The Tolman Surface Brightness Test for the Reality of the Expansion. V. Provenance of the Test and a New Representation of the Data for Three Remote HST Galaxy Clusters

A new reduction is made of the HST photometric data for E galaxies in three remote clusters at redshifts near z=0.85 in search for the Tolman surface brightness (SB) signal for the reality of the expansion. Because of the strong variation of SB of such galaxies with intrinsic size, and because the Tolman test is about surface brightness, we must account for the variation. In an earlier version of the test, Lubin&Sandage calibrated the variation out. In contrast, the test is made here using fixed radius bins for both the local and remote samples. Homologous positions in the galaxy image at which to compare the surface brightness values are defined by radii at five Petrosian eta values ranging from 1.0 to 2.0. Sersic luminosity profiles are used to generate two diagnostic diagrams that define the mean SB distribution across the galaxy image. A Sersic exponent, defined by the r^n family of Sersic profiles, of n=0.46 fits both the local and remote samples. Diagrams of the dimming of thewith redshift over the range of Petrosian eta radii shows a highly significance Tolman signal but degraded by luminosity evolution in the look-back time. The expansion is real and a luminosity evolution exists at the mean redshift of the HST clusters of 0.8 mag in R_cape and 0.4 mag in the I_cape photometric rest-frame bands, consistent with the evolution models of Bruzual and Charlot.


INTRODUCTION
In one of the outstanding ironies in the history of observational cosmology, Hubble, even in his last years, expressed doubts about the reality of the expansion. His reasons were based on what he considered to be anomalies in the correlation of apparent magnitudes and redshifts (the Hubble diagram) and in his galaxy counts (Hubble 1934(Hubble , 1936, when both are corrected for the effects of redshifts on the apparent magnitudes. These corrections, the K terms, are calculated by shifting an assumed spectral energy curve of the mean mixture of galaxy types through the fixed photometric band pass of the detector. Applying his calculated corrections to his redshift diagram and his N(m) count data gave Hubble what seemed to be unacceptable results if the expansion is real.
His most direct statements were these.
(1). "It is evident that the observed result, [of applying a blue K correction of 2.94 z mag] is accounted for if the redshifts are not velocity shifts". . . . [the data are consistent but only if] "the expansion and spatial curvature are either negligible or zero" (Hubble 1936, p. 542).
(2). In considering the redshift-distance relation, "The inclusion of recession factors [to the magnitudes] would displace all the points [in the Hubble diagram] to the left [higher redshifts at a given magnitude], thus destroying the linearity of the law of redshifts" (Hubble 1937).
(3). . . . "if redshifts are not primarily due to velocity shifts . . . [then] the velocitydistance relation is linear, the distribution of nebulae is uniform, there is no evidence of expansion, no trace of curvature, no restriction of the time scale." [But] "the unexpected and truly remarkable features are introduced by the additional assumption that redshifts [actually do] measure recession. The velocity-distance relation deviates from linearity by the exact amount of the postulated recession. The distribution departs from uniformity by the exact amount of the recession. The departures are compensated by curvature which is the exact equivalent of the recession. Unless the coincidences are evidence of an underlying necessary relation between the various factors, they detract materially from the plausibility of the interpretation. . . . the small scale of the expanding model both in space and time is a novelty, and as such will require rather decisive evidence for its acceptance" (Hubble 1936, p. 553/4).
(4). And finally in his Darwin Lecture in 1953, only a few months before his death: "When no recession factors are included, the law will represent approximately a linear relation between redshift and distance. When recession factors are included, the distance relation [becomes] non-linear." [If no recession factor is included] "the age of the universe is likely to be between 3000 and 4000 million years, and thus [again with no recession factor] comparable with the age of the rock crust of the Earth" (Hubble 1953). (Here concerning the time scale, Hubble clearly was factoring in the beginning of the major correction to his distance scale that would eventually reach a ratio between 7 and 10 for the new scale to the old. Clearly, in 1953 he was beginning to accept that his value of the Hubble constant must be considerably revised downward, based, as it was in 1953, on Baade's revision of Hubble's distance to M31 by a scale factor close to 2).
It is now known that these arguments against a true recession are incorrect. Details of why have been variously set out elsewhere, the most recent in several reviews (Sandage 1995(Sandage , 1998. The essence of the case is threefold. (A). The K corrections for the effects of redshifts on apparent magnitudes used by Hubble are incorrect because he assumed a black body spectral energy distribution (SED) of 6000 K temperature whereas Greenstein (1938) had shown already in 1938 that the correct 'color temperature' of the bulge of M31 is closer to 4200 K. A directly observed template SED for E galaxies was not measured until beginning in 1968 (Oke & Sandage 1968;Whitford 1971;Schild & Oke 1971;Code & Welch 1979), and eventually for galaxies of other morphological types. Examples include the works by Wells (1973), Pence (1976), Coleman, Wu, &Weedman (1980), andYoshii &Takahara (1988), and thereafter by many others in various pass bands, summarized elsewhere (Sandage 1988, § 4.2).
(B). Hubble's magnitude scales were extrapolations of the Mount Wilson Catalogue of Photographic Magnitudes (Seares, Kapteyn, & van Rhijn 1930), which themselves needed systematic corrections fainter than about 16 apparent blue magnitude (Stebbins, Whitford, & Johnson 1950;Sandage 2001). Baade had often called Hubble's approximations "enthusiastic magnitudes", not out of derision but as a tribute to Hubble's skill in devising by practical methods what he needed in his reconnaissance studies of difficult problems.
(C). Hubble's definition of redshift distance as D = cz/H 0 is the intuitive choice but is incorrect in the standard model of cosmological parameters. The exact formulation was not made until the fundamental paper by Mattig (1958). The difference between Hubble's assumption and the exact formulation of metric coordinate distance for different geometries and redshifts is large, as seen in Figure 4 of Sandage (1998).
The consequence of these developments has been that Hubble's reticence to accept the redshift-distance relation as due to real recession is no longer valid. There is almost universal acceptance that the redshift phenomenon which increases linearly with distance is due to a real recession. Of the several reasons, the most direct is the excellent agreement in the three time scales of (a) the expansion age once the Hubble constant is known, (b) the age of the oldest globular clusters in the Galaxy, and (c) the age of the chemical elements. All three average about 14 × 10 9 years.
Other less direct tests, less direct because they require more complicated assumptions concerning a hot early universe giving arise to the 3 • MWB, include the observation that the temperature of the MWB is hotter at high redshift as we look back in time (Songaila et al. 1994;Srianand, Petitjean, & Ledoux 2000;Molaro et al. 2002), the acoustic waves in the correlation of the fluctuation in the MWB, and, of course the MWB itself. Although in a minority, there are still astronomers who offer alternate explanations of these latter tests in their questioning of the standard model of a hot early universe; yet the time scale agreement remains the decisive test (Sandage 1968) on which there is now such a large literature as to deny an adequate summary here.
Nevertheless, a true expansion where the redshifts are cosmological, not due to "some unknown law of nature" as favored by Hubble, is itself such a remarkable proposition that decisive proofs are still of interest, even if only as academic curiosities now.
The Tolman surface brightness test is particularly interesting because its principle is so clear as to give a major predicted difference in observational data between an expanding manifold and a stationary one where the redshift would then be due to "an unknown law of nature". Tolman (1930,1934) discovered the effect that the surface brightness of a "standard" radiating object that is receding with redshift z will be fainter than a similar stationary "standard" object at rest by (1 + z) 4 . However, if the manifold is stationary but nevertheless has a redshift due to an "unknown law of nature", the factor is only (1 + z). The test, as set out in theory by Tolman, is described again in Hubble & Tolman (1935) and is derived heuristically elsewhere (eg. Sandage 1961Sandage , 1974Sandage , 1995. The test, simple in principle is difficult in practice because there are no simple "standard" galaxies to compare with each other, one at high redshift and the other at small. The difficulty of using E galaxies for the test has been discussed elsewhere (Sandage & Perelmuter 1990a,b, 1991Sandage & Lubin 2001;Sandage & Lubin 2001a,b,c, hereafter LS01a,b,or c) and is not repeated here.
The first attempt at the test was made by Sandage & Perelmuter using ground based data in the references just cited (hereafter SP90a,b; SP91). The purpose of the present paper is to continue the discussion made in a second attempt of the test by Sandage & Lubin using HST data (LS01c) by showing more directly that a strong Tolman signal exists in three remote clusters observed with HST by Oke, Postman, & Lubin. Their photometric data are published in papers by various permutations of the order of the authors (OPL 1998;PLO 1998PLO , 2001Lubin et al. 1998Lubin et al. , 2001. The analysis by LS01c remains valid that the Tolman signal exists, but the reductions and the representation of the data are different here.

