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Cosmology's Missing Mass Problems

Robert S. Fritzius
Shade Tree Physics

Installed as a web page on 27 Jun 2003. Latest update 10 Apr 2019.
Text additions or changes are in bold.
Parts 1 - 7 have been incorporated into a single file.

This research has made use of NASA's Astrophysics Data System (NADS) Bibliographic Services.

The first Missing Mass Problem

In 1933 Fritz Zwicky was the first to find a need to invoke the idea of missing mass or dark matter. He looked at eight Coma galaxies. By assuming visual equilibrium,* he calculated the mass-to-light ratio and determined that about 90% of the mass necessary to account for the observed ratio was missing and therefore invisible. or "dark." Here, the apparent rapid velocities of the galaxies, with respect to their common center of mass, suggested that much more mass (than could be seen) was required to keep the galaxies from flying out of the cluster. (BVSD) [Link no longer works.]

*Visual equilibrium means that the amount of light (of stars) is proportional to amount of mass (number of stars).

In 1936 Sinclair Smith found similar evidence for the existence of invisible mass in the Virgo cluster. [An outline of the Virgo cluster can be seen in the lower left polar plot of the Star Map in Galactic Perspective]

In 1940 Oort estimated (based on the Mass-to-light ratio of spiral nebulae) that 90% of the mass in the local group of spiral nebulae is "missing." Oort didn't cite Zwicky's 1933 paper (BSVD).

The Second Missing Mass Problem

In 1970 Vera Rubin & Kent Ford, and in 1975; Roberts & White determined that the radial velocity curve plot (radius vs. velocity) for the Milky Way Galaxy flattens out rather than trailing down. The implication was that mass continues to increase with radius. Many other "galaxies" show the same effect. (BSVD)

It was commonly assumed that stars in galaxies would follow Kepler's laws like planets in the solar system. One of the earliest papers on this idea was published in 1927 by Bertil Lindblad, the newly appointed director of the Stockholm Observatory. (LB27) - NADS. [Added 15 Oct 2003.] Since the stars in the Milky Way galaxy and in spiral nebulae didn't follow Kepler's laws, then it was assumed that there was a whole lot of distributed invisible mass affecting them. (DJ)

Below is a diagram that demonstrates how Keplerian motions compare to the dynamics of stars in sparsely populated disk strutures that have no invisible masses. A computer program was used to model free-running Newtonian interactions between stars themselves and a Central Object (CO). Star masses, radii, and speeds are on arbitrary scales. All stars were assigned masses of 1.0, and in each run the mass of the Central Object was set to equal 32 stellar masses minus the summed mass of the stars in orbit. (The first run, with 4 stars and CO mass=28, (red curve) approximates Keplerian conditions. In the final run, with 32 stars, the mass of the CO was zero, and the blue curve resembles the radial velocity curves for big stellar disk systems. [Added 04 Jun 2010]

sparsely populated stellar disks
Stellar Linear Speeds versus System Configuration and Central Object Mass

In any given simulation run the initial speeds of the stars in each ring were set manually, so as to cause them to travel (for a while, at least) with nearly constant radii. Usually, after one or two orbits of the innermost star(s), the orbits begin gradually spiralling into one another and the traffic pattern becomes more and more like a square dance. Finally, they transition into a state of bedlam. I think this tendency toward disorder arises from not including magnetism in the modelling, i.e., gravitational interactions, by themselves, are incapable of leading to long term group stability. [Last sentence was added on 25 Aug 2012.]

The ring radii for these runs are 2.0, 3.18, 5.06 and 8.03 (The ratio between adjacent ring sizes is 1.59:1.) [Added 31 May 2010. The graph and descriptive material were moved here from Part 2 on 04 Jun 2010.]

Where the author is going in this article

A brief outline of most of the missing mass explanations, with links to internet sources of more in-depth information is provided. But then, the article will move on to what the author considers to be the real problems (and what to do about them).

First Proposed Solutions to Missing Mass Problem

(Including some "known" or suspected problems in selected cases)

The basic outline for this section is derived from the articles Dark Matter versus MOND URL was http://www.astro.umd.edu/~ssm/mond/mondvsDM.html (DMVM), and What is the Missing Mass problem? Science Net - Physics (SNP). URL was at http://www.sciencenet.org.uk/database/Physics/Cosmology/p00717c.html. Other sources are identified as appropriate. This list is under construction.

CDM (Cold Dark Matter)
    WIMPs (Weakly Interacting Massive Particles) (SNP) These move slowly.
        Axions, 6-15 Mev?, Seishi Matsuki ca 1983 - None visualized yet. - (SMKU)
        Massless neutrinos (CXO)
        Photinos (CXO)
        Neutralinos (DJ) [Link to www.astro.queensu.ca no longer works.]
        No WIMPS detected yet. (CXO)
    Hydrogen Gas - Difficult for it to hide. (CXO)

WDM (Warm Dark Matter) - Baryonic
    Warm Intergalactic Fog - (NF03) - The Jury is out. - [Added 16 August 2003.]

HDM (Hot Dark Matter) -Non-Baryonic (SNP)
    (Hot, means traveling at or near the speed of light.)
    Neutrinos - If their mass was = 92 eV it would make Omega = 1.0 (DJ)
    Link to (DJ) was: http://www.astro.queensu.ca/~dursi/dm-tutorial/dm0.html
        Apparently Neutrinos have mass but not enough to fit the bill. (SNP)
    The author's Sub-quantum Positive and negative chargelets (FR93) would fit
        into the HDM category, but he is of the opinion that trying to capture them
        is equivalent to trying to capture virtual photons. He does, however, apply
        them to the missing mass problem. See A Variable Charge Explanation for
        Cosmological Redshift
in Part 6 of this article. [Added 21 Jan 2004.]
BDM (Baryonic Dark Matter)
    MACHOs (Massive Astronomical Compact Halo Objects) (SMKU)
        Dead stars/white dwarfs, in galactic halo (BBCN) (SMKU)
        Not enough helium, which should accompany white dwarfs. (CXO)
        Brown Dwarves (Dwarfs) / Jupiters - Not enough of them. (CXO)
        Red Dwarfs - Not enough of them. (SEV94) - [Added 3 Aug 2003.]
        Neutron Stars - Scarcer than white dwarfs -
            No evidence of released energy and heavy elements. (CXO)
        Unborn stars (SMKU)
    Black Holes (SMKU) (CXO)
        Predicted by Einstein's GTR
    Interloper galaxies may account for all the missing mass. (DJ)

