# Linear recurrences over a finite field with exactly two periods

Ghurumuruhan Ganesan

Institute of Mathematical Sciences, HBNI, Chennai E-Mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the periodicity structure of finite field linear
recurring sequences whose period is not necessarily maximal and determine
necessary and sufficient conditions for the characteristic polynomial $f$ to
have exactly two periods in the sense that the period of any sequence
generated by $f$ is either one or a unique integer greater than one.

Key words: Linear recurring sequences, characteristic polynomial, exactly two
periods.

AMS 2010 Subject Classification: 11T71, 12Y05, 68R01;

##  1 Introduction

Sequences generated from linear recursions are extensively used in practice
due to their periodic nature. The characteristic polynomial which results in
finite field sequences with maximal periods have been studied before (Golomb
(1967), Lidl and Niederreiter (1986)) in various contexts (see for example
Tsaban and Vishne (2002), Goltvanitsa et al. (2012)). Recently Quijada (2015)
and Bush and Quijada (2018) have investigated recurrent sequences over rings
and finite fields and studied decomposition of the set of sequences into
periodic and null sequences and determine the set of all possible least
periods associated with polynomials of a given degree and a finite field.

In this paper, we study biperiodic characteristic polynomials that generate
sequences whose period is either one or a unique integer larger than one,
irrespective of the initial state. We begin with a discussion of linear
recurrent sequences.

###  1.1 Model Description

Let $GF(q)$ denote the finite field containing $q$ elements and consider a
sequence $\\{a_{n}\\}_{n\geq 0}$ in $GF(q)$ obtained recursively as

$a_{n}=\sum_{i=1}^{r}c_{i}a_{n-i},$ (1.1)

where $\\{c_{i}\\}_{1\leq i\leq r},c_{r}\neq 0$ are elements of $GF(q)$ and
$\mathbf{s}_{0}:=(a_{-r},a_{-r+1},\ldots,a_{-1})$ is an $r$-tuple forming the
_initial state_. The summation in (1.1) is the addition corresponding to the
elements of $GF(q)$ and throughout a state always refers to an $r$-tuple with
entries in $GF(q).$ The set ${\cal S}:=(GF(q))^{r}$ of all $r$-tuples with
entries in $GF(q)$ is the _state space_.

In what follows, we derive the cycle structure of the state space ${\cal S}$
using intuitive arguments and refer to Lemma 2 of Section 2 for formal proofs.
Letting $\mathbf{s}_{1}=(a_{-r+1},\ldots,a_{0}),$ we say that the state
$\mathbf{s}_{0}$ _leads to_ the state $\mathbf{s}_{1}$ and denote it as
$\mathbf{s}_{0}\rightarrow\mathbf{s}_{1}.$ Because the recurrence in (1.1) is
linear, each state $\mathbf{s}$ of ${\cal S}$ leads to a unique state
$\mathbf{t}$ and is in turn obtained from another unique state $\mathbf{u};$
i.e. for each $\mathbf{s}\in{\cal S}$ there are unique states
$\mathbf{u},\mathbf{t}$ such that
$\mathbf{u}\rightarrow\mathbf{s}\rightarrow\mathbf{t}$ (see Lemma 2 $(i)$ of
Section 2).

Suppose
$\mathbf{s}_{0}\rightarrow\mathbf{s}_{1}\rightarrow\mathbf{s}_{2}\rightarrow\cdots$
is the sequence of states obtained while generating the sequence
$\\{a_{n}\\}_{n\geq 0}$ with initial state $\mathbf{s}_{0}.$ Since the number
of states in ${\cal S}$ is finite, the sequence $\\{\mathbf{s}_{i}\\}_{i\geq
1}$ repeats itself and so there are integers $i,k\geq 1$ such that
$\mathbf{s}_{i}=\mathbf{s}_{i+k}$ and all the states $\mathbf{s}_{j},i\leq
j\leq i+k-1$ are distinct. We then define

$\mathbf{s}_{i}\rightarrow\mathbf{s}_{i+1}\rightarrow\dots\rightarrow\mathbf{s}_{i+k-1}\rightarrow\mathbf{s}_{i+k}=\mathbf{s}_{i}$
(1.2)

to be a _cycle_ and denote it by ${\cal C}.$

Because each state leads to a unique state and is obtained from another unique
state, we deduce from the above paragraph that each state of ${\cal S}$
belongs to a unique cycle and there are disjoint cycles ${\cal C}_{i},1\leq
i\leq L$ such that every state $\mathbf{s}\in{\cal S}$ belongs to exactly one
of the cycles in $\\{{\cal C}_{i}\\}_{1\leq i\leq L}$ (see also Lemma 2 $(ii)$
of Section 2). Therefore if $\mathbf{s}=\mathbf{s}_{0}$ belongs to the cycle
${\cal C}_{i}$ containing $k$ states, then the resulting sequence of states
$\mathbf{s}_{0}\rightarrow\mathbf{s}_{1}\rightarrow\mathbf{s}_{2}\rightarrow\cdots$
obtained by applying the recursion (1.1) is periodic and
$\mathbf{s}_{i}=\mathbf{s}_{i+k}$ for all $i\geq 0$ (Lemma 2 $(ii),$ Section
2).

Due to the periodicity of the state sequence $\\{\mathbf{s}_{i}\\}_{i\geq 0},$
repetitions occur in the symbol sequence
$A(\mathbf{s}_{0}):=\\{a_{n}\\}_{n\geq 0}$ as well and so we define the period
$P(\mathbf{s}_{0})$ of the sequence $A(\mathbf{s}_{0})$ to be the smallest $p$
such that there exists an integer $l=l(p)$ satisfying $a_{n}=a_{n+p}$ for all
$n\geq l$ (Lidl and Niederreiter (1986)). Recalling that $\mathbf{s}_{0}$
belongs to the cycle ${\cal C}_{i}$ containing $k$ states, the period of the
symbol sequence $A(\mathbf{s}_{0})$ must therefore also be $k$ (see Lemma 2
$(iii),$ Section 2).

Let ${\cal P}$ be the set of all possible periods of sequences with initial
states in ${\cal S}$ and satisfying recursion (1.1). The set ${\cal P}$ is
defined to be the _period set_ and based on the discussion in the previous
paragraphs, we see that

${\cal P}:=\\{\\#{\cal C}_{i}:1\leq i\leq L\\}$ (1.3)

where $\\#{\cal C}_{i}$ denotes the number of states in the cycle ${\cal
C}_{i}.$ The zero state $\mathbf{0}$ always belongs to the cycle
$\mathbf{0}\rightarrow\mathbf{0}$ and so $1\in{\cal P}.$ Clearly, the
structure of the period set ${\cal P}$ depends on the coefficients
$\\{c_{i}\\}_{1\leq i\leq r}$ in the recursion (1.1). The main question we are
interested in is this: What restrictions must be placed on $\\{c_{i}\\}_{1\leq
i\leq r}$ so that ${\cal P}$ contains exactly two elements ?

For convenience, we state our results in terms of the _characteristic
polynomial_ (Golomb (1967))

$f(x):=1-\sum_{i=1}^{r}c_{i}x^{i}.$ (1.4)

Throughout all polynomials have coefficients in $GF(q).$

We have the following definition.

######  Definition 1.

For integer $T\geq 2$ we say that the characteristic polynomial $f$ is
_biperiodic_ with _biperiod_ $T$ if the period set ${\cal P}=\\{1,T\\}.$

From (1.3) we see that $f$ is biperiodic with biperiod $T$ if and only if for
each $1\leq i\leq L,$ the cycle ${\cal C}_{i}$ has either $T$ states or a
single state.

To state our main result describing characteristic polynomials with a given
biperiod $T,$ we begin with a couple of definitions. For integers $a,b\geq 1$
we use the notation $a\mid b$ to denote that $a$ divides $b$ and let
$\gcd(a,b)$ denote the greatest common divisor of $a$ and $b.$ We say that $a$
and $b$ are relatively prime if $\gcd(a,b)=1$ and let $\varphi(a)$ be the
Euler function counting the number of positive integers less or equal to $a$
that are relatively prime to $a.$

We say that a polynomial $g(x)$ is _irreducible_ if $g$ cannot be written as a
product of two polynomials, each with degree strictly larger than one. The
_order_ $\theta(g)$ of $g(x)$ is the smallest integer $j\geq 1$ such that
$g(x)$ divides $1-x^{j}$ but does not divide $1-x^{i}$ for any integer $1\leq
i<j$ (Lidl and Niederreiter (1986)). Letting $q$ be a power of a prime $p$ we
consider two cases below depending on whether the desired biperiod $T$ is
relatively prime to $q$ or not.

######  Theorem 1.

Let $q$ be a power of a prime number and let $T\geq 1$ be an integer.

  1. (i)

Suppose $\gcd(T,q)=1.$ There exists a polynomial $f$ with degree $r$ and
biperiod $T$ if and only if

$T\mid(q^{r-d_{0}}-1)\text{ and }\varphi(T)\geq r-d_{0}\text{ for some
}d_{0}\in\\{0,1\\}.$ (1.5)

The condition (1.5) holds either for $d_{0}=0$ or $d_{0}=1$ but not both and
if (1.5) holds, then the polynomial $f$ is of the form

$f(x)=(1-x)^{d_{0}}\cdot g_{1}(x)\cdot g_{2}(x)\cdots g_{l}(x),$

where $l$ is the largest divisor $j$ of $r-d_{0}$ such that
$T\mid(q^{\frac{r-d_{0}}{j}}-1)$ and the $\\{g_{i}\\}_{1\leq i\leq l}$ are
distinct irreducible polynomials, each with degree $\frac{r-d_{0}}{l}$ and
order $T$ and satisfying $\prod_{j=1}^{l}g_{j}(0)=1.$ Moreover letting
$N:=\frac{q^{r}-q^{d_{0}}}{T},$ there are $L:=N+q^{d_{0}}$ cycles $\\{{\cal
C}_{i}\\}_{1\leq i\leq L}$ with $N$ of the cycles having $T$ states each and
the remaining $q^{d_{0}}$ cycles having one state each.

  2. (ii)

Suppose $T=p^{u}\cdot M$ with $\gcd(M,q)=1$ and $u\geq 1.$ There exists a
polynomial $f$ with biperiod $T$ and degree $r$ if and only if

$M=u=1\text{ and }r\leq p.$ (1.6)

If (1.6) holds, then $f(x)=(1-x)^{r}$ and letting $N:=\frac{q^{r}-q}{p},$
there are $L:=N+q$ cycles $\\{{\cal C}_{i}\\}_{1\leq i\leq L}$ with $N$ of the
cycles having $p$ states each and the remaining $q$ cycles having one state
each.

As a consequence of Theorem 1, a polynomial $f$ with degree $r\geq 2$ has
biperiod $q^{r-1}-1$ if and only if $f(x)=(1-x)\cdot g(x),$ where $g(x)$ is a
primitive polynomial of degree $r-1$ (i.e. the order of $g(x)$ is $q^{r-1}-1$)
and satisfies $g(0)=1.$ In this case, there are exactly $2q$ cycles: $q$
cycles each containing $q^{r-1}~{}-~{}1$ states each and $q$ cycles containing
one state each. In particular, for $q=2$ this characterises all polynomials
with the maximum possible biperiod _strictly less_ than $2^{r}-1.$ Indeed if
there existed a biperiod $T$ with value between $2^{r-1}$ and $2^{r}-2,$ then
from (1.5) we have that $T$ must be a factor of $2^{r}-1$ and so we would have
$\gamma\cdot T=2^{r}-1$ for some integer $\gamma\geq 2.$ But this would mean
that $\gamma\cdot T\geq 2\cdot T\geq 2\cdot 2^{r-1}=2^{r},$ a contradiction.

We illustrate Theorem 1 with an example.