THE PROVENANCE OF THE TOLMAN TEST USING HST
Extensive planning related to constructing and deploying a large space telescope had its beginnings in the early 1960s, although the concept had already been set out 20 years earlier (Spitzer 1946) in the United States and two decades earlier by Oberth (1923) in Germany. The project, accepted by NASA in the late 1950s, was first called the LST for Large Space Telescope, but more appropriately known to the underground as the Lyman Spitzer Telescope because Spitzer had done so much to promote the idea.
Many planning sessions occurred to define the scientific goals, and the means to achieve them. By the mid 1960s a NASA department for the LST project was formed with Nancy Roman as its head. The chief scientist had also been appointed, the first of which was Charles O'Dell. In these early days of planning and of selling the project to the astronomical community and especially to Congress for funding, the importance of Roman and O'Dell cannot be overstated.
When it became apparent that an LST could be built and operated remotely, based on the experience and successes of the Orbiting Astronomical Observatories project led by the Grumman Aerospace Corporation with their chief engineer F. P. Simmons as coordinator, serious scientific and technical workshops were convened to advance the project. One such conference was held in 1974 at the Park-Sheraton Hotel in Washington DC, organized by Simmons, then at the McDonald-Douglas Astronautics Company. In a three day meeting, astronomers and engineers made presentations to Roman, O'Dell, and the astronomical and technical community on a variety of scientific projects ripe for the LST. Space engineers from 12 aerospace companies also discussed the means to achieve the goals. The astronomers giving papers were Spitzer, J. L. Greenstein, I. King, E. M. Burbidge, L. W. Fredrick, G. Neugebauer, G. Herbig, J. Bahcall, Harlan Smith, and the writer. Together with other projects for LST in observational cosmology, I discussed the requirements of spatial resolution and input flux needed to make the Tolman surface brightness test. Details were set out for a particular form of the test, comparing the ratio of a suitably defined isophotal diameter (the angular diameter to a particular isophotal level) in an E galaxy to a suitably defined metric diameter (the angular diameter to a given number of parsecs) for the same galaxy. I concluded that the LST would be an ideal instrument to make the test with its proposed 3-meter aperture and a spatial resolution of 0.05 to 0.1 arc sec. Of course, not known at the time was the way that surface brightness over the image of E galaxies at some standard diameter varies systematically with intrinsic size and absolute magnitude, or indeed how to define a series of "standard diameters" within which to measure the surface brightness. These complications were to be encountered later. Nevertheless, the project with LST was set out as a serious observing plan that was ideal for the capabilities of the proposed telescope. It has worked out that way, where the Tolman test with HST has been done (LS01a,b,c), 25 years beyond the initial proposal for a space experiment.
The photometric data for three remote clusters observed by Oke, Lubin, and Postman, cited earlier as OPL 1998;PLO 1998PLO , 2001Lubin et al. 1998Lubin et al. , 2001, and reduced in the way described in LS01a,b,c are again used here. What is different in the analysis is threefold.
(1). A more comprehensive calibration of the local data on E galaxy surface brightness as function of intrinsic radius is made in § 3, reaching to smaller radii than was done in LS01c. This is necessary because the galaxies in the HST clusters are at the faint end of the luminosity function of the first few ranked galaxies in the calibrating sample taken from Postman & Lauer (1995, hereafter PL95) and from Table 1 of SP91 used in a first study of the test. This new calibration of the SB -radius correlation avoids a less certain extrapolation to the necessary smaller radii for the three remote clusters than we made in SL01 (Table 3 there).
(2). The variation of surface brightness across the E galaxy image for particular Petrosian (1976) sizes, for both the local calibrating galaxies and the remote sample, is explicitly shown as a function of intrinsic Petrosian radii. The data are divided into groups of different mean intrinsic radii so as to isolate the large systematic variation of the mean surface brightness with radius (or absolute magnitude for M brighter than M V = −20, eg. see Figs. 1 and 2 of SP91). This dependence of the SB zero point on intrinsic radius at a given standard size degrades a raw Tolman signal and must be calibrated out. Much of § 3 is devoted to determining these properties of the Sérsic profiles.
(3). We show in § 3.4 that the Sérsic profile with an exponent of 0.46, rather than a de Vaucouleurs value of 0.25, fits our local sample over its range of intrinsic radii. The systematic deviation of the profile exponent from a de Vaucouleurs r 0.25 value with intrinsic radius is strong and is mapped. The variation of n with radius is the same as discovered by Binggeli & Jerjen (1998, Fig. 2).

