🚀 go-pugleaf

-From: marc@r-node.gts.org (Marc Fournier - Admin)
-Subject: Re: Dyson's Spheres
-Date: 28 Oct 92 22:16:02 GMT

-In article <1992Oct26.202658.154145@zeus.calpoly.edu> jgreen@zeus.calpoly.edu (James Thomas Green) writes:
->Why not make the radius of the DS larger than 1 AU to avoid heating 
->up the inside too much?

-	I'm not a scientist, but how would that help??  The energy would still
-be trapped, wouldn't it?  Supposing a solid sphere, that is.

-Marc G. Fournier      | R-node Public Access Unix running UnixBBS 1.10
-Etobicoke, Ontario    | 416-249-5366   24hrs 7 days/week   network email

If the energy is really trapped, your sphere will melt or explode. A long-term
stable sphere must radiate over its entire surface at a rate equal to the
heat production of the central star. For a uniform radiating surface, the
radiation per square meter must equal the intensity of incident sunlight
at that distance from the sun. Therefore, the greater the radius of the
sphere, the fewer watts per square meter have to be radiated from the outer
surface.

For a given radius, you can get the most effective radiation (and thus the
lowest outer temperature) if the outer surface of your sphere approximates
a "black body" (completely nonreflective at all wavelengths). The formula
for blackbody radiation is:   power radiated = k * (Ts^4 - Ta^4), where
k is the Stephan-Boltzmann constant (which is listed in the faq list as 
5.6697E-8 W / (m^2 * k) (should be k^4)), Ts is the temperature of the radiating
surface (absolute temperature, measured in kelvins), and Ta is the ambient
temperature. The background radiation of space is about 3K, but light from
stars, etc. heats it up a little more. I think Henry posted the effective
temperature of the sky a few months ago, and I seem to recall it as being
about 15K. In any event, the Ta term is negligible when we want Ts to be
comfortable for humans.

As I posted before, a sphere around the sun of 1AU radius would be much too
hot for humans to live on (except maybe in refrigerated compartments).
I calculated the temperature at 1AU radius (given 1370 W/m^2 at 1AU) as
394K = 121 C = 250 F (very uncomfortable!). If you want the outer surface
of the sphere to be a comfortable 25 C, the power radiated from the outer
surface must be ~447 W/m^2, so the radius of the sphere must be increased
to 1.75 AU.

There's a further complicating factor - a true black body surface is
unobtainable. Given a flat spectral response (again unrealistic), efficiency
of radiation is directly proportional to the emissivity(?), which is 
equal to one minus the reflectivity. For instance, a 90% reflective 
surface will only radiate 10% as much energy at a given temperature 
as a black body would, so for a given amount of power radiated, the 
temperature of the surface is greater by the fourth root of the 
inverse of the emissivity. So a 1AU Dyson sphere with a 90% reflective 
outer surface would have an outer temperature of 
394 * 10^.25 = 701K = 428 C = 802 F. To get back down to 25 C, you'd have to
move the radius out to 5.5 AU. Fortunately, I believe most metals have
pretty low reflectivity at thermal infrared wavelengths, and black paint
has emissivity of over 90%.

John Roberts
roberts@cmr.ncsl.nist.gov