diff --git a/docs/cluster.md b/docs/cluster.md index b4d83983..9e3eaaac 100644 --- a/docs/cluster.md +++ b/docs/cluster.md @@ -76,14 +76,14 @@ $$ \frac{3 H_0^2}{8 \pi G}. $$ -Parameters for the HERNQUIST BCG are controlled via: +Parameters for the `HERNQUIST` BCG are controlled via: ``` m_bcg_s = 0.001 # in code_mass r_bcg_s = 0.004 # in code_length ``` -where a HERNQUIST profile adds a gravitational acceleration defined by +where a `HERNQUIST` profile adds a gravitational acceleration defined by $$ g_{BCG}(r) = G \frac{ M_{BCG} }{R_{BCG}^2} \frac{1}{\left( 1 + \frac{r}{R_{BCG}}\right)^2} @@ -216,7 +216,7 @@ triggering_mode = COLD_GAS # or NONE, BOOSTED_BONDI, BONDI_SCHAYE ``` where `triggering_mode=NONE` will disable AGN triggering. -With BOOSTED_BONDI accretion, the mass rate of accretion follows +With `BOOSTED_BONDI` accretion, the mass rate of accretion follows $$ \dot{M} = \alpha \frac { 2 \pi G^2 M^2_{SMBH} \hat {\rho} } { @@ -235,7 +235,7 @@ m_smbh = 1.0e-06 # in code_mass accretion_radius = 0.001 # in code_length bondi_alpha= 100.0 # unitless ``` -With BONDI_SCHAYE accretion, the `$\alpha$` used for BOOSTED_BONDI accretion is modified to depend on the number density following: +With `BONDI_SCHAYE` accretion, the $\alpha$ used for `BOOSTED_BONDI` accretion is modified to depend on the number density following: $$ \alpha = @@ -252,7 +252,7 @@ bondi_n0= 2.9379989445851786e+72 # in 1/code_length**3 bondi_beta= 2.0 # unitless ``` -With both BOOSTED_BONDI and BONDI_SCHAYE accretion, mass is removed from each +With both `BOOSTED_BONDI` and `BONDI_SCHAYE` accretion, mass is removed from each cell within the accretion zone at a mass weighted rate. E.g. the mass in each cell within the accretion region changes by ``` @@ -264,7 +264,7 @@ unchanged. Thus velocities and temperatures will increase where mass is removed. -With COLD_GAS accretion, the accretion rate becomes the total mass within the accretion zone equal to or +With `COLD_GAS` accretion, the accretion rate becomes the total mass within the accretion zone equal to or below a defined cold temperature threshold divided by a defined accretion timescale. The temperature threshold and accretion timescale are defined by ``` @@ -360,13 +360,17 @@ velocity $v_{jet}$ can be set via kinetic_jet_temperature = 1e7 # K ``` However, $T_{jet}$ and $v_{jet}$ must be non-negative and fulfill + $$ -v_{jet} = \sqrt{ 2 \left ( \epsilon c^2 - (1 - \epsilon) \frac{k_B T_{jet}}{ \mu m_h \left( \gamma - 1 \right} \right ) } +v_{jet} = \sqrt{ 2 \left ( \epsilon c^2 - (1 - \epsilon) \frac{k_B T_{jet}}{ \mu m_h \left( \gamma - 1 \right) } \right ) } $$ + to ensure that the sum of rest mass energy, thermal energy, and kinetic energy of the new gas sums to $\dot{M} c^2$. Note that these equations places limits on $T_{jet}$ and $v_{jet}$, specifically + $$ -v_{jet} \leq c \sqrt{ 2 \epsilon } \qquad \text{and} \qquad \frac{k_B T_{jet}}{ \mu m_h \left( \gamma - 1 \right} \leq c^2 \frac{ \epsilon}{1 - \epsilon} +v_{jet} \leq c \sqrt{ 2 \epsilon } \qquad \text{and} \qquad \frac{k_B T_{jet}}{ \mu m_h \left( \gamma - 1 \right) } \leq c^2 \frac{ \epsilon}{1 - \epsilon} $$ + If the above equations