Changes between Version 28 and Version 29 of u/johannjc/scratchpad

Timestamp:

07/05/16 06:08:13 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad

@@ -4 +4 @@
 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($d\mathbf{p}=d \rho \mathbf{\mathcal{V}}(\theta)$)]]
+Let's assume that the density and velocity have the following form
 
-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
+$\rho_{\pm} = \alpha_{\pm} \rho(x)$
 
-[[latex($\displaystyle{\sum_{top}{d \rho_i}}=\displaystyle{\sum_{bottom}{d \rho_ i}} = \frac{f \dot{M}}{2}$)]]
+$p_{\pm} =\alpha_{\pm} \left ( a_\pm v_r(x) + dv_{\perp,\pm} + b_\pm R_\pm v_\Omega \right)$
 
+$\alpha$, $a$, and $b$ are scalar adjustment factors, $R$ is a rotation matrix to align the spin, and $dp$ is a vector normal to the spin axis used to control the total momentum.
 
-And since we also must have the total momentum be balanced, we need to have a magnitude constraint
+We begin by transforming the coordinates of the cell centers into the coordinate system of the star using a rotation matrix.
 
-[[latex($\displaystyle{\sum_{top}{|dp^j_i|}}+\displaystyle{\sum_{bottom}{|dp^j_i|}} = f \dot{M}\mathcal{V}_0$)]]
+We then calculate the mass correction factors $\alpha_\pm$ trivially by summing up the predicted mass injection into each hemisphere and then calculating the normalization factors.
 
-as well as the symmetry constraint
+We then estimate the velocity of each cell using 
 
-[[latex($\displaystyle{\sum_{top}{dp^j_i}}+\displaystyle{\sum_{bottom}{dp^j_i}} = \mathbf{0}$)]]
+$a_\pm v_r(x) + dv_{\perp,\pm} + b_\pm R_\pm v_\Omega$
 
-and directional constraint
+with the scalars set to 1, the rotation matrix set to the identity, and dp=0.
 
-[[latex($\displaystyle{\sum_{top}{\mathbf{dp}_i}} \times \mathbf{S} = \mathbf{0}$)]]
+We then calculate the spin and momentum in the top and bottom hemispheres.  We then adjust $b_\pm$ to adjust the magnitude of the spin, and we calculate the rotation matrix $R$ necessary to rotate the spin to the jet axis which we apply directly to the $v_\Omega$.  We also calculate the total momentum in the top and bottom hemispheres and update the magnitude of the z component using $a_\pm$, and zero out the transverse components using the shift $dv_{\perp,\pm}$.
 
-It is easier to solve the system if the magnitude constraint takes the form of 
+This process is iterated a few times and then a final velocity scaling and shift is applied to make the momentum injection correct to machine precision
 
-[[latex($\displaystyle{\left | \sum_{top}{\mathbf{dp_i}}\right | }+\displaystyle{\left | \sum_{bottom}{\mathbf{dp}_i} \right | } = \gamma f \dot{M}\mathcal{V}_0$)]]
+$v_\pm = a_{\pm,f} \left [ \left (a_\pm v_r(x) + dv_{\perp,\pm} + b_\pm R_\pm v_\Omega \right)  + dv_{\perp,\pm,f} \right ]$
 
-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