# On perfect symmetric rank-metric codes

Usman Mushrraf and Ferdinando Zullo

###### Abstract

Let $\mathrm{Sym}_{\\_}q(m)$ be the space of symmetric matrices in
$\mathbb{F}_{\\_}q^{m\times m}$. A subspace of $\mathrm{Sym}_{\\_}q(m)$
equipped with the rank distance is called a symmetric rank-metric code. In
this paper we study the covering properties of symmetric rank-metric codes.
First we characterize symmetric rank-metric codes which are perfect, i.e. that
satisfy the equality in the sphere-packing like bound. We show that, despite
the rank-metric case, there are non trivial perfect codes. Also, we
characterize families of codes which are quasi-perfect.

MSC2020: 94B05; 94B65; 94B27
Keywords: symmetric matrix; covering density; perfect code

##  1 Introduction

Rank metric codes were first introduced by Delsarte in [5] and independently
by Gabidulin in [10] and Roth in [19]. These codes have been extensively
researched due to their applications in crisscross error correction [19],
cryptography [11], and network coding [24]; see [2] for more applications. The
coding-theoretic properties of these codes have been thoroughly studied, and
optimal codes with respect to the Singleton-like bound, known as Maximum Rank
Distance (MRD for short) codes, have been constructed. For more details,
interested readers can refer to [14, 23].

The study of subsets of _restricted_ matrices equipped with the rank metric
began in 1975 by Delsarte and Goethals in [6], where they considered sets of
alternating bilinear forms. The theory developed in [5] and [6] also found
applications in classical coding theory. Specifically, the evaluations of the
forms found in [6] give rise to subcodes of the second-order Reed-Muller
codes, including the Kerdock codes and the chain of Delsarte–Goethals codes;
see also [20].

Using the theory of association schemes, bounds, constructions, and structural
properties of restricted rank metric codes have been investigated in symmetric
matrices [3, 8, 12, 13, 17, 21, 27], alternating matrices [1, 6], and
Hermitian matrices [22, 25].

In this paper, we will focus on symmetric rank-metric codes in
$\mathrm{Sym}_{\\_}q(m)$, i.e. subspaces of the vector space of symmetric
matrices over $\mathbb{F}_{\\_}q$ of order $m$ equipped with the rank
distance. The rank distance between two matrices $A$ and $B$ in
$\mathrm{Sym}_{\\_}q(m)$ is defined as

$d(A,B)=\mathrm{rk}(A-B).$

The minimum distance of a symmetric rank-metric code $C$ is defined as

$d(C)=\min\\{d(A,B)\colon A,B\in C,A\neq B\\}.$

The parameters of a symmetric rank-metric code in $\mathrm{Sym}_{\\_}q(m)$ are
$(m,|C|,d(C))$ and they are related by the following Singleton-like bound.

######  Theorem 1.1 ([21, Theorem 3.3]).

Let $C$ be a symmetric rank-metric code in $\mathrm{Sym}_{\\_}q(m)$ with
minimum distance $d$ then we have

$\dim(C)\leq\begin{cases}\frac{m(m-d+2)}{2}&\text{if }m-d\text{ is even},\\\
\frac{(m+1)(m-d+1)}{2}&\text{if }m-d\text{ is odd}.\\\ \end{cases}$

The above bounds turn out to be sharp, as proved in [21], and this allows us
to give the following definition.

######  Definition 1.2.

Let $C$ be a symmetric rank-metric code in $\mathrm{Sym}_{\\_}q(m)$. We say
that $C$ is a symmetric Maximum Rank Distance (or shortly a symmetric MRD)
code if its parameters satisfy the equality in Theorem 1.1.

Following the classical arguments in the Hamming metric, in this paper we
first prove a sphere-packing like bound and then we call perfect those codes
satisfying the equality in this bound. Then we characterize the perfect
symmetric rank-metric codes by proving that, despite the rank-metric (see
[16]) and the alternating rank-metric cases (see [1]), there exist nontrivial
perfect codes. Indeed, apart from the entire space, we show that the only
other family of perfect symmetric rank-metric codes is given by the symmetric
MRD codes of odd order matrices and minimum distance three. Then we study the
covering density of a symmetric rank-metric code $C$, which can be seen as a
measure of how much the spheres centered in the codewords and of radius
$\lfloor(d(C)-1)/2\rfloor$ cover the ambient space. We give some bounds and
then we define families of symmetric rank-metric codes that are quasi-perfect
as those whose covering density goes to one. We characterize families of
quasi-perfect codes, by proving that they exist for special sets of
parameters.

##  2 Bounds on sphere size

Given a matrix $M\in\mathrm{Sym}_{\\_}q(m)$, we define the sphere of radius
$t\in\mathbb{N}_{\\_}0$ centered in $M$ as

$S(M,t)=\\{N\in\mathrm{Sym}_{\\_}q(m)\colon\mathrm{rk}(M-N)=t\\},$

and the ball of radius $t\in\mathbb{N}_{\\_}0$ centered in $M$ as

$B(M,t)=\cup_{\\_}{i=0}^{t}S(M,i).$

It is easy to check that the size of a sphere and of a ball does not depend on
the center and so we can denote the size of a sphere of radius $t$ by
$S_{\\_}t$ and the size of a ball of radius $t$ by $B_{\\_}t$. Also, note that

$B_{\\_}t=\sum_{\\_}{i=0}^{t}S_{\\_}i,$

and $S_{\\_}i$ represents the number of symmetric matrices of order $m$ having
rank $i$ and $B_{\\_}i$ represents the number of symmetric matrices of order
$m$ having rank at most $i$. The values of $S_{\\_}t$ and $B_{\\_}t$ depend on
$q$, $m$ and $t$, more precisely the following hold (see also [18, Theorem
2]).

######  Theorem 2.1 ([4, Theorem 3]).

Let $0\leq t\leq m$ be an integer. We have

$S_{\\_}t=|\\{M\in\mathrm{Sym}_{\\_}m(q)\mid\mathrm{rk}(M)=t\\}|=\prod_{\\_}{s=1}^{\lfloor
t/2\rfloor}\frac{q^{2s}}{q^{2s}-1}\,\prod_{\\_}{s=0}^{t-1}\left(q^{m-s}-1\right).$

As a consequence of the above theorem, we obtain that

$B_{\\_}t=\sum_{\\_}{i=0}^{t}\left(\prod_{\\_}{s=1}^{\lfloor
i/2\rfloor}\frac{q^{2s}}{q^{2s}-1}\,\prod_{\\_}{s=0}^{i-1}\left(q^{m-s}-1\right)\right).$

We will now provide some upper and lower bounds on the size of a sphere and of
a ball.

######  Proposition 2.2.

For any $i\in\\{0,\ldots,m\\}$ and for any $q$, we have

$q^{(m-1)i-\frac{i(i-1)}{2}}<S_{\\_}i\leq
q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor},$

and

$q^{(m-1)i-\frac{i(i-1)}{2}}\leq B_{\\_}i\leq
q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor+1}.$

###### Proof.

The assertion is clearly true when $i=0$. So, assume that $i\geq 1$ and let us
start by determining the bounds on

$S_{\\_}{i}=\prod^{\lfloor
i/2\rfloor}_{\\_}{s=1}\frac{q^{2s}}{q^{2s}-1}\prod^{i-1}_{\\_}{s=0}(q^{m-s}-1).$

Since

$\prod^{i-1}_{\\_}{s=0}(q^{m-s}-1)\leq\prod^{i-1}_{\\_}{s=0}q^{m-s}=q^{mi-\frac{i(i-1)}{2}}$

and

$\frac{q^{2s}}{q^{2s}-1}\leq q,$

it is easy to see that

$\prod^{\lfloor\frac{i}{2}\rfloor}_{\\_}{s=1}\frac{q^{2s}}{q^{2s}-1}\leq\prod^{\lfloor\frac{i}{2}\rfloor}_{\\_}{s=1}q=q^{\lfloor\frac{i}{2}\rfloor}.$

So,

$S_{\\_}{i}\leq q^{\lfloor\frac{i}{2}\rfloor}\cdot
q^{mi-\frac{i(i-1)}{2}}=q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor}.$
(1)

For lower bound on $S_{\\_}i$, observe that

$q^{m-s}-1\geq q^{m-s-1},$

from which

$\prod^{i-1}_{\\_}{s=0}(q^{m-s}-1)\geq\prod^{i-1}_{\\_}{s=0}q^{m-s-1}=q^{(m-1)i-\frac{i(i-1)}{2}}.$

Using that,

$\frac{q^{2s}}{q^{2s}-1}>1\,\,\,\text{and}\,\,\,\prod^{\lfloor
i/2\rfloor}_{\\_}{s=1}\frac{q^{2s}}{q^{2s}-1}>1,$

we get

$S_{\\_}{i}=\prod^{\lfloor\frac{i}{2}\rfloor}_{\\_}{s=1}\frac{q^{2s}}{q^{2s}-1}\prod^{i-1}_{\\_}{s=0}(q^{m-s}-1)>1\cdot
q^{(m-1)i-\frac{i(i-1)}{2}}=q^{(m-1)i-\frac{i(i-1)}{2}}$ (2)

From (1) and (2) we obtain the desired bounds

$q^{(m-1)i-\frac{i(i-1)}{2}}<S_{\\_}i\leq
q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor}.$

Let us know analyze the size of the balls. Since
$B_{\\_}i=\sum_{\\_}{s=0}^{i}S_{\\_}s\geq S_{\\_}i$, the lower bound on
$B_{\\_}i$ comes directly from the lower bound on the size of sphere of radius
$i$. So, we only need to prove the upper bound. Because of the upper bound on
$S_{\\_}i$, we have

$\displaystyle B_{\\_}{i}$
$\displaystyle\leq\sum^{i}_{\\_}{s=0}q^{ms-\frac{s(s-1)}{2}+\lfloor\frac{s}{2}\rfloor}$
$\displaystyle=q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor}\left(1+\sum^{i-1}_{\\_}{s=0}\frac{q^{ms-\frac{s(s-1)}{2}+\lfloor\frac{s}{2}\rfloor}}{q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor}}\right)$
$\displaystyle=q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor}\left(1+\sum^{i-1}_{\\_}{s=0}q^{ms-
mi+\frac{i^{2}-s^{2}}{2}-\frac{i-s}{2}+\lfloor\frac{s}{2}\rfloor-\lfloor\frac{i}{2}\rfloor}\right).$

As, for every $s\leq i-1$,

$ms-
mi+\frac{i^{2}-s^{2}}{2}-\frac{i-s}{2}+\left\lfloor\frac{s}{2}\right\rfloor-\left\lfloor\frac{i}{2}\right\rfloor\leq(i-s)\left(i-m\right),$

since

$\left\lfloor\frac{s}{2}\right\rfloor\leq\left\lfloor\frac{i}{2}\right\rfloor\,\,\,\text{and}\,\,\,s\leq
i,$

we have that

$B_{\\_}{i}\leq
q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor}\left(1+\sum^{i-1}_{\\_}{s=0}q^{(i-s)(i-m)}\right)\leq
q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor}\left(1+\sum^{i}_{\\_}{j=1}q^{j(i-m)}\right).$
(3)

Observe that

$1+\sum^{i}_{\\_}{j=1}q^{j(i-m)}=\frac{1-q^{(i+1)(i-m)}}{1-q^{i-m}}\leq\frac{1}{1-q^{i-m}}\leq
q,$

so that, together with (3), we have

$B_{\\_}{i}\leq q^{mi-\frac{i(i-1)}{2}+\lfloor\frac{i}{2}\rfloor}\cdot q,$

which concludes the proof. ∎

######  Remark 2.3.

The above bounds can be certainly improved. Because of our purposes of the
next section, we need to upper bound $B_{\\_}i$ by a power of $q$.

In the next section we will use these bounds to characterize perfect codes in
the theory of symmetric rank-metric codes.

##  3 Perfect symmetric rank-metric codes

In this section we will explore the sphere-packing bound and we will
investigate the parameters for which perfect codes exist in the symmetric
matrices framework.

