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Astronomy

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    $\begingroup$ There is a reasonably well-defined mid plane and the Sun is above it and being accelerated back towards it. The thickness isn't really relevant (other than to make SHM a better approximation). $\endgroup$
    – ProfRob
    Commented Jun 10 at 19:10
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    $\begingroup$ Yes "reasonably" well defined. But still the uncertainty in the location of the mid plane is a lot more than a few AU. And there is nothing particularly on or around the mid plane. "Above" is of course conventional, not in any sense absolute. If we choose to view the galaxy from the other size, it would be rotating in the oppose sense and the sun would be below the plane. $\endgroup$
    – James K
    Commented Jun 10 at 19:37
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    $\begingroup$ @ProfRob SHM is not a good approximation at all, except perhaps for orbits that don't move more than a few pc out from the mid plane. $\endgroup$
    – Walter
    Commented Jun 12 at 1:44
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    $\begingroup$ @Walter How so? The epicyclic approximation for near circular orbits is a totally standard bit of galactic dynamics isn't it? The SHM approx is good so long as you can assume the density is constant. Hence as long as the vertical amplitude is less than the vertical scale on which the density varies. galaxiesbook.org/chapters/II-03.-Orbits-in-Disks.html (Sec 10.3.1). Which is what I said. $\endgroup$
    – ProfRob
    Commented Jun 12 at 7:10
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    $\begingroup$ @ProfRob At one scale height you are already making significant errors with SH. For a vertically exponential density $\rho(z)=(\Sigma/2h)\exp(-\zeta)$ with $\zeta=|z|/h$, the potential is $\Phi=2\pi G\Sigma h[\zeta-1+\exp(-\zeta)] = \tfrac12\omega^2z^2[1-\tfrac13\zeta+O(\zeta^2)]$. Thus, the deviation from SH is first not second order and becomes significant early. The force $F=-\omega^2z[1-\tfrac12\zeta+O(\zeta^2)]$ deviates by ~50% from SH by $|z|=h$. At low $|z|$, the ISM layer becomes important, which has a scale height of $\sim 40\,$pc. So, the SH is perhaps okay for $|z|<20\,$pc. $\endgroup$
    – Walter
    Commented Jun 19 at 10:33