# Visualization of Morse flow with two saddles on 3-sphere diagrams

Svitlana Bilun and Alexandr Prishlyak

###### Abstract

We describe all possible topological structures of Morse-Smale flows without
closed trajectories on a three-dimensional sphere, which have two sources, two
sinks, one saddle of Morse index 1, one saddle of Morse index 2, and no more
than 10 saddle connections. To classify such flows, a generalized Heegaard
diagram or Pr-diagram is used, which in this case consists of a sphere and two
closed curves, the intersection points of which correspond to saddle
connections. We have found all possible, up to homeomorphism, ways to embed
two circles in a 2-sphere with no more than 10 points of transversal
intersection and construct its planar visualisations.

## Introduction

In the paper, a topological classification of Morse flows on the three-
dimensional sphere $S^{3}$ is carried out using diagrams (configurations)
consisting of 2 closed curves without self-intersections, which intersect
transversally on the sphere in $R^{3}$. These diagrams are generalized
Heegaard diagrams for closed 3-manifolds and Pr-diagrams for 3-manifolds with
the boundary. We consider them as a graph of valence 4 with two selected
cycles. Since a Morse flow given a consistent ordering of fixed points defines
the structure of a Morse function, the given studies can be used to classify
these functions. At the same time, the Morse flow has the structure of a
gradient flow of the Morse function for some Riemannian metric [51].

The use of graph theory to classify dynamical systems is typical in two and
three dimensions. For Morse flows on surfaces, the first invariants were
called distinguishing graphs [2, 17, 50, 23, 24, 21, 22, 6]. After [19],
three-color graphs became popular [46, 1]. Then these invariants were
generalized to a wider class of dynamical systems, and other invariants were
also constructed [47, 44, 42, 40, 53, 58, 29, 27, 30, 28, 16, 44, 20, 7, 5,
14, 25].

For simple Morse functions on closed surfaces, the main invariant is the Reeb
graph[48, 15]. These graphs are also generalized to a wider class of functions
[34, 13, 12, 39, 36, 33, 45, 18, 59, 60, 56, 3, 52, 49].

In dimension 3, diagrams consisting of a surface and the curves embedded in
them are often used to classify flows and functions [55, 32, 57, 35, 38, 41,
37, 43, 26, 10, 4]. For polar Morse flows, they coincide with Heegaard
diagrams.

The main invariants of graphs and their embeddings in surfaces can be founded
in [31, 9, 54, 61, 11, 8].

The first section gives the basic definitions and topological properties
associated with the diagrams (configurations) of two closed curves on the
sphere. For each such diagram, an adjacency graph and its summary matrix are
considered, on the basis of which the classification of diagrams
(configurations) is carried out. The number of non-equivalent configurations
with 2, 4, 6, 8, 10 intersection points is counted, which are equal to 1, 1,
2, 4, 13, respectively.

In the final section, it is shown that the number of topologically non-
equivalent Morse flow on the 3-dimensional sphere, which have one fixed point
of index 1 and 2 and two sinks and sources each, is equal to the corresponding
number of non-equivalent diagrams. In this way, the number of topologically
non-equivalent Morse flow is calculated. At the same time, each intersection
point on the diagram corresponds to a single integral trajectory connecting
special points of index 1 and 2.

We express our gratitude to the students Eugenie Buket, Alex Danciger,
Christian Hatamian and Julia Pilina for their essential assistance in
performing the calculations and to the president of AUI Stepan Moskalyuk for
supporting these studies.

##  1 Basic constructions

We consider the classification of diagrams ( configurations) of two closed
curves without self-intersections on the sphere in $R^{3}$ that intersect
transversally in the section. Two configurations are considered equal or
equivalent if there is a homeomorphism of the sphere onto itself, which
translates the curves of one configuration into the curve of the other. For
each configuration, a scan can be constructed - a stereographic projection of
the sphere onto a plane from an arbitrary point of the sphere that does not
belong to any of the curves. It is easy to show that from the equivalence of
sweeps follows the equivalence of configurations, but not vice versa. An arc
of a curve is a part of a curve bounded by two points of intersection with
another curve. We call a region a part of the surface of the sphere, which is
bounded by arcs of curves and to which other regions do not belong. Each
configuration can have at most different sweeps than regions on the sphere.

Each configuration of two closed curves on the sphere is matched by an
adjacency graph constructed as follows: each region corresponds to a vertex of
the graph, the arc of the curve between two regions corresponds to an edge
incident to both vertices of the regions.

Example.

Figure 1:  sweep and the correspondent graph

Regions of the sphere can always be colored in 2 colors so that each arc of
the curve separates regions of different colors. Indeed, first we draw the
first curve on the sphere and color the two resulting areas. Next, we draw the
second curve and repaint the areas bounded by it. Therefore, we can decolorize
the sweep pattern into 2 colors so that each circle arc separates areas of
different colors. It follows that the adjacency graph is bichromatic and
bipartite according to the terminology [9, 61, 11, 8], i.e. the set of graph
vertices can be divided into two subsets (destinies) in such a way that the
ends of each edge belong to different destinies. Each edge of the graph
(circle arc) can also be assigned responsibility for one of the two colors
according to whether the arc belongs to one or another circle. Therefore, we
will introduce an additional marking of the graph:

Figure 2:  marked sweep and the correspondent marked graph

It follows from the construction that the degree of each vertex is even and
the same number of edges of different colors are incident to it - the main
invariant of the considered graphs.

Each bipartite graph corresponds to and is given by the composite adjacency
matrix of the bipartite graph [9], the matrix of graph or just a matrix. Let
$x_{1},\dots,x_{n}$ be the vertices of one color (of one fate), and
$y_{1},\dots,y_{m}$ be the vertices of another color (of the second fate).
Then in the matrix $M=M_{i,j}$:

$M_{i,j}=1$ if $x_{i}$ and $y_{j}$ is connected by an edge of first color,

$M_{i,j}=-1$ if $x_{i}$ and $y_{j}$ is connected by an edge of second color,

$M_{i,j}=-1$ if $x_{i}$ and $y_{j}$ is not connected by an edge.

Regarding matrices, the main invariant looks like this: the number of rows and
columns of the matrix is even, the sum of the elements of each row and column
is zero.

The matrix of the graph that was given in the example:

In our case, the graph is determined by its matrix with accuracy up to
arbitrary permutation of rows and columns or rotations by 90 o (corresponds to
recoloring of areas) or replacement sign of all variables (corresponds to the
recoloring of closed curves). Also, to simplify identification, let’s
introduce the defining vectors of the graph matrix - two degree vectors of
vertices of different colors. For the graph considered above

$(4,4,2,2,2),(6,4,4,2)\ \text{ or }\ \ (4^{2},2^{3}),(6,4^{2},2).$

The equivalence of two configurations implies the equivalence of their graphs,
which is equivalent to the equivalence of the matrices, and the equality of
the corresponding determinant vectors follows from it. And vice versa: from
the inequality of defining vectors follows the non-equivalence of matrices and
graphs, and from it follows the inequality of configurations. We will use
determinant vectors to differentiate matrices.

Now let’s consider a single crossing operation, with the help of which there
is a transition from 2 n points of intersection to 2( n+1 ) points of
intersection of curves on the sphere .

Consider an arbitrary region 1 and two adjacent regions 2 and 3, such that
edges 1 2 and 1 3 are of different colors (belong to different curves). The
operation of crossing edges 1 2 and 1 3 in region 1 is called intersection 2 1
3. It follows from the construction that operations 2 1 3 and 3 1 2 are the
same.

Figure 3:

When crossing, a new area 4 (from two edges) is formed , which is adjacent to
areas 3 and 2 and areas 1’ and 1’’ (obtained as a result of the splitting of
region 1 ), which are also adjacent to regions 3 and 2 each. Other areas that
were adjacent to area 1 are redistributed (hereinafter this redistribution
will be called ? - distribution) between areas 1’ and 1’’. At the same time,
areas 1’ , 1’’ , 4 retain the color of area 1 .

The following happens with the edges: instead of edges 1 2 and 1 3 , edges 1’
2, 1’’ 2, 1’ 3, 1’’ 3 appear , and edges 2 4 and 3 4 also appear , while the
color 1 2 remains at edges 1’ 2, 1’’ 2, 3 4 , and color 1 3 - at edges 1’ 3,
1’’ 3, 2 4 . Therefore, we have: when intersecting, the number of edges
increases by 4, and the number of regions - by 2.

Let’s consider what happens to the adjacency graph when intersecting:

Figure 4:

Vertex 4 is added and instead of vertex 1 two vertices 1’ and 1’’ are added
(while all three new vertices are of the same color) and two pairs of edges of
different colors are added. The invariant of parity of degrees of all vertices
and the same size of edges of different colors that are incident to one vertex
is naturally preserved during interpolation.

Now let’s consider the operation of interpolation on the graph matrix:

In any column (row), we will choose two arbitrary non-zero elements of
different signs $x_{i}$ and $x_{j}$ in rows (columns) $i$ and $j$,
respectively. Let’s cross out this column (row). Instead of it, we add 3
columns (rows) to the matrix as follows: in one of them, there are zeros in
all places except $i$ and $j$, in the i-th place there is $x_{j}=-x_{i}$, in
the j-th place there is $x_{i}=-x_{j}$; in the other two columns (rows) on the
i-th and j-th places are $x_{i}$ respectively and $x_{j}$, and other elements
of the crossed out column (row) are distributed ( x -distribution ) in these
two columns (rows) in their places according to the invariant.