######  Theorem 3.1 (Sphere-packing bound).

Let $C\subseteq\mathrm{Sym}_{\\_}{q}(m)$ be a symmetric rank metric code of
minimum distance $d$ and $|C|=M$. If $t=\lfloor\frac{d-1}{2}\rfloor$, then

$MB_{\\_}{t}\leq q^{\frac{m^{2}+m}{2}}.$

###### Proof.

Let $c_{\\_}{1},c_{\\_}{2},...,c_{\\_}{M}$ be the codewords of $C$. Using the
triangle inequality of the rank metric, we can see that

$B_{\\_}{t}(c_{\\_}{i})\cap B_{\\_}{t}(c_{\\_}{j})=\varnothing,$

for any $i,j\in\\{1,\ldots,M\\}$ with $i\neq j$. So,

$\left|\bigcup^{M}_{\\_}{i=1}B_{\\_}{t}(c_{\\_}{i})\right|=\bigcup^{M}_{\\_}{i=1}\left|B_{\\_}{t}(c_{\\_}{i})\right|\leq|\mathrm{Sym}_{\\_}{q}(m)|=q^{\frac{m^{2}+m}{2}}$

implying that

$\sum^{M}_{\\_}{i=1}B_{\\_}{t}\leq q^{\frac{m^{2}+m}{2}},$

and hence the desired bound. ∎

As for the already studied metrics, we give the following definition of
perfect code.

######  Definition 3.2.

Let $C$ be a symmetric rank-metric code in $\mathrm{Sym}_{\\_}q(m)$. We say
that $C$ is a perfect if its parameters satisfy the equality in the sphere-
packing bound.

A trivial example is obtained when considering $C=\mathrm{Sym}_{\\_}q(m)$,
indeed in this case its minimum distance is $1$ and so $t=(d-1)/2=0$ and
$B_{\\_}0=1$. We will now show, through some steps, that a code in
$\mathrm{Sym}_{\\_}q(m)$ with minimum distance $d$ is perfect if and only if
either it is the trivial code (i.e. $\mathrm{Sym}_{\\_}q(m)$) or it is a
symmetric MRD code, with $m$ odd and $d=3$. We first analyze the case in which
$\lfloor\frac{d-1}{2}\rfloor=1$.

######  Lemma 3.3.

Let $C\subseteq\mathrm{Sym}_{\\_}{q}(m)$ be a perfect code with minimum
distance $d$ and let $t=\lfloor\frac{d-1}{2}\rfloor=1$. Then $d=3$, $m$ is odd
and $C$ is a symmetric MRD code.

###### Proof.

Note that in this case, $d\in\\{3,4\\}$. Since $t=1$ and $B_{\\_}{1}=q^{m}$,
we have that

$M=q^{\frac{m^{2}-m}{2}}.$

When $m-d$ is even, Theorem 1.1 implies

$q^{\frac{m^{2}-m}{2}}=M\leq q^{\frac{m(m-d+2)}{2}},$

that is

$d\leq 3,$

and so $d=3$ and $m$ is odd. When $d=3$ and $m$ is odd, we have the equality
in the sphere-packing bound if and only if $C$ is a symmetric MRD code. When
$m-d$ is odd, Theorem 1.1 implies

$q^{\frac{m^{2}-m}{2}}=M\leq q^{\frac{(m+1)(m-d+1)}{2}},$

that is

$d(m+1)\leq 3m+1,$

a contradiction to $d\in\\{3,4\\}$. ∎

In the next lemma we will analyze the case in which
$\lfloor\frac{d-1}{2}\rfloor=2$.

######  Lemma 3.4.

Let $C\subseteq\mathrm{Sym}_{\\_}{q}(m)$ be a symmetric rank-metric code with
minimum distance $d$ and let $t=\lfloor\frac{d-1}{2}\rfloor=2$ then $C$ is not
perfect.

###### Proof.

Suppose on contrary that $C$ is perfect, then

$MB_{\\_}{2}=q^{\frac{m^{2}+m}{2}}.$

Note that $M=q^{i}$ for some natural number $i<(m^{2}+m)/2$, as $t=2$ and $C$
cannot be equal to $\mathrm{Sym}_{\\_}{q}(m)$. Therefore, we get that

$B_{\\_}{2}=q^{\frac{m^{2}+m}{2}-i}.$ (4)

Since $B_{\\_}{2}=q^{m}+\frac{q^{2}(q^{m}-1)(q^{m-1}-1)}{q^{2}-1}$ then if
$m>2$ we have $B_{\\_}2/q^{2}\equiv-1\pmod{q}$, so we get a contradiction from
(4). When $m=2$ we still get a contradiction from (4) and the size of
$B_{\\_}2$. ∎

We can now completely determine the parameters of a perfect code in
$\mathrm{Sym}_{\\_}q(m)$.

######  Theorem 3.5.

Let $C\subseteq\mathrm{Sym}_{\\_}{q}(m)$. Then $C$ is a perfect code if and
only if $C=\mathrm{Sym}_{\\_}{q}(m)$ or $d(C)=3$, $m$ is odd and $C$ is a
symmetric MRD code.

###### Proof.

Suppose that $m-d$ is even, then from Theorem 1.1 we have

$M\leq q^{\frac{m(m-d+2)}{2}},$

and using the fact that $C$ is perfect we have

$q^{\frac{m(m-d+2)}{2}}B_{\\_}{t}\geq q^{\frac{m^{2}+m}{2}},$

with $t=\lfloor(d(C)-1)/2\rfloor$. Since, by Proposition 2.2 $B_{\\_}{t}\leq
q^{mt-\frac{t(t-1)}{2}+\frac{t}{2}+1}$, then

$q^{\frac{m^{2}+m}{2}}\leq
q^{mt-\frac{t(t-1)}{2}+\frac{t}{2}+1+\frac{m(m-d+2)}{2}},$

that is

$\frac{m^{2}+m}{2}\leq mt-\frac{t(t-1)}{2}+\frac{t}{2}+1+\frac{m(m-d+2)}{2}.$
(5)

Assume that $d=2t+1$, then (5) reads as follows

$0\leq-t^{2}+2t+2,$

which holds only for $t\in\\{0,1,2\\}$.
Now, consider $d=2t+2$, then (5) implies that

$m\leq-t^{2}+2t+2.$

Since $m\geq 2$, again we have that $t\in\\{0,1,2\\}$. Arguing as before in
the case in which $m-d$ is odd, we have again that $t\in\\{0,1,2\\}$. If
$t=0$, then $d=1$ and $C$ is perfect if and only if
$C=\mathrm{Sym}_{\\_}{q}(m)$. The remaining part follows by Lemmas 3.3 and
3.4. ∎

###  3.1 Examples of non-trivial perfect codes

In this section we will recall the known examples of symmetric MRD codes.
Their description is given in terms of linearized polynomials, which is an
equivalent framework as we will see in the next lines. Let
$\mathcal{L}_{\\_}{m,q}$ denote the quotient $\mathbb{F}_{\\_}q$-algebra of
all $q$-polynomials (or linearized polynomials) over $\mathbb{F}_{\\_}{q^{m}}$
with degree smaller than $q^{m}$, namely,

$\mathcal{L}_{\\_}{m,q}=\left\\{\sum\limits_{\\_}{i=0}^{m-1}a_{\\_}ix^{q^{i}}\colon
a_{\\_}i\in\mathbb{F}_{\\_}{q^{m}}\right\\}.$

It is well known that the $\mathbb{F}_{\\_}{q}$-algebra
$\mathcal{L}_{\\_}{m,q}$ is isomorphic to the $\mathbb{F}_{\\_}{q}$-algebra
$\mathbb{F}_{\\_}q^{m\times m}$; see e.g. [26]. In [17] symmetric matrices are
described in terms of linearized polynomials. Indeed, the vector space
$\mathrm{Sym}_{\\_}q(m)$ can be identified to

${\mathcal{S}}_{\\_}q(m)=\left\\{\sum\limits_{\\_}{i=0}^{m-1}c_{\\_}ix^{q^{i}}\colon
c_{\\_}{m-i}=c_{\\_}i^{q^{m-i}}\,\text{ for
}\,i\in\\{0,\ldots,m-1\\}\right\\}\subseteq\mathcal{L}_{\\_}{m,q}.$

In [21, Theorem 4.4] the author proved that for all the admissible values of
the minimum distance, $q$ and $m$ there are symmetric MRD codes (see also [17]
for the description in terms of linearized polynomials).

######  Theorem 3.6 ([21, Theorem 4.4]).

Let $m$ and $d$ be two positive integers such that $1\leq d\leq m$ and $m-d$
is even. The set

$\mathcal{S}_{\\_}{q,m,d}=\left\\{b_{\\_}0x+\sum\limits_{\\_}{j=1}^{\frac{m-d}{2}}\left(b_{\\_}jx^{q^{j}}+(b_{\\_}jx)^{q^{m-j}}\right)\colon
b_{\\_}0,\ldots,b_{\\_}{\frac{m-d}{2}}\in\mathbb{F}_{\\_}{q^{m}}\right\\}$ (6)

defines a symmetric MRD code in $\mathrm{Sym}_{\\_}q(m)$.

In [21, Theorem 4.1] it has been shown that constructions of symmetric MRD
codes with minimum distance $d$ in $\mathrm{Sym}_{\\_}q(m)$ with $m-d$ odd can
be obtained by puncturing (in terms of matrices, removing rows and columns)
the above examples $\mathcal{S}_{\\_}{q,m,d}$ of symmetric MRD codes. When
$d=3$ and $m$ is odd, $\mathcal{S}_{\\_}{q,m,d}$ is a perfect code. Moreover,
for this families of codes fast encoding and decoding algorithms have been
developed; see [8], [9], [12], [13] and [15]. In [17, Section 5], also another
construction of symmetric MRD codes has been found, but in this case the codes
are not perfect.

##  4 Covering density

In the previous sections, we showed that existence of perfect codes in
symmetric rank-metric context depends strongly on the order of matrices, the
dimension of the code and minimum distance of code. However, a natural
investigation regards the study of covering density of a symmetric rank-metric
code $C$, which gives a measure of how much of the ambient space is covered by
the spheres centered in the codewords and of radius
$\lfloor\frac{d(C)-1}{2}\rfloor$.

######  Definition 4.1.

Let $C$ be a symmetric rank-metric code in $\mathrm{Sym}_{\\_}q(m)$ of size
$M$. The covering density of $C$ is defined as

$D(C)=\frac{MB_{\\_}{t}}{q^{\frac{m^{2}+m}{2}}},$

where $t=\lfloor\frac{d(C)-1}{2}\rfloor$.

As a consequence of the bounds in Proposition 2.2, we have the following bound
for the covering density of a symmetric MRD code.

######  Proposition 4.2.

Let $C$ be symmetric MRD code in $\mathrm{Sym}_{\\_}q(m)$ with minimum
distance $d$ and let $t=\lfloor\frac{d(C)-1}{2}\rfloor$. The covering density
$D$ of $C$ is upper bounded as follows

$D\leq\begin{cases}q^{-t-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor+1}&\text{if
}m\text{ is even and }d=2t+1,\\\
q^{-t-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor-\frac{m}{2}+\frac{1}{2}}&\text{if
}m\text{ is odd and }d=2t+2,\\\
q^{-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor+1}&\text{if }m\text{ is odd and
}d=2t+1,\\\
q^{-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor-\frac{m}{2}+1}&\text{if
}m\text{ is even and }d=2t+2,\end{cases}$

and lower bounded as follows

$D\geq\begin{cases}q^{-2t-\frac{t(t-1)}{2}}&\text{if }m\text{ is even and
}d=2t+1,\\\ q^{-2t-\frac{t(t-1)}{2}-\frac{m}{2}-\frac{1}{2}}&\text{if }m\text{
is odd and }d=2t+2,\\\ q^{-t-\frac{t(t-1)}{2}}&\text{if }m\text{ is odd and
}d=2t+1,\\\ q^{-t-\frac{t(t-1)}{2}-\frac{m}{2}}&\text{if }m\text{ is even and
}d=2t+2.\end{cases}$

It is worth to mention that when the minimum distance is odd, the upper and
lower bounds do not depend on the order of the matrices involved but only on
the minimum distance. Clearly, using Theorem 1.1, the upper bounds in
Proposition 4.2 immediately reads as follows for a symmetric rank-metric code.

######  Corollary 4.3.

Let $C$ be a symmetric rank-metric code in $\mathrm{Sym}_{\\_}q(m)$ with
minimum distance $d$ and let $t=\lfloor\frac{d(C)-1}{2}\rfloor$. The covering
density

$D\leq\begin{cases}q^{-t-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor+1}&\text{if
}m\text{ is even and }d=2t+1,\\\
q^{-t-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor-\frac{m}{2}+\frac{1}{2}}&\text{if
}m\text{ is odd and }d=2t+2,\\\
q^{-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor+1}&\text{if }m\text{ is odd and
}d=2t+1,\\\
q^{-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor-\frac{m}{2}+1}&\text{if
}m\text{ is even and }d=2t+2.\end{cases}$

We did not mention the lower bounds as they strongly depends on the size of
the codes considered. As the exponents appearing in the bounds of Proposition
4.2 are relatively small, these bounds also suggest that there should not
exist quasi-perfect codes, apart from the cases in which we know that there
are perfect codes. As classically done, see e.g. [7] and [16] (for the rank-
metric version), we give the definition of quasi-perfect families of codes.

