{M}^4$ and in the TF limit $M\to +\infty$, we get ${\rho }_0\sim 2.533\times {10}^{-2}M$. The system is always stable. We have also represented in a dotted line the analytic expression obtained from the Gaussian ansatz. It is given by ${\rho }_0=M/{\pi }^{3/2}{R}^3$ where $R$ is related to $M$ by the equation given in the caption of Fig. 4 yielding ${\rho }_0=({\nu }^3{M}^4/{\pi }^{3/2}{\sigma }^3)(1+\sqrt{1+6\pi \zeta \nu M^2/{\sigma }^2}{)}^{-3}$. In the noninteracting limit $M\to 0$, we get ${\rho }_0\sim ({\nu }^3/8{\sigma }^3{\pi }^{3/2})M^4$ yielding ${\rho }_0^{\mathrm{Gauss}}\sim 0.003378M^4$ and in the TF limit $M\to +\infty$, we get ${\rho }_0=(\nu /6{\pi }^2\zeta {)}^{3/2}M$ yielding ${\rho }_0^{\mathrm{Gauss}}\sim 0.03456M$.Reuse & Permissions

Figure 13

${\rho }_0$ as a function of $M$ for fixed $a<0$. The central density is normalized by ${\rho }_a$ and the mass by $M_a$. In the noninteracting limit $M\to 0$ with ${\rho }_0\to 0$, we get ${\rho }_0\sim 4.400\times {10}^{-3}{M}^4$. The central density at the point of maximum mass $M_{\max }\simeq 1.012$ is $({\rho }_0{)}_{*}\simeq 0.04$. The configurations with high central density ${\rho }_0>({\rho }_0{)}_{*}$ (corresponding to $\mu <{\mu }_{*}$), located after the mass peak, are unstable. We have also represented in a dotted line the analytic expression obtained from the Gaussian ansatz. It is given by ${\rho }_0=M/{\pi }^{3/2}{R}^3$ where $R$ is related to $M$ by the equation given in the caption of Fig. 6 yielding ${\rho }_0=({\nu }^3{M}^4/{\pi }^{3/2}{\sigma }^3)(1\pm \sqrt{1-6\pi \zeta \nu M^2/{\sigma }^2}{)}^{-3}$. In the noninteracting limit $M\to 0$ with ${\rho }_0\to 0$, we get ${\rho }_0\sim ({\nu }^3/8{\sigma }^3{\pi }^{3/2})M^4$ yielding ${\rho }_0^{\mathrm{Gauss}}\sim 0.003378M^4$. In the nongravitational limit $M\to 0$ with ${\rho }_0\to +\infty$, we get ${\rho }_0\sim (\sigma /3{\pi }^{3/2}\zeta {)}^3{M}^{-2}$ yielding ${\rho }_0^{\mathrm{Gauss}}\sim 0.3535/M^2$. At the point of maximum mass, the central density is $({\rho }_0{)}_{*}^{\mathrm{Gauss}}=\sigma \nu /{\pi }^{3/2}(6\pi \zeta {)}^2=0.03751$Reuse & Permissions

Figure 14

Density profiles $\rho (r)$ along the series of equilibria for fixed $a>0$. The density is normalized by ${\rho }_a$ and the radius by $R_a$. The central density increases as $\mu$ increases. The noninteracting limit corresponds to $\mu \to 0$ and the TF limit to $\mu \to +\infty$. The different curves correspond to $\mu =0.078,0.34,0.69,1$.Reuse & Permissions

Figure 15

Same as Fig. 14 for $\mu =2.86,4.1,5.3,7.3,9.66$. In the TF limit $\mu \to +\infty$, the normalized density profile tends to the asymptotic distribution $\rho (r)=(1/4{\pi }^2)\sin (r)/r$.Reuse & Permissions

Figure 16

Density profiles $\rho (r)$ along the series of equilibria for fixed $a<0$. The density is normalized by ${\rho }_a$ and the radius by $R_a$. The central density increases as $\mu$ decreases. The noninteracting limit corresponds to $\mu \to 0$ and the nongravitational limit to $\mu \to -\infty$. The different curves correspond to $\mu =-0.053$, $-0.55$, $-1.6$, $-2.86$, $-4.4$, $-7.83$, $-14.2$, $-23$. The distribution becomes unstable when $\mu <{\mu }_{*}\simeq -1.000$ corresponding to the turning point of mass (see Sec. 3b).Reuse & Permissions

