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title: Bonnor-Ebert sphereの構造(5)

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# Bonnor-Ebert sphereの構造(5)
KrumHoltz p.143 Problem Set 2 (f)
小問ごとのリンク[(a)](https://hackmd.io/IdrIWDqPQpOodQxyYvXnhw)[(b)(c )](https://hackmd.io/Ij09i_P-RH-WlOyI1Rda4g)[(d)](https://hackmd.io/ebZUoLp7StGS25T0oHaWmA)[(e)](https://hackmd.io/NxfRWzPnTiKauQMTGgRKZA)[(f)](https://hackmd.io/EkWtW1DTSVaUu140MZ3A2g)[(g)](https://hackmd.io/aEX4mHAHSuqE7AAc3k5voA)

## (f)Plot the dimensionless mass m versus the dimensionless density contrast $\rho_c/\rho_s$

$m=M/(c_s^4/(G^3P_s)^{1/2})=M/(c_s^4/(G^3\rho_s c_s^2)^{1/2})$
$=MG^{3/2}\rho_s^{1/2}/c_s^3=MG^{3/2}P_s^{1/2}/c_s^4$
(e1)を代入すると、
$m=\frac{e^{-\psi_s/2}}{\sqrt{4\pi}}\xi_s^2 \frac{d\psi}{d\xi}|_{\xi_s}=\sqrt{\frac{\rho_s}{4\pi\rho_c}}\xi_s^2 \frac{d\psi}{d\xi}|_{\xi_s}=\frac{1}{\sqrt{4\pi(\rho_c/\rho_s)}}\xi_s^2 \frac{d\psi}{d\xi}|_{\xi_s}$

$exp(-\psi)と\xi$のグラフ
![](https://i.imgur.com/O54BFfz.png)

$\xi^2\frac{d\psi}{d\xi}と\xi$のグラフ
![](https://i.imgur.com/UkfrO3A.png)


$mと\rho_c/\rho_s$のグラフ
![](https://i.imgur.com/yDAsHAs.png)

$\rho_c/\rho_s=14.0$のとき最大値$m_{max}=1.18$を取る



