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 Schwarzschild radius. 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 radiation. 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.