PROPERTIES OF SÉRSIC LUMINOSITY PROFILE FUNCTIONS
IMPORTANT FOR THE TOLMAN TEST

The Petrosian Radius Function
In the original 1974 proposal, the test was to be made by comparing the ratio of isophotal to metric diameters for E galaxies at zero redshift to those with redshift z. For the high redshift galaxies, the ratio of r(isophotal) to r(metric) at a given metric diameter should decrease with increasing redshift if a Tolman signal exists. However, the test is difficult to apply because the metric diameters depend on the world model. We would also have to apply the K correction for the effect of redshift on the isophotal surface brightness values. We wish to avoid these problems at this stage in the data reductions. A formulation of the test using Petrosian radii which avoids these problems is used here instead.
Some of the remarkable properties of the Petrosian radius function have been discussed elsewhere (SP90a; Kron 1995;Sandage 1995;SL01) and are only summarized here.
Two equivalent definitions are these. (1) A Petrosian radius is a distance from the center of an image where the surface brightness (SB), averaged over the image that is interior to that radius is greater than the profile SB at that radius by a fixed number of magnitudes.
(2) It can be shown (Djorgovski & Spinrad 1981;SP90a) that this is the same number that is calculated from the slope of the growth curve, as, η (in mag) = 2.5 log(5d log r/d mag), where the growth curve is the summed intensity (in magnitudes) out to that r and its slope at r is d mag/d log r. Expressed as the slope in intensity units, as in PL95, α(= d log L/d log r), which is used by some authors. Of course, η(in mag) = 2.5 log 2/α. The Petrosian η function for radii was used throughout the first and second attempts to make the Tolman test (SP90a,b, SP91; SL01 and LS01a,b,c).

The Variation of Petrosian η Radii Along the Hubble Sequence From E to Sm Types
It is of interest to display the η function at various ratios of radii to the effective (the half-light) radius, r e . Because the luminosity profiles of galaxies are so different along the Hubble sequence, there is a systematic variation of η with morphological type, as calculated using equation (1) using the growth curves from Table 11 of the Third Reference Catalogue (de Vaucouleurs et al. 1991, the RC3). The results are listed in Table 1 as function of log r/r e , and shown in Figure 1. The T coding for the morphological types is the same as in the RC3. The even numbered columns of Table 1 are the growth curve magnitudes. The odd numbered columns are the η radii at the listed log r/r e values.
In the next subsection we show that the entries in Table 1 for the E (T = −5) galaxies is close to, but not identical with the de Vaucouleurs r 0.25 profile. We also show there that the η values for the Sm (T = 9) galaxy type is nearly identical with the exponential intensity profile whose Sérsic exponential index is n = 1.
3.3. The η, log r/r e and log r/r(η = 2) Relations for the Family of Sérsic Luminosity Profiles for E Galaxies It is now well known (citations later in this section) that the de Vaucouleurs r 0.25 luminosity profile only fits E galaxies near the bright end of the E galaxy luminosity function that themselves are not cD subtypes (Oemler 1974(Oemler , 1976. Sérsic (1968) generalized the de Vaucouleurs r 0.25 profile with the function, where a is a fixed radius to make the radius factor scale free, n is the Sérsic exponent, and b n is calculated to make the half-light radius occur at the value of a = r e . The de Vaucouleurs profile is a special case with n = 0.25 and b n = 3.33.
The adopted b n values, listed in Table 2, were calculated by numerical integration of equation (2) to generate growth curves in r/a, such that a = r e for each of Sérsic profiles with n values of 0.15, 0.2, 0.25, 0.4, 0.46, 0.6, 1.0, and 1.5. The growth-curve magnitudes and Petrosian η values in Table 3 are at the listed r/r e values for these n values. The listings for T = −5, and n = 0.25 and 0.46 are the same as in Table 1. Figure 2 shows the variation of η with r/r e for the various n values from Table 3. If all E galaxies had a fixed luminosity profile, such as that for the de Vaucouleurs r 0.25 special case, Table 3 would be germane only to the E galaxy types for the T = −5 column. However, in papers cited below, it has become known that the Sérsic n exponent varies systematically with absolute magnitude in E and dE galaxies, ranging from 0.15 to 1.5 as the absolute magnitude varies between M B = −23 and −13.
The progression of SB with absolute magnitude was discovered by many authors (eg. Oemler 1973Oemler , 1974Kormendy 1977Kormendy , 1987Strom & Strom 1978a,b,c;Michard 1979;Thomsen & Frandsen 1983;Binggeli, Sandage, & Tarenghi 1984;Choloniewski 1985;Schombert 1986;Ichikawa, Wakamatsu, & Okamura 1986;Caldwell & Bothun 1987;Djorgovski & Davis 1987;Ferguson & Sandage 1988;Impey, Bothun, & Malin 1988;Caon, Capaccioli, & D'Onofrio 1993) and we suspect many others. If the deviation from the classical Hubble or de Vaucouleurs E galaxy profiles can be described by Sérsic functions, which we prove in the next section, it is this deviation that is listed in Table 3. 1 But before giving the proof, we first rearrange Table 3 to use η rather than r/r e as the independent variable. The table is also renormalized to change the fiducial radius from the half-light radius to the radius at η = 2, hereafter called r(2). The change is made because it will be more convenient in § 5 to use r/r(2) to compare surface brightness levels at various points rather than r e . To this end, Table 4 is reconstituted by a graphical interpolation of Table 3, and then defines the radius at η = 2 to be the zero point of the coordinate radius. Figure 3 shows the data from Table 4 for the ratio of the various r(η) radii to r(2). This diagnostic diagram will be often used later to compare observational data for local and the HST cluster galaxies.

