Erik Reuter wrote:
>
...
> > The same as in case 1.
>
> Yes, I agree.
>
> P/P0 = exp[ - ( h / R )^2 / 3.45 ]
>
> Since h/R = 1/5 = 0.2, P/P0 = 0.988
>
> > (Although a pressure of .988 bar seems a bit high--a kilometer of
> > height makes a much larger pressure difference on Earth.)
>
> As I said before, it does not make sense to make direct numerical
> comparisons with Earth. Earth has a different potential gradient and is
> much larger than a 5km habitat. You have a better physical intuition
> than I do, David, but I think your refusal to work with actual equations
> and numbers is hampering you here.
Thank you. O.K., maybe next week...
> The potential energy at a height h above the Earth is
>
> U = m g h / ( 1 + h / R_e )
Agreed.
>
> and the resulting equation for pressure
>
> P/P0 = exp[ -( h / R_e )( R_e m g / k / T ) / ( 1 + h / R_e ) ]
Sorry, didn't check.
>
> but since R_e = 6370km, and h = 1km, (1 + h / R_e) = 1 is an excellent
> approximation so the formula becomes
>
> P/P0 = exp[ -( h / R_e )( R_e m g / k / T ) ]
> = exp[ -739 ( h / R_e )]
>
I can't find the post where you derived the potential
energy at a height h above the rim of a habitat of radius R.
So here's mine, assuming artificial gravity on the rim of 1 g.
The radius from the axis is R-h, and centrifical force goes as
radius, so the force must be (mg/R)*(R-h). We choose the zero
of potential energy to be when h = 0, just as in your formula
for the Earth. We get this potential U by integrating the
force, so we have:
U = Integral(0,h) of (mg/R)*(R-t) dt
= (mg/R)*[Rt - t^2/2] Evaluate(0,h)
= (mg/R)*[Rh - h^2/2]
= mgh*[1 - (h/2R)]
As you point out above in the case of the Earth, this is
also approximately equal to mgh for small h.
> At 1km on Earth, P/P0 = 0.89, but it is worth repeating again that the
> formula is different, exp[-h] dependence instead of exp[-h^2], and the
> radius used in each formula is vastly different. So it is a bad idea to
> make direct numerical comparisons of pressure gradients between Earth
> and small, spinning habitats.
We're doing a habitat with R = 5, and are considering h = 1.
But then [1 - (h/2R)] is .9, which is not that far from 1, and I have
trouble seeing why it should make such a huge difference...
---David
More math than usual, at least.
_______________________________________________
http://www.mccmedia.com/mailman/listinfo/brin-l
