```Scalefactor Function of Time
--------------------------
Integrate the energy equation to get R = R(t)

1der(R,t)^2 / 2 = 4 * pi * gamma * rho0 / (3 * R)

1der(R,t)^2 = 8 * pi * gamma * rho0 / (3 * R)

R * 1der(R,t)^2 = 8 * pi * gamma * rho0 / 3

R^(1/2) * 1der(R,t) = (8 * pi * gamma * rho0 / 3)^(1/2)

R^(1/2) * dR = (8 * pi * gamma * rho0 / 3)^(1/2) * dt

intl(R^(1/2),R) = (8 * pi * gamma * rho0 / 3)^(1/2) * intl(1,t)

(2/3) * R^(3/2) = (8 * pi * gamma * rho0 / 3)^(1/2) * t+C

C [large-c] is the integration constant and establishes the
scalefactor of the universe at t = 0; (2/3)*R^(3/2) = C. By the
bigbang hypothesis the scalefactor is identicly zero at zero time, so
C = 0.

(2/3) * R^(3/2) = (8 * pi * gamma * rho0 / 3)^(1/2) * t

R^(3/2) = (3/2) * (8 * pi * gamma * rho0 / 3)^(1/2) * t

R = (2/3)^(3/2) * (8 * pi * gamma * rho0 / 3)^(1/3) * t^(2/3)

+-------------------------------------------------------------+
| SCALEFACTOR EQUATION                                        |
|                                                               |
| R = (3/2)^(2/3 ) *( 8 *pi * gamma * rho0 / 3)^(1/3) * t^(2/3) |
+---------------------------------------------------------------+

If we normalize the scalefactor and elapsed time to the values at
the present we get

+-------------------------+
| NORMALIZED SCALEFACTOR  |
|                         |
| R / R0 = (t / t0)^(2/3) |
|                         |
| R = R0 * (t / t0)^(2/3) |
|   = (t / t0)^(2/3)      |
+-------------------------+

The scalefactor increases with the 2/3 power of the elapsed time
since the bigbang. R increases at a declining rate. H in the past is
greater than H0 and t0 is less than T0.

True Age of the Universe
----------------------
Retrosolve the scalefactor equation for t

R = (3/2)^(2/3) * (8 * pi * gamma * rho0 / 3) * (1/3) * t^(2/3)

(t / t0)^(2/3) = (3/2)^(2/3) * (8 * pi * gamma * rho0 / 3)
* (1/3) * t^(2/3)

t^(2/3) * t0^(-2/3) = (3/2)^(2/3) * (8 * pi * gamma * rho0
/ 3)^(1/3) * t^(2/3)

t0^(-2/3) = (3/2)^(2/3) * (8 * pi * gamma * rho0 / 3)^(1/3)

t0 = (3/2)^(-1) * (8 * pi * gamma * rho0 / 3)^(-1/2)
= (2/3) * (8 * pi * gamma * rho0 / 3)^(-1/2)
= (2/3) * (1 / H0)
= (2/3) * T0

t0 = 2 * T0 / 3

+-----------------------------------+
| AGE OF THE UNIVERSE               |
|                                   |
| t0 = (2 / (3 * H0)                    |
|    = 2 * T0 / 3                       |
|                                   |
| t0[lo] = 13.2Gy,   t0[hi] = 8.8Gy |
+-----------------------------------+

Deceleration Parameter
--------------------
The acceleration 2der(R,t) is more conveniently rolled into a
dimensionless number, the deceleration parameter q

+----------------------------------+
| DECELERATION PARAMETER           |
|                                  |
| q = -R*2der(R,t) / (1der(R,t))^2 |
|   = -2der(R,t) / (H * R)         |
+----------------------------------+

For the standard bigbang scenario q is

q = -R * 2der(R,t) / (1der(R,t))^2
= (-R * (-4 * pi * gamma * rho0 / (3 * R^2))
/ (8 * pi * gamma * rho0 / (3 * R))
= -3 * R * R * (-4) * pi * gamma * rho0
/ (8 * pi * gamma * rho0 * 3 * R^2)
= -(-4) / 8
= 1/2

Density Parameter
---------------
Until now we allowed that the observed H0 and rho0 are concordant.