Figure 17

$R_{99}$ as a function of $a$ for a fixed value of the mass $M$. The radius is normalized by $R_Q$ and the scattering length by $a_Q$. Stable solutions exist for $a\ge a_{\min }\simeq -1.025$ and their radius $R_{99}\ge R_{99}^{*}\simeq 5.6$ is monotonically increasing with $a$. In the noninteracting case $a=0$, $R_{99}\simeq 9.946$ and in the TF limit $a\to +\infty$, $R_{99}\sim 2.998a^{1/2}$. The nongravitational limit corresponds to $a\to 0$ and $R_{99}\to 0$ but these solutions are unstable. The dotted line corresponds to the approximate analytical radius vs scattering length relation $a=(\nu R^2-2\sigma R)/6\pi \zeta$ (with $R_{99}=2.38167R$) based on the Gaussian ansatz [42]. The radius can be expressed in terms of the scattering length as $R=(\sigma /\nu )(1\pm \sqrt{1+6\pi \zeta \nu a/{\sigma }^2})$. In the noninteracting case $a=0$, we get $R\sim 2\sigma /\nu$, i.e., $R_{99}^{\mathrm{Gauss}}\simeq 8.955$ and in the TF limit $a\to +\infty$, we get $R\sim (6\pi \zeta /\nu {)}^{1/2}{a}^{1/2}$, i.e., $R_{99}^{\mathrm{Gauss}}\sim 4.125a^{1/2}$. In the nongravitational limit $R\to 0$, we get $R\sim (3\pi \zeta /\sigma )|a|$, i.e., $R_{99}^{\mathrm{Gauss}}\sim 1.900|a|$. On the other hand $a_{\min }^{\mathrm{Gauss}}=-{\sigma }^2/6\pi \zeta \nu \simeq -1.178$ and $R_{*}=\sigma /\nu$, i.e., $(R_{99}^{*}{)}^{\mathrm{Gauss}}\simeq 4.477$. The analytical radius versus scattering length relation has the same qualitative shape as the numerical curve and provides a good quantitative agreement except in the TF limit where the density profile differs sensibly from a Gaussian.Reuse & Permissions

Figure 18

$R_{99}$ as a function of $a$ in log-log plot for a fixed value of the mass $M$. The dashed line corresponds to $R_{99}\sim 2.998a^{1/2}$ valid in the TF limit.Reuse & Permissions

Figure 19

$E$ as a function of $a$ for a fixed value of the mass $M$. The eigenenergy is normalized by $E_Q^{\prime }$ and the scattering length by $a_Q$. In the noninteracting case $a=0$, we have $E\simeq -0.1628$ and in the TF limit $a\to +\infty$, we get $E\sim -0.3183/a^{1/2}$. The eigenenergy at the point of minimum scattering length $a_{\min }\simeq -1.025$ is $E_{*}\simeq -0.35$. We have also represented in a dotted line the analytic expression obtained from the Gaussian ansatz. It is given by $E=\sigma /R^2+4\pi \zeta a/R^3-2\nu /R$ where $R$ is related to $a$ by the equation given in the caption of Fig. 17. This yields $E=-(\sigma +4\nu R)/3R^2$ or, equivalently, $E=-({\nu }^2/3\sigma )(5\pm 4\sqrt{1+6\pi \zeta \nu a/{\sigma }^2})/(1\pm \sqrt{1+6\pi \zeta \nu a/{\sigma }^2}{)}^2$. In the noninteracting case $a=0$, we get $E\sim -(3{\nu }^2/4\sigma )$ yielding $E^{\mathrm{Gauss}}\simeq -0.1592$ and in the TF limit $a\to +\infty$, we get $E\sim -(4{\nu }^{3/2}/(3(6\pi \zeta {)}^{1/2}))a^{-1/2}$ yielding $E^{\mathrm{Gauss}}\sim -0.3071/a^{1/2}$. In the nongravitational limit $M\to 0$ with $R\to 0$, we get $E\sim -({\sigma }^3/(3(3\pi \zeta {)}^2))a^{-2}$ yielding $E^{\mathrm{Gauss}}\sim -0.3927/a^2$. The eigenenergy corresponding to the minimum scattering length $a_{\min }^{\mathrm{Gauss}}\simeq -1.178$ is $E_{*}^{\mathrm{Gauss}}=-5{\nu }^2/3\sigma \simeq -0.3537$. The analytical eigenenergy versus scattering length relation has the same qualitative shape as the numerical curve and provides a good quantitative agreement for all values of the scattering length $a$.Reuse & Permissions

Figure 20

Eigenenergy $E$ vs scattering length $a$ relation (for $a>0$) in log-log plot for a fixed value of the mass $M$. The dashed line corresponds to $E\sim -0.3183/a^{1/2}$ characterizing the TF limit.Reuse & Permissions