Both CDM and HDM have their problems; HDM can't form small structures like galaxies, and CDM has problems forming large scale structure (DJ)

Other ways of looking at the problem

Changes to Gravity
    Quantum Gravity
        A distortion in the quantum vacuum energy leads to an additional "Bubble Force"
        which may explain the constant galactic-rotation curves. (NRV)
            But.... the Hubble Space Telescope is taking pictures that are too clear.
            They show no evidence of the hazy effects that quantum foam should produce.
            Looks like trouble for quantum foam and hence quantum gravity. ( BR03)
    Peebles ICDM (Isocurvature Cold Dark Matter) model (PICDM-1, PICDM-2)
      "[T]he BOOMERANG measurement of the height and the position of the
        first "acoustic peak" in the CMB fluctuations has ruled out the IDCM
        model as it was originally proposed." - (MW02)
    Conformal Weyl gravity (No details)
    Non-symmetric gravity (No details)
    MOND (Modified Newtonian Dynamics)
        One proponent of MOND says that all of the dark matter theories fail. (MS1)
        Link to (MS1) was http://www.astro.umd.edu/~ssm/mond/mondvsDM.html.
        See the MOND section below.
    Inertial Induction (Noriaki Namba) (NN02)
        Stars are hypothesized to exert inertial induction on each other which tends to
        produce coherent group accelerations. This coherence tendency leads to constant
        rotation speeds in the outer regions of rotating stellar systems, the Milky Way
        galaxy for example. [Added 20 April 2004.]

Electrodynamic Considerations
    Plasma cosmology (Hannes Alfvén) (No Ref.)
    Electric Stars (Ralph Juergens) (No Ref.)
        These models complement one another. Large-scale electrodynamic processes
        moderate the interactions between stars and lead to "group transport" behavior
        along the lines of fluid dynamics).

Part 2

New Force Laws

MOND (MOdified Newtonian Dynamics) Mordehai Milgrom, 1983. (MS1)

MOND is a modification of the usual Newtonian force law, hypothesized in 1983 by Moti Milgrom of the Weizmann Institute, as an alternative to Dark Matter. It explains the flat rotation curves of some* spiral galaxies without resorting to the use of dark matter. It applies to Dwarf and Low Surface Brightness galaxies only. (MS2) (NADS)
*(Acceleration is critical factor - Low acceleration disk galaxies needed.)

MOND can be interpreted as either a modification of gravity through a change to the Poisson equation, or as a modification of inertia through a breaking of the equivalence of inertial and gravitational mass. (It's basically a mathematical exercise.) The modification occurs at very small accelerations.

Stacy McGaugh, a proponent of MOND, says "MOND is difficult to test, and does not constitute a falsifiable theory." (MS1). With that in mind, we move on to new territory.

Chandra observations of the hot gas halo of NGC 720 (a flattened gas distribution) is not consistent with the MOND predictions.
(NNR02) - Was http://www1.msfc.nasa.gov/NEWSROOM/news/releases/2002/02-264.html

Soap-Box Statement

In the writer's opinion, the first missing mass problem (not enough visible mass to account for observed spiral nebulae motions) is inextricably tied to the scale of the universe problem, which is intertwined with the controversy spawned by the idea that Hubble's increasing redshifts should be interpreted as evidence of actual radial velocity increases, or of an expansion of space. [Re-phrased on 01 Aug 2003, and again on 18 Mar 2006.]

* * *

Einstein's equations of GTR did not permit a static universe. The Universe must either be expanding or contracting. The de Sitter solution to the GTR equations predicted an expanding universe.* De Sitter is also the mathematician who almost buried the ideas of Einstein's biggest competitor on relativity theory, Swiss Physicist Walter Ritz. See: (SV87) The author happens to be a fan of Ritz's Newtonian-like electrodynamics, some of which can be seen at (RW08).
*(This statement may need some modifications. Checking! - 17 July 2003)

The Friedman-Lemaitre Standard Cosmological Model (which came to be accepted as a fix to GTR) also predicts an expanding universe. Hubble's increasing redshifts for galaxies at greater and greater distances came to be interpreted as being evidence of the required expansion of the universe, and thus the Big Bang came to be the paradigm.

The author contends that cosmological redshift, as a velocity/distance indicator, has led to a universe whose size is grossly overestimated, and that the inflated "scale of the universe" has contributed to the first of the missing mass problems. Such a view demands a re-interpretation of cosmological redshift. The author's re-interpretation will be respectfully offered for consideration. [Modified 18 Mar 2006.]

The second ... missing mass problem (the flat rotation velocity curves of galaxies/spiral nebulae) is hypothesized to be related to our failure to take into account the electrodynamic interactions of moving interstellar plasma as it affects the group transport of visible galactic matter (stars and gases). This paper will not dwell on the electrodynamic interactions just mentioned, but astrometric evidence consistent with flat galactic velocity curves was copiously published in the early part of the twentieth century. That evidence came to be rejected by mainstream cosmologists. The author maintains that the real reason for the rejection (of the astrometric evidence) was because the evidence was ... contrary to the growing consensus regarding the idea of a large expanding universe. This article calls for renewed attention to the rejected evidence. [Paragraph was "tweaked" on 1 August 2003.}

The Shapley-Curtis Debate (SC21)

Words inserted into these quotes, to clarify matters, are enclosed in [square brackets].
This procedure will be used in other quotations below.

By 1920, the nature of spiral nebulae had become one of the major questions in astronomy. The National Academy of Sciences organized a Scale of the Universe debate on the topic between Harlow Shapley and Heber Curtis.

Shapley defended the "conventional" view that spiral nebulae were objects associated with the galaxy, rather than large, independent stellar systems. (He did want a bigger Milky Way; 300,000 light-years in diameter.)

At that time the estimates for the diameter of "our" galaxy ranged from 7,000 to 60,000 light years. (SC21)

Curtis wanted to restrict the diameter of the Milky Way to about 30,000 light-years, and championed the hypothesis of a large universe in which spiral nebulae were independent stellar systems, e.g., other island universes.

Among Shapley's arguments:

    "Another consequence of the conclusion that the galactic system is of the order of
    300,000 light-years in greatest diameter, is the previously mentioned difficulty it gives
    to the 'comparable-galaxy' theory of spiral nebulae. I shall not undertake a description and
    discussion of this debatable problem. Since the theory probably stands or falls with the
    hypothesis of a small galactic system, there is little point in discussing other material on
    the subject, especially in view of the recently measured rotations of spiral nebulae which
    appear fatal to such an interpretation."

