# The origin of the MOND critical acceleration scale.

D. F. Roscoe
Dept of Mathematics
The Open University
Milton Keynes MK7 6AA
<EMAIL_ADDRESS>

ORCID: 0000-0003-3561-7425

###### Abstract

The irrefutable successes of MOND are predicated upon the idea that a critical
gravitational acceleration scale, $a_{0}$, exists. But, beyond its role in
MOND, the question _Why should a critical gravitational acceleration scale
exist at all?_ remains unanswered. There is no deep understanding about what
is going on.

Over roughly the same period that MOND has been a topic of controversy,
Baryshev, Sylos Labini, Pietronero and others have been arguing, with equal
controversy in earlier years, that, on medium scales at least, material in the
universe is distributed in a quasi-fractal $D\approx 2$ fashion. There is a
link: if the idea of a quasi-fractal $D\approx 2$ universe on medium scales is
taken seriously then there is an associated characteristic mass surface
density scale, $\Sigma_{F}$ say, and an associated characteristic
gravitational acceleration scale, $a_{F}\equiv 4\pi G\,\Sigma_{F}$. If,
furthermore, the quasi-fractal structure is taken to include the inter-
galactic medium, then it is an obvious step to consider the possibility that
$a_{0}$ and $a_{F}$ are the same thing.

Through the lens of very old ideas rooted in a Leibniz-Mach worldview we
obtain a detailed understanding of the critical acceleration scale which,
applied to the SPARC sample of Lelli, McGaugh & Schombert (2016A) with a
stellar MLR, $\Upsilon_{*}\in(0.5,\,1.0)$, and using the mass models of Lelli,
McGaugh & Schombert (2016B), provides a finite algorithm to recover the
information that $a_{F}\approx 1.2\times 10^{-10}\,mtrs/sec^{2}$. This,
combined with the fact that the Baryonic Tully-Fisher Relationship (BTFR)
arises directly from the same source, but with $a_{0}$ replaced by $a_{F}$,
leads to the unambiguous conclusion that $a_{0}$ and $a_{F}$ are, in fact, one
and the same thing.

In summary, $a_{F}\approx 1.2\times 10^{-10}\,mtrs/sec^{2}$ is embedded in
SPARC data, and the Leibniz-Mach worldview provides the tools with which to
recover it directly. In this way, a window is opened onto the physics
underlying MOND.

##  1 Introduction:

Over the years since Milgrom conceived MOND (Milgrom (1983a, b, c, d, e)),
significant effort has been expended on devizing various variational
principles designed to reproduce its details. The very first of these to
appear was AQUAL (Bekenstein and Milgrom (1984)) although there are now
several competing flavours of such derivative theories.

It is striking that in all of these theories there has been virtually no
discussion about the significance of Milgrom’s primary insight - the existence
of a critical gravitational acceleration scale, $a_{0}$ say, in the first
place. Instead, $a_{0}$ is routinely consigned to the passive role of simply
signposting the boundary where one gravitational regime gives way to another.

But if we are ever to come to a fundamental understanding of the mechanisms
which MOND emulates to some degree, then the question _Why should a critical
gravitational acceleration scale exist at all?_ must necessarily be posed and
answered. It is precisely this question which connects Milgrom’s work to that
of Baryshev et al (1995) and many others (see appendix §B for an overview)
which has shown how the distribution of galaxies on the medium scale
($\approx$ 100Mpc at least) is quasi-fractal, $D\approx 2$. The connection is
simply this: if the idea of a quasi-fractal universe on medium scales is taken
seriously to the extent that the distribution of material within the IGM
itself (routinely assumed to consist largely of the canonical Dark Matter of
modern astrophysics) has the same quasi-fractal quality then, on the galactic
scale, there is an associated characteristic mass surface density scale,
$\Sigma_{F}$ say, and an associated characteristic gravitational acceleration
scale, $a_{F}=4\pi G\,\Sigma_{F}$. It is then a natural step to consider the
possibility that $a_{0}$ of MOND and $a_{F}$ of the postulated quasi-fractal
IGM are one and the same thing.

Regardless of the precise material properties of this postulated IGM, a
quantitative tool with the potential to consider the possibility that
$a_{0}\equiv a_{F}$ already exists in the mainstream literature as Roscoe
(2002) (or Roscoe (2020) in the arXiv). The relevant point is simply that this
tool - a Lagrangian-based synthesis constructed from a Leibniz-Mach worldview
- has an irreducible equilibrium state in which universal material necessarily
exists as a non-trivial fractal $D=2$ distribution, thereby providing the
theoretical link to the work of Baryshev, Sylos Labini, Pietronero et al and
hence, potentially, to that of Milgrom. The Leibniz-Mach worldview itself is
encapsulated in two related statements: firstly, that there is no such thing
in the physical world as _an empty metrical space_ ; secondly, that there is
no such thing as _abstract time_ (all we can ever do is to measure change in
one system against change in a second system which has been defined as a
standard). The synthesis then derives from an intensive analysis of how clocks
and rods are to be consistently defined in this worldview.

Given the irreducible equilibrium state of the synthesis, the way forward is
easily recognized: a galactic object is modelled as a finite bounded
perturbation of the fractal $D=2$ equilibrium state, characterized by the mass
surface density parameter $\Sigma_{F}$. Whilst the bounded perturbation can be
any topological isomorphism of a three-dimensional spherical surface (for
example, a thick pancake), we develop the model for the simplest possible case
for which the perturbation is spherical and the perturbation boundary,
$R=R_{0}>0$ say, is finite but initially undetermined. We then consider the
special case which refers to purely circular motions, and which integrates to
give the scaling relation

$\left[\frac{V_{rot}(R)}{V_{flat}}\right]^{2}\Sigma_{R}=\Sigma_{F};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ R<\infty,$ (1)

where $\Sigma_{R}\equiv\Sigma(R,R_{0},M_{0})$ represents the mass surface
density of material around the galactic object at radius $R$, and $M_{0}$ is
the mass contained within the perturbation boundary $R_{0}$.

According to (1), the dynamics within any given galaxy are tightly constrained
by the external matter field, characterized by $\Sigma_{F}$. This is no more
than a direct consequence of the underlying Leibniz-Mach worldview, in fact.
But, in the terminology of MOND, it means that an External Field Effect (EFE)
is the dominating control which determines disk dynamics.

When (1) is supplemented by a condition of dynamical equilibrium on the
$R_{0}$ boundary ($g_{0}=a_{F}$ where $g_{0}$ is the centripetal acceleration
on $R_{0}$) then there are five primary consquences:

  * •

firstly, the EFE manifests itself as a set of symmetries that leave (1)
invariant and which together provide a finite algorithm for recovering the
value of $\Sigma_{F}$, and hence of $a_{F}$, from rotation curves &
photometry. Applying the algorithm to the SPARC sample of Lelli, McGaugh &
Schombert (2016A) with a fixed stellar MLR, $\Upsilon_{*}\in(0.5,\,1.0)$, and
using the mass models of Lelli, McGaugh & Schombert (2016B), we find
$a_{F}\approx 1.2\times 10^{-10}\,mtrs/sec^{2}$;

  * •

secondly, the Baryonic Tully-Fisher Relation (BTFR) emerges automatically, but
with $a_{0}$ replaced by $a_{F}\equiv 4\pi G\Sigma_{F}$. This incorporation of
$\Sigma_{F}$ into the BTFR implies that the BTFR itself is a manifestation of
an EFE;

  * •

thirdly, the conclusion that $a_{0}\equiv a_{F}$ follows directly from the two
results above;

  * •

fourthly, MOND is seen to work as successfully as it does because, in
practice, the far-field form of the MOND force-law is actually emulating
significant aspects of the $\Sigma_{F}$ External Matter Field;

  * •

fifthly, the whole process provides a dynamical means for estimating absolute
radial scales for disks and hence their absolute distance scales.

Finally, the Leibniz-Mach synthesis contains no new physics, but simply gives
revitalized expression to very old ideas using 20th Century analytical methods
and, in the process, provides the tools which open a window onto the physics
underlying MOND.

##  2 Toy model: a working hypothesis for $a_{0}$

For the purpose of de-mystifying the critical acceleration parameter, $a_{0}$,
of MOND and our approach to it, we describe a toy model which is our nominal
starting point and use it to justify, as a working hypothesis, the equivalence
$a_{0}\equiv a_{F}$.