### Example

Let $q=2$ and $r=4$ so that $2^{r}-1=15$ and $2^{r-1}-1=7.$ From condition
(1.5) we therefore see that the desired biperiod $T$ must either be a factor
of $15$ and satisfy $\varphi(T)\geq 4$ or must satisfy $T=7$ and
$\varphi(T)\geq 3.$ We first consider the case that $T$ is a factor of $15.$
The non-trivial factors of $15$ are $3,5$ and $15$ and we have
$\varphi(3)=2<r,\varphi(5)=4=r$ and $\varphi(15)=2\cdot 4>r.$ Thus there
exists degree $4$ polynomials with biperiod $5$ and $15$ and there does not
exist a degree $4$ polynomial with biperiod $3.$ In what follows, we give one
example of a polynomial in each of the above mentioned valid cases and also
describe the corresponding cycle structure of the states.

For $T=5,$ the largest divisor $j$ of $4$ such that $T\mid(2^{\frac{4}{j}}-1)$
is $j=1$ and so any irreducible polynomial of degree $4$ and order $5$ also
has biperiod $5.$ As an example, the polynomial
$f_{1}(x)=x^{4}+x^{3}+x^{2}+x+1$ is irreducible with degree $4$ and order $5$
and therefore has biperiod $5.$ Correspondingly, there are $3$ cycles
containing $5$ states each and one cycle containing one state as enumerated
below (states are of the form $(a_{j-r},\ldots,a_{j-1})$):
${\cal C}_{1}:0001\rightarrow 0011\rightarrow 0110\rightarrow 1100\rightarrow
1000\rightarrow 0001.$
${\cal C}_{2}:0111\rightarrow 1111\rightarrow 1110\rightarrow 1101\rightarrow
1011\rightarrow 0111.$
${\cal C}_{3}:1001\rightarrow 0010\rightarrow 0101\rightarrow 1010\rightarrow
0100\rightarrow 1001.$
${\cal C}_{4}:0000\rightarrow 0000.$

For $T=15,$ any polynomial with degree $4$ and order $15$ also has biperiod
$15.$ For illustration, the polynomial $f_{2}(x)=x^{4}+x^{3}+1$ is irreducible
and has order $15$ and so correspondingly, there is one cycle containing $15$
states and one cycle containing one state as enumerated below (states are of
the form $(a_{j-r},\ldots,a_{j-1})$):
${\cal C}_{1}:0001\rightarrow 0010\rightarrow 0100\rightarrow 1001\rightarrow
0011\rightarrow 0110\rightarrow 1101\rightarrow 1010\rightarrow
0101\rightarrow 1011\rightarrow 0111\rightarrow 1111\rightarrow
1110\rightarrow 1100\rightarrow 1000\rightarrow 0001.$
${\cal C}_{2}:0000\rightarrow 0000.$

We now study the case $T=7.$ Since $\varphi(7)=6\geq 3=r-1,$ there exists a
polynomial $f$ with biperiod $7$ and degree $4$ and since $r-1=3$ is prime,
the largest divisor $j$ of $r-1$ such that $7\mid(2^{\frac{r-1}{j}}-1)$ is
one. From Theorem 1, we therefore get that $f$ is of the form $(1+x)\cdot
g(x),$ where $g(x)$ is irreducible with degree $3$ and order $7.$ For example,
the polynomial

$f_{3}(x)=x^{4}+x^{3}+x^{2}+1=(x^{3}+x+1)(x+1)$

has an irreducible factor $x^{3}+x+1$ of order $7.$ Correspondingly, there are
two cycles containing $7$ states each and two cycles containing one state each
as enumerated below (states are of the form $(a_{j-r},\ldots,a_{j-1})$):
${\cal C}_{1}:0001\rightarrow 0010\rightarrow 0101\rightarrow 1011\rightarrow
0110\rightarrow 1100\rightarrow 1000\rightarrow 0001.$
${\cal C}_{2}:0111\rightarrow 1110\rightarrow 1101\rightarrow 1010\rightarrow
0100\rightarrow 1001\rightarrow 0011\rightarrow 0111.$
${\cal C}_{3}:1111\rightarrow 1111.$
${\cal C}_{4}:0000\rightarrow 0000.$

The paper is organized as follows. In Section 2, we collect preliminary
results used in the proof of Theorem 1 and in Section 3, we prove Theorem 1.

##  2 Preliminary results

To prove Theorem 3 we collect a few preliminary results in the form of three
Lemmas. For completeness we provide small proofs. We recall that ${\cal
S}=(GF(q))^{r}$ is the state space consisting of all $r$-tuples with entries
in $GF(q).$

######  Lemma 2.

Let $\\{c_{i}\\}_{1\leq i\leq r}$ be as in the recursion (1.1). The following
properties hold:

  1. (i)

For every $\mathbf{y}\in{\cal S},$ there exists unique
$\mathbf{z}_{1},\mathbf{z}_{2}\in{\cal S}$ such that
$\mathbf{z}_{1}\rightarrow\mathbf{y}\rightarrow\mathbf{z}_{2}.$

  2. (ii)

There are cycles ${\cal C}_{i},1\leq i\leq L$ such that every state
$\mathbf{s}\in{\cal S}$ belongs to exactly one of the cycles in $\\{{\cal
C}_{i}\\}_{1\leq i\leq L}.$

  3. (iii)

Pick a cycle ${\cal C}_{i},1\leq i\leq L$ with $p_{i}$ states and let
$\\{a_{n}\\}_{n\geq 0}$ be the sequence generated with initial state
$\mathbf{s}\in{\cal C}_{i}.$ The period $P(\mathbf{s})$ of the sequence
$\\{a_{n}\\}_{n\geq 0}$ equals $p_{i},$ the terms $a_{n}=a_{n+p_{i}}$ for all
$n\geq 0$ and the generating function

$G(x)=G(x;\mathbf{s}):=\sum_{n\geq
0}a_{n}x^{n}=\frac{D(x;\mathbf{s})}{f(x)}=\frac{a(x)}{1-x^{p_{i}}}$ (2.1)

where $f(x)=1-\sum_{j=1}^{r}c_{j}x^{j}$ is the characteristic polynomial as
defined in (1.4),

$D(x;\mathbf{s})=\sum_{j=0}^{r-1}d_{j}x^{j}\text{ with
}d_{j}=\sum_{i=1}^{r-j}a_{-i}c_{i},0\leq j\leq r-1.$ (2.2)

and $a(x):=\sum_{n=0}^{p_{i}-1}a_{n}x^{n}.$

_Proof:_ [Proof of Lemma 2] Property $(i)$ is true since if
$\mathbf{y}=(y_{1},\ldots,y_{r})\in{\cal S}$ then
$\mathbf{z}_{1}\rightarrow\mathbf{y}\rightarrow\mathbf{z}_{2}$ where
$\mathbf{z}_{1}=(b,y_{1},\ldots,y_{r-1})$ with

$b:=c_{r}^{-1}\left(y_{r}-\sum_{i=2}^{r}c_{r-i}y_{i-1}\right)$

and $\mathbf{z}_{2}=(y_{2},\ldots,y_{r},d)$ with
$d:=\sum_{i=1}^{r}c_{i}y_{r-i}.$

To prove $(ii),$ let $\mathbf{s}_{0}:=\mathbf{s}$ be any state. From property
$(i)$ there exists a unique state $\mathbf{s}_{1}$ such that
$\mathbf{s}_{0}\rightarrow\mathbf{s}_{1}.$ Similarly, there exists a unique
state $\mathbf{s}_{2}$ such that $\mathbf{s}_{1}\rightarrow\mathbf{s}_{2}.$
Because the number of states in ${\cal S}$ is finite, we eventually get that
$\mathbf{s}_{j}\rightarrow\mathbf{s}_{i}$ for some integers $j\geq 1$ and
$0\leq i\leq j-1.$ The integer $i=0$ necessarily else we would have
$\mathbf{s}_{i-1}\rightarrow\mathbf{s}_{i}$ in addition to
$\mathbf{s}_{j}\rightarrow\mathbf{s}_{i},$ a contradiction to $(i).$ By the
same argument we get that if ${\cal C}(\mathbf{s})$ is the cycle containing
the state $\mathbf{s},$ then for any two distinct states $\mathbf{s}_{1}$ and
$\mathbf{s}_{2}$ the corresponding cycles ${\cal C}(\mathbf{s}_{1})$ and
${\cal C}(\mathbf{s}_{2})$ satisfy the following property: Either ${\cal
C}(\mathbf{s}_{1})={\cal C}(\mathbf{s}_{2})$ or ${\cal
C}(\mathbf{s}_{1})\cap{\cal C}(\mathbf{s}_{2})=\emptyset.$ The distinct cycles
in $\\{{\cal C}(\mathbf{s})\\}_{\mathbf{s}\in{\cal S}}$ form the desired set
${\cal C}_{1},\ldots,{\cal C}_{L},$ proving $(ii).$

To prove $(iii),$ let
$\mathbf{s}_{0}\rightarrow\mathbf{s}_{1}\rightarrow\ldots$ be the sequence of
states generated starting with $\mathbf{s}_{0}\in{\cal C}_{i}.$ Recalling that
$\mathbf{s}_{j}:=(a_{j-r},\ldots,a_{j-1},a_{j})=\mathbf{s}_{j+p_{i}}$ for all
$j\geq 0,$ we get that $a_{n}=a_{n+p_{i}}$ for all $n\geq 0.$ Thus
$P(\mathbf{s})\leq p_{i}.$ If it so happens that $P=P(\mathbf{s})\leq p_{i}-1$
so that $a_{j}=a_{j+P}$ for all $j$ sufficiently large, then there exists $J$
sufficiently large so that $\mathbf{s}_{j}=\mathbf{s}_{j+P}$ for all $j\geq
J.$ This implies that $\mathbf{s}_{j}\in{\cal C}_{i}$ belongs to a cycle of
length at most $P\leq p_{i}-1,$ a contradiction. Finally, we substitute the
recursion (1.1) into the definition of $G(x)$ in (2.1) to get the second
relation in (2.1) (see Golomb (1967)). Writing

$(1-x^{p_{i}})\cdot
G(x)=a(x)+\sum_{n=0}^{\infty}(a_{n+p_{i}}-a_{n})x^{n+p_{i}}$

and using the fact that the sequence $\\{a_{n}\\}_{n\geq 0}$ also has period
$p_{i},$ we get the final relation in (2.1).

The next Lemma relates the period set ${\cal P}$ to the orders of factors of
the characteristic polynomial $f(x).$

######  Lemma 3.

Let $\\{c_{i}\\}_{1\leq i\leq r}$ be as in the recursion (1.1) and let ${\cal
C}_{i},1\leq i\leq L$ be the cycle decomposition of the state space ${\cal S}$
as in Lemma 2. If $p_{i}$ denotes the number of states in the cycle ${\cal
C}_{i},$ then the period set

${\cal P}={\cal P}(f):=\\{p_{1},\ldots,p_{L}\\}=\\{\theta(g):g\mid f\\}$ (2.3)

is precisely the set of orders of all possible factors of $f.$

We recall that the state $\mathbf{0}$ always belongs to the cycle
$\mathbf{0}\rightarrow\mathbf{0}$ and so $1\in{\cal P}.$ The rest of the
period set is determined from the non-trivial factors of the characteristic
polynomial $f.$

As an example, we let $q=2$ and consider the characteristic polynomial
$f_{0}(x)=1+x^{2}+x^{4}=(1+x+x^{2})^{2}.$ The order of the factor $1+x+x^{2}$
of $f_{0}$ is $3$ and the order of $f_{0}$ is $6$ and so the period set for
the sequences generated by $f_{0}$ is ${\cal P}=\\{1,3,6\\}.$ Indeed, the
degree of $f_{0}$ is $4$ and the sixteen states in $\\{0,1\\}^{4}$ are present
in the following four cycles where each state $\mathbf{s}$ is of the form
$(a_{j-r},a_{j-r+1},\ldots,a_{j-1}):$
${\cal C}_{1}:0001\rightarrow 0010\rightarrow 0101\rightarrow 1010\rightarrow
0100\rightarrow 1000\rightarrow 0001.$
${\cal C}_{2}:0011\rightarrow 0111\rightarrow 1111\rightarrow 1110\rightarrow
1100\rightarrow 1001\rightarrow 0011.$
${\cal C}_{3}:0110\rightarrow 1101\rightarrow 1011\rightarrow 0110.$
${\cal C}_{4}:0000\rightarrow 0000.$

The period set in Lemma 3 could also be studied using minimal polynomials (see
for example, Chapter $6$ of Lidl and Niedereitter (1986)). For completeness,
we give a small proof.

