A study on the formation of field, binary or multiple stars: a 2D approach through dynamical system

Abstract

A dynamical model has been developed to envisage the star formation scenario in filamentary molecular clouds. In the present work a double well potential has been considered to find the number of stable stationary points as the origin of field star, binary stars or stellar associations. It is found that low density structure can lead to the formation of field stars while an intermediate density structure is favourable for formation of binary stars. Stellar associations are generally prevalent in dense filamentary molecular clouds.

Appendix

1.1 6.1 Sufficient conditions for existence of the stationary points (iv)

Case 1: Let $\lambda C >0$ Therefore either $\lambda >0$ and $C >0$ i.e., $\lambda >0$, and $a^{2}_{x y} - a_{x x} a_{y y} > 0$ or, $\lambda < 0$ and $C < 0$ i.e., $\lambda < 0$, and $a^{2}_{x y} - a_{x x} a_{y y} < 0$. Then these stationary points (iv) are real iff $A > 0$ and $B>0$.

(i) When $\lambda >0$, and $a^{2}_{x y} - a_{x x} a_{y y} > 0$, then $A > 0$ implies $Z_{y} a_{x y} - Z_{x} a_{y y} > 0$, that means $\frac{Z_{x}}{Z_{y}} < \frac{a_{x y}}{a_{y y}}$, if $Z_{x}$, $Z_{y}$, $a_{x y}$, and $a_{y y}$ are positive

Similarly, $B > 0$ implies $-Z_{y} a_{x x}+ Z_{x} a_{x y} > 0$, that means $\frac{Z_{x}}{Z_{y}} > \frac{a_{x x}}{a_{x y}}$, if $Z_{x}$, $Z_{y}$, $a_{x y}$, and $a_{x x}$ are positive. So, combining these two we have $\lambda$, $Z_{x}$, $Z_{y}$, $a_{x x}$, $a_{x y}$, and $a_{y y}$ are all positive, and $\frac{a_{x x}}{a_{x y}} < \frac{Z_{x}}{Z_{y}} < \frac{a_{x y}}{a_{y y}}$.

(ii) When $\lambda < 0$, and $a^{2}_{x y} - a_{x x} a_{y y} < 0$, then $A > 0$ and $B >0$ implies $\lambda > 0$, and $\frac{a_{x x}}{a_{x y}} < \frac{Z_{x}}{Z_{y}} < \frac{a_{x y}}{a_{y y}}$, which does not satisfy the condition $a^{2}_{x y} - a_{x x} a_{y y} < 0$. So, here we can not get real points.

Case 2: Let $\lambda C < 0$. Therefore either $\lambda >0$ and $C < 0$ i.e., $\lambda >0$, and $a^{2}_{x y} - a_{x x} a_{y y} < 0$ or, $\lambda < 0$ and $C > 0$ i.e., $\lambda < 0$, and $a^{2}_{x y} - a_{x x} a_{y y} > 0$. Then these stationary points (iv) are real iff $A < 0$ and $B < 0$.

(i) When $\lambda >0$, and $a^{2}_{x y} - a_{x x} a_{y y} < 0$, then $A < 0$ implies $Z_{y} a_{x y} - Z_{x} a_{y y} < 0$, that means $\frac{Z_{x}}{Z_{y}} > \frac{a_{x y}}{a_{y y}}$, if $Z_{x}$, $Z_{y}$, $a_{x y}$, and $a_{y y}$ are positive.

Similarly, $B < 0$ implies $-Z_{y} a_{x x}+ Z_{x} a_{x y} < 0$, that means $\frac{Z_{x}}{Z_{y}} < \frac{a_{x x}}{a_{x y}}$, if $Z_{x}$, $Z_{y}$, $a_{x y}$, and $a_{x x}$ are positive. So, combining these two we have $\lambda > 0$, and $\frac{a_{x y}}{a_{y y}} < \frac{Z_{x}}{Z_{y}} < \frac{a_{x x}}{a_{x y}}$, if $Z_{x}$, $Z_{y}$, $a_{x x}$, $a_{x y}$, and $a_{y y}$ are all positive

(ii) When $\lambda < 0$, and $a^{2}_{x y} - a_{x x} a_{y y} > 0$, then $A < 0$ and $B < 0$ implies $\lambda < 0$ and, $\frac{a_{x y}}{a_{y y}} < \frac{Z_{x}}{Z_{y}} < \frac{a_{x x}}{a_{x y}}$, which does not satisfy the condition $a^{2}_{x y} - a_{x x} a_{y y} > 0$. So, here we can not get real points. So, final two conditions for positive $\lambda$, $Z_{x}$, $Z_{y}$, $a_{x x}$, $a_{x y}$, and $a_{y y}$ are $\frac{a_{x x}}{a_{x y}} < \frac{Z_{x}}{Z_{y}} < \frac{a_{x y}}{a_{y y}}$, or $\frac{a_{x y}}{a_{y y}} < \frac{Z_{x}}{Z_{y}} < \frac{a_{x x}}{a_{x y}}$.

1.2 6.2 Scaling

$\ddot{X}= 2 Z_{X}X -4\lambda (A_{X X}X^{3}+A_{X Y}X Y^{2})+\Omega ^{2} X +2\Omega \dot{Y}$. Let us put, $X=x \times 10^{17}~\text{cm}$, $Y= y \times 10^{17}~\text{cm}$, and $T = t \times 10^{13}~s$. Then the above equation reduces to

$\frac{10^{17}}{10^{26}} \ddot{x} = 2 Z_{X}(x \times 10^{17}) -4 \lambda (A_{X X }x^{3}+A_{X Y}x y^{2}) \times 10^{51} +\Omega ^{2} x \times 10^{17} +2\Omega \dot{y} \times \frac{10^{17}}{10^{13}}$.

Or, $\ddot{x} = 2 (Z_{X} \times 10^{26})x -4( \lambda \times 10^{60})(A_{X X}x^{3}+ 4A_{X Y}x y^{2}) +\Omega ^{2} x \times 10^{26} +2\Omega \dot{y} \times 10^{13}$.

Or, $\ddot{x} = 2 z_{x}x -4\lambda (a_{xx}x^{3}+a_{x y}x y^{2})+(\Omega \times 10^{13})^{2} x +2(\Omega \times 10^{13} )\dot{y}$.

Where $Z_{x} = Z_{X} \times 10^{26}$, $a_{x x}= A_{X X} \times 10^{60}$ and $a_{x y}= A_{X Y} \times 10^{60}$. Let $\Omega = \omega \times 10^{-13}~s^{-1}$. Then the equation of motion along the x axis in terms of the dimensionless variables (x, y, t) and the dimensionless parameters ($\lambda$, $a_{x x}$, $a_{x, y}$, $a_{x, y}$) is

$$ \ddot{x} = 2 Z_{x}x -4\lambda (a_{xx}x^{3}+a_{x y}x y^{2}) +\omega ^{2} x +2\omega \dot{y}. $$

To reduce the equation of motion along y axis in terms of dimensionless variables and parameters we followed the similar manner.