######  Definition 4.4.

Let $\mathcal{F}=\\{C_{\\_}i\\}_{\\_}i$ be a family of
$(m_{\\_}i,|C_{\\_}{i}|,d)$ symmetric rank-metric codes and let $D_{\\_}{i}$
be the covering density of code $C_{\\_}{i}$. If

$\lim_{\\_}{i\to\infty}D_{\\_}{i}=1,$

then we say that the family $\mathcal{F}$ is quasi-perfect.

We will now show that, apart from the parameters from which we know there
exist perfect codes, there are no quasi-perfect families of codes.

######  Theorem 4.5.

There are no quasi-perfect families of codes with minimum distance $d$ such
that

$\left\lfloor\frac{d-1}{2}\right\rfloor\geq 3.$

###### Proof.

Let $\mathcal{F}=\\{C_{\\_}i\\}_{\\_}i$ be a family of
$(m_{\\_}i,|C_{\\_}{i}|,d)$ symmetric rank-metric codes and let

$D_{\\_}{i}=\frac{|C_{\\_}{i}||B_{\\_}{t}|}{q^{\frac{i^{2}+i}{2}}}$

be the covering density of code $C_{\\_}{i}$, where
$t=\lfloor(d(C_{\\_}i)-1)/2\rfloor$. We first show that $D_{\\_}{i}\leq
1/q=q^{-1}$ for any $i$. From Corollary 4.3, when $d=2t+1$ and $i$ is even we
have

$D_{\\_}{i}\leq q^{-t-\frac{t(t-1)}{2}+\lfloor\frac{t}{2}\rfloor+1}\leq
q^{-1},$

if and only if $t\geq 2$. Arguing in a similar way for the remaining cases, we
have that $D_{\\_}{i}\leq q^{-1}$ for any $i$.

Therefore if $\lim_{\\_}{i\to\infty}D_{\\_}i$ exists, it is at most $1/q$, and
so it cannot be one. ∎

We will now analyze the cases in which
$t=\left\lfloor\frac{d-1}{2}\right\rfloor\in\\{0,1,2\\}$. When $t=0$ then we
have either $d=1$ or $d=2$. In this case we can consider the family of codes
$\\{\mathrm{Sym}_{\\_}q(i)\\}_{\\_}i$, and since the covering densities of the
considered codes are $1$, this family defines a (trivial) quasi-perfect family
of codes. When $d=2$ the covering density is given by the ratio of the size
code and size of the ambient space, and it is easy to see that in this case we
cannot have quasi-perfect families of codes.

When $t\in\\{1,2\\}$ we need a deeper analysis of the covering density.
Indeed, using Theorem 2.1 and the fact that

$B_{\\_}2=\frac{-q^{m}+q^{2m+1}-q^{m+1}+q^{2}}{q^{2}-1},$

we obtain the following.

######  Proposition 4.6.

Let $C$ be a symmetric rank-metric code in $\mathrm{Sym}_{\\_}q(m)$ with
minimum distance $d$ and let $t=\left\lfloor\frac{d-1}{2}\right\rfloor$. The
covering density $D$ of $C$ is upper bounded as follows for $t=1$

$D\leq\begin{cases}1&\text{if }m\text{ is odd and }d=3,\\\ q^{-1}&\text{if
}m\text{ is even and }d=3,\\\ q^{-\frac{m}{2}}&\text{if }m\text{ is even and
}d=4,\\\ q^{-\frac{m+3}{2}}&\text{if }m\text{ is odd and }d=4,\end{cases}$

and for $t=2$

$D\leq\begin{cases}\frac{-q^{-m}+q-q^{-m+1}+q^{2-2m}}{q^{2}-1}&\text{if
}m\text{ is odd and }d=5,\\\
\frac{-q^{-m-2}+q^{-1}-q^{-m-1}+q^{-2m}}{q^{2}-1}&\text{if }m\text{ is even
and }d=5,\\\
\frac{-q^{\frac{-3m}{2}}+q^{\frac{-m+2}{2}}-q^{\frac{-3m+2}{2}}-q^{\frac{-5m+4}{2}}}{q^{2}-1}&\text{if
}m\text{ is even and }d=6,\\\
\frac{-q^{\frac{-3m-5}{2}}+q^{\frac{-m-3}{2}}-q^{\frac{-3m-3}{2}}-q^{\frac{-5m-1}{2}}}{q^{2}-1}&\text{if
}m\text{ is odd and }d=6.\end{cases}$

If $C$ is a symmetric MRD code, then the equality holds.

Using the above proposition we can exclude the existence of family of quasi-
perfect codes having minimum distance either $5$ or $6$.

######  Proposition 4.7.

There are no quasi-perfect families of symmetric rank-metric codes with
minimum distance $d$ such that

$\left\lfloor\frac{d-1}{2}\right\rfloor=2.$

###### Proof.

Let $\\{C_{\\_}{i}\\}_{\\_}i$ be a family of symmetric rank-metric codes with
minimum distance $d$ where $C_{\\_}i$ has covering density $D_{\\_}{i}$. A
case by case analysis shows that, using the upper bounds in Proposition 4.6,
$D_{\\_}i$ is strictly less than $1/q$. So, when considering the limit of the
$D_{\\_}i$’s, if it exists, it is bounded by $1/q$ and it cannot be one.
Therefore it can be concluded that there are no quasi-perfect families of
codes with minimum distance $d$ such that $\lfloor(d-1)/2\rfloor=2$. ∎

When $t=1$, quasi-perfect families of codes exist. Indeed, if $C_{\\_}i$ is a
symmetric MRD code with parameters $(2i+1,q^{\frac{i(i-1)}{2}},3)$, then
$D(C_{\\_}i)=1$ for any $i$ by Theorem 3.5 and so

$\lim_{\\_}{i\to\infty}D_{\\_}i=1.$

######  Proposition 4.8.

Let $\\{C_{\\_}i\\}_{\\_}i$ be a family of symmetric rank-metric codes where
$C_{\\_}i$ has parameters $(m_{\\_}i,|C_{\\_}i|,d)$ and $d\in\\{3,4\\}$.

  * •

If $d=3$ and the sequence $\\{m_{\\_}i\\}_{\\_}i$ has at least one subsequence
of even numbers, then $\\{C_{\\_}i\\}_{\\_}i$ is not quasi-perfect.

  * •

If $d=4$ then $\\{C_{\\_}i\\}_{\\_}i$ is not quasi-perfect.

###### Proof.

Suppose that $d=3$. Let $\\{m_{\\_}j\\}_{\\_}{j\in\mathcal{J}}$ be a
subsequence of $\\{m_{\\_}i\\}_{\\_}i$ with the property that $m_{\\_}j$ is an
even number for any $j\in\mathcal{J}$. From Proposition 4.6, when $d=3$ and
$m_{\\_}j$ is even the covering density is bounded by $1/q$ and therefore if
$\lim_{\\_}{j\to\infty}D(C_{\\_}j)$ exists then it is bounded by $1/q$ and so
either $\lim_{\\_}{i\to\infty}D(C_{\\_}i)$ does not exist or it is at most
$1/q$. Assume now that $d=4$, then by Proposition 4.6 for any $i$ we have that

$D(C_{\\_}i)\leq q^{-\frac{m_{\\_}i}{2}},$

and therefore $\lim_{\\_}{i\to\infty}D(C_{\\_}i)=0$. ∎

Combining the above results we have the following classification of quasi-
perfect families of codes.

######  Corollary 4.9.

There are quasi-perfect families of symmetric rank-metric codes with minimum
distance $d$ if and only if $d\in\\{1,3\\}$. The latter case only happens when
for the families of codes that does not contain infinitely many code of even
order.

## Acknowledgements

The research of the last two authors was partially supported by the project
COMBINE of “VALERE: VAnviteLli pEr la RicErca" of the University of Campania
“Luigi Vanvitelli” and was partially supported by the INdAM - GNSAGA Project
_Tensors over finite fields and their applications_ , number E53C23001670001.

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Usman Mushrraf and Ferdinando Zullo
Dipartimento di Matematica e Fisica,
Università degli Studi della Campania “Luigi Vanvitelli”,
I– 81100 Caserta, Italy
<EMAIL_ADDRESS>


Title The $\kappa$-model, a minimal model alternative to dark matter.
Application to the galactic rotation problem

G. Pascoli

emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
Faculté de sciences, Département de physique

Université de Picardie Jules Verne (UPJV)

33 rue saint Leu, Amiens, France

Abstract The determination of the velocities, accelerations and the
gravitational field intensity at a given location in a galaxy could
potentially be achieved in an unexpected manner with the environment of the
observer, for instance, the local mean mass density in the galaxy. This idea,
mathematically supported by the asymmetric distance concept, is illustrated
here by a study regarding the rotation of spiral galaxies. This suggestion is
new in the astrophysics field (in the following, it is called the
$\kappa$-model) and could help to mimic the main effects seen in modified
Newtonian dynamics (MOND) theory, modified gravity (MOG) models, or other
related models built with the aim of eliminating dark matter that are already
well-established theories. Thus, starting from selected examples of galaxies,
in sections 5, 6 and 7 we show that there is an equivalence between MOND and
the $\kappa$-model. In particular, on the opposite side, we have the
speculative nature of the dominant paradigm, the elusive dark matter, a matter
whose properties always remain undefined despite intense theoretical,
experimental and observational efforts for over 50 years.

Keywords: galaxy, asymmetric distance, galactic rotation, spiral pattern, dark
matter

##  1 Introduction

Alternative theories to dark matter attribute the flatness of the galactic
rotation curve either to a departure from the law of inertia (modified
Newtonian dynamics (MOND)) in the very low acceleration regime (Milgrom, 1983,
2009) or to a failure of Newtonian gravity at a large scale, emulated by a
variable gravitational constant $G$ (Moffat, 2006, 2008), and neither invoke
an additional very large quantity of dark matter. Many other models that do
not involve dark matter have also been proposed (Capozziello and De Laurentis
M., 2012; O’Brien and Moss, 2015; Ludwig, 2021). One of them, conformal
gravity (CG), has generated extensive interest since it claims to explain the
flat rotation curve of galaxies without dark matter. The majority of the CG is
based on the Mannheim-Kazanas vacuum solution of the CG field equations for a
static, spherically symmetric spacetime (Mannheim and Kazanas, 1989).
Nevertheless, the conformal equivalence of the Mannheim-Kazanas and
Schwarzschild–de Sitter metrics strongly suggests that the prediction of a
flat rotation curve for the galaxies by this type of model may be a gauge
artifact since performing a similar analysis in the Schwarzschild–de Sitter
metric yields rotation curves without any flat region (Hobson and Lasenby,
2021).

Negative masses have also been invoked (Farnes, 2018; Benoit-Lévy and Chardin,
2012; Manfredi, Rouet, Miller and Chardin, 2018). However, a careful analysis
of the negative mass concept identifies a number of incompatibilities with
existing observations (Socas-Navarro, 2019). Regardless, we can ask what is
the benefit of replacing an unobserved exotic entity, i.e., dark matter, with
another unobserved, even more exotic entity, for instance, a fluid composed of
negative mass. On the other hand, a correlation between the radial
acceleration traced by rotation curves and that predicted by the observed
distribution of baryons has been reported (McGaugh, Lelli and Schombert,
2016). This radial acceleration relation seems tantamount to a natural law for
rotating galaxies and is not compatible with the hidden presence of a large
amount of exotic mass.

There are other channels that have yet to be explored. The assumption
developed in the present paper, namely, that both gravitational field
intensity and the measurement of the velocities and accelerations of a
particle could depend on the mean mass density estimated in a given location,
has not been considered thus far. However, this idea, even though very simple,
offers new perspectives because of its originality and its various
consequences (Pascoli, and Pernas, 2020). The key strengths of this concept
are as follows:

  1. 1.