_Proof:_ [Proof of Lemma 3]: Let ${\cal P}=\\{p_{1},\ldots,p_{L}\\}$ be the
period set as defined in (2.3) where $p_{i}$ denotes the number of states in
the cycle ${\cal C}_{i}$ and let ${\cal Q}:=\\{\theta(g):g\mid f\\}$ be the
set of orders of all possible factors of the characteristic polynomial $f.$ We
first see that ${\cal P}\subseteq{\cal Q}.$ Let
$\mathbf{s}=(a_{-r},\ldots,a_{-1})$ be any state in ${\cal S}$ and let
$\\{a_{n}\\}_{n\geq 0}$ be the corresponding sequence generated using (1.1).
If $G(x)$ is the generating function of $\\{a_{n}\\}_{n\geq 0}$ then
cancelling the common factors in the second relation of (2.1) and using the
final relation of (2.1), we get that
$G(x)=\frac{u(x)}{b(x)}=\frac{a(x)}{1-x^{W}}$ where $b(x)$ is a factor of
$f(x)$ and $W$ is the period of $\\{a_{n}\\}_{n\geq 0}.$ Thus $(1-x^{W})\cdot
u(x)=b(x)\cdot a(x)$ and since $u(x)$ and $b(x)$ have no factors in common,
$b(x)$ divides $1-x^{W}.$ Consequently, the order $\theta$ of $b(x)$ is at
most $W.$ Next using the fact that

$(1-x^{\theta})\cdot
G(x)=\sum_{n=0}^{\theta-1}a_{n}x^{n}+\sum_{n=\theta}^{\infty}(a_{n}-a_{n-\theta})x^{n}$

has finite degree, we get that $a_{n}=a_{n-\theta}$ for all $n$ sufficiently
large. Thus the period $W$ of $\\{a_{n}\\}_{n\geq 0}$ is at most $\theta.$
Thus $W=\theta$ and since $W$ is also the number of states in the cycle ${\cal
C}(\mathbf{s})$ (see Lemma 2), this completes the proof of ${\cal
P}\subseteq{\cal Q}.$

To see that ${\cal Q}\subseteq{\cal P},$ let $g$ be a factor of $f$ so that
$g(x)\cdot h(x)=f(x)$ for some polynomial $h(x)=\sum_{j=0}^{r-1}h_{j}x^{j}$ of
degree at most $r-1.$ We first see that choosing
$\mathbf{s}=(a_{-r},\ldots,a_{-1})$ appropriately, the polynomial
$D(x;\mathbf{s})$ defined in (2.1) equals $h(x).$ Indeed we solve (2.2)
backwards; first we choose $a_{-1}$ such that $a_{-1}=c_{r}^{-1}\cdot h_{r-1}$
and having determined $a_{-1},\ldots,a_{-i+1},$ we set

$a_{-i}:=c_{r}^{-1}\left(h_{r-i}-\sum_{j=1}^{i-1}a_{-j}c_{r-j}\right).$

The generating function of the resulting sequence $\\{a_{n}\\}_{n\geq 0}$ then
equals $\frac{1}{g(x)}.$ If $\omega$ is the number of states in the
corresponding cycle ${\cal C}(\mathbf{s}),$ then from Lemma 2 we get that the
sequence $\\{a_{n}\\}_{n\geq 0}$ also has period $\omega$ and arguing as in
the previous paragraph, we get that $\omega=\theta(g),$ the order of the
polynomial $g.$

The final ingredient used in the proof of Theorem 1 concerns the order of
powers of polynomials and number of polynomials with a given degree. We say
that a polynomial $g(x)=\sum_{i=0}^{w}c_{i}x^{i},c_{w}\neq 0$ is _monic_ if
the leading coefficient $c_{w}=1$ and have the following result.

######  Lemma 4.

  1. (a)

If $g$ is an irreducible polynomial of order $a$ with $g(0)\neq 0,$ then for
integer $b\geq 1$ the polynomial $g^{b}$ has order $a\cdot p^{t},$ where $t$
is the smallest integer such that $p^{t}\geq b.$

  2. (b)

Let $e\geq 2$ be an integer satisfying $\gcd(e,q)=1.$ For integer $m\geq 1,$
let $\delta(m,e)$ be the number of monic irreducible polynomials of degree $m$
and order $e.$ We have that $\delta(m,e)=\frac{\varphi(e)}{m}$ if $m$ is the
smallest integer $j\geq 1$ such that $q^{j}\equiv 1\mod{e}.$ For all other
$m,$ we have that $\delta(m,e)=0.$

_Proof:_ [Proof of Lemma 4] Part $(a)$ follows from Theorem $3.8$ of Lidl and
Niederreiter (1986) and part $(b)$ follows from Theorem $3.5$ of Lidl and
Niederreiter (1986).

##  3 Proof of Theorem 1

_Proof:_ [Proof of Theorem 1$(i)$] Let

$f=(1-x)^{d_{0}}\cdot g_{1}^{d_{1}}\cdots g_{t}^{d_{t}}$ (3.1)

be biperiodic of degree $r$ and biperiod $T$ where $\\{g_{i}\\}_{1\leq i\leq
t}$ are distinct irreducible polynomials having order
$\theta_{i}:=\theta(g_{i})\geq 2.$ Since $f(0)=1$ we must have that
$\prod_{j=1}^{t}g_{j}^{d_{j}}(0)=1$ and so in particular we get that
$g_{j}(0)\neq 0$ for each $1\leq j\leq t.$

From Lemma 3, we know that $\\{1,\theta_{1},\ldots,\theta_{t}\\}\subseteq{\cal
P},$ the period set. But since $f$ is biperiodic with biperiod $T,$ the period
set ${\cal P}=\\{1,T\\}$ and so we must have that the $g_{i}$’s all have the
same order $T.$ Moreover, if $d_{i}\geq 2$ for some $i\geq 1,$ then from Lemma
4$(a),$ we have that the order of $g_{i}^{d_{i}}$ is at least $T\cdot p$ and
so we must have $d_{i}=1$ for all $1\leq i\leq t.$ Similarly, if $d_{0}\geq 2$
then the order of $(1-x)^{d_{0}}$ is of the form $p^{u}$ for some integer
$u\geq 1.$ Since $\gcd(T,q)=\gcd(T,p)=1,$ we have that $T\neq p^{u}$ and so we
must have that $d_{0}\in\\{0,1\\}.$

From the above paragraph we get that $f=(1-x)^{d_{0}}\cdot g_{1}\cdots g_{t}$
where
$d_{0}\in\\{0,1\\}$ and all the $g_{i}$ are irreducible and have the same
order $T.$ Also since $f(0)=1$ we must have that $\prod_{j=1}^{t}g_{j}(0)=1.$
For convenience we now write

$f=(1-x)^{d_{0}}\cdot\alpha\cdot h_{1}\cdots h_{t}$ (3.2)

where $\alpha\in GF(q)$ and each $h_{i}$ is monic. From Lemma 4$(b)$ we know
that there exists a monic irreducible polynomial of order $T\geq 2$ and degree
$m$ if and only if $m$ is the smallest integer $j$ such that $T\mid(q^{j}-1).$
For such an $m,$ all the polynomials $\\{h_{i}\\}_{1\leq i\leq t}$ must have
the same degree $m$ which in turn must equal $\frac{r-d_{0}}{t}.$ Thus in
effect, the integer $m$ must satisfy two conditions:
$(1)$ $m$ is the smallest integer $j$ such that $T\mid(q^{j}-1).$
$(2)$ $m$ is a divisor of $r-d_{0}.$

We show that an integer $m$ satisfying conditions $(1)-(2)$ above exists if
and only if $T\mid(q^{r-d_{0}}-1).$ Indeed if $m$ satisfies $(1)-(2),$ then
$m$ divides $r-d_{0}$ and so $r-d_{0}=a\cdot m$ for some integer $a.$ We then
have that

$q^{r-d_{0}}=q^{am}=(q^{m})^{a}\equiv 1\mod{T}.$

Conversely, suppose that $T\mid(q^{r-d_{0}}-1)$ and let $m$ be the smallest
integer $j$ such that $T\mid(q^{j}-1).$ Thus $m$ satisfies condition $(1)$
above and we would like to see that $m$ also satisfies $(2).$ Indeed if $m$
does not divide $r-d_{0},$ we write $r-d_{0}=mw+s$ for $1\leq s\leq m-1$ and
get that

$q^{r-d_{0}}=\left(q^{m}\right)^{w}\cdot q^{s}\equiv q^{s}\mod{T}.$

Since $q^{r-d_{0}}\equiv 1\mod{T},$ we then get that $q^{s}\equiv 1\mod{T},$ a
contradiction to the definition of $m.$

So far we have shown that if $f$ is biperiodic and of the form (3.2), then all
the polynomials $h_{i}$ must have the same degree $m=\frac{r-d_{0}}{t}.$ Such
an integer $m$ exists if and only if $T\mid(q^{r-d_{0}}-1)$ and moreover since
$r$ and $r-1$ have no common factors, there cannot exist an integer $m$ that
is a divisor of both $r-1$ and $r.$ Consequently, the above condition
$T\mid(q^{r-d_{0}}-1)$ holds either for $d_{0}=0$ or $d_{0}=1$ but not both.

Setting $\frac{r-d_{0}}{t}=m,$ we again use Lemma 4$(b)$ to get that the
number of monic irreducible polynomials of order $T$ and degree
$\frac{r-d_{0}}{t}$ equals $\varphi(T)\cdot\frac{t}{r-d_{0}}.$ Since the
number of polynomials in (3.2) equals $t,$ we must have
$\varphi(T)\cdot\frac{t}{r-d_{0}}\geq t$ which is possible if and only if
$\varphi(T)\geq r-d_{0}.$ This proves (1.5).