Curtis had difficulty dealing with Shapley's proposed 300,000 light-year diameter galaxy. In his words:

    "If the spirals are island universes it would seem reasonable and most probable to assign
    to them dimensions of the same order as our galaxy. If, however, their dimensions are
    as great as 300,000 light-years, the [other] island universes must be placed at such enormous
    distances that it would be necessary to assign what seem impossibly great absolute magnitudes
    to the novae which have appeared in these objects." (SC21)

Currently, the galaxy is thought to be about 110,000 light-years in diameter, [roughly four times larger
than Curtis wanted but about one third the diameter estimated by Shapley.] with the solar system being
about 30,000 light-years from the center. (DT71) p. 290)

Here are some further thought provoking remarks that Curtis made with respect to spirals.
From (SC21)

    Many edge-on spirals show a dark, obscuring belt in the middle of the disk. A similar belt
    in the Galaxy would explain why spiral nebulae aren't seen near the plane [of our galaxy].

    Most spiral nebulae have large radial velocities, [based on their redshifts] so they would
    probably escape from the Galaxy's gravity. [This is what lead Zwicky and later researchers
    to postulate missing mass.]

    "The spirals are found in greatest numbers just where the stars are fewest (at the galactic
    poles), and not at all where stars are most numerous (in the galactic plane).... No spiral
    has yet been found actually within the structure [within the confines of the disk] of the
    Milky Way."

    "Their abhorrence of the regions of greatest star density can only be explained on
    the hypothesis that they are, in some unknown manner, repelled by the stars [in our galaxy]."

    "Why should this repulsion have invariably acted essentially at right angles to our galactic plane?"

    "Why have not some been repelled in the direction of our galactic plane?"

    "Should the results of the next quarter-century show close agreement among different
    observers to the effect that the annual motions of translation or rotation of the spirals
    equal or exceed 0.01 arc-seconds in average value, it would seem that the island
    universe theory must be definitely abandoned."

This last remark seems to be in reference to van Maanen's work, who had been regularly reporting annual spiral rotations on the order of 0.02 arcseconds, and in one case as high as 0.038. (Curtis didn't mention van Maanen by name.)

    "It is improbable that our galaxy should, by mere chance, be placed about half way
    between the two great groups of island universes."

Here, he was getting close to admitting a Copernican problem, i.e., a special position for us in the universe.

    "Their space velocity [based on redshifts?] is one hundred times that of the galactic
    diffuse nebulosities, about thirty times the average velocity of the stars, ten times that
    of the planetary nebulae, and five times that of the clusters."

Part 3

Adriaan van Maanen's Evidence for a Small Universe

(Or, Five Pounds of Stuff for a Five Pound Bag)

Adriaan van Maanen's monumental work on of spiral nebulae, as it stood in 1921, offers a long overlooked (read "rejected") solution to the flat rotational velocity curves for "galaxies." In the author's opinion, van Maanen's investigations have been wrongfully relegated to oblivion. Evidence will be presented below to hopefully help rectify the situation, and to commend van Maanen's findings as an already-accomplished solution to one half of the missing mass problem.

From 1916 through 1927 van Maanen, who was a recognized authority on precision astronomical position measurements, produced twelve papers dealing with astrometrically determined internal motions of spiral nebulae. Shapley used his findings as ammunition in the great debate, but later on lost the faith in a groundswell of criticism directed at van Maanen's findings.

In his 1921 paper on the spiral nebula M 81, van Maanen summarized his findings on the internal motions of four spirals, M33, M51, M81 and M101. He states,

    "As in the case of M101, 33, and 51, there is a question as to whether the [nearly circular]
    displacements [of M 81] are best represented by a rotation or by a motion along the arms
    of the spiral. (MA21)

See the tabular summary: Internal Proper Motions for M101, M33, M51 and M81.
[Added 27 July 2003.]

According to van Maanen, this motion of matter along the arms was river-like, i.e., gaseous and/or unresolved-stars spiraling outward in stream-like fashion.

See: Messier 81 Internal Motions. (Added 19 July 2003.)
       Messier 33 Internal Motions. (Added 9 June 2005.)

For each of the spirals he subtracted the average proper motion (in milliarcseconds per year) of the spiral from the measurements for individual data points to arrive at his rotational versus stream angular displacements.

He divided his measured displacements by the number of years between photographs to obtain the annual rates for angular displacements.

There is no direct information in van Maanen's 1921 paper (MA21) that identifies his estimated sizes of, or the distances to, the spirals, but if we assume, with Curtis, that the spirals should have similar tangential linear rotation speeds to our own, i.e., 320 km/sec at our location in the Milky Way disk, and use that speed to convert van Maanen's rotational periods to spiral diameters, we get the following results.

Rotations & Periods

Table 1

In van Maanen's 1923 study on M33 ( MA23), he states, "All this material seems to point to parallaxes for the larger spiral nebulae between a few ten-thousandths and a few thousandths of a second of arc. With such values the diameters of the spirals range from a few light years to several hundred light years. ... it is clear that the present material indicates that the spirals, while enormous in size compared with our solar system, are not at all comparable with the Milky Way system. [Added 7 July 2005.]

These "starting point" spiral diameters are quite contrary to Curtis' idea for other island universes, where diameters on the order of 30,000 light-years were desired. That kind of information had to be available to his colleagues, and van Maanen's train was headed down the wrong track!

The average of recent spectroscopic determinations of rotational velocities for the four objects in Table 1 is 178 km/sec. That's within a factor of two of the 320 km/sec starting point velocity used above. See Table 2.

Rotational Velocities

Table 2


A Cornell University astronomy course webpage shows an edge-on rotation curve for the spiral "galaxy" UGC 9242. The Doppler shift rotation velocities, in the "flat" part of the curve, vary from 195 to 235 Km/sec, and average approximately 220 Km/sec. [Added 8 Aug 2003.]

* * *

Table 3 shows how van Maanen's linear proper motions for a very limited sample of spirals and globular clusters compared to other astronomical objects.

