Changes between Version 11 and Version 12 of SinkParticlesDevel

Timestamp:

05/06/15 17:28:34 (10 years ago)

Author:

Jonathan

Comment:

—

SinkParticlesDevel

@@ -78 +78 @@
 
 
+== Outflows ==
+Particles can launch outflows following the description in [http://arxiv.org/pdf/1406.3625.pdf Federrath et al 2014]
+
+The basic problem is to apply the following outflow function, but in a discrete way that is numerically symmetric and exact.  
+
+The change in rho has the following functional form
+[[latex($d \rho=\rho_0 \mathcal{R}(r) \Theta(\theta)$)]]
+
+as does the change in momentum
+[[latex($d\mathbf{p}=d \rho \mathbf{\mathcal{V}}(\theta)$)]]
+
+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
+
+[[latex($\displaystyle{\sum_{top}{d \rho_i}}=\displaystyle{\sum_{bottom}{d \rho_ i}} = \frac{f \dot{M}}{2}$)]]
+
+
+And since we also must have the total momentum be balanced, we need to have
+
+[[latex($\displaystyle{\sum_{top}{|dp^j_i|}}+\displaystyle{\sum_{bottom}{|dp^j_i|}} = f \dot{M}\mathcal{V}_0$)]]
+
+as well as
+
+[[latex($\displaystyle{\sum_{top}{dp^j_i}}+\displaystyle{\sum_{bottom}{dp^j_i}} = \mathbf{0}$)]]
+
+If we introduce scaling parameters for the density and momentum for the top and bottom...
+
+[[latex($d \rho_\pm=\alpha_{\pm} \rho_0 \mathcal{R}(r) \Theta(\theta)$)]]
+
+
+[[latex($d\mathbf{p}_\pm=d \rho_\pm \beta_\pm \mathbf{\mathcal{V}}(\theta)$)]]
+
+and plug these into the above equations, we can solve for [[latex($\alpha_\pm$)]] and [[latex($\beta_\pm$)]]
+
+
+[[latex($\alpha_\pm = \frac{\displaystyle{\sum_\mp{\mathcal{R}_i \Theta_i}}}{\displaystyle{\sum_+{\mathcal{R}_i \Theta_i}\sum_-{\mathcal{R}_i \Theta_i}}} \frac{f \dot{M}}{2 \rho_0}$)]]
+
+
 For reference on another way astrobear has implemented sinks in the past, see SinkParticle
 
-