This matrix shows the 2 1 3 opposition discussed above.

Next, consider the construction of intersections of closed curves on the
sphere for 2, 4, 6, 8, 10 intersection points. The number of configurations
will be denoted by N(2),…, N(10), respectively.

##  2  Two intersection points

For this case, we have a single configuration of closed curves on the sphere

Figure 5:

The corresponding matrix has the form :

Matrix 2-1

Thus, the following theorem holds.

###### Theorem 1

There is a unique configuration with 2 intersections: $N(2)=1.$

Next, we will stick to the following algorithm for building configurations:

1\. for each constructed matrix, we will consider all possible contradictions;

2\. for each extension, we consider all cases of the x-distribution using a
sweep-distribution;

3\. based on the new matrix, we construct a new matrix, a new graph and
defining vectors, and with the help of the previous sweep we construct a new
sweep;

4\. we use defining vectors and sweeps to determine the equivalent
configuration.

##  3  Fore intersection points

From the appearance of the previous matrix, the unity of the matrix, which
will be constructed from the previous one by means of multiplication,
immediately follows:

In this case, the x-distribution is uniform. Two views of the corresponding
graph :

Figure 6:

This graph corresponds to a single configuration of the intersection of closed
curves on the sphere and the following two equivalent sweeps.

Figure 7:

Thus, the following theorem holds.

###### Theorem 2

There is a unique configuration with 4 intersections: $N(4)=1.$

##  4  Six intersection points

In matrices and 4-1 it is possible to construct, without limiting the
generality, two intersections.

For each of the cases, the x -distribution is unique. The corresponding graphs
look like this:

Figure 8:

The corresponding sweeps

Figure 9:

Thus, the following theorem holds.

###### Theorem 3

There are 2 configurations with 6 intersections: $N(6)=2$.

##  5  Eight intersection points

###  5.1  Let’s consider all intersections of matrix 6-1

For all considered for this matrix , there are 4 cases of friction
x-distribution is uniform.

####  5.1.1 Case 1

Corresponding graph and sweep:

Figure 10:

The dotted line shows the sweep before stretching.

####  5.1.2 Case 2

Corresponding graphs and examples of sweeps of one configuration:

Figure 11:

####  5.1.3  Case 3

Corresponding graphs and examples of sweeps of one configuration:

Figure 12:

####  5.1.4 Case 4

Corresponding graphs and examples of sweeps of one configuration:

Matrix 8-4 is equal to matrix 8-1 up to the permutation of the columns, so
this case does not add new configurations.

###  5.2 Now consider the cross-section of matrix 6-2.

####  5.2.1  Case 1

Corresponding graphs and examples of sweeps of one configuration:

Figure 13:

It is clear that in this case the x-distribution is unique.

####  5.2.2 Case 2

For this case, the x-distribution is not unique.

a)

Corresponding graphs and examples of sweeps of one configuration:

Figure 14:

Corresponding graphs and examples of sweeps of one configuration:

b)

Corresponding graphs and examples of sweeps of one configuration:

Figure 15:

Matrices 8-6 and 8-7 do not form new configurations, because they are equal to
matrices 8-1 and 8-2 up to the permutation of the columns.

Now we can write out all possible configurations for 8 intersection points:

matrix 8-1

matrix 8-2, matrix 8-3 and matrix 8-5 (reversed)

, , ,

Figure 16:  All possible configurations with 8 intersections

Thus, the following theorem holds.

###### Theorem 4

There are 4 configurations with 8 intersections: $N(8)=4$.

##  6  Ten intersections

Later, we use the determinant vectors of the matrices in order to distinguish
or check the equality of the studied matrices.

###  6.1 We consider the intersection of matrix 8-1

####  6.1.1 Case 1

x -distribution is uniform.

The initial vectors and its matrix.

Figure 17:

####  6.1.2 Case 2

x -distribution is uniform.

The initial vectors and its matrix. Corresponding Count and sweep:

Figure 18:

####  6.1.3 Case 3

x -distribution is uniform.

The initial vectors and its matrix. Corresponding Count and sweep:

Figure 19:

####  6.1.4 Case 4

x -distribution is not uniform.

a)

The initial vectors and its matrix. Corresponding Count and sample sweeps.

Figure 20:

b)

The initial vectors and its matrix. Corresponding Count and an example sweep.

Figure 21:

####  6.1.5 Case 5

The initial vectors and its matrix. Corresponding Count and an example sweep.

Figure 22:

####  6.1.6 Case 6

x -distribution is not uniform.

a )

The initial vectors and its matrix. Corresponding Count and an example sweep.

Figure 23:

b )

The initial vectors and its matrix. Corresponding Count and sample sweeps.

Figure 24:

###  6.2  We consider the intersection of matrix 8-3

####  6.2.1  Case 1

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and examples of
sweeps.

Figure 25:

####  6.2.2  Case 2

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and examples of
sweeps.

Figure 26:

####  6.2.3  Case 3

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and sample sweeps.

Figure 27:

####  6.2.4  Case 4

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and sample sweeps.

Figure 28:

####  6.2.5  Case 5

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and examples of
sweeps.

Figure 29:

###  6.3 We consider the intersection of the 8-2 matrix

####  6.3.1  Case 1

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and examples of
sweeps.

Figure 30:

####  6.3.2  Case 2

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and examples of
sweeps.

Figure 31:

####  6.3.3  Case 3

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and examples of
sweeps.

Figure 32:

####  6.3.4  Case 4

x -distribution is uniform.

The initial vectors and its matrix. Corresponding graph and examples of
sweeps.

Figure 33:

####  6.3.5  Case 5

x -distribution is uniform.

The initial vectors. Corresponding graph and examples of sweeps.

Figure 34:

####  6.3.6  Case 6

x -distribution is not uniform.

a)

The initial vectors. Corresponding graph and examples of sweeps.

Figure 35:

b)

The initial vectors. Corresponding graph and examples of sweeps.

Figure 36:

####  6.3.7  Case 7

x -distribution is uniform.

The initial vectors. Corresponding graph and examples of sweeps.

Figure 37:

####  6.3.8  Case 8

x -distribution is uniform.

The initial vectors. Corresponding graph and examples of sweeps.

Figure 38:

###  6.4  Now consider the intersection of the 8-5 matrix.

####  6.4.1  Case 1

x -distribution is uniform.

The initial vectors 10-23. Corresponding graph and examples of sweeps.

Figure 39:

####  6.4.2  Case 2

x -distribution is not uniform.

a)

The initial vectors 10-24. Corresponding graph and examples of sweeps.

Figure 40:

b)

The initial vectors. Corresponding graph and examples of sweeps.

Figure 41:

###  6.5  Calculation of results

If the defining vectors of the configurations differ, then the configurations
also differ. Therefore, among the configurations given above, we will look for
equivalent ones only among those that have the same defining vectors.

Upon direct verification (one can equate to the non-equivalence of a graph or
matrix, or to equate to the equivalence of a sweep) we have: matrices 10-1 and
10-24 are equivalent; 10-2 and 10-9; 10-10 and 10-15; 10-3, 10-14 and 10-25;
10-4 and 10-13; 10-5, 10-12, 10-17 and 10-19; 10-7 and 10-16; 10-18 and 10-21.
Now we can write out all configurations from 10 points: 10-1 (fig. 17), 10-2
(fig. 18),10-3 (fig. 19),10-4 (fig. 20),10-5 (fig. 21),10-6 (fig. 22),10-7
(fig. 23),10-8 (fig. 24),10-11 (fig. 27),10-18 (fig. 34),10-20 (fig. 36),10-22
(fig. 38),10-23 (fig. 39).

Thus, the following theorem holds.

###### Theorem 5

There are 13 configurations with 10 intersections: $N(10)=13$.

##  7 Morse flow and the intersection of two closed curves on the sphere

We consider Morse flow ( gradient-like Morse-Smale vector fields) on a
3-dimensional sphere, which have one singular point of index 1 and 2 and two
sinks and sources each.

Figure 42:

One trajectory from each of the sources enters into the point of index 1. From
the point of index 2, one trajectory leads to each of the sources. Let $F_{1}$
be the regular neighborhood circle of the first two trajectories. Its boundary
is topologically equivalent to $S^{2}$. Let $F_{2}$ be the regular
neighborhood of their last two trajectories. Its boundary is also
topologically equivalent to $S^{2}$. Intersections of the trajectories
starting from the point of index 1 with the boundary of $F_{1}$ forms a closed
curve. In the same way, a closed curve is formed on the boundary of $F_{2}$.
We have: from each point of $\partial F_{1}$ a certain trajectory is going
out, to each point of $\partial F_{2}$ some trajectory is included .

Figure 43:

Thus, we can identify $\partial F_{1}$ and $\partial F_{2}$. Two closed curves
corresponding to points with indices 1 and 2 remain on the formed sphere. The
intersections of these curves were considered in the previous sections of the
paper. Each point of intersection of two curves corresponds to a saddle
connection, trajectory connecting fixed points of index 1 and index 2.
According to [55, 32, 57, 35, 38, 37, 43], the configuration of two closed
curves on the sphere defines the field with accuracy up to topological
equivalence.