That is, the equation H0 = (8*pi*gamma*rho0/3)^(1/2) in fact balances
with substitution of the H0 and rho0 we actually assess. Sadly it does
not. The H0 and rho0 are altogether out of joint. Fixing H0 yields a
rho0 that's two orders more than observed. Fixing rho0 gives an H0 two
orders too small.
in the 2-thous with observations of dark energy and dark matter
the discrepancy is substantially closed. Here we continue the
historical development to appreciate how and why the discord affected
cosmology.
Being that H0 seems more certain by far than rho0 we set H0.
There derives then a rho0 that would in fact concord with the accepted
H0, rho@. rho@ IS NOT EQUAL TO rho0!! But it is, never the less, a
constant and not a function of time. That is

(1der(R,t) / R)^2 = H0^2
= 8 * pi * gamma * rho@ / 3

rho@ = 3 * H0^2 / (8 * pi * gamma)

This rho@ is that density which satisfies H0 in the flat universe
model. We call this density the critical or closure density. Its value
is governed by the value taken for H0.

+------------------------------------+
| CLOSURE OR CRITICAL DENSITY        |
|                                    |
| rho@ = 3 * H0^2 / (8 * pi * gamma) |
|                                    |
| rho@[lo] = 4.6E-27kg/m^3           |
| rho@[hi] = 10.3E-27kg/m^3          |
+------------------------------------+

This rho@ amounts to 3 to 6 protons per cubic meter. So important
is it to distinguish rho0 from rho@ that a new factor, the density
parameter OMEGA, is defined

+--------------------------------------------+
| DENSITY PARAMETER                          |
|                                            |
| OMEGA = rho0 / rho@                        |
|       = rho0 / (3 * H0^2 / 8 * pi * gamma) |
|       = 8 * pi * gamma * rho0 / (3 * H0^2) |
+-----------------------------------------7--+

Under the standard bigbang model

rho0 = rho@

OMEGA = rho0 / rho@
= rho@  / rho@
= 1

This, with q = 1/2 from above, yields

+--------------------------+
| VALUE OF q AND OMEGA     |
| FOR THE STANDARD BIGBANG |
|                          |
| q = 1/2                  |
|                          |
| OMEGA = 1                |
|       = 2 * q            |
+--------------------------+

Radius of the Universe
--------------------
A common query is for the 'size' of the universe or 'how much' it
expanded so far. The answer comes from the peculiar model at hand but
there is a plausible upper size the universe can have. If the universe
expanded at the speed of light since its creation, then

r0 = c * t0

This c, lightspeed, is not a physical limit on the expansion rate.
Nothing, neither mass nor energy, actually moves at this speed in the
outswell of the universe. Space can expand at any arbitrary rate with
no violation of Einstein physics. We set delr/delt to c as a
reasonable most rapid rate under the extant models of the universe.
The t0 is the age of the universe, the time elapsed since the
bigbang. This is 2/3 of the Hubble time T0, which in turn is 1/H0.
Given the range of H0 from 1.6E-18/s to 2.4E-18/s we have

t0[lo] = 4.17E17s

t0[hi] = 2.78E17s

Thus the upper limit on the radius of the universe is

r0 = c  * t0
= (2.998E8m/s) * [4.17:2.78]E17s
= [12.51 : 833]E25m

This is the range that most astronomers cite when asked about the
'size' or 'radius' of the universe.

The volume is found by

V0 = (4 * pi / 3) * r0^3
= (4 * pi / 3) * ([12.51 : 8.33]E25]m)^3
= (4 * pi / 3) * [19.58 : 5.78]E77]m^3
= [8.20 : 2.42]E78]m^3

+-------------------------------------------+
| RADIUS, OR SIZE, AND VOLUME OF UNIVERSE   I
|                                           |
| r0[lo] = 12.51E25m,  r0[hi] = 8.33E25m    |
|        = 13.2Gly,           = 8.8Gly
|                                           |
| V0[lo] = 8.20E78m^3,  V0[hi] = 2.42E78m^3 |
|        = 9.7E3Gly^3,         = 2.8E3Gly^3 |
+-------------------------------------------+

Mass of the Universe
------------------
We do not know at all how much matter there is in the universe.
