Appendix/Ramblings/BdHN

Binary-driven Hpernovae

The material presented here builds on our separate discussion of close binary stars.

Setup

Consider the simple model of two spherical stars in circular orbit about one another, as depicted here on the right. In addition to the physical parameters explicitly labeled in this diagram, we adopt the following variable notation:

The total system mass is, $M_{\mathrm{t}\mathrm{o}\mathrm{t}}\equiv M+M^{\text{'}};$ The ratio of the primary to secondary mass is, $\lambda \equiv \frac{M}{M^{\text{'}}};$ And the separation between the two centers is, $d\equiv r_{\mathrm{c}\mathrm{m}}+r_{\mathrm{c}\mathrm{m}}^{\text{'}}.$

For a circular orbit, the angular velocity is related to the the system mass and separation via the Kepler relation,

${\omega }^2{d}^3=G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}},$

and the distances, $r_{\mathrm{c}\mathrm{m}}$ and $r_{\mathrm{c}\mathrm{m}}^{\text{'}}$, between the center of each star and the center of mass (cm) of the system must be related to one another via the expression,

$\frac{r_{\mathrm{c}\mathrm{m}}^{\text{'}}}{r_{\mathrm{c}\mathrm{m}}}$

$=$

$\frac{M}{M^{\text{'}}}=\lambda .$

Note that the following relations also hold:

$M=M_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(\frac{\lambda }{1+\lambda }\right)$	and	$M^{\text{'}}=M_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(\frac{1}{1+\lambda }\right)$
$r_{\mathrm{c}\mathrm{m}}=d\left(\frac{1}{1+\lambda }\right);$	and	$r_{\mathrm{c}\mathrm{m}}^{\text{'}}=d\left(\frac{\lambda }{1+\lambda }\right).$

Hence, the orbital angular momentum is,

$L_{\mathrm{o}\mathrm{r}\mathrm{b}}$	$=$	$[M{r}_{\mathrm{c}\mathrm{m}}^2+M^{\text{'}}(r_{\mathrm{c}\mathrm{m}}^{\text{'}}{)}^2]\omega$
	$=$	$M_{\mathrm{t}\mathrm{o}\mathrm{t}}{d}^2\left[\right(\frac{\lambda }{1+\lambda }\left)\right(\frac{1}{1+\lambda }{)}^2+\left(\frac{1}{1+\lambda }\right)\left(\frac{\lambda }{1+\lambda }{)}^2\right][\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{d^3}{]}^{1/2}$
	$=$	$(G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3}d{)}^{1/2}[\frac{\lambda }{(1+\lambda {)}^2}].$

Assuming that both stars are rotating synchronously with the orbit, their respective spin angular momenta are,

$L_M=I_M\omega$	$=$	$\frac{2}{5}M{R}^2\omega$
	$=$	$\frac{2}{5}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(\frac{\lambda }{1+\lambda }\right)R^2[\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{d^3}{]}^{1/2}$
	$=$	$\frac{2}{5}(G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3}d{)}^{1/2}(\frac{\lambda }{1+\lambda }\left)\right(\frac{R}{d}{)}^2,$
$L_{M^{\text{'}}}=I_{M^{\text{'}}}\omega$	$=$	$\frac{2}{5}{M}^{\text{'}}(R^{\text{'}}{)}^2\omega$
	$=$	$\frac{2}{5}{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(\frac{1}{1+\lambda }\right)(R^{\text{'}}{)}^2[\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{d^3}{]}^{1/2}$
	$=$	$\frac{2}{5}(G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3}d{)}^{1/2}(\frac{1}{1+\lambda }\left)\right(\frac{R^{\text{'}}}{d}{)}^2.$

Hence, the total angular momentum of the system is,

$L_{\mathrm{t}\mathrm{o}\mathrm{t}}=L_{\mathrm{o}\mathrm{r}\mathrm{b}}+L_M+L_{M^{\text{'}}}$	$=$	$(G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3}d{)}^{1/2}[\frac{\lambda }{(1+\lambda {)}^2}]+\frac{2}{5}(G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3}d{)}^{1/2}\left(\frac{\lambda }{1+\lambda }\right)(\frac{R}{d}{)}^2$
		$+\frac{2}{5}(G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3}d{)}^{1/2}(\frac{1}{1+\lambda }\left)\right(\frac{R^{\text{'}}}{d}{)}^2$
	$=$	$(G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{3}R{)}^{1/2}\{\left[\frac{\lambda }{(1+\lambda {)}^2}\right](\frac{d}{R}{)}^{1/2}+\frac{2}{5}(\frac{1}{1+\lambda }\left)\right[\lambda +\left(\frac{R^{\text{'}}}{R}{)}^2\right]\left(\frac{d}{R}{)}^{-3/2}\right\}$

See Also