# The icosahedral line configuration and Waldschmidt constants

Sebastian Calvo  Department of Mathematics, The Pennsylvania State University,
University Park, PA 16802<EMAIL_ADDRESS>
###### Abstract.

There is a highly special point configuration in $\mathbb{P}^{2}$ of 31
points, naturally arising from the geometry of the icosahedron. The 15 planes
of symmetry of the icosahedron projectivize to 15 lines in $\mathbb{P}^{2}$,
whose points of intersections yield the 31 points. Each point corresponds to
an opposite pair of vertices, faces or edges of the icosahedron. The symmetry
group of the icosahedron is $G=A_{5}\times\mathbb{Z}_{2}$, one of finitely
many exceptional complex reflection groups. The action of $G$ on the
icosahedron descends onto an action on the line configuration. We blow up
$\mathbb{P}^{2}$ at the 31 points to study the line configuration. The
Waldschmidt constant is a measure of how special a collection of points in
$\mathbb{P}^{2}$. In this paper, we study negative $G$-invariant curves on
this blow-up in order to compute the Waldschmidt constant of the ideal of the
$31$ singularities.

During the preparation of this article the author was partially supported by
the NSF FRG grant DMS-1664303 and NSF DMS-2142966.

##  1\. Introduction

In this paper, we study a line configuration $\mathcal{A}$ of $15$ lines and
$31$ points in $\mathbb{P}^{2}=\mathbb{P}^{2}_{\mathbb{C}}$ that naturally
arise from studying the symmetries of the icosahedron, the platonic solid
studied heavily by Klein [Kle58]. The icosahedron has 15 mirror planes that
projectivize to the 15 lines of $\mathcal{A}$. The 15 lines intersect at 6
quintuple points, 10 triple points and 15 double points. Each pair of opposite
vertices, faces and edges correspond to a quintuple, triple and double point
respectively.

We may consider $I\subseteq S=\mathbb{C}[x,y,z]$ to be the ideal of a reduced
collection of points in $\mathbb{P}^{2}$. Define the $m$-th symbolic power
$I^{(m)}$ to be $\cap_{i}I_{p_{i}}^{m}$ [Har18, pg 6]. Geometrically, this
ideal corresponds to curves having multiplicity at least $m$ at each $p_{i}$.
The Waldschmidt constant of an ideal $I$ is defined to be

$\widehat{\alpha}(I)=\lim_{m\rightarrow\infty}\frac{\alpha(I^{(m)})}{m}$

where $\alpha(I)$ is the least positive integer $t$ such that the graded piece
$I_{t}\neq 0.$ In this paper, we compute the Waldschmidt constant for
$I_{\mathcal{A}}$, the homogeneous ideal of singularities of $\mathcal{A}$.
The singularities are highly non-general points of $\mathbb{P}^{2}$. We
demonstrate similar techniques to those in [BDRH19] in order to calculate
$\widehat{\alpha}(I_{\mathcal{A}})$. The following is the main result of the
paper.

######  Theorem (Main Result).

Let $I_{\mathcal{A}}$ be the homogeneous ideal of singularities of
$\mathcal{A}$. Then

$\widehat{\alpha}(I_{\mathcal{A}})=\frac{11}{2}.$

Determining the Waldschmidt constant of a collection of points is difficult in
general. The Nagata conjecture [Nag59] states that for $s\geq 10$ very general
points in $\mathbb{P}^{2}$, a curve $C$ passing through each $p_{i}$ with
multiplicity at least $m_{i}$ satisfies

$\deg C\geq\frac{1}{\sqrt{s}}\sum_{i=1}^{s}m_{i}.$

The formulation of Nagata’s conjecture in terms of Waldschmidt constants
states $\widehat{\alpha}(I)=\sqrt{s}.$ From this perspective, the Waldschmidt
constant measures how special an arrangement of points in $\mathbb{P}^{2}$ is.
The Waldschmidt constant can be regarded as the reciprocal of a multi-point
Seshadri constant, a well-known measure of local positivity of a line bundle.
It is always true that $1\leq\widehat{\alpha}(I)\leq\sqrt{s}$. For special
configurations of points, such calculations of $\widehat{\alpha}(I)$ are non-
trivial but we expect $\widehat{\alpha}(I)$ to be smaller than $\sqrt{s}$. We
exploit a relationship between the 31 points and the group
$A_{5}\times\mathbb{Z}_{2}$ to alleviate the difficulty.

