은하 실종 질량의 수학적 기초: 디스크, 구, 밀도, 포텐셜 및 방사형 스케일링

디스크 영역 요소:

\[ dA = R\,dR\,d\phi \]

구형 체적 요소:

\[ dV = 4\pi r^2\,dr \]

\[ \시그마(R)=\Sigma_0 e^{-R/R_d} \]

\[ dM_{\rm 디스크}=2\pi R\시그마(R)\,dR \]

\[ dM_{\rm disk}=2\pi R\Sigma_0e^{-R/R_d}\,dR \]

\[ M_{\rm disk}(R)=2\pi\int_0^R R’\Sigma_0e^{-R’/R_d}\,dR’ \]

\[ M_{\rm disk}(R)=2\pi\Sigma_0R_d^2 \왼쪽[ 1-e^{-R/R_d} \왼쪽( 1+\frac{R}{R_d} \오른쪽) \오른쪽] \]

\[ M_{\rm 디스크}(R)\우측 화살표 2\pi\Sigma_0R_d^2 \]

\[ M(r)=4\pi\int_0^r \rho(r’)r’^2\,dr’ \]

\[ \rho(r)=\frac{1}{4\pi r^2}\frac{dM}{dr} \]

\[ \frac{v(r)^2}{r}=\frac{GM(r)}{r^2} \]

\[ M_{\rm dyn}(r)=\frac{v(r)^2r}{G} \]

\[ V(R)\약 V_0 \]

\[ M_{\rm dyn}(r)\약 \frac{v_0^2}{G}r \]

\[ M_{\rm dyn}(r)\propto r \]

\[ \rho_{\rm dyn}(r)\propto \frac{1}{r^2} \]

\[ M_{\rm miss}(r)=M_{\rm dyn}(r)-M_{\rm vis}(r) \]

\[ M_{\rm miss}(r)= \frac{v(r)^2r}{G} – 2\pi\Sigma_0R_d^2 \왼쪽[ 1-e^{-r/R_d} \왼쪽( 1+\frac{r}{R_d} \오른쪽) \오른쪽] \]

\[ M_{\rm miss}(r)\approx \frac{v_0^2}{G}r – 2\pi\Sigma_0R_d^2 \왼쪽[ 1-e^{-r/R_d} \왼쪽( 1+\frac{r}{R_d} \오른쪽) \오른쪽] \]

\[ \Phi(r)=-\frac{GM}{r} \]

\[ g(r)=-\frac{d\Phi}{dr} \]

\[ g(r)=-\frac{GM}{r^2} \]

\[ \nabla^2\Phi=4\pi G\rho \]

\[ \frac{1}{r^2}\frac{d}{dr} \왼쪽( r^2\frac{d\Phi}{dr} \오른쪽) = 4\pi G\rho(r) \]

\[ \rho(r) \긴직선 M(r) \긴직각시선 \피(r) \긴직각시선 v(r) \]

\[ \Phi(\mathbf{x}) = -G\int \frac{\rho(\mathbf{x}’)}{|\mathbf{x}-\mathbf{x}’|} \,d^3x’ \]

\[ \Phi(R) = -G \int_0^\인프티 \int_0^{2\pi} \frac{\Sigma(R’)R’\,dR’\,d\phi} {\sqrt{R^2+R’^2-2RR’\cos\phi}} \]

\[ d(R,R’,\phi) = \sqrt{R^2+R’^2-2RR’\cos\phi} \]

\[ g_R(R)=-\frac{\partial \Phi}{\partial R} \]

\[ v^2(R)=R\,|g_R(R)|. \]

\[ \코스\세타 = \frac{R-R’\cos\phi}{d(R,R’,\phi)} \]

\[ K_{\rm disk}(R,R’,\phi) = K(d) \frac{R-R’\cos\phi}{d} \]

\[ K(d)\propto \frac{1}{d^2} \]

\[ K_{\rm 디스크}\프로토 \frac{R-R’\cos\phi}{d^3} \]

\[ K(d)\propto e^{-d/\lambda} \]

\[ K_{\rm disk}\propto e^{-d/\lambda} \frac{R-R’\cos\phi}{d} \]

\[ e^{-r/\람다} \]

\[ \Phi_Y(r)\propto -\frac{e^{-r/\lambda}}{r} \]

\[ g_Y(r) = -\frac{d}{dr} \왼쪽( \frac{e^{-r/\lambda}}{r} \오른쪽) \]

\[ g_Y(r)\propto e^{-r/\lambda} \왼쪽( \frac{1}{r^2} + \frac{1}{\람다 r} \오른쪽) \]

\[ \rho(r)\propto \frac{1}{r^2} \]

\[ M(r)=4\pi\int_0^r \rho(r’)r’^2dr’ \]

\[ M(r)\프로토 r \]

\[ v^2(r)=\frac{GM(r)}{r} \]

\[ v^2(r)\근사 상수 \]

\[ v(r)\근사 상수 \]

\[ M_{\rm 디스크}(R)\우측수렴 M_d \]

\[ M_{\rm dyn}(r)\propto r \]

\[ M_{\rm miss}(r)=M_{\rm dyn}(r)-M_{\rm disk}(r) \]

\[ M_{\rm miss}(r)\approx \frac{v_0^2}{G}r-M_d \]

\[ M_{\rm miss}(r)\sim r \]

\[ \rho_{\rm miss}(r)\sim \frac{1}{r^2} \]

\[ M(r)=\frac{v(r)^2r}{G} \]

디스크 표면 밀도:

\[ \Sigma(R)=\Sigma_0e^{-R/R_d} \]

디스크 질량 요소:

\[ dM_{\rm 디스크}=2\pi R\시그마(R)dR \]

가시 디스크 질량:

\[ M_{\rm disk}(R)=2\pi\Sigma_0R_d^2 \왼쪽[ 1-e^{-R/R_d} \왼쪽( 1+\frac{R}{R_d} \오른쪽) \오른쪽] \]

구형 부피 질량:

\[ M(r)=4\pi\int_0^r\rho(r’)r’^2dr’ \]

밀폐된 질량으로부터의 밀도:

\[ \rho(r)=\frac{1}{4\pi r^2}\frac{dM}{dr} \]

동적 질량:

\[ M_{\rm dyn}(r)=\frac{v(r)^2r}{G} \]

누락된 질량:

\[ M_{\rm miss}(r)=M_{\rm dyn}(r)-M_{\rm vis}(r) \]

뉴턴 포텐셜:

\[ \Phi(r)=-\frac{GM}{r} \]

푸아송 방정식:

\[ \nabla^2\Phi=4\pi G\rho \]

3D 포텐셜 적분:

\[ \Phi(\mathbf{x}) = -G\int \frac{\rho(\mathbf{x}’)}{|\mathbf{x}-\mathbf{x}’|} d^3x’ \]

씬 디스크 전위:

\[ \Phi(R) = -G \int_0^\infty \int_0^{2\pi} \frac{\Sigma(R’)R’dR’d\phi} {\sqrt{R^2+R’^2-2RR’\cos\phi}} \]

투영된 방사형 커널:

\[ \cos\theta= \frac{R-R’\cos\phi} {\sqrt{R^2+R’^2-2RR’\cos\phi}} \]

지수 3D 스크린 전위:

\[ \Phi_Y(r)\propto -\frac{e^{-r/\lambda}}{r} \]