As we have noted, the work of Baryshev et al (1995) and many others (see
appendix §B for an overview) has shown how the distribution of galaxies on the
medium scale ($\approx$ 100Mpc at least) is quasi-fractal, $D\approx 2$. If
the idea of a quasi-fractal universe on medium scales is idealized to include
all physical scales then we can say that, about any point chosen as the
centre, mass is distributed according to

$\mathcal{M}_{F}(R)=4\pi R^{2}\,\Sigma_{F};\,\,\,R<\infty,$ (2)

where $\Sigma_{F}$ is the characteristic mass surface density of the
distribution.

Since this idealized matter distribution is isotropic (by definition) about
the chosen centre, then the notional Newtonian gravitational acceleration
imparted to a particle at radius $R$, and generated by the material contained
within $R$, is directed towards the chosen centre and has magnitude given by

$a_{F}\equiv\frac{\mathcal{M}_{F}(R)\,G}{R^{2}}=4\pi
G\,\Sigma_{F};\,\,\,R<\infty.$

On this basis, we note that:

  * •

The _net_ actual gravitational acceleration imparted to a material particle
immersed in the global distribution (2) is zero;

  * •

If a finite spherical volume, radius $R_{0}$, is imagined emptied of all
material, then the net actual gravitational acceleration of any material
particle placed on $R_{0}$ will be $a_{F}$ directed radially outwards from the
centre of the empty volume;

  * •

A material particle placed on $R<R_{0}$ will experience an outward
acceleration $a_{F}^{*}<a_{F}$ which satisfies $a_{F}^{*}=0$ when $R=0$, and
which is a primitive form of an external field effect;

  * •

A material particle placed on $R>R_{0}$ will experience an outward
acceleration $a_{F}^{*}<a_{F}$ which is such that $a_{F}^{*}\rightarrow 0$ as
$R\rightarrow\infty$;

  * •

The empty spherical volume is unstable since all accelerations on $R_{0}$ are
outward; it follows that stability requires the volume to be occupied by a
stablizing mass, a proto-galaxy, say.

On this basis, our working hypothesis is that $a_{0}\equiv a_{F}$. Whilst the
foregoing toy model has no power to predict or quantify anything (beyond
global equilibrium), it has the benefit of providing a de-mystified
interpretation of $a_{0}$ and creating a rational basis upon which this
interpretation can be explored.

##  3 Galaxy dynamics in the $\Sigma_{F}$ environment

The rational account of the MOND critical acceleration scale arising from the
working hypothesis $a_{0}\equiv a_{F}$, and developed in the following, has
its source in a Leibniz-Mach worldview which can be summarized in two related
statements: firstly, that there is no such thing in the physical world as _an
empty metrical space_ ; secondly, that there is no such thing as _abstract
time_. The Leibniz-Mach synthesis then arises from an intensive analysis of
how clocks and rods can be consistently defined in this worldview.

The synthesis appeared in the mainstream literature in an early form as Roscoe
(2002). This early form is mathematically complete, but only partially
interpreted around the meaning of clocks in the equilibrium state. The fully
interpreted form exists in the arXiv as Roscoe (2020) and a brief outline is
given in §A.

###  3.1 The galaxy model and the IGM:

We describe the simplest case of the Leibniz-Mach galaxy model, for which the
integrated equation of motion (26) gives the scaling relationship (5) below. A
detailed understanding of the workings of External Field Effects (EFEs) in
this simple model emerges directly from this relationship, and is described in
§3.2.

The irreducible basic expression of the Leibniz-Mach synthesis is a world of
dynamic equilibrium within which material is necessarily non-trivially present
in the form of a $D=2$ fractal distribution. Consequently, about any point
chosen as the centre, mass is distributed according to (2) which is consistent
with the modern perspective that, on medium scales at least, material in the
universe is distributed in a $D\approx 2$ quasi-fractal fashion, Baryshev et
al (1995) and many others. To progress, it is necessary to further assume that
(2) is a fair description of the local universe down to at least the level of
the IGM. This latter assumption raises obvious questions concerning the nature
of the material populating the IGM which are discussed in detail in §7.

Given the foregoing, an individual galaxy is then modelled as a finite bounded
spherical perturbation of (2) so that it has the general form:

$\displaystyle\mathcal{M}(R)$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{g}(R),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ R\leq R_{0};$ (3)
$\displaystyle\mathcal{M}(R)$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{g}(R_{0})+4\pi(R^{2}-R_{0}^{2})\,\Sigma_{F},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ R>R_{0},$

where $\mathcal{M}_{g}(R)$ represents the galactic mass contained within
radius $R\leq R_{0}$ and $R_{0}$ is the finite, but otherwise unspecified,
radial boundary of the perturbation.

For later clarity, we write

$\mathcal{M}_{g}(R)\equiv M_{0}\,F(R/R_{0});\leavevmode\nobreak\
\leavevmode\nobreak\ F(0)=0,\leavevmode\nobreak\ F(1)=1,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ R\leq R_{0},$ (4)

where $F(R/R_{0})$ describes the relative distribution of mass in $R\leq
R_{0}$ and $M_{0}$ specifies the total mass contained within $R=R_{0}$.

Whilst the case of general orbits for unspecified $\mathcal{M}_{g}(R)$ is
given by the equations of motion (27), the special case of purely circular
orbits is given by the integrated equation of motion (26) which, with galaxy
model (3) and notation (4), gives the scaling relationship:

$\displaystyle\left[\frac{V_{rot}(R)}{V_{flat}}\right]^{2}\Sigma_{R}$
$\displaystyle=$ $\displaystyle\Sigma_{F},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ R<\infty;$ (5)
$\displaystyle\Sigma_{R}$ $\displaystyle\equiv$
$\displaystyle\frac{M_{0}\,F(R/R_{0})}{4\pi R^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ R\leq R_{0};$
$\displaystyle\Sigma_{R}$ $\displaystyle\equiv$
$\displaystyle\frac{M_{0}+4\pi(R^{2}-R_{0}^{2})\,\Sigma_{F}}{4\pi
R^{2}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ R>R_{0}$

where $V_{flat}$ is the flat rotation velocity and $M_{0}$ is the mass
contained within the perturbation boundary $R=R_{0}$. This is the expanded
form of (1).

###  3.2 EFEs & model symmetries

The following unambiguously demonstrates the deep connection between the
external field (characterized by $\Sigma_{F}$) and the internal dynamics of
the galactic object, and provides the theoretical basis for the algorithm used
to recover the value of $\Sigma_{F}$ from SPARC rotation curves & photometry.

The scaling relationship (5) is invariant under three distinct symmetry
transformations:

$\Sigma_{F}\rightarrow\alpha\,\Sigma_{F},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ M_{0}\rightarrow\alpha\,M_{0};$ (6)
$M_{0}\rightarrow\beta\,M_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ R\rightarrow\beta^{1/2}\,R;$ (7)
$\Sigma_{F}\rightarrow\gamma\,\Sigma_{F},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ R\rightarrow\gamma^{-1/2}\,R,$ (8)

so that linear scales, local mass scales and the global $\Sigma_{F}$ scale are
tightly bound together. Using $a_{F}=4\pi G\Sigma_{F}$ and defining $g_{0}$ as
the centripetal acceleration on $R=R_{0}$, then (8) gives:

$a_{F}\rightarrow\gamma\,a_{F};\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
g_{0}\rightarrow\gamma^{1/2}\,g_{0}\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm where}\leavevmode\nobreak\ \leavevmode\nobreak\
g_{0}\equiv\frac{V_{0}^{2}}{R_{0}}$ (9)

from which it is immediate that

$\Gamma\equiv\frac{g_{0}^{2}}{a_{F}},$ (10)

which has dimensions of $mtrs/sec^{2}$, is an invariant of (5) under the
symmetry (8). If we further make the _assumption_ that every object is in
dynamical equilibrium with its environment on the $R=R_{0}$ boundary, so that

$g_{0}\equiv\frac{V_{0}^{2}}{R_{0}}=a_{F},$ (11)

then (10) gives

$g_{0}=a_{F}=\Gamma$ (12)

which completes the required theory. We are now in a position to compute
$a_{F}$ from SPARC rotation curves and photometry.