Proper Motion

Table 3

*The largest proper motion for four globular clusters that van Maanen found was 8.6 milliarcseconds per year for the Globular Cluster M2. The total motion of the Hercules Globular Cluster, M13, as well as its internal motion was about one tenth that of the spirals studied. (MA27)

In their paper Internal Motions in Globular Clusters (KA00), King and Anderson report astrometric measurements of internal proper motions in globular cluster 47 Tucanae, using HST's WFPC2 camera with photographic epochs separated by two years. In 47 Tuc, the dispersion of internal proper motions has been found to be about 0.6 milli-arcsec/year in each coordinate. URL for (KA00) was http://arxiv.org/PS_cache/astro-ph/pdf/0007/0007028.pdf.

Referring back to a radial velocity study of 47 Tuc (Meylan & Mayor 1986, get ref.) which showed evidence of rotation of the cluster, King and Anderson report "We can in fact see rotation clearly in the proper motions too..." [No quantitaive value given yet.]

Based on asymmetry [a slight flattening] found in several globular clusters (Pease and Shapley), Van Maanen (MA27) stated "...it follows that the motions resulting from a possible rotation of the clusters are small. Tangential components of the motions are derived therefore only because evidence of such motions had been found in the measures of spiral nebulae." [Emphasis added.]

Van Maanen's findings included the following. For the plates taken at the 25-foot focus of the Mt. Wilson 60-inch reflector, the mean [internal proper motion] tangential component for globular clusters M 13 and M2 was 3 milli-arcsecs/year, compared to the average for seven spirals of 18.4 milli-arcsecs/year.

Comparing findings from 47 Tuc to those of M13 and M2 is not quite the same as comparing apples and oranges, but it does leave much to be desired. Even so, King and Anderson's 0.6 milli-arcsecond/year dispersion for 47 Tuc and van Maanen's 3 milli-arcsecond/year average tangential component for M13 and M2 are within a factor of ten-to-one of another. It might be worthwhile to bring HST's astrometric capabilities to bear on one or both of these latter objects.

[Added 10 August 2003.]

* * *

The proper motions of Quasi-Stellar Objects (QSOs) that the author has found so far, have mostly been attributed to gravitational lensing by intervening objects. It will be of interest to see if the gravitational lenses (which have to be moving in order to induce apparent QSO proper motions) move on, and let the QSOs return to their stationary outposts.

The following comments are from van Maanen's 1921 paper. (MA21) Certain words or phases have been italicized or bolded for emphasis.

    For M 101 there is no appreciable change in the measured rotational component
    with distance from the center, but for M 33, 51, and 81 there appears to be some
    increase of [rotational] motion with distance."

Was 1970 a year for echoes - a' la flat galactic-rotation curves?

    "In all cases, however, the measured displacements agree better with the hypothesis of
    outward motion along the arms of the spiral than they do with an assumed rotation
    of the nebulae as a whole."

Those colleagues of van Maanen who were pushing for an expanding universe, were "unable" to repeat his findings. They dismissed his work as it being a case of his finding what he wanted to find. One of the primary, and often repeated, (and to this writer, fallacious) criticisms levied against van Maanen was that the precision of his measurements was smaller than normally encountered measurement errors. He was said to be reading positional variations much smaller than the sizes of the atmospherically blurred photographic images. References (BA02), (GK02), (IU), and (TV95) provide evidence of how his work was demeaned.
URL for (BA02) was http://nchalada.org/archive/WaveLng21.html.
URL for    (IU)   was http://physics.syr.edu/courses/AST104.99Spring/Island.htm

Here's a quote from the first of the four references above. It is typical of the precision arguments made against van Maanen. The quote is taken from Andrew Bell's section on Suggested Discussion Topics (Part II) under the second sub heading of Achievable precision. The other three references are left to the reader.

"By any standard, I think that Adriaan van Maanen's internal rotations* ought to have been set aside as unsupportable, right from the outset. All of his measured displacements corresponded to a half an arc-second or less between each pair of plates, and usually less than a quarter of an arc-second. Ignoring the fact that he claimed a hundredth of an arc-second as his measured measurement precision, I fail to understand how even a quarter of an arc second could be measured off photochemical plates from a ground-based telescope, then or now [c.f., Baade** (1963) at pp. 46-47, for a discussion of how difficult it was to achieve half arc-second photographic resolution, working in the mid-1930s]."

* Strictly speaking, van Maanen reported on internal motions in spiral nebulae.
These motions were not synonymous with rotations. See: Messier 81 Internal Motions.

**Baade was one of van Maanen's detractors.

Part 4

van Maanen's "Unsupportable" Internal Motions

[Added on 26 July 2003]

van Maanen's critics err! Object image sizes are not the limiting factor regarding the precision in astrometrically locating salient points on astronomical plates, for example, the centroids of object images.

According to the History of Leander McCormick Observatory web page, in 1916, the Gaertner Single Screw Measuring Engine allowed an operator to read positions down to about 0.003 arcseconds on plates taken with a 40 inch telescope (HPA) Van Maanen used a 60 inch telescope for the majority of his work. (MA21). URL for (HPA) was http://www.astro.virginia.edu/~afs5z/gaertner1.html

Figure 1 shows that position measurements can be made down to dimensions much less than the diameter of blurred stellar images. Figure 1a represents a section of an early photographic image, and Figure 1b represents the image taken a suitable number of years later. It should be obvious that delta X and delta Y can be much smaller than the atmospherically blurred diameters of the stellar images.

Astrometric precision

Figure 1
Illustration of Astrometric Measurements Smaller than Atmospheric Seeing Limit

Present day. - CCD astrometry has demonstrated 1 milliarcsecond positional accuracy (Monet et al. 1992). This level of precision means that proper motions of distant objects can be determined on time scales of 10 years or less. (MD&SP) (CCD photography does not get rid of atmospheric blurring. It allows you digitally dig into the fuzzy stuff without the mechanical idiosyncrasies of the measuring engines.)

See: On Spiral Nebulae, van Maanen et al..

* * *

Contradictory Reports on Lundmark's Findings

Added on 13 July 2003

There are unresolved differences of opinion about the results of Knut Lundmark's re-reading of van Maanen's plates for M33. Listed here are two quotes from recent critics of van Maanen's "rotations."

In his section on "The Island Universe Theory," Kurtiss Gordon (GK02) says:

    "Although his results correlated well with van Maanen's with respect to both
    direction and relative size of proper motion, the absolute scale of Lundmark's
    proper motions was less than 1/10 as large as those van Maanen had measured."