are not satified then an exception will be thrown at initialization. If neither $T_{jet}$ nor $v_{jet}$ are specified, then $v_{jet}$ will be computed assuming $T_{jet}=0$ and a warning will be given @@ -429,9 +433,9 @@ where the injected magnetic field follows $$ \begin{align} -\mathcal{B}_r &=\mathcal{B}_0 2 \frac{h r}{\ell^2} \exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right )} \\\\ -\mathcal{B}_\theta &=\mathcal{B}_0 \alpha \frac{r}{\ell} \exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right ) } \\\\ -\mathcal{B}_h &=\mathcal{B}_0 2 \left( 1 - \frac{r^2}{\ell^2} \right ) \exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right )} \\\\ +\mathcal{B}\_r &= \mathcal{B}\_0 2 \frac{h r}{\ell^2} \exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right )} \\ +\mathcal{B}\_\theta &= \mathcal{B}\_0 \alpha \frac{r}{\ell} \exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right ) } \\ +\mathcal{B}\_h &= \mathcal{B}\_0 2 \left( 1 - \frac{r^2}{\ell^2} \right ) \exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right )} \end{align} $$ @@ -439,9 +443,9 @@ which has the corresponding vector potential field $$ \begin{align} -\mathcal{A}_r &= 0 \\\\ -\mathcal{A}_{\theta} &= \mathcal{B}_0 \ell \frac{r}{\ell} \exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right )} \\\\ -\mathcal{A}_h &= \mathcal{B}_0 \ell \frac{\alpha}{2}\exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right )} +\mathcal{A}\_r &= 0 \\ +\mathcal{A}\_{\theta} &= \mathcal{B}\_0 \ell \frac{r}{\ell} \exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right )} \\ +\mathcal{A}\_h &= \mathcal{B}\_0 \ell \frac{\alpha}{2}\exp{ \left ( \frac{-r^2 - h^2}{\ell^2} \right )} \end{align} $$ @@ -463,7 +467,7 @@ $$ where $\dot{\rho}_B$ is set to $$ -\dot{\rho}_B = \frac{3 \pi}{2} \frac{\dot{M} \left ( 1 - \epsilon \right ) f_{magnetic}}{\ell^3} +\dot{\rho}_B = \frac{3 \pi}{2} \frac{\dot{M} \left ( 1 - \epsilon \right ) f\_\mathrm{magnetic}}{\ell^3} $$ so that the total mass injected matches the accreted mass propotioned to magnetic feedback. @@ -485,13 +489,17 @@ according to their ratio. #### Simple field loop (donut) feedback Magnetic energy is injected according to the following simple potential + $$ -A_h(r, \theta, h) = B_0 L \exp^\left ( -r^2/L^2 \right)$ for $h_\mathrm{offset} \leq |h| \leq h_\mathrm{offset} + h_\mathrm{thickness} +A_h(r, \theta, h) = B_0 L \exp^\left ( -r^2/L^2 \right) \qquad \mathrm{for} \qquad h_\mathrm{offset} \leq |h| \leq h_\mathrm{offset} + h_\mathrm{thickness} $$ + resultig in a magnetic field configuration of + $$ -B_\theta(r, \theta, h) = 2 B_0 r /L \exp^\left ( -r^2/L^2 \right)$ for $h_\mathrm{offset} \leq |h| \leq h_\mathrm{offset} + h_\mathrm{thickness} +B_\theta(r, \theta, h) = 2 B_0 r /L \exp^\left ( -r^2/L^2 \right) \qquad \mathrm{for} \qquad h_\mathrm{offset} \leq |h| \leq h_\mathrm{offset} + h_\mathrm{thickness} $$ + with all other components being zero. @@ -562,4 +570,4 @@ temperature_threshold = 2e4 # in K ``` Note that all parameters need to be specified explicitly for the feedback to work -(i.e., no hidden default values). \ No newline at end of file +(i.e., no hidden default values).