Changes between Version 27 and Version 28 of u/johannjc/scratchpad

Timestamp:

05/10/15 12:26:00 (10 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad

@@ -1 @@
-[[latex($M'=M+\sum{\Delta m_i}$)]]
+== Outflows ==
+Particles can launch outflows following the description in [http://arxiv.org/pdf/1406.3625.pdf Federrath et al 2014]
 
-[[latex($M'V'=MV+\sum{\Delta m_i v_i}$)]]
+The basic problem is to apply the following outflow function, but in a discrete way that is numerically symmetric and exact.  
 
-[[latex($V' = V \left ( \frac{1}{1+\epsilon} \right ) + $)]]
+The change in rho has the following functional form
+[[latex($d \rho=\rho_0 \mathcal{R}(r) \Theta(\theta)$)]]
 
-[[latex($M'R' = M'R + \sum{\Delta m_i \left ( r_i - R \right )}=MR + \sum{\Delta m_i r_i }$)]]
+as does the change in momentum
+[[latex($d\mathbf{p}=d \rho \mathbf{\mathcal{V}}(\theta)$)]]
 
-[[latex($S'=S+\sum{\Delta m_i \left ( r_i -R \right ) \times v_i}$)]]
+Since we want the total mass to equal some fraction of the accreted mass, as well as the top and bottom mass injection to be symmetric, we must have
 
-However, a better treatment is to conserve angular momentum.
-
-[[latex($L=MR \times V$)]]
-
-[[latex($J=L+S$)]]
-
-[[latex($J'=J +\sum{\Delta m_i r_i  \times v_i}$)]] 
-
-[[latex($S'=S + MR \times V - M'R' \times V' + \sum{\Delta m_i r_i  \times v_i}$)]]
-
-[[latex($S'=S + MR \times V - M'R' \times V' + \sum{\Delta m_i r_i  \times v_i}$)]]
+[[latex($\displaystyle{\sum_{top}{d \rho_i}}=\displaystyle{\sum_{bottom}{d \rho_ i}} = \frac{f \dot{M}}{2}$)]]
 
 
-[[latex($S' = S+MR \times V - \left( MR + \sum{\Delta m_i r_i} \right ) \times \frac{MV + \sum{\Delta m_i v_i}}{M+\sum{m_i}}+\sum{\Delta m_i r_i  \times v_i}$)]]
+And since we also must have the total momentum be balanced, we need to have a magnitude constraint
 
-which if we approximate to first order in $\frac{\sum{\Delta m_i}}{M}$
+[[latex($\displaystyle{\sum_{top}{|dp^j_i|}}+\displaystyle{\sum_{bottom}{|dp^j_i|}} = f \dot{M}\mathcal{V}_0$)]]
 
-[[latex($S' \approx S+MR \times V - \left( MR + \sum{\Delta m_i r_i} \right ) \times \left ( V + \frac{\sum{\Delta m_i v_i}}{M} \right ) \left (1 -\frac{\sum{\Delta m_i}}{M} \right )+\sum{\Delta m_i r_i  \times v_i}$)]]
+as well as the symmetry constraint
 
-[[latex($S' \approx S+MR \times V -  MR \times V - \sum{\Delta m_i r_i } \times V - R \times \sum{\Delta m_i v_i} + R \times \sum{\Delta m_i} V+\sum{\Delta m_i r_i  \times v_i}$)]]
+[[latex($\displaystyle{\sum_{top}{dp^j_i}}+\displaystyle{\sum_{bottom}{dp^j_i}} = \mathbf{0}$)]]
 
-[[latex($S' \approx S +\sum{\Delta m_i \left (r_i - R \right )  \times \left (v_i - V \right )}$)]]
+and directional constraint
 
-which implies our method is only accurate if $ V << v_i$ and $\sum{\Delta m_i} << M$
+[[latex($\displaystyle{\sum_{top}{\mathbf{dp}_i}} \times \mathbf{S} = \mathbf{0}$)]]
+
+It is easier to solve the system if the magnitude constraint takes the form of 
+
+[[latex($\displaystyle{\left | \sum_{top}{\mathbf{dp_i}}\right | }+\displaystyle{\left | \sum_{bottom}{\mathbf{dp}_i} \right | } = \gamma f \dot{M}\mathcal{V}_0$)]]
+
+where [[latex($\gamma$)]] is the analytic solution for the fraction of the scalar momentum in the z direction.
+
+
+
+If we introduce scaling parameters for the density and momentum for the top and bottom...
+
+[[latex($d \rho_\pm=\alpha_{\pm} d \rho$)]]
+
+
+[[latex($d p^j_\pm=\beta^j_\pm d p^j $)]]
+
+and plug these into the above equations, we can solve for [[latex($\alpha_\pm$)]] and [[latex($\beta_\pm$)]]
+
+
+[[latex($\alpha_\pm =  \frac{f \dot{M}}{2\displaystyle{\sum_\pm{d \rho}}}$)]]
+
+
+The momentum equations give us 4 constraints for two variables.  
+
+For reference on another way astrobear has implemented sinks in the past, see SinkParticle