###  1.1.  Emphasis of the group action on the point configuration

The $60$ rotational symmetries of the icosahedron form a group isomorphic to
$A_{5}$. The full symmetry group of the icosahedron
$A_{5}\times\mathbb{Z}_{2}$ has order $120$, obtained by introducing the
reflection through the center of the icosahedron. This group is one of
finitely many exceptional complex reflection groups. For a $(n+1)$-dimensional
complex vector space $V$, a complex reflection group
$G\subseteq\textrm{GL}(V)$ is a finite group that is generated by
pseudoreflections: those elements that fix a hyperplane pointwise. These
groups were studied and classified in [ST54]. Considering the collection of
hyperplanes, given by pseudoreflections of $G$, and projectivizing $V$ yields
a hyperplane configuration in $\mathbb{P}^{n}$. In addition, the natural
action of $G$ on $\mathbb{P}^{n}$ permutes the hyperplanes. Configurations
admitted by complex reflection groups exhibit interesting characteristics
[DS21]. For $n=2$, we obtain a line configuration in $\mathbb{P}^{2}$. Line
configurations have been studied extensively [Har18, OT13] yet continue to be
subject of recent research [BDRH19, AD09].

The group $G$ then naturally acts on the homogeneous coordinate ring $S$ of
$\mathbb{P}^{2}$. An important characteristic of comflex reflection groups is
that the invariant ring of such groups is finitely generated [DK15]. Let
$G=A_{5}\times\mathbb{Z}_{2}$ The invariant ring $S^{G}$ is generated as a
$\mathbb{C}$-algebra by homogeneous forms $\phi_{2},\phi_{6},\phi_{10}$ where
the subscript denotes the degree of the form. The polynomials $\phi_{d}$ turn
out to be vital in considering curves vanishing to points of the point
configuration. This will allow us to construct curves of high multiplicities
at the singularities of $\mathcal{A}$ in order to bound
$\widehat{\alpha}(I_{\mathcal{A}})$ from above.

###  1.2. The blow-up of $\mathbb{P}^{2}$ at the 31 singularities.

To give a lower bound of $\widehat{\alpha}(I_{\mathcal{A}})$, we consider the
Picard group of the blow-up $X_{\mathcal{A}}$ of $\mathbb{P}^{2}$ at the $31$
singularities and we show certain divisor classes are not effective. This is
equivalent to showing certain divisor classes are nef. In addition, the action
of $G$ on $\mathbb{P}^{2}$ extends to Pic$(X_{\mathcal{A}})$. We will see that
the divisor class $40H-5E_{5}-7E_{3}-8E_{2}$ is $G$-invariant and nef, where
$H$ is the divisor class of a line and $E_{k}$ is the sum of exceptional
divisors over the points of multiplicity $k$ of $\mathcal{A}$.

###  1.3. Outline

In section 2, we review group theoretic facts and some representation theory
of $G$. We construct the corresponding line configuration $\mathcal{A}$ while
emphasizing its geometry with respect to the group action on $\mathbb{P}^{2}$.
In Section 3, we exhibit an upper bound for
$\widehat{\alpha}(I_{\mathcal{A}})$. We assume nefness of $D$ to give a lower
bound. In Section 4, we focus on studying $G$-invariant curves on
$X_{\mathcal{A}}$ in order to prove $D$ is nef. Lastly, in Section 5 we
compute the Waldschmidt constants of sub point-configurations naturally found
in the configuration $\mathcal{A}$.