The basic elements of this assumption are very natural and reside within the
framework of the Newtonian mechanics. Let us recall that Newtonian mechanics
is relevant only when the velocities are low and the gravitational field is
weak, but this is the case here, even though the model can still be derived
from a relativistic context111See(Pascoli, and Pernas, 2020) for an attempt to
develop the $\kappa$-model in a relativistic context.. The concept,
hereinafter named the $\kappa$-model, postulates an enhancement, resp. a
diminution, of the self-gravity in the regions where the mean mass density is
weak, resp. high, in a galaxy. An associated effect is that the measured
distances become apparent and asymmetric. This statement is not explicit in
the context of either the MOND paradigm (Milgrom, 1983, 2019) or modified
gravity (MOG) theories (Moffat, 2006, 2008). In addition, the $\kappa$-model
remains in the strict framework of a preserved Newtonian law of gravity, at
least from a formal point of view. More generally, all the physical laws,
locally expressed, remain unmodified in the $\kappa$-model.

  2. 2.

The persistence of a spiral substructure in ”grand design” galaxies and the
flatness of the rotation curves are shown to be interrelated in an unexpected
manner.

  3. 3.

Some problems such as the bullet cluster that are seemingly difficult to solve
in the framework of the various alternatives to the dark matter paradigm (for
instance, MOND) could perhaps be naturally explained under this framework.
This is because gravity, where it acts, is now modulated by the local mean
density. Another issue is related to the accelerating expansion of the
Universe (Frieman, Turner and Dragan, 2008; Weinberg, 2008). However, this
problem could also be solved in the framework of the $\kappa$-model,
eventually leading from two distinct paradigms (dark matter and dark energy)
to a single one. All these important issues, where a possible density-
dependent aspect of the gravitational force is put in evidence, are considered
later.

##  2 The $\kappa$-model

The equations of this framework are the usual Newtonian (nonrelativistic)
equations, except they are weighted by a coefficient $\kappa$, as explained
immediately below 222The relativistic expression is not useful here, even
though the relativistic transcript would be easy to form in the kappa model
framework.. For a system composed of $N$ identical particles of mass $m$
(index $i=1...,N)$, we write

$\frac{d}{dt}\left({\kappa{}}_{i}[\bar{\rho{}}]\frac{d{\boldsymbol{\sigma{}}}_{i}}{dt}\right)=-Gm\sum_{j=1,\
\
j\not=i}^{N}\frac{{\kappa{}}_{i}[\bar{\rho{}}]({\boldsymbol{\sigma{}}}_{i}-{\boldsymbol{\sigma{}}}_{j})}{{[{\kappa{}}_{i}[\bar{\rho{}}]\
\left\|{}{\boldsymbol{\sigma{}}}_{i}-\boldsymbol{\sigma{}}_{j}\right\|{}]}^{3}}$
(1)

###  2.1 A formal deduction of equation (1) from the gravitational Newtonian
law

Here, we start with the Newton gravitational law, as it is confirmed to be
accurate at the local level (the solar system). Let us define two very distant
particles $M$ and ${M}^{\prime}$ with respective masses $m$ and
${m}^{\prime}$, denoted by the vectors $\bf{R}$ and ${\bf{R}}^{\prime}$
(arbitrary origin). We write formally for the particle $M$, assuming that the
gravitational constant $G$ is universal

$\frac{d\bf{P}}{dt}=-Gmm^{\prime}\frac{(\bf{R}-\bf{R}^{\prime})}{{\
\left\|{}\bf{R}-\bf{R}^{\prime}\right\|{}}^{3}}$

where ${\bf{P}}$ is the momentum of particle $M$, and we can write a similar
equation for ${M}^{\prime}$. At this stage, there is no observer (no reference
frame) to measure the kinematic quantities. As is well known in mechanics,
without a reference frame, both the velocity and acceleration are
undetermined. Likewise, $\frac{d\bf{P}}{dt}$ cannot be added to
$\frac{d\bf{P}^{\prime}}{dt}$ apart from in a purely formal manner. In
addition, in the $\kappa$-model, the norms of $\bf{R}$ and $\bf{R}^{\prime}$
are now themselves left undetermined. Thus, in To solve this pair of
equations, we must choose a representation: a fictitious inertial observer $A$
is assumed to be located near $M$ (resp. $B$ near ${M}^{\prime}$), each of
these observers is equipped with a scale factor $\kappa{}$ (resp.
$\kappa{}^{\prime}$). Then, $A$ locally expresses ${\bf{R}},{\bf{R}^{\prime}}$
and ${\bf{P}}$: ${\bf{R}}\longrightarrow{}\kappa{}{\boldsymbol{\sigma{}}},\
{\bf{R}^{\prime}}\longrightarrow{}\kappa{}{\boldsymbol{\sigma{}}}^{\prime}$
and ${\bf{P}}\longrightarrow{}\kappa{}(m\frac{d{\boldsymbol{\sigma{}}}}{dt})$,
and $B$ does the same thing but with a scale factor $\kappa{}^{\prime}$. We
can note that there exists some – even though obviously very remote –
mathematical analogy with quantum mechanics when we pass from the operational
notation to the Schrödinger representation for the position $\bf{R}$ and
momentum $\bf{P}$ of a point particle ($\hat{\bf{R}}\longrightarrow{}\bf{R},\
\hat{\bf{P}}\longrightarrow{}-i\hslash{}\boldsymbol{\nabla{}}$).

The scale factor (the mathematics) is then linked to the mean density by
equation (2) introduced below (the physics). Eventually, we assume that the
fictitious (inertial) observers $A$ and $B$ are motionless with respect to
each other, an assumptions that is easily verified by spectroscopic
measurements. Another deduction of equation (1) issued from the variational
principle is given in appendix A.

###  2.2 coefficient $\kappa$

In equation (1), the coefficient $\kappa{}$ is no longer a constant (equal to
$\kappa_{E}$) as in the basic Newtonian equations. In contrast, $\kappa{}$ is
now defined as a functional of the mean local density $\bar{\rho{}}$333Note
that the densities in equation (2) are the densities estimated by the same
observer or, more precisely, that $\rho_{0}$ and $\bar{\rho}$ are
simultaneously measured by this observer.. Obviously, at the local scale from
our perspective, i.e., at the scale of the Earth or the solar system, we can
take ${\kappa{}}_{i}=\kappa_{E}=Const,\ \forall{}i$ and analyze a group of
particles contained in the solar system (planets, asteroids, etc.); these
equations exactly express the usual Newtonian law.

By considering the fact that $\kappa{}$ is assumed to be a smooth linear
function and that $\bar{\rho{}}$ is an exponential function in a typical
galaxy (Binney and Merrifield, 1998), it appears intuitive to impose a natural
and simplest form, i.e., a logarithmic relationship between $\kappa{}$ and
$\bar{\rho{}}$. Let

$\frac{\kappa{}[{\rho{}}_{0}]}{\kappa{}\left[\bar{\rho{}}\right]}=1+Ln(\frac{{\rho{}}_{0}}{\bar{\rho{}}})$
(2)

where the index $0$ labels the maximum value of the density distribution ($Ln$
is the symbol for the Napierian logarithm). This law is assumed to be
universal and available for any galaxy.

At this level, the system of equations (1) and (2) can be admitted from the
outset as a postulate of the model, and we could simply end here, leaving the
deduction from first principles aside. However, the reader can refer to
appendix A for a synthetic demonstration of the dynamics formula represented
by equation (1).

Let us examine equation (1). The inertia term is modulated by the factor
$\kappa{}$ (as in MOND). On the other hand, from the right-hand side, we can
see that wherever the mean density is high (resp. low), the gravitational
”sensation” between two masses is low (resp. high). Two masses are more
strongly gravitationally linked with each other in the outer regions (low
density, low $\kappa{}$) of a galaxy than in the inner regions (high density,
high $\kappa{}$). This bears some resemblance to electromagnetism in a
situation where two charged particles are placed in a medium of relative
dielectric permittivity ${\epsilon{}}_{r}$ different from unity
(${\epsilon{}}_{r}>1$). The electric field between these charges is lower than
that of vacuum. However, this analogy must not be taken too far. There exists
an important difference: no background medium exists in the $\kappa$-model,
and the coefficient $\kappa{}$ induces no refractive effect. The light still
propagates in a straight line with the same speed $c$ for all frequencies, and
$\kappa{}$ is frequency-independent, contrary to $\ {\epsilon{}}_{r}$, which
depends on the frequency. Likewise, the trajectory of any free particle is
rectilinear in the $\kappa$ model (when the gravitational force is
eliminated).

##  3 Computational details

Equation (1) seems to be ”easily” treated by any standard and well-known
method of smoothed particle hydrodynamics (SPH), for instance, by using a code
available online. Unfortunately, these codes are most often based on the
default 4th order Runge-Kutta algorithm. In the present situation, the process
of solving is more complex given that the factor $\kappa{}$ is now a
functional of the mean density. Then, we are facing a self-consistent problem,
and a more stable numerical algorithm than the 4th order Runge-Kutta order
method must be used, even though using a higher order Runge-Kutta algorithm
considerably increases the CPU runtime.

Thus, to solve the system of equations (1) and (2), an SPH code based on a 6th
order Runge-Kutta ordinary differential equation (ODE) algorithm in MATLAB was
used throughout. This package is currently implemented on an SGI Altix UV 100
(MatriCS Platform) at UPJV.

A damping term is added to each equation of the system of equations (1) in
order to simulate a radial pseudoviscosity effect when the density is larger
than a fixed threshold (the inner region). This term has the simple form

$-{\alpha{}}_{i}[{\kappa{}}_{i}\frac{d{{\boldsymbol{\sigma{}}}}_{i}}{dt}.\frac{{{\kappa{}}_{i}{\boldsymbol{\sigma{}}}}_{i}}{\left\|{}{{\kappa{}}_{i}{\boldsymbol{\sigma{}}}}_{i}\right\|{}}]\frac{{{\kappa{}}_{i}{\boldsymbol{\sigma{}}}}_{i}}{\left\|{}{{\kappa{}}_{i}{\boldsymbol{\sigma{}}}}_{i}\right\|{}}$
(3)

where the damping coefficient
${\alpha{}}_{i}=\alpha{}\sqrt{\frac{4GM}{{\left\|{}{{\kappa{}}_{i}{\boldsymbol{\sigma{}}}}_{i}\right\|{}}^{3}}}$
($M$ is defined in the following, and $\alpha{}$ is a numerical coefficient,
possibly adjusted, which we have taken equal to $1$)444From a physical point
of view, we can still imagine that when the system of particles shrinks, the
energy of the free fall is rapidly transformed in infrared radiation which are
on the spot evacuated from the system. .

The characteristic time is taken to be equal to the free fall of an outer
particle, $\sim{}{10}^{8}$ years (reference unit taken for the time in figures
1, 3, and 6). We start with an initial discoidal configuration of radius
$\sim{}10\ r_{G}$, where ${r}_{G}\sim{}10\ kpc$ (reference unit taken for the
distances in figures 1, 3, and 6), this radius is estimated by a terrestrial
observer provided with a coefficient $\kappa{}={\kappa{}}_{E}$. The total mass
of gas $M$ is set to be equal to ${10}^{44}\ g\ $(this mass is only baryonic
in the $\kappa$-model). The initial disk configuration of thickness
$\sim{}0.6\ kpc,$ is assumed to be cold and homogeneous. The mean density
$\bar{\rho{}}$ is automatically recalculated after 10 time steps. This means
that the density is obtained by counting the number of particles of mass $m$
in a volume of ${(0.6\ kpc)}^{3}$. Eventually, an initial low shear velocity
field of the form $v_{x}=$ 10 $\frac{{{\kappa{}}_{E}\sigma{}}_{x}}{10}$
${{(\kappa{}}_{E}\sigma{}}_{x}\ $in $kpc$ and $v_{x}$ in $km/s$) is assumed to
pervade the disk. The coefficient $\kappa$ is a constant $={\kappa{}}_{ini}$
in the initial, assumed to be homogeneous, distribution of baryonic matter and
is, for example, $\frac{{\kappa{}}_{ini}}{k_{E}}\sim{}0.15\ $, which appears
to be a reasonable value according to trial. This is for the physics content;
then, the equations are properly normalized.

The Cartesian coordinates $(x,y)$ are used. An initial uniform distribution of
$10\ 000=100\times{}100$ of identical particles of mass $m$ are selected555The
study should be resumed with a larger number of particles, but the excessive
CPU time due to the self-consistent process currently precludes this.. To
distinctly exhibit the asymmetric substructures, the quantity reported in
figures 3.a, 3.b and 6.a, 6.b is not directly $\bar{\rho{}}\left(x,y\right)$
but rather the difference

$\delta{}\bar{\rho{}}(x,y)=\bar{\rho{}}(x,y)-\bar{\rho{}}\left(\sqrt{x^{2}+y^{2}}\right)$
(4)

where $\bar{\rho{}}\left(\sqrt{x^{2}+y^{2}}\right)$ is the density averaged
over the polar angle. The latter gives a circular distribution in the galactic
plane. Eventually, only one particle out of 10 is reported in the figures to
clearly distinguish the substructures.