##  4 Recovery of $\Sigma_{F}$ from SPARC data

Our basic resource is the SPARC sample of Lelli, McGaugh & Schombert (2016A),
for which the object selection criteria is described in §9. Following Lelli,
McGaugh & Schombert (2016B), all required baryonic mass determinations in
galaxy disks are then provided by the model

$M_{bar}=M_{gas}+\Upsilon_{*}\,L_{[3.6]},$ (13)

where $M_{gas}$ is the gas mass, $L_{[3.6]}$ is the $[3.6]$ luminosity and
$\Upsilon_{*}$ is the stellar MLR, assumed here to be equal for the disk and
the bulge (where present), and constant across the whole SPARC sample.

Lelli et al find that the optimal choice for $\Upsilon_{*}$ (which minimizes
scatter in the BTFR) approximately satisfies $\Upsilon_{*}\geq 0.5$. For
current purposes we restrict the choice to $\Upsilon_{*}\in(0.5,\,1.0)$
finding a ‘sweet spot’ of $\Upsilon_{*}\approx 0.8$ which, using the
symmetries of §3.2, leads to $\Sigma_{F}\approx 0.14\,kg/mtrs^{2}$ so that
$a_{F}\approx 1.2\times 10^{-10}\,mtrs/sec^{2}$.

###  4.1 The numerical model

The primary objective is the recovery of $\Sigma_{F}$ (and hence of $a_{F}$)
from SPARC rotation curves and photometry using the model (5) and, to this
end, we need the best possible estimates for the three parameters
$(R_{0},M_{0},V_{flat})$ for each object.

There is now a choice: do we use SPARC photometry with mass model (13) to
represent $\mathcal{M}_{g}(R)\equiv M_{0}\,F(R/R_{0})$ in (5) directly (the
MOND approach in practice), or do we replace $\mathcal{M}_{g}(R)\equiv
M_{0}\,F(R/R_{0})$ by a highly parameterized empirical mass-model, and then
use the parameter space of this model to help optimize the rotation curve fit
for each object which, by definition, will then optimize the estimates of
$(R_{0},M_{0},V_{flat})$?

We choose the empirical mass modelling route and so, writing
$F(R/R_{0})\approx f(R/R_{0},\mathbf{a})$ where $\mathbf{a}$ represents the
five-dimensional parameter space of the $f$-model defined at (29), then (5)
becomes

$\displaystyle\left[\frac{V_{rot}(R)}{V_{flat}}\right]^{2}\Sigma_{R}$
$\displaystyle=$ $\displaystyle\Sigma_{F},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ R<\infty;$ (14)
$\displaystyle\Sigma_{R}$ $\displaystyle\equiv$
$\displaystyle\frac{M_{0}\,f(R/R_{0},\mathbf{a})}{4\pi
R^{2}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ R\leq R_{0};$ $\displaystyle\Sigma_{R}$
$\displaystyle\equiv$
$\displaystyle\frac{M_{0}+4\pi(R^{2}-R_{0}^{2})\,\Sigma_{F}}{4\pi
R^{2}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ R>R_{0},$

which is now the basis for the computation.

###  4.2 The $\Sigma_{F}$ recovery algorithm

In the following, $R_{0}^{S}$ denotes the radius of the perturbation boundary
expressed in terms of SPARC’s radial scales, whilst $M_{0}^{S}$ denotes the
mass contained within $R_{0}^{S}$ estimated using SPARC photometry and the
mass model (13). Absence of the superscript $S$ indicates that the scaling of
the quantity concerned is not calibrated against SPARC data.

We begin with the basic constraint that, at the end of the computation, the
total computed mass within $R_{0}^{S}$ for each object must be rescaled using
symmetry (7) to equal $M_{0}^{S}$.

The $\Sigma_{F}$ recovery algorithm now proceeds as follows:

  * •

Since the true value of the mass surface density parameter,
$\Sigma_{F}=\Sigma^{1}_{F}$ say, is initially unknown then, in the first
instance, set $\Sigma_{F}=\Sigma_{F}^{0}$ where $\Sigma_{F}^{0}$ is some
arbitrarily chosen trial value. The realities around making a reasonable
choice for $\Sigma_{F}^{0}$ are discussed in appendix §C;

  * •

Then, for each SPARC object, optimize the fit of (14) to the rotation curve by
varying the eight local parameters $(V_{flat},R_{0}^{S},M_{0},\mathbf{a})$
where $\mathbf{a}\equiv(a_{0}...a_{4})$ are the parameters of the empirical
mass-model (29). Since $\Sigma_{F}^{0}$ is a trial value, then symmetry (6)
makes it plain that the computed $M_{0}$ for any given object cannot generally
coincide with its photometric counterpart, $M_{0}^{S}$, which is initially
unknown since $R_{0}^{S}$ is also initially unknown;

  * •

However, once the value of $R_{0}^{S}$ for each SPARC object has been
estimated via the optimization process, we can use SPARC photometry with the
mass-model (13) to estimate $M_{0}^{S}$;

  * •

Using the symmetry (7), we now map $M_{0}\rightarrow M_{0}^{S}$ for each
object according to

$M_{0}\rightarrow\beta M_{0}=M_{0}^{S},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
R_{0}^{S}\rightarrow\beta^{1/2}R_{0}^{S}\equiv R_{0}\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm where}\leavevmode\nobreak\ \leavevmode\nobreak\
\beta=\left(\frac{M_{0}^{S}}{M_{0}}\right)$ (15)

so that, as required, the masses in the $\Sigma_{F}^{0}$ trial solution are
now calibrated against SPARC photometry using mass model (13). The radial
scaling has also necessarily changed;

  * •

Symmetry (8) maps $\Sigma^{0}_{F}\rightarrow\Sigma^{1}_{F}$ and also has
$\Gamma\equiv g_{0}^{2}/a_{F}$ of (10) as an invariant. Consequently, $\Gamma$
can be evaluated from either of the $\Sigma^{0}_{F}$ or $\Sigma^{1}_{F}$
solutions. But the $\Sigma^{1}_{F}$ solution remains unknown, and so we
evaluate $\Gamma$ from the $\Sigma^{0}_{F}$ solution. Once $\Gamma$ is
evaluated, we can use the assumed equilibrium condition of (11),
$g_{0}=a_{F}$, to evaluate the required final state using $g_{0}=a_{F}=\Gamma$
from (12);

  * •

For the computation: using (15) the $\Sigma^{0}_{F}$ solution gives
$a_{F}=4\pi G\Sigma_{F}^{0}$ and $g_{0}=V_{0}^{2}/R_{0}$ so that $\Gamma\equiv
g_{0}^{2}/a_{F}$ becomes evaluated;

  * •

In practice, since $g_{0}$ exhibits statistical variation across the SPARC
sample (principally because of uncertainties in estimating $R_{0}^{S}$ in the
curve-fitting process) then, across the whole sample with a globally applied
$\Upsilon_{*}\in(0.5,\,1.0)$, we get distributions of values for $\Gamma$
similar to that plotted in figure 1 for which $\Upsilon_{*}=0.8$. These
distributions are sharply modal and, using the modal values of these
distributions as the estimate for $\Gamma$ in each case, we find:

$\Gamma\in(1.4,\,0.9)\times 10^{-10}\,mtrs/sec^{2}$ (16)

so that, finally, from the equilibrium condition (12):

$g_{0}=a_{F}=\Gamma\in(1.4,\,0.9)\times 10^{-10}\,mtrs/sec^{2}$

as required. In particular, the choice $\Upsilon_{*}=0.8$ yields $a_{F}\approx
1.2\times 10^{-10}\,mtrs/sec^{2}$ which corresponds to $\Sigma_{F}\approx
0.14\,kg/mtrs^{2}$ for the mass surface density of the External Matter Field.

Figure 1: Solid black curve = density distribution of $\log(\Gamma)$. Here,
$\Upsilon_{*}=0.8$.

###  4.3 Absolute linear scale determinations

We show that the mappings of §4.2 provide the means for the (albeit noisy)
estimation of the absolute radial dimensions of SPARC objects and hence, by
implication, allows the absolute distance scale for those objects to be
determined.

