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Stolmar's cosmological model uses a tired light redshift. The Earth is located
in the center of a uniform density sphere of stars extending out to some
maximum radius *R _{max}* which is about 7 or 8 Hubble radii.
The redshift factor

In his CMB page
on 27 July 2001
Stolmar gave the following equation for the solid angle covered by stars
in a shell with radii between *R _{-}* and

where *A* is the area of
a star and *n* is the number density of stars.
Stolmar gave this equation for the energy in the background radiation:

where the index *j* is the frequency in GHz.
It is hard to determine exactly what is meant by this equation,
since it appears to give the odd combination of energy density per unit
wavelength divided by the frequency, but
Stolmar also gave the equation for a blackbody:

which can be used to normalize the previous equation.
Now Stolmar also assumes *(1+z) = exp(HR/c)*. If I take the limit of
infinitesimally small shells in radius, I get a normalized
integral for the specific intensity of the CBR which is

where *y = H R/c*, *f* is the frequency in Hz, and
*I _{f}* is in
[erg/cm

Figure 1: Stolmar's model with *T _{*} = 4000* and

Since Stolmar's cosmology career on sci.astro started with the announcement
of the DIRBE far IR background, it is interesting to plot his model on a
much wider frequency range and compare not just to the CMB but also to
the IR and optical backgrounds.
This is shown in Figure 2.
The long *I _{f}* proportional to

Figure 2: Stolmar's model compared to FIRAS, DIRBE, HST, groundbased and
far UV measurements of the cosmic background.
Since Stolmar's graphs do not extend to such short wavelengths, I have
evaluated my integral version of his equations numerically.
Green curve: Stolmar's model. Black curve, FIRAS BB fit.
Black IR points: Hauser *et al.* (1998) on the CIRB.
Red points: Wright *et al.* (2000, 2001) on the CIRB.
Magenta upper limits from lack of TeV gamma-ray absorption.
Blue points: Bernstein *et al.* (2001) on the optical background.
Black optical and UV points: Toller; Dube *et al.*;
and Hurwitz *et al.*

The stars at the edge of the sphere that produce the peak of the CMB are radiating at a time that is 7 or 8 Hubbles times ago. That is nearly 100 billion years, and only very low mass red dwarfs last that long. But low mass M dwarfs produce very little radiation, and Stolmar's model requires a lot of radiation. On 24 July 2001 Stolmar changed his H to 160 km/sec/Mpc, which alleviates this problem but disagrees with the data on the Hubble constant.

If we are not in the center of the Universe, a large dipole anisotropy is
produced that has the spectrum of a graybody at temperature
*T _{*}/exp(y_{max})*. The observed dipole anisotropy
has a different spectrum. Thus we must be nearly exactly in the center of
the sphere of stars. A rough analysis of the FIRAS dipole anisotropy
spectrum suggests that we must be centered within Stolmar's sphere of stars
to within 1 part in 100,000 of

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© 2001 Edward L. Wright. Last modified 29 Jul 2001