##  2\. Preliminaries

###  2.1. The icosahedron

We return to the icosahedron. There are $15$ mirror planes of the icosahedron
in $\mathbb{R}^{3}$. On the icosahedron, five planes meet at each vertex,
three planes meet at the center of each face and two planes meet at the mid-
point of each edge. Figure 1 displays the marked icosahedron alongside the
line configuration $\mathcal{A}$ in $\mathbb{P}^{2}$. To obtain $\mathcal{A}$,
fix a pair of opposite vertices $\\{p,-p\\}$ and identify the plane $P$ wedged
between the pair of vertices that slices the icosahedron in half. Project the
mirror planes onto the plane containing $p$ parallel to $P$. These are the
lines of $\mathcal{A}$ and they meet at $6$ quintuple points, $10$ triple
points and $15$ double points. We say that $\mathcal{A}$ is the line
configuration corresponding to the group $G.$

Alternatively, one may construct $\mathcal{A}$ in $\mathbb{R}^{2}$ as follows.
Fix a regular pentagon $R$. Extend the edges of $R$ to lines. Draw the
diagonals of $R$ and line segments between the center of $R$ and each
vertex111Instead of connecting the center to a vertex of $R$, one may also
draw a line segment from the center to each midpoint of a side of $R$. This is
known to be called an apothem. . Extend these to lines. These are the $15$
lines of $\mathcal{A}$. In the following subsection, we construct the line
configuration algebraically.

Figure 1. An icosahedron marked with quintuple (white), triple (black) and
double (grey) points alongside the line configuration in the affine patch
$y-\omega z\neq 0$. Note that $5$ double points lie on the line at infinity.

###  2.2. The group $A_{5}\times\mathbb{Z}_{2}$

The group $G$ is generated by idempotent elements $g,h,i$. Let $\omega$ be the
golden ratio. There is a 3-dimensional representation $\rho$ given by

$\rho(g)=\begin{pmatrix}-1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\quad\rho(h)=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&1\end{pmatrix},\quad\rho(i)=\frac{-1}{2}\begin{pmatrix}\omega-1&\omega&1\\\
\omega&-1&\omega-1\\\ 1&\omega-1&-\omega\end{pmatrix}.$

with the following character values

$\begin{array}[]{c|rrrrrrrrrr}\rm class&\rm 1&\rm 2A&\rm 2B&\rm 2C&\rm 3&\rm
5A&\rm 5B&\rm 6&\rm 10A&\rm 10B\cr\rm
size&1&1&15&15&20&12&12&20&12&12\cr\hline\cr\chi_{\rho}&3&-3&1&-1&0&\omega&1-\omega&0&\omega-1&-\omega\cr\end{array}.$

There are 15 pseudoreflections as observed from the character values of
$\rho$. Indeed there $15$ elements of conjugacy class $2B$ whose trace is $1$.
This implies for each such element, the dimension of the eigenspace $E_{1}$ is
$2$. This is an equivalent condition to fixing a hyperplane pointwise. Now
considering the arrangement of these $15$ hyperplanes, the projectivization of
the arrangement in $\mathbb{P}^{2}$ admits $\mathcal{A}$. Let
$\widetilde{G}=A_{5}$, the image of $G$ in $\textrm{PGL}(3,\mathbb{C})$.

######  Remark 2.1.

The action of $\widetilde{G}$ on $\mathbb{P}^{2}$ with respect to the line
configuration is the following. Notice that the lines and points may be
defined over $\mathbb{Q}(\omega)$.

  1. (1)

The line $x=0$ is a line of $\mathcal{A}$. The 15 lines of $\mathcal{A}$ form
a single orbit of size $15$.

  2. (2)

The point $p_{2}=[0:0:1]$ is a double point and has stabilizer
$\widetilde{G}_{p_{2}}=D_{4}$, the Klein four-group. The double points of
$\mathcal{A}$ form a single orbit of size $15$ and each double point has
stabilizer isomorphic to $D_{4}$.