However, we know how much there should be for a given model. For the
standard bigbang, where OMEGA = 1 or rho0 = rho@, the density of this
mass is

rho@ = 3  * H0^2 / (8 * pi * gamma)
= (3 / (8 * pi * (6.672E-11n.m^2/kg^2))
* ([1.6:2.4]E-18/s)^2
= (1.789E9kg^2/n.m^2) * ([1.6:2.4]E-18/s)^2
= [4.58 : 10.31]E-27kg/m^3

Recall that the observed ordinary matter in the universe amounts
to only 4E-28kg/m^2, or from 3% to 10% of the closure density. This
constitutes the missing-mass problem if one declares that rho0 must
equal rho@.In the 2-thous the missing mass problem evolved into the
acceptance and observation of dark matter an d ark energy.. Ordinary
matter is in fact about 4% of these combined. We have no general
consensus of what dark energy or dark matter is.
The mass of the universe is the density times the volume,

m0 = rho0 * V0
= ([4.58 : 10.31]E-27kg/m^3) * ([8.20 : 2.42]E78m^3)
= [3.76 : 2.50]E52kg
-> [1.9 : 1.3]E22 suns

This is in fact the working value of the mass of the universe as
cited in other treatises from more elaborate calcs.

+-------------------------------------------------------+
| DENSITY AND MASS OF THE UNIVERSE                      |
|                                                       |
| rho0[lo] = 4.58E-27kg/m^3,  rho0[hi] = 8.21E-27kg/m^3 |
|                                                       |
| m0[lo] = 3.76E52kg,  m0[hi] = 2.50E52kg               |
|        = 1.9E22 suns,       = 1.3E22 suns             |
+-------------------------------------------------------+

Note that the compensating action of the density and volume make
the range of mass estimates rather narrow. One could say that for any
value of H0 the mass of the universe is about 2E22 suns.

Is the Universe a Blackhole?
--------------------------
A common inquiry is whether the universe is itself a gigantic
blackhole. That may be, is all of the mass in the universe contained
within the Schwarzschild radius for that mass? A blackhole is created
when the enclosing radius around a given mass is within the
with our understanding of what's inside the Schwarzschile radius
of a blackhole, it is obvious that we are NOT in a blackhole. All
matter within that radius would be pulled into the central
singularity. We would have utterly no material  existence to
appreciate the universe, such as by studying this article.
Given the approximations for the radius and the mass, we can play
with the blackhole question. The question can not be positively
answered. Yet, the possibilities are interesting. The Schwarzschild
radius for a given mass m0 is

r| = 2 * m0 * gamma / c^2
= 2 * ([3.78 : 2.50]E52kg)
* (6.672E-11n.m^2/kg^2) /( 2.998E8m/s)^2
= [5.61 : 3.71]E25m

If we compare these to the radii obtained from the density-volume
approach we have

| density-volume | Schwarzschild |
| -------------- | ------------ |
H0[lo] = 1.6E-18/s | 12.51 E 25 m   | 5.61 E 25 m  |
H0[hi] = 2.4E-18/s |  8.33 E 25 m   | 3.71 E 25 m  |

At first glance it appears that the mass of the universe spills
beyond the Schwarzschild radius and therefore the universe is not a
blackhole. But we calcked the radius based on an expansion at the
speed of light from t = 0 to today. This is unrealistic in as much as
all the bigbang models derive a continuously declining rate of
expansion until the very tiny rate today. So reasonably the true
'radius of the universe' is considerable less than that from the
density-volume method. By detailed means it turns out that the
Friedmann model with k = 0, OMEGA = 1 expands ultimately right up to
but never reaching the Schwarzschild limit. The flat and open universe
is a blackhole, as is the spherical and closed one. The hyperbolic and
open case allows for a universe to exist beyond its Scwarzschild
limit. From this we gather that the universe could in fact be one
humongous blackhole.

Proper or Einstein Distance
-------------------------
The expansion phaenomenon, based on Einstein physics, forces us to
revamp our notions of time and space within the universe. The Euclidrs
regime of Newton physics does not work. In Einstein physics the
distance between two points is a four-dimensional measure and not the
three-dimensional one of Newton physics. Specificly

delD^2 = (delx^2 + dely^2 + delz^2) - (c * delt)^2

as contrasted to the Newton delD^2 = delx^2+dely^2+delz^2. Placing the
one point at us and the other at the remote site, the x,y,z component
is the radial distance of the remote from us

delx^2 + dely^2 + delz^2 = delr^2

delD^2 = delr^2 - (c * delt)^2

The delr is a function of time thru R(t)

delD^2 = (R(t) * delr)^2 - (c * delt)^2

We are attached to the remote galaxies by the photons of light
(and other electromagnetic radiation) that procede from them to us.