##  4 Results

When the simulation is running, the disk rapidly shrinks by self-gravity, and
a weak spiral substructure naturally quickly appears, just after $\sim{}6\
{10}^{8}$ years666This characteristic time is almost twice as long in the dark
matter model, where the shrinkage is lower. The galaxies form much more
rapidly with the $\kappa$-model than with the dark matter paradigm.. This weak
substructure is shown in figure 1, with the axisymmetric background excluded.
In figure 2, the trajectories of some individual particles are also displayed,
and we can observe a rapid and chaotic falling of these particles toward the
center777The irregularities seen along on the trajectories are due to close
encounters between particles over the free fall process. During this process,
the particles that are ejected from the galaxy are no longer taken into
account. and the consecutive circularization of the orbits.

Figure 1 Formation of the spiral substructure

|
---|---

Figure 2

The axisymmetric disk on which a spiral pattern is superimposed then
stabilizes; the system becomes quasi-steady after approximately ${2\ 10}^{9}$
years ($t=20$). An impressive large-scale coherent ”grand design” galaxy
appears. This is evidenced in figures 3.b and 3.c.

Figure 3 The quasi-permanent spiral in the $\kappa$-model

That the spiral pattern and the central bulge are self-maintained with time is
also clearly seen in these figures. This result is quite remarkable, and this
is not the case within the framework of the dark matter paradigm, where the
spiral substructures are just transient phenomena over a few rotational
periods of the galaxy. However, some local and relatively strong deformations,
such as those due to self-gravity, persist even after two milliards of years,
and the galaxy after formation does not rotate as a rigid body, in contrast to
what one might think in the framework of the $\kappa$-model. There remain a
very large number of stars that exhibit elliptic, not fully circularized,
motions, a phenomenon that also contributes to the deformations. In the
framework of the $\kappa$-model, a galaxy is a ”living” object, and it should
not be seen as a perfect and immutable wheel. Accordingly, the substructure
spiral is always evolving, albeit much more slowly than in other contexts,
such as, for instance, in density wave theory, where the lifetime of the
spiral substructure appears very short unless the density wave is constantly
fed.

In parallel, a quasi-flat rotation curve for the velocities is obtained
(figure 4; $r={\kappa{}}_{E}\sigma{}$). Let us note that for practical
(observational) reasons, we do not report the velocities in this figure but
rather the measured shift of the frequencies $\nu{}$. Let us specify that the
radial velocity measurements supply the true velocities (in fact, the radial
part), i.e., those measured by an inertial observer on site888The curve that
is displayed is not properly covered for $r\leq 0.3$. A three dimensional
model for the bulge is needed. Curve(2) is obtained from the mass density
derived by resolving equations (1) and (2) and then recalculating the
velocities.. We have also reported Keplerian velocity curve (2) in the same
figure, which makes $\kappa{}=\kappa_{E}$ everywhere. However, for $r\gtrsim
5,\ \kappa{}$ becomes constant again, and rotation curve (1) falls off in a
Keplerian manner (not shown in the figure).

Figure 4 Galactic rotation curve: |
---|---
(1) $\kappa$-model (2) pure Newtonian model |  Figure 5 Variation in $\kappa_{E}/\kappa{}$ along a galactic radius

###  4.1  The resolution of the winding dilemma

The winding problem has always been recurrent in the simulation of ”grand
design” galaxies (Shu, 2016) and remains partially unsolved. However, against
all odds, the two interlinked effects, i.e., the flatness of the rotation
curve and the almost steady nature of the spiral design, can easily be
understood in the framework of the $\kappa$-model. Let us define two inertial
observers $A$ and $B$ situated along a galactic radius, each of them measuring
the (true) velocity of one particle, resp. $M$ for $A$ and ${M}^{\prime}$ for
$B$, located near them. We have

$v_{A}\left(M\right)=v_{B}\left(M^{\prime}\right)$

expressing the flatness of the rotation curve as measured by spectroscopy999In
the Universe, the radial velocities are estimated from spectroscopic
measurements. These velocities are the same as those measured by a local
inertial observer. However, the tangential velocities are deduced from proper
motions and parallax measurements. These apparent quantities are linked to the
terrestrial observer who measures them by trigonometry, postulating a unique,
and most likely imaginary, background. These two methods are very different,
and this difference is expressed in the framework of the $\kappa$-model. The
first method (spectroscopy) supplies true (radial) velocities, whereas the
second method (observations of proper motions) supplies apparent (tangential)
velocities.).

However, $v_{A}\left(M\right)={\kappa{}}_{A}\sigma{}(M)\dot{\theta{}}(M)$ and
$v_{B}\left(M^{\prime}\right)={\kappa{}}_{B}\sigma{}(M^{{}^{\prime}})\dot{\theta{}}(M^{\prime})$,
where $\theta{}$ is the polar angle defined from a radial baseline with origin
at the galaxy center (in the $\kappa$-model, a given direction is well
identified and is the same for all observers). Following figure 3, the
coefficient $\kappa{}$ (or more rigorously, the measurable ratio
$\frac{\kappa}{\kappa_{E}}$) is approximately proportional to
$\frac{1}{\kappa_{E}\sigma{}}$ in the outer regions of a galaxy101010More
rigorously, this coefficient is a mean $\kappa{}$ obtained by curve fitting
(the curve in figure 5). ; thus, we obtain

$\dot{\theta{}}(M)=\dot{\theta{}}(M^{\prime})$ (5)

for any pair of points located along a galactic radius. The points remain
steadily aligned with time along the galactic radius (a straight line plotted
from the galactic center). This noteworthy outcome enables us to now
understand why we can simultaneously observe a quasi-steady ”grand design”
substructure and a flat rotation curve, two effects that seemingly to
contradict each other. Let us note, however, that the persistence of the
spiral substructure is not absolute and that ultimately, the latter may slowly
distort with time. The main reason for this is that the orbits of stars are
not perfect circles but rather ellipses with slight eccentricities. On the
other hand, the factor $\kappa{}$ fluctuates with the variation in the density
within the spiral substructure.

In this respect, comparison with the results obtained in the framework of the
dark matter paradigm is quite interesting. In the latter simulations, we
obviously take the same initial conditions for the baryonic matter. We can
conclude from this comparison that the $\kappa$-model yields something similar
to the results from the dark matter paradigm in many aspects but without dark
matter; however the spiral substructure is self-maintained and is much more
impressive.

To perform this comparison, we add a halo of dark matter with a density
distribution proportional to $r^{-2}$ in the outer regions (halo mass) and we
take $M_{DM}\sim{}10\ M_{B}$ and $\kappa{}=\kappa_{E}{}$ everywhere, i.e., the
orthodox strategy for rehabilitating the common background, where we can set
$r=\kappa_{E}\sigma{}$. Then, with the dark matter superimposed on the
baryonic component taken into account, we can see that a spiral pattern with a
central bar effectively appears to be essentially due to self-gravity, but
there are now are now multiple, transient and discontinuous outer arms with
very short lifetimes (figures 6.a, 6.b).

Figure 6 The transient spiral in the dark matter model

In any case, a flocculent galaxy pattern seems to emerge from the simulations,
but definitely not a permanent ”grand design”. Certainly, at $t=20$, figure 6a
clearly mimics a spiral-type structure but with four arms instead of two;
however, at $t=30$, the situation becomes confusing; the galaxy core is
getting denser, and the arms are more spiraled and are unrecognizable. This
effect is well known, and the same conclusion is drawn by the other published
N-body simulations within spherical dark matter models, even those using very
sophisticated SPH, grid-based procedures, including complicated physics such
as cooling, star formation, and energy injection from supernovae (Guedes,
Callegari, Madau and Mayer, 2011). We can compare figure 6 of the present
paper and figure 2 of (Guedes, Callegari, Madau and Mayer, 2011) obtained
after a series of state-of-the-art simulations; on the one hand, we see that
they are fairly similar, but on the other hand, we are far from the set
objective for a realistic ”grand design” galaxy. Obviously, additional aspects
can be artificially introduced in the calculations such as a permanent density
wave within the gas driven by a rotating central bar (Shu, 2016), a triaxial
distribution of dark matter forming a rotating halo with angular momentum
(Martinez-Medina, Bray and Matosa, 2015), or a Yukawian gravitational
potential (Brandao and de Araujo, 2012). However, even though all these
proposals are interesting, a solid experimental/observational basis is still
lacking. The interaction with a galaxy companion to sustain a ”grand design”
substructure may be much more realistic (Dobbs, Theis, Pringle. and Bate,
2010), even though there exist cases of ”grand design” galaxies without the
presence of a companion.

By contrast, starting from an initial homogeneous cold gas pervaded with a
shear velocity field, it is remarkable that, in a very simple way, the
$\kappa$-model naturally directly leads to a quasi-steady ”grand design”
spiral galaxy with very clearly well-formed and distinct arms. A bar also
appears in the central region, and the arms are formed by self-gravity, which
is much higher in the outer regions (weak mean density) than in the inner
regions (high mean density) of the galaxy. Another noticeable point is that
the spiral substructure becomes stronger with time instead of the weakening
phenomenon generated in other models with dark matter or even in MOND or in
MOG. These results provide strong support for the self-consistent
$\kappa$-model, the important conclusion being that the knowledge of the sole
distribution of visible baryonic matter is sufficient to understand the
dynamics of a galaxy and that the introduction of exotic particles is not
needed.

##  5 Two selected examples

The preceding simulation can, however, appear to be a toy model. Most
interesting is the practical application. For this, we choose two examples, a
high luminosity galaxy (New General Catalogue of Nebulae and Clusters of Stars
(NGC) 6946) and a low luminosity galaxy (NGC 1560). These two examples are
taken from the review of Famaey and McGaugh (2012, see the references therein
for the original data sources).

Figure 7 Surface density profiles (stars + gas) of two galaxies: the high
surface brightness (HSB) spiral NGC 6946 and the low surface brightness (LSB)
galaxy NGC 1560 (from Famaey and McGaugh, 2012, figure 13)

We assume here that the galaxies to be studied are stabilized and that the
orbits of the stars are circular. The Newtonian velocities $v_{Newt}$ could
then be obtained from equation (1), making $\kappa_{i}\equiv\kappa_{E}$, ,
$\forall i$ in it111111We can see that in equation (1), the factor
$(\frac{\kappa_{E}}{\kappa_{i}})$ is factorizable in front of the sum over
$j$. This sum expresses the Newtonian force acting on a particle with index
$i$.. The true velocities $v$ (measured by spectroscopy) are then deduced from
the Newtonian velocities $v_{Newt}$ by the following: ”magnification” relation
(this deduction is made in appendix B)

$v={\left(\frac{{\kappa{}}}{\kappa{}_{E}}\right)}{\left(\frac{{\kappa{}}_{E}}{\kappa{}}\right)}^{\frac{3}{2}}v_{Newt}={\left(\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}\right)}^{\frac{1}{2}}{\left(\frac{{\kappa{}}_{0}}{\kappa{}}\right)}^{\frac{1}{2}}v_{Newt}$
(6)

where

$\frac{{\kappa{}}_{0}}{\kappa{}}=1+Ln(\ \frac{{\Sigma{}}_{0}}{\Sigma{}}\
\frac{\delta{}}{{\delta{}}_{0}})$ (7)

Equation (7) directly derives from equation (2), where $\Sigma{}$ represents
the surface density and $\delta{}$ the mean thickness of the matter (stars +
gas) along the line of sight. The index $0$ represents the maximum value of
$\Sigma{}$ for a given galaxy. Assuming a uniform thickness, we take
$\delta{}={\delta{}}_{0}=Const$ throughout the calculations. Figure 8 displays
the ratio $(\frac{{\kappa{}}_{0}}{\kappa{}})^{\frac{1}{2}}$ for the two
selected galaxies calculated with equation (7).