When classifying the intersections of closed curves on the sphere, we
considered the configurations to be equivalent even if there was a
homeomorphism that translates the first curve into the second and vice versa.
When applied to flow classification, such configurations must be
distinguished.

Therefore, the results of the previous section can be adjusted as follows:
when calculating the number of topologically non-equivalent flows, each
configuration from the first section is checked for symmetry with respect to
the permutation of the curves. If the configuration turns out to be symmetric,
then (by construction) one flow corresponds to it, otherwise, an asymmetric
configuration corresponds to two topologically non-equivalent flows. Checking
for symmetry of a configuration is checking the reduced matrix of the
adjacency graph of this configuration. If the matrix remains equal to itself
when multiplied by -1 (exchange of curves by places) with accuracy up to
arbitrary permutation of rows and columns and rotation by $\pi/2$, then the
matrix corresponds to a symmetric configuration, otherwise the matrix
corresponds to an asymmetric configuration.

For cases with 2, 4, 6, and 8 intersection points (saddle connections), all
configurations are symmetrical and the number of topologically non-equivalent
fields is 1, 1, 2, 4, respectively. For the case with 10 intersection points,
only the 10-18 matrix is asymmetric, which corresponds to two configurations:

Figure 44:

Thus, there are 14 topologically non-equivalent Morse flows with 10 saddle
connections on $S^{3}$, in which there is one fixed point of index 1 and 2 and
two sinks and sources each.

## Conclusion

We found all possible configurations of two nested circles on a sphere with 2,
4, 6, 8 and 10 points of transversal intersection. The corresponding
configuration numbers are: 1,1,2,4, 13. They specify the Morse flows on the
three-dimensional sphere, respectively 1,1, 2, 4 and 14 flows. We hope that
the obtained results can be generalized to other flows.

## References

  * [1] O. Akchurin, S. Bilun, and A. Prishlyak.  Three-color graph as the 1-skeleton of the 2-sphere triangulation.  arXiv.org, arXiv:2209.05737 [math.CO], 25 p., 2022.  doi: 10.48550/arXiv.2209.05737.
  * [2] A. Andronov and L. Pontryagin.  Rough systems.  Doklady Akademii Nauk SSSR, 14(5):247–250, 1937.
  * [3] S. Bilun and A. Prishlyak.  The closed morse 1-forms on closed surfaces.  Visn., Mat. Mekh., Kyv. Univ. Im. Tarasa Shevchenka, 2002(8):77–81, 2002.
  * [4] S. Bilun, A. Prishlyak, and A. Prus.  Morse flows with fixed points on the boundary of 3-manifold.  arXiv.org, arXiv:2209.04019 [math.GT], 16 p., 2022.  doi: 10.48550/arXiv.2209.04019.
  * [5] A.V. Bolsinov and A.T. Fomenko.  Integrable Hamiltonian systems. Geometry, Topology, Classification.  A CRC Press Company, Boca Raton London New York Washington, D.C., 2004\.  724 p.
  * [6] G. Fleitas.  Classification of gradient-like flows on dimensions two and three.  Bol. Soc. Brasil. Mat., 6(2):155 –183, 1975.
  * [7] O.A. Giryk.  Classification of polar morse-smale vector fields on two-dimensional manifolds.  Methods Funct. Anal. Topology, 2(1):23 – 37, 1996.
  * [8] J. Gross and T. Tucker.  Topological graph theory.  Wiley, 1987.
  * [9] F. Harary.  Graph Theory.  Westview Press, 1969.
  * [10] Ch. Hatamian and A. Prishlyak.  Heegaard diagrams and optimal morse flows on non-orientable 3-manifolds of genus 1 and genus 2.  Proc. Int. Geom. Cent., 13(3):33–48, 2020.  doi: 10.15673/tmgc.v13i3.1779.
  * [11] P. Hilton and S. Wylie.  Homology theory: An introduction to algebraic topology.  Cambridge University Press, 1968.
  * [12] B.I. Hladysh and A.O. Prishlyak.  Topology of functions with isolated critical points on the boundary of a 2-dimensional manifold.  SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 13:050, 2017.  doi: 0.3842/SIGMA.2017.050.
  * [13] B.I. Hladysh and A.O. Prishlyak.  Simple morse functions on an oriented surface with boundary.  Журнал математической физики, анализа, геометрии, 15(3):354–368, 2019.  doi: 10.15407/mag15.03.354.
  * [14] O.A. Kadubovskyj.  Classification of morse-smale vector fields on 2-manifolds.  Visn., Mat. Mekh., Kyïv. Univ. Im. Tarasa Shevchenka, 14:85–88, 2005.
  * [15] A.S. Kronrod.  Functions of two variables.  Russian Mathematical Surveys, 5:24–134, 1950.
  * [16] Z. Kybalko, A. Prishlyak, and R. Shchurko.  Trajectory equivalence of optimal Morse flows on closed surfaces.  Proc. Int. Geom. Cent., 11(1):12–26, 2018.  doi: 10.15673/tmgc.v11i1.916.
  * [17] E. Leontovich and A. Mayer.  On a scheme defining topological structure of decomposition into trajectories.  Dokl. Akad. Nauk SSSR, 103(4):557–560, 1955.
  * [18] D.P. Lychak and A.O. Prishlyak.  Morse functions and flows on nonorientable surfaces.  Methods of Functional Analysis and Topology, 15(03):251–258, 2009\.
  * [19] A.A. Oshemkov and V.V. Sharko.  Classication of morse-smale flows on two-dimensional manifolds.  Matem. Sbornik, 189(8):93–140, 1998.
  * [20] J. Palis.  On morse-smale dynamical systems.  Topology, 8:385–404, 1968.
  * [21] J. Palis and W. de Melo.  Geometric theory of dynamical systems. An introduction.  Springer-Verlag, New York-Berlin,, 1982.  xii+198 p.
  * [22] J. Palis and S. Smale.  Structural stability theorems.  Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), 1970.
  * [23] M.M. Peixoto.  On structural stability.  Annals of Mathematics, 69(1):199 – 222, 1959.  doi: 10.2307/1970100.
  * [24] M.M. Peixoto.  On the classication of flows of 2-manifolds.  Dynamical Systems (Proc. Symp. Univ. of Bahia, Salvador, Brasil, 1971), pages 389–419, 1973.
  * [25] D. Poltavec.  Equivalent polar morse-smale systems on two dimensional manifolds of genus 3.  In Abstract of International Conference on Topology and its Applications, Kiev, page 29, 1995.
  * [26] A. Prishlyak.  Complete topological invariants of morse-smale flows and handle decompositions of 3-manifolds.  J. Math. Sci., New York, 144(5):4492–4499, 2007.  doi: 10.1007/s10958-007-0287-y.
  * [27] A. Prishlyak and L. Di Beo.  Flows with minimal number of singularities in the boy’s surface.  Proceedings of the International Geometry Center, 15(1):32–49, 2022\.  doi: 10.15673/tmgc.v15i1.2225.
  * [28] A. Prishlyak and M. Loseva.  Topology of optimal flows with collective dynamics on closed orientable surfaces.  Proc. Int. Geom. Cent., 13(2):50–67, 2020.  doi: 10.15673/tmgc.v13i2.1731.
  * [29] A. Prishlyak and A. Prus.  Morse-smale flows on the torus with a hole.  Proc. Int. Geom. Cent., 10(1):47–58, 2017.  doi: 10.15673/tmgc.v1i10.549.
  * [30] A. Prishlyak, A. Prus, and S. Guraka.  Flows with collective dynamics on a sphere.  Proc. Int. Geom. Cent., 14(1):61–80, 2021.  doi: 10.15673/tmgc.v14i1.1902.
  * [31] A.O. Prishlyak.  On graphs embedded in a surface.  Russian Mathematical Surveys, 52(4):844, 1997.  doi:10.1070/RM1997v052n04ABEH002074.
  * [32] A.O. Prishlyak.  Equivalence of morse function on 3-manifolds.  Methods of Func. Ann. and Topology, 5(3):49–53, 1999.
  * [33] A.O. Prishlyak.  Conjugacy of morse functions on surfaces with values on a straight line and circle.  Ukrainian Mathematical Journal, 52(10):1623–1627, 2000.  doi: 10.1023/A:1010461319703.
  * [34] A.O. Prishlyak.  Conjugacy of morse functions on 4-manifolds.  Russian Mathematical Surveys, 56(1):170–171, 2001.  doi: 10.1070/RM2001v056n01ABEH000370.
  * [35] A.O. Prishlyak.  Morse–smale vector fields without closed trajectories on 3 -manifolds.  Mathematical Notes, 71(1):230–235, 2002.  doi: 10.1023/A:1013963315626.
  * [36] A.O. Prishlyak.  Morse functions with finite number of singularities on a plane.  Meth. Funct. Anal. Topol, 8:75–78, 2002.
  * [37] A.O. Prishlyak.  Topological equivalence of morse–smale vector fields with beh2 on three-dimensional manifolds.  Ukrainian Mathematical Journal, 54(4):603–612, 2002.
  * [38] A.O. Prishlyak.  Topological equivalence of Morse-Smale vector fields with beh 2 on three-dimensional manifolds.  Ukrainian Mathematical Journal, 54:603–612, 2002.  doi: 10.1023/A:1021035327909.
  * [39] A.O. Prishlyak.  Topological equivalence of smooth functions with isolated critical points on a closed surface.  Topology and its Applications, 119(3):257–267, 2002.  doi: 10.1016/S0166-8641(01)00077-3.
  * [40] A.O. Prishlyak.  On sum of indices of flow with isolated fixed points on a stratified sets.  Zhurnal Matematicheskoi Fiziki, Analiza, Geometrii [Journal of Mathematical Physics, Analysis, Geometry], 10(1):106–115, 2003.
  * [41] A.O. Prishlyak.  Regular functions on closed three-dimensional manifolds.  Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 8:21–24, 2003.
  * [42] A.O. Prishlyak.  Topological classification of m-fields on two-and three-dimensional manifolds with boundary.  Ukrainian Mathematical Journal, 55(6):966–973, 2003.
  * [43] A.O. Prishlyak.  Complete topological invariants of morse–smale flows and handle decompositions of 3-manifolds.  Fundamentalnaya i Prikladnaya Matematika, 11(4):185–196, 2005.
  * [44] A.O. Prishlyak and M.B. Loseva.  Optimal morse–smale flows with singularities on the boundary of a surface.  J. Math. Sci., New York, 243:279–286, 2019.  doi: 10.1007/s10958-019-04539-9.
  * [45] A.O. Prishlyak and K.I. Mischenko.  Classification of noncompact surfaces with boundary.  Methods of Functional Analysis and Topology, 13(01):62–66, 2007\.
  * [46] A.O. Prishlyak and A.A. Prus.  Three-color graph of the morse flow on a compact surface with boundary.  Journal of Mathematical Sciences, 249(4):661–672, 2020.  doi: 10.1007/s10958-020-04964-1.
  * [47] A. Prislyak and A. Prus.  Morse-smale flows on torus with hole.  Proc. Int. Geom. Cent., 10(1):47–58, 2017.  doi: 10.15673/tmgc.v1i10.549.
  * [48] G. Reeb.  Sur les points singuliers d’une forme de pfaff complétement intégrable ou d’une fonction numérique.  C.R.A.S. Paris, 222:847—849, 1946.
  * [49] V.V. Sharko.  Functions on manifolds. Algebraic and topological aspects., volume 131 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1993.
  * [50] S. Smale.  Morse inequalities for dynamical systems.  Bull. Amer. Math. Soc., 66:46–49, 1960.
  * [51] S. Smale.  On gradient dynamical systems.  Ann. of Math., 74:199–206, 1961.
  * [52] С.В. Білун, А.О. Гагай, О.О. Ворончук, and M.В. Лосєва.  Деформації замкнених 1-форм Морса на замкнених поверхнях.  Праці Міжнародного геометричного центру, 6(4):52–56, 2013.
  * [53] В.М. Кузаконь, В.Ф. Кириченко, and О.О. Пришляк.  Гладкi многовиди. Геометричнi та топологiчнi аспекти.  Працi Iнcтитуту математики НАН України.—2013.—97.—500 с, 2013.
  * [54] Л. С. Понтрягин.  Основы комбинаторной топологии.  Наука, 1986.
  * [55] А.О. Пришляк.  Векторные поля Морса–Смейла с конечным числом особых траекторий на трехмерных многообразиях.  Доповiдi НАН України, 6:43–47, 1998.
  * [56] А.О. Пришляк.  Сопряженность функций Морса.  Некоторые вопросы совр. математики. Институт математики АН Украины, Киев, 1998.
  * [57] А.О. Пришляк.  Топологическая эквивалентность функций и векторных полей Морса—Смейла на трёхмерных многообразиях.  Топология и геометрия. Труды Украинского мат. конгресса, pages 29–38, 2001.
  * [58] А.О. Пришляк.  Векторные поля Морса–Смейла без замкнутых траекторий на трехмерных многообразиях.  Математические заметки, 71(2):254–260, 2002\.
  * [59] О.О. Пришляк.  Теорія Морса.  Київський університет, 2002.
  * [60] О.О. Пришляк.  Топологія многовидів.  Київський університет, 2015.
  * [61] У. Татт.  Теория графов.  Мир, 1988.