These rays travel along geodesics such that the speed of light c is a
constant and the total distance delD is zero. That is, a photon is
everywhere  and everywhen. Setting delD = 0

(R(t) * delr)^2 - (c * delt)^2 = delD^2
= 0

(c * delt)^2 = (R(t) * delr)^2

c * delt =  +-]R(t) * delr

The choice of the plus or minus sign is made by the sign convention
for r and t. As the ray procedes from the remote to us, t increases
and r decreases. So we take the minus sign

c * delt = -R(t) * delr

c * delt / R(t) = -delr

This can be evaluated only if we know the actual function of time for
the scalefactor R. In the standard bigbang model we have

R(t) = (t / t0)^(2/3)

c * (1 / (( t / t0)^(2/3)) * delt = -delr

c * t^(-2/3) * t0^(2/3) * delt = -delr

c * t0^(2/3) * intl(t^(-2/3),t,t..t0) = intl(-1,r,r..0)

where these are definite integrals over the span r..0 and t..t0. r is
the distance to the remote; r0 = 0 is the distance to our own selfs; t
is the time of emission of the photon from the remote; t0 is the time
of reception of the photon at us.

3 * c * t0^(2/3) * t^(1/3)[t..t0] = -r[r..0]

3 * c * t0^(2/3) * (t^(1/3) - t0^(1/3)) = -(r- 0 )

3 * c * t0 * ((t / t0)^(1/3) - 1) = -r

t0 = 2 * T0 / 3
= 2 / (3 * H0)

(t0 / t)^(2/3) = Z+1

(t0 / t) = (Z+1)^(3/2)

3 * c * (2 / (3 * H0) * ((t / t0)^(1/3) - 1) = -r

3 * c * (2 / (3 * H0) * ((Z+1)^(-3/2))^(1/3)) - 1 = -r

(2 * c / H0) * ((Z+1)^(-1/2) - 1) = -r

r = (2 * c / H0) * (1 - (Z+1)^(-1/2))

+----------------------------------------+
| PROPER OR EINSTEIN DISTANCE            |
|                                        |
| r! = (2 * c / H0) * (1 - (Z+1)^(-1/2)) |
+----------------------------------------+

It is this proper distance, as it's called in Einstein physics,
that separates us from the remote site. In Newton physics with Z=0 it
degenerates to the ordinary distance in 3D space. Manipulations of
this r! produces several new features of the expanding universe.

Hubble Horizon
------------
The proper or Einstein distance between us and the remote is a
function of Z and H0. For a set H0 there is an absolute maximum
distance away beyond which we can never see!  This occurs at Z =
infinity and it is called the Hubble horizon. That is

rH = (2 * c / H0) * (1 - (Z+1)^(-1/2))
= (2 * c / H0) * (1 - (infy + 1)^(-1/2))
= (2 * c / H0) * (1 - infy^(-1/2))
= (2 * c / H0) * (1 - 0)
= (2 * c / H0) * 1
= 2 * c / H0

+------------------+
| HUBBLE HORIZON   |
|                  |
| rH = 2 * c / H0  |
|                  |
| rH[lo] = 40.0Gly |
| rH[hi] = 26.1Gly |
+------------------+

Enclosed Volume
-------------
It is routine to fathom the universe in boxes of equal volume, as
to count galaxies and determine rho0. The volumes are shells centered
on the Earth, with front and rear faces of suitably chosen radii. But
the radii are a function of Z. Hence the enclosed volume is that
between spheres of radius r(Z) and r(Z+delZ) or between r! and
(r+delr)!.

r! = (2 * c / H0) * (1 - (Z+1)^(-1/2))

1der(r!,Z) = (2 * c / H0) * 1der((1-(Z+1)^(-1/2)),Z)
= (2 * c / H0) * 1der((Z+1)^(-1/2),Z)
= (2 * c / H0) * (1/2) * (Z+1)^(-3/2)

v! = 4 * pi * (r!^3) / 3

1der(v!,Z) = (4 * pi / 3) * 1der(r!^3,r) * 1der(r!,Z)
= (4 * pi * (r!^2) * 1der(r!,Z)
= (4 * pi * ((2 * v / H0) * (1 - (Z+1)^(-1/2)))^2
* (2 * c / H0) * (1/2) * (Z+1)^(-3/2)
= (16 * pi * (c^3) / (H0^3))
* (1 - (Z+1)^(-1/2))^2 * ((Z+1)^(-3/2))

+--------------------------------------------------+
| VOLUME OF THE UNIVERSE                           |
|                                                  |
| v! = (4 * pi / 3) * (r!^3)                       |
|                                                  |
| 1der(v!,Z) = (2 * c / H0) * (1/2) * (Z+1)^(-3/2) |
+--------------------------------------------------+

In practice one finds r! for the inner, nearer, front boundary of
the shell and adds to it the increment delr! corresponding to the
desired delv!, all worked out thru the redshift Z+1.