If we look at the §4.2 solution in detail we see that the symmetry
transformation (15) is central to the process. Broadly speaking, this
transformation ensures that object masses are calibrated against SPARC
photometry so that the mass contained within $R_{0}^{S}$ is given by
$M_{0}^{S}$ but, in the process, it necessarily recalibrates radial scales
according to $R_{0}^{S}\rightarrow R_{0}$ so that, in general, these will
differ from the original SPARC radial scalings.

We then find that this radial rescaling has the automatic consequence of
mapping the centripetal acceleration $g_{0}\equiv V_{0}^{2}/R_{0}$ directly
onto the acceleration scale $\Gamma\approx 1.2\times 10^{-10}\,mtrs/sec^{2}$.
The only possible conclusion is that $R_{0}$ must be giving the absolute
radial dimension for the perturbation boundary of the object concerned. It
follows immediately that a dynamical method for the determination of absolute
astrophysical distances is available.

##  5 The BTFR

From (5a) we have, on the $R=R_{0}$ boundary:

$V_{0}^{2}=V_{flat}^{2}\,\frac{\Sigma_{F}}{\Sigma_{0}}=V_{flat}^{2}\left(\frac{a_{F}R_{0}^{2}}{GM_{0}}\right)$
(17)

where the external field relation $a_{F}=4\pi G\Sigma_{F}$ has been used.
Eliminating $R_{0}$ between the equilibrium condition (11) and (17) gives
directly:

$V_{flat}^{4}=a_{F}\,G\,\left[\left(\frac{V_{flat}}{V_{0}}\right)^{2}M_{0}\right]$
(18)

Defining $M_{flat}$(theory) according to the scaling relation

$M_{flat}{\rm(theory)}\equiv\left(\frac{V_{flat}}{V_{0}}\right)^{2}M_{0},$
(19)

then (18) becomes

$V_{flat}^{4}=a_{F}\,G\,M_{flat}{\rm(theory)}$ (20)

which has the exact structure of Milgrom’s form of the empirical BTFR, except
for the replacement of $a_{0}$ by $a_{F}$. So, everything hinges on the extent
to which $M_{flat}$(theory) tracks $M_{flat}$(photometric), where estimates of
the latter are supplied for a significant number of objects in the SPARC
database.

However, before continuing, there is an issue which must be resolved:
specifically, on those RCs for which $V_{0}>V_{flat}$ then the rotation
velocity $V=V_{flat}$ occurs _twice_ on the RC - once on the rising part
_before_ $V=V_{0}$, and once asymptotically on the falling part _after_
$V=V_{0}$. Consequently, in such cases, there _appears_ to be an ambiguity
around what is meant by $M_{flat}$(theory). However since, for such cases,
(19) shows that we necessarily must have $M_{flat}$(theory)$<M_{0}$ then it is
clear that $M_{flat}$(theory) must be defined to mean the mass contained
within the _first_ occurence of $V=V_{flat}$.

With this understanding, the scatter plot given in figure 2 of
$\log{M_{flat}}({\rm photometry})$ against $\log M_{flat}$(theory) makes it
clear that these two quantities are in an almost perfect statistical
correspondence, and a _least-area_ linear regression (a method which treats
$x$ and $y$ data identically, see §D) gives:

$\log M_{flat}({\rm photometry})=\left(0.96\pm 0.10\right)\log M_{flat}({\rm
theory})+\left(0.21\pm 1.00\right)$ (21)

so that, via (18) and (19), the BTFR is implicitly satisfied. To make the
point that the BTFR itself is explicitly satisfied, we find

$\log M_{flat}({\rm photometry})=\left(3.80\pm 0.58\right)\log
V_{flat}+\left(1.94\pm 1.20\right).$ (22)

Figure 2:  $M_{flat}$(theory) vs $M_{flat}$(SPARC).

##  6 MOND & External Field Effects

The foregoing exposition of the dominant role EFEs play in disk dynamics
appears to conflict very strongly with the MONDian view which, whilst
considering EFEs to be a phenomonological necessity (Sanders & McGaugh
(2002)), views them as subtle and very difficult to detect (see Chae et al
(2020)).

This apparent conflict is resolved by recognizing that the MOND far-field
force law itself works as it does because it is actually emulating significant
aspects of the $\Sigma_{F}$ External Matter Field, giving rise to primary EFEs
which are not recognizable as such from within the MONDian context. This
interpretation becomes self-evident when we consider how the BTFR arises from
each point of view.

In the Leibniz-Mach synthesis, the BTFR arises directly from the integrated
equation of motion (1) constrained by the equilibrium condition $g_{0}=a_{F}$
on the $R_{0}$ perturbation boundary ($g_{0}$ is the centripetal
acceleration), as per the analysis of §5, to give

$V_{flat}^{4}=a_{F}\,G\,M_{flat},$

where $M_{flat}$ is the total estimated mass out to $V_{flat}$. The fact that
this is an effect of the $\Sigma_{F}$ External Matter Field is manifested by
the appearance of $a_{F}\equiv 4\pi G\Sigma_{F}$.

By contrast, in the MONDian analysis, the far-field force law is chosen to
_ensure_ that the BTFR, in Milgrom’s form:

$V_{flat}^{4}=a_{0}\,G\,M_{flat},$

arises automatically. In other words, the MOND far-field force law is directly
emulating significant aspects of the $\Sigma_{F}$ External Matter Field.

In summary, it is self-evident that the chosen form of MOND’s far-field force
law works as it does because it emulates significant aspects of the
$\Sigma_{F}$ External Matter Field giving rise to a primary EFE in the form of
the BTFR. We are left to conclude that Chae et al (2020) are in fact detecting
secondary EFEs.

##  7 The inferred nature of the unseen IGM

The successes of the model (5) described here are predicated upon the
hypothesis that the IGM itself consists of a non-trivial $D\approx 2$ quasi-
fractal material distribution. Given that IGM material is not visible in any
usual sense, an obvious question is raised: specifically, _What is the nature
of this material?_ The results of §4.2 & §4.3 provide a working answer to this
question.

As before, $R_{0}^{S}$ represents the radius of the perturbation boundary
expressed in terms of SPARC radial scaling, whilst $M_{0}^{S}$ represents the
mass contained within $R_{0}^{S}$ estimated using SPARC photometry amd mass
model (13).

###  7.1 The details

When reflecting upon the mysteries of the internal motions of virialized
astrophysical systems, it has always been the case that, within the Newtonian
context in particular, the same evidence supports two quite distinct
responses, which can be briefly stated as:

  1. 1.

Leave radial scales unchanged and expand the masses (the Dark Matter
hypothesis);

  2. 2.

Leave masses unchanged and contract the radial scales (an undeveloped
hypothesis),

and exactly the same is true within the Leibniz-Mach synthesis, as we now
show.

#### 1\. Leave radial scales unchanged

If (5) is applied to the SPARC sample with $\Sigma_{F}\approx
0.14\,kg/mtr^{2}$ being set apriori, together with the assumption that SPARC
radial scales are correct, so that masses, $M_{0}$ for each object, are
determined by the curve-fitting process, then we find, generally speaking, the
result:

$M_{0}^{S}<M_{0}<100\,M_{0}^{S},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ R_{0}\equiv R_{0}^{S}$ (23)

so that there has been a systematic expansion of the mass scale, as per Virial
Theorem expectations. This approach leads inevitably to a version of the Dark
Matter hypothesis and provides no understanding of either MOND’s critical
acceleration scale, or of the BTFR.

#### 2\. Leave masses unchanged

For the solution of §4.2, the assumption is that masses set by SPARC
photometry with mass models (13) are correct, so that the radial scales,
$R_{0}$ for each object, are determined by the curve-fitting process. We find,
generally speaking, the result:

$M_{0}\equiv M_{0}^{S},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 0.1\,R_{0}^{S}<R_{0}<R_{0}^{S}$ (24)

so that there has been a systematic contraction of the radial scale, as per
Virial Theorem expectations. But, as discussed in §4.3, it is this approach
which has mapped $V_{0}^{2}/R_{0}$ onto $\approx 1.2\times
10^{-10}\,mtrs/sec^{2}$, from which fact the only possible conclusion is that
$R_{0}$ itself must be an estimate of the absolute radial dimension for the
perturbation boundary of the object concerned. This makes it virtually
impossible to dismiss (24) as an objective result. So the pertinent question
is: _How are we to understand it?_

It is critical to the answer that the IGM consists of a $D\approx 2$ quasi-
fractal material distribution. Given this, then the systematic radial-scale
contraction of (24) provides difficult-to-ignore circumstantial evidence to
the effect that the material concerned must consist largely of near-perfect
blackbody absorbers. Such a material distribution would give rise exactly to a
systematic and undetectable _exaggeration_ of photometric distances, and hence
of radial scales whilst, simultaneously, allowing a high degree of
transparency in the IGM. Consequently, generally speaking, we would find
$R_{0}<R_{0}^{S}$ where $R_{0}$ is the actual radial measure and $R_{0}^{S}$
is the photometric estimate of the same.