Taras Shevchenko National University of Kyiv

Email address<EMAIL_ADDRESS>Orcid ID: 0000-0003-2925-5392

Email address<EMAIL_ADDRESS>Orcid ID: 0000-0002-7164-807X


# The 320 EeV Fly’s Eye event: a key messenger or a statistical oddity?

Thomas Fitoussi  Gustavo Medina-Tanco  Juan Carlos D’Olivo

###### Abstract

Almost three decades ago, the Fly’s Eye experiment recorded the most energetic
cosmic-ray ever observed. With an energy of 320 EeV, this event is well beyond
the suppression region of the ultra-high energy cosmic rays (UHECR) spectrum.
Modern and larger observatories, with an exposure up to 60 times larger, have
never observed an event with even remotely comparable energy. Thus, if the
energy of the Fly’s Eye event was indeed well measured, as strongly suggested
by the data, then it remains a great mystery or an unbelievable stroke of
luck. At such high energies, the Universe is very opaque to electromagnetic
interacting particles, whether photons, protons or heavy nuclei, and therefore
its source must be relatively close. Using numerical simulations for the
propagation of protons and nuclei, we reexamine the problem by testing
different possibilities for the nature of the primary, the injection spectrum
and the location of the source. Based on these calculations, we show that the
most feasible scenario corresponds to a nearby ($\sim 2-3$ Mpc) bursting
source of heavy nuclei in the northern sky, which injected a hard spectrum
($\gamma\leq 1.5$) with an energy cut-off between $300$ and $1000$ EeV. Such
scheme generates a significant probability for the observation of one event by
Fly’s Eye combined with a null result of Telescope Array at the same energy.

##  1 Introduction

On October 15, 1991, the Fly’s Eye (FE) experiment in Utah recorded the most
energetic cosmic-ray ever observed. The so called Fly’s Eye event has a very
well determined energy of $320^{+92}_{-94}$ EeV with a clean fluorescence
profile. Moreover, the sky was clear that night and the atmospheric cascade
created by this event did not point towards the detector, which minimizes the
uncertainty due to contamination by direct Cherenkov light. Therefore, the
estimate of 220 EeV seems to be a rather solid lower limit for the energy of
the event, putting it far beyond the GZK cut-off [1]. Indeed, protons with
such a high energy cannot travel more than few tens of Mpc due to photo-pion
production interaction with the CMB [2, 3]. Heavier nuclei, on the other hand,
are effectively absorbed on an even shorter distance because of photo-
disintegration [4].

The nature of this cosmic ray with extremely high energy has not been properly
identify. The first study of the event [1] was not able to determine the exact
nature of the primary. More recently, the depth of the shower maximum
$X_{max}$ and the longitudinal profile of the event were recomputed, with
equally inconclusive results [5, 6]. The primary cosmic-ray can be either a
proton or a heavier nuclei (C, Fe). In fact, even a photon event cannot be
completely ruled-out at more than two sigmas. The arrival direction of the FE
event is instead reasonably well established at $\sim 20^{\circ}$ from the
anti-galactic center (right ascension: $85.2^{\circ}\pm 0.48^{\circ}$,
declination: $48^{\circ}\pm 6^{\circ}$, galactic latitude: $9.6^{\circ}$ ,
galactic longitude: $163.4^{\circ}$ [1]). On the other hand, the location of
the source on the sky is poorly constrained, due to the deflections of the
particle by the intergalactic magnetic fields which are not well known [7].
From the probability of absorption of a cosmic ray, it has been shown that the
source can not be further away than few Mpc for heavy nuclei and a hundred of
Mpc for protons [8, 9], or not more than tens of Mpc at 95% CL, according to
our own calculations (see below).

Unless the event was a statistical fluke and the cosmic ray travelled from the
source to the Earth without interacting with the photon background, the
primary particle must have left the source at an even higher energy. This make
the observation of the air shower produced by it more intriguing in the
absence of a counterpart in detectors with a much larger exposure. Even more,
non obvious source in other high energy channels, e.g., X-rays, gamma-rays or
neutrinos, is recognizable in the region of the sky around the direction of
the FE event, which could be plausibly associated with it. Much the same can
be said on radio galaxies, which are promising candidates, and of which there
are not many inside a 100 Mpc. Nonetheless, some candidates were considered in
the literature, including quasars and AGNs [8, 10], as well as a strong gamma-
rays burst observed by BATSE near the location of the event [11]. In a more
recent study [12], the particle is backtracked for a specific realization of
the Galactic and extragalactic magnetic fields, with cataclysmic events (like
the birth of a millisecond pulsar or a magnetar flare) in galaxies UGC04874 or
UGC03394 suggested as possible sources for a C, N, O or Fe primary.

To the extent that its nature and origin remain unknown, the FE event
continues to be challenging for physics and astrophysics after almost three
decades of its observation. Particularly puzzling is the fact that no other
event of such energy has been detected since then, even by subsequent larger
and more sensitive detectors: HiRes, Pierre Auger Observatory (PAO) or TA. The
total exposure of FE was of only 820 km2 sr yr [13, 14], while the one of
HiRes was of $\sim$ 5000 km2 sr yr [15, 16, 17, 18], and those of TA and PAO
are of 8300 and 67000 km2 sr yr [19, 20], respectively. The UHECR sky in the
South may differ from the one in the North, as indeed the data of PAO and TA
seem to indicate [21], rendering the potential source invisible to the PAO.
However, the problem still persists for HiRes and TA, which are at the same
location as FE and have not seen an event with such high energy. Also
intriguing is the fact that FE never observed a single event at 100 EeV
despite the reported detection at 320 EeV.