Assuming (1.5) holds, let $\\{{\cal C}_{i}\\}_{1\leq i\leq L}$ be the set of
all resulting cycles as determined in Lemma 3, each having either $T$ states
or a single state. A cycle ${\cal C}_{i}$ is of the form
$\mathbf{s}\rightarrow\mathbf{s}=(a_{-r},a_{-r+1},\ldots,a_{-1})$ if and only
if

$a_{-r}=a_{-r+1}=\ldots=a_{-1}=\beta$

for some $\beta\in GF(q).$ Therefore if $d_{0}=0$ then
$f(1)=1-\sum_{i=1}^{r}c_{i}\neq 0$ and so the only cycle with containing a
single state is the cycle $\mathbf{0}\rightarrow\mathbf{0}.$ If $d_{0}=1,$
then $\sum_{i=1}^{r}c_{i}=1$ and there are $q$ cycles of the form
$\beta\mathbf{1}\rightarrow\beta\mathbf{1},\beta\in GF(q)$ each containing one
state, where $\mathbf{1}$ denotes the all ones state $(1,1,\ldots,1).$

_Proof:_ [Proof of Theorem 1$(ii)$] We first eliminate the case $M\geq 2$ by
contradiction. Assume there exists a biperiodic polynomial $f$ with degree $r$
and biperiod $T$ of the form (3.1) where $\\{g_{i}\\}_{1\leq i\leq t}$ are
distinct irreducible polynomials having order $\theta(g_{i})\geq 2.$ If
$d_{0}\geq 2,$ then from Lemma 4$(a),$ we get that the order of
$(1-x)^{d_{0}}$ is of the form $p^{b}$ for some $b\geq 1.$ From Lemma 3 we
know that $p^{b}\in{\cal P}=\\{1,T\\}$ and this is a contradiction since
$T=p^{u}\cdot M$ where $M\geq 2$ is not a multiple of $p.$ Thus $d_{0}\leq 1$
and arguing as in the proof of Theorem 1$(i)$ above, we have that $d_{i}=1$
for all $1\leq i\leq t$ and that each $g_{i}$ has order $\theta(g_{i})=T.$

Next if $g_{i}$ has degree $m_{i},$ then its order $T\mid(q^{m_{i}}-1)$ and so

$q^{m_{i}}-1=p^{n_{i}}-1\equiv 0\mod T$

for some integer $n_{i}\geq 1.$ In particular this means that $T$ cannot
contain $p$ as a divisor since $\gcd(p,p^{n_{i}}-1)=1$ and this is a
contradiction. This completes the case $M\geq 2.$

Now suppose $T=p^{u}$ and let $f$ be any biperiodic polynomial with degree $r$
and biperiod $T$ of the form (3.1) where $\\{g_{i}\\}_{1\leq i\leq t}$ are
distinct irreducible polynomials having order $\theta(g_{i})\geq 2.$ If
$d_{i}\geq 1$ for some $i,$ then from Lemma 3 we get that
$\theta(g_{i})\in{\cal P}$ and arguing as in the previous paragraph we get
that $\gcd(\theta(g_{i}),p)=1.$ This is a contradiction since ${\cal
P}=\\{1,p^{u}\\}.$ Thus $d_{i}=0$ for all $1\leq i\leq t$ and so
$f=(1-x)^{d_{0}}.$ Moreover, since $f$ is of degree $r,$ we must have that
$d_{0}=r.$

From Lemma 4$(a)$ we have that the order of $(1-x)^{r}$ is $p^{b}$ where $b$
satisfies $p^{b-1}<r\leq p^{b}.$ If $b\geq 2,$ then there exists an integer
$e_{0}\leq r-1$ such that $p^{b-2}<e_{0}\leq p^{b-1}$ and so $(1-x)^{e_{0}}$
has order $p^{b-1}.$ By Lemma 3, we then have that both $p^{b}$ and $p^{b-1}$
belong to ${\cal P},$ a contradiction. Thus $b=1$ and we must have that
$1<r\leq p.$ In this case the characteristic polynomial is $f(x)=(1-x)^{r}$
with period set $\\{1,p\\}.$ Arguing as in the last paragraph of proof of
Theorem 1 $(i),$ we get that there are $N$ cycles containing $p$ states each
and $q$ cycles containing one state each.

#### Acknowledgement

I thank Professors V. Arvind, C. R. Subramanian and the referees for crucial
comments that led to an improvement of the paper. I also thank IMSc for my
fellowships.

## References

  * [1] M. R. Bush and D. Quijada. (2018).  Period sets of linear recurrences over finite fields and related commutative rings.  Arxiv Link: https://arxiv.org/abs/1805/03238.
  * [2] D. Quijada. (2015).  Periods of linearly recurring sequences.  Bachelor Thesis, Washington and Lee University.
  * [3] S. W. Golomb. (1967).  Shift Register Sequences.  Holden-Day, San Francisco.
  * [4] M. A. Goltvanitsa, S. A. Zaitsev and A. A. Nechaev. (2012).  Skew linear recurring sequences of maximal period over Galois rings.  Journal of Mathematical Sciences, 187, pp. 115–128.
  * [5] R. Lidl and H. Niederreiter. (1986).  Introduction to Finite Fields and Their Applications.  Cambridge University Press.
  * [6] B. Tsaban and U. Vishne. (2002).  Efficient Linear feedback shift registers with maximal period.  Finite Fields and Their Applications, 8, pp. 256–267.


#  Chandra view of Abell 407: the central compact group of galaxies and the
interaction between the radio AGN and the ICM

Chao Geng1,2 ID , Chong Ge1 ID , Dharam V. Lal3 ID , Ming Sun4 ID , Li Ji1 ID
, Haiguang Xu5, Wenhao Liu1 ID , Martin Hardcastle6 ID , William Forman7 ID ,
Ralph Kraft7 ID , Christine Jones6 ID
1Purple Mountain Observatory, Chinese Academy of Science, Nanjing 210034,
People’s Republic of China
2Department of Astronomy, University of Science and Technology of China, Hefei
230026, People’s Republic of China
3National Centre for Radio Astrophysics, Tata Institute of Fundamental
Research, Post Box 3, Ganeshkhind P.O., Pune 411007, India
4Department of Physics and Astronomy, University of Alabama in Huntsville,
Huntsville, AL 35899, USA
5School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan
Road, Minhang, Shanghai 200240, People’s Republic of China
6Centre for Astrophysics Research, School of Physics, Astronomy $\&$
Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK
7Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
gengchao@pmo.ac.cnastrochongge<EMAIL_ADDRESS>
(Accepted. Received; in original form)

###### Abstract

Abell 407 (A407) is a unique galaxy cluster hosting a central compact group of
nine galaxies (named as ‘Zwicky’s Nonet’; G1 - G9 in this work) within a 30
kpc radius region. The cluster core also hosts a luminous radio active
galactic nucleus (AGN), 4C 35.06 with helically twisted jets extending over
200 kpc. With a 44 ks Chandra observation of A407, we characterize the X-ray
properties of its intracluster medium (ICM) and central galaxies. The mean
X-ray temperature of A407 is 2.7 keV and the $M_{200}$ is $1.9\times
10^{14}\,{M_{\odot}}$. We suggest that A407 has a weak cool core at $r<60$ kpc
scales and at its very center, $<1$-2 kpc radius, a small galaxy corona
associated with the strong radio AGN. We also conclude that the AGN 4C 35.06
host galaxy is most likely G3. We suggest that the central group of galaxies
is undergoing a ‘slow merge’ procedure. The range of the merging time-scale is
$0.3\sim 2.3$ Gyr and the stellar mass of the future brightest cluster galaxy
(BCG) will be $7.4\times 10^{11}M_{\odot}$. We find that the regions which
overlap with the radio jets have higher temperature and metallicity. This is
consistent with AGN feedback activity. The central entropy is higher than that
for other clusters, which may be due to the AGN feedback and/or merging
activity. With all these facts, we suggest that A407 is a unique and rare
system in the local universe that could help us to understand the formation of
a massive BCG.

###### keywords:

galaxies: clusters: individual: Abell 407 – galaxies: clusters: intracluster
medium – X-rays: galaxies: clusters – galaxies: ISM – galaxies: elliptical and
lenticular, cD

††pubyear: 2021

##  1 Introduction

Galaxy clusters are the most massive virialized systems in the Universe. They
form hierarchically through the merger of smaller substructures (Voit, 2005).
Clusters not only provide cosmological constraints but also provide a test of
structure formation scenarios (Kravtsov & Borgani, 2012). The cores of massive
clusters are one of the densest galaxy environments in the Universe, and thus
are expected to host the strongest dynamical evolution processes. The
brightest cluster galaxy (BCG) is usually located at the minimum potential
well of the cluster and lies close to the peak of the X-ray emission (Beers &
Geller, 1983; Jones & Forman, 1984). They are the largest and most massive
galaxies in the Universe and usually an order of magnitude brighter than
typical elliptical galaxies (Von Der Linden et al., 2007). They may also host
the most massive super-massive black holes (SMBHs) in the Universe (McConnell
et al., 2011; Thomas et al., 2016). Hence, they have a distinctive formation
history (Dressler, 1978).

There are several different formation scenarios for BCGs: (1) the continuous
merging of galaxies in the cluster potential, known as ‘galactic cannibalism’
(Ostriker & Tremaine, 1975; Hausman & Ostriker, 1978); (2) direct cooling from
the hot-gas halo onto a preexisting central galaxy (Cowie & Binney, 1977;
Fabian, 1994); (3) tidal stripping of cluster galaxies that pass near the
cluster center (Tovmassian & Andernach, 2012); (4) early merging during
cluster collapse (Merritt, 1985; Tremaine, 1990; West, 1994; Dubinski, 1998).
However, previous works have shown that mergers of smaller non-star-forming
early-type galaxies (so-called ‘dry mergers’) are the primary mechanism
through which BCGs grow from $z\sim 1$ to the present day (van Dokkum, 2005;
Bell et al., 2006; De Lucia & Blaizot, 2007). Liu et al. (2009) find that the
merging rate increases with both the galaxy mass and age, which leads to the
large fraction of dry mergers in the nearby universe. Khochfar & Silk (2009)
suggest there are about $10\%\sim 20\%$ of massive galaxies with a dry merger
in the last Gyr.

Abell 407 (A407; $z=0.0462$) contains a compact group of elliptical galaxies
in a luminous matrix (Fig. 1). Several bright galaxies appear to be embedded
in a diffuse optical halo within a region of diameter $\sim$60 kpc (Schneider
& Gunn, 1982). It possibly provides a unique laboratory for the ongoing
formation of a giant cD galaxy. In addition, the BCG-hosted active galactic
nucleus (AGN) has a strong influence via ‘AGN feedback’ on its surroundings.
In the radio bands, A407 shows 400-kpc scale helically twisted jets (Shulevski
et al., 2015; Biju et al., 2017). Thus, A407 also provides a great opportunity
to study the interaction between the galaxy nuclear activity and the cluster
environment.

In this article, we make a detailed study of this remarkable system. This
paper is organized as follows. In Section 2, we summarise the previous optical
and radio observations of A407. Section 3 presents the Chandra data reduction
methods. Section 4 is the X-ray observation results. Section 5 is the
discussion of the results. Section 6 is our summary. Throughout this article,
we assume a cosmology with $H_{0}$ = 70 km s-1 Mpc-1, $\Omega_{\rm m}=0.3$,
and $\Omega_{\Lambda}=0.7$. The redshift of A407 $z=0.0462$ gives
$1^{\prime\prime}=0.907$ kpc. Uncertainties quoted in this paper are
1$\sigma$.

