| 92 | | The ambient is acting as the unperturbed jet, and the velocity perturbations come from the left edge of the domain. The perturbations range from 0 to 100 km/s, and the period can be adjusted. The first simulations went through 10 periods, while the second set of simulations only went through 5 periods. Having a longer period leads to fewer but larger shocks and rarefactions. This perhaps makes it easier to watch how the perturbations evolve. |
| | 94 | Note that these runs feature a stationary ambient, and they do not include MHD or jet perturbations. |
| | 95 | In order to make some predictions based on these initial parameters, we need some equations... |
| | 96 | [[BR]] |
| | 97 | ==== Important Equations ==== |
| | 98 | A strong shock implies that the mach number M >> 1. We will also be assuming an ideal gas, so [[latex($\gamma = \frac{5}{3}$)]]. The following equations can be derived from the usual shock jump conditions: |
| 108 | | [[CollapsibleStart(Velocity Lineouts (longer period))]] |
| 109 | | [[Image(nocool_vel_lesspert.gif,width=500)]] |
| 110 | | [[CollapsibleEnd]] |
| | 104 | [[latex($n_{ps} = 4n_{a} \hspace{1in} (3)$)]] |
| | 105 | |
| | 106 | Where s stands for shock, ps for post-shock, and a for ambient. T is temperature, v is velocity, n is number density, mH is the mass of a hydrogen atom, and k is Boltzmann's constant. |
| | 107 | [[BR]] |
| | 108 | |
| | 109 | In order to figure out these post-shock values, we need to somehow relate them to the jet. This is where these equations come in: |
| | 110 | |
| | 111 | [[latex($v_{ps} = v_{j}(1 + \eta^{-\frac{1}{2}})^{-1} \hspace{1in} (4)$)]] |
| | 112 | |
| | 113 | [[latex($\eta = \frac{n_{j}}{n_{a}} \hspace{1in} (5)$)]] |
| | 114 | |
| | 115 | Where j stands for jet, and [[latex($\eta$)]] is basically the density contrast of the jet material to the ambient material. Now we can combine these equations and constants to come up with an equation for the post-shock temperature in terms of jet parameters: |
| | 116 | |
| | 117 | [[latex($T_{ps} \approx (40.41 \mathrm{\frac{K s^{2}}{km^{2}}}) v_{j}^{2} (1+\eta^{-\frac{1}{2}})^{-2} \hspace{1in} (6)$)]] |
| | 118 | |
| | 119 | Where vj is more conveniently in units of km/s. Now we need equations for the characteristic cooling time and cooling length: |
| | 120 | |
| | 121 | [[latex($t_{cool} = \frac{3kT_{ps}}{n_{ps}\Lambda} \hspace{1in} (7)$)]] |
| | 122 | |
| | 123 | [[latex($l_{cool} = v_{ps} t_{cool} \hspace{1in} (8)$)]] |
| | 124 | |
| | 125 | Where [[latex($\Lambda$)]] is the cooling rate in units of cm^3^erg/s. Using equations (3) and (4), these cooling equations can be rewritten in terms of our known initial parameters, and the post-shock temperature which we will want to have calculated already: |
| | 126 | |
| | 127 | [[latex($t_{cool} = \frac{3kT_{ps}}{4n_{a}\Lambda} \hspace{1in} (9)$)]] |
| | 128 | |
| | 129 | [[latex($l_{cool} = v_{j}(1 + \eta^{-\frac{1}{2}})^{-1} \frac{3kT_{ps}}{4n_{a}\Lambda} \hspace{1in} (10)$)]] |
| | 130 | [[BR]] |
| | 131 | ==== Some Predictions ==== |
| | 132 | Based on all these equations, we can make some predictions about the post-shock temperature and what resolution we might want. Using the aforementioned initial parameters, you get the following predictions: |
| | 133 | ||= Tps (10^3^ K) =||= tcool (10^7^ s) =||= lcool (AU) =|| |
| | 134 | ||= 58.3123 =||= 1.26155 =||= 3.20345 =|| |
| | 135 | |
| | 136 | So for example, if we have a domain that is 30 AU and we want 10 cells per cooling length, then we would need a resolution of about 94 cells. That doesn't seem too bad, so I will round up to 100 and start from there. |