Having said that, we have not been able to find in the literature any strong
argument against the physical reality of the FE event. Therefore, it may be
worth having a new look at the problem, examining the implications of this
event in the context of the scenario that has been developing in the last
years as a result of the measurements of the largest experiments currently in
operation.

The organization of the paper is as follows. In Sec. 2, we explore with the
aid of a simple model the likelihood of the FE event and the simultaneous null
results of TA and HiRes, assuming that the true UHECR background spectrum is
well described by TA. We extend this analysis by adding a secondary component
and check the impact of the normalization and the hardness of the spectrum on
the before mentioned probability. In Sec. 3 and 4, we use numerical
simulations to analyze the impact of the propagation of iron nuclei and
protons on the observation of the secondary component. Then, we use that
information to propose a plausible scenario. Finally, our conclusion are
presented in Sec. 5.

##  2 Exploratory model

Some preliminary insight into the problem can be gained by using a simple
model, in which the effects of propagation are neglected.

For the purpose of the paper, we assume that the UHECR spectra published by
each experiment, regarding shape, is a good estimator of the incoming flux.
Nevertheless, we apply an energy rescaling to account for systematics
uncertainties and bring them into agreement in the energy region between
$10^{19}$ and $10^{19.3}$ eV. We start by applying the rescaling done in the
joint analysis lead by PAO and TA (-5.2% for TA). The FE spectrum is then
rescaled by a factor of -20%, which is compatible with the published
systematic errors [14], while the HiRes spectrum is rescaled by -5%.

Fig. 1 shows the number of events as a function of energy for each experiment,
computed by multiplying the corresponding rescaled spectra by its exposure in
each energy bin [data taken from 13, 22, 19]. Blue dots are for TA surface
detectors, red crosses for HiRes monocular and green squares for FE monocular.

Consider the all-sky spectrum as fitted by TA [23]:

$J_{TA}(E)\propto\left\\{\begin{array}[]{rcl}E^{-\gamma_{1}}\,,&&E<E_{\rm
ankle}\,,\\\ E^{-\gamma_{2}}\,,&&E_{\rm ankle}<E<E_{\rm break}\,,\\\
E^{-\gamma_{3}}\,,&&E>E_{\rm break}\,,\end{array}\right.$ (2.1)

where $E_{\rm ankle}=5.2\pm 0.2$ EeV, $E_{\rm break}=60\pm 7$ EeV,
$\gamma_{1}=3.226\pm 0.007$, $\gamma_{2}=2.66\pm 0.02$ and $\gamma_{3}=4.7\pm
0.6$.

Figure 1: Number of events expected for the exposures of TA, HiRes and FE
(blue, red and green dashed lines, respectively), by assuming that the
northern sky UHECR flux is well described by the TA spectrum. A hypothetical
secondary component is depicted in dotted lines for different power law
indexes $\gamma_{EE}$: $\gamma_{EE}=1$ for HiRes (blue) and TA (red),
$\gamma_{EE}=1$, 1.5, 2, 2.5 for FE (green from dark to light). The combined
spectra are depicted with solid lines (same colors).

Assuming that, to a considerable degree, TA, HiRes and FE see the same sky and
that the energy spectrum is well described by TA, using Eq. 2.1 we compute the
expected number of events for the three observatories. The results, normalized
according to the respective exposures, are presented in Fig. 1 (TA in blue,
HiRes in red, and FE in green). Let us denote by $P(\geq 1)$ the probability
of detecting at least one particle. If such a component were the only one
present in the UHECR flux and could be extrapolated to higher energies with
the same power law index, then we see that $P(\geq 1)\sim 5\times 10^{-3}$,
which means that the occurrence of the FE event would be a hundred times
beyond expectations. Even more, although TA has an exposure an order of
magnitude greater, so far it has not detected a single event in the same
spectral region. These facts highlight the peculiarity of the FE event,
seeming to point towards a statistical fluke. One can envision at least two
kinds of outliers: either a particle that traveled from a distant source
several mean free paths without, or with very few, interactions with the
background photons, or a nearby source that is responsible for increasing the
probability of FE observing an event of 300 EeV and, at the same time, allows
a non-negligible probability that TA has not seen anything comparable.

Although the first scenario can never be completely ruled out, as Fig. 2
shows, for protons the mean free path increases rapidly as the energy
decreases and it is therefore difficult to understand how, for a given
distance to the source, particles of lower energy have not being detected. In
the event that the primary was an iron nuclei, the mean free path is so small
that the source cannot be far away; indeed, it should probably be inside or
near the Local Group. Besides, Fig. 4a indicates that, regardless of the
energy injected at the source, the energy of the leading fragment degrades
below the energy range of the FE event after propagating a few Mpc. In the
case of a nearby source, it should be transient, presumably bursting, so that
its activity in other messengers, like high energy X-ray or $\gamma$-ray
photons, declined to undetectability long ago and only charged particles are
arriving at present, delayed by their deflection in the magnetic fields
involved.

In order to roughly model the second scenario, we introduce into the spectrum
an additional harder component, which would be associated to a rather local
source within the region of the sky that corresponds to the field of vision
common to both experiments, FE and TA. The flux of this secondary component
extends to extreme energies (EE) and is featured by a power law with an
spectral index $\gamma_{EE}$:

$J_{EE}(E)\propto E^{-\gamma_{EE}}.$ (2.2)

The normalization is done in such a way that TA observations are not violated,
namely, requiring that $J_{TA}+J_{EE}$ does not exceed the flux inferred by TA
at $10^{20.15}$ eV and does not predict more than one event at higher
energies. Accordingly, we impose the condition $J_{EE}/J_{TA}(\log E=20)\leq
0.1$.

The dotted lines in Fig. 1 show the number of expected events for FE due to
the secondary component for different hardness of the spectrum:
$\gamma_{EE}=1$, 1.5, 2, 2.5. The blue and red dotted lines represent this
component for $\gamma_{EE}=1$, normalized to HiRes and TA, respectively. The
combined contributions of both components are plotted using solid lines.

Figure 2: Mean free path of protons and Fe nuclei due to photo-pion production
and photo-disintegration, respectively, as a function of energy due to
interaction with the photon backgrounds involved.

From each predicted number of events, we can estimate the Poissonian
probabilities $P_{TA}(0)$ ($P_{HiRes}(0)$) of TA (HiRes) detecting zero events
and $P_{FE}(\geq 1)$ of FE detecting at least one event. Since these
measurements are independent, the probabilities for the combined observations
are given by the product of the individual probabilities. They are shown in
Fig. 3 for different values of $\gamma_{EE}$. The solid lines corresponds to
the combined probabilities for TA and FE, while those for the observations of
the three detectors, TA, FE and HiRes, are plotted in dashed lines.

As seen from Fig. 3, without a secondary component (blue lines), the combined
probability has a single strong peak at $\sim 10^{20.3}$ eV, but it flattens
at energies above this value when a sufficiently hard secondary spectrum
($1\leq\gamma_{EE}<1.5$) is incorporated. Therefore, such additional component
seems to be a necessary ingredient to bring the joint probability of the
results of all experiments to a level compatible with their observations. It
should be noted that, although small, the resulting probability is not
negligible.

Figure 3: Poissonian probability that FE see at least one event ($P_{FE}(\geq
1)$) while HiRes and TA see none ($P_{HiRes}(0)$ and $P_{TA}(0)$) versus the
energy, considering that they see the same spectrum ($J_{TA}+J_{EE}$). A power
law spectrum with spectral index $\gamma_{EE}=1$, 1.5, 2, 2.5 is injected by
the source.

The simple model considered here does not include the propagation of the
particles between the source and the observer and, therefore, the conclusions
are mainly qualitative. In what follows, we use numerical simulations to
analyze the impact that propagation have on the observation of the secondary
component. First we consider that the primary particle is an iron nuclei and
the we examine the case of a proton.

##  3 FE event as an iron nuclei

As already mentioned, in order to properly interpret observations on Earth, it
is important to assess the effects of the UHECR propagation in the
intergalactic medium, which are largely dependent on the nature of the
injected particle. The composition of cosmic rays at the highest energies is
still an open question. Above $10$ EeV, the PAO claims a decrease in the
elongation rate accompanied by a decrease in its shower-to-shower
fluctuations. Both effects suggest a gradual increase in the average mass
number up to $A\sim 14-20$ at $E\sim 30$ EeV. What is the composition at even
higher energies is anybody’s guess. Although initial TA data seemed to favor a
lighter, even pure proton, composition [24], the Auger-TA joint working group
has recently concluded that the results of the two experiments are compatible
at all energies once systematic and statistical uncertainties have been taken
into account [25, 26]. To better understand the influence that the identity of
the initial particle may have, we will analyse the extreme cases of an iron
nucleus (56Fe) and a proton. First, we will examine tthe possibility that the
FE event originated as an iron nucleus and lost energy and nucleons through
photo-disintegration during its journey to the Earth. The case in which the
emitted particle could have been a proton will be considered in the next
section.