We expand upon the issues raised by this statement in §7.2, §7.3 and §7.4.

###  7.2 Blackbody absorber material in the IGM ?

Modern photometric methods of distance estimation are predicated upon one
relevant implicit assumption in particular: that there is no such thing in
nature as a perfect (or near-perfect) blackbody absorber. However, recent
developments in the material sciences have shown how near-perfect blackbody
absorbers can be created via exotic carbon chemistry in industrial processes
for everyday usage. Consequently, the idea that such materials cannot exist in
nature can no longer be sustained.

In particular, Mizuno et al (2009) were the first to show how to fabricate,
from agglomerations of single-walled carbon nanotubes (SWCNTs), material
distributions having specific bulk statistics which act as near-perfect
blackbody absorbers (emissivity $>0.98$) across a very wide range of incident
wavelengths from UV at 200$nm$ to the far IR at 200$\mu m$. This behaviour has
been shown to be independent of the specific properties of the individual
SWCNTs, but is rather a consequence of the bulk statistical characteristics of
the fabricated SWCNT distributions.

We know that many allotropes of carbon exist in interstellar space and these
must to some extent be blown into the IGM from the generality of galactic
interiors. It is a short step to visualizing the possible existence of clouds
of SWCNTs dispersed throughout the IGM containing sub-populations which, when
viewed in projection along any given line of sight, possess the bulk
statistical characteristics required to mimic the fabricated SWCNT
distributions of Mizuno et al (2009). In this way, it is possible to conceive
how SWCNT clouds within the IGM have the potential to act as ‘dispersed near-
perfect blackbody objects’.

###  7.3 The distribution of material in the IGM

The model (5) explicitly requires the existence of a non-trivial $D\approx 2$
quasi-fractal IGM of unspecified material. In consequence, the quality &
precision of the results derived in §3, §4 and §5 stand as strong evidence in
support of such an IGM.

There is additional independent evidence: in a very recent paper Hong et al
(2021) construct the ‘unseen matter’ map of the local universe (the local
cosmic web) using a novel neural-network machine-learning algorithm. It is
immediately striking from the images in this paper that this map shows all the
voids, sheets and filaments that are such familiar features in the maps of
galaxy distributions in the local universe, and the authors explicitly make
this exact point. But, as Baryshev et al (1995) and many others have shown
(see appendix §B for an overview), it is precisely these features which
characterize the $D\approx 2$ quasi-fractal nature of galaxy distributions in
the local universe.

Thus, although Hong et al (2021) do not talk in terms of fractal distributions
of unseen matter, the clear inference from their results, and the work of the
astrophysical fractal community in general, is that their predicted
distribution of unseen matter in the local cosmic web can be characterized as
$D\approx 2$ quasi-fractal - in other words, the idea of a $D\approx 2$ quasi-
fractal unseen IGM is independently supported.

###  7.4 The implications of a quasi-fractal SWCNT cloud IGM

Suppose that the $D\approx 2$ quasi-fractal IGM consists substantially of
‘dispersed near-perfect blackbody objects’ in the form of SWCNT clouds. What
are the consequences arising?

By virtue of its $D\approx 2$ quasi-fractal distribution, such an IGM would,
to a significant extent, be transparent to radiation, which mirrors the
primary reason why Charlier (1908, 1922, 1924) suggested the ‘hierarchical
universe’ as an early answer to the question _Why is the sky dark at night?_

But whilst a $D\approx 2$ quasi-fractal SWCNT IGM would, to a significant
extent, be transparent to radiation, it would by no means be totally
transparent; broadly speaking and in addition to the usual inverse square
dimming process, light from a source at distance $R$ would experience dimming
via a process of near-perfect blackbody absorption by SWCNT clouds in a way
which would be proportional to $R^{2}$, again because this material is
distributed quasi-fractally, $D\approx 2$.

This dimming mechanism would be indistinguishable in its effects from the
ordinary inverse-square distance dimming process so that the total of observed
dimming would be interpreted entirely as a distance effect. There are two
consequences:

  * •

The principles underlying the process by which standard candles are used to
estimate the absolute luminosities of distant objects are unchanged so that
such estimates would not be affected by SWCNT cloud absorption, should the
phenomenon actually exist;

  * •

The photometric distance scale would be systematically exaggerated with the
effect that objects of a given absolute luminosity would generally be
estimated as being further away and larger than they actually are. Such an
unrecognized exaggeration of linear scales taken at face value would
automatically give rise to a ‘missing mass’ problem within any given galaxy.

On this basis, it follows that for the SPARC data (for example) we can expect
the photometric estimates of mass to remain unaffected and the length scales
to be systematically exaggerated.

##  8 MOND and radial scale contraction?

MOND assumes that the photometric distance scale is secure and then, in its
most simple form, defines the critical acceleration radius, $R_{c}$, as that
radius at which $V_{c}^{2}/R_{c}=a_{c}$ where $a_{c}$ is the critical
acceleration scale. $R_{c}$ is then the nominal radius at which the MOND
gravitation law switches to its far-field form.

However, if the radial-scale contraction (24) of the Leibniz-Mach synthesis is
taken at face value, then the magnitude of $R_{c}$ will contract accordingly
so that $V_{c}^{2}/R_{c}>a_{c}$. This means that the MOND force-law switch is
actually occurring in the Newtonian regime, and not at the correct critical
acceleration radius. Since this undermines the details of the MOND algorithm,
then a significant question is raised: _How does one then explain that MOND
still works very effectively?_

The answer is as follows: the force-law switch (even at the ‘wrong’ radius)
still ensures that the BTFR is satisfied so that the correct far-field form of
the rotation curve remains guaranteed. The interior deep-Newtonian part of the
rotation curve is also still calculated correctly. Furthermore, at the (now
‘wrong’) switching radius, MOND actually deploys an interpolation function
which ensures a smooth transition between the force-laws along the computed
rotation curve.

The net result is that the computed rotation curve is still constructed as a
smooth interpolation between a correct near-field Newtonian segment and a
correct far-field BTFR segment.

##  9 Object selection from the SPARC sample

The SPARC sample compiled by Lelli, McGaugh & Schombert (2016A) consists of
175 nearby galaxies with modern surface photometry at $3.6\,\mu m$ and high
quality rotation curves. The sample has been constructed to span very wide
ranges in surface brightness, luminosity, rotation velocity and Hubble type,
thereby forming a representative sample of galaxies in the nearby Universe. To
date, the SPARC sample is the largest collection of galaxies with both high-
quality rotation curves and NIR surface photometry.

As we have noted, the galaxy model (14) contains eight free parameters in all.
Bearing this in mind, then we select a subsample of SPARC objects according to
the following criteria. Objects must have:

  * •

a rotation curve quality flag Q = 1 or 2;

  * •

an inclination $>30^{o}$;

  * •

a SPARC estimate of $V_{flat}$ for the rotation curve. This is simply to
ensure that the measured rotation curve is probably sufficiently sampled to
allow a good fit of (14) to it;

  * •

at least 10 rotation velocity measurements remaining after rejecting all those
velocity measurements with a relative error $>10\%$.

These constraints whittle the original 175 SPARC objects down to a potentially
usable sample of 90 objects. Having fitted (14) to each of these 90 objects,
we retain only those for which the rotation curves satisfy:

  * •

the condition that $R_{0}<R_{max}$ where $R_{max}$ is the outermost radial
measurement on the rotation curve. This is to ensure that $R_{0}$ is not
obtained by extrapolating beyond the range of the rotation curve data;

  * •

the condition that there are at least _seven_ data points on the interior
region $0<R\leq R_{0}$. The reasoning here is simply that of the eight model
parameters, $(V_{flat},R_{0},M_{0},a_{0}...a_{4})$, seven are directly
associated with quantities that are either on or interior to the perturbation
boundary $R=R_{0}$ and should therefore be determined by interior
measurements.

