The prolate Bok globules evidence for the existence of dark matter sub-halo

Abstract

Although the existence of dark matter is generally accepted by the mainstream scientific community, there is no generally agreed direct detection of it. Also, observations show that some Bok globules are prolate in some regions without suitable explanation for its cause. In this paper, we investigate the effect of dark matter sub-halo in transformation of the Bok globules from spherical to the prolate shape. We limit the investigation to a particular case that the magnetic field and turbulent effects are negligible through the Bok globule. We consider the gravitational effect of dark matter sub-halo on the isothermal Bok globule that is exposed to suitable distance of it. The results show that the dark matter sub-halo can justify the transformation of Bok globules in some regions. In this paper, we introduce a new method for proving the existence of dark matter sub halo.

Appendix: The finite difference method

In this appendix, we describe the numerical technique, which is used to solve Eq. (11). Here, we use an equally spaced mesh points. The number of points for r-component are M+1, and for θ-component are N+1, and spacing between points are h _r and h _θ , respectively. Also, index j is used for r-component and n is used for θ-component. In $\tilde{r}=0$, we have

$$ \tilde{\rho}(0,n)=1, $$

(14)

and since its gradient at $\tilde{r}=0$ is zero, we have

$$ \begin{aligned} &\frac{\tilde{\rho}(1,n)-\tilde{\rho}(0,n)}{h_r}=0 \\ &\therefore\tilde{\rho}(1,n)=1. \end{aligned} $$

(15)

For other mesh points, density can be obtained from Eq. (11) as follows

$$\begin{aligned} &{\tilde{\rho}(j+1,n)} \\ &{\quad=2\tilde{\rho}(j,n)-h_r^2 \biggl(\frac{2}{r_j} \biggl[\frac{\tilde{\rho}(j,n) -\tilde{\rho}(j-1,n)}{h_r} \biggr]} \\ &{\qquad{} -\frac{1}{\tilde{\rho}(j,n)} \biggl[\frac{\tilde{\rho}(j,n)-\tilde{\rho}(j-1,n)}{h_r} \biggr]^2 \biggr)} \\ &{\qquad{}-h_r^2\frac{\partial \tilde{\phi^*}(j,n)}{\partial \tilde{r}_j} \biggl[ \frac{\tilde{\rho}(j,n)-\tilde{\rho}(j-1,n)}{h_r} \biggr]} \\ &{\qquad{} - \frac{h_r^2}{r_j^2} \biggl(\frac{\partial\tilde {\phi^*}(j,n)}{\partial \theta_n}\biggl( \frac{\tilde{\rho}(j,n+1)-\tilde{\rho}(j,n)}{h_{\theta}}\biggr) \biggr)} \\ &{\qquad{}-h_r^2\alpha\bigl(\tilde{\rho}(j,n) \bigr)^2} \\ &{\qquad{} +\frac{h_r^2}{r_j^2\tilde{\rho}(j,n)} \biggl(\frac {\tilde{\rho}(j,n+1)-\tilde{\rho}(j,n)}{h_{\theta}} \biggr)^2} \\ &{\qquad{} -\frac{h_r^2}{r_j^2} \biggl[\frac{\tilde{\rho}(j,n+1) -2\tilde{\rho}(j,n)+\tilde{\rho}(j,n-1)}{h^2_{\theta}} \biggr]} \\ &{\qquad{} -h_r^2\tilde{\rho}(j,n) \biggl[ \frac{\partial^2\tilde{\phi^*}(j,n)}{\partial\tilde{r_j}^2} +\frac{2}{\tilde{r_j}}\frac{\partial\tilde{\phi^*}(j,n)}{\partial\tilde{r_j}}} \\ &{\qquad{} +\frac{1}{\tilde{r_j}^2} \frac{\partial^2\tilde{\phi^*}(j,n)}{\partial\theta_n^2} +\frac{\cot\theta_n}{\tilde{r_j}^2}\frac{\partial\tilde{\phi ^*}(j,n)}{\partial\theta_n} \biggr]-\tilde{ \rho}(j-1,n)} \\ &{\qquad{} -h_r^2\frac{\cot\theta_n}{r_j^2} \biggl[ \frac{\tilde{\rho}(j,n+1)-\tilde{\rho}(j,n)}{h_{\theta}} \biggr].} \end{aligned}$$

(16)

For example, we can write $\tilde{\rho}(2,n)$ as follows

$$\begin{aligned} \tilde{\rho}(2,n) =&1-h_r^2 \biggl[\alpha+ \frac{\partial^2\tilde{\phi^*}(1,n)}{\partial\tilde{r_1}^2} +\frac{2}{\tilde{r_1}}\frac{\partial\tilde{\phi^*}(1,n)}{\partial\tilde{r_1}} \\ & {} +\frac{1}{\tilde{r_1}^2} \frac{\partial^2\tilde{\phi^*}(1,n)}{\partial\theta_n^2} +\frac{\cot \theta_n}{\tilde{r_1}^2}\frac{\partial\tilde{\phi^*}(1,n)}{\partial \theta_n} \biggr], \end{aligned}$$

(17)

where is related only to $\tilde{\phi^{*}}(1,n)$. Knowing $\tilde{\rho}(2,n)$, we can find $\tilde{\rho}(3,n)$ and so on.