Using CRPropa 3 [27] in 1D mode, we propagated $10^{5}$ iron nuclei for each
injected energy at the source, discarding the cosmological effects. Photo-
disintegration was computed off the photons of the cosmic microwave radiation
and the infrared background (model from ref. [28]). Figure 4 shows the
evolution of (a) the energy and (b) the mass number of the leading particle
versus the traveled distance, while losing nucleons through photo-
disintegration. The Fe nuclei were injected at 250 (purple), 320 (blue), 500
(yellow) and 800 (red) EeV. The 68% and 95% confidence level are shown to
highlight fluctuations and in Fig. 4b the isoenergy lines at 100, 226, 320 and
442 EeV are represented, where the second (fourth) value corresponds to the
lower (upper) limit of the energy range of the FE event.

Figure 4: Evolution of the energy (a) (top) and the mass number (b) (bottom)
of nuclei due to photo-disintegration for a primary iron nucleus injected at
energies of 250 EeV (mauve), 320 EeV (blue), 500 EeV (yellow) and 800 EeV
(red) versus traveled comoving distance ($D_{travel}=c\,t$, where $t$ is the
comoving time). The solid lines are the median values, while the dark and
light areas correspond to the 68% and 95% C.L. regions, respectively, for each
injected energy. The isolines show where the particle reaches 100, 150, 226,
320 and 442 EeV, where 226 (420) EeV is the lower (upper) limit of the FE
event.

The first thing to be noted (see Fig. 4a) is that, independently of the
initial energy, the leading particle loses energy rapidly as a function of
distance. Then, given the uncertainty in the energy for the FE event, as long
as the injection energy is greater than the value measured on Earth, the
source cannot be much farther than 5 Mpc, even taking into account
fluctuations. Moreover, the median behavior favors a source at around 3 Mpc or
less.

Fig. 4b, on the other hand, shows that the change in mass composition has a
strong dependence on the initial energy. An iron core emitted with an energy
of 320 EeV, or less, can travel several tens of Mpc without the mass number of
the nuclear fragments falling below 14. In contrast, above this energy, the
mass of the leading particle degrades rapidly in less than 10 Mpc and, taking
into account fluctuations, the mass of the arriving particle has a poor
correlation with the injected energy. In addition, the isoenergy lines of the
incoming particles indicate the existence of a pile-up, which implies a
distance to the source of the FE event of no more than 5 Mpc and, most likely,
about 3 Mpc. As discussed in Fig. 4, propagation results in a pile-up of
events inside the uncertainty energy range of the FE event around 320 EeV.

To establish this result in a more quantitative way and emphasize its
relevance, we run a new set of simulations with CRPropa. This time we used a
3D mode with and without a magnetic field. We consider a magnetic field with
an intensity of 1 nG and a coherence length of 1 Mpc, which are consistent
with the extreme values that one can expect for the Extragalactic Magnetic
Field. The Galactic magnetic field was discarded because at these energies it
can barely deflect a particle, all the more true for the FE event, which is
almost located towards the direction of the anti-galactic center. In order to
reduce the computation time and increase the number of observed events, a
source that emits isotropically was placed in the center of a sphere of radius
equal to the distance from the source to the Earth. All the events reaching
the sphere were recorded. The simulation was repeated for $2\times 10^{5}$ Fe
nuclei drawn from a power-law injection spectrum, between $10^{19}$ eV and
$10^{22}$ eV. One simulation was performed for every set of parameters of
distance, power-law index and magnetic field ($0$ or $1$ nG).

In addition to the simulated secondary component, we generated a large set of
events corresponding to the main diffuse component (the one observed so far by
TA) so that the sum of both components produces a combined spectrum that fits
the TA spectrum at lower energies (Eq. 2.1). The combined spectrum is scaled
at $\log E=20.25$, the energy of the last point of TA measurements [19]. In
order to account for the uncertainty in the energy reconstruction, the events
from the simulation are diluted with a Gaussian error
$\sigma=\sqrt{0.21^{2}+0.10^{2}}=0.23$ for TA (systematic and energy
resolution errors [19]).

For each experiment, we drew randomly a set of events corresponding to the
real number of events observed with $\log E\geq 19.3$. To take into account
the energy reconstruction for HiRes and FE, the energy of the events are
convolved with Gaussian errors corresponding to the difference between the
statistical error from the given experiment and TA. This gives
$\sigma=\sqrt{\sigma_{FE}^{2}-\sigma_{TA}^{2}}=\sqrt{0.28^{2}-0.21^{2}}=0.14$
for FE [1] and $\sigma=0.12$ for HiRes [18]. One thousand independent
realizations are produced in this way.

Fig. 5 shows (a) the predicted number of events at the detector (averaged over
the 1000 realizations), as well as the probability as a function of energy of
observation by FE and simultaneous non-observation by either (b) TA alone or
(c) TA and HiRes combined (see the figure caption for details). In accordance
with the previous discussions, we assume that the observed spectrum includes a
heavy (Fe) component of extreme energy originated in a bursting source located
at a distance of 2 Mpc, as suggested in Figure 4. This Fe component escapes
the source as a power low with varying spectral indexes $\gamma_{EE}=$ 1, 1.5
and 2. We performed 1000 independent realizations for each detector.

From the results of these simulations, we can compute the Poissonian
probabilities $P_{TA}(0)\times P_{FE}(\geq 1)$ and $P_{TA}(0)\times
P_{HiRes}(0)\times P_{FE}(\geq 1)$ as functions of energy, by counting for
each energy bin the number of times, among the 1000 independent realizations
of the observed spectra, that 0 events were observed by TA and HiRes and at
least 1 event was observed by FE. This gives us directly the probability of
reproducing the observations of these three experiments. As shown in Figs. 5b
and 5c, for each value of the parameter $\gamma_{EE}$, a peak appears in the
region corresponding to the energy uncertainty interval of the FE event, which
renders the observations plausible. If only TA and FE are considered, the
probabilities of TA non-observation and FE observation is $\sim 5.5$% for a
very hard spectrum, $\gamma_{EE}=1$, $\sim 4.5$% for $\gamma_{EE}=1.5$ and
$\sim 4$% for $\gamma_{EE}=2$. While the probabilities decrease when HiRes is
added, their respective values are still non-negligible: $\sim 4$%, 4% and
3.5%.

Figure 5:  (a) (top): Mean of the expected number of events for TA and FE
(blue and green lines, respectively) under the assumption that besides the
diffuse spectrum ($J_{D}$) there is an additional extreme energy component
($J_{EE}$) originated from an iron source located at a distance of 2 Mpc. A
power law spectrum with spectral index $\gamma_{EE}=1.0,1.5,2.0$ is injected
by the source, with a maximum energy at $E_{cut}=10^{21}$eV, and propagated
through the diffuse photon background. This component at Earth is shown by the
orange curves, normalized so that the combined spectrum ($J_{D}+J_{EE}$) fits
the TA spectrum at lower energies. (b) (middle): Poissonian probability that
FE detects at least one event ($P_{FE}(\geq 1)$) and TA none ($P_{TA}(0)$)
versus energy, considering that both experiments see the same spectrum for
different spectral indexes of the heavy component: $\gamma_{EE}=1.0$ (solid
lines), $1.5$ (dotted lines) and $2.0$ (broken lines). (c) (bottom): Idem as
(b) but including also the probability of simultaneous non-observation by
HiRes.

It must be noted that the injection of hard spectra at the source is not
unreasonable. In fact, a combined spectrum and composition fit with 5 nuclear
species performed by the Auger Collaboration [29], favors low spectral indexes
($\gamma\lesssim 1$). This may seem in contradiction with the expectations
from the most plausible acceleration mechanisms, e.g., 1st order Fermi
acceleration, which predicts power-laws with $\gamma_{acc}\gtrsim 2$. But
there is no contradiction if a distinction is made between the acceleration
spectrum and the spectrum of the particles that are actually able to escape
from the source into the intergalactic medium. In fact, photo-disintegration
in the surroundings of UHECR sources has been shown to harden the spectrum to
values of the spectral index as small as $\gamma_{esc}\sim 1$ [30], while
acceleration models involving Gamma-Ray Bursts (GRBs), unipolar inductors,
magnetic reconnection, and tidal disruption events have been suggested as
producing hard spectra with $0\leq\gamma_{acc}\leq 1.5$ [30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42].

Hence, the previous result strongly argues for a northern sky source of heavy
nuclei, injecting a hard spectrum (either due to acceleration or escape
mechanisms) located at $\sim 2-3$ Mpc, as a plausible explanation for the FE
event combined with a null result of TA at the same energy. Such a potent
source in the immediate neighborhood of the Local Group must be transient, and
far enough in the past as to be unobservable in high energy photons at
present. Therefore, stellar mass range candidates, e.g. GRB, are favored.

In this scenario, the source of the FE event could be just the same type of
source responsible for the general UHECR flux seen in both hemispheres, but
this particular one just happened to burst nearby and in a position of the sky
where only a northern instrument could see it. Therefore, the existence of
such source would not challenge the current interpretation of observations
either by TA or PAO: other sources of the same population would be just far
away and reproduce collectively a spectrum with the suppression and other
spectral features consistent with observations by both modern experiments.

A preliminary search for potential sources in the vicinity of the event did
not produce any galaxy that could host such event. The closest galaxy at less
than 15∘ from the FE event location is PGC 019156, which is located at 13 Mpc,
too far according to our analysis. Galaxies in the Local Group at less than 5
Mpc are at least 25∘ from the FE event. The absence of obvious candidate
hosting the source may be an indication of a deeper physical process or a poor
knowledge of the magnetic field inside the Local Group. A detailed study about
this latter possibility is in progress.