Depending on how $\Upsilon_{*}\in(0.5,1.0)$ is chosen, this gives a final
sample of between 60 and 70 objects which are used to determine the invariant
$\Gamma$ of (10) in the manner described in §4.2.

##  10 Conclusions

Whilst MOND, and its various derivatives, are now very familiar to the
astrophysical community, the fact remains that there is no understanding of
the underlying fundamentals and specifically none around the question _Why
does a critical acceleration scale exist at all?_

For all its familiarity, MOND remains a very odd construct - in essence it
consists of two Newtonian force-laws bolted together at a particular
acceleration scale in a way which allows the internal motions of virialized
astrophysical systems to be substantially reconciled with their
photometrically determined mass contents. It is the very oddness of this
construct which alerts us to the idea that the underlying theory which
provides for its fundamentals can be expected to present significant novelty.

According to the arguments presented here, this novelty has its roots in two
very old but related ideas. Firstly, and in modern terms, that there is no
such thing in the physical world as _an empty metrical space_ \- an idea that
arguably can be traced to Aristotle, but certainly to Leibniz. Secondly, that
there is no such thing as _abstract time_ \- Leibniz & Mach had similar views
on this topic: all we can ever do, Mach argued, is to measure change in one
system against change in a second system which has been defined as a standard
- for example, it takes half of one complete rotation of the earth about its
own axis to walk thirty miles. These two ideas provide the foundations for the
Leibniz-Mach synthesis described briefly in §A and in the mainstream
literature as Roscoe (2002).

Our basic assumption is then simply that the mysteries illuminated by MOND’s
startling successes are rooted in a cosmos which performs its evolutions
according to this synthesis, the fundamental state of which is an equilibrium
world consisting of a fractal $D=2$ material distribution with characteristic
mass surface density $\Sigma_{F}$, and an associated characteristic
acceleration $a_{F}=4\pi G\Sigma_{F}$. An isolated galaxy is then modelled as
a spherically symmetric _bounded_ perturbation of the equilibrium world, with
perturbation boundary $R=R_{0}$. We then consider the special case which
refers to purely circular motions, and for which the equation of motion
integrates to give the scaling relation:

$\left[\frac{V_{rot}(R)}{V_{flat}}\right]^{2}\Sigma_{R}=\Sigma_{F};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ R<\infty,$

where $\Sigma_{R}\equiv\Sigma(R,R_{0},M_{0})$ is the mass surface density
around the isolated galaxy at radius $R$ and where $M_{0}$ is the mass
contained within $R_{0}$. It is clear that the dominating control for the
internal dynamics of this model galaxy is the $\Sigma_{F}$ External Matter
Field (EMF) - which, in the language of MOND, amounts to an External Field
Effect. The details show that the EMF exerts its influence via three internal
symmetries which bind together local mass scales, the global $\Sigma_{F}$
scale and radial scales. Thus, for example, in order that the scaling relation
remains invariant for a given object under change in $\Sigma_{F}$, then there
must be a corresponding re-balancing of local mass scales and/or radial
scales.

The direct connection to MOND is through the working hypothesis that MOND’s
critical acceleration scale and the characteristic acceleration scale of the
$\Sigma_{F}$ External Matter Field are one and the same thing, so that
$a_{0}\equiv a_{F}$. This connection is validated by showing how the scaling
relation, supplemented by an equilibrium condition on the $R_{0}$ boundary:

  * •

reveals that $a_{F}\approx 1.2\times 10^{-10}\,mtrs/sec^{2}$ is embedded in
the SPARC data of Lelli, McGaugh & Schombert (2016A);

  * •

recovers Milgrom’s form of the BTFR, but with $a_{0}$ replaced by $a_{F}$;

  * •

demonstrates that MOND works as it does because the far-field form of its
force-law actually emulates significant aspects of the $\Sigma_{F}$ External
Matter Field.

In summary, the Leibniz-Mach synthesis contains no new physics, but simply
gives revitalized expression to very old ideas using 20th Century analytical
methods and, in the process, provides the tools which open a window onto the
physics underlying MOND.

##  Appendix A The Leibniz-Mach synthesis: Overview

The Leibniz-Mach synthesis appeared in the mainstream literature in a
primitive form as Roscoe (2002). This early form is mathematically complete,
but only partially interpreted around the meaning of clocks in the equilibrium
state. The fully interpreted form exists in the archive as Roscoe (2020). The
following provides an outline of the synthesis.

###  A.1 Briefly, Leibniz on space

In the debate of Clarke-Leibniz (1715$\sim$1716) (Alexander (1984)) concerning
the nature of space, and in which Clarke argued for Newton’s conception of an
absolute space, Leibniz made three arguments of which the second was:

> _Motion and position are real and detectable only in relation to other
> objects … therefore empty space, a void, and so space itself is an
> unnecessary hypothesis._

Of Leibniz’s three arguments, this latter was the only one to which Clarke had
a good objection - essentially that _accelerated motion_ can be perceived
without reference to external bodies and is therefore, he argued, necessarily
perceived with respect to the absolute space of Newton. Given Clarke’s
objection, which was pertinent, and whatever Leibniz’s actual intention, it is
clear that the world implicit to his argument above is a world in which motion
is everywhere uniform and unaccelerated - otherwise Clarke’s objection must
stand. So, Leibniz’s implied equilibrium state leads us to consider whether or
not such a thing can actually exist, and hence we are led to that which we
nominate as _The Leibniz Question:_

> _Is it possible to conceive a non-trivial global mass distribution,
> isotropic about every spatial origin, which is such that motions are
> everywhere uniform and unaccelerated and, if so, what are the properties of
> this distribution?_

The Leibniz Question was a primary driver in the development of the Leibniz-
Mach synthesis and, as we shall see, the answer to the Leibniz Question is

> _Yes, and the distribution of material is necessarily fractal, $D=2$. _

### Briefly, Leibniz & Mach on time

Mach (1919) was equally clear in expressing his views about the nature of time
which are, in effect, very similar to those expressed by Leibniz. They each
viewed _time_ (specifically Newton’s _absolute time_) as a meaningless
abstraction. All that we can ever do, Mach argued, is to measure _change_
within one system against _change_ in a second system which has been defined
as the standard (eg it takes half of one complete rotation of the earth about
its own axis to walk thirty miles). So, on this Machian view, the ‘clock’ used
to quantify the passage of time for a physical system A is simply an
independent physical system B which has been arbitrarily defined as the
standard clock.

It follows from the foregoing considerations that any definition of _time_
based upon the Machian argument must be based upon the ideas of:

  * •

every physical system having its own subjective and private internal time-
keeping, any one of which can be chosen as the _standard clock_ against which
all the other systems reckon the passage of time;

  * •

in practice, one such system being chosen as the standard clock, the choice
being merely one of convention usually involving a natural cyclic process.

This ultimately leads to the qualitative result that:

> _Every particle and every system of particles is no more than an angular
> momentum conserving clock. So, in particular, every disk galaxy is nothing
> more than a giant angular momentum conserving clock._

###  A.2 Synthesis of quantitative model

In modern terms, the net conclusions arising from the detailed analysis of how
clocks and rods are to be defined in the Leibniz-Mach worldview are that:

  * •

there is no such thing as a physical space which is metrical _and_ empty;

  * •

physical time is no more than a metaphor for sequential physical process.