Figure 8 Variation in $(\frac{{\kappa{}}_{0}}{\kappa{}})^{0.5}$ as a function
of the radial distance $R$

The global ratios $\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}$ are unknown, but we
can attempt to supply an estimate by again employing equation (2)121212The
relation that has then been used is
$\frac{{\kappa{}}_{1}}{{\kappa{}}_{2}}=1+Ln(\
\frac{{\Sigma{}}_{1}}{{\Sigma{}}_{2}}\ \frac{{\delta{}}_{2}}{{\delta{}}_{1}})$
respecting the condition
${\Sigma{}}_{1}{\delta{}}_{2}>{\Sigma{}}_{2}{\delta{}}_{1}$.. With
${\Sigma{}}_{\odot}=60\ M_{\odot}\ {pc}^{-2}$ (from Famaey, and McGaugh, 2012,
figure 19; Moni Bidin et al, 2012, figure 5), we find that
${\left(\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}\right)}^{0.5}=0.36$ for NGC 6946
and $1.24$ for NGC 1560. We, however, note that a better fit of the $v$-curves
is realized by taking
${\left(\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}\right)}^{0.5}=0.46$ (instead of
$0.36$) for NGC 6946 and $1.30$ (instead of $1.24$) for NGC 1560. In any case,
the calculation of these global magnifications factors is certainly vitiated
by various biases.

For instance, we know that the inclination angle $i$ of a galaxy can strongly
impact the velocity curves $\left[26\right]$. For NGC 6946, a variation of
$7^{\circ}{}$ in the inclination angle, starting from $i=38^{\circ}{}$ for
this galaxy $\left[26\right]$, is equivalent to increasing the factor
${\left(\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}\right)}^{0.5}$ from the value
predicted by the $\kappa$ model ($0.36$) to its fit value ($0.46$).
Eventually, even though to a lesser extent, the distance of this galaxy is
highly uncertain (Elridge and Xiao, 2019). This important parameter intervenes
in the estimate of the absolute luminosity, subsequently in that of the
surface density, and eventually in that of the ratio
${\left(\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}\right)}^{0.5}$.

The estimate of ${\Sigma{}}_{\odot}$ can also play a role. Thus, for NGC 1560,
passing from ${\Sigma{}}_{\odot}=60$ to $70\ M_{\odot}\ {pc}^{-2}$ increases
the numerical value for
${\left(\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}\right)}^{0.5}$ from $1.24$ to
the value taken for the fit in figure 10, $1.30$ (with an insignificant change
for NGC 6946). Note that a value $\sim 90\ M_{\odot}\ {pc}^{-2}$ is given in
[Sofue, Honma and Omodaka, 2009, Table 4]. However, this value is well above
the estimates of other authors (Famaey and McGaugh, 2012; Moni Bidin et al,
2012) and is very likely overestimated.

Fortunately, while some uncertainty is associated with the global ratio
${\left(\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}\right)}^{0.5}$, the resulting
effect is a global shift of the mean height of the velocity curve131313In the
literature, the mean height of the observational velocity curve can also vary
substantially; for instance, for the Milky Way, we can compare (Sofue, Honm
and Omodaka, 2009, figure 1) and (McGaugh, 2016, figures 5,6], which does not
heavily affect the details of the curve, such as bumps, wiggles, or the
typical flatness in the outer regions. The final results are presented in
figure 9 (NGC 6946) and figure 10 (NGC 1560).

Figure 9 NGC 6946: the observed rotation curve (black circles) is displayed
together with those predicted by the baryonic mass model (black line), MOND
(red line) and $\kappa{}$-model (blue line)

By examining the curves reproduced in figure 9, we can see that MOND and the
$\kappa{}$-model give very similar values for the velocities between $6>r>13\
kpc.$ For $r<6\ kpc$, MOND is slightly better than the $\kappa{}$-model, in
contrast to $r>13\ kpc$, where the MOND curve does not initiate a slow
decrease, while this decrease is very noticeable in the curve of the
$\kappa{}$-model. However, the differences between MOND and the
$\kappa{}$-model are small.

Regarding galaxies of lower baryonic surface densities, it is well known that
Newtonian prediction is very poor everywhere, even at smaller radii [29]. The
galaxy NGC 1560 illustrates this situation, whereas both MOND and the
$\kappa{}$-model are fairly consistent with the observed curve (figure 10).

Figure 10 NGC 1560: the observed rotation curve (black circles) is displayed
together with the theoretical curves predicted by the baryonic mass model
(black line), MOND (red line) and $\kappa{}$-model (blue line)

The results from both MOND and the $\kappa{}$-model are superimposed well
between $3>r>6\ kpc$, even though outside this interval, MOND seems slightly
better. However, an optimal overlap of the $\kappa{}$-model versus the
observation could eventually be realized by an adjustment of the thickness
$\delta{}$. Thus, the $\kappa{}$-model could predict that the thickness along
the line of sight, $\delta{},$ is larger than ${\delta{}}_{0}$ for $1>r>3\
kpc$ by a factor $1.5$ and conversely smaller by the same factor for $6>r>10\
kpc$.

However, let us note that a comparison with the observational curve derived
from other sources (Sánchez-Salcedo and Hidalgo-Gámez, 1999) can lead to
another interpretation of the results. The x-axis of figure 10 has been
adjusted for figure 11. In this case, we can see that within the intervals
$0>r>3\ kpc$ and $8>r>10\ kpc$, the observational data are now located in the
area bounded by MOND and the $\kappa$ model, namely, resp. above by MOND and
below by the $\kappa$-model for $0>r>3\ kpc$ and the opposite for $6>r>10\
kpc$.

Figure 11 NGC 1560: Another observed rotation curve (black circles), derived
from (Sánchez-Salcedo and Hidalgo-Gámez, 1999), is displayed together with the
theoretical curves predicted by MOND (red line, reproduced from Sánchez-
Salcedo and Hidalgo-Gámez (1999)) and the $\kappa{}$-model (blue line)

##  6 AGC 114905: an ultradiffuse galaxy without dark matter?

The ultradiffuse galaxies are LSB galaxies with an extended light distribution
(Conselice, 2018; Mancera Piña et al, 2021). Belonging to this peculiar
category of galaxies, AGC 114905 seems to pose a challenge to the standard
dark matter model as well as to MOND. Figures 12 and 13 show the high-
resolution HI interferometric observations obtained by Mancera Piña et al
(2021) for AGC 114905.

Figure 12 SMD profile (stars + gas) of AGC 114905 (from figure 1 of Mancera
Piña et al, 2021).

Figure 13 The rotation curve of AGC 114905 (from figure 5 of Mancera Piña et
al, 2021). The red points show the observational data. The inclination angle
is equal to $26^{\circ}$. The dark matter halo is represented by a red line,
MOND by a green dashed line and the $\kappa$-model by a blue line.

As shown in figure 13, we note that MOND and the $\kappa$-model give very
similar profiles. The agreement between MOND and the $\kappa$-model, already
identified in the preceding section, naturally underlines that the domain of
weak accelerations (MOND) is related to the domain of weak mean densities (the
$\kappa$-model).

Nevertheless, we see that all three theoretical models, dark matter, MOND and
$\kappa$, fail to represent the observational curve. Mancera Piña et al (2019)
suggest that a drastic modification of the inclination angle, from
$26^{\circ}$ within the range $10^{\circ}-15^{\circ}$, would be needed in
order to find agreement between the dark matter or MOND prediction and the
observational profile of AGC 114905. The same reasoning can be applied to the
$\kappa$-model. These authors further stated that a radially varying
inclination could potentially alleviate the tension between the rotation curve
shapes, especially for MOND. However, Mancera Piña et al, (2021) also specify
that inclinations as low as $10^{\circ}-15^{\circ}$ are very likely
inconsistent with the observational data.

Thus, even though the discrepancy between the $\kappa$-model and the
observational data could be reduced by lowering the inclination angle, Mancera
Piña et al (2021) find this proposal unreasonable. However, apart from the
inclination parameter, the thickness of the galaxy plays a role in the
$\kappa$-model because the latter factor influences the global magnification
ratio ${\left(\frac{{\kappa{}}_{E}}{{\kappa{}}_{0}}\right)}^{0.5}$ (equation
(7)). For instance, a drastic reduction in the mean thickness $\delta$ by a
factor $10$ (at constant $\Sigma$ given this quantity is measured) would bring
the $\kappa$-model curve within the error bars of the observational data
(figure 13). Thus, rather than seeing AGC 114905 as a galaxy deprived of dark
matter, such an explanation, i.e., by considering the mean thickness, would
lead to envisaging this galaxy as a very smeared and hence ultrathin object.

##  7 The Milky Way

The curve of rotation of the Milky Way has been extensively studied, but the
observational curves appear to strongly vary with the type of disk tracer. We
can compare, for instance, figure 2 of Bhattacharjee, Chaudhury and Kundu
(2014), figure 11 of Huang et al. (2016) or figure 1(d) of Sofue (2020), where
we can see that the data exhibit extensive scattering. Here, our aim is to fit
the panel of observational rotation curves of the Milky Way with the
$\kappa{}-$model. For that, the baryonic surface mass density (SMD) is needed.
This is the sole component given that we exclude the dark matter halo. We use
two $SMD$ profiles, one from Famaey and Mc Gaugh (2012$)$ and the other very
close to the $SMD$ profile of Sofue (2020, figure 2).

  * •

The SMD for the distances $8<r<12\ kpc$

Figure 14 reproduces the SMD of Famaey and Mc Gaugh (2012, figure 29). Other
profiles are also supplied in Mc Gaugh (2016), but these profiles are mostly
the same except for a few details. The results are displayed in figure 15. We
adopted a global magnification factor of $0.46$ (against $0.43$ for NGC 6946).
We observe that both MOND and the $\kappa{}-$model adequately reproduce the
sequence of wiggles of observational curve. These wiggles, which are already
present in the Newtonian curve, are found in the observational curve following
Sencisi’s rule (Sancisi, 2004). In the framework of the $\kappa{}-$model, the
interpretation of this effect is that the observational curve is simply a
magnification of the Newtonian curve, a phenomenon that is very well
illustrated here. Let us note that this patent fact remains unexplained in the
framework of dark matter models.

Figure 14 The SMD of the Milky Way between $8$ and $12\ kpc$ (from figure 29
of Famaey and Mc Gaugh, 2012)

Figure 15 The observed rotation curve of the Milky Way between $8$ and $12\
kpc$ (from figure 29 of Famaey and Mc Gaugh, 2012). The MOND curve is
represented by the green dashed line, and the profile issued from the
$\kappa$-model is displayed by the blue line.

  * •

The SMD profile for the distances $0<r<100\ kpc$

The SMD profile for the distances $0<r<100\ kpc$ is taken from Sofue (2020).
It is reproduced in figure 16. With this SMD, the velocity of the Sun
$v_{\bigodot}$, located at $r_{\odot}=8\ kpc$ from the galactic center, is
$\sim{}240\ kms^{-1}$ and the total mass of the Miky Way without dark matter
(i.e. bulge + disk) is $\sim{}\ 7\ {10}^{10}\ M_{\odot{}}$. The latter value
is very close to that one obtained elsewhere (Licquia and Newman, 2015).

Figure 16 The SMD of the Milky Way between $0$ and $100\ kpc$ obtained from
figure 3 of Sofue (2020) deprived of the dark matter halo

The Newtonian profile in figures 17 and 18 is produced from the SMD (figure
16) following the same methodology as that described in Sofue (2020).

In figure 17, the Newtonian and $\kappa{}-$ model curves are superimposed on
various observational curves obtained using various disk tracer samples for
$0<r<25\ kpc$ (figure 2 of Bhattacharjee, Chaudhury and Kundu, 2014). The
$\kappa$-model curve is produced by still using the same global magnification
factor (namely $0.46$) than in figure 15. The observed rotation velocity shows
obvious deviation from the predicted Newtonian $r^{-1/2}$ law, but the
$\kappa{}-$model curve fits the data, i.e., all error bars overlap the
$\kappa{}-$model curve. The error bars are, however, very large, which is
linked to the difficulty of accurately estimating the distances and measuring
the velocities from inside the Milky Way.

Figure 17 A collection of data for the rotation curve of the Milky Way between
$0$ and $25\ kpc$ obtained using various disk tracer samples (from figure 2 of
Bhattacharjee, Chaudhury and Kundu, 2014). The $\kappa$-model curve is
represented by the blue line.