##  4 FE event as a proton

Protons don’t seem to be favored at present as a major component of the UHECR
flux at the highest energies. Nevertheless, they cannot be ruled out as part
of the flux coming from some specific source mechanism. In any case, they
indeed must be present at least as fragments of photo-disintegration.
Therefore, it is worth exploring protons as a candidate for the FE event.

The same analysis made in Sec. 3 is repeated here for a source injecting
protons. Fig. 6 shows the evolution of the energy of a proton due to photo-
pion production versus traveled distance. The median value and the
corresponding 68 and 95 % C.L. are shown for four different initial energies:
250, 320, 500 and 800 EeV. The method is the same as the one used for Fig. 4a.
The energy limits of the FE event are also plotted.

Figure 6: Energy evolution due to photo-pion production of protons with
initial energies of 100 EeV (mauve), 320 EeV (blue), 500 EeV (yellow) and 800
EeV (red) versus the traveled cosmological distance ($D_{travel}=c\,t$, where
$t$ is the cosmological time). The solid lines are the median values, while
the dark and light areas correspond to the 68% and 95% C.L. regions,
respectively, for each injected energy.

First, one can note that, due to statistical fluctuations, to the energies of
the FE event, 16% of the protons can travel between 10 and 20 Mpc without
interaction, and 2.5% an even longer distance. Particles injected at higher
energies still have a considerable probability to reach Earth inside the FE
event energy uncertainty band, from distances of several tens of Mpc.
Increasing the energy with which the proton is injected only steadily
increases the distance at which the FE event energy is reached (fluctuations
included). No particular distance scale emerges as favored, with the possible
exception of 20 Mpc, which englobes 95% probability of particles arriving
without interaction.

Fig. 7 is the proton equivalent to Fig. 5 but for a source at 20 Mpc. One can
easily recognize that, unlike the Fe injection, no pattern appears. In
particular, from Fig. 7b it can be seen that the only maximum of
$P_{TA}(0)\times P_{FE}(\geq 1)$ occurs at $E=10^{20.1}$ eV, independently of
the hardness of the injected spectrum. This a remarkable difference with
respect to the Fe injection, which has a peak around $\sim 320$ EeV,
consistent with the FE event.

Figure 7:  (a) (top): Mean of the expected number of events for TA and FE
(blue and green lines, respectively) under the assumption that besides the
diffuse spectrum ($J_{D}$) there is an additional extreme energy component
($J_{EE}$) originated from a proton source located at a distance of 20 Mpc. A
power law spectrum with spectral index $\gamma_{EE}=1.0,1.5,2.0$ is injected
by the source, with a maximum energy at $E_{cut}=10^{21}$eV, and propagated
through the diffuse photon background. This component at Earth is shown by the
orange curves, normalized so that the combined spectrum ($J_{D}+J_{EE}$) fits
the TA spectrum at lower energies. (b) (middle): Poissonian probability that
FE detects at least one event ($P_{FE}(\geq 1)$) and TA none ($P_{TA}(0)$)
versus energy, considering that both experiments see the same spectrum for
different spectral indexes of the heavy component: $\gamma_{EE}=1.0$ (solid
lines), $1.5$ (dotted lines) and $2.0$ (broken lines). (c) (bottom): Idem as
(b) but including also the probability of simultaneous non-observation by
HiRes.

Thus, contrary to the heavy nuclei scenario, no obvious specific pattern,
natural distance or energy scale arises which can favor the FE event at 320
EeV and the simultaneous non-observation of events between 100 and 320 EeV by
TA. Furthermore, the source should be relatively far away, tens of Mpc, making
it harder to justify the absence of another such source in the field of view
of PAO. Additionally, if it were indeed a separate kind of source emitting
high energy protons, those protons should also be observable at both
hemispheres, as a distinct light component, in the general composition
measurement at lower energies. Thus, though not conclusively, protons do not
seem to be favored as a candidate for the FE event.

##  5 Discussion

Extrapolating the current measurements, we show that the TA spectrum by itself
cannot provide a reasonable explanation for the observation of the FE event. A
secondary component with a hard spectrum is required. But this alone, without
considering the nature of the primary particle involved, is not enough to
explain why FE detected an event with extremely high energy ($\sim 320$ EeV)
and, at the same time, why the TA detector, with an exposure 10 times greater
than the one of FE, has not observed anything at those energies.

With the aim of taking into account these important characteristics of the
observations, in this work we study the propagation of iron nuclei and
protons. Based on the calculations, we show that iron nuclei injected with
energies greater than 320 EeV tend to reach this energy after traveling
distances between 2 and 3 Mpc, while the mass of the main fragment vary from N
to Fe. As a consequence, a source located at such distance and that emits a
power law spectrum with a limit of $10^{21}$ eV generates a spectrum on Earth
with an excess of events in the region around the energy of the FE event.

Since the incoming direction of the event is within the field of vision of
both experiments, besides the mere existence of the 320 EeV event, we faced
the problem of understanding why more advanced experiments like TA have not
observed a similar event. Contrary to normal intuition, we demonstrated that
the Fe pile-up results in a combined probability of both results,
$P_{TA}(0)\times P_{FE}(\geq 1)$ which can be as high as 5% for a hard
spectrum with $\gamma_{EE}\sim 1$. In this sense, it is important to remark
that a like hard spectrum ($\gamma_{EE}=1$) and a low cut-off energy
($10^{21}$ eV) are compatible with recent experimental results [43, 44]. The
small power law index does not necesarily have to be a consequence of the
acceleration mechanism, but rather the result of the interaction of the nuclei
with the environment of the source during the escape process.

A prominent Galactic source able to accelerate particle beyond $10^{20}$ eV is
not apparent inside the error box of the FE event, neither in $\gamma$-rays
nor $X$-rays. Therefore, it is more likely to be a transient source whose
activity ended in the past, a time long enough that only the component of
charged particles is arriving today. At such high energies the Galactic
magnetic field would only add a few degrees to an already large FE error box,
inside of which there are no conspicuous X-ray sources which could be capable
to accelerate particles up to 300 EeV [c.f., the Chandra Catalog 45] and, in
this way, significantly change our previous statement. The bursting source
should probably be related to an object that is the product of stellar
evolution. Due to its spatial and angular location, the source would only be
visible in the Northern Hemisphere, outside the field of view of the PAO,
explaining the fact that there is no counterpart of the FE event in this
detector.

In the event that the primary was a proton, we demonstrate that the highest
probability of detecting a particle with FE and not with TA is at $\sim$ 100
EeV instead of the $320$ EeV observed, which makes it difficult to justify the
existence of the FE event together with a null results from HiRes and TA.
Moreover, there is no reason why a source of protons should be located only in
the Northern Hemisphere and at a short distance, which conflicts with the lack
of detection by the PAO. In addition, the associated neutrino and gamma-rays
productions would also be important and can exceed the limits already set on
the neutrino and gamma-ray background [44, 46]. As a consequence, even though
the possibility of a secondary pure proton component with a hard spectrum
extending at higher energy cannot be completely ruled out, it looks much more
complicated to implement.

Finally, let’s briefly comment on the viability of a photon as a primary
particle. In principle, this cannot fully disregarded. We can envision two
possible scenarios: a conservative bottom-up and a more exotic top-down. For a
bottom-up scenario, an extragalactic source seems unlikely due to the opacity
of the Universe to high energy photons. If the source were Galactic, then it
would be very difficult to justify that the only messengers produced are UHE
photons. In fact, an energetic source should also be a very luminous source of
photons at lower energies and be easily identifiable by several astronomical
observatories within the error box of the FE event. To our knowledge this is
not the case. Regarding to the top-down mechanism, in theory, superheavy dark
matter is not excluded. Nevertheless, the FE event error box is clearly
located toward the outskirts of the Galaxy, where the dark matter density is
much smaller that in the center. Since the center of the Galaxy is in the full
view of the PAO, with a much larger exposure and no clear signature of UHE
photons, it does not seem that this scenario is favored.

## Acknowledgments

This work was supported by DGAPA-UNAM Grant PAPIIT IN103207 and by CONACYT
Grants CB 239660 and CB 240666. The authors acknowledge Mtro. Juan Luciano
Díaz González for his technical assistance with the computational cluster
infrastructure. One of the authors (TF) would like to thank Hélène Courtois
for her help in trying to locate a potential source of the FE event. TF also
acknowledges support from DGAPA-UNAM.