  3. (3)

The point $p_{3}=[1:1:1]$ is a triple point and has stabilizer
$\widetilde{G}_{p_{3}}=D_{6}.$ The triple points of $\mathcal{A}$ form a
single orbit of size $10$ and each triple point has stabilizer isomorphic to
$D_{6}$.

  4. (4)

The point $p_{5}=[\omega+1:0:2]$ is a quintuple point and has stabilizer
$\widetilde{G}_{p_{5}}=D_{10}$. The quintuple points of $\mathcal{A}$ form a
single orbit of size $6$ and each quintuple point has stabilizer isomorphic to
$D_{10}$.

The group $G$ acts naturally on the homogeneous coordinate ring $S$ of
$\mathbb{P}^{2}$. We may then consider the ring of invariants $S^{G}$, the
polynomials that are invariant under the action of $G$. The curves given by
these polynomials are $G$-invariant. The ring $S^{G}$ is generated by
particular polynomials $\phi_{2},\phi_{6},$ and $\phi_{10}$ of degrees $2,6,$
and $10$ [Kle58]. The invariant ring $S^{\widetilde{G}}$ is generated by these
polynomials in addition to a fourth invariant $\phi_{15}$. The polynomials
$\phi_{2},\phi_{6}$ and $\phi_{10}$ are algebraically independent, while
$\phi_{15}^{2}$ can be expressed in terms of the other generators. To obtain
the generating polynomials $\phi_{d}$, note that $G$ is an orthogonal group
since it is the symmetry group of the regular icosahedron. Since each element
of $G$ preserves distance, we have $\phi_{2}=x^{2}+y^{2}+z^{2}$ to be a degree
$2$ invariant. Recall from Section 2.1 the pair of opposite vertices and the
unique plane $P$ corresponding to this pair. There are $6$ such pairs of
opposite vertices, thus $6$ unique planes. We take $\phi_{6}$ to be the
product of the defining equations of these planes

$\phi_{6}=x^{4}y^{2}+y^{4}z^{2}+x^{2}z^{4}+4\,\omega\,x^{2}y^{2}z^{2}-\left(\omega+1\right)(x^{2}y^{4}+y^{2}z^{4}+x^{4}z^{2}).$

Note that each plane $P$ contains $5$ double points and so for each double
point $p_{2}$, we have $\phi_{6}\equiv 0$ modulo $\mathfrak{m}_{p_{2}}^{2}$,
the maximal ideal corresponding to $p_{2}$. Similarly, there ten pairs of
opposite faces with opposite centers. There is a unique plane sandwiched
between a pair of opposite faces that cuts the icosahedron in half. We take
$\phi_{10}$ to be the product of the defining equations of the planes

$\displaystyle\phi_{10}=$ $\displaystyle
x^{8}y^{2}+x^{2}z^{8}+y^{8}z^{2}+(3\omega-5)(x^{2}y^{8}+x^{8}z^{2}+y^{2}z^{8})+(3\omega-7)(x^{6}y^{4}+x^{4}z^{6}+y^{6}z^{4})$
$\displaystyle-(6\omega-11)(x^{4}y^{6}+x^{6}z^{4}+y^{4}z^{6})-(30\omega-40)(x^{6}y^{2}z^{2}+x^{2}y^{6}z^{2}+x^{2}y^{2}z^{6})$
$\displaystyle+(45\omega-60)(x^{2}y^{4}z^{4}+x^{4}y^{2}z^{4}+x^{4}y^{4}z^{2})$

Similarly, we have $\phi_{10}\equiv 0\mod\mathfrak{m}_{p_{2}}^{2}$ for each
double point. The polynomial ${\phi}_{15}$ is the defining polynomial of the
line configuration $\mathcal{A}$. It may be obtained as the Jacobian
determinant

$\phi_{15}=\begin{vmatrix}\partial\phi_{2}/\partial
x&\partial\phi_{6}/\partial x&\partial\phi_{10}/\partial x\\\
\partial\phi_{2}/\partial y&\partial\phi_{6}/\partial
y&\partial\phi_{10}/\partial y\\\ \partial\phi_{2}/d\partial
z&\partial\phi_{6}/\partial y&\partial\phi_{10}/\partial z\par\end{vmatrix}.$

