Schroedinger Equation for Warped Accretion Discs

Model

star

disc

r_o
ror_o
\beta r_o
βro\beta r_o

Zoom in on neighbouring slices

r_{j+1} - r_{j} \approx \beta r_j
rj+1−rj≈βrjr_{j+1} - r_{j} \approx \beta r_j

Model

\Delta z
Δz\Delta z
r_j
rjr_j
r_{j-1}
rj−1r_{j-1}
r_{j+1}
rj+1r_{j+1}
\frac{d^2}{d t^2} \Delta z \approx G \rho_j \Delta z
d2dt2Δz≈GρjΔz\frac{d^2}{d t^2} \Delta z \approx G \rho_j \Delta z

Force per unit mass

Energy per unit mass

G \rho_j \Delta z^2
GρjΔz2G \rho_j \Delta z^2

Model

\gamma
γ\gamma
\hat{n} = \left(\sin \theta_j \cos \phi_j , \sin \theta_j \sin \phi_j, \cos \theta_j\right)
n^=(sin⁡θjcos⁡ϕj,sin⁡θjsin⁡ϕj,cos⁡θj)\hat{n} = \left(\sin \theta_j \cos \phi_j , \sin \theta_j \sin \phi_j, \cos \theta_j\right)
\cos \gamma_{j,j+1} = \cos \theta_j \cos \theta_{j+1} + \cos \left(\phi_j - \phi_{j+1}\right) \sin \theta_j \sin \theta_{j+1}
cos⁡γj,j+1=cos⁡θjcos⁡θj+1+cos⁡(ϕj−ϕj+1)sin⁡θjsin⁡θj+1\cos \gamma_{j,j+1} = \cos \theta_j \cos \theta_{j+1} + \cos \left(\phi_j - \phi_{j+1}\right) \sin \theta_j \sin \theta_{j+1}

Model

\gamma_{j,j+1}^2 \approx \theta_j^2 + \theta_{j+1} -2 \theta_j \theta_{j+1} \cos \left(\phi_j - \phi_{j+1}\right)
γj,j+12≈θj2+θj+1−2θjθj+1cos⁡(ϕj−ϕj+1)\gamma_{j,j+1}^2 \approx \theta_j^2 + \theta_{j+1} -2 \theta_j \theta_{j+1} \cos \left(\phi_j - \phi_{j+1}\right)
\Delta U_{j,j+1} \approx G \rho_j r_j^2 \left(\theta_j^2 + \theta_{j+1}^2-2 \theta_j \theta_{j+1} \cos \left(\phi_{j} - \phi_{j+1} \right) \right)
ΔUj,j+1≈Gρjrj2(θj2+θj+12−2θjθj+1cos⁡(ϕj−ϕj+1))\Delta U_{j,j+1} \approx G \rho_j r_j^2 \left(\theta_j^2 + \theta_{j+1}^2-2 \theta_j \theta_{j+1} \cos \left(\phi_{j} - \phi_{j+1} \right) \right)
j
jj
j+1
j+1j+1

Model

\mathcal{H}_j/ G \rho_j m_j r_j^2 \approx 2 \theta_j^2 - \theta_j \theta_{j+1} \cos \left(\phi_j - \phi_{j+1}\right) - \theta_j \theta_{j-1} \cos \left(\phi_j-\phi_{j+1} \right)
Hj/Gρjmjrj2≈2θj2−θjθj+1cos⁡(ϕj−ϕj+1)−θjθj−1cos⁡(ϕj−ϕj+1)\mathcal{H}_j/ G \rho_j m_j r_j^2 \approx 2 \theta_j^2 - \theta_j \theta_{j+1} \cos \left(\phi_j - \phi_{j+1}\right) - \theta_j \theta_{j-1} \cos \left(\phi_j-\phi_{j+1} \right)
\Psi_j = \theta_j \exp \left(i \phi_j\right)
Ψj=θjexp⁡(iϕj)\Psi_j = \theta_j \exp \left(i \phi_j\right)
\mathcal{H}_j/ G \rho_j m_j r_j^2 \approx 2 \Psi_j \bar{\Psi}_j - \Psi_j \bar{\Psi}_{j+1} - \Psi_{j} \bar{\Psi}_{j-1} - \Psi_{j-1} \bar{\Psi}_{j}
Hj/Gρjmjrj2≈2ΨjΨˉj−ΨjΨˉj+1−ΨjΨˉj−1−Ψj−1Ψˉj\mathcal{H}_j/ G \rho_j m_j r_j^2 \approx 2 \Psi_j \bar{\Psi}_j - \Psi_j \bar{\Psi}_{j+1} - \Psi_{j} \bar{\Psi}_{j-1} - \Psi_{j-1} \bar{\Psi}_{j}