Figure 1:  X-ray and optical images of Abell 407 and its central region. The
central region hosts nine closely packed galaxies which are embedded within a
diffuse stellar halo. This system is possibly a rare cD galaxy precursor
assembling in a galaxy cluster environment. (a): Chandra background-subtracted
and exposure corrected image of A407 in the 0.5 - 2 keV band. (b): SDSS three-
color composite image (blue: $g$-band; green: $r$-band; red: $i$-band) of the
same field as the upper left panel. (c): Soft X-ray (0.5-2 keV) band image of
the central galaxies within a circle of 30′′ radius. (d): Hard X-ray (2-7 keV)
band image of the central galaxies. (e): SDSS three-color composite image of
the central galaxies. All the X-ray images are binned with a size of two
0.492′′ pixels and smoothed with a uniform Gaussian kernel of
1.97$\textquoteright^{\prime}$.  Table 1: Galaxy properties in the central
region of Abell 407

Galaxya | RA, DEC | czb |  $L_{Ks}$c | Massd | SMBH Masse | Net counts | log$L$ | log$L$ | HRf | log$L_{X}$g
---|---|---|---|---|---|---|---|---|---|---
| (J2000) | (${\rm km\ s^{-1}}$) | ($10^{10}$$L_{\odot}$) | ($10^{10}$$M_{\odot}$) | ($10^{8}$$M_{\odot}$) | (0.5-7keV) | (0.5-2keV) | (2-7keV) |  | (0.5-2keV)
G1 | 03h 01m 51.5s +35d 50m 28.2s | 14339 (481) | 8.5 | 12.63 | $3.88\pm 1.23$ | $23\pm 8$ | $40.02_{-0.33}^{+0.19}$ | $39.60_{-0.33}^{+0.19}$ | $-0.61_{-0.31}^{+0.31}$ | 40.82
G2 | 03h 01m 51.2s +35d 50m 22.0s | 13600 (258) | 2.7 | 3.07 | $0.53\pm 0.44$ | $14\pm 7$ | $40.25_{-0.24}^{+0.15}$ | $39.83_{-0.24}^{+0.15}$ | $<-0.16$ | 39.07
G3 | 03h 01m 51.8s +35d 50m 19.6s | 14119 (261) | 8.1 | 10.71 | $9.83\pm 2.96$ | $41\pm 9$ | $40.41_{-0.13}^{+0.10}$ | $39.99_{-0.13}^{+0.10}$ | $-0.46_{-0.20}^{+0.20}$ | 40.74
G4 | 03h 01m 51.7s +35d 50m 12.0s | 15030 (1172) | 4.7 | 6.32 | $0.40\pm 0.45$ | $13\pm 7$ | $39.29_{-0.18}^{+0.13}$ | $38.87_{-0.18}^{+0.13}$ | $-0.39_{-0.52}^{+0.52}$ | 39.93
G5 | 03h 01m 52.9s +35d 50m 12.0s | 13199 (658) | 2.2 | 2.95 | $4.52\pm 0.74$ | $42\pm 9$ | $40.38_{-0.12}^{+0.09}$ | $40.51_{-0.11}^{+0.09}$ | $0.21_{-0.21}^{+0.21}$ | 38.79
G6 | 03h 01m 52.4s +35d 50m 28.9s | 13299 (558) | 0.6 | 0.57 | $0.52\pm 0.65$ | $<19$ | $<40.68$ | $<41.06$ | – | 36.73
G7 | 03h 01m 53.2s +35d 50m 25.7s | 13990 (132) | 7.6 | 9.96 | $3.08\pm 1.00$ | $17\pm 7$ | $40.07_{-0.28}^{+0.17}$ | $39.65_{-0.28}^{+0.17}$ | $<-0.25$ | 40.65
G8 | 03h 01m 53.8s +35d 50m 26.5s | – | – | – | – | $<17$ | $<40.72$ | $<40.84$ | – | –
G9 | 03h 01m 54.5s +35d 50m 17.7s | 13520 (338) | 1.9 | 2.22 | $1.34\pm 0.68$ | $12\pm 7$ | $39.99_{-1.61}^{+0.30}$ | $39.43_{-1.61}^{+0.30}$ | $<-0.05$ | 38.56

  * •

a Galaxy names are the same as Biju et al. (2017).

  * •

b Galaxy velocity values are from NED. The values in parentheses are the
absolute value of difference between the galaxy velocities and the HI
absorption velocity (13858${\rm km\ s^{-1}}$) detected by Shulevski et al.
(2015). The HI gas has a similar velocity to that of the host galaxy for 4C
35.06.

  * •

c The $K_{s}$ band luminosity is derived from the 2MASS point source catalog
(PSC). We also measure the total $K_{s}$ band luminosity within 35′′ radius of
the cluster center, which contains all nine galaxies and their extended
emission. The total luminosity is $8.52\times 10^{11}L_{\odot}$.

  * •

d The galaxy stellar masses are derived from $L_{Ks}$ with the stellar mass-
to-light ratio $\log_{10}(M/L)=a_{\lambda}+b_{\lambda}\times\text{ color }$
(Bell et al., 2003).

  * •

e The SMBH masses are calculated by Biju et al. (2017) from bulge stellar
velocity dispersion.

  * •

f HR = (H - S)/(H + S), HR: Hardness ratio; H: count rate at the 2 - 7 keV
band; S: count rate at the 0.5 - 2 keV band.

  * •

g The predicted log($L_{\rm 0.5-2keV}$ / erg s-1) from the $L_{Ks}$-$L_{X}$
relation (Sun et al., 2007).

##  2 Previous Observations

###  2.1 Optical detection

Fig. 1(b) shows the optical image of the central region of A407 from the Sloan
Digital Sky Survey (SDSS). It shows a complex ensemble of at least nine
galaxies within $\sim$1 arcmin. The central galaxies are embedded in a diffuse
and low surface brightness stellar halo. Table 1 lists the optical $K_{s}$
band magnitudes, line of sight velocities, and masses of these nine galaxies.

The compact group of galaxies in A407 was first noticed by Zwicky & Zwicky
(1971). Biju et al. (2017) proposed to name this extraordinary system as
‘Zwicky’s Nonet’. It was later studied in more detail by Schneider & Gunn
(1982) who suggest this puzzling system is a snapshot of a forming cD galaxy.
They consider that it is unlikely to be a pre-existing central cD capturing
multiple galaxies. Instead, the cD is being created as a result of tidal
disruption/merger. In summary, a recent merger generated the large-scale
envelope and dynamical friction caused the remaining galaxies to merge.

###  2.2 Radio detection

The compact group also hosts a moderately luminous radio source, 4C 35.06. The
earliest detection of this source was at 1.4 GHz with the Cambridge one-mile
telescope (Riley, 1975). Bondi et al. (1993) studied this source with the Very
Large Array (VLA) at 1.4 $\&$ 5 GHz and found a structure with two lobes. The
Very Long Baseline Array (VLBA) at 5 GHz found that the large-scale radio
galaxy is most likely to be identified as the inner galaxy G3 (Liuzzo et al.,
2010). Later, Shulevski et al. (2015) studied this source with the Low-
Frequency Array (LOFAR) at 62 MHz and found the radio jets have a helical
morphology. Biju et al. (2017) has also taken multifrequency radio
observations of this source with the Giant Meter-wave Radio Telescope (GMRT)
at 610, 235, and 150 MHz. They revealed that the large-scale radio structures
show 400-kpc scale helically twisted radio jets and outer diffuse lobes.

The radio observations present an interesting helically twisted radio jet
structure that could be triggered by the merger of galaxies. To reveal more
details of the merging progress and identify the progenitor galaxy of the
large-scale jet structure firmly, higher spatial resolution radio or X-ray
data are still needed. X-ray observations can also help us to study the
interaction signatures between the hot gas and the radio AGN.

##  3 Observation and Data Reduction

The X-ray observation used in this paper was performed with the Chandra
Advanced CCD Imaging Spectrometer (ACIS). The observation ID is 18267 (PI:
Dharam V. Lal). The exposure time is 43.5 ks. We use Chandra Interactive
Analysis of Observation (CIAO, version 4.11) and calibration database (CALDB,
version 4.7.7) to reprocess the data following a procedure similar to that
described in Sun et al. (2009) and Ge et al. (2019). We reprocess a new level
2 event file using the chandra_repro script with the VFAINT mode correction.
We investigate the light curve in the band $0.5\sim 9.5$ keV from source-free
regions to remove the intervals affected by flares, using the deflare script.
Such intervals are defined as those deviating more than 3$\sigma$ from the
mean rate. We use the wavdetect script to detect discrete sources with the
point spread function (PSF) map created by the mkpsfmap script. Blank-sky
background files are processed with the blanksky script.

To obtain the background-subtracted and exposure corrected cluster X-ray
image, we scale the blank-sky background for each CCD according to the
exposure time and count rate in the 9.5-12 keV band. The corresponding
exposure maps are generated with the fluximage script. The final X-ray image
is shown in Fig. 1.

As explained in the Appendix of Sun et al. (2009), there are two basic
components in the quiescent Chandra background, the instrumental background,
and the cosmic X-ray background (CXB), which are not separately recorded in
the blank-sky background file. The CXB would be unphysically scaled if we
scale the blank-sky spectrum according to the flux of the instrumental
background. After subtracting the scaled blank-sky background, there would
still be some residual X-ray sky background in the spectrum.

To get a clean spectrum of the cluster, we use the method of ‘double
subtraction’ to remove the residual sky background. Based on the surface
brightness profile (SBP) in Fig. 2, we first choose an off-source region in
ACIS-I3 where the cluster emission is negligible and the sky background
dominates. Then the spectra are extracted with specextract script for both the
off-source region and the cluster region. The blank-sky background is rescaled
according to the count rate in the 9.5-12 keV band. Finally, the spectra of
the off-source region and the cluster region are jointly fitted with the
TBABS*(APEC+PL)+APEC model, with the normalizations connected by the ratio of
their areas. The APEC model with absorption represents the cluster emission
component. The power-law model with absorption represents the cosmic hard
X-ray background composed of unresolved X-ray point sources (Sun et al.,
2009). The second APEC model without absorption represents the residual sky
background (Vikhlinin et al., 2005). The spectra are fitted jointly with the
XSPEC package. We use the AtomDB (version 3.0.8) database of atomic data and
the solar abundance tables from Anders & Grevesse (1989). For the hydrogen
column density $N_{\rm H}$, we adopt the value $1.72\times
10^{21}\rm{cm^{-2}}$ from the NHtot tool (Willingale et al., 2013).

##  4 Analysis and Results

###  4.1 X-ray properties of central galaxies

Fig. 1(a) is the background-subtracted and exposure corrected image of A407 in
the 0.5-2.0 keV band. The central galaxies in a region with a radius of
$30^{\prime\prime}$ are shown in Fig. 1(c,d). We use the circles with a
$5^{\prime\prime}$ radius to mark the location of each galaxy. Fig. 1(b) is
the corresponding SDSS image. There are seven out of nine optical bright
galaxies detected in the soft X-ray (0.5-2.0 keV) image. Among the seven
galaxies, the soft X-ray emission from G1, G2, G4, G7, and G9 is weak, while
G3 and G5 host the brightest soft X-ray sources in the group. In the hard
X-ray (2.0-7.0 keV) image in Fig. 1(d), G5 is the only X-ray bright galaxy.

We derive net X-ray counts for the galaxies in both soft and hard X-ray bands.
For the count measurement, we use the local background to exclude the
contribution from the surrounding hot intracluster medium (ICM). Uncertainties
for net counts are given as $1\sigma$ and upper limits are given as $3\sigma$.
The results are listed in Table 1. As shown in Table 1, only G6 and G8 are not
detected. We give the upper limits of their net counts and exclude them in the
following HR and spectral analysis.

From the counts of the central galaxies, we calculate the hardness ratio for
each galaxy. The hardness ratio is defined as HR = (H - S)/(H + S), where S
and H are the net counts in the soft and hard X-ray bands (Wang et al., 2004).
The results are shown in Table 1. In the hard X-ray band, since the counts of
G2, G7 and G9 are close to the background, we could only give the upper limits
of their HR. G5 is the only source that has more counts in the hard X-ray band
than the soft band, which may suggest intrinsic absorption for its AGN. The
X-ray emission of other galaxies is soft.

We combine the spectra of G1, G2, G3, G4, G7, and G9 as they have similar
hardness ratios. We fit the combined spectra with an absorbed APEC thermal
model. The redshift and the abundance we adopt are 0.0462 and 1 solar value
(Schneider & Gunn, 1982). The fitting result shows that the mean temperature
of the X-ray emission from these six galaxies is $1.6^{+1.2}_{-1.3}$ keV. To
examine the potential contribution from AGNs and LMXBs, we also add a power-
law component with a photon index of 1.7 to fit the combined spectra, as did
in Sun et al. (2007). We find the 0.5-2 keV luminosity of the power-law
component is at most 30% of the total 0.5-2 keV luminosity, which suggests
that most of the flux arises from thermal coronae.

Due to the limited counts of each galaxy, we estimate their luminosities by
fitting a model with fixed parameters. The luminosities are given both in the
soft and hard X-ray bands. For G1, G2, G3, G4, G7, and G9, we use the single
absorbed APEC model with a temperature of 1.6 keV and 1 solar abundance. For
G5, we adopt the single absorbed power-law model with the typical AGN photon
index of 1.7. For G6 and G8, we give the estimated upper limits. All results
are listed in Table 1.

