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January, 2007 PROGRESS IN PHYSICS Volume 1
Quantization State of Baryonic Mass in Clusters of Galaxies
Franklin Potter∗and Howard G. Preston†
∗Sciencegems.com, 8642 Marvale Dr., Huntington Beach, CA 92646 USA
†15 Vista del Sol, Laguna Beach, CA 92651 USA
E-mail: ∗drpotter@lycos.com and †hpres@cox.net
The rotational velocity curves for clusters of galaxies cannot be explained by
Newtonian gravitation using the baryonic mass nor does MOND succeed in reducing
this discrepancy to acceptable differences. The dark matter hypothesis appears to
offer a solution; however, non-baryonic dark matter has never been detected. As
an alternative approach, quantum celestial mechanics (QCM) predicts that galactic
clusters are in quantization states determined solely by the total baryonic mass of the
cluster and its total angular momentum. We find excellent agreement with QCM for ten
galactic clusters, demonstrating that dark matter is not needed to explain the rotation
velocities and providing further support to the hypothesis that all gravitationally bound
systems have QCM quantization states.
1 Introduction
The rotational velocity curves of galaxy clusters [1] are very
similar to the rotational velocity curves of individual gala-
xies, with the rotational velocity value rising rapidly at very
small radial distances only to quickly reach an approximately
constant velocity for all greater radial distances from about
200 kpc to out beyond 1500 kpc. Newtonian gravitation using
only the observed baryonic mass fails to explain the curves
both for galaxies and for clusters of galaxies. In clusters, the
baryonic mass is predominantly due to the hot intracluster
gas that is observed by its free-free X-ray emissions. This
gas fraction plus the stellar masses make up the observed
baryonic mass of about 10%–15% of the dynamic mass
required to explain the rotational velocity curves using New-
tonian gravitation, an enormous discrepancy.
Three interesting possible explanations for galactic rota-
tion curves have been proposed: (1) the dark matter hypo-
thesis (DM) introduces non-baryonic matter that is insensi-
tive to all interactions except gravitation, but there has been
no detection of any possible form of dark matter; (2) a modi-
fied Newtonian dynamics (MOND) effective at all distance
scales when the accelerations are less than 1.2×10−10 m/s2,
which has been very successful in explaining the rotation
and luminosity curves of individual galaxies but has large
discrepancies for galaxy clusters [2] in both the cluster core
and in the outer regions; (3) quantum celestial mechanics
(QCM) derived [3] from the general relativistic Hamilton-
Jacobi equation which dictates that all gravitationally bound
systems have quantization states. The QCM states are deter-
mined by two physical quantities only — the system’s total
baryonic mass and its total angular momentum. QCM agrees
with MOND and the baryonic Tully-Fisher relation for indi-
vidual galaxies.
In this paper, we compare the QCM predictions for the
baryonic mass for ten galaxy clusters to the detected bary-
onic masses. Our new result is that the QCM baryonic mass
values agree with the measured baryonic values even where
DM succeeds and MOND fails. No dark matter is required
to explain the observed rotation curves. The baryonic matter
in a single QCM quantization state produces the correct
rotational velocity for the cluster.
2 Conceptual review of QCM
In a series of papers [3, 4, 5], we derived a Schr ¨
odinger-like
scalar wave equation from the general relativistic Hamilton-
Jacobi equation via a tranformation that utilizes the total an-
gular momentum of the gravitationally bound system instead
of an angular momentum proportional to Planck’s constant.
We have shown agreement of its quantization state solutions
with the energy states of the planets of the Solar System,
of the satellites of the Jovian planets, and of the disk states
of galaxies. In a preliminary table-top investigation with a
torsion bar system that is now being modified to minimize
possible extraneous influences, the QCM predicted quanti-
zation states with quantized energy per mass and quantized
angular momentum per mass have been detected. The results
from the improved apparatus will be reported.
According to QCM, the quantization state energies per
orbiting particle mass μare
En
μ=−
G2M4
2n2H2
Σ
(1)
where Gis the gravitational constant, Mis the total mass
of the gravitationally bound system, HΣis the system’s total
angular momentum, and n is an integer. Typically, Enis
on the order of 10−6μc2. Unlike the quantum mechanics of
atomic states whereby each electron is in its own quantum
state, in QCM there can be billions of stars (i.e., particles)
F.Potter and H. G. Preston. Quantization State of Baryonic Mass in Clusters of Galaxies 61
Volume 1 PROGRESS IN PHYSICS January, 2007
Cluster kT , keV R200, kpc M200,×1014 MvkT ,×106m/s M,×1013MHΣ,×1070 kg×m2/s
A1983 2.18±0.09 1100±140 1.59 ±0.61 0.65±0.03 1.12 ±0.21 5.10±1.65
MKW9 2.43±0.24 1006 ±84 1.20±0.30 0.68±0.08 1.34±0.63 7.00±5.76
A2717 2.56±0.06 1096±44 1.57±0.19 0.70±0.02 1.50±0.17 8.57±1.71
A1991 2.71±0.07 1106±41 1.63 ±0.18 0.72±0.02 1.68 ±0.19 10.4±2.0
A2597 3.67±0.09 1344±49 3.00±0.33 0.84±0.02 3.11±0.30 30.7±5.1
A1068 4.67±0.11 1635±47 5.68 ±0.49 0.95±0.03 5.09 ±0.64 72.7±16.1
A1413 6.62±0.14 1707±57 6.50±0.65 1.13±0.03 10.2±1.1 245.±46
A478 7.05±0.12 2060±110 10.8±1.8 1.16±0.02 11.3±0.8 294. ±36
PKS0745 7.97±0.28 1999 ±77 10.0±1.2 1.24±0.05 14.8±2.4 469.±132
A2204 8.26±0.22 2075±77 11.8±1.3 1.26±0.04 15.7 ±2.0 525.±116
Table 1: QCM predicted galactic cluster baryonic mass Mand angular momentum HΣ.
in the same QCM state. Also notice that there is no explicit
distance dependence in this energy state expression, in sharp
contrast to classical mechanics, because the state radial wave
function extends over a large range. QCM tells us that grav-
itationally bound systems, such as planetary systems and ga-
laxies, are quantized systems and that their behavior cannot
be fully understood by classical general relativistic dynamics.