Light Travel or Lookback Time
---------------------------
An other useful feature is the time it takes the light to traverse
the geodesic from the remote to us.

t! = t0 - t
= t0-(1 - (t / t0))
= t0 - (1 - (Z+1)^(-3/2))
= 2 * (1 - (Z+1)^(-3/2)) / (3 * H0)
= (2 /(3 * H0)) * (1-(Z+1)^(-3/2))

+------------------------------------------+
| LIGHT TRAVEL OR LOOKBACK TIME            |
|                                          |
| t! = (2 / (3 * H0)) * (1 - (Z+1)^(-3/2)) |
+------------------------------------------+

The notion of lookback time comes from the fact that the
information content of the light arriving at us was put into that ray
at its emission. Since the light was emitted t! ago, the information
is ediurnate by t! or we are seeing the remote now as it was t! in the
past. This lookback time is NOT the same as the proper distance.
However, astronomers usually think of the lookback time when citing
distances to the remote targets even tho they may not be calculating
this correct light travel time. They may use the r = c*Z/H0 formula
and deem this r to be the distance in lightyears and the lookback time
in years.

Inverse-Square Law
----------------
Distance is altered also for the use of the inverse-square law, by
which the remotes illuminate us. The light (or other EMR) comes to us
via photons of energy = h*c/lambda as emitted and come at the rate of
so  many per second. This is diffused by distance onto a sphere on
which we sit. The radius of this sphere is the proper distance r!.
However, as received the light has wavelength lambda0 =
lambda*(Z+1) and the increment of time delt is increased to delt0 =
delt*(Z+1). Hence, the flux -- astronomers express this as apparent
magnitude -- f0 produced by a remote of power P at distance r! is

f0 = P / (4 * pi * r!^2)
= (h * c / lambda0) * (1 / delt0) / (4 * pi * r!^2)
= (h * c / (lambda0 * delt0)) / (4 * pi * r!^2)
= (h * c / (lambda * (Z+1) * (delt * (Z+1))) / (4 * pi * r!^2)
= (h * c / (lambda * delt)) / (4 * pi * r!^2*(Z+1)^2)
= P / (4  * pi * (r! * (Z+1))^2)
The r!*(Z+1) in the denominator behaves like an ordinary distance
in the inverse-square law and is called the luminosity distance r&.

r! = (2 * c / H0) * (1 - (Z+1)^(-1/2))

r& = r! * (Z+1)
= (2 * c / H0) * (1 - (Z+1)^(-1/2))*(Z+1)
= (2 * c / H0) * ((Z+1) - (Z+1)^(1/2))

+-------------------------------------------+
| LUMINOSITY DISTANCE                       |
|                                           |
| r& = r!* ( Z+1)                           |
|    = (2 * c / H0) * ((Z+1) - (Z+1)^(1/2)) |
+------------------------------------------+

Some treatments derive the luminosity distance first and use it as
the primary measure of remoteness, being that in the end we know of
the remote's existence by the flux they raise up at us. They then do
not separately derive the proper distance, but disguise it as r&/(Z+1)
in other equations.

Spectroillumination
-----------------
We can not capture all the wavelengths of radiation from a target
with equal sensitivity. This is a shortfall of our instrumental
ability and the blockage (for ground observatories) of various
spectral zones by Earth's atmosphere. Measurements are made between
two particular wavelengths bounding a zone of width dellambda0 and at
a central wavength lambda0. Hence the illumination is really a
spectroillumination cited in flux0/(receiving area*dellambda0)
centered at lambda0. The inverse-square law developed above is rarely,
if ever, applicable because it presumes the capture of the entire
spectrum from the target.