The domain $0<r<100\ kpc$ is supplied in figure 18. This time, the Newton and
$\kappa{}-$model curves are superimposed on four dark matter profiles
calculated by Lin and Li (2019). Each of these dark matter profiles have two
free parameters. The observational data are taken from Huang et al. (2016). We
can see that the dark matter profiles and the $\kappa{}-$model profile fit
equally the data, except around $20-40\ kpc$, where the observational results
are above all the curves. We can conclude that the rotation curve can be
reproduced in the framework of the $\kappa{}-$model, that is, an overall value
of $\sim{}210-250\ kms^{-1}$ for $4-20\ kpc$ (with a higher margin of error),
beyond which the value declines steadily to $175\ kms^{-1}$ at $100\ kpc.$
Obviously, we can claim that the same conclusion is drawn from the dark matter
models, but the key difference is that in the latter case, a conspiracy is
needed between the dark matter halo and the baryonic distribution of matter to
reproduce the quasi-flat part of the rotation curve.

Figure 18 The observational rotation curve of the Milky Way between $0$ and
$100\ kpc$ compared to various dark matter profiles (from figure 2 of Lin and
Li, 2019). The $\kappa$-model curve is represented by a thick blue line.

None of the theoretical profiles seen in figure 17 and figure 18 take into
account the spiral arms and rings, which also affect the rotation velocity.
The presence of two prominent bumps on the rotation curve has been reported
(Huang et al., 2016). Let us note that these two bumps, located at $12-13$ and
$19-20\ kpc$, are not explained by the dark matter halo (figure 12 of Huang et
al, 2016). Huang et al. suggest that two quite massive caustic rings of dark
matter on the order of ${10}^{10}$ $M_{\odot}$ are needed to explain these
bumps (Duffy and Sikivie, 2008; Huang et al, 2016). Despite this interesting
proposal, we cannot exclude the fact that these bumps could be simply an
artifact of the measurements. We may then wonder whether the addition of such
structures composed of dark matter is somewhat artificial. As acknowledged by
Huang et al (2016), at present, the observational evidence linking the dips
seen in the rotation curve to hypothetical caustic rings of dark matter is
still marginal. Unfortunately, this type of methodology is typical of the dark
matter paradigm, which eventually suits all situations but which is ultimately
not really predictive. Nor should it be forgotten that the rotation curve of
the Milky Way still remains more poorly described than the curves of other
galaxies, where no such intriguing bumps are apparent. Thus, Sofue (2020), in
a review on the current status of the study of rotation curve of the Milky
Way, presents an SMD distribution that is fitted by taking into account just
the bulge, the disk composed of stars and gas, and the Navarro-Frenk-White
(NFW) dark halo with no mention of a hypothetical series of dark matter rings.
Ultimately, the $\kappa{}-$model is an even more economical solution, as it
takes in account only the baryonic matter; i.e., the Milky Way may be
conveniently parameterized by a bulge and a disk to produce an acceptable fit
of the observational curve as it is known today.

In conclusion, taking into account the various uncertainties regarding the
measured velocities, the surface density profiles, the inclination of the
galaxies and the estimate of the distances, the analysis of the individual
examples performed clearly shows that with the sole consideration of the
observed distribution of baryonic matter, both MOND and the $\kappa$-model can
satisfactorily predict the observed rotation curves of galaxies. Such a
prediction is not possible using the dark matter paradigm, unless we take a
fine-tuned distribution of dark matter especially suited to each type of
galaxy.

In addition, there is observational evidence that the peculiarities seen in
the Newtonian curve at a location are nicely transcribed to the observed curve
at the same location, as attested in figures 10 and 15. This is a rather
astonishing fact that has already been pointed out by other authors (Famaey
and McGaugh, 2012; McGaugh, 2016) and that is named Sancisi’s law (Sancisi,
2004)141414The mass discrepancy-acceleration relation (MDAR) formula (McGaugh,
2016) was a first attempt to quantify the phenomenon. Unfortunately, even
though very noteworthy, this relation is purely empirical and is not based on
a specific theoretical background.. Here again, this phenomenon seems to be
very difficult to mimic using the dark matter paradigm, given that the
gravitational force is a long-range force. Mass removal from a particular
location does not create a trough in the observed velocity curve at the same
location. An unlikely conspiracy between dark matter and baryonic mass is then
needed for the trough in the Newtonian curve to be replicated on the observed
velocity curve at the same place. This fact is clearly favors models such as
MOND and the $\kappa{}$-model, the latter assuming a ”simple” magnification of
the velocity curve produced on the spot from the baryonic model. Thus,
following a pair of equations (6 and 7), this magnification increases as the
mean density (or, in an equivalent manner, the factor $\kappa{}$) in the
galaxy weakens. This circumstance is very well illustrated when comparing a
high luminosity galaxy (e.g., NGC 6946), where the magnification is relatively
weak, and a low luminosity galaxy (e.g., NGC 1560), where the magnification is
strong.

Ultimately, let us note that in spiral galaxies, the surface brightness
profile typically varies in an exponential manner (Binney and Merrifield,
1998). Following the $\kappa$-model, this feature naturally implies a very
extended radial range with near-constant rotation velocity. A remarkable
example is Malin 1, which is a spiral galaxy exhibiting an extremely large low
surface brightness disk of gas that is five times wider than the Milky Way
(figure 19). Thus, Lelli, Fraternali, and Sancisi (2010) provide a flat
rotation curve up to $100\ kpc$ for Malin 1 using HI data. However, Malin 1 is
a relatively isolated galaxy, and it is very likely that the flat profile does
not continue forever for any galaxy, especially when in the outermost regions
of the disk, interaction with the environment, i.e., the influence of a
neighboring galaxy, begins to be detected.

Figure 19 The observed rotation curve of Malin 1 between $0$ and $100\ kpc$
compared to the dark matter, MOND and $\kappa$-model profiles (from figures 7
and 9 of Lelli, Fraternali and Sancisi, 2010). The $\kappa$-model curve is
represented by a blue line.

##  8 Conclusion

The $\kappa$-model is a nice application of the notion of the - here apparent
- asymmetric distance well known in mathematics that essentially leads to
privileging local physics rather than global physics. Undoubtedly, the
$\kappa$-model is no longer the whole of the story and might even ultimately
be a pure mathematical construction or a simple exercise of the mind.
Nevertheless, regardless of these considerations, it should not be discarded
too hastily, mainly due to the strong support of its conclusions, i.e., the
unification of some properties of galaxies under a sole umbrella, mainly an
unexpected, but however possible, correlation between the appearance of a
quasi-steady spiral substructure and the flat rotation curve in galaxies.
Another greatly interesting aspect of the $\kappa$-model is that it is not
artificially parametrized and is sufficient by itself, with the presence of
only baryonic matter, which is a major issue. In this sense, it is more easily
refutable than any other theory postulating the existence of a large variety
of exotic particles with unknown characteristics that can be changed at will.

An extension of this work is required. For elliptical galaxies and globular
clusters, a three-dimensional model is needed. However, a forthcoming analysis
of the data obtained for both the train wreck and bullet clusters should
enhance the understanding of the density-dependent character of the measured
gravitational force, which has been assumed. Perhaps the $\kappa$-model could
also help to eliminate the necessity of dark energy. After the formation of
the galaxies, the mean value of the factor $\kappa{}$ increases151515The
harmonic mean of $\kappa$ for a set of particles (a spiral galaxy, an
elliptical galaxy or a cluster of galaxies) can be defined by
$\frac{\kappa_{E}}{<\kappa>}=\sqrt{\frac{\int{\kappa_{E}^{3}}d^{3}{\sigma}\
\rho(\kappa_{E}{\boldsymbol{\sigma}})\
(\frac{\kappa_{E}}{\kappa})^{2}}{{\int{\kappa_{E}^{3}}{d^{3}{\sigma}}\
\rho({\kappa_{E}\boldsymbol{\sigma}})}}}$ The two integrations under the
square root must be performed over the volume containing the set of
particles.. This phenomenon leads to a slow decline in the mean gravitational
forces between the galaxies, which could eventually be interpreted as an
apparent acceleration of the Universe.

Data availability statement: The author confirms that the data supporting the
findings of this study are available within the article and the reference
list.

Acknowledgements: I would like to thank the reviewer for his fruitful
comments, which helped to improve the manuscript.

Conflicts of Interest: The author declares no conflicts of interest.

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## Appendix A

Here, we explain the origin of form (1) for the dynamics equation. We
introduce the formal action

$S=\int
dt\left[\frac{1}{2}m\left(\frac{{d{\bf{M}}}}{dt}\right)^{2}-V({M})\right]$ (8)

where $m$ is the mass of a test particle and $V(M)$ the potential experienced
by this particle located at a given point $M$. An arbitrary variation
$\delta{}{\bf{M}}$ from $M$ gives

$\delta{}S=\delta{}\int
dt\left[\frac{1}{2}m\left(\frac{{d{\bf{M}}}}{dt}\right)^{2}-V\left(M\right)\right]=\int
dt\
\delta{}\left[\frac{1}{2}m{\left(\frac{{{d\bf{M}}}}{dt}\right)}^{2}-V\left(M\right)\right]$
(9)

$=\int dt\
\left[m\frac{d{\bf{M}}}{dt}\delta{}\left(\frac{{d{\bf{M}}}}{dt}\right)-\delta{}V\left(M\right)\right]$
(10)

To continue the calculations, we must now exchange $d$ and $\delta{}$.

The exchange of $d$ and $\delta{}$

Let three observers $A$, $B$ and $C$ be located in a plane (figure 12). We can
define this plane by imagining a common direction perpendicular to
${\bf{M}}_{0}{\bf{M}}_{1}$ and ${\bf{M}}_{0}{\bf{M}}_{2}$. This operation is
possible because any orientation is well defined in the $\kappa$-model. With
the help of this figure, we write161616Notation: The arrow $\longrightarrow{}$
indicates that a given observer measures the corresponding bipoint. The symbol
$\triangleq{}$ indicates a definition, and the symbol $\equiv{}$ signifies
that the two compared vectors have the same orientation and the same norm but
that each of them is seen by a distinct observer.

$d{\bf{M}}\triangleq{}{\bf{M}}_{0}{\bf{M}}_{1}[d{\boldsymbol{\sigma{}}}]|{}_{A}\equiv{}d{\bf{M}}_{\parallel{}}={\bf{M}}_{2}{\bf{M}}_{1\parallel{}}[d{\boldsymbol{\sigma{}}}]|{}_{C}\longrightarrow{}\kappa{}d{\boldsymbol{\sigma{}}}\
\ (a)$

The first expression signifies that the observer $A$ measures
${\bf{M}}_{0}{\bf{M}}_{1}$ and obtains $\kappa{}d{\boldsymbol{\sigma{}}}$ and
that observer $C$ measures ${\bf{M}}_{2}{\bf{M}}_{1\parallel{}}$ and obtains
the same value. Other very similar relations follow

$\delta{}{\bf{M}}\triangleq{}{\bf{M}}_{0}{\bf{M}}_{2}\left[\delta{}{\boldsymbol{\sigma{}}}\right]|{}_{A}\equiv{}{\delta{}{\bf{M}}}_{\parallel{}}\triangleq{}{{\bf{M}}_{1}{\bf{M}}_{2\parallel{}}\left[\delta{}{\boldsymbol{\sigma{}}}\right]|{}}_{B}\longrightarrow{}\kappa{}\delta{}{\boldsymbol{\sigma{}}}\
\ (b)$

${{\bf{M}}_{2}{\bf{M}}_{4}^{"}\
\left[d{\boldsymbol{\sigma{}}}^{\prime}\right]|{}}_{C}\longrightarrow{}(\kappa{}+\delta{}\kappa{})d{\boldsymbol{\sigma{}}}^{\prime}\equiv{}{{\bf{M}}_{2}{\bf{M}}_{4}\left[d{\boldsymbol{\sigma{}}}+\delta{}d{\boldsymbol{\sigma{}}}\right]|{}}_{A}\longrightarrow{}\kappa{}\left(d{\boldsymbol{\sigma{}}}+\delta{}d{\boldsymbol{\sigma{}}}\right)\
\ (a^{\prime})$

${\bf{M}}_{1}{\bf{M}}_{4}^{{}^{\prime}}\
\left[\delta{}{{\boldsymbol{\sigma{}}}^{\prime}}\right]|{}_{B}\longrightarrow{}(\kappa{}+d\kappa{})\delta{\boldsymbol{\sigma{}}}^{\prime}\equiv{}{{\bf{M}}_{1}{\bf{M}}_{4}\left[\delta{}{\boldsymbol{\sigma{}}}+d\delta{}{\boldsymbol{\sigma{}}}\right]|{}}_{A}\longrightarrow{}\kappa{}\left(\delta{}{\boldsymbol{\sigma{}}}+d\delta{}{\boldsymbol{\sigma{}}}\right)\
\ (b^{\prime})$

We must remark that ${d{\bf{M}}}_{\parallel{}}$ is $d{\bf{M}}$ parallelly
displaced with respect to itself respecting the conservation of the length
during the transport (likewise for ${\delta{}{\bf{M}}}_{\parallel{}}$ vs
$\delta{}\bf{M}$). The observer $C$ located at ${{{M}}}_{2}$ sees
${d{\bf{M}}}_{\parallel{}}$ exactly as observer $A$ located at ${M}_{0}$ sees
$d{\bf{M}}$; thus, $d\bf{M}$ and ${d{\bf{M}}}_{\parallel{}}$ have the same
orientation and the same norm but their origins are distinct. However, these
two vectors are perceived differently by observer $A$ located at $M_{0}$.
Figure 20 is the projection of the full set of vectors on the background of
this observer. Let us note that ${\bf{M}}_{0}{\bf{M}}_{1}$ is the real path
and ${\bf{M}}_{2}{\bf{M}}_{4}^{"}$ is the corresponding varied path. Now, we
set

${d\delta{}{\bf{M}}\triangleq{}{\bf{M}}}_{2\parallel{}}{\bf{M}}_{4}^{{}^{\prime}}\
\ \ \ \ \
{\delta{}d{\bf{M}}\triangleq{}{\bf{M}}}_{1\parallel{}}{\bf{M}}_{4}^{"}$

Figure 20 Diagram of real and virtual paths: the thick full segment represents
the real path, the virtual path is indicated by a thick-dotted line.

