\(f_{res}=\cfrac{1}{2\pi}\sqrt{\cfrac{gradient|_{\pi}}{m}}=\cfrac{1}{2\pi}\sqrt{\cfrac{G^2sech^2(\pi)}{m}}=\cfrac{sech(\pi)}{2\pi}\cfrac{G}{\sqrt{m}}\)
\(G=\cfrac{\pi \sqrt { 2{ mc^{ 2 } } } }{a_e}\)
\(a_E=6371e3m\)
\(m_E=5.972e24 kg\)
\(G=\)pi*sqrt(2*5.972*(299792458)^2*10^(24))/(6371e3)
\(G=5.109e14\)
If instead we use half the gradient at \(\theta_{\psi\,\pi}=3.135009\)
\(gradient|_{\pi}=sech^2(3.135009)\)
without the constant \(G\).
Approximating the tangent with a line with gradient \(tan(\beta)\),
\(tan(\beta)=sech^2(3.135009)\)
\(\cfrac{1}{2}\beta=\cfrac{1}{2}tan^{-1}(sech^2(3.135009))\)
\(tan(\cfrac{1}{2}\beta)=0.00377\)
\(f_{res}=\cfrac{1}{2\pi}\sqrt{\cfrac{G^2*0.00377}{m}}\)
\(f_{res}=0.009772\cfrac{G}{\sqrt{m}}\)
\(f_{res}=2.0430 Hz\)
which is low....
If we instead do not half the gradient at \(\theta_{\psi\,\pi}=3.135009\),
\(tan(\beta)=sech^2(3.135009)=0.00754\)
\(f_{res}=\cfrac{1}{2\pi}\sqrt{\cfrac{G^2*0.00754}{m}}\)
\(f_{res}=0.01382\cfrac{G}{\sqrt{m}}\)
\(f_{res}=2.8893 Hz\)
If we take the gradient, without \(G\) at the origin instead,
\(sech^2(0)=1\)
\(\cfrac{1}{2}\beta=0.39270\)
\(tan(\cfrac{1}{2}\beta)=0.41421\)
\(f_{res}=\cfrac{1}{2\pi}\sqrt{\cfrac{G^2*0.41421}{m}}\)
\(f_{res}=21.41 Hz\)
What happen when \(\psi\) is pushed along its profile? Maybe \(a_{\psi}\) increases...chicken! Big chicken!
Or,
\(tan(\beta)=sech^2(0)=1\)
\(\cfrac{1}{2}tan(\beta)=0.5\)
\(f_{res}=\cfrac{1}{2\pi}\sqrt{\cfrac{G^2*0.5}{m}}\)
\(f_{res}=0.11254\cfrac{G}{\sqrt{m}}\)
\(f_{res}=23.53 Hz\)
We are comparing \(\psi\) to the energy curve of a SHM system, \(U=\cfrac{1}{2}kx^2\) the gradient of which \(U^{'}=kx\).