Oemler Profiles
But before comparing the entries in Tables 3 and 4 with observed E galaxy profiles, it is of interest to compare the Sérsic profiles with those of Oemler (1974Oemler ( , 1976). Oemler's function embrace a family based on a modified Hubble profile with various strengths of a luminosity turndown at large radii. The Oemler profiles have the equation, where α and β are fitting parameters. This function is illustrated in SP90a where the resulting family, relative to the Hubble profile, is shown in Figures 2 and 3 of that paper. The profiles are generated by varying the α/β ratio, and is made scale free by using r/β as the radius measure.
Note that the denominator on the right of equation (3) is the Hubble law. The numerator is the added decay factor making the integrated growth curve finite, which the raw Hubble law does not. The similarity and the differences with the Sérsic family that is used here is in Figure 4, taken from Table 5 where the η, r/r(η = 2) Oemler profiles are listed for α/β ratios of 10, 30, 60, and 100. The T = −5 relation is copied from Table 1 but is modified in an obvious way to be a function of r/r(2) rather than r/r e . Three Oemler profiles from Table 5 are shown and are compared with the T = −5 profile from Table 1 and with the galaxy surface photometry remarkably well. However, the treatment differs in the two works. We develop the properties of the Sérsic functions in terms of Petrosian η radii to define homologous regions of the image for galaxies of different absolute magnitude. Kormendy et al. do not use Petrosian radii because they have no need in their study of origins to define similar points over an E galaxy image, which we must for the Tolman test. Such points are naturally given by Petrosian radii. The sometimes parallel conclusions, here and by Kormendy et al., of the properties of Sérsic functions, agree. n = 0.46 Sérsic profile from Table 3. The fit of the T = −5 data with the Oemler profile for α/β = 30 for η between 1 and 2 is excellent, after which the fit must be made to an α/β ratio of 100 for η between 2 and 3.5. The fit of the n = 0.46 Sérsic profile is excellent for an Oemler profile of α/β = 10 for η < 2, after which an α/β ratio of 30 must be used. Table 5 has been calculated in an obvious way from equation (1) using Table 1 of SP90a for the slope of the Oemler growth curves.
Our reason for discussing the Oemler profiles here is that they are a natural generalization of the Hubble single parameter profile that was standard for many years, and it was the Hubble profile with its scale factor, a, that was discussed in proposing the Tolman test at the LST meeting in 1974.
Return now to the Sérsic profiles to display the surface brightness-η properties of the Sérsic function.

Surface Brightness vs. η for Sérsic Profiles Normalized to η = 2
Because the Tolman test concerns surface brightness, it is necessary to cast the Sérsic profiles in Tables 3 & 4 and Figure 3 in terms of surface brightness. The relative shape of the SB profiles in the Sérsic family is the subject of this section.
Once the growth curve of a particular intensity profile is known, the surface brightness averaged over a given radius follows from, where the first term on the right is the area expressed in mag per square arc sec, and the other is the magnitude from the growth curve (GC) at the particular value of η. In practice, if we want only the run of SB with η, normalized to say η = 2, it is sufficient to use Mean SB relative to that at η = 2 for the Sérsic family using equation (5) are listed in Table 6 and shown in Figure 5 for η values from 1.0 to 4.0. The SB values for E galaxies, calculated from the observed T = −5 growth curve, are also listed in column (5). Figure 5 is the second diagnostic diagram that will be used in later sections.
The close agreement of the listed SB for T = −5 with the n = 0.25 de Vaucouleurs profile is one of the proofs we are seeking that Sérsic profiles have relevance for E galaxies. A generalization of the proof for different absolute magnitudes for E galaxies is in the next section.

THE VARIATION OF THE SURFACE BRIGHTNESS WITH INTRINSIC RADIUS AT VARIOUS η VALUES FOR LOCAL GALAXIES IN THE VIRGO, FORNAX, AND COMA CLUSTERS
The relevance of the Sérsic family of profiles for observations of real E galaxies was shown in the last section, but only for the case of galaxies with the T = −5 growth curve. The E galaxies used to define the T = −5 growth curve (Table 11 of the RC3) are those of the average absolute magnitude for such galaxies that exist in the catalogs of aperture photometry that were available when the RC3 was compiled. However, because the luminosity profile of E galaxies, and hence the Sérsic n value, is such a strong function of linear size, we must generalize the RC3 T = −5 growth curve over a large range of absolute magnitude and intrinsic size.
Data on the observed profiles for local galaxies that span a range of five magnitudes are in the literature for the Virgo, Fornax, and Coma clusters. The photometry is by Fraser (1977), King (1978), Hodge (1978), Ichikawa, Wakamatsu, & Okamura (1986), Schombert (1986), Jedrzejewski (1987), andVigroux et al. (1988) for Virgo, by Schombert (1986) and Caldwell (1987) for Fornax, and by Oemler (1976) for Coma. η was calculated in SP90b from these data using equation (2). A diagram similar to Figure 5 in SP90a was drawn for each galaxy showing the published profile, the growth curve, and η calculated at the listed radii. The log r (arc sec) for η values of 1.0, 1.3, 1.5, 1.7, 2.0, 2.5, 3.0, 3.5, and 4.0 were determined graphically from these diagrams. The surface brightness averaged over the area interior to the listed radii is listed in Tables 1-3 in SP90b at each η value.
Data for 153 cluster galaxies were determined in this way; 83 in Virgo, 20 in Fornax, and 42 in Coma. The data cover a range in absolute magnitude from M B (T ) = −23 to −13 and a range of intrinsic radii from log R(η = 2) of 3.3 to 5.1 (pc), and are listed in Tables 1-3 in SP90b. Where needed, we adopted distance moduli of 31.7 for Virgo, 31.9 for Fornax, and 35.5 for Coma.
Calculations were made from these data for the log r(η)/r(2) radius ratios and the SB(η) − SB(η = 2) surface brightness differences relative to the values at η = 2 over the range of η values from 1.0 to 4.0. These data permit the observations to be compared with the expectations in the diagnostics Figures 3 & 5 using the family of Sérsic profiles.
Based on the strong correlation of the Sérsic n exponent with absolute magnitude discovered by Binggeli & Jerjen (1998, their Fig. 2), there must be a correlation of log r(η)/r(2) with intrinsic radius if the Sérsic theoretical profiles are good fits to the observed profiles. This expectation for the Virgo, Fornax, and Coma cluster galaxies is verified in Figures 6 & 7 where both log r(1)/r(2) and SB(2) − SB(1) are plotted against log R(η = 2). Be-cause these ratios change systematically with log R(2), and because they are both functions of Sérsic n at any given η (Figs. 3 & 5 here), n must itself be a function of log R(2), and therefore of absolute magnitude. Figure 2 of Binggeli & Jerjen (1998) is explicit, showing that the Sérsic n varies between 0.15 to 1.5 as the apparent B T magnitude in Virgo cluster E galaxies change between 9 and 18, or M B(T ) between −22.7 and −13.7. Table 4 by using Figure 6 for log r(1)/r(2) = −0.5, we determine that the Sérsic exponent is 0.46±0.02 at log R(2) = 3.9, which is the average for the HST clusters. This interpolation shows that n increases as R(2) decreases, reaching the exponential value of n = 1 near apparent magnitude B(T ) = 16 (M T = −15.7) in the Virgo cluster (see also Binggeli & Jerjen 1998), well into the dE morphological types.