The quantitative theory was synthesized, using first-principle arguments, to
capture these ideas. It has the following structure:

  * •

The level surface $\Phi=k$: in the most simple case of the synthesis this
represents a three-dimensional spherical surface but, in general, it
represents any topological isomorphism of a three-dimensional spherical
surface;

  * •

The primitive mass function $\mathcal{M}(\Phi)$: this represents the amount of
material contained within the level surface $\Phi=k$;

  * •

The metric structure of physical three-space is projected out of its mass
content $\mathcal{M}(\Phi)$ according to:

$\displaystyle g_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi\Sigma_{F}}\nabla_{a}\nabla_{b}\mathcal{M}\equiv\frac{1}{8\pi\Sigma_{F}}\left(\frac{\partial^{2}\mathcal{M}}{\partial
x^{a}\partial x^{b}}-\Gamma_{ab}^{k}\frac{\partial\mathcal{M}}{\partial
x^{k}}\right),$ $\displaystyle\Gamma_{ab}^{k}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}g^{kj}\left(\frac{\partial g_{bj}}{\partial
x^{a}}+\frac{\partial g_{ja}}{\partial x^{b}}-\frac{\partial g_{ab}}{\partial
x^{j}}\right).$

This represents a non-linear differential equation to be solved for $g_{ab}$
in terms of $\mathcal{M}$. This is exactly resolvable for the spherical case.
No other cases have been attempted. Note that if $\mathcal{M}(R)\equiv 0$,
then there is no metric space;

  * •

Physical time is not yet defined, and so there can be no WEP to fix particle
trajectories since the WEP is a dynamical principle and relies on a system of
clocks;

  * •

In place of the WEP we can note that the _shape_ of a particle’s orbit is also
independent of particle properties;

  * •

So, using this shape-principle, any particle trajectory minimizes:

$I(p,q)=\int^{q}_{p}{\cal
L}\,dt\equiv\int^{q}_{p}\sqrt{g_{ij}\dot{x}^{i}\dot{x}^{j}}\,dt$

where $t$ is simply an ordering parameter, since physical time is not yet
defined.

  * •

The definition of $I(p,q)$ is homogeneous degree zero in the parameter $t$,
which means that $I(p,q)$ is invariant under arbitrary monotonic
transformations of the $t$-parameter. This means that the parameter $t$ cannot
be physical time without further conditions being attached and, without those
conditions (as a standard result), the Euler-Lagrange equations are not a
linearly independent set. This is formally identical to the situation in GR,
where the same problem is resolved by defining particle proper time. In the
present case, once further conditions are imposed to close the EL equations,
the closure information has the effect of constraining $t$ to be what is meant
by physical time. In practice, the system is closed by specifying that the
choice of $t$ must remove the dissipative terms of the EL equations, thereby
making the system conservative. We refer to the required condition as the
_clock constraint_ condition.

This model has an equilibrium state which is necessarily associated with a
non-trivial fractal $D=2$ distribution of material so that, for a
characteristic mass surface density $\Sigma_{F}$, then

$\mathcal{M}(R)=4\pi R^{2}\Sigma_{F}$ (25)

valid about any centre.

###  A.3 General equations of motion

There are two distinct cases which arise from the Euler-Lagrange equations:

####  A.3.1 The degenerate case of purely circular motions

In this case $\dot{R}\equiv 0$ and so the radial EL equation

$\frac{d}{dt}\left(\frac{\partial{\cal
L}}{\partial\dot{R}}\right)-\frac{\partial{\cal L}}{\partial R}=0$

integrates directly to give

${\cal L}=const$

leading to

${\cal L}^{2}=v_{0}^{2}\leavevmode\nobreak\ \leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\left(\frac{V_{rot}(R)}{V_{flat}}\right)^{2}=\frac{\Sigma_{F}}{\Sigma_{R}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm where}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Sigma_{R}\equiv\frac{\mathcal{M}(R)}{4\pi R^{2}}$ (26)

and where $\Sigma_{F}$ is the mass surface density of the fractal $D=2$
equilibrium state, $\mathcal{M}(R)$ is any non-trivial finite perturbation of
the equilibrium state, (25) and $V_{flat}$ is the asymptotic flat rotation
velocity for the system.

####  A.3.2 General case for arbitrary orbits in the spherical system

The full set of Euler-Lagrange equations for arbitrary orbits (assumed to be
equatorial with no loss of generality), together with the _clock constraint_
condition which defines what is meant by physical time (discussion of §A.2),
can be rearranged as:

$\mathbf{\ddot{R}}=-\frac{d\mathcal{V}}{dR}\,\mathbf{\hat{R}},$
$\mathcal{V}=-\frac{1}{2}\left[4\,d_{0}^{2}v_{0}^{2}\left(\frac{\mathcal{M}}{4\pi\Sigma_{F}}-\frac{h^{2}}{d_{0}v_{0}^{2}}\right)\left(\frac{\,\mathcal{M}^{2}}{R^{4}\mathcal{M}^{\prime}\mathcal{M}^{\prime}}\right)+\frac{h^{2}}{R^{2}}\right],$
(27)

_clock constraint:_

$\frac{1}{2}\left(\dot{R}^{2}+R^{2}\dot{\theta}^{2}\right)\,=\,-\mathcal{V}$

where the _clock constraint_ (which defines the system clock in the Leibniz-
Mach worldview) is the condition required to ensure that a dissipation term in
the original EL system becomes identically zero.

It is clear that this choice of the _clock constraint_ is also identical to
the first integral with respect to $t$ of (27a) with the condition that the
constant of integration which arises from this process is explicitly set to
zero. In classical terms, the clock constraint is identical to the energy
equation but, of course, in the classical case ‘time’ is pre-ordained to be
Newton’s idea of time.

Since angular momentum is trivially conserved by (27), it follows immediately
that the total independent content of these equations is, as stated, an
equation (the clock constraint) defining elapsed physical time in terms of
_process_ or _change_ and angular momentum conservation. In this way, the
proposition, that every particle in the ensemble is a _system clock_ is
demonstrated.

It follows, immediately, that the orbits are given by

$\displaystyle V_{rot}(R)$ $\displaystyle=$ $\displaystyle\frac{h}{R}$
$\displaystyle V_{rad}^{2}(R)+V_{rot}^{2}(R)$ $\displaystyle=$
$\displaystyle-2\mathcal{V}.$ (28)

These are the orbits which form the basis of the general case of the Leibniz-
Mach galaxy model in a spherical world.

###  A.4 Brief note on general Newtonian orbits

In order to reproduce the generality of Newtonian orbits, the clock constraint
must have the form

$\frac{1}{2}\left(\dot{R}^{2}+R^{2}\dot{\theta}^{2}\right)=\left(\frac{GM_{S}}{R}-\omega^{2}\right),$

where $\omega$ is either real, zero or imaginary for elliptical, parabolic or
hyperbolic orbits respectively. In other words, the potential is necessarily
defined to be

$\mathcal{V}=-\left(\frac{GM_{S}}{R}-\omega^{2}\right)$

exactly, so that distinct orbits have distinct potentials - the potential
function becomes a function of dynamical state, so that any Newtonian orbit
has its own unique potential state. This means that $\mathcal{M}$ must be
defined to ensure that this is the case which means that it must satisfy (27b)
with $\mathcal{V}$ defined as above.

##  Appendix B A fractal universe: the observations & the debate

A basic assumption of the Standard Model of modern cosmology is that, on some
scale, the universe is homogeneous; however, in early responses to suspicions
that the accruing data was more consistent with Charlier’s conceptions
Charlier (1908, 1922, 1924) of an hierarchical universe than with the
requirements of the Standard Model, De Vaucouleurs (1970) showed that, within
wide limits, the available data satisfied a mass distribution law $M\approx
r^{1.3}$, whilst Peebles (1980) found $M\approx r^{1.23}$. The situation, from
the point of view of the Standard Model, continued to deteriorate with the
growth of the data-base to the point that, Baryshev et al (1995) were able to
say

> _…the scale of the largest inhomogeneities (discovered to date) is
> comparable with the extent of the surveys, so that the largest known
> structures are limited by the boundaries of the survey in which they are
> detected._

For example, several redshift surveys of the late 20th century, such as those
performed by Huchra et al (1983), Giovanelli and Haynes (1986), De Lapparent
et al (1988), Broadhurst et al (1990), Da Costa et al (1994) and Vettolani et
al (1993) etc discovered massive structures such as sheets, filaments,
superclusters and voids, and showed that large structures are common features
of the observable universe; the most significant conclusion drawn from all of
these surveys was that the scale of the largest inhomogeneities observed in
the samples was comparable with the spatial extent of those surveys
themselves.