###  4.2 X-ray properties of cluster hot gas

We extract a SBP from the ICM in Fig. 2. SHERPA is used to fit the SBP with
the $\beta$-model (Cavaliere & Fusco-Femiano, 1976):

$I=I_{0}\left(1+x^{2}\right)^{1/2-3\beta},$ (1)

where $x=R/r_{\mathrm{c}}$, and $I_{0},r_{c}$, and $\beta$ are free
parameters. The ICM distribution can be reasonably fit by the $\beta$-model,
which is shown in Fig. 2. The best-fitting parameters are in Table 2. We also
check the SBP in the sector regions shown in Fig. 4 and the SBPs are similar
in different directions.

Figure 2: Surface brightness profile of A407 in 0.5-2 keV band. The dashed line marks the $r_{500}$ of A407, which is 774 kpc. The brightness profile is fitted with the $\beta$-model. The best fitting parameters are listed in Table 2.  Table 2: The $\beta$-model fitting results of A407’s surface brightness profile Parameters | Value
---|---
$I_{0}$ ($10^{-8}\text{ photons }\mathrm{s}^{-1}\mathrm{cm}^{-2}\operatorname{arcsec}^{-2}$) | $1.75\pm 0.07$
$r_{c}$ (arcsec) | $66.7\pm 2.9$
$\beta$ | $0.42\pm 0.01$

A global spectrum, extracted excluding the central region, gives a mean X-ray
temperature of 2.7 keV. This implies a cluster total mass of
${M_{500}}=1.4\times 10^{14}{M_{\odot}}$ and a radius of $r_{500}=774$ kpc
according to the $M_{500}-T_{500}$ relation from Sun et al. (2009). Here
$r_{500}$ refers to the radii within which the cluster mass density is 500
times the universe critical density and $M_{500}$ is the mass within
$r_{500}$. We also extract the spectrum from the annuli centered on the X-ray
peak to produce the temperature profile as shown in Fig. 3. The region of the
central galaxies is excluded to avoid their contamination. The radii of the
annuli gradually increase from 30′′ to 675′′. The radius is increased by steps
of 10′′ from 30′′ to 110′′ and 15′′ from 110′′ to 185′′. For the outer 4 large
regions, their radii are increased by 30′′, 60′′, 100′′, and 300′′
respectively. As detailed in Section 3, we use the double subtraction method
to fit the cluster’s spectrum. The energy range is chosen as 0.5-8.0 keV. The
best-fitting results are shown in Fig. 3. The temperature profile indicates
that A407 has a cool core (CC) and the gas temperature reaches a maximum,
$4.18^{+0.58}_{-0.50}$ keV, at 85 kpc. We also notice that the temperature
rises at $\sim$ 250 kpc, which implies there may exist hot regions.

Figure 3:  Fitting results of radial annular regions. Left-hand panel: Chandra
0.5-2 keV image overlaid with GMRT 610 MHz contours. The GMRT observation code
is 21_066 (PI: Joydeep Bagchi) with $5^{\prime\prime}$ FWHM resolution. The
X-ray image is binned with a size of two 0.492′′ pixels and smoothed with a
uniform Gaussian kernel of 1.97$\textquoteright^{\prime}$. The radius of the
rings gradually increases from 30′′ to 675′′ i.e. 27.2 to 612.2 kpc. Right-
hand panel: the temperature as a function of radius shows that A407 has a weak
cool core and the gas temperature reaches a maximum, $4.18^{+0.58}_{-0.50}$
keV, at 85 kpc. The temperature and abundance values are given in Table 3.

Figure 4: Fitting results of seven panda regions. Left-hand panel: Chandra
0.5-2 keV image overlaid with GMRT 610 MHz contours. The X-ray image is binned
with a size of two 0.492′′ pixels and smoothed with a uniform Gaussian kernel
of 1.97$\textquoteright^{\prime}$. The inner radius of the sector regions is
30′′ and the outer radius is 115′′. Right-hand panel: the temperature is
plotted against azimuthal region number. The regions that contain the radio
jets (plotted in red) have the maximum temperatures which are
$4.00_{-0.36}^{+0.36}$ and $3.57_{-0.41}^{+0.45}$ keV respectively. The
temperature and abundance values are listed in Table 4.

Alternatively, the spectra from sector regions are fitted to analyze the
interaction of radio jets and cluster hot gas. The inner radius of the sector
regions is 30′′ and the outer radius is 115′′. The fitting results are shown
in Fig. 4. The regions that contain the radio jets (region 1 and 4) have the
maximum temperatures which are $4.00_{-0.36}^{+0.36}$ and
$3.57_{-0.41}^{+0.45}$ keV respectively. Region 1 and 7 have the highest
abundances, $0.99_{-0.30}^{+0.39}$, and $0.68_{-0.21}^{+0.28}$, which
indicates that the higher abundance value may be produced as the jets uplift
high abundance gas from the galaxy core.

To analyze the gas structure of the cluster, we derive the ICM temperature
map. We extract spectra from ACIS-S3 where the cluster mainly fits. We
generate adaptive spatial bins to fit the cluster temperature using ‘Contour
binning’ (Sanders, 2006), an algorithm for binning X-ray data by following the
surface brightness of the X-ray image. After masking the point sources of the
exposure corrected X-ray image, we set the signal-to-noise ratio to 30, which
requires $\sim 900$ background-subtracted counts per bin. We set the geometric
constraint value to 1.3, ensuring the shape of bins is not too elongated. As
shown in Fig. 5, we use these regions to extract the spectra and response
files and fit the gas temperature. We use two regions (region 14 and 15) to
cover the radio jet regions and add the fitting result in the temperature map.
The detailed temperature of each region are listed in Table 5. We note that
regions 14 and 15, where the radio jets lie, have relatively higher
temperatures than others.

Figure 5: The temperature map generated by Contour Binning and overlaid with
GMRT 610 MHz contours. The temperature values are listed in Table 5. The white
numbers marked on the picture are the region identifiers. Regions 14 and 15,
where the radio jets lie, have relatively higher temperature than others.

Entropy is also a useful quantity to study the thermal gas history of a
cluster as it reflects ICM global properties. It is defined as
$K=T_{X}n_{e}^{-2/3}$ (Ponman et al., 1999; Lloyd-Davies et al., 2000), where
$T_{X}$ is X-ray temperature and $n_{e}$ is electron density. With the SBP and
temperature profile, we can calculate the entropy profile of A407. Assuming
spherical symmetry, the cluster’s deprojected hydrogen density profile as a
function of the physical off-center radius is (e.g. Cavaliere & Fusco-Femiano
1976; Sarazin 1988):

$\centering
n_{\mathrm{H}}=n_{0}\left(1+\frac{r^{2}}{r_{\mathrm{c}}^{2}}\right)^{-\frac{3}{2}\beta}\@add@centering$
(2)

where $n_{0}$ represents the central density and we use the equation (4) of Ge
et al. (2018) to derive it:

$\centering n_{0}=\frac{3600\times
180}{\pi}\sqrt{\frac{10^{14}4\sqrt{\pi}I_{0}\Gamma(3\beta)}{\left(\frac{n_{e}}{n_{H}}\right)\frac{f_{x}}{\eta}r_{c}\Gamma(3\beta-1/2)}}\@add@centering$
(3)

This equation uses parameters ($I_{0},\beta,r_{c}$) listed in Table 2 from the
$\beta$-model fit of the cluster image. $n_{e}$ and $n_{H}$ are the electron
and hydrogen number densities and $n_{e}\simeq 1.2n_{H}$. $\Gamma$ is the
gamma function. $\frac{f_{x}}{\eta}$ is the ratio between the X-ray photon
flux and the normalization of the APEC model. With the density profile and the
temperature map, A407’s entropy profile is generated and shown in the middle
panel of Fig. 6. We also calculate the central entropy of A407, which is 108
$\mathrm{keV}\mathrm{cm}^{2}$.

As detailed in Hudson et al. (2010), from the density profile and temperature
profile, we can estimate the average cooling time profile of the gas using:

$t_{\mathrm{cool}}=\frac{3}{2}\frac{\left(n_{\mathrm{e}}+n_{\mathrm{i}}\right)kT}{n_{\mathrm{e}}n_{\mathrm{H}}\Lambda(T,Z)}$
(4)

where $n_{\mathrm{i}}$ and $n_{\mathrm{e}}$ are the number density of ions and
electrons respectively. $\Lambda(T,Z)$ is determined from the cooling curve in
Schure et al. (2009). The cooling time profile is shown in the bottom panel in
Fig. 6. From the cooling time profile, we note that the cooling time of the
central region is 4.4 Gyr.

Hudson et al. (2010) test several parameters to distinguish cool core (CC),
weak CC and non-cool-core (NCC) clusters. They find that the central cooling
time (measured at 0.048 $r_{500}$) is the best parameter to distinguish among
the classes for low redshift clusters. Weak CC are characterized as having
central cooling times between 1 and 7.7 Gyr. They also suggest the entropy of
weak CC clusters is between $\sim 22$ and $\sim 150$
$\mathrm{keV}\mathrm{cm}^{2}$.

As the central cooling time of A407 is 4.4 Gyr and the central entropy is 108
$\mathrm{keV}\mathrm{cm}^{2}$, A407 is a typical weak CC cluster. The weak CC
size of A407 is $\sim 60$ kpc where the cooling time equals the time from
$z=1$ to $z=0$.

Figure 6: Top panel: Hydrogen density profile of A407. Middle panel: Entropy
profile of A407. Bottom panel: cooling time profile of A407. The central
region has an entropy of 108 $\rm{keV\ cm^{2}}$ and a cooling time of 4.4 Gyr.
See Section 4.2 for details.

##  5 Discussion

###  5.1 Merging activity of central galaxies

Two broad classes of physical processes are responsible for the interaction of
galaxies in high-density environments. The first class includes the
gravitational processes which include all sorts of tidal interactions. The
second class includes the hydrodynamic interactions between the galaxy ISM and
the hot ICM, such as viscous stripping, ram pressure stripping, and thermal
evaporation (Boselli & Gavazzi, 2006).

Previous observations of A407 indicate that it possibly provides unique
evidence for the formation of a BCG by mergers. Schneider & Gunn (1982)
suggest this system is a forming cD by merging events as it harbors a
remarkably compact group of galaxies and has an extended stellar envelope.
Biju et al. (2017) present the GMRT low-frequency radio images at 610, 235,
and 150 MHz and they find a very peculiar twisted jet system which may be
caused by the galaxy merging activities. Shulevski et al. (2015) reveal the
two episodes of AGN activity with the LOFAR data. The jet precession implies
that the AGN is intermittently active as it moves in the dense cluster core.
Edwards et al. (2020) investigate the integral spectroscopy of 23 BCGs and
determine the age, metallicity, velocity, and velocity dispersion of stars in
each. Their results are consistent with the scenario that BCG cores formed
long ago (with an average age of $13.3\pm 2.8$ Gyr) and the outer intracluster
light (ICL) formed more recently. However, they also notice some local BCGs
are still assembling and A407 is one such example.

Chandra’s high spatial resolution data reveal more information about the
merging process of A407. If it is a rapid merging, we would see signs of gas
stripping or no X-ray coronae at all. On the other hand, if the galaxies are
in a process of a slow ‘merger of equals’, we might expect to see X-ray
coronae associated with several of the bright galaxies, following the
$L_{Ks}$-$L_{X}$ relation (Sun et al., 2007). As detailed in Section 4, seven
out of nine optical bright galaxies are detected in the soft X-ray band.
However, the detection limit of Chandra with an exposure time of 43 ks is
$\sim 10^{40}\ \rm{erg\ s^{-1}}$. The $L_{X}$ of galaxies are close to this
limit. Therefore, the two remaining galaxies, G6 and G8, may also have faint
X-ray corona which could be confirmed with longer exposures.