Model

L_j \approx m_j \sqrt{G M r_j}
Lj≈mjGMrjL_j \approx m_j \sqrt{G M r_j}
\bar{\Psi}_{j} \leftrightarrow P_j=i L_j \Psi_j
Ψˉj↔Pj=iLjΨj\bar{\Psi}_{j} \leftrightarrow P_j=i L_j \Psi_j
i \frac{d \Psi_j}{dt} \approx \frac{G \rho_j}{\sqrt{GM/r_j^3}} \left(2 \Psi_j -\Psi_{j-1} - \Psi_{j+1} \right)
idΨjdt≈GρjGM/rj3(2Ψj−Ψj−1−Ψj+1)i \frac{d \Psi_j}{dt} \approx \frac{G \rho_j}{\sqrt{GM/r_j^3}} \left(2 \Psi_j -\Psi_{j-1} - \Psi_{j+1} \right)

Angular momentum

Conjugate momentum

Hamilton's equations

Model

v_r \propto 1/\sqrt{r}
vr∝1/rv_r \propto 1/\sqrt{r}
\dot{M} \approx r^2 \rho v_r
M˙≈r2ρvr\dot{M} \approx r^2 \rho v_r
\rho \propto r^{-3/2}
ρ∝r−3/2\rho \propto r^{-3/2}
\rho \approx \frac{m_d}{\beta r_o^3} \left(\frac{r_o}{r}\right)^{3/2}
ρ≈mdβro3(ror)3/2\rho \approx \frac{m_d}{\beta r_o^3} \left(\frac{r_o}{r}\right)^{3/2}

Model

i \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \frac{1}{\beta} \frac{m_d}{M} \left(2 \Psi_j -\Psi_{j-1} - \Psi_{j+1} \right)
idΨjdt≈GMro31βmdM(2Ψj−Ψj−1−Ψj+1)i \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \frac{1}{\beta} \frac{m_d}{M} \left(2 \Psi_j -\Psi_{j-1} - \Psi_{j+1} \right)
\Psi_{j+1} - \Psi_{j} \approx \beta \frac{\partial \Psi}{\partial \ln r}
Ψj+1−Ψj≈β∂Ψ∂ln⁡r\Psi_{j+1} - \Psi_{j} \approx \beta \frac{\partial \Psi}{\partial \ln r}
i \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \beta \frac{m_d}{M} \frac{\partial^2 \Psi}{\partial \ln r^2}
idΨjdt≈GMro3βmdM∂2Ψ∂ln⁡r2i \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \beta \frac{m_d}{M} \frac{\partial^2 \Psi}{\partial \ln r^2}

QED

Quite easily done

Robin Boundary condition

i \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \frac{1}{\beta} \frac{m_d}{M} \left(\Psi_j -\Psi_{j+1} \right)
idΨjdt≈GMro31βmdM(Ψj−Ψj+1)i \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \frac{1}{\beta} \frac{m_d}{M} \left(\Psi_j -\Psi_{j+1} \right)
\Psi_{j+1} - \Psi_{j} \approx \beta \frac{\partial \Psi}{\partial \ln r}
Ψj+1−Ψj≈β∂Ψ∂ln⁡r\Psi_{j+1} - \Psi_{j} \approx \beta \frac{\partial \Psi}{\partial \ln r}
i \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \frac{m_d}{M} \frac{\partial \Psi}{\partial \ln r}
idΨjdt≈GMro3mdM∂Ψ∂ln⁡ri \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \frac{m_d}{M} \frac{\partial \Psi}{\partial \ln r}

Epilogue

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