An "Anonymous Critic" (IU) says:

    "The Swedish astronomer Knut Lundmark also acquired evidence that van Maanen's
    evidence was faulty. He was allowed to remeasure the very same plates used by
    van Maanen to determine the motion of M33. In 1924 he found completely different
    results, namely that there was no measurable rotation of M33's image in the five
    year interval between the plates. In other words, van Maanen's claims were completely
    bogus. Lundmark told some of his colleagues privately of his results. He also published
    a paper to the same effect in 1925, but it was either too obscure or too polite to cause
    many other astronomers to notice."

The following table shows a comparison of van Maanen and Lundmark's measurements of motions in M33.

van Maanen & Lundmark M33 motions
Table added 26 Oct 2003.

Readers may read Lundmark's obscure 1926 report (LK26) in the Astrophysical Journal.
[Added 05 October 2003; updated 26 October 2003.]

- - -

See Gustav Holmberg's article Astronomy in Sweden 1860-1940, Uppsala Newsletter: History of Science nr 26 (1997), for a commentary on Lundmark's work on spiral nebulae. (Comments are in the fourth and third paragraphs from the end.) [Added 27 September 2003]

Hubble's Demolition of van Maanen?

[Added 1 August 2003]

Michael Hoskin summarizes the warfare between Hubble and van Mannen (HM97). He says: "The van Maanen problem became acute as the Dutchman pointedly persevered with his comparisons of pairs of photographs of spirals, concluding in every case that the nebula was rotating at a speed that made it physically impossible for it to be an island universe."

The warfare seemingly ended in 1935, with Hubble winning. Hoskin reports "A compromise was imposed, and in 1935 readers of the Astrophysical Journal were no doubt intrigued to find there a two-page paper by Hubble delicately outlining his conclusions, immediately followed by a two-page paper by van Maanen, conceding that 'it is desirable to view the motions [the rotations in the spirals] with reserve.'

The author would hope that Hoskin's purpose was to encourage readers to read the papers for themselves, because although what he says about the papers is true, may be misleading. Hoskin makes it sound as though Hubble demolished van Maanen's findings and that van Maanen's new tests put him in his proper place. That's not quite like it was!

Hubble's four paragraphs and single ambiguous tabular summary (HE35) offers no substantive data to back his claim for non-rotations. He fails to explain the relation between his total observed displacements and what he calls extrapolated rotations. The text reads as though he vectorially added the motion vectors of individual nebular points to get his total observed displacements. (Such a process would, in fact, tend toward zero, but the result would be meaningless.) In three cases he relates negative total observed displacements to positive extrapolated rotations (which were supposedly of a much smaller order than the values van Maanen had been "finding."). In four other cases null values for total observed displacements are related to positive extrapolated rotations. It is very hard to tell just what Hubble was reporting, other than that he still didn't like fast nebular rotations. [This paragraph will in all likelihood come to be toned down.]

In regard to van Maanen's supposed declaration of surrender (MA35), he stated "The measures give motions for both nebulae [M33 and M74] in the same direction as those found for the spirals measured previously; but the value of M33, mean of 114 nebular points, is considerably smaller (+0.013 milliarcsec as compared with +0.020 milliarcsec) than that found before. " The phrase considerably smaller was the politically correct thing to say, and was probably part of the imposed settlement. On the other hand his new smaller value for M33's rotation was a hefty 65% of his 1921 measure. (Reading between the lines is encouraged.)

This writer has not had opportunity to do a good literature search to compare current findings on spiral nebula rotations, compared to those reported by van Maanen, but is of the opinion that the findings will match up just fine. If they do, then the constant rotation curves for spiral nebulae, a.k.a. "galaxies" can most likely be explained in terms of plasma dynamics.

[A paragraph, duplicate to that preceding the section on "Contradictory Reports on Lundmark's Findings," above, was removed on 18 Sep 2015.]

* * *

Part 5

But What About Hubble's Redshifts?

When Hubble and Humason first announced their systematic redshift-distance findings for spiral nebulae, Humason cautioned their colleagues that what they were calling apparent radial velocities might not really be velocities. He explained that because they didn't have a new word or phrase to apply to the newly discovered phenomena, they took the already-in-use phrase apparent radial velocity and used it in conjunction with their redshift measurements. [This may not be said right.]

[Humason's actual words on the subject:]   "It is not at all certain that the large red-shifts observed in the spectra are to be interpreted as a Doppler effect, but for convenience they are expressed in terms of velocity and referred to as apparent velocities." (HM31) Page 35. [Added 25 Oct 2003.]


When Hubble's redshift-distance relation is applied to the Virgo "Cluster" of galaxies, the cluster appears to be elongated along a line that passes through our Solar neighborhood. Goodman says this causes the "Copernican Problem." (GJ), i.e., it puts us into a preferential reference frame. (GJ Link no longer works.)

Copernican Problem

Figure 2
The Copernican Problem

"Instead of immediately recognizing this as a problem, the mainstream adopted these configurations, calling them the "fingers of God." (GJ) This effect is sometimes called a redshift-space distortion.

Louis Desroches says it this way.

"Two important effects occur in redshift space. Although redshift corresponds to true distance according to the Hubble Law, small peculiar velocities not associated with the Hubble flow can cause distortions in redshift space. The most evident of these is the Fingers-of-God effect, where long thin filaments in redshift space point directly back at [an] observer. We should know by now that we are not privileged observers, this effect must be unphysical." (DL).
URL for (DL) was http://astron.berkeley.edu/~louis/astro228/redshift.html.

According to P.J.E. Peebles, Zwicky and Smith found the velocities of individual galaxies in the Coma and Virgo Clusters were about a factor of ten to 100 larger than they expected. (DJ) - (Link to www.astro.queensu.ca no longer works.)

This "Fingers-of-God" effect can be seen in Figure 3, which shows the elongated shape of the Virgo Cluster. (The Coma Cluster might be in there too.)

Slice of Universe

Figure 3
Virgo Cluster in Redshift Space

The idea behind cosmological redshift is that spectral elements from remote locations (in other galaxies/spiral nebulae, for example) were generated at essentially the same wavelengths as for local sources, but were somehow shifted. (Originally it was supposed to be a Doppler shift; nowadays it's politically correct to say that it's caused by an expansion of space.)

