Schroedinger Equation for Warped Accretion Discs
Model
star
disc
r_o
ro
\beta r_o
βro
Zoom in on neighbouring slices
r_{j+1} - r_{j} \approx \beta r_j
rj+1−rj≈βrj
Model
\Delta z
Δz
r_j
rj
r_{j-1}
rj−1
r_{j+1}
rj+1
\frac{d^2}{d t^2} \Delta z \approx G \rho_j \Delta z
dt2d2Δz≈GρjΔz
Force per unit mass
Energy per unit mass
G \rho_j \Delta z^2
GρjΔz2
Model
\gamma
γ
\hat{n} = \left(\sin \theta_j \cos \phi_j , \sin \theta_j \sin \phi_j, \cos \theta_j\right)
n^=(sinθjcosϕj,sinθjsinϕj,cosθj)
\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}
cosγj,j+1=cosθjcosθj+1+cos(ϕj−ϕj+1)sinθjsinθj+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)
\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))
j
j
j+1
j+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)
\Psi_j = \theta_j \exp \left(i \phi_j\right)
Ψj=θjexp(iϕj)
\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
Model
L_j \approx m_j \sqrt{G M r_j}
Lj≈mjGMrj
\bar{\Psi}_{j} \leftrightarrow P_j=i L_j \Psi_j
Ψˉj↔Pj=iLjΨj
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)
idtdΨj≈GM/rj3Gρj(2Ψj−Ψj−1−Ψj+1)
Angular momentum
Conjugate momentum
Hamilton's equations
Model
v_r \propto 1/\sqrt{r}
vr∝1/r
\dot{M} \approx r^2 \rho v_r
M˙≈r2ρvr
\rho \propto r^{-3/2}
ρ∝r−3/2
\rho \approx \frac{m_d}{\beta r_o^3} \left(\frac{r_o}{r}\right)^{3/2}
ρ≈βro3md(rro)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)
idtdΨj≈ro3GMβ1Mmd(2Ψj−Ψj−1−Ψj+1)
\Psi_{j+1} - \Psi_{j} \approx \beta \frac{\partial \Psi}{\partial \ln r}
Ψj+1−Ψj≈β∂lnr∂Ψ
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}
idtdΨj≈ro3GMβMmd∂lnr2∂2Ψ
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)
idtdΨj≈ro3GMβ1Mmd(Ψj−Ψj+1)
\Psi_{j+1} - \Psi_{j} \approx \beta \frac{\partial \Psi}{\partial \ln r}
Ψj+1−Ψj≈β∂lnr∂Ψ
i \frac{d \Psi_j}{dt} \approx \sqrt{\frac{G M}{r_o^3}} \frac{m_d}{M} \frac{\partial \Psi}{\partial \ln r}
idtdΨj≈ro3GMMmd∂lnr∂Ψ
Epilogue
schoedinger equation for warped discs
By almog yalinewich
schoedinger equation for warped discs
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