Traditionally the various spectral zones are divvied by wavelength
or by frequency. Optical astronomers, for example, exclusively work
with wavelength. Radioastronomers usually think in frequency. Being
that c = frequency*wavelength it is trivial to crosswalk between the
two. In all but the most exact work the diminution of lightspeed
outside of vacuum is ignored; c is always 2.998E8m/s (about 3E8m/s).
Here we use wavelength as being the more familiar to home astronomers,
who deal virtually only with the photozone of the spectrum.

+-------------------------------+
| WAVELENGTH-FREQUENCY EQUATION |
|                               |
| c = nu * lambda               |
+-------------------------------+

Hubble Expansion of lambda
------------------------
Radiation from a remote centered at lambda and bounded within
dellambda is redshifted by the Hubble expansion. The received central
wavelength is lambda0 and the received bandwidth is dellambda0.
We have two ways to measure this flux. One is to keep the
spectrometer set to lambda and dellambda as emitted by the target. The
other is to tune the spectrometer to lambda0 and dellambda0. In the
first case the emitted radiation misses the spectrometer and other
radiation, redshifted into the set lambda and dellambda, is received
in the stead. In the second case the emitted radiation at lambda and
dellambda is captured at lambda0 and dellambda0. Both methods are used
in astronomy and they must be carefully distinguished.
Consider the case of the spectrometer set to lambda0 and
dellambda0.

P = Plambda * lambda

f0 = flambda0 * dellambda0
= P / (4 *pi * r!^2 * (Z+1)^2)
= Plambda * dellambda / (4 *pi * r!^2 * (Z+1)^2)

dellambda0 = dellambda * (Z+1)

flambda0 = f0 / dellambda0
= f0 / (dellambda0 * (Z+1))
= P / (4 * pi * r!^2 * (Z+1)^2 * dellambda * (Z+1)))
= P / (4 * pi * dellambda * r!^2 * (Z+1)^3)
= Plambda / (4 * pi * r!^2 * (Z+1)^3)

Notice that the spectroillumination flambda0 is the illumination
f0 divided by an extra (Z+1) factor; it diminishes as the cube -- not
the square -- of the (Z+1) of the target.

+------------------------------------------------+
| INVERSE-POWER LAWS OF ILLUMINATION              |
|                                                |
| f0 = P / (4 * pi * r!^2 * (Z+1)^2)             |
|                                                |
| flambda0 = Plambda / (4 * pi * r!^2 * (Z+1)^3) |
+------------------------------------------------+

By substituting the definition of r! into the above we get the
power laws directly in the measured parameters H0 and (Z+1). H0 is
expressed in 1/s, NOT in the more usual km/Mly.s or km/Mpc.s. This
vastly simplifies the maths.

f0 = P / (4 * pi * r!^2 * (Z+1)^2)
= P / (4 * pi * (Z+1)^2 * ((2 * c / H0)
* (1 - (Z+1)^(-1/2)))^2)
= P / (4 * pi * (Z+1)^2 * (4 * c^2 / H0^2)
* ((1 - (Z+1)^(-1/2))^2)
= P * H0^2 / ((16 * pi * c^2 ) * ((Z+1) - (Z+1)^(1/2))^2)

flambda0 = Plambda / (4 * pi * r!^2 * (Z+1)^3)
= Plambda / (4 * pi * (Z+1)^3 * ((2 * c / H0)
* (1 - (Z+1)^(-1/2)))^2)
= Plambda / (4 * pi * (Z+1)^3 * (4 * c^2 / H0^2)
* ((1 - (Z+1)^-((1/2))^2)
= Plambda * H0^2 / ((16 * pi * c^2 * (Z+1))
* ((Z+1) - (Z+1)^(1/2))^2)

+-------------------------------------------------------------+
| INVERSE-POWER LAWS OF ILLUMINATION IN H0 AND (Z+1)          |
|                                                             |
| f0 = P * H0^2 / ((16 * pi * c^2) * ((Z+1) - (Z+1)^(1/2))^2) |
|                                                             |
| flambda0 = Plambda * H0^2 / ((16 * pi * c^2 * (Z+1))        |
|            * ((Z+1)  - (Z+1)^(1/2))^2)                      |
+-------------------------------------------------------------+

Magnitude-Distance Relation
-------------------------
Astronomers traditionally assessed the received fluxes from
sources on the Hipparchus 'magnitude' scale, not on the actual
photometric scale. This magnitude scale is formally defined by

mag1-mag2 = 2.5 * log(f02 / f01)

The logarithm is on base 10, the Brigg or common system.  This
definition is analogous to the decibel scale in acoustics and
electronics; a difference of magnitude values corresponds to a
[inverse] ratio of illuminations.