In 1938 Hubble (HE38) wrestled with two viewpoints on the origin of cosmological redshifts. The first was that the universe (the small part of it that we could see) was stationary and homogenous and the redshift-distance relation was linear, however we did not know the non-Doppler cause of the observed redshifts. The second viewpoint was that the universe was expanding (the spiral nebulae were receding from us) but that the distribution of matter was no longer uniform (the density increases outward), and the law of red shifts is no longer linear (redshifts increase with distance at an accelerated rate).

A Variable Charge Explanation for Cosmological Redshift

Regardless of the manner of how light gets cosmologically redshifted, the redshift, as currently understood, is tied to the unverifiable assumption that the unit electrical charge (the charge associated with electrons and protons) is a space-and-time constant, even in the remotest parts of space and at the longest times ago.

In 1988 the author published a (non-mathematical) hypothesis (FR88) that electrical charge is not constant, rather that it is a matter-density-dependent variable.

In quantum mechanics, the wavelengths of emission or absorption lines in spectral series are proportional to the inverse fourth power of the unit electrical charge If electrical charge in a given region of space is different than locally then we should expect to see all of the spectral elements from that region to be shifted in a systematic manner. (The fine-structure constant, which is proportional to the fourth power of the unit electrical charge, may also come into play here. This is because the internal spacing of individual lines within spectral multiplets is proportional to the fine structure constant.) An observed set of redshifted lines thus becomes an indirect measure of the ambient ... matter density in the source's region of space. [The word local was replaced by ambient on 14 Apr 2007.] (See the redshift derivation in Figure 4.)

Redshift versus Charge

Figure 4
Variable Charge Redshift Z as a Function of e(remote) and e(local)

[Figure was revised on 13 Apr 2007.]

A "GIGO" function for the absorption-driven emission of force carrying particles, positive and negative charglets, in the author's ... Emission-Absorption-Scattering sub-quantum physics model (FR93), provides a theoretical framework, which allows (demands) charge to be a material density dependent variable.

"Ambient matter density" as used above, is related to the total mass in a volume of space on the order of one cubic light year or less.

* * *
Part 6

The [pdf] article, Does the proton-to-electron mass ratio mu = Mp/Me vary in the course of cosmological evolution? (IPRV), may bear a relation to the prospect of electrical charge being a cosmological variable. It touches on some of the same consequences, for example, anomalies in quasar redshifts, which would ensue if electrical charge were to be cosmologically variable. [Added 07 Aug 2003.]

* * *

The author's variable electrical charge hypothesis for explaining anomalous redshifts is similar to the Narlikar-Das Variable Mass Hypothesis (VMH), based on the Hoyle-Narlikar theory of gravitation. Arp has incorporated the latter approach in his reasonings on anomalous quasar redshifts. See: (ND80), (AH98), and (NV02).

One primary difference between the two approaches is that the author's variable charge wavelength changes are proportional to the inverse fourth power of the local-to-remote differential in unit electrical charge, whereas for the variable mass hypothesis the changes would only be proportional to the inverse of the first power of the local-to-remote elementary particle mass differentials. (See the redshift derivation in Figure 4, Part 5.) The two approaches are not mutually exclusive. [Added 20 October 2003.]

See Bill Keel's commentary (KW03) that touches on Arp's handling of possible close relations between Active Galactic Nuclei, galaxies, quasars, and discordant (non Hubble flow) redshifts. [Added 28 October 2003.]

* * *

Figure 5 shows a "Milky Way" Cosmos. In 5a a density profile line is shown piercing a globular cluster, a spiral nebula, and the Milky Way galactic disk. A density plot, along the profile line, is shown in 5b. The Solar System is located near the peak for the galactic disk.

Variable Charge Cosmos

Figure 5
Material Density as a function of Galactic Location

A Note of Caution About Using Cepheid Variables as Standard Candles

[Added 23 January 2004.]

According to the author's EAS model of physics, not only is the unit electrical charge an environmenally determined variable, but so is the gravitational constant (G). (In the EAS model gravity is a shielding side effect of elementary electrical interactions.)

In this article Cepheid variables are being portrayed as Ritzian objects, i.e., binary stars where c+v effects (with respect to us as remote observers) produce periodic brightness variations. (See: A Ritzian Interpretation of Variable Stars and Non-Pulsating Cepheid Variables.)

What the above two paragraphs are leading to, is the idea that using Cepheid variables as standard candles for galactic versus so-called extra-galactic environments may not be what it's cracked up to be. For Ritzian Cepheids, which happen to be located in or near the equatorial part of the galactic disk (where the gravitational constant G may be considered as more or less normal), we observe Type I (first studied) Cepheids. For Cepheids located on the fringes of the galactic disk, or further out into the galactic halo we should expect to see, kilogram-for-kilogram, longer periods giving us a transition to Type II Cepheids and perhaps beyond. [This needs more work.]

The following, overly simplified, diagram is meant to illustrate the postulated interplay between G and the periods of Cepheids as binary stars. (For circular orbits the separation between the stars will be a constant R + r.)

Cepheid period as 
function of  gravitational constant

It may be that the lengths of the orbital radii R and r will also be functions of G.


Van Maanen's local spirals may make a comeback. If so, they will be relatively small (planetary-nebula-like in some cases) near by members of the Milky Way structure, probably located just outside the outer-most layer of the disk. Their internal motions will be found to not be principally subject to Keplerian dynamics but most their behavior will likely be in accordance with the conventions of plasma and magnetohydrodynamics (PA88) (GJ2), (ionized gaseous matter moving en-masse; sweeping electrically charged stars along in the stream.) - Refs added 06 August 2003. - (GJ2) link no longer works.]

If the spirals are clustered near the galactic plane then there is no requirement for missing mass to make them behave the way we want them to. We should learn to accept their behavior as it is, and adjust our theoretical outlooks so as to conform to the way the universe is.

Cosmological redshift may turn out to not be a measure of an expanding universe. If so, according to the variable electric charge model, it can be used as a measure of density differences between our observing vantage point and the remote objects observed. Galaxy clusters should at least become short, stubby fingers of God.