One possible explanation for the existence of the galaxy corona is that the
galaxies are in the process of a ‘slow merge’ which doesn’t strip all the
galaxy gas halos. Another explanation is that the galaxies with coronae are
far from each other along the line of sight. This leads to the survival of the
coronae while some other galaxies, such as G6 and G8, likely lost their halos
in the merging progress. However, considering that the velocities of the
galaxies (Table 1) vary little, the ‘slow merge’ model is preferred in A407.
We also calculate the $L_{X}$ of each galaxy based on the $L_{Ks}$-$L_{X}$
relation. However, the merging activities may produce the diffuse stellar
halos of each galaxy. Therefore, we scale the $L_{Ks}$ of each galaxy with a
value of 2.35, which is the ratio of the total $Ks$ band luminosity inside the
radius of 35′′ and the sum of $L_{Ks}$ of the central nine galaxies. The
predicted $L_{X}$ values from $L_{Ks}$ are shown in the last column of Table
1, which are close to the observed $L_{\rm 0.5-2keV}$ listed in Table 1.
Despite the large scatter in the $L_{Ks}$-$L_{X}$ relation, this result
supports the ‘slow merge’ model that coronae of the galaxies are not disrupted
entirely in the merging progress.

We calculate the required mass to bind the central galaxies gravitationally
with the equation:

$M=\frac{3R\sigma^{2}}{G}$ (5)

where $\sigma$ is the line of sight velocity dispersion of central galaxies in
the rest-frame and $\sigma=584$ ${\rm km\ s^{-1}}$. The radius R is 35′′ (31.7
kpc) which contains all nine galaxies and their extended emission. The
required binding mass is $M_{\rm bind}=7.5\times 10^{12}M_{\odot}$. As a
comparison, we remove galaxy G4, which has a most different velocity from
those of other central galaxies. The value of $\sigma$ reduces to 413 ${\rm
km\ s^{-1}}$ and the binding mass reduces to $M_{\rm bind}=3.8\times
10^{12}M_{\odot}$.

We also calculate the total mass within 35′′ with the dynamical mass-to-light
($M/L$) relation, which includes the dark matter. Cappellari et al. (2006)
investigated the dynamical $M/L$ ratio of elliptical galaxies. They fitted the
correlation between the $I$-band $M/L$ and the $Ks$-band luminosity:

$(M/L_{I})=(1.88\pm
0.20)\left(\frac{L_{K}}{10^{10}L_{K,\odot}}\right)^{0.32\pm 0.05}$ (6)

The $I$-band luminosity within 35′′ of A407 is measured from SDSS data
($L_{I}=3.1\times 10^{11}L_{\odot}$) and the $Ks$-band luminosity is from the
2MASS data ($L_{ks}=8.5\times 10^{11}L_{\odot}$). Thus, the dynamical mass
within 35′′ is $M_{\rm d}=2.4\times 10^{12}M_{\odot}$. This value is close to
the result that calculated by Biju et al. (2017). They calculated the
dynamical mass within the radius of 30′′ as $2.2\times 10^{12}M_{\odot}$ from
the halo radial velocity dispersion. Thus, the central dynamical mass of A407
is close to the binding mass calculated above.

Many early-type galaxies (ETGs) have distinctive tidal tails. These tidal
tails may be produced during the merging process (Malin & Carter, 1983; van
Dokkum, 2005; Bennert et al., 2008; Janowiecki et al., 2010). We might also
expect such features in the optical band of A407. However, there are no
stellar trails detected in the SDSS $r$ band image in Fig. 1(c). Deeper
optical observations will help to reveal more details of the merger by
detecting such features.

###  5.2 The future BCG

One interesting question is, when the central galaxies merge, what will be the
mass of the central BCG? The stellar mass of a galaxy can be calculated from
the stellar $M/L$ ratio relation (Bell et al., 2003). We choose a circle with
a radius of 35′′, which contains all nine galaxies and their extended
emission. From the 2MASS $K_{s}$ band image, the total $K_{s}$ band luminosity
within this circle is $(8.5\pm 0.5)\times 10^{11}L_{\odot}$. Thus, the
corresponding stellar mass of the future BCG is $(7.4\pm 0.4)\times
10^{11}M_{\odot}$ using the stellar $M/L$ ratio relation (Bell et al., 2003).

There is a correlation between BCG stellar mass and the host cluster mass. It
can be fit with a power-law of the form $M_{\mathrm{BCG}}=\beta
M_{\mathrm{Cluster}}^{\alpha}$ where $M_{\mathrm{BCG}}$ is the BCG stellar
mass and $M_{\mathrm{Cluster}}$ is $M_{200}$ which represents the cluster mass
within $r_{200}$ (Lidman et al., 2012; Bellstedt et al., 2016). Accordingly,
we can check whether A407 and its BCG follow this mass relation. The cluster
mass $M_{500}=1.4\times 10^{14}{M_{\odot}}$ (from the $M_{500}-T_{500}$
relation; Sun et al. (2009)). We then convert ${M_{500}}$ to ${M_{200}}$ by
assuming the cluster follows an NFW mass profile (Navarro et al., 1997) which
gives $r_{500}/r_{200}=0.669$. The resulting ${M_{200}}$ of A407 is $1.9\times
10^{14}{M_{\odot}}$. We find the masses of A407 and its future BCG are
consistent with this relation. Thus the predicted value of the mass of the
future BCG is reasonable.

To estimate the merging time of this system, we use the mean merging time-
scale relation proposed by Kitzbichler & White (2008). They investigate the
major merger rates based on the Millennium N-body simulation and calibrate the
results from deep galaxy surveys. Their work shows that, for samples with
stellar masses $\mathrm{M}_{*}>5\times 10^{9}{M}_{\odot}$ and radial velocity
differences $\Delta v<3000\mathrm{~{}km}\mathrm{~{}s}^{-1}$, the
characteristic mean merging time-scale of galaxy pairs in the low-redshift
region ($z\leq 1$) could be fitted well by:

$\left\langle T_{\text{merge
}}\right\rangle=3.2\mathrm{Gyr}\frac{r_{\mathrm{p}}}{50\mathrm{kpc}}\left(\frac{M_{*}}{4\times
10^{10}h^{-1}{M}_{\odot}}\right)^{-0.3}\left(1+\frac{z}{20}\right)$ (7)

$\left\langle T_{\text{merge }}\right\rangle$, $r_{\mathrm{p}}$ and ${M}_{*}$
represent the mean merging time of galaxy pairs, projected separation (kpc),
and stellar mass of the galaxy pairs (${M}_{\odot}$). The projected distance
of A407’s central galaxies can be obtained from the optical image (Fig. 1).
The masses of galaxies are obtained from their $L_{Ks}$ which are derived from
the 2MASS point source catalog (PSC) (Table 1). However, galaxies are extended
in the 2MASS optical image. Thus, these $L_{Ks}$ are underestimated. As
detailed in Section 5.1, we apply a scale value of 2.35 to each galaxy to
derive $L_{Ks}$ and stellar mass. We obtain the total mass of each galaxy pair
and calculate the merging time-scale by equation 7. The merging time-scale
range is $0.3\sim 2.3$ Gyr which provides a rough estimate of the central
merging time of A407. We note that the estimation is based on the two body
merging time-scale, the merging process of this compact group is much more
complicated and the final merging time is likely larger.

###  5.3 Host of the Radio AGN 4C 35.06

Which galaxy is the host of 4C 35.06? The VLBA observations by Liuzzo et al.
(2010) show pc-scale radio emission centered at the galaxy G3, which suggests
that G3 is radio active now. Shulevski et al. (2015) derived the radio
luminosity of 4C 35.06 to be $L_{1.4\ \mathrm{GHz}}=2.5\times
10^{24}\,\mathrm{W}\,\mathrm{Hz}^{-1}$, which makes it a strong radio AGN.
They suggested that the rapid movement of galaxy G3 and its episodic AGN radio
activities are the reasons for the observed peculiar radio morphology. They
also suggest that G3 switched off its radio emission and then restarted while
it was moving to its current position, which leads to the offset inner double-
lobed morphology. Shulevski et al. (2015) also detected a broad HI absorption
feature (FWHM of 288 ${\rm km\ s^{-1}}$) on top of the strong radio core of 4C
35.06 at a velocity of $\sim$ 13858 ${\rm km\ s^{-1}}$. Assuming the HI gas
has a similar velocity to that of the host galaxy for 4C 35.06, only G2, G3,
and G7 have velocities within 300 ${\rm km\ s^{-1}}$ of that of the HI
absorption feature, as shown in Table 1. Moreover, strong radio AGN are
typically associated with massive galaxies. As listed in Table 1, G1, G3, and
G7 have the largest galaxy masses. Thus, G3 would be the most likely host from
these constraints, as G7 seems to be too far away from the radio center.
However, Biju et al. (2017) suggested that the host of the radio AGN is G6
which lies at the center of the principal jet direction. Though G6 is faint in
both optical and infrared light, they suggest it may have been stripped of the
majority of its outer halo stars in multiple tidal encounters, while still
retaining a dense stellar core at the center.

With the Chandra data, we could reveal more information of the central
galaxies. Best et al. (2005) and Hardcastle et al. (2007) suggest that radio
AGN activity is associated with the cooling of the hot gas from the halos of
elliptical galaxies and clusters. Sun (2009) presented a systematic analysis
of 152 groups and clusters, and claimed that every BCG with a strong radio AGN
($L_{1.4\ \mathrm{GHz}}>2\times 10^{23}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ ) has
an X-ray CC either in the large cluster CC class or the galaxy corona class.
Since the radio luminosity of 4C 35.06 is $L_{1.4\ \mathrm{GHz}}=2.5\times
10^{24}\,\mathrm{W}\,\mathrm{Hz}^{-1}$, which is larger than $2\times
10^{23}\,\mathrm{W}\,\mathrm{Hz}^{-1}$, it should have an X-ray CC. As
mentioned in Section 4, A407 is a weak CC cluster and G3 has the most
prominent X-ray corona and lies closer to the radio core. Therefore, we
suggest G3 is the most likely host of the luminous radio AGN. Of course, an
ambiguous determination of the host would require radio images at higher
angular resolution. As G5 hosts the brightest X-ray AGN in this group and
likely hosts a massive SMBH from the analysis by Biju et al. (2017), we cannot
exclude the possibility that G5 also contributes to the radio emission of 4C
35.06.

Figure 7: Chandra 0.5-2 keV image overlaid with VLA 1.5 GHz (white) and 4.9
GHz (green) contours.

###  5.4 Feedback from AGN

AGN can release tremendous amounts of energy to their surroundings. Boehringer
et al. (1993) noted the existence of radio-filled cavities in NGC1275 based on
ROSAT observations. McNamara et al. (2000) found a close association between
the radio lobes and X-ray cavities in Hydra A which could suggest the gas was
displaced as the radio lobes expanded. Thus X-ray cavities can provide a
direct way to measure the energy injected by AGN (Churazov et al., 2000,
2001).

The energy from radio jets are near those required to offset radiative cooling
(Fabian, 2012; McNamara & Nulsen, 2012).Nulsen et al. (2009) studied a sample
of nearby giant elliptical galaxies from Chandra observations. Their results
show that the average power injected by AGN is sufficient to stop the hot
atmospheres from cooling. Not only the temperature, heavy elements can also be
transported from the core by the large-scale outflows and change the ICM
abundance (McNamara & Nulsen, 2007; Simionescu et al., 2008; Kirkpatrick et
al., 2011; Kirkpatrick & McNamara, 2015).

Biju et al. (2017) suggested the possibility that radio jets in this source
might be heating and interacting with the ICM via a jet-mode feedback. The
temperature map of A407 does indicate the presence of feedback features
including heating and abundance enrich. As shown in Fig. 4, the regions that
overlap with the radio jets (region 1 and 4) have the maximum temperatures
i.e. $4.0_{-0.4}^{+0.4}$ and $3.6_{-0.4}^{+0.5}$ keV. This implies that the
gas around the radio jets is heated by the feedback. As can be seen from Table
4, region 1 and 7, which include the north-west radio jet, have the highest
abundances, $1.0_{-0.3}^{+0.4}$, and $0.7_{-0.2}^{+0.3}$ respectively. The
increase in temperature and abundance along the jet direction is also seen in
Fig. 5 and Table 5.

