Module kidou::orbit::calc::potentials

This module provides models for components of the Galactic potential

Here’s a list of available potentials and their R and Z derivatives. Note that $ r^2 = X^2 + Y^2 + Z^2 = R^2 + Z^2 $.

1. Plummer (P) potential with parameters $ (M, b) $:

$$ \Phi(r(R, Z)) = - \frac{M}{(r^2 + b^2)^{1/2}}; $$

$$ \frac{\partial \Phi(R, Z)}{\partial R} = \frac{M R}{(R^2 + Z^2 + b^2)^{3/2}}; $$

$$ \frac{\partial \Phi(R, Z)}{\partial Z} = \frac{M Z}{(R^2 + Z^2 + b^2)^{3/2}}. $$

2. Miyamoto & Nagai (MN) potential with parameters $ (M, a, b) $:

$$ \Phi(R, Z) = - \frac{M}{\left[ R^2 + \left( a + \sqrt{Z^2 + b^2} \right)^2 \right]^{1/2}}; $$

$$ \frac{\partial \Phi(R, Z)}{\partial R} = \frac{M R}{\left[ R^2 + \left( a + \sqrt{Z^2 + b^2} \right)^2 \right]^{3/2}}; $$

$$ \frac{\partial \Phi(R, Z)}{\partial Z} = \frac{M Z \left( a + \sqrt{b^2 + Z^2} \right)}{ \sqrt{b^2 + Z^2} \left( R^2 + \left( a + \sqrt{b^2 + Z^2} \right)^2 \right)^{3/2}}. $$

3. Navarro-Frenk-White (NFW) potential with parameters $ (M, a) $:

$$ \Phi(r(R, Z)) = - \frac{M}{r} \ln{\left( 1 + \frac{r}{a} \right)}; $$

$$ \frac{\partial \Phi(R, Z)}{\partial R} = \frac{M R \ln{\left( \sqrt{R^2 + Z^2} / a + 1 \right)}}{\left( R^2 + Z^2 \right)^{3/2}} - \frac{M R}{\left( R^2 + Z^2 \right) \left( \sqrt{R^2+ Z^2} + a \right)}; $$

$$ \frac{\partial \Phi(R, Z)}{\partial Z} = \frac{M Z \ln{\left( \sqrt{R^2 + Z^2} / a + 1 \right)}}{\left( R^2 + Z^2 \right)^{3/2}} - \frac{M Z}{\left( R^2 + Z^2 \right) \left( \sqrt{R^2+ Z^2} + a \right)}; $$

Modules

miyamoto_nagai

Miyamoto & Nagai potential

navarro_frenk_white

Navarro-Frenk-White potential

plummer

Plummer potential