AH98 - Arp, Halton C., Seeing Red: Redshifts, Cosmology and Academic Science, Apeiron, Montreal, pp. 20, 42, 208, 214 (1988). (Part 6)

AR79 - Annual Reviews - Astron. Astrophys, 17, 135-187 (1979)
Section 3.1 - Table 1. Galaxies with extended rotation curves
http://nedwww.ipac.caltech.edu/level5/Faber/Faber3_1.html (Part 3, Table 2)

BA02 - Bell, Andrew - Wavelengths - Nineteenth-Century Spectroscopy and the birth of Modern Astrophysics (Part II) -
Link was http://nchalada.org/archive/WaveLng21.html (Part 4)

BBCN - BBC News - Dead stars could be 'missing mass' - 22 March 2001
http://news.bbc.co.uk/1/hi/sci/tech/1236460.stm (Part 1)

BR03 - Britt, Robert Roy, "Hubble Pictures Too Crisp, Challenging Theories of Time and Space," Falun Dafa - Minghui.org (Originally was Space.com/science/astronomy, 02 April 2003,) http://en.minghui.org/emh/articles/2003/4/8/34268.html (Part 2)
[Added 24 March 2004. - Updated 02 Aug 2016.]

BSVD - van den Bergh, Sidney, Dark Matter in the Local Group,
http://www.macalester.edu/astronomy/courses/physics50/spring2002.html [Link no longer works - 20 Jan 2012] (Part 1)

CE01 - The Columbia Encyclopedia, Sixth Edition, (2001)
Link was http://www.bartleby.com/65/pr/propermo.html (Part 3, Table 3)

CXO - Chandra X-Ray Observatory - Dark Matter Mystery
http://chandra.harvard.edu/xray_astro/dark_matter.html (Part 1)

DJ - Dursi, Jonathan, Cosmology is the study of the evolution of the Universe
Link was www.astro.queensu.ca - (Parts 1, 5)

DL - Desroches, Louis, Redshift-space distortions
URL was http://astron.berkeley.edu/~louis/astro228/redshift.html (Part 5)

DMVM - Dark Matter vs MOND
URL was: http://www.astro.umd.edu/~ssm/mond/mondvsDM.html

DP00 - Durrell, Patrick R., Ciardullo, Robin, Laychak, Marry Beth, Jacoby, George H., Feldmeier, John J., and Moody, Ken, Kinematics of M33's Disc Planetary Nebulae - Figure 4: The 'smoothed' rotation curve derived from the PNe velocities compared to the HI curve from Corbelli & Salucci, (2000). (Curves very similar.) Radial velocities obtained for 142 planetary nebulae.
URL was: http://www.astro.psu.edu/users/pdurrell/M33_aas_poster.pdf (Part 3, Table 2)
See Ciardullo, Robin, et al, The Planetary Nebula System of M33 (2008)

DT71 - Dixon, Robert T., Dynamic Astronomy, Prentice Hall, Inc. Englewood Cliffs, NJ, (1971). (Part 3)

FR88 - Fritzius, Robert, Cosmological Redshift, Magnolia Scientific Tech Note 1-88, (1988)
http://www.shadetreephsyics.com/cosmo_rs.htm (Part 6)

FR93 - Fritzius, Robert, Emission-Absorption-Scattering (EAS) Sub-quantum Physics, Presented at the International Conference on Sir Isaac Newton, St. Petersburg, Russia, 22-27 March, (1993) http://www.shadetreephysics.com/eas.htm (Parts 1, 6)

GJ - Goodman, Jason - The Case Against the Big Bang
URL was: http://www.geocities.com/kingvegeta80/BBT.html (Part 5)

GJ2 - Goodman, Jason - The Case for Plasma Cosmology
URL was: http://www.geocities.com/kingvegeta80/plasma.html (Part 6)

GK02 - Gordon, Kurtiss J., History of our Understanding of as Spiral Galaxy: Messier 33, Quarterly Journal of the Royal Astronomical Society, 10, 293-307 (1969). http://nedwww.ipac.caltech.edu/level5/March02/Gordon/Gordon_contents.html (Parts 4, 5)

HB92 - Hetherington, N., Brashear, R., Journal for the History of Astronomy, 23, 53-56 (1992).

HE35 - Hubble, Edwin, ApJ, 81, 334-335 (1935). NADS (Part 5)

HE38 - Hubble, Edwin, ASPL, 3, 120-123 (1983). NADS (Part 6) - [Added 24 January 2004.]

HM31 - Humason, Milton L., Apparent Velocity-Shifts in the Spectra of Faint Nebulae, ApJ 74, 35-42 (1931) - NADS (Part 5) - [Added 25 October 2003.] Part 7

HM97 - Hoskin, Michael, The Cambridge Illustrated History of Astronomy,, Cambridge University Press, (1997), page 333. (Part 5)

HPA - History of Leander McCormick Observatory - Hall of Precision Astrometry - Measuring Engines and Blink Comparators - Gaertner Single Screw Measuring Engine, (1916)
URL was: http://www.astro.virginia.edu/~afs5z/gaertner1.html (Part 4)

IPRV - Ivanchik, A., Petitjean, P., Rodriguez, E., and Varshalovich, D.,
Does the proton-to-electron mass ratio mu = Mp/Me vary in the course of cosmological evolution? - arXiv:astro-ph/0210299 v1, 14 Oct (2002) - http://arxiv.org/PS_cache/astro-ph/pdf/0210/0210299v1.pdf" (Part 6)
[Added 07 Aug 2003.]

IU - "Island Universes"/ Galaxies: Conflict and Resolution
URL was: http://physics.syr.edu/courses/AST104.99Spring/Island.htm (Parts 4, 5)

KA00 - King, Ivan R., and Anderson, Jay, Internal Motions in Globular Clusters, Dynamics of Star Clusters and the Milky Way ASP Conference Series, Vol. 000, (2000), Deiters, Fuchs, Spurzem and Wielen, eds. URL was: http://arxiv.org/PS_cache/astro-ph/pdf/0007/0007028.pdf (Part 4)

KW03 - Keel, William, Alternate Approaches and the Redshift Controversy - http://astr.ua.edu/keel/galaxies/arp/html (Part 6) - [Added 28 Oct 2003]


LB27 -Lindblad, Bertil, On the Nature of the Spiral Nebulae, MNRAS, 87, 420-426 (1927). - NADS (Part 1)

LJ99 - Lucey, John, Distance to the Hyades via Moving-Cluster Parallax -
https://community.dur.ac.uk/physics.astrolab/hyades_cp.html (Part 3, Table 3)

LK26 - Lundmark K, Internal Motions of Messier 33, Ap.J., 63, 67 (1926). - NADS
(Part 5) [Added 05 October 2003.]