Interpolating in
We have also tested the adequacy of the Sérsic profiles using the much larger sample of 178 galaxies in 78 local Abell clusters studied by SP91. The data are in the V band for this independent sample. They were analyzed in the way just described for the Virgo/Fornax/Coma clusters. The observed profiles are from Oemler (1976), Thuan & Romanishin (1981, Malumuth & Kirshner (1985), and Schombert (1987). Here, the intrinsic radius at η = 2 varies from log R(2) = 4.0 to 5.5. From § 6 later, we see that this range is larger than the average log R(2) = 3.9 for the three remote HST clusters, and the difference must be accounted for, as is done in § 5 and 6.
A third independent sample of local cluster galaxies is from PL95 who again used first ranked E galaxies in local Abell clusters. There are 128 galaxies in the PL sample whose intrinsic radii are again large, ranging from log R(2) of 4.0 to 5.5.
Using these data from the three independent sources, we drew diagrams (not shown) similar to Figures 6 & 7 over the additional range of η values from 1.3, 1.5, to 1.7. Comparing the variation of log r(η)/r(2) and SB(η) − SB (2)  This completes the proofs that the Sérsic family of profiles fit the observed E galaxy data over the total range of E galaxy absolute magnitudes, and that the curves in Figures 3 & 5 for the Sérsic family can be used to model the profiles of real galaxies. The systematically varying n values can be determined by fitting the data to these two diagnostic diagrams, as we do in Figure 15 later.

CALIBRATION OF THE RELATION BETWEEN SURFACE BRIGHTNESS AND INTRINSIC RADIUS USING LOCAL E GALAXIES
That the surface brightness of E galaxies varies strongly and systematically with intrinsic radius, becoming brighter as the radius decreases, has variously been rediscovered in the many papers cited earlier, perhaps earliest seen in photographs by Burbidge, Burbidge, & Crampin (1964, BBC). The sense of fainter SB for larger R(2) values of E galaxies, seen so well in the BBC photographs, holds until log R(2) = 3.8, where the relation reverses (Binggeli et al. 1984;SP90b, Fig. 14) for the faint dE galaxies.
We showed in SL01 that the variation of SB with intrinsic radius for E galaxies is also well defined at the η values of 1.0, 1.3, 1.5, 1.7, and 2.0. The spread at a given radius is one magnitude whereas the total range of SB between log R(2) of 4.2 and 5.4 is 4 magnitudes (Fig. 2 of SL01). Because the Tolman test is about surface brightness, we must either calibrate out this variation, or we must make the test by comparing the SB of the data set at the same R(η) radii. We chose the latter here.
The analysis in SL01 provides the first step in determining the zero point of the SBintrinsic radius relation for local galaxies. The data listed in Table 1 of that reference are η, SB , and intrinsic size for 118 first ranked local Abell clusters. These have been calculated from equations (1), (4), and (5) using the observed growth curves of PL95 with H 0 = 50.
The R magnitudes by Postman & Lauer are on the Cape/Cousins near red photometric system as realized by Landolt (1983Landolt ( , 1992. The R cape band pass differs from R J of Johnson (1965), which is identical to the r(S20) system of Sandage & Smith (1963) that I used for all the Mount Wilson/Palomar near-red photometry of first ranked cluster galaxies with S20 photocathods in the 1970s (eg. Sandage 1972Sandage , 1973. That photometric R system has now been largely replaced in the literature by the Cape/Cousins (RI) cape system because of the work of Landolt with his all-sky UBV (RI) cape standards. The difference in zero point between the R cape and R J systems is 0.26 mag at the color of local E galaxies, with R cape being fainter at this color (Sandage 1997). The SB Rcape and log R(η) data for the PL sample are in Table 1 of SL01.
Least squares linear fits to the SB , log R data of Postman & Lauer are in Table 2 of SL01 and shown in Figure 2 there for η values of 1.3, 1.5, 1.7, and 2.0. These linear fits are valid only for log R > 4.0. The SB data have been reduced to zero redshift by making them brighter by 0.16 mag than the values measured by PL95, determined by the method described in SL01.
The extension of equation (6) to log R(2) < 4.3 is nonlinear, bending toward fainter SB at the smaller radii, shown in Figures 1 & 2 of SL01. It was accounted for there by a nonlinear addition to equation (6) for log R(2) < 4.4. A first approximation for the correction was made from Table 3 in SL01. It is revised in Table 7 here. The correction is smaller than in SL01 by 0.3 mag at log R(2) = 3.4, the difference decreasing to zero at log R(2) = 4.4. The new corrections here are derived from the combined SB , log R correlations in the Virgo, Fornax, and Coma clusters, discussed above, by a better spline connection of the cluster data from SP90b above and below R(2) = 4.4 (pc).
A second calibration, independent of equation (6), is made using the extensive V band data from Table 1 of SP90b, discussed earlier. The least squares regression of SB V on log R(2) 50 for the 178 galaxies in this earlier sample is, SB V (2) = 2.89 log R(2) + 9.53, reduced to zero redshift and valid for log R > 4.3. The nonlinearity for log R < 4.3 (H 0 = 50) is clearly seen (not shown) in these data, needing again the Table 7 correction for the smaller radii.
again valid only for log R(2) > 4.3(H 0 = 50). With these changes, equation (8) is everywhere within 0.08 mag of equation (7) over the range of log R between 4.0 and 5.0.
To compare equation (8) in V with equation (6) in R we need the color index, (V −R) cape , for E galaxies with linear sizes of log R(2) between 4.0 and 5.0. The (V − R) color for such giant E galaxies have been determined from the catalog of growth curves in R and I by de Vaucouleurs & Longo (1988) for a variety of photometric systems. Choosing from those that are on the Cape/Cousins system (listed as R ′ and I ′ in their catalog) for a sample of 17 local T = −5 galaxies with 66 measurements gave the mean color indices of (V −R) cape = 0.57 ± 0.003, and (R−I) cape = 0.64 ± 0.003. These are nearly identical with the values by Poulain & Nieto (1994) from a larger sample.
Applying V − R = 0.57 to equation (8) gives, SB(2) Rcape = 2.97 log R(2) + 8.60, for local E galaxies with log R(2) > 4.3 (H 0 = 50). This is within 0.08 mag of the SB , log R(2) calibration in equation (6) which is based on the independent photometry of Postman & Lauer. In what follows we adopt equation (9), and correct it by Table 7 for smaller radii. Table 8, based on equation (9), corrected by Table 7, is our final adopted calibration for local zero redshift E galaxies for the nonlinear relation between log R(2) and SB . The entries are calculated from the SB R values just described using (V − R) cape = 0.57 and (R − I) cape = 0.64.

SURFACE BRIGHTNESS PROFILES FOR THE THREE HST CLUSTER GALAXIES COMPARED WITH THE LOCAL SAMPLE: THE TOLMAN TEST
Surface brightness magnitudes averaged over R(η), the R(η) sizes, and absolute magnitudes are listed in Tables 2-4 of LS01c for 34 galaxies in the three HST clusters. The data in LS01b show that the accuracies of η and SB are better than 2% at radii between 0.1 and 1 arc sec. The intrinsic R sizes for the database range from log R(2) = 3.45 to 4.20 (H 0 = 50, q 0 = 1/2, Λ = 0), with a mean of 3.9.
Because of the strong variation of SB with intrinsic size, which we have been emphasizing, we have divided the galaxies in the HST clusters into three bins of different intrinsic radii, and make the Tolman test in each bin separately to compensate for the variation. Table 9 lists the SB , log R(η) data at the η values of 1.0, 1.3, 1.5, 1.7, and 2.0 for the three radius bins in HST Cl 1324+3011. The root data are from Table 3 of LS01c. The bottom entries are the mean values.
To make the Tolman test we need the SB , log R(η) relation for galaxies of zero redshift at the mean log R(2) for the three HST radius bins in Table 9. An example, reading from Table 9, is that the mean radius at η = 2 for the largest radius group in Cl 1324+3011 is log R(2) = 4.087 (in parsecs).
The SB I -radius relation at η = 2.0 and log R(2) = 4.087 for local galaxies is interpolated for the standards in Table 8. The surface brightness of SB I (2) = 22.86 mag/arcsec 2 at η = 2.0 is then spread to the η values of 1.0, 1.3, 1.5, and 1.7 using the surface brightness ratios in the n = 0.46 column of Table 6. This procedure gives the standard SB , η curve for zero redshift at the five fiducial η values for local E galaxies.
The result for Cl 1324+3011 (z = 0.7565) is shown in Figure 8. The zero-redshift SB I , η curves from Table 10, valid at the marked mean R(η = 2) radii for each group, are shown in the upper part of the diagram. The data from Table 9 are plotted in the lower part for the three radius bins. The smooth curves are interpolated between the points. The bin number is marked at the left. Clearly, a Tolman signal is present. The family of observed curves is fainter than the family of zero-redshift calibration curves.
To see this signal more clearly, the data in each radius bin are plotted separately in Figure 9. Error bars are put on the observed points. Zero-redshift standard curves from Table 10, are shown as dashed. The shape of the Sérsic curve for n = 0.46 is brought down from the dashed curve near the top of each panel in Figure 9 and is zero-pointed to the data at η = 1.5.
The fit of HST data to the standard curve is excellent. The SB difference between the zero-redshift standard (upper curves) and the observed points is the effect we are seeking. It is the Tolman signal as modified by the luminosity change in the remote 1324 + 3011 cluster due to evolution in the look-back time.
The same analysis is made in Figures 10 & 11 for Cl 1604+4304 (z = 0.8967) from the data in Tables 11 & 12 (there are only  The surface brightness differences between the remote and the local galaxies, read from Figures 9, 11, and 13, are collected in Table 15. The entries can be compared with Tables 5, 6, & 7 of LS01c. They are similar but not identical. The small difference between the studies is because the method of search here for the Tolman signal is not the same. There we accounted for the change of SB with intrinsic size by comparing the zero points of the SB , radius curve at different η values (Figs. 1 and 2 there). Here we compare the local and remote data at fixed radii as η is varied (Figs. 8-12 here). Part of the difference is also due to the slight change in the zero redshift standard SB -log R relation in Table 8. Each method uses the same observational data and with each we reach the same conclusion, which is this.
The differences in SB are large between remote and local galaxies over parts of E galaxy images defined by Petrosian η radii, but in each case are smaller than the (1 + z) 4 Tolman prediction. We interpret this to mean that a Tolman (1 + z) 4 cosmological signal exists but is degraded by luminosity evolution that amounts to approximately 0.8 mag in R and 0.4 mag in I. The exact values at the bottom of Table 15 are listed for each of the five fiducial η positions in the mean of the E galaxy images for each radius bin.

The Estimated Errors
Before the error budget can be determined, two caveats are necessary.
(1) In Figures 8-13 we have used an n = 0.46 Sérsic exponent to define the shape of the zero redshift SB -η relation in every radius bin for the mean radius in each. However, following Binggeli & Jerjen (1998, their Fig. 2), we showed in Figures 6 & 7 using Tables 4 and 6 that the Sérsic n exponent is a strong function of the intrinsic R(2) size. Hence, we should have used different standard zero-redshift curves with different n values that are relevant for the mean radius for each of the particular bins. We have ignored this detail because it makes a negligible difference in Table 15 and Figures 8-13.
Nevertheless it is of interest to estimate the variation of effective n values for the HST cluster galaxies. Figure 14 with Table 4 is useful in making the estimate. These show that, as the size varies between log R(2) = 3.5 and 4.2, the Sérsic exponent varies between 0.4 and 0.6 for the HST galaxies. The mean over all the HST galaxies is n = 0.46, shown in Figure 15 from the data listed in Table 16.
(2) The second caveat is this. We have calculated the intrinsic sizes using a world model with q 0 = 1/2, H 0 = 50, Λ = 0. Different radii would be obtained for different q 0 models. Table 8 of LS01c shows that if q 0 = 0, the radii would be about 25% larger than we have used. If q 0 = 1, the R values would be 15% smaller. Therefore, using the slope of equation (9), the surface brightness of the standard zero redshift curves would be displaced relative to the HST points by 0.3 mag brighter and 0.18 mag fainter from the offsets we use here for q 0 = 1/2.
We have not calculated the effect on R using a finite value of Λ, taken from the current "concordance world model", with assigned values of Λ and q 0 because the uncertainties here are at the level just stated for the range of q 0 from 0 to 1, and are bracketed by them.
This amounts to a systematic error over which we have no control unless we know the correct world model, which we do not. We ignore this systematic uncertainty in what follows, giving only the statistical errors for the q 0 = 1/2 case.
Ignoring these two caveats, the statistical errors are estimated as follows. We require the mean SB offsets in the SB vs. η curves between the zero redshift standard curves and the data points in Figures 9, 11, and 13. There are two components. (a) One is the difference between the observed points and the standard curves brought down from the zero redshift SB curves. (b) The second is the accuracy with which we know the position of the standard curves at zero redshift for particular radius values.
(a). Table 15 lists the individual SB offsets for each fiducial η Petrosian homology position in each of the radius bins. The rms differences from the standard curves are small, averaging 0.06 mag for each cluster in each radius bin. In addition, there is no correlation of the residues with η, showing that the data have the same shape, on average, as the standard n = 0.46 Sérsic profile. The mean rms of the SB differences with the standard curve averages 0.12 mag over the range of η from 1.0 to 2.0. With five data points in each group, the average error of the offset is ±0.06 mag.
(b). As read from the error in the least squares intercept in Table 2 of SL01, the accuracy with which we know the placement of the zero-redshift standard SB -η curve at fixed radius is ±0.08 mag. However, the radius in each size bin itself varies within the bin, and its mean radius has an rms variation that also introduces a contribution to the placement error. These variations in R are listed within the body of Tables 9, 11, and 13. They average rms = 0.100, giving a mean error of ±0.05 in log R. From equation (9) this translates to an mean error in the placement of the standard SB curves at zero redshift of ±0.15 mag. Adding this in quadrature to ±0.08 mag gives a total mean placement error of the zero-redshift curves of ±0.17 mag. Adding this in quadrature to the ±0.06 mag mean error of the HST data points gives a total uncertainty of the mean SB offsets in Table 15 as ±0.18 mag.
From Table 15 we adopt the mean SB offsets between the HST galaxies and the local E galaxies of the same size as 2.04 ± 0.18 mag for Cl 1324+3011 (z = 0.7565) in the rest frame I band, 2.51 ± 0.18 mag for Cl 1604+4304 (z = 0.8967) in the rest frame I band, and 1.99 ± 0.18 mag for Cl 1604+4321 (z = 0.9243) in the rest frame R band.
These offsets correspond to Tolman plus luminosity evolution signals of p = 3.33 ± 0.30 for Cl 1324+3011 in the I band, p = 3.61 ± 0.26 for Cl 1604+4304, also in the I band, and p = 2.80 ± 0.25 for Cl 1604+4321 in the R band, where p is the exponent on (1 + z). Combining the p values for the two clusters in the I band, and repeating the answer in the R band for Cl 1604+4321 gives the final answers as, ∆ SB = 2.5 log(1 + z) 2.80±0.25 mag in the R band 2.5 log(1 + z) 3.48±0.14 mag in the I band. (10)

Evolution in the Look-Back Time
If the true Tolman signal is (1 + z) 4 , then the component due to luminosity evolution is (1 + z) 4−p , which is, M evol = 2.5 log(1 + z) 1.20±0.25 mag in the R band 2.5 log(1 + z) 0.52±0.14 mag in the I band.
For redshifts of z = 0.86, which is the mean for the three HST clusters, these luminosity evolutions are 0.81 mag in rest frame R and 0.35 mag in rest frame I. Such luminosity changes at this mean redshift are consistent with the stellar evolution models of Bruzual & Charlot (1993) if the initial star formation was a burst near the beginning of the creation of the galaxy (LS01c, § 4.2). Equations (10) and (11) are closely the same as Lubin and I (LS01c) found, using a different representation of the HST data. Q.E.D.
I have been told by my son John, who majored in physics at University of California, Davis that the professor of advanced mechanics would post the solutions to the weekly five assigned problems a week after they were due, and always signed the solutions Q.E.D. As the difficulty of the problems increased week by week, some of the students grew increasingly disturbed by what they conceived to be a mocking by the professor, because they thought that Q.E.D. meant "Quite Easily Done" rather than "Quod Erat Demonstrandum", meaning "As Was To Be Demonstrated".
The solution of the Tolman test given by LS01c and differently here has not been quite as easily done as I set out at the planning meeting for the LST in 1974. It has required many developments not yet made at the time. However, the test seems to have been successful. The Tolman prediction is verified. The expansion would seem to be real.

Preliminaries
One of the major problems that hampered progress was how to define a homologous radius at which to compare the surface brightness of local and remote E galaxies. An obvious early choice was the half-light radius r e , but this was hard to measure before the advent of area detectors. To determine r e requires a growth curve that extends to "infinite" radius so that it can be backed off by 0.75 mag to the half-light value. Finding the asymptotic "total" magnitude depends on an assumed luminosity profile which is generally not the standard de Vaucouleurs r 1/4 curve.
Another scale-free radius is the Hubble a fitting parameter, used in my first proposal for the test with an LST. This is the radius where the measured surface brightness is 1/4 of the central value. However, this is even harder to measure. It requires knowing the central intensity, which is elusive because of insufficient spatial resolution of the telescope, the detector, and from the ground, the seeing.
The solution has been to use Petrosian radii, defined as the comparison of the SB at a particular radius to the average surface brightness, SB , inside that radius. We have formulated the Tolman test, both in SP90a,b, LS01a,b,c and here by using Petrosian radii throughout.
The properties of various ratios of measurable parameters at Petrosian η values of 1.0, 1.3, 1.5, 1.7, and 2.0 mag is the subject of § 3. The two diagnostic diagrams of Figures 3 & 5 are central to the discussion. They are related to the luminosity profile, permitting the assigning of a particular Sérsic profile to the data but also providing a way to calculate the SB over a range of η radii when the SB is known at log R at η = 2. Based on these two diagrams, we have used 11 steps to complete the Tolman test.

The Eleven Steps
Only the shape of the SB vs. η standard curve is given by the second diagnostic diagram in Figure 5. Once the Sérsic n exponent is known, the shape of its standard curve must be calibrated in zero point. We proceed in eight steps for this calibration.
(1). By the method in § 5 we determine the zero point of the SB vs. log R zero redshift curve at the R radius corresponding to η = 2.0. The results are in equations (6)-(9) in the V and R cape band passes, valid for log R > 4.3.
(2). For smaller radii, non-linear corrections to these equations have been determined from photometric data in the Virgo, Fornax, & Coma clusters. These are listed in Table 7.
(3). Applying Table 7 to equation (9) gives the SB averaged over R at zero redshift vs. the log R values in Table 8, tabulated in V , R, and I for intrinsic radii at η = 2.0 over the range of log R from 5.4 to 3.4 (pc).
(4). The calibration of SB vs. R at η values other than 2.0 is found by spreading the calibration of Table 7 to the four fiducial η values between 2.0 to 1.0 by using the diagnostic diagram of Figure 5 (Table 6) with a Sérsic exponent of n = 0.46, determined as follows.
(5). Proof that the Sérsic family of profiles is appropriate and that the correct Sérsic n exponent can be found is made by recovering the discovery, made by the many authors cited, that n varies strongly with absolute magnitude. From Figures 7 and 8 in  § 4, it is shown that both the log r(η = 1)/r(2) radii ratios in Figure 3 and the SB(2) − SB(1) differences in Figure 5 vary systematically with log R. That this must be so, provided that the Sérsic family is a good fit to E galaxy profiles, follows from the discovery by Binggeli & Jerjen (1998) that the Sérsic n exponent varies between 0.15 and 1.5 as the absolute magnitude of E galaxies becomes fainter between M B = −23 and −13 for Virgo cluster galaxies.
(6). Reading Figure 6 at log R = 3.9, which is the average for galaxies in the in the HST sample, and interpolating with log r(1)/r(2) = −0.50 between the Sérsic n values, gives n = 0.46.
(7). Reading Table 6 at this n gives the SB differences with SB(η = 2.0) needed to spread the calibration from η = 2 to the other four fiducial η values we are using.
(8). This calibration of the SB -log R relation for local E galaxies at zero redshift at these five η values at the mean radii of the size bins used for the HST clusters is listed in Tables 10, 12, & 14 of § 6. This completes the calibration steps. The remaining steps to the Tolman test itself are three.
(9). Figures 8-13 show the comparison of the SB of local E galaxies with the three HST clusters, broken into radius bins to compensate for the variation of SB with intrinsic radii. The Tolman signals, degraded by luminosity evolution, are listed in Table 15. The errors are put at ±0.18 mag in each radius bin by the accounting in § 6.1.
(11). That a Sérsic profile with n = 0.46 is appropriate for the local standards and for the three HST cluster galaxies at these absolute magnitudes, is shown in Figures 6, 7  The two conclusions are that the universe expands, and that there is luminosity evolution in the look-back time. Although Q.E.D., it has not been quite so easily done as the way we tried to sell it for the LST in 1974.
I am grateful to G. A. Tammann for reading an early draft of the paper and for making comments that have clarified a number of the arguments. Bernd Reindl's skill is greatly appreciated in preparing the diagrams in digital form, and in preparing the text in the proper format with dispatch. John Grula, editorial chief for the Observatories, formed again the liaison with the press, for which I am grateful. I thank the Carnegie Institution for its support with post retirement facilities and publication charges. This preprint was prepared with the AAS L A T E X macros v5.2.                      Fig. 6.-Variation of the log r(η = 1)/r(2) radius ratio with intrinsic diameter in pc for the Virgo (skipping jack crosses), Fornax (Roman crosses), and Coma cluster galaxies (triangles) from the data in Tables 1-3 of SP90b. The region of the dwarf dE galaxies for log R < 3.6 is hatched. Interpolating in the diagnostic Table 4, or using Figure 3, shows that n for E and dE galaxies varies between 0.15 and 1.0 over the range of log R(2) between 3.3 and 4.8.  Figure 6 but for the surface brightness differences between η = 1 and 2 for Virgo, Fornax, and Coma from the data in Tables 1-3 in SP90b.   Table 10. Smooth curves with the same shape are put through the observed data (Table 9) in the lower part of the diagram. Data are from Tables 9 and 10.   Figure 8 plotted separately for the three radius bins and for the mean for Cl 1324+3011. The shape of the standard profile at each mean radius is dropped onto the data from the upper curves, zero pointed at η = 1.5.       Table 4, are shown for comparison. From this, we deduce that the mean Sérsic exponent, averaged all radii for the HST cluster galaxies, is n = 0.46.