inkliing

Hi. In 3 dimensional Euclidean space with the usual metric, d=[(delta x)^2+(delta y)^2+(delta z)^2]^1/2, I'm trying to figure out the average distance between nearest neighbors in a randomly distributed sample of particles. My best initial guess for the average distance from any given particle to its nearest neighbor is d_nearest neighbor_mean=(volume/n)^1/3 where n particles are randomly distributed in a 3 dimensional volume. The question originated when I wondered what was the average distance between stars in the solar neighborhood. atlasoftheuniverse.com gives 35 stars (including the Sun) within 12.5 light-years, and the above formula yields 6.16 ly as the avg distance from any given star to its closest neighbor. This seemed a little high to me, since the distance from the Sun to its nearest neighbor (Proxima Centauri) is 4.4 ly. But perhaps the Sun has a closer-than-avg nearest neighbor, since, after all, the distribution should be very close to random. Let us assume that the stars are randomly distributed. I originally thought it would be easy to figure this out, but after trying unsuccessfully for an hour to work out a better formula, then another hour trying to google one, I gave up. Thanks in advance

i'm confused about how COBE (the Cosmic Background Explorer) measured the frequency peak of the Cosmic Microwave Background (CMB). the following seems clear to me and relatively straightforward: for black body radiation, i.e., for an ideal photon gas in local thermodynamic equilibrium with matter, e.g., the surface of last scattering of the CMB, the spectral radiance, I_nu (T) = [2h/c^2][nu^3/(exp(h nu/[k T])-1)] or I'_lamba (T) = [2hc^2][1/lamba^5(exp(h c/[lambda k T])-1)] <forgive my clumsy notation, i don't know how to do LaTeX notation>, peaks at nu_max or lambda_max, respectively, such that c/lambda_max = 1.76*nu_max. although the lambda_max peak is the actual energy peak at which a black body radiates the maximum energy [photons at lambda_max are 1.76 times as energetic as photons at nu_max], radio astronomers seem to prefer the frequency peak. in fact, doing a google image search of "cosmic microwave background," then picking out spectral graphs [spectral radiance vs frequency or wavelength], will find almost exclusively graphs depicting the frequency peak. examples include: http://map.gsfc.nasa.gov/media/ContentMedia/990015b.jpg http://www.phy.duke.edu/~kolena/cmbspectrum1.gif which show the frequency peak of 384 MJy/sr = 384 megaJanskys per steradian = 3.84e-18 W/(m^2-Hz-sr) = I_nu (2.725 K) at 1.87mm (160 GHz). http://en.wikipedia.org/wiki/File:Firas_spectrum.jpg which shows the frequency peak of 1.15e-4 erg/(s-cm^2-cm^-1-sr) = 3.84e-18 W/(m^2-Hz-sr) = I_nu (2.725 K) at 1.87mm = 5.34 cm^-1 (160 GHz). my understanding is that the FIRAS interferometer on board COBE compared the spectral radiance of the CMB to an on-board black body. what is NOT clear to me is exactly how these measurements were made. i would think that if u measured the CMB's [or any black body's] spectral radiance, u would measure the wavelength peak, not the frequency peak, since the wavelength peak is the physical maximum, that is, the point on the EM spectrum where the black body radiates the maximum energy. the frequency peak seems to me to have no physical significance and to only be a mathematical tool. i don't see how physical measurements can measure any energy peak other than the wavelength peak. i would very much appreciate it if some1 could clear this up for me. thx in advance. ö¿ö¬ E=mc² ~