In the closing years of the century, several quantitative analyses of both
pencil-beam and wide-angle surveys of galaxy distributions were performed:
three examples are given by Joyce, Montuori & Sylos Labini et al (1999) who
analysed the CfA2-South catalogue to find fractal behaviour with
$D\,$=$\,1.9\pm 0.1$; Sylos Labini & Montuori (1998) analysed the APM-Stromlo
survey to find fractal behaviour with $D\,$=$\,2.1\pm 0.1$, whilst Sylos
Labini, Montuori & Pietronero (1998) analysed the Perseus-Pisces survey to
find fractal behaviour with $D\,$=$\,2.0\pm 0.1$. There are many other papers
of this nature, and of the same period, in the literature all supporting the
view that, out to $30-40h^{-1}Mpc$ at least, galaxy distributions appeared to
be consistent with the simple stochastic fractal model with the critical
fractal dimension of $D\approx D_{crit}=2$.

This latter view became widely accepted (for example, see Wu, Lahav & Rees
(1999)), and the open question became whether or not there was transition to
homogeneity on some sufficiently large scale. For example, Scaramella et al
(1998) analyse the ESO Slice Project redshift survey, whilst Martinez et al
(1998) analyse the Perseus-Pisces, the APM-Stromlo and the 1.2-Jy IRAS
redshift surveys, with both groups claiming to find evidence for a cross-over
to homogeneity at large scales.

At around about this time, the argument reduced to a question of statistics
(Labini & Gabrielli (2000), Gabrielli & Sylos Labini (2001), Pietronero &
Sylos Labini (2000)): basically, the proponents of the fractal view began to
argue that the statistical tools (that is, two-point correlation function
methods) widely used to analyse galaxy distributions by the proponents of the
opposite view are deeply rooted in classical ideas of statistics and
implicitly assume that the distributions from which samples are drawn are
homogeneous in the first place. Hogg et al (2005), having accepted these
arguments, applied the techniques argued for by the pro-fractal community
(which use the _conditional density_ as an appropriate statistic) to a sample
drawn from Release Four of the Sloan Digital Sky Survey. They claimed that the
application of these methods does show a turnover to homogeneity at the
largest scales thereby closing, as they see it, the argument. In response,
Sylos Labini, Vasilyev & Baryshev (2006) criticized their paper on the basis
that the strength of the conclusions drawn is unwarrented given the
deficencies of the sample \- in effect, that it is not big enough.

More recently, Tekhanovich & Baryshev (2016) have addressed the deficencies of
the Hogg et al analysis by analysing the 2MRS catalogue, which provides
redshifts of over 43,000 objects out to about 300Mpc, using conditional
density methods; their analysis shows that the distribution of objects in the
2MRS catalogue is consistent with the simple stochastic fractal model with the
critical fractal dimension of $D\approx D_{crit}=2$.

To summarize, the proponents of non-trivially fractal large-scale structure
have won the argument out to medium distances and the controversy now revolves
around the largest scales encompassed by the SDSS.

##  Appendix C Some computational details

There are various details which are necessary to reliably reproduce the
results of this paper.

### The empirically defined mass distribution

In practice, it is found that the five-parameter model

$\displaystyle f(R/R_{0},\mathbf{a})$ $\displaystyle\equiv$ $\displaystyle
Y^{a_{0}},$ (29) $\displaystyle Y$ $\displaystyle\equiv$
$\displaystyle\left[(1-a_{1}-a_{2}-a_{3}-a_{4})\,X+a_{1}\,X^{3}+a_{2}\,X^{5}+a_{3}\,X^{7}+a_{4}\,X^{9}\right],$
$\displaystyle X$ $\displaystyle\equiv$ $\displaystyle\frac{R}{R_{0}},$
$\displaystyle\mathbf{a}$ $\displaystyle\equiv$
$\displaystyle\left(a_{0},a_{1},a_{2},a_{3},a_{4}\right)$

provides for an extremely accurate fit of the theory to the SPARC rotation
curves and hence, by inference, provides a high fidelity modelling of the mass
distributions.

### The minimization metric

For minimization problems involving noisy data, it is generally considered
best to use a metric based on the $L_{1}$-norm. So, for a rotation curve with
measured velocities $V_{sparc}(R_{i}),\,i=1..N$, we seek to determine the disk
parameters $(R_{0},M_{0},V_{flat})$ by minimizing:

$metric=\sum^{N}_{i=1}\left|\frac{V_{sparc}(R_{i})-V_{theory}(R_{i})}{V_{sparc}(R_{i})}\right|$

with respect to variation in them. This gives far superior results to those
arising from use of the $L_{2}$-norm, for example.

### The Nelder-Mead iteration

Because the data is very noisy, we estimate the eight free parameters
($R_{0},V_{flat},M_{0},\mathbf{a}$), where
$\mathbf{a}\equiv(a_{0},a_{1},a_{2},a_{3},a_{4})$, as follows:

  1. 1.

Choose a set of randomly generated initial guess for each of the free
parameters;

  2. 2.

Run the Nelder-Mead minimization process for, typically, about 25000
($2.5\times 10^{4}$) times per disk before the results completely settle down.
For most objects, this is far more than is necessary, but there are a few
awkward objects.

### Choice of the trial value $\Sigma_{F}^{0}$

Whilst in principle $\Sigma_{F}^{0}$ _can_ be chosen to be an arbitrary
positive number, in practice the very noisy nature of the SPARC data means
that the process fails if the choice is ‘too’ extreme. So, in practice, we
proceed as follows:

Assume that the required value for $\Sigma_{F}$ is similar to

$\Sigma_{F}=\frac{a_{F}}{4\pi\,G}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{where}\leavevmode\nobreak\ \leavevmode\nobreak\ a_{F}\approx 1.2\times
10^{-10}\,mtrs/sec^{2}.$

Then, to make the point that the theory works for a given choice of
$\Upsilon_{*}\in(0.5,\,1.0)$, we begin by choosing $\Sigma_{F}^{0}$ to satisfy

$0.1\,\Sigma_{F}\leq\Sigma_{F}^{0}\leq 10.0\,\Sigma_{F}.$

Because of the noisy nature of the calculation, this will produce an estimate
of $a_{F}$ which is very close to its ‘theoretical’ value which depends on the
choice of $\Upsilon_{*}$. If this latter estimate is now used as the starting
point for one more iteration, then the resulting estimate for $a_{F}$ will be
the best possible on the data.

##  Appendix D Least-areas linear regression

In the following, the quantity being minimised is independent of how the
predictor/response pair is chosen for the regression. The result is a linear
model which can be algebraically inverted to give the exact linear model which
would also arise from regressing on the interchanged predictor/response pair.
So, any inference drawn about the relationship between the predictor and
response is independent of how the predictor/response pair is chosen.

Imagine a straight line drawn through a two-dimensional scatter diagram. Every
point in this diagram subtends a right-angled triangle onto the straight line
such that the two short sides of the triangle are parallel to the coordinate
axes and meet at the point concerned. The least-areas linear regression is
then defined to be the particular line which minimizes the total area of all
the triangles summed over all the points.

Suppose that we have the data $(X_{i},Y_{i}),\leavevmode\nobreak\ i=1..N$ to
which we fit the model

$y=Ax+B.$

It is a simple matter to show that the least-areas regression simply requires
that:

$A=\pm\sqrt{\frac{\left(\Sigma Y_{i}\right)^{2}-N\,\Sigma
Y_{i}^{2}}{\left(\Sigma X_{i}\right)^{2}-N\,\Sigma
X_{i}^{2}}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
B=\frac{\left(\Sigma Y_{i}-A\,\Sigma X_{i}\right)}{N}.$

To demonstrate the algebraically invertable property, fit the model $x=\alpha
y+\beta$ to the same data, to get:

$\alpha=\pm\sqrt{\frac{\left(\Sigma X_{i}\right)^{2}-N\,\Sigma
X_{i}^{2}}{\left(\Sigma Y_{i}\right)^{2}-N\,\Sigma
Y_{i}^{2}}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \beta=\frac{\Sigma X_{i}-\alpha\,\Sigma Y_{i}}{N}.$

Comparing the two models quickly shows that $\alpha=1/A$ and $\beta=-B/A$ so
that $x=\alpha y+\beta$ is the algebraic inverse of $y=AX+B$.

### Data availablity statement

The data underlying this article were provided by Stacy McGaugh
<EMAIL_ADDRESS>under licence / by permission. Data will be shared on
request to the corresponding author with permission of Stacy McGaugh.

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