| 2 | |
| 3 | == Units in Astrobear == |
| 4 | |
| 5 | Astrobear can use arbitrary computational units for length, mass, time, etc... These are derived from specifying 4 of the following in physics.data |
| 6 | |
| 7 | * nScale - Number density |
| 8 | * rScale - Mass density |
| 9 | * !TempScale - Inverse Gas constant $\frac{\rho T}{P}$ |
| 10 | * pScale - Pressure |
| 11 | * !TimeScale - Time |
| 12 | * lScale - Length |
| 13 | * Xmu - Mean atomic mass (in amu) |
| 14 | |
| 15 | The 4 specified must be independent. Specifying nScale, rScale and Xmu will result in an error - since there is a single equation linking just those three. Likewise specifying nScale, pScale, and !TempScale will result in an error. |
| 16 | |
| 17 | |
| 18 | === Electromagnetism === |
| 19 | |
| 20 | For electromagnetic units, Astrobear uses something like rationalized electromagnetic units (extra factor of $\sqrt{4 \pi}$ in the electric and magnetic fields) - or Lorentz-Heaviside but scaling the $E$ field by $c$ and the charge density $\rho$ (and current $J$ ) by $\frac{1}{c}$. This avoids the need for any additional multiplication by factors of $c$. |
| 21 | |
| 22 | || Computational || Lorentz-Heaviside || Gaussian || |
| 23 | || $E$ || $c E^{LH}$ || $\frac{c }{\sqrt{4\pi}}E^{G}$ || |
| 24 | || $\rho$ || $\frac{1}{c}\rho^{LH}$ || $ \frac{\sqrt{4 \pi} }{c}\rho^{G}$ || |
| 25 | || $J$ || $\frac{1}{c} J^{LH}$ || $\frac{\sqrt{4 \pi}}{c} J^{G}$ || |
| 26 | || $B$ || $ B^{LH} $ || $ \frac{1}{\sqrt{4\pi}} B^{G}$ || |
| 27 | |
| 28 | Using the approach in the appendix of Jackson, we have |
| 29 | || $k_1 = \frac{c^2}{4 \pi} $ || |
| 30 | || $k_2 = \frac{1}{4\pi}$ || |
| 31 | || $k_3 = 1$ || |
| 32 | || $\alpha = 1$ || |
| 33 | || $\mu_0 = 1$ || |
| 34 | || $\epsilon_0 = \frac{1}{c^2}$ || |
| 35 | This allows us to write Maxwell's equations as |
| 36 | |
| 37 | || $\nabla \cdot \mathbf{E} = c^2 \rho$ || |
| 38 | || $\nabla \times \mathbf{B} = \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$ || |
| 39 | || $\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$ || |
| 40 | || $\nabla \cdot \mathbf{B} = 0$ || |
| 41 | |
| 42 | as well as |
| 43 | || Lorentz Force Law || $\mathbf{F} = q \left ( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right)$ || |
| 44 | || Coulomb's Law || $\mathbf{F} = -\frac{c^2}{4 \pi} \frac{q_1 q_2}{r^2}\hat{\mathbf{r}}$ || |
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