QCM has been used also to derive the general expression
for the MOND acceleration a0=1.2×10−10 m/s2, this speci-
fic MOND value being an average value for many galaxies.
Our general expression is
a0=G3M7
n4H4
Σ
,(2)
a result which suggests that a0would be slightly different
for each galaxy instead of being taken as a universal value.
We combine these equations algebraically to solve for
Mand HΣin terms of the measured asymptotic rotational
velocity and the MOND acceleration. Assuming that the
virial theorem holds for galaxies, the velocity vis derived
from Eq. 1 to yield
M=v4
Ga0
, HΣ=v7
nGa2
0
.(3)
These expressions hold true for galaxies. In the next
section they will be applied to clusters of galaxies and the
predicted baryonic mass values will be compared to the
dynamic mass values determined from observational data.
3 Galaxy cluster QCM masses
QCM is assumed to have universal application to isolated
gravitationally bound systems. To a good approximation,
clusters of galaxies are isolated gravitationally bound sys-
tems and therefore should demonstrate the quantization sta-
tes dictated by QCM. In many cases the galaxy clusters
have no dominant central mass, with the intragalactic gas
dispersed throughout the cluster. For simplicity, we assume
that the cluster system is in the n= 1 state, that the virial the-
orem applies, that a0=1.2×10−10 m/s2, and that the cluster
is approximately a flattened ellipsoid similar to the Local
Group [6] that includes our Galaxy and M31. The latter as-
sumption is not strictly required but allows an easy analogy
to disk galaxies where we know that QCM and MOND apply
extremely well.
We use the ten galaxy clusters analyzed by Arnaud et
al. [7] to determine the QCM predicted baryonic mass and
angular momentum via Eqs. 3 above. Their radial distance
R200 is the distance where inside that radius the mean mass
density is 200 times the critical density of the universe, and
their M200 is the total mass within this radius in solar masses
Mas determined by a standard NFW universal density
profile for a dark matter halo as determined by Navarro et al.
[8] from N-body simulations. The kT (keV) represents the
spectroscopic temperature of the 0.1 6r60.5 R200 region,
and the velocity vkT comes from these temperatures. Table
1 lists our results for the total baryonic mass Mand the total
angular momentum HΣ.
4 Discussion
Our predicted QCM baryonic masses Min Table 1 for the
clusters are about a factor of ten smaller (1013 vs. 1014 )
than the dynamic masses M200 which were determined by
assuming a dark matter NFW profile. There is reasonable
agreement between our QCM baryonic mass values and the
baryonic masses from the spectroscopic data. There is no
need to invoke the gravitational consequences of DM. The
galactic cluster is in a QCM quantization state. This result
indicates that quantum celestial mechanics determines cer-
tain dynamic behavior of galaxies and galactic clusters.
One additional physical quantity we know now is the
total baryonic angular momentum of each galactic cluster.
This angular momentum value determines all the quantiza-
62 F. Potter and H. G.Preston. Quantization State of Baryonic Mass in Clusters of Galaxies
January, 2007 PROGRESS IN PHYSICS Volume 1
tion states of the system in which the gas and the individual
galaxies (i.e., particles) can occupy. Particles at all radii
from the cluster center are in the same angular momentum
quantization state. Note that we have assumed that n= 1 for
each cluster; however, some clusters could have baryonic
mass in the n= 2 state as well.
QCM has been applied successfully to solar systems,
galaxies and clusters of galaxies. The results strongly suggest
that the known baryonic mass in each system is sufficient
to explain the rotational velocity values without invoking
the gravitational consequences of dark matter. As expected
from QCM, these gravitationally bound systems all behave
as non-classical systems exhibiting quantization states deter-
mined by the total mass and the total angular momentum.
References
1. Pointecouteau E., Arnaud M. and Pratt G. W. The structural
and scaling properties of nearby galaxy clusters: I — The uni-
versal mass profile. arXiv: astro-ph/0501635.
2. Pointecouteau E. and Silk J. New constraints on MOND from
galaxy clusters. arXiv: astro-ph/0505017.
3. Preston H. G. and Potter F. Exploring large-scale gravitational
quantization without ˉhin planetary systems, galaxies, and the
Universe. arXiv: gr-qc/0303112.
4. Potter F. and Preston, H.G. Gravitational lensing by galaxy
quantization states. arXiv: gr-qc/0405025.
5. Potter F. and Preston H. G. Quantum Celestial Mechanics:
large-scale gravitational quantization states in galaxies and the
Universe. 1st Crisis in Cosmology Conference: CCC-I, Lerner
E.J. and Almeida J. B., eds., AIP CP822, 2006, 239–252.
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7. Arnaud M., Pointecouteau E. and Pratt G. W. The structural
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arXiv: astro-ph/0502210.
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F.Potter and H. G. Preston. Quantization State of Baryonic Mass in Clusters of Galaxies 63
