Bondi flow revisited

Abstract

Newtonian spherically symmetric transonic accretion is studied by including the mass of the accreting matter, while considering the growth of the accretor itself to be negligibly small. A novel iterative method is introduced to accomplish that task. It is demonstrated that the inclusion of the mass of the fluid changes the critical properties of the flow as well as the topological phase portraits of the stationary integral solution. The changes are small in the framework of this methodology. It is shown that to get large changes one has to develop a new method.

Appendix A

For adiabatic flow the changes in the critical parameters are found to be small in the frame of method of iteration. Mathematically

$$\begin{aligned} \frac{\delta x_{b}}{x_{b}}\ll 1,\qquad \frac{\delta\lambda_{b}}{\lambda_{b}}\ll 1,\qquad \frac{\delta z_{b}}{z_{b}}\ll 1 \end{aligned}$$

Now using expression $\lambda_{b}^{2}=z_{b}^{(\gamma+1)}x^{4}$ in Sect. 3,

$$ 2\frac{\delta\lambda_{b}}{\lambda_{b}}=(\gamma+1)\frac{\delta z_{b}}{z_{b}}+4\frac{\delta x_{b}}{x_{b}} $$

(56)

Multiplying the above expression by 100,

$$ 2h=(\gamma+1)g-4f $$

(57)

This the relation among the percentage change in critical parameters. Rearranging,

$$ g=\frac{2}{(\gamma+1)}(h+2f) $$

(58)

In the range of $\gamma$ (1.33 to 1.66), $g$ is always greater than $f$. Numerical results show that $g$ is also greater than $h$. Numerical results show that $f$ does not depend on $\gamma$ and $h$ decreases with $\gamma$ and from the above expressions, $g$ decreases with $\gamma$.

Similarly for isothermal flow using $\lambda_{b}=x_{b}^{2}z_{b}$,

$$ h=2f+g $$

(59)

Our previous analysis shows that $f$ is negligibly small, so

$$ h\approx g $$

(60)