The polynomial $\phi_{15}$ satisfies the following relation

$\displaystyle c\phi_{15}^{2}=$ $\displaystyle
125\phi_{10}^{3}+(1300\omega-2275)\phi_{2}^{2}\phi_{6}\phi_{10}^{2}+(-12\omega+16)\phi_{2}^{5}\phi_{10}^{2}+(46800\omega-75600)\phi_{2}\phi_{6}^{3}\phi_{10}$
$\displaystyle-(6360\omega-10335)\phi_{2}^{4}\phi_{6}^{2}\phi_{10}+(200\omega-320)\phi_{2}^{7}\phi_{6}\phi_{10}+(343872\omega-556416)\phi_{2}^{5}$
$\displaystyle-(84624\omega-136912)\phi_{2}^{3}\phi_{6}^{4}+(6916\omega-11193)\phi_{2}^{6}\phi_{6}^{3}-(188\omega-304)\phi_{2}^{9}\phi_{6}^{2}$

for an appropriate constant $c\in\mathbb{C}^{*}$.

######  Remark 2.2.

Continuing the theme of Remark 2.1, we record how $\widetilde{G}$ acts on
orbits of $\mathbb{P}^{2}$. The representation $\rho$ is irreducible and so
$G$ fixes no point of $\mathbb{P}^{2}$.

  1. (1)

The double points of $\mathcal{A}$ form an orbit of $15$ points.

  2. (2)

The triple points of $\mathcal{A}$ form an orbit of 10 points.

  3. (3)

The quintuple points of $\mathcal{A}$ form an orbit of $6$ points.

  4. (4)

The curves $\phi_{2}=0$ and $\phi_{6}=0$ intersect at and form an orbit of
$12$ points. These points do not lie on $\mathcal{A}$. One of the points is

$\left[\sqrt{\frac{1}{2}(5+\sqrt{5}}):1:1-\omega\right]$

and this point is not defined over $\mathbb{Q}(\omega)$.

  5. (5)

The curves $\phi_{2}=0$ and $\phi_{10}=0$ intersect at and form an orbit of
$20$ points. These points do not lie on $\mathcal{A}$. One of these points is

$[\zeta:\overline{-\zeta}:1]$

where $\zeta=e^{2\pi i/3}$. Similarly, one would need to extend to the field
$\mathbb{Q}(\omega,\zeta)$ to define these intersections.

  6. (6)

The curves $\phi_{6}=0$ and $\phi_{10}=0$ intersect at the double points of
$\mathcal{A}$. Each curve passes through a double point twice, therefore by
Bezout’s theorem, the curves only meet at the double points.

  7. (7)

A non-singular point of the line configuration has an orbit of $30$ points.

  8. (8)

Otherwise a point has an orbit of $60$ points.

##  3\. Waldschmidt Computations on the blow-up of $\mathbb{P}^{2}$

In this section, we compute an upper bound for
$\widehat{\alpha}(I_{\mathcal{A}})$ and discuss our approach to proving a
lower bound. To give such bounds, we consider the blow-up of $\mathbb{P}^{2}$
at the singularities of $\mathcal{A}$. In particular, we study certain divisor
classes on this blow-up. These divisor classes correspond to curves on
$\mathbb{P}^{2}$ with incidence relations to $\mathcal{A}$. To obtain a lower
bound, we rely on the nefness of a specific divisor $D$. We prove the nefness
of $D$ in Section 4. In Subsection LABEL:ss-sub-point, we consider the 3 point
configurations corresponding to the three types of $k$-uple points and
directly compute the Waldschmidt constant of these configurations.