## References

  * [1] D. J. Bird et al., _Detection of a cosmic ray with measured energy well beyond the expected spectral cutoff due to cosmic microwave radiation_ , _ApJ_ 441 (1995) 144.
  * [2] K. Greisen, _End to the cosmic-ray spectrum?_ , _Phys. Rev. Lett._ 16 (1966) 748.
  * [3] G. T. Zatsepin and V. A. Kuz’min, _Upper limit of the spectrum of cosmic rays_ , _Soviet. Phys. - JETP Letters_ 4 (1966) 78.
  * [4] D. Allard, _Extragalactic propagation of ultrahigh energy cosmic-rays_ , _Astroparticle Physics_ 39-40 (2012) 33.
  * [5] M. Risse, _Primary particle type of the most energetic Fly’s Eye air shower_ , _Astroparticle Physics_ 21 (2004) 479.
  * [6] M. Risse et al., _On the primary particle type of the most energetic Fly’s Eye event_ , _Nuclear Phys. B - Proc. Sup._ 151 (2006) 96.
  * [7] R. Durrer and A. Neronov, _Cosmological magnetic fields: their generation, evolution and observation_ , _A &ARv_ 21 (2013) 62.
  * [8] G. Sigl, D. N. Schramm and P. Bhattacharjee, _On the origin of highest energy cosmic rays_ , _Astroparticle Physics_ 2 (1994) 401.
  * [9] J. W. Elbert and P. Sommers, _In search of a source for the 320 EeV Fly’s Eye cosmic ray_ , _ApJ_ 441 (1995) 151.
  * [10] J. P. Rachen, _Possible extragalactic sources of the highest energy cosmic rays_ , _Annals of the New York Academy of Sciences_ 759 (1995) 468.
  * [11] M. Milgrom and V. Usov, _Possible association of ultra–high-energy cosmic-ray events with strong gamma-ray bursts_ , _The Astrophysical Journal_ 449 (1995) L37.
  * [12] R. Gnatyk, Y. Kudrya and V. Zhdanov, _Search for the astrophysical sources of the Fly’s Eye event with the highest to date cosmic ray energy e= $3.2\cdot 10^{20}$ eV_, _Advances in Astronomy and Space Physics_ 6 (2016) 41.
  * [13] D. J. Bird, S. C. Corbat , H. Y. Dai, B. R. Dawson, J. W. Elbert, T. K. Gaisser et al., _The cosmic-ray energy spectrum observed by the fly’s eye_ , _The Astrophysical Journal_ (1994) .
  * [14] D. J. Bird et al., _Results from the Fly’s Eye experiment_ , in _AIP Conference Proceedings_ , pp. 839–854, AIP, 1995, DOI.
  * [15] D. R. Bergman and J. W. Belz, _Cosmic rays: the second knee and beyond_ , _J. Phys. G: Nucl. Part. Phys._ 34 (2007) R359.
  * [16] D. R. Bergman, _Observation of the GZK cutoff using the HiRes detector_ , _Nuclear Physics B - Proceedings Supplements_ 165 (2007) 19.
  * [17] M. Abbasi, R. U. Abe, T. Abu-Zayyad et al., _First observation of the Greisen-Zatsepin-Kuzmin suppression_ , _Phys. Rev. Lett._ 100 (2008) .
  * [18] J. Belz, _Overview of recent HiRes results_ , _Nuclear Physics B - Proceedings Supplements_ 190 (2009) 5.
  * [19] D. Ivanov, _Energy spectrum measured by the telescope array surface detector_ , 2012.  PhD thesis.
  * [20] T. T. Collaboration and T. P. Collaboration, _Pierre Auger observatory and Telescope Array: Joint contributions to the 35th international cosmic ray conference (ICRC 2017)_ , _arXiv:1801.01018 [astro-ph]_ (2018) [1801.01018].
  * [21] J. Biteau et al., _Covering the celestial sphere at ultra-high energies: full-sky cosmic-ray maps beyond the ankle and the flux sup- pression_ , _Proceedings of Ultra High Energy Cosmic Rays 2018_ (2018) 8.
  * [22] P. Sokolsky, _Observation of the GZK cutoff by the HiRes experiment_ , _Nuclear Physics B - Proceedings Supplements_ 196 (2009) 67.
  * [23] V. Verzi, D. Ivanov and Y. Tsunesada, _Measurement of energy spectrum of ultra-high energy cosmic rays_ , _ArXiv e-prints_ (2017) arXiv:1705.09111.
  * [24] M. Abbasi, R. U. Abe, T. Abu-Zayyad et al., _Indications of proton-dominated cosmic ray composition above 1.6 EeV_ , _Phys. Rev. Lett._ 104 (2010)  [0910.4184].
  * [25] Pierre Auger and T. Collaborations, _Report of the working group on the composition of ultrahigh energy cosmic ray_ , _JPS Conf. Proc. 9, 010016 (2016)_ (2016) doi:10.7566/JPSCP.9.010016 [arXiv:1503.07540].
  * [26] H. Hanlon et al., _Report of the working group on the mass composition of ultrahigh energy cosmic rays_ , _JPS Conf. Proc. 19, 011013 (2018)_ (2018) doi:10.7566/JPSCP.19.011013.
  * [27] R. A. Batista, A. Dundovic, M. Erdmann et al., _CRPropa 3 - a public astrophysical simulation framework for propagating extraterrestrial ultra-high energy particles_ , _J. Cosmol. Astropart. Phys._ 2016 (2016) 038.
  * [28] T. M. Kneiske, T. Bretz, K. Mannheim and D. H. Hartmann, _Implications of cosmological gamma-ray absorption: II. modification of gamma-ray spectra_ , _A &A_ 413 (2004) 807.
  * [29] P. Collaboration, _Combined fit of spectrum and composition data as measured by the Pierre Auger observatory_ , _Journal of Cosmology and Astroparticle Physics 4 (Apr., 2017) 038_ (2017) arXiv:1612.07155.
  * [30] M. Unger, G. R. Farrar and L. A. Anchordoqui, _Origin of the ankle in the ultrahigh energy cosmic ray spectrum and of the extragalactic protons below it_ , _Physical Review D 92 (Dec., 2015) 123001_ (2015) arXiv:1505.02153.
  * [31] N. Globus, D. Allard, R. Mochkovitch and E. Parizot, _UHECR acceleration at GRB internal shocks_ , _Monthly Notices of the Royal Astronomical Society 451 (July, 2015) 751?790_ (2015) arXiv:1409.1271.
  * [32] P. Baerwald, M. Bustamante and W. Winter, _UHECR escape mechanisms for protons and neutrons from GRBs, and the cosmic ray-neutrino connection_ , _Astrophys. J. 768 (2013) 186_ (2013) arXiv:1301.6163.
  * [33] D. Biehl, D. Boncioli, A. Fedynitch and W. Winter, _Cosmic-ray and neutrino emission from gamma-ray bursts with a nuclear cascade_ , _Astron. Astrophys. 611 (2018) A101_ (2017) arXiv:1705.08909.
  * [34] B. T. Zhang, K. Murase, S. S. Kimura, S. Horiuchi and P. Mészáros, _Low-luminosity gamma-ray bursts as the sources of ultrahigh-energy cosmic ray nuclei_ , .
  * [35] P. Blasi, R. I. Epstein and A. V. Olinto, _Ultra-high-energy cosmic rays from young neutron star winds_ , _The Astrophysical Journal Letters 533 (Apr., 2000) L123?L126_ (2000) arXiv:9912240.
  * [36] K. Fang, K. Kotera and A. V. Olinto, _Newly born pulsars as sources of ultrahigh energy cosmic rays_ , _The Astrophysical Journal 750 (May, 2012) 118_ (2012) arXiv:1201.5197.
  * [37] G. Kowal, E. M. de Gouveia Dal Pino and A. Lazarian, _Magnetohydrodynamic simulations of reconnection and particle acceleration: Three-dimensional effects_ , _The Astrophysical Journal 735 (July, 2011) 102_ (2011) arXiv:1103.2984.
  * [38] G. Kowal, E. M. de Gouveia Dal Pino and A. Lazarian, _First-order fermi acceleration driven by magnetic reconnection_ , _Monthly Notices of the Royal Astronomical Society 422 (May, 2012) 2474?2476_ (2012) arXiv:1201.6612.
  * [39] R. Alves Batista and J. Silk, _Ultrahigh-energy cosmic rays from tidally-ignited white dwarfs_ , _Physical Review D 96 (Nov., 2017) 103003_ (2017) arXiv:1702.06978.
  * [40] D. Biehl, D. Boncioli, C. Lunardini and W. Winter, _Tidally disrupted stars as a possible origin of both cosmic rays and neutrinos at the highest energies_ , _Sci. Rep. 8 (2018) 1082_ (2018) arXiv:1711.03555.
  * [41] B. T. Zhang, K. Murase, L. Oikonomou and Z. Li, _High-energy cosmic ray nuclei from tidal disruption events: Origin, survival and implications_ , _Phys. Rev. D96 (2017) 063007_ (2018) arXiv:1706.00391.
  * [42] C. Guépin, K. Kotera, E. Barausse, K. Fang and K. Murase, _Ultra-high energy cosmic rays and neutrinos from tidal disruptions by massive black holes_ , _Astron. Astrophys. 616 (2018) A179_ (2018) arXiv:1711.11274.
  * [43] T. P. Collaboration, _Combined fit of spectrum and composition data as measured by the Pierre Auger observatory_ , 1612.07155.
  * [44] R. A. Batista, R. M. de Almeida, B. Lago and K. Kotera, _Cosmogenic photon and neutrino fluxes in the Auger era_ , _JCAP_ 2019 (2019) 002 [1806.10879].
  * [45] I. N. Evans and others, _The CHANDRA Source Catalog_ , _The Astrophysical Journal Supplement Series_ 189 (2010) 37.
  * [46] A. van Vliet, R. A. Batista and J. R. H randel, _Determining the fraction of cosmic-ray protons at ultra-high energies with cosmogenic neutrinos_ , _arXiv:1901.01899 [astro-ph]_ (2019) [1901.01899].