Figure 21

$E_{\mathrm{tot}}$ (obtained from the Gaussian ansatz) as a function of $a$ for a fixed value of the mass $M$. The total energy is normalized by $E_Q$ and the scattering length by $a_Q$. This yields $E_{\mathrm{tot}}=\sigma /R^2+2\pi \zeta a/R^3-\nu /R$ where $R$ is related to $a$ by the equation given in the caption of Fig. 17. This yields $E_{\mathrm{tot}}=(\sigma -2\nu R)/3R^2$ or, equivalently, $E_{\mathrm{tot}}=-({\nu }^2/3\sigma )(1\pm 2\sqrt{1+6\pi \zeta \nu a/{\sigma }^2})/(1\pm \sqrt{1+6\pi \zeta \nu a/{\sigma }^2}{)}^2$. In the noninteracting case $a=0$, we get $E_{\mathrm{tot}}=-{\nu }^2/(4\sigma )\simeq -0.05305$ and in the TF limit $a\to +\infty$, we get $E_{\mathrm{tot}}\sim -(2{\nu }^{3/2}/(3(6\pi \zeta {)}^{1/2}))a^{-1/2}\simeq -0.1536/\sqrt{a}$. In the nongravitational limit $a\to 0$, we get $E_{\mathrm{tot}}\sim ({\sigma }^3/(3(3\pi \zeta {)}^2))a^{-2}\simeq 0.3927/a^2$. The total energy at the point of minimum scattering length is $E_{\mathrm{tot}}=-{\nu }^2/3\sigma \simeq -0.07074$.Reuse & Permissions

Figure 22

${\rho }_0$ as a function of $a$ for a fixed value of the mass $M$. The density is normalized by ${\rho }_Q$ and the scattering length by $a_Q$. In the noninteracting case $a=0$, we get ${\rho }_0\sim 4.400\times {10}^{-3}$ and in the TF limit $a\to +\infty$, we get ${\rho }_0\sim 2.533\times {10}^{-2}/a^{3/2}$. The central density at the point of minimum scattering length $a_{\min }\simeq -1.025$ is $({\rho }_0{)}_{*}\simeq 0.04$. We have also represented in dotted line the analytic expression obtained from the Gaussian ansatz. It is given by ${\rho }_0=1/{\pi }^{3/2}{R}^3$ where $R$ is related to $a$ by the equation given in the caption of Fig. 17. This yields ${\rho }_0=({\nu }^3/{\pi }^{3/2}{\sigma }^3)/(1\pm \sqrt{1+6\pi \zeta \nu a/{\sigma }^2}{)}^3$. In the noninteracting case $a=0$ we get ${\rho }_0\sim ({\nu }^3/8{\sigma }^3{\pi }^{3/2})$ yielding ${\rho }_0^{\mathrm{Gauss}}\sim 0.003378$, in the TF limit $a\to +\infty$ we get ${\rho }_0=(\nu /6{\pi }^2\zeta {)}^{3/2}{a}^{-3/2}$ yielding ${\rho }_0^{\mathrm{Gauss}}\sim 0.03456/a^{3/2}$ and in the nongravitational limit $a\to 0$ with ${\rho }_0\to +\infty$ we get ${\rho }_0\sim (\sigma /3{\pi }^{3/2}\zeta {)}^3/|a{|}^3$ yielding ${\rho }_0^{\mathrm{Gauss}}\sim 0.3535/|a{|}^3$. Finally, the central density at the point of minimum scattering length is $({\rho }_0{)}_{*}^{\mathrm{Gauss}}={\nu }^3/{\pi }^{3/2}{\sigma }^3=0.02703$. The approximate analytical curve has the same qualitative shape as the numerical curve and provides a good quantitative agreement for all values of the scattering length. The most severe deviation occurs close to the minimum scattering length value but it remains weak.Reuse & Permissions

Figure 23

${\rho }_0$ as a function of $a$ (for $a>0$) in log-log plot for a fixed value of the mass $M$. The dashed line corresponds to ${\rho }_0\sim 2.533\times {10}^{-2}/a^{3/2}$ characterizing the TF limit.Reuse & Permissions

Figure 24

Density profiles $\rho (r)$ along the series of equilibria for a fixed value of the total mass $M$. The density is normalized by ${\rho }_Q$ and the radial distance by $R_Q$. The central density decreases as $\mu$ increases. The nongravitational limit corresponds to $\mu \to -\infty$, the noninteracting case to $\mu =0$ and the TF limit to $\mu \to +\infty$. We have represented $\mu =-2.86$, $-2.55$, $-1.6$, $-0.55$, $-0.053$ (negative scattering lengths). The distribution becomes stable when $\mu >{\mu }_{*}\simeq -1.000$ corresponding to the turning point of scattering length (see Sec. 3b).Reuse & Permissions

Figure 25

Same as Fig. 24 for $\mu =0,0.078,0.34,0.69,1$ (positive scattering lengths).Reuse & Permissions

Figure 26

Same as Fig. 24 for $\mu =4.1,4.4,5.3,7.3$ (positive scattering lengths).Reuse & Permissions