Plan June 2023

Tomorrow: Create fork

Add enable/disable flag Finish gravity change

notes on gravity

Note $$\begin{align*} g &= G\frac{m_1m_2}{r^2} \\ &= G\frac{m_1m_2}{(R_0+z)^2} \\ &= G\frac{m_1m_2}{R_0^2} \left(\frac{R_0}{R_0+z}\right)^2\\ &= g_0 \left(\frac{R_0}{R_0+z} \right)^2 \end{align*}$$ Alright let's do geopotential $$\begin{align*} \phi &= zg(z)\\ &= zg_0 \left(\frac{R_0}{R_0 + z}\right)^2\\ \implies (R_0+z)^2 \phi &= zg_0R_0^2 \\ \implies \phi R_0^2 + 2\phi zR_0 + \phi z^2 &= zg_0R_0^2 \\ \implies \phi z^2 + (2\phi R_0 - g_0R_0^2)z + \phi R_0^2 = 0 \\ \end{align*}$$ and applying the quadratic formula we get $$\begin{align*} z &= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\ &= \frac{(g_0R_0^2- 2\phi R_0 ) + \sqrt{(2\phi R_0 - g_0R_0^2)^2 - 4\phi^2 R_0^2}}{2\phi}\\ &= \frac{(g_0R_0^2-2\phi R_0 ) + \sqrt{R_0^2(2\phi- g_0R_0)^2 - 4\phi^2R_0^2}}{2\phi}\\ &= \frac{(g_0R_0^2-2\phi R_0 ) + R_0 \sqrt{(2\phi- g_0R_0)^2 - 4\phi^2}}{2\phi}\\ \end{align*}$$ verify: