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 HUBBLE SPACE TELESCOPE GALAXY CLUSTERS

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, 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 bandpass 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 velocity–distance relation is linear, the distribution of nebulae is uniform, there is no evidence of expansion, no trace of curvature, no restriction of the timescale." (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. 553/554).
• 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) nonlinear." (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 timescale, 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, 1998). The essence of the case is threefold.

• 1.  
  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 the beginning of 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 et al. (1980), and Yoshii & Takahara (1988), and thereafter by many others in various pass bands, summarized elsewhere (Sandage 1988, Section 4.2).
• 2.  
  Hubble's magnitude scales were extrapolations of the Mount Wilson Catalog of photographic magnitudes (Seares et al. 1930), which themselves needed systematic corrections fainter than about 16 apparent blue magnitude (Stebbins et al. 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.
• 3.  
  Hubble's definition of redshift distance as D = cz/H₀ 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 timescales of (1) the expansion age once the Hubble constant is known, (2) the age of the oldest globular clusters in the Galaxy, and (3) the age of the chemical elements. All three average about 14 × 10⁹ years.

Other less direct tests (less direct because they require more complicated assumptions concerning a hot early universe giving arise to the 3° microwave background (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 et al. 2000; Molaro et al. 2002), the acoustic waves in the correlation of the fluctuation in the MWB, and, of course the MWB itself which not only has a near perfect Planck shape (zero chemical potential), but also the correct surface brightness (SB) zero point for the temperature that is derived from the shape. 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 timescale 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 proofs of various kinds are still of interest, if for nothing else than that they form part of scientific literacy.

The Tolman SB 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 SB 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 (e.g., Sandage 1961, 1974, 1995). The dimming factor is the same for all world models regardless of world geometry if the redshift luminosity relation is that given by Robertson (1938).

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, 1990b, 1991; Sandage & Lubin 2001; Lubin & Sandage 2001a, 2001b, 2001c, hereafter LS01a, LS01b, or LS01c) and is not repeated here.

The first attempt at the test was made by Sandage & Perelmuter (1990a, 1990b, 1991) using ground-based data in the references just cited (hereafter SP90a, SP90b; SP91). The purpose of the present paper is to continue the discussion made in a second attempt of the test by Lubin & Sandage using Hubble Space Telescope (HST) data (LS01c) by showing more directly that a Tolman signal exists in three remote clusters observed with HST by Oke, Postman, and Lubin. Their photometric data are published in papers by various permutations of the order of the authors (Oke et al. 1998; Postman et al. 1998, 2001; Lubin et al. 1998; L. M. Lubin et al. 2001, in preparation). The analysis by LS01c remains valid but the representation of the data is different here.

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 author.

Together with other projects for LST in observational cosmology, I discussed the requirements of spatial resolution and input flux needed to make the Tolman SB 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 m aperture and a spatial resolution of 0.05–0.1 arcsec. Of course, not known at the time was the way that SB over the image of E galaxies at some standard diameter varies systematically with intrinsic size and absolute magnitude, or how to define a series of "standard diameters" within which to measure the SB. 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, LS01b, LS01c), 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 Oke et al. (1998); Postman et al. (1998, 2001); Lubin et al. (1998); L. M. Lubin et al. (2001, in preparation), and reduced in the way described in LS01a, LS01b, LS01c are again used here. What is different in the analysis here is threefold.

• 1.  
  A more comprehensive calibration of the local data on E galaxy SB as a function of intrinsic radius is made in Section 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 SB 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 SB with radius (or absolute magnitude for M brighter than M_V = −20, e.g., see Figures 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 Section 3 is devoted to determining these properties using Sérsic profiles.
• 3.  
  We show in Section 3.4 that the Sérsic profile with a mean exponent of 0.46, rather than a de Vaucouleurs value of 0.25, fits our local sample over the small range of intrinsic radii that is encompassed by the 34 galaxies in the remote HST sample. The systematic deviation of the profile exponent from a de Vaucouleurs r^0.25 value with intrinsic radius is strong and mapped. The variation of n with radius is the same as discovered by Binggeli & Jerjen (1998; Figure 2), showing why the value differs from 0.25 for the HST sample at the radii that are encountered here.

3.1. 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 deceleration parameter, q₀, of a particular world model (Mattig 1958). We would also have to apply the K-correction for the effect of redshift on the isophotal SB 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. A particular value of q₀ is assumed, and the variation of the results for a range of q₀ is discussed.

Some of the remarkable properties of the Petrosian radius function have been discussed elsewhere (SP90a; Kron 1995; Sandage 1995; SL01) and are not repeated here.

Two equivalent definitions of these radii are these. (1) A Petrosian radius is a distance from the center of an image where the 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 (GC), as,

$$\begin{equation} \eta \ (\mbox{in mag}) = 2.5\log\, (5 d\log\, r/d\mbox{\;mag}), \end{equation} \tag{ 1 }$$

where the GC is the summed intensity (in magnitudes) out to that r and its slope at r is d mag/dlog  r. Expressed as the slope in intensity units, as in PL95, α(=dlog  L/dlog  r), which is used by some authors. Of course, η(inmag) = 2.5log  2/α.

The Petrosian η function for radii was used throughout the first and second attempts to make the Tolman test (SP90a, SP90b, SP91; SL01; and LS01a, LS01b, LS01c).

3.2. 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 GCs from Table 11 of the Third Reference Catalog (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 GC magnitudes. The odd numbered columns are the η radii at the listed log  r/r_e values.

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Table 1. Growth-curve Magnitudes and Petrosian η Radii as Function of log  r/r_e for Various Hubble Types and for the de Vaucouleurs r^1/4 law and a Sérsic profile with n = 0.46

log  r/re			Sérsic				Sérsic
	T = −5: E		n = 0.25		T = 1: Sa		n = 0.46		T = 3: Sb		T = 5: Sc		T = 7: Sd		T = 9: Sm
	Δm	η	Δm	η	Δm	η	Δm	η	Δm	η	Δm	η	Δm	η	Δm	η
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)	(17)
−1.0	2.97	...			3.22	...			3.41	...	3.65	...	3.99	...	4.43	...
−0.9	2.69	0.65	2.70	0.61	2.95	0.67	3.45	0.36	3.13	0.61	3.33	0.54	3.63	0.39	4.01	0.22
−0.8	2.42	0.69	2.36	0.77	2.68	0.61	3.00	0.40	2.84	0.59	3.04	0.59	3.29	0.48	3.61	0.27
−0.7	2.16	0.73	2.04	0.83	2.38	0.61	2.62	0.44	2.55	0.67	2.77	0.67	2.99	0.57	3.23	0.34
−0.6	1.91	0.80	1.80	0.89	2.11	0.73	2.30	0.49	2.30	0.71	2.50	0.59	2.70	0.47	2.88	0.28
−0.5	1.68	0.89	1.61	0.97	1.87	0.75	2.00	0.55	2.03	0.65	2.19	0.56	2.34	0.40	2.46	0.27
−0.4	1.47	0.94	1.41	1.03	1.61	0.80	1.69	0.62	1.75	0.69	1.90	0.57	2.01	0.47	2.10	0.34
−0.3	1.26	1.02	1.23	1.10	1.39	0.92	1.42	0.72	1.50	0.73	1.60	0.56	1.68	0.45	1.73	0.34
−0.2	1.08	1.14	1.06	1.19	1.18	0.92	1.21	0.82	1.24	0.73	1.30	0.59	1.35	0.48	1.37	0.42
−0.1	0.91	1.20	0.90	1.28	0.96	0.92	0.92	0.94	0.99	0.78	1.02	0.65	1.04	0.65	1.05	0.52
0.0	0.75	1.31	0.75	1.39	0.75	0.89	0.74	1.08	0.75	0.89	0.75	0.73	0.75	0.69	0.75	0.65
0.1	0.61	1.46	0.64	1.50	0.57	1.14	0.60	1.23	0.55	1.05	0.53	1.00	0.51	0.94	0.50	0.89
0.2	0.49	1.64	0.50	1.63	0.40	1.38	0.45	1.41	0.37	1.31	0.35	1.27	0.33	1.31	0.31	1.27
0.3	0.39	1.80	0.41	1.80	0.29	1.69	0.31	1.61	0.25	1.64	0.22	1.60	0.21	1.60	0.19	1.64
0.4	0.30	1.99	0.32	1.97	0.19	1.92	0.22	1.87	0.15	1.99	0.12	2.06	0.10	2.06	0.09	2.14
0.5	0.23	2.12	0.25	2.14	0.12	2.22	0.15	2.19	0.09	2.40	0.07	2.61	0.06	2.89	0.05	2.89
0.6	0.17	2.50	0.18	2.38	0.06	2.89	0.09	2.58	0.04	3.06	0.03	3.25	0.03	3.25	0.02	3.50
0.7	0.13	2.24	0.14	2.60	0.05	3.81	0.03	3.00	0.03	4.25	0.02	4.25	0.01	3.81	0.01	4.25
0.8	0.09	2.89	0.10	2.90	0.03	3.50	0.01	3.52	0.02	...	0.01	...	0.01	4.25
0.9	0.06	3.25	0.07	3.27	0.01	4.25	0.00	4.20	0.01	...
1.0	0.04	3.81	0.05	3.70	0.00	...

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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, 1976). Sérsic (1968) generalized the de Vaucouleurs r^0.25 profile with the function,

$$\begin{equation} \log\, (B(r/r_{e})/B(r_{e})) = -b_{n}((r/a)^{n} -1)), \end{equation} \tag{ 2 }$$

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 the 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 GCs 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 GC magnitudes¹ and Petrosian η values in Table 3 are at the listed r/r_e values for these n values. The listings for n = 0.25 and 0.46 are the same as in Table 1.

Table 2. Calculated b_n Values for the Sérsic Family of Luminosity Profiles

Sérsic n	bn	Sérsic n	bn
(1)	(2)	(3)	(4)
0.15	5.62	0.46	1.72
0.20	4.17	0.60	1.28
0.25	3.33	1.00	0.73
0.40	2.00	1.50	0.44

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Table 3. Growth Curves and η Values for the Sérsic Function at the Listed log  r/r_e Ratios

log r/re	n = 0.15		n = 0.20		n = 0.25		n = 0.40		n = 0.46		n = 0.60		n = 1.00		n = 1.50
	Δm	η	Δm	η	Δm	η	Δm	η	Δm	η	Δm	η	Δm	η	Δm	η
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)	(17)
−0.4	1.26	1.26	1.34	1.13	1.41	1.03	1.60	0.74			1.79	...	2.05	...	2.30	...
−0.3	1.14	1.35	1.19	1.20	1.23	1.10	1.32	0.80			1.55	...	1.70	...	1.81	...
−0.2	1.00	1.41	1.02	1.29	1.06	1.19	1.10	0.88	1.21	0.82	1.28	0.66	1.40	0.40	1.42	0.21
−0.1	0.86	1.51	0.89	1.38	0.90	1.28	0.94	1.00	0.92	0.94	1.00	0.76	1.05	0.54	1.08	0.32
0.0	0.75	1.60	0.75	1.47	0.75	1.39	0.75	1.14	0.74	1.08	0.75	0.90	0.75	0.72	0.75	0.50
0.1	0.64	1.70	0.63	1.60	0.64	1.50	0.61	1.26	0.60	1.23	0.55	1.10	0.50	0.94	0.44	0.73
0.2	0.56	1.80	0.52	1.71	0.50	1.63	0.48	1.44	0.45	1.41	0.38	1.30	0.32	1.20	0.25	1.06
0.3	0.48	1.91	0.43	1.88	0.41	1.80	0.33	1.63	0.31	1.61	0.26	1.56	0.20	1.52	0.11	1.53
0.4	0.40	2.08	0.36	2.01	0.32	1.97	0.22	1.86	0.22	1.87	0.15	1.90	0.11	2.10	0.04	2.36
0.5	0.33	2.19	0.29	2.19	0.25	2.14	0.16	2.14	0.15	2.19	0.08	2.31	0.04	2.80	0.01	4.10
0.6	0.27	2.32	0.21	2.37	0.18	2.38	0.10	2.43	0.09	2.58	0.03	2.80	0.00	3.91
0.7	0.21	2.51	0.16	2.53	0.14	2.60	0.06	2.80	0.03	3.00	0.00	3.38
0.8	0.15	2.67	0.11	2.77	0.10	2.90	0.04	3.24	0.01	3.52
0.9	0.11	2.88	0.09	2.99	0.07	3.27	0.02	3.75	0.00	4.20
1.0	0.07	3.10	0.04	3.23	0.05	3.70	0.00	...
1.1	0.05	3.32	0.02	3.53	0.03	...	0.00	...
1.2	0.02	3.50	0.01	3.88

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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.

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The progression of 〈SB〉 with absolute magnitude was discovered by many authors (e.g., Oemler 1973, 1974; Kormendy 1977, 1987; Strom & Strom 1978a, 1978b, 1978c; Michard 1979; Thomsen & Frandsen 1983; Binggeli et al. 1984; Choloniewski 1985; Schombert 1986; Ichikawa et al. 1986; Caldwell & Bothun 1987; Djorgovski & Davis 1987; Ferguson & Sandage 1988; Impey et al. 1988; Caon et al. 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 3.²

But before giving the proof, we first rearrange Table 3 as Table 4 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 Section 5 to use r/r(2) to compare SB levels at various points rather than r_e. To this end, Table 4 has been 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.

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Table 4. log r/r(η = 2) at Various η Values for the Sérsic Family from Table 3

η/n	0.15	0.20	0.25	T = −5	0.40	0.46	0.60	1.00	1.50
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
1.0	...	(−1.0)	−0.82	−0.72	−0.65	−0.50	−0.34	−0.27	−0.15
1.3	−0.78	−0.58	−0.51	−0.42	−0.31	−0.29	−0.21	−0.15	−0.10
1.5	−0.51	−0.38	−0.32	−0.28	−0.21	−0.18	−0.14	−0.09	−0.06
1.7	−0.29	−0.21	−0.20	−0.16	−0.11	−0.10	−0.06	−0.06	−0.03
2.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2.5	0.33	0.28	0.22	0.21	0.17	0.15	0.09	0.07	0.05
3.0	0.59	0.50	0.41	0.40	0.28	0.26	0.18	0.13	0.09
3.5	0.81	0.70	0.56	0.54	0.40	0.36	0.26	0.19	0.13
4.0	...	0.83	0.69	0.62	0.50	0.44	0.30	0.22	0.17

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3.4. Oemler Profiles

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 (1974, 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,

$$\begin{equation} B(r)/B_{0}=\frac{\mbox{exp}(-r/\alpha)^{2}}{(1 + r/\beta)^{2}}, \end{equation} \tag{ 3 }$$

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 GC 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, log r/r(η = 2) Oemler profiles are listed for α/β ratios of 10, 30, 60, and 100. The T = −5 and n = 0.46 relations are copied from Table 4. Three Oemler profiles from Table 5 are shown and are compared with the T = −5 profile and the n = 0.46 Sérsic profile. 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 GCs.

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Table 5. log r/r(η = 2) as Function of η and the α-to-β Ratio for Four Oemler Profiles and the Observed T = −5 and a Sérsic Profile With n = 0.46

η	α-to-β Ratio				E	Sérsic
	10	30	60	100	T = −5	0.46
(1)	(2)	(3)	(4)	(5)	(6)	(7)
1.0	−0.44	−0.70	−0.88	−0.91	−0.72	−0.50
1.3	−0.25	−0.42	−0.58	−0.67	−0.42	−0.29
1.5	−0.18	−0.26	−0.38	−0.46	−0.28	−0.18
1.7	−0.10	−0.14	−0.21	−0.25	−0.16	−0.10
2.0	0.00	0.00	0.00	0.00	0.00	0.00
2.5	0.10	0.15	0.20	0.26	0.21	0.15
3.0	0.18	0.24	0.31	0.40	0.40	0.26
3.5	0.22	0.31	0.39	0.48	0.54	0.36

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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 SB–η properties of the Sérsic function.

3.5. SB versus η for Sérsic Profiles Normalized to η = 2

Because the Tolman test concerns SB, it is necessary to cast the Sérsic profiles in Tables 3 and 4 and Figure 3 in terms of SB. The relative shape of the SB profiles in the Sérsic family is the subject of this section.

Once the GC of a particular intensity profile is known, the SB averaged over a given radius follows from,

$$\begin{equation} \langle \mbox{SB}(\eta)\rangle =2.5\log\, (\pi r_{\eta })^{2} + m(\eta)_{\rm GC}, \end{equation} \tag{ 4 }$$

where the first term on the right is the area expressed in mag per square arcsec, and the other is the magnitude from the 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

$$\begin{equation} \Delta \langle \mbox{SB}\rangle _{\eta =2}=5\log\, r/r(2)+m(\eta)_{\rm GC}. \end{equation} \tag{ 5 }$$

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 GC, are also listed in column (5). Figure 5 is the second diagnostic diagram that will be used in later sections.

Table 6. Surface Brightness Normalized to η = 2 for Eight Sérsic Profiles, Averaged Over the Area Interior to the Listed η Radii

η/n	0.15	0.20	0.25	T = −5a	0.40	0.46	0.60	1.00	1.50
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
1.0	...	−3.30	−2.82	−2.78	−2.02	−1.70	−1.38	−0.93	−0.52
1.3	...	−2.15	−1.82	−1.78	−1.18	−1.10	−0.83	−0.58	−0.35
1.5	−1.80	−1.50	−1.20	−1.25	−0.78	−0.73	−0.60	−0.38	−0.26
1.7	−1.07	−0.85	−0.68	−0.73	−0.48	−0.40	−0.38	−0.20	−0.14
2.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2.5	1.52	1.37	1.10	0.97	0.68	0.64	0.48	0.32	0.25
3.0	2.66	2.45	1.90	1.58	1.30	1.25	1.13	0.59	0.46
3.5	3.62	3.30	2.60	2.20	1.85	1.72	1.54	0.80	0.66
4.0	...	3.90	...	2.72	2.30	2.21	2.05	1.00	0.84

Notes. The unit is magnitudes (relative to η = 2 values) per unit area. ^aCalculated from the observed T = −5 growth curve from Table 11 of the RC3.

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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 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 GC. The E galaxies used to define the T = −5 GC (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 RCT3 = −5 GC over a large range of absolute magnitude and intrinsic size.

Data on the observed profiles for local galaxies that span a range of 5 mag are in the literature for the Virgo, Fornax, and Coma clusters. The photometry is by Fraser (1977), King (1978), Hodge (1978), Ichikawa et al. (1986), Schombert (1986), Jedrzejewski (1987), and Vigroux 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 (1). A diagram similar to Figure 5 in SP90a was drawn for each galaxy showing the published profile, the GC, and η calculated at the listed radii. The log  r (arcsec) 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 SB 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) SB 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 and 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 Figure 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 and 7 where both log  r(1)/r(2) and 〈SB(2)〉 − 〈SB(1)〉 are plotted against log  R(η = 2). Because these ratios change systematically with log  R(2), and because they are both functions of Sérsic n at any given η (Figures 3 and 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 and 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.

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Interpolating in 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.

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 Section 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 Section 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 and 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) with Figures 3 and 5 for the Sérsic family as the n exponent is varied shows that the Sérsic models are excellent fits to actual E galaxy profiles over the entire range of η and Sérsic n values, consistent with Figure 2 of Binggeli & Jerjen (1998) and with the conclusion of Kormendy et al. (2009).

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 and 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 to determine a mean value of n that applies to the HST data.

That the SB 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 et al. (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; Figure 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 1 mag whereas the total range of SB between log  R(2) of 4.2 and 5.4 is 4 mag (Figure 2 of SL01). Because the Tolman test is about SB, 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 SB–intrinsic 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 GCs of PL95 with H₀ = 50.

The R magnitudes by Postman & Lauer are on the Cape/Cousins near-red photometric system as realized by Landolt (1983, 1992). The R_cape bandpass 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 photocathodes in the 1970s (e.g., Sandage 1972, 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 $\it UBV(RI)_{\rm cape}$ 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$\rangle _{R_{\rm cape}}$ 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 linear correlation of 〈SB〉 and size for η = 2.0 for zero redshift is,

$$\begin{equation} \langle \mbox{SB}(2)\rangle _{R_{\rm cape}}=2.97\log R(2)_{50}+8.53, \end{equation} \tag{ 6 }$$

valid for log R(2) (pc) >4.3(H₀ = 50). The redshifts are small enough that the value of the spatial curvature, q₀, is inconsequential.

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 and 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,

$$\begin{equation} \mbox{SB}_{V}(2) = 2.89 \log R(2) + 9.53, \end{equation} \tag{ 7 }$$

reduced to zero redshift and valid for log  R>4.3. The nonlinearity for log  R < 4.3(H₀ = 50) is clearly seen (not shown) in these data, needing again the Table 7 correction for the smaller radii.

To equate the slope in Equation (7) with Equation (6), we change it to 2.97 and adjust the zero point, giving,

$$\begin{equation} \mbox{SB}_{V}(2) = 2.97 \log R(2) + 9.17, \end{equation} \tag{ 8 }$$

again valid only for log  R(2)>4.3(H₀ = 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 local 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 GCs 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 of local E and S0 galaxies.

Applying V − R = 0.57 to Equation (8) gives,

$$\begin{equation} \mbox{SB}(2)_{R_{\rm cape}} = 2.97 \log\, R(2) + 8.60, \end{equation} \tag{ 9 }$$

for local E galaxies with log  R(2)>4.3(H₀ = 50). This is within 0.07 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.

SB 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 arcsec and 1 arcsec. The intrinsic sizes for the database range from log R(2) = 3.45 to 4.20 (H₀ = 50, q₀ = 1/2, Λ = 0), with a mean of 3.9.

The photometry was done in the HST F702W and F814W photometric system and was transformed to the ground-based Cape/Cousins R and I system as described in Paper III of this series (LS01b; Section 2.1.2). The K-corrections for the effects of redshift and passive evolution were calculated using Keck spectra, as shown in Figure 7 of LS01b.

It is of course crucial for the test that the high-redshift HST galaxies be of the same type as the local galaxies to which they are to be compared. The sample-selection procedure of the HST galaxies insures this identity with local E and S0 galaxies.

As explained in Section 2.2 of LS01b, the tests for similarity of the local and the high-redshift HST galaxies were threefold. (1) All the HST galaxies were chosen to be of morphological type E or S0 by inspection of the HST images by Lubin and Oke as part of their cluster program, and verified again by the author. (2) Keck spectra confirmed the typing by showing typical early-type spectra with normal H and K lines and a prominent 4000 Å break. (3) R − I broadband colors, listed in the last column of Tables 6–8 of LS01b (Paper III) average 〈R − I〉_K(cape) = 0.67 ± 0.08, after the individual K-corrections from Table 4 of Paper III are applied individually to each of the 34 HST galaxies (the large error is due to the large rms spread in the mean K-corrections listed in Table 4 of Paper III). This K-corrected mean (R − I)_cape color for the HST galaxies is clearly consistent with the G, K spectral types found from the Keck spectroscopy because the 〈R − I〉_cape color of the local E galaxies is 0.64 (see below).

The method of calculating the K-corrections, discussed in Section 3 of Paper III, is the usual way of numerically redshifting an adopted SED (in this case, that of a local E galaxy but changed for evolution of the SED of the mean of the HST galaxies according to the evolution models of Bruzual & Charlot 1993) through the spectral energy transmission bands of the detector and then calculating the intensity that would have been observed in the rest frame of the particular photometric bands. This is the same method as set out in Appendix B of Humason et al. (1956, HMS) and is the same as used by Oke & Sandage (1968) in their first determination of the K-term for local E galaxies. The bandwidth-shrinking factor of 2.5log(1 + z) is included in the definition of the K-correction in Equation (B7) of HMS.

Because of the strong variation of 〈SB〉 with intrinsic size, which we have been emphasizing and as listed in Table 8, 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. We have binned the data rather than working with a continuum to reduce the scatter in using individual galaxies.

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 SB of 〈SB_I(2)〉 = 20.18 mag/arcsec² at η = 2.0 is then spread to the η values of 1.0, 1.3, 1.5, and 1.7 using the SB 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. These are listed in Table 10.

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.

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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. The shape of the Sérsic curve for n = 0.46 is brought down from the zero-redshift curve near the top of each panel in Figure 9 and is zero-pointed to the data at η = 1.5.

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The fit of the shape of the HST data to the standard curve is generally good. The average 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 and 11 for Cl 1604+4304 (z = 0.8967) from the data in Tables 11 and 12 (there are only two radius bins), and in Figures 12 and 13 for Cl 1604+4321 (z = 0.9243) from Tables 13 and 14.

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The SB 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 5–7 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 (Figures 1 and 2 there). Here we compare the local and remote data at fixed radii as η is varied (Figures 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.

6.1. The Estimated Errors

Before the error budget can be determined, two caveats are necessary.

• 1.  
  In Figures 8–13, we have used an average value of n = 0.46 for the 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 Figure 2), we showed in Figures 6 and 7 using Tables 4 and 6 that the Sérsic n exponent is a 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. It makes a negligible difference in Table 15 and Figures 8–13, because the variation of n in this range of radii is small. As the size varies between log R(2) = 3.5 and 4.2, the Sérsic exponent varies only between 0.4 and 0.6 for the HST galaxies, as shown in Figure 14 using the top row of Table 4. 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 q₀ = 1/2, H₀ = 50, and Λ = 0. Different radii would be obtained for different q₀ models. Table 8 of LS01c shows that if q₀ = 0, the radii would be about 25% larger than we have used. If q₀ = 1, the R values would be 15% smaller. Therefore, using the slope of Equation (9), the SB 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 q₀ = 1/2.

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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 q₀ because the uncertainties here are at the level just stated for the range of q₀ 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 q₀ = 1/2 case.³

It might be thought that a systematic error could be present due to the effect on the Petrosian η radii of a color gradient in the E or S0 galaxy image. Such an error would only exist if a color gradient is different between the local standard galaxies and those in the remote HST clusters. This is deemed unlikely because color gradients are due to events in the early formation processes of E and S0 galaxies, and these are expected to be the same for the modest (for this problem) redshift HST galaxies and the local sample. In any case, the gradients are very small compared to the large Tolman signal that is observed here, and any such effect is clearly negligible.

Table 16. Average log r(η)/r(2) vs. η Listed Separately for Each of the HST Clusters and the Average Over All HST Galaxies

〈log r(η)/r(2)〉
Cluster/η	1.0	1.3	1.5	1.7	2.0	n
(1)	(2)	(3)	(4)	(5)	(6)	(7)
1324+3011	−0.489	−0.291	−0.195	−0.091	0.000	11
1604+4304	−0.454	−0.273	−0.169	−0.096	0.000	6
1604+4321	−0.510	−0.310	−0.213	−0.115	0.000	13
Mean	−0.491	−0.296	−0.198	−0.102	0.000	30

Note. Mean values are shown in Figure 15.

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Ignoring these two caveats and the non-existent color gradient problem, the statistical errors are estimated as follows. We require the mean 〈SB〉 offsets in the 〈SB〉 versus η curves between the zero-redshift standard curves and the data points in Figures 9, 11, and 13. There are two components. (1) One is the difference between the observed points and the standard curves brought down from the zero-redshift 〈SB〉 curves. (2) The second is the accuracy with which we know the position of the standard curves at zero redshift for particular radius values.

• 1.  
  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.
• 2.  
  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. There average rms = 0.100, giving a mean error of ±0.05 in log R. From Equation (9), this translates to a 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 mean radius 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,

$$\begin{equation} \Delta \langle \mbox{SB}\rangle =\left\lbrace \begin{array}{@{}l@{\quad }l} 2.5\log (1+z)^{2.80\pm 0.25}, & \mbox{mag in the\ } R\ \mbox{band} \\ \ms 2.5\log (1+z)^{3.48\pm 0.14}, & \mbox{mag in the\ } I\ \mbox{band}. \end{array}\right. \end{equation} \tag{ 10 }$$

6.2. 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,

$$\begin{equation} M_{\rm evol} =\left\lbrace \begin{array}{@{}l@{\quad }l} 2.5\log (1+z)^{1.20\pm 0.25}, & \mbox{mag in the\ } R\ \mbox{band} \\ \ms 2.5\log (1+z)^{0.52\pm 0.14}, & \mbox{mag in the\ } I\ \mbox{band}. \end{array}\right. \end{equation} \tag{ 11 }$$

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; Section 4.2). A more complete discussion of the expected size of the passive evolution of the stellar population via the change in the turnoff luminosity of the main sequence and its consequences is given in Paper IV of this series (LS01c) and is not repeated here. 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 at Davis, who majored in physics at the University of California at 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 differently here then by LS01c 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.

7.1. Preliminaries

7.2. The Eleven Steps

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. 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.