The power of cavities can provide a lower limit to the jet energy as it is a
direct evidence for AGN heating (Bîrzan et al., 2008). Bîrzan et al. (2008),
Cavagnolo et al. (2010), and O’Sullivan et al. (2011) investigate the scaling
relationship between mechanical jet power and radio luminosity. Based on their
work, we estimate the jet power’s lower limit in A407 from its 1.4 GHz radio
luminosity ($L_{1.4\ \mathrm{GHz}}=2.5\times
10^{24}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ ). The lower limit of the mechanical
jet power is $1.0\times 10^{44}\,\mathrm{erg}\,\mathrm{s}^{-1}$.

As detailed in Section 4.2, no cavity structure has been found in A407 yet. It
may need deeper X-ray observation to confirm. Assuming there exist two
cavities with the same scale of the 610 MHz radio contour in Fig. 4 and their
shape could be approximated by spheres with the radius of 40 kpc, we can
estimate the mechanical pV energy of the cavities based on the gas temperature
and density profile (McNamara & Nulsen, 2012). The total mechanical energy of
the two cavities is $5.8\times 10^{59}$ erg. As the mechanical jet power is
$1.0\times 10^{44}\,\mathrm{erg}\,\mathrm{s}^{-1}$, it would take 0.18 Gyr for
the radio AGN to supply the cavities’ mechanical pV energy. For comparison,
the central cooling time of A407 is 4.4 Gyr.

Entropy can also reflect ICM global properties. Cavagnolo et al. (2009)
present radial entropy profiles for 239 clusters from the Chandra X-ray
Observatory Data Archive. They find that most entropy profiles are well fitted
by a power law at large radii and a constant value at small radii:
$K(r)=K_{0}+K_{100}(r/100\mathrm{kpc})^{\alpha}$. $K_{0}$ is the typical
excess of core entropy above the best-fitting power law found at larger radii.
For clusters with temperature $T_{X}<4\,\mathrm{keV}$, the $K_{0}$
distribution peaks at $K_{0}\sim 15\,\mathrm{keV}\,\mathrm{cm}^{2}$. For
clusters with temperature $T_{X}>8\,\mathrm{keV}$, the $K_{0}$ distribution
peaks at $K_{0}\sim 150\,\mathrm{keV}\,\mathrm{cm}^{2}$. The mean temperature
of A407 is 2.7 keV and the entropy profile is shown in Fig. 6. From the
figure, we notice that the entropy profile of A407 is relatively higher
compared to other clusters with temperatures lower than $T_{X}<4$ keV in
Cavagnolo et al. (2009). The entropy at small radii is $\sim
200\,\mathrm{keV}\,\mathrm{cm}^{2}$. The higher entropy in A407 at small radii
might be caused by the AGN feedback.

In summary, A407 shows evidence that its ICM temperature, abundance, and
entropy are affected by AGN feedback, most likely from the radio jets.

##  6 summary

A407 contains a compact group of galaxies in its central region. We study A407
with the Chandra observation combined with previous radio and optical
observations. Our major results and conclusions are:

  * •

A407 has a weak CC within 60 kpc. The central region of A407 has a cooling
time of 4.4 Gyr and an entropy of 108 $\mathrm{keV}\mathrm{cm}^{2}$. The
${M_{200}}$ of A407 is $1.9\times 10^{14}\,{M_{\odot}}$.

  * •

Due to the existence of the central galaxy X-ray coronae, we suggest the
central galaxies are in the process of a ‘slow merge’. When these galaxies
merge, we predict the future BCG’s stellar mass would be $7.4\times
10^{11}{M_{\odot}}$ and the merging time-scale range is $0.3\sim 2.3$ Gyr. The
predicted BCG mass follows the mass relation between BCGs and host clusters
(e.g. Lidman et al. 2012; Bellstedt et al. 2016).

  * •

For galaxies in the central region of A407, G3 has the most prominent X-ray
thermal corona. Combining the results of X-ray and radio observations, we
suggest galaxy G3 is the host of radio AGN 4C 35.06.

  * •

We also find evidence of AGN feedback in the hot gas of A407. The regions that
overlap with the radio jets have higher temperatures and abundance than other
regions. The entropy of A407’s center is also relatively higher than other
clusters which may be due to the effects of AGN feedback.

In summary, A407 has a very unusual cluster core in optical, which is
populated by a cluster of galaxies instead of a single dominant galaxy as in
most clusters. We witness it in a crucial formation epoch accompanied by the
helically twisted jets and merging features which have the potential of
clarifying how BCGs originate. We present a study focused on the X-ray band to
reveal merging details of the central galaxies and implications of the AGN
feedback activities. A407 suggests the scenario in which AGN activity can be
triggered during the formation of the cluster BCG. Deeper X-ray observations
would reveal more details of the merging and AGN feedback process. Further
optical observation would help to confirm if there exist any stellar tidal
trails. More cases, from both low and high redshifts, are needed to reveal
more detail about the BCG formation processes.

## Acknowledgements

We thank the referee, Dr. Joydeep Bagchi, for comments. The scientific results
reported in this article are based on the observation made by the Chandra
X-ray Observatory, ObsIDs 18267, which is operated by the Smithsonian
Astrophysical Observatory (SAO) for and on behalf of the National Aeronautics
Space Administration (NASA). This publication makes use of data products from
the Two Micron All Sky Survey, which is a joint project of the University of
Massachusetts and the Infrared Processing and Analysis Center/California
Institute of Technology, funded by the National Aeronautics and Space
Administration and the National Science Foundation. We also present data
obtained with the Giant Meter-wave Radio Telescope (GMRT), which is run by the
National Centre for Radio Astrophysics of the Tata Institute of Fundamental
Research. We use the images from SDSS and the funding for SDSS is provided by
Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science,
and the Participating Institutions. We also made use of the NASA/IPAC
Extragalactic Database (NED) which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with NASA.
D.V.L. acknowledges the support of the Department of Atomic Energy, Government
of India, under project no.12-R&D-TFR-5.02-0700. L.J. is supported by NSFC
grant U1531248 and BICCAS grant 114332KYSB20180013. W.F. and C.J. acknowledge
support from the Smithsonian Institution and the Chandra High Resolution
Camera Project through NASA contract NAS8-03060.

## DATA AVAILABILITY

The Chandra raw data used in this paper are available to download at the
Chandra Data Archive111https://cxc.harvard.edu/cda/ and HEASARC Data
Archive222https://heasarc.gsfc.nasa.gov/docs/archive.html website. The reduced
data underlying this paper will be shared on reasonable requests to the
corresponding authors.

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Table 3: The fitting results of the radial annular regions.

Radius (arcsec) | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 | 100-110 | 110-125
---|---|---|---|---|---|---|---|---|---
Radius (kpc) | 27.2-36.2 | 36.2-45.3 | 45.3-54.4 | 54.4-63.4 | 63.4-72.5 | 72.5-81.6 | 81.6-90.7 | 90.7-99.7 | 99.7-113.3
Temperature (keV) | $2.89_{-0.30}^{+0.34}$ | $2.85_{-0.29}^{+0.35}$ | $3.25_{-0.35}^{+0.40}$ | $2.88_{-0.27}^{+0.31}$ | $3.67_{-0.41}^{+0.45}$ | $4.18_{-0.50}^{+0.58}$ | $3.68_{-0.42}^{+0.46}$ | $2.64_{-0.31}^{+0.37}$ | $2.99_{-0.30}^{+0.33}$
Abundance | $0.32_{-0.16}^{+0.23}$ | $0.31_{-0.15}^{+0.21}$ | $0.18_{-0.14}^{+0.19}$ | $0.47_{-0.19}^{+0.25}$ | $0.60_{-0.26}^{+0.35}$ | $0.76_{-0.32}^{+0.46}$ | $0.67_{-0.28}^{+0.38}$ | $0.17_{-0.13}^{+0.19}$ | $0.50_{-0.19}^{+0.25}$
Radius (arcsec) | 125-140 | 140-155 | 155-170 | 170-185 | 185-215 | 215-275 | 275-375 | 375-675 |
Radius (kpc) | 113.3-126.9 | 126.9-140.5 | 140.5-154.1 | 154.1-167.7 | 167.7-195.1 | 195.0-249.4 | 249.4-340.1 | 340.1-612.2 |
Temperature (keV) | $2.46_{-0.23}^{+0.30}$ | $2.43_{-0.27}^{+0.32}$ | $2.62_{-0.33}^{+0.39}$ | $2.29_{-0.33}^{+0.42}$ | $2.51_{-0.25}^{+0.28}$ | $2.78_{-0.29}^{+0.30}$ | $1.89_{-0.29}^{+0.36}$ | $1.82_{-1.18}^{+0.92}$ |
Abundance | $0.18_{-0.11}^{+0.14}$ | $0.54_{-0.23}^{+0.34}$ | $0.18_{-0.13}^{+0.19}$ | $0.13_{-0.11}^{+0.19}$ | $0.26_{-0.12}^{+0.16}$ | $0.53_{-0.17}^{+0.22}$ | $0.24_{-0.18}^{+0.30}$ | $0.23_{-0.23}^{+0.90}$ |

Table 4: The fitting results of the 7 panda shape regions. region | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
Temperature (keV) | $4.00_{-0.36}^{+0.36}$ | $3.08_{-0.30}^{+0.31}$ | $3.19_{-0.32}^{+0.36}$ | $3.57_{-0.41}^{+0.45}$ | $3.19_{-0.33}^{+0.42}$ | $2.62_{-0.24}^{+0.33}$ | $3.12_{-0.28}^{+0.27}$
Abundance | $0.99_{-0.30}^{+0.39}$ | $0.20_{-0.13}^{+0.16}$ | $0.31_{-0.16}^{+0.22}$ | $0.31_{-0.18}^{+0.23}$ | $0.30_{-0.16}^{+0.22}$ | $0.17_{-0.10}^{+0.14}$ | $0.68_{-0.21}^{+0.28}$
Table 5: The fitting results of the regions generated by Contour Binning. region | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
Temperature (keV) | $3.36_{-0.30}^{+0.37}$ | $3.18_{-0.30}^{+0.33}$ | $3.44_{-0.26}^{+0.27}$ | $3.04_{-0.27}^{+0.31}$ | $2.66_{-0.21}^{+0.26}$ | $3.09_{-0.36}^{+0.38}$ | $2.94_{-0.31}^{+0.36}$ | $2.81_{-0.37}^{+0.40}$
Abundance | $0.48_{-0.20}^{+0.28}$ | $0.25_{-0.13}^{+0.17}$ | $0.62_{-0.17}^{+0.20}$ | $0.42_{-0.17}^{+0.22}$ | $0.40_{-0.21}^{+0.32}$ | $0.39_{-0.16}^{+0.22}$ | $0.19_{-0.12}^{+0.21}$ | $0.30_{-0.11}^{+0.14}$
region | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Temperature (keV) | $2.01_{-0.22}^{+0.31}$ | $3.07_{-0.38}^{+0.46}$ | $2.99_{-0.28}^{+0.29}$ | $2.40_{-0.37}^{+0.45}$ | $2.36_{-0.34}^{+0.53}$ | $4.16_{-0.45}^{+0.52}$ | $3.53_{-0.40}^{+0.47}$ |
Abundance | $0.41_{-0.14}^{+0.18}$ | $0.10_{-0.07}^{+0.12}$ | $0.40_{-0.20}^{+0.29}$ | $0.21_{-0.14}^{+0.19}$ | $0.27_{-0.15}^{+0.20}$ | $1.55_{-0.56}^{+0.87}$ | $0.38_{-0.20}^{+0.28}$ |