Changes between Version 5 and Version 6 of u/erica/angularmomentumoutflows


Ignore:
Timestamp:
08/01/16 16:34:25 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/angularmomentumoutflows

    v5 v6  
    22
    33[[latex($m_{acc}=\sum \Delta m_i $)]] [[br]]
    4 [[latex($com_{acc}=\sum \Delta m_i r_i $)]] [[br]]
    5 [[latex($p_{acc}=\sum \Delta m_i  v_i$)]] [[br]]
    6 [[latex($L_{acc}=\sum \Delta m_i r_i \times v_i $)]]
     4[[latex($\bold{com}_{acc}=\sum \Delta m_i \bold{r}_i $)]] [[br]]
     5[[latex($\bold{p}_{acc}=\sum \Delta m_i  \bold{v}_i$)]] [[br]]
     6[[latex($\bold{L}_{acc}=\sum \Delta m_i \bold{r}_i \times \bold{v}_i $)]]
    77
    88these represent the total *accreted* mass (m_acc), center of mass (COM_acc), momentum (p_acc) and angular momentum (L_acc) from all the cells within the accretion volume surrounding the sink particle.
     
    1111
    1212[[latex($m'=m + m_{acc}$)]] [[br]]
    13 [[latex($com'=com + com_{acc}$)]] [[br]]
    14 [[latex($p'=p + p_{acc}$)]] [[br]]
     13[[latex($\bold{com}'=\bold{com} + \bold{com}_{acc}$)]] [[br]]
     14[[latex($\bold{p}'=\bold{p} + \bold{p}_{acc}$)]] [[br]]
    1515
    1616where primes (') denote post-accretion quantities (i.e. m'-m represents the change in particle mass after an accretion event).
     
    1818Now, angular momentum does not have the same form as above, namely:
    1919
    20 [[latex($L'\neq L+L_{acc}$)]]
     20[[latex($\bold{L}'\neq \bold{L}+\bold{L}_{acc}$)]]
    2121
    2222which is somewhat unexpected. This is because the position of the sink can change ever so slightly over the course of the accretion step, and thus, the angular momentum can change slightly as well (to maintain conservation of the com).. Thus, we instead write the angular momentum, post-accretion, as:
    2323
    24 [[latex($L'=m'r'\times v'$)]]
     24[[latex($\bold{L}'=m'\bold{r}'\times \bold{v}'$)]]
    2525
    2626
    2727That is, the cross product of the updated particle's com and velocity. Given this may be a slight over (or under) shoot of the updated angular momentum, due to the fact that the particle may have lost (or gained) some angular momentum over the accretion time step, we create a new vector to absorb this difference, the 'spin' vector (S). We want the total angular momentum (L+S) to maintain conservation over the accretion time step. That is, we want:
    2828
    29 [[latex($L'+S'=(L+S)+L_{acc}$)]]
     29[[latex($\bold{L}'+\bold{S}'=(\bold{L}+\bold{S})+\bold{L}_{acc}$)]]
    3030
    3131and thus, we can define the spin vector using:
    3232
    33 [[latex($S'=S+L-L'+L_{acc}$)]]
     33[[latex($\bold{S}'=\bold{S}+\bold{L}-\bold{L}'+\bold{L}_{acc}$)]]
    3434
    3535Thus, we can say the conserved accreted angular momentum per time step is:
    3636
    37 [[latex($S'-S=(L-L')+L_{acc}$)]]
     37[[latex($\bold{S}'-\bold{S}=(\bold{L}-\bold{L}')+\bold{L}_{acc}$)]]
    3838
    39 Notes: calling it 'spin' might be slightly confusing.
     39Next, we would like to launch a fraction of the accreted angular momentum back along the spin axis of the particle, into the jets. To do this, we want to select only those components that are parallel to the jet, so we have the following prescription for the angular momentum for the outflow:
     40
     41[[latex($\bold{L}_{out}=f_a (\bold{S}'-\bold{S})\cdot\hat{\bold{S}'} ~\hat{\bold{S}'}$)]]
     42
     43where f_a is the fraction of accreted angular momentum that should be launched in the jet.
     44
     45Notes: calling it 'spin' might be slightly confusing.