From: perlman@qso.Colorado.EDU (Eric S. Perlman)
Subject: Re: Apparent brightness of Sun's surface vs distance
Organization: University of Colorado, Boulder
Date: Mon, 11 Jan 1993 15:04:54 GMT

In article <73361@cup.portal.com> Van@cup.portal.com (Sylvan Van Jacques) writes:
> I posted this in the Space and Science section of Genie, and was met with 
>disbelief. The more I think about this, the more it seems to make sense. I 
>would appreciate any comments.
>   ======
>  I was once told that the apparent brightness of the Sun would be the same
>from Venus, Earth, Mars, or at any distance. It would only appear smaller.
>

This is correct.  

>Is this true? How does one reconcile this with the 1/r^2 decrease in the
>intensity of radiation received by a given area, whether it be the eye or a
>planet? (Venus gets a lot more radiation than the Earth.)

Ignoring cosmological effects (standard Friedmann cosmology also
includes a (1+Z)^-4 decrease of surface brightness with redshift), 
surface brightness remains a constant.  Think of it this way:  as an
object is moved away from you, the amount of (angular) area it takes up
on the sky decreases as 1/r^2.  But the amount of area this corresponds
to on the object *increases* as 1/r^2.  Therefore if all areas of the
appropriate surface are equal in brightness, then surface brightness 
remains constant.

> One could say that the reason the flux goes down as
> 1/r^2 is that the apparent area of the sun goes down as 1/r^2, but
> the apparent brightness of the sun is the same.

I think you're getting things confused here.
 
> Does this breakdown when we get so far way that we can't resolve the source?
> Point sources, like stars, do obey the 1/r^2 law for apparent brightness,
>don't they?

Certainly (though interferometry techniques can resolve stars, and are
coming into reasonably large-scale use).

>  How about this? Suppose that the Sun had twice the radius, but the same
>brightness. Any surface would receive 4 times as much radiation (wouldn't it?).
> But it would not appear any brighter to our eyes (again, unless it were so far
>away that we couldn't resolve the surface). Isn't this the inverse of what I 
>posted above?

No.  Then you have a third factor coming into effect:  the limited
resolving power of your instrument, be it the human eye, a telescope,
or whatever.  The other two factors are independent of these.  What
then occurs is that you still receive less and less light (with 1/r^2),
but because of the poorer resolution of your instrument (i.e. the
Rayleigh diffraction limit is greater than the angular size of your
star, planet, galaxy, whatever), you perceive it as a point source. This
doesn't affect brightness.  Only distance and intrinsic luminosity
(and of course any intervening absorbing matter, etc) do that.
Similarly, on the surface of the earth, seeing is rarely better than 1
arcsecond.  Therefore if the angular size of your object is <1 arcsec,
it still appears as 1 arcsec, and its light gets spread out (in roughly
a gaussian) over that arcsecond.  

> Does this make sense to anyone?
> Has anyone else heard this about the sun?
>  Does anyone see this as an apparent paradox?

No paradox, just a bit of confusion, I think.

-- 
"How sad to see/A model of decorum and tranquillity/become like any other sport
A battleground for rival ideologies to slug it out with glee." -Tim Rice,"Chess"
     Eric S. Perlman 				 <perlman@qso.colorado.edu> 
  Center for Astrophysics and Space Astronomy, University of Colorado, Boulder