After subtraction $({a}^{\prime})-(a)$ and $({b}^{\prime})-(b)$, we have

${\bf{M}}_{1}{\bf{M}}_{4}^{{}^{\prime}}\
\left[\delta{}{{\boldsymbol{\sigma{}}}^{\prime}}\right]|{}_{B}\longrightarrow{}(\kappa{}+d\kappa{})\delta{\boldsymbol{\sigma{}}}^{\prime}\equiv{}{{\bf{M}}_{1}{\bf{M}}_{4}\left[\delta{}{\boldsymbol{\sigma{}}}+d\delta{}{\boldsymbol{\sigma{}}}\right]|{}}_{A}\longrightarrow{}\kappa{}\left(\delta{}{\boldsymbol{\sigma{}}}+d\delta{}{\boldsymbol{\sigma{}}}\right)\
\ (b^{\prime})$

${\bf{M}}_{1\parallel{}}{\bf{M}}_{4}^{"}\left[{\delta{}d{\boldsymbol{\sigma{}}}^{\prime}}\right]|{}_{C}\
\ \ \longrightarrow{}\kappa{}\delta{}d{\boldsymbol{\sigma{}}}\ \ \ \ \ \ \ \ \
\ \ \ \
{\bf{M}}_{2\parallel{}}{\bf{M}}_{4}^{{}^{\prime}}\left[d\delta{}{{\boldsymbol{\sigma{}}}^{\prime}}\right]|{}_{B}\longrightarrow{}\kappa{}d\delta{}{\boldsymbol{\sigma{}}}$

We naturally have
$\delta{}d{\boldsymbol{\sigma{}}}=d\delta{}{\boldsymbol{\sigma{}}}$, thus
(omitting the indexes)

$d\delta{}{\bf{M}}=\delta{}d{\bf{M}}$

Let us specify, however, that for observer $A$ located at $M_{0}$, the vectors
$d\delta{}{\bf{M}}$ and $\delta{}d{\bf{M}}$ (projected on the proper
background) are parallel, but their (apparent) lengths are different, resp.
$\frac{\kappa{}}{\kappa{}+d\kappa{}}d\delta{}\boldsymbol{\sigma{}}$ and
$\frac{\kappa{}}{\kappa{}+\delta{}\kappa{}}\delta{}d\boldsymbol{\sigma{}}$;
for observer $A$,
${\bf{M}}_{3}{\bf{M}}_{4}[d\delta{}\boldsymbol{\sigma{}}]\longrightarrow{}\kappa{}d\delta{}\boldsymbol{\sigma{}}$,
even though these three vectors differ only by a small third order term. Let
us remark that
$\left(\kappa{}+\delta{}\kappa{}\right)d{\boldsymbol{\sigma{}}}^{\prime}-\kappa{}d\boldsymbol{\sigma{}}=$
$\kappa{}\left(d{\boldsymbol{\sigma{}}}^{\prime}-d{\boldsymbol{\sigma{}}}\right)+\delta{}\kappa{}d\boldsymbol{\sigma{}}^{\prime}=\kappa{}\delta{}d{\boldsymbol{\sigma{}}}^{\prime}+\delta{}\kappa{}d{\boldsymbol{\sigma{}}}^{\prime}.$
This quantity is equal to $\kappa{}\delta{}d\boldsymbol{\sigma{}}$ (in both
orientation and norm), but
$\kappa{}\delta{}d{\boldsymbol{\sigma{}}}^{\prime}+\delta{}\kappa{}d{\boldsymbol{\sigma{}}}^{\prime}$
(origin $M_{1\parallel{}}$) is evaluated by $C$, and
$\kappa{}\delta{}d\boldsymbol{\sigma{}}\ $(origin $M_{3}$) is evaluated by
$A$.

After exchanging $d$ and $\delta{}$ in equation (10), we obtain

$\int dt\
\left[m\frac{d{\bf{M}}}{dt}\frac{d}{dt}\delta{}{\bf{M}}-{\boldsymbol{\nabla{}}}_{M}V\left(M\right)\delta{}{\bf{M}}\right]$

$=\left.m\frac{{d{\bf{M}}}}{dt}\delta{}M\right|{}_{extremities}+\int dt\
\left[-\frac{d}{dt}\left(m\frac{{d{\bf{M}}}}{dt}\right)-{\nabla{}}_{M}V\left(M\right)\right]\delta{}M\
\ \ $ (11)

For a stationary value of $S$, we have $\delta{}S=0$. We also take
$\delta{}M=0$ at both extremities of the portion of the real trajectory.
Eventually, we obtain

$\frac{d}{dt}\left(m\frac{{d{\bf{M}}}}{dt}\right)+{{\boldsymbol{\nabla{}}}}_{M}V\left(M\right)={\bf{0}}\
\ \ \ $ (12)

This is the same expression as the usual dynamics equation in Newtonian
mechanics. The physics is left formally unchanged. This is very interesting.
For practical (computational) reasons, however, we rewrite this equation

$\frac{d}{dt}\left(m\kappa{}\frac{d{\boldsymbol{\sigma{}}}}{dt}\right)+{\boldsymbol{\nabla{}}}_{(\kappa{}{\boldsymbol{\sigma{}})}}V\left(\kappa{}\boldsymbol{\sigma{}}\right)=0$
(13)

This is equation (1). The potential $V\left(M\right)$ is the gravitational
potential. The dressed potential, experienced by an observer located at a
point $M$ and produced by a point source (mass $\sf{M}$) located at an
arbitrary origin (labeled by ${\boldsymbol{\sigma{}}}=\bf{0}$), is171717For
two masses $m$ and ${m}^{\prime}$ located at $M$ and ${M}^{\prime}$, the
interaction potential is asymmetric and
$V_{M}=-Gm{m}^{\prime}\frac{1}{({\kappa{}}_{M}n\|{}\sigma{}-{{\boldsymbol{\sigma{}}}}^{\prime}\|{})}\
\not=\ V_{{M}^{\prime}}=-Gmm^{\prime}\frac{1}{({\kappa{}}_{{M}^{\prime}}\
\|{}{\boldsymbol{\sigma{}}}-{{\boldsymbol{\sigma{}}}}^{\prime}\|{})}$. The
principle of reciprocal action seems to be altered, but it must be kept in
mind that the two masses are not isolated and are not located in the same
environment. Thus, this fundamental principle is not violated, but it is not
directly applicable here. In contrast, the bare potential,
$V_{b}=-Gm{m}^{\prime}\frac{1}{\|{}{\boldsymbol{{\boldsymbol{\sigma{}}}}}-\sigma{}^{\prime}\|{}}$,
is symmetric (even though it is not measurable, given that
$\|{}{\boldsymbol{\sigma{}}}-{\boldsymbol{\sigma{}}}^{\prime}\|{}$ is hidden.

$V\left({\kappa{}}_{M}{\boldsymbol{\sigma{}}}\right)=-G{\sf{M}}m\frac{1}{{(\kappa{}}_{M}{\boldsymbol{\sigma{}}})}$
(14)

We explicitly indexed the point where the potential is measured by
$\kappa{}$181818Let us specify that we cannot directly measure a potential
(this quantity is defined up to a constant), but its gradient (the force on a
test particle of unit mass). However, this remark is simply a detail here..
For a shifted origin (at $O$), we likewise have

$V\left({\kappa{}}_{M}\sigma{}\right)=-G{\sf{M}}m\frac{1}{({\kappa{}}_{M}\
\left\|{}{\boldsymbol{\sigma{}}}-{{\boldsymbol{\sigma{}}}}_{O}\right\|{})}$
(15)

Equations (14) and (15) need to be explained. The coefficient
${\kappa{}}_{M}$, which is linked to the observer located at $M$, determines
the intensity of the potential measured by this observer. It would seem that
it is the measurement process itself that imposes the distance that separates
the observer from the attractive mass $\sf{M}$ on the observer. This circular
reasoning may sound irrational. In reality, both the measured distance and the
apparent gravitational potential felt by the observer depend on the mean
density $\bar{\rho{}}$ at $M$. It is the environment of the observer (and
obviously not the observer himself) that affects the measurements. There is
nothing strange about this. Thus, we can equivalently reason that by admitting
that the attractive mass $\sf{M}$ is perceived by the observer as
$\frac{\sf{M}}{{\kappa{}}_{M}}$, the smaller ${\kappa{}}_{M}$ is, the higher
the apparent attractive mass and vice versa (however, the true mass is always
$\sf{M}$).

## Appendix B

Magnification formula

We assimilate a galaxy into a steady and axisymmetric thin disk. The stars
travel in pure, uniform circular motion, and the coefficient $\kappa{}$ is
independent of time. After removing the index $i$ for a test particle of unit
mass and taking ${\kappa{}}_{E}=1$191919The full equations with the
coefficient ${\kappa{}}_{E}$ are
${\frac{d}{dt}(\kappa{}}_{E}\frac{d{\boldsymbol{\sigma{}}}}{dt})={(\frac{{\kappa{}}_{E}}{\kappa{}})}^{3}\bf{F}_{New}$
and
${\bf{F}}_{New}=-Gm\sum_{j=1}^{N-1}\frac{{\kappa{}}_{E}({\boldsymbol{\sigma{}}}-{{\boldsymbol{\sigma{}}}_{j}})}{[{\kappa{}}_{E}\|{}{\boldsymbol{\sigma{}}}-{\boldsymbol{\sigma{}}]_{j}\|{}]}^{3}}$
, can simplify equation (1) to

$\frac{d}{dt}(\frac{d{\boldsymbol{\sigma{}}}}{dt})=\frac{{\bf{F}}_{New}}{{\kappa{}}^{3}}$
(16)

where the Newtonian force ${\bf{F}}_{New}$ acting on the test particle is

${\bf{F}}_{New}=-Gm\sum_{j=1}^{N-1}\frac{({\boldsymbol{\sigma{}}}-{\boldsymbol{\sigma{}}}_{j})}{{\
\|{}{\boldsymbol{\sigma{}}}-{\boldsymbol{\sigma{}}}_{j}\|{}}^{3}}$ (17)

Because the trajectory of the test particle is circular and the force acting
on it is purely radial, equation (16) immediately gives

$\frac{{(\sigma{}\dot{\theta{}})}^{2}}{\sigma{}}=\frac{F_{New}}{{\kappa{}}^{3}}$
(18)

where $\theta{}$ designates the polar angle in the galactic plane from a
reference direction taken in this plane. This leads to

$\sigma{}\dot{\theta{}}=\frac{1}{{\kappa{}}^{\frac{3}{2}}}{{(F}_{New}\
\sigma{})}^{\frac{1}{2}}=\frac{1}{{\kappa{}}^{\frac{3}{2}}}{\ v}_{New}$ (19)

where ${v}_{New}$ is the Newtonian velocity. Eventually, the true
velocity202020The true velocity refers to the local velocity that is
observable by spectroscopy. $v$ is obtained by multiplying equation (19) by
$\kappa{}$. We obtain magnification formula 6 after reinserting in it the
coefficient ${\kappa{}}_{E}$.