The flux used here is the total flux over all wavelengths, the
bolometric magnitude. Altho we can not directly measure the flux over
the entire spectrum we can calculate it given several observable
points in the spectrum and knowledge of how the source emits
When the magnitude system was mathematicly formalized in the late
1800s the zeropoint was Polaris, alpha Ursae Minoris, set to 2.0
magnitude. Polaris was thneafter found to vary its illumination as a
delta Cephei star and was let go as the magnitude standard.
We then set up Vega, alpha Lyrae, as the new zeropoint as 0.0
magnitude. In due time we photometricly measured Vega's flux was
2.65E-6 lumen/meter^2. Vega was in the 1980s found to have a dust disc
around it that could disturb the star's illumination.

mag0 - (0.0) = 2.5 * log((2.65E-6) / f0)
mag0 = 2.5 * log((2.65E-6) / f0)
= (2.5 * log(2.65E-6)) - (2.5 * log(f0))
= (-13.94) - (2.5 * log(f0))

Because the received flux is a function of both distance and
innate power, astronomers long ago normalized the magnitude rating of
sources to a standard distance of 10 parsecs. This, the absolute or
normalized or reduced magnitude is retrocalcked from an assessed
distance and the flux. In certain cases the nature of the source
suggests a power output; this is then combined with the flux to yield
a distance. Thus,

mag0 - magabs = 2.5 * log(fabs / f0)
= 2.5 * log((r / rabs)^2)
= 5 * log(r / rabs)
=  5 * log(r) - 5 * log(rabs)
= 5 * (log(r) - 5 * log(10)
= 5 * log(r) - 5 * 1
= 5 * log(r ) -5

This becomes

mag0 = magabs - 5 + 5 * log(r)

which is one of the most handy formulae in astronomy. It is nothing
more than the inverse-square law expressed in terms of the Hipparchus
magnitude scale.

+-------------------------------+
| MAGNITUDE-DISTANCE EQUATION   |
|                               |
| mag0 = magabs - 5 + 5 *log(r)|
+------------------------------+

In assessing the mag0 for remotes, the r must be replaced by r&

mag0 = magabs - 5 + 5 * log(r&)
= magabs - 5 + 5 * log(((2 * c / H0)
* ((Z+1) - (Z+1)^(1/2)))
= magabs - 5 + 5 * log(2 * c / H0)
+ 5 * log(((Z+1) - (Z+1)^(1/2))

=--------------------------------------------+
| MAGNITUDE-DISTANCE EQUATION IN H0 AND (Z+1)|
|                                            |
| mag0 =  magabs - 5 + 5 *log(2 * c / H0)    |
|         + 5 * log((Z+1) - (Z+1)^(1/2))     |
+--------------------------------------------+

Because this equation was derived using parsec as the distance
unit, H0 must involve parsecs, not lightyears or inverse seconds.

Angular Diameter
--------------
The angular size of a remote target as seen by us behaves in a
most strange manner. Considering a face-on length s on the remote, we
have that due to the expansion effect s0 = s*(Z+1) and the distance is
the Einstein distance r!. Using radian angles for simplicity

theta0 = s0 / r!
= s * (Z+1) / r!
= s * (Z+1) / (2 * c * (1 - (Z+1)^(-1/2)) / H0)
= H0 * s * (Z+1) / (2 * c * (1 - (Z+1)^(-1/2)))

+--------------------------------------------------------+
| ANGULAR SIZE IN THE STANDARD BIGBANG MODEL             |
|                                                        |
| theta0 = H0 * s * (Z+1 ) /(2 * c * (1 - (Z+1)^(-1/2))) |
+--------------------------------------------------------+

At small Z theta0 behaves like the Euclid angular size theta =
s/r. As Z increases, the angular size INCREASES and at very large Z
the angular size goes to infinity. A very large Z target can actually
fill the sky. In fact this is exactly what the cosmic background
radiation does at its Z of about 900.
