Thanks to everyone for their kind words and recommendations.  The orb itself
was quite simple to construct:

difference{
  sphere{
    0,1
  }
  #declare l2=-80;
  #while (l2<90)
    #declare l1=0;
    #while (l1<360)
      box{
        <.95,-.4,-.4>,<1.1,.4,.4>
        rotate <0,0,l2>
        rotate <0,l1,0>
      }
      #declare l1=l1+22.5;
    #end
    #declare l2=l2+20;
  #end
  texture{
    pigment { rgbt < 1,1,1,1 > }
    finish { specular .1 diffuse .1 reflection .4}
  }
  interior{
    ior 2.4
    dispersion .9
    dispersion_samples 50
  }
  photons{
    target 1.0
    refraction on
    reflection on
  }
  bounded_by{
    sphere{
      0,1
    }
  }
  rotate <-12,0,18>
}

Note that I did not set the ambient level to 0, so it becomes a light source
itself with radiosity.

I am quite certain that the strange disk of points in the orb's shadow is at
least an artefact, it's suprising that there weren't more of them in a
scene with millions of bouncing photons.   I got a warning that the
max_trace_level had been reached, even though it had been set (in the
global photon block) to what is to me an  incredible 41.  It would be nice
if POV could pop up a dialog box when this happens, allowing you the option
of increasing it.  While this was rendering, windows increased the size of
my page file to over one gig, no suprise considering the huge photon map.
What was a suprise at first was the erratic CPU usage.  While doing the
area above the orb the CPU ran flat out, with a render speed well over 100
PPS.  When it started on the orb, the speed slowly dropped, eventually to
under 1 PPS near the center.  CPU usage dropped dramatically, often to just
an idle with occasional spikes.  A friend pointed out that the hard disk
was in constant use, even though no other major apps were running.  I would
guess than in situations like this the speed of the CPU is not the
bottleneck, hard disk access speed is.  A possible solution I'll try is
relocating the swap file on a second hard drive.

A question that comes to mind is how random is dispersion without jitter?
Suppose I am doing an animation (I truly am a cruel task master) where the
orb spins.  The facets are 22.5 degrees apart.  If I use no anti-alias and
no photon jitter the orb rotated 22.5 degrees should produce exactly the
same scene as the orb with no rotation, but I have a sinking feeling this
will not be true.  More tests on the way - I'll let you know.

An armillary (or spherical astrolabe) makes an excellent sundial but is by
no means limited to that.  Their history goes back to ancient times and
they have been used for real research, teaching,  garden ornamants (very
popular now, often totally non-functional) and as VERY elaborate fakes
intended
to fill the pockets of their makers and empty the pockets of his un-educated
patron.

My model will be as functional and astronomically accurate as I can make it,
with rings to show celestial equator and poles, latitude, longitude, the
tropics, the arctic and antarctic circles, and the plane of the ecliptic
with the zodiac constellations.  I promise lots of brass :)

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