BA Elias Saarmann

• Diss auf Overleaf

Literature

• Radiating Bondi Flows I
• PLUTO manual

Bondary Conditions

in Radiating Bondi Flows I

For the outer boundry conditions \(r \rightarrow \infty\) \[ (\tilde{\rho},\mathcal{M}, s, \tilde{L}, \tilde{E}_\text{r}) \rightarrow (1, 0, s_\infty, \tilde{L}_\infty, 1) \]

With dimensionless density \( \tilde{\rho} = \frac{\rho(r)}{\rho_\infty}\), Mach number \(\mathcal{M} = \frac{v(r)}{c_s(r)}\), dimensionless entropy \( s = \frac{S}{c_\text{V}}\) and dimensionless radiation Energy density \(\tilde{E}_\text{r} = \frac{E_\text{r}}{a_\text{r}T_\infty^4}\).

in PLUTO/belt

General independent paramters:

Density at infinity \(\rho_\infty\)

Temperature at infinity \(T_\infty\)

Constant opacity \(\kappa\)

Adiabatic constant \(\gamma\)

Central mass \(M\)

molecular Mass of gas molecules \(m_\text{molec}\)

Useful dependent parameters:

Sound velocity:

\[c_\text{s} = \sqrt{\frac{T \gamma k_\text{B}}{m_\text{molec}}}\]

Bondi Radius:

\[r_\text{B} = \frac{GM}{2c_{\text{s},\infty}^2} = \frac{GM m_\text{molec}}{2T\gamma k_\text{B}}\]

Parameters for belt:

Central Mass:

\[M = M_\odot\]

Optical density:

\[\kappa = \]

Adiabatic constant/degrees of freedom

\[gamma = \frac{7}{5} \rightarrow \text{degrees of freedom} = 5\]

inner radius:

\[r_\text{s} = 10^{-2} r_B\]

outer radius:

\[r_\text{s} = 10^{-2} r_B\]

molecular mass:

\[m_\text{molec} = \]

Boundary conditions for fields in belt...

...at outer radius \(r_\text{o}\):

\[ p(r_\text{o}) = \frac{\rho_\infty k_\text{b} T_\infty}{m_\text{molec}}\]

\[ \rho (r_\text{o}) = rho_\infty\]

\[\partial_r v(r_\text{o}) = 0\]

\[E_\text{r}(r_\text{o}) = a_\text{r} T_\infty^4\]

...at inner radius \(r_\etxt{s}\):

\[\partial_r p(r_\text{s}) = 0\]

\[\partial_r \rho (r_\text{s}) = 0\]

\[\partial_r v(r_\text{s}) = 0\]

\[\partial_r E_\text{r}(r_\text{s}) = 0\]

(While the zero gradient conditions do not match theoretical curves they should still be appropriate as the will only lead to local devastation near the respective boundary.)

Code-Anpassungen für konstante Zentralmasse

(Lothar)

• M_centr (o.Ä.) als weiteren Parameter in user_defined_parameters.h definieren und die Anzahl USER_DEF_PARAMETERS entsprechend erhöhen
• den Wert von M_centr in pluto.ini unter [Parameters] setzen (in Gramm)
• body-force.c aus belt/src/Misc in den Run-Folder kopieren und in Zeile 64 die M_X1_BEG durch (g_inputParam[M_centr]/ReferenceMass) ersetzen