Z for which theta0 is a minimum, when it ceases to decrease with
distance and starts to increase with farther distance, is

1der(theta0,Z) = H0 * s / (2 * c)
* 1der((Z+1) / (1 - (Z+1)^(-1/2)),Z)

For ease of handling the derivative let z = Z+1

1der(theta0,z) = H0 * s / ( 2 * c) * 1der(z / (1 - z^(-1/2)),z)
= H0 * s / (2 * c) * ((1 - z^(-1/2))
* 1der(z,z) - z * 1der(1 - z^(-1/2)),z)
/ (1 - z^(-1/2))^2
= H0 * s / (2 * c) * ((1 - z^(-1/2))
* 1 - z * (-1/2 ) * -z^(-3/2) /                (1-z^(-1/2))^2
= H0 * s / (2 * c) * (1 - z^(-1/2) - z
* (-1/2) * z^(-3/2)) /(1 - z^(-1/2))^2
= H0*s/(2*c)*(1-3*z^-1/2)
/ 2 ) / (1 - z^(-1/2))^2

Setting 1der(theta0,z) = 0

H0*s / (2 * c) * (1 - 3 * z^(-1/2) / 2) / (1 - z^(-1/2))^2
= 0

1 - 3 * z^(-1/2) / 2 = 0

-3 * z^(-1/2) / 2 = -1

z^(-1/2) = 2/3

z^(1/2) = 3/2

z = 9/4
= 2.25
= Z+1

Z = z - 1
= 2.25 - 1
= 1.25

theta0[min] is then itself

theta0 = H0 * s / (2 * c) * (Z+1) / (1 - (Z+1)^(-1/2))
= H0 * s / (2 * c) * (2.25 / (1 - 2.25^(-1/2))
= H0 * s / (2 * c) * (2.25 / 0.333)
= 3.375 * H0 * s / c

+----------------------------------+
| Z AND theta0 AT MINIMUM          |
|                                  |
| Z = 1.25                         |
|                                  |
| theta0[min] = 3.375 * H0 * s / c |
+----------------------------------+

Angular Spectroillumination
-------------------------
Galaxies and many quasars are extended targets with appreciable
angular radius. Radiation is sent to us from an area on the heavens and
is thus diffused relative to the same radiation sent from a star. For
a spectroillumination flambda and angular radius theta the angular
spectroillumination philambda in Euclid space is

philambda = flmabda / (pi * theta^2)
= Plambda / (4 * pi * r^2 * pii * theta^2)
= Plambda / (4 * pi^2 * r^2 * theta^2)

Recalling that in Euclides geometry theta = s/r,

flambda = Plambda / (4 * pi^2 * s^2)

Due to the Hubble expansion r is r!; flambdda, flambda0; theta,
theta0.

r! = (2 * c / H0) * (1 - (Z+1)^(-1/2)) * (Z+1)

theta0 = H0*s * (Z+1) / (2 * c * (1 - (Z+1)^(-1/2)))

flambda0 = Plambda / (4 * pi * r!^2 * (Z+1)^2)
= Plambda / (4 *p i * ((2 * c / H0)
* (1 - (Z+1)^(-1/2))^2 * (Z+1)^2)
= Plambda / ((4 * pi * 4 * c^2 / H0^2)
* (1 - (Z+1)^(-1/2))^2 * (Z+1)^2)
Then

philambda0 = flambda0 / (pi * theta0^2)
= flambda0 * (2 * c * (1 - (Z+1)^(-1/2))^2
/ (pi * (H0 * s * (Z+1))^2)
= Plambda * (2 * c * (1 - (Z+1)^(-1/2))^2
/ (pi * 4 * c^2 / H0^2) * (1 - (Z+1)^(-1/2))^2
* (Z+1)^2 * pi * (H0 * s * (Z+1))^2)
= Plambda * (2 * c * (1 - (Z+1)^(-1/2))^2
/ ((16 * pi^2 * c^2 * s^2 * (Z+1))^2)
* (1 - (Z+1)^(-1/2))^2 * (Z+1)^2
= Plambda / (4 * pi^2 * (Z+1)^2 * s^2 * (Z+1)^2)
= Plambda / (4 * pi^2 * s^2 * (Z+1)^4)

This is a function of (Z+1)^(-4)! The angular spectroillumination
falls off very rapidly at large Z+1 so that remoter targets are
exponentially fainter and harder to discern.

+---------------------------------------------------+
| ANGULAR SPECTROILLUMINATION                       |
|                                                   |
| philambda0 = Plambda / (4 * pi^2 * s^2 * (Z+1)^4) |
+---------------------------------------------------+

Note that for small Z the equation loses its dependence on distance.
The angular spectroillumination in Euclid space is constant with
distance. ```