MA21 - Maanen, Adriaan van - "Investigations on Proper Motion - Fifth Paper: The Internal Motion in the Spiral Nebula Messier 81" Ap.J., 54, 347-356 (1921). NADS (Parts 3, 4)

MA23 - Maanen, Adriaan van - "Investigations on Proper Motion - Tenth Paper: Internal Motion in the Spiral Nebula Messier 33, N.G.C. 598" - Ap.J., 57, 264 (1923). - NADS (Part 3) [Added 7 July 2005.]

MA27 - Maanen, Adriaan van, - Investigations on Proper Motion - Twelfth Paper: The Proper Motions and Internal Motions of Messier 2, 13, and 56. Ap.J., 66, 89-112 (1927). - NADS (Parts 3, 4)

MA30 - Maanen, Adriaan van, - Investigations on Proper Motion - Sixteenth Paper: The Proper Motion of Messier 51, MGC 5194. Contributions from the Mount Wilson Solar Observatory, 408, 311-314 (1930).

MA35 - Maanen, Adriaan van, ApJ, 81, 336-337 (1935). NADS (Part 5)

MD&SP - Méndez, Bryan and Seitzer, Patrick, "Precision CCD Astrometry",
http://cse.ssl.berkeley.edu/bmendez/html/figures.html (Part 5)

MS1 - McGaugh, Stacy, The MOND pages
URL was: http://www.astro.umd.edu/~ssm/mond/mondvsDM.html. (Part 2)

MS2 - McGaugh, Stacy - Dwarf and Low Surface Brightness Galaxies
NADS (Part 2)

MW02 - Mathis, H. and White, S.D.M., "Numerical simulations of Peebles's Isocurvature Cold Dark Matter model
http://www.mpa-garching.mpg.de/~virgo/virgo/hmathis/ICDM (Part 2) [Added 11 March 2004.]

ND80 - Narlikar, J.V., and Das, P.K., Anomalous Redshifts of Quasi-Stellar Objects, ApJ 240, 401-414 (1980). - NADS (Part 6)

NF03 - Nicastro, Fabrizio, et al - Missing Mass Exists as Warm Intergalactic Fog, (2003)
http://www.spacedaily.com/news/darkmatter-03c.html (Part 1) - [Added 16 Aug 2003]

NN02 - Noriaki Namba - Stellar Movement in the Galaxy Explained by Inertial Induction, (2002) Physics Essays 15, 156-161. (Part 2) [Added 20 April 2004.]

NNR02 - NASA News Release - Chandra casts cloud on alternative to dark matter, (2002)
URL was: http://www1.msfc.nasa.gov/NEWSROOM/news/releases/2002/02-264.html (Part 2) This article is probably archived. Haven't located it yet.

NRV - Nieuwenhove, Rudi van, Is the missing mass really missing?

NV02 - Narlikar, J.V., Vishwakarma, R.G., Banerjee, S.K., Das, P.K., Arp, H.C., Dynamics of Ejection From Galaxies and the Variable Mass Hypothesis, International Journal of Modern Physics D, 11, 245-257 (2002) - World Science Abstract (Part 6)

PA88 - Peratt, A. "Hannes Alfvén (1908-1995)" - The World and I, pp.190-197, (1988). URLwas: http://public.lanl.gov/alp/plasma/people/alfven.html (Part 6)
- [Added 06 August 2003.] - Thanks to John M. Fritzius of Arnold Missouri for the link.

PICDM-1 - Peebles, P. J. E. "An Isocurvature Cold Dark Matter Cosmonogy. I A Worked Example of Evolution Through Inflation" - ApJ 510, 523-530 (1999). - NADS (Part 2)

PICDM-2 - Peebles, P. J. E., "An Isocurvature Cold Dark Matter Cosmonogy. II Observational Tests" - ApJ 510, 531-540 (1999). - NADS (Part 2)

RW08 - Ritz, Walter, Recherches critiques sur l'Électrodynamique Générale, Annales de Chimie et de Physique, 13, 145-278 (1908). English translation online at:
http://www.shadetreephysics.com/crit/1908a.htm (Part 2)

SC21 - Harlow Shapley & Heber D. Curtis - The Scale of the Universe, NRC Transcripts of the "Great Debate" Bulletin of the National Research Council, Volume 2, Part 3, Number 11, pp 171-217 (May 1921).
http://antwrp.gsfc.nasa.gov/diamond_jubilee/1920/cs_nrc.html (Parts 2, 3)

SEV94 - Savage, D., Elliot, J., and Villard, R. - Hubble Rules Out a Leading Explanation For Dark Matter" - NASA Release 94-188 (Part 1)

SMKU - Seishi Matsuki, Kyoto University, Dark matter - the mystery of missing mass
Link was http://www.oxford-instruments.com/SCNRMP13.htm - 15 January 2003. (Part 1)

SNP - ScienceNet - Physics - What is the Missing Mass problem?
URL was: http://www.sciencenet.org.uk/database/Physics/Cosmology/p00717c.html (Part 1)

SRR90 - Sharma, Rati Ram, Unified Physical Theory, A Falcon Book from Cosmo Publications, New Delhi, India (1990). (Part 1) [Added 25 March 2004.]

SV87 - Vladimir Sekerin - Gnosiological Peculiarities in the Interpretation of Observations (For Example the Observation of Binary Stars), Contemporary Science and Regularity in its Development, 4, 119-123, Tomsk University (1987) - English translation at: http://www.shadetreephysics.com/sekerin.htm (Part 2)

TV95 - Trimble, Virginia The 1920 Shapley-Curtis Discussion: Background, Issues, and Outcome - (Prepared for the 1995 75th Anniversary Astronomical Debate and for Publications of the Astronomical Society of the Pacific. - http://antwrp.gsfc.nasa.gov/diamond_jubilee/papers/trimble.html (Part 4)


Related Webpages and Articles

The Mystery of the Missing Mass - History.NASA,gov - The problem of the missing mass has gotten to the point where it is more than just a problem. It is an embarrassment, an obstacle to understanding such things as the structure of galaxies, the evolution of clusters of galaxies, and the ultimate fate of the universe. [Added 20-Dec 2017.]

Cosmology looks beyond the standard model - Astronomy Magazine - Tuesday July 7, 2015 [Added 02 DEC 2015.]

Clouds gather over 'dark matter' - Astronomers challenge assumptions on galaxies' formation - Keay Davidson, San Francisco Chronicle - Monday September 8, 2003. [Added 30 November 2004.]

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