###  3.1. Divisor classes on the blow-up

Denote the blow-up of $\mathbb{P}^{2}$ at the singularities of $\mathcal{A}$
by $X_{\mathcal{A}}$. Then $\textrm{Pic}(X_{\mathcal{A}})$ is generated by the
pullback of a line $H$ and 31 exceptional divisors. The divisor class of the
proper transform of the line configuration on the blow-up is

$A=15H-5E_{5}-3E_{3}-2E_{2}$

where $E_{m}$ is the sum of the exceptional divisors lying over the points of
multiplicity $m$ for $m\geq 2$. These divisor classes on the blow-up are
subject to the following intersections

$H^{2}=1,\hskip 17.07182ptE_{5}^{2}=-6,\hskip 17.07182ptE_{3}^{2}=-10,\hskip
17.07182ptE_{2}^{2}=-15,\hskip 5.69046pt\textrm{ and }\hskip 5.69046ptH\cdot
E_{i}=E_{i}\cdot E_{j}=0$

for $i\neq j$. For example, $A^{2}=-75$. With this, we now are able to bound
$\widehat{\alpha}(I_{\mathcal{A}})$ from above.

######  Lemma 3.1.

Let $I_{\mathcal{A}}$ be the homogeneous ideal corresponding to the
singularities of $\mathcal{A}$. Then

$\widehat{\alpha}(I_{A})\leq\frac{11}{2}.$

###### Proof.

Consider the divisor class

$D=40H-5E_{5}-7E_{3}-8E_{2}.$

For $k\geq 1$, we calculate the expected dimension of the linear series
$|kD+2H|$ to be

$\chi(kD+2H)={40k+4\choose 2}-6{5k+1\choose 2}-10{7k+1\choose
2}-15{8k+1\choose 2}=6+30k>0.$

Therefore $kD+2H+kA=(55k+2)H-10kE_{5}-10kE_{3}-10kE_{2}$ is an effective
divisor class, and there is some element in $I^{(10k)}_{\mathcal{A}}$ of
degree $55k+2$. This gives

$\widehat{\alpha}(I_{\mathcal{A}})=\lim_{m\rightarrow\infty}\frac{\alpha(I^{(m)})}{m}\leq\lim_{k\rightarrow\infty}\frac{55k+2}{10k}=\frac{11}{2}.\qed$

To give a lower bound for $\widehat{\alpha}(I_{\mathcal{A}})$, we assume the
nefness of $D$.

######  Lemma 3.2.

If $D=40H-5E_{5}-7E_{3}-8E_{2}$ is a nef divisor, then
$\widehat{\alpha}(I_{\mathcal{A}})=\frac{11}{2}$.

###### Proof.

Assume $D$ is nef and suppose for contradiction there exists a
$\beta\in\mathbb{Q}$ such that

$\widehat{\alpha}(I_{\mathcal{A}})<\beta<\frac{11}{2},$

so that $F=\beta H-E_{5}-E_{3}-E_{2}$ is effective. Then

$F\cdot D=40\beta-30-70-120=40\left(\beta-\frac{11}{2}\right)<0.$

This contradicts the nefness of $D$. Therefore
$\widehat{\alpha}(I)\geq\frac{11}{2}$. By the preceding lemma, we have
equality. ∎

##  4\. $G$-irreducible and $G$-invariant curves

In this section, we prove the divisor $D$ is nef. Together with Lemma 3.2,
Theorem 4.4 proves the main result of the paper.

###  4.1. A decomposition of the divisor class $D$

A curve $\mathcal{C}$ is $G$-irreducible if its irreducible components are in
a single orbit under the action of $G$. For example, $G$ acts transitively on
the lines of $\mathcal{A}$ and so the curve $\phi_{15}=0$ is $G$-irreducible.
A curve $\mathcal{C}$ is $G$-invariant if for every $g\in G$, $g(C)=C$.
Observe that $G$-irreducibility of a curve implies the curve is $G$-invariant.
We first consider the point-configuration of quintuple points.

######  Remark 4.1.

The group $G$ acts on $\mathbb{P}^{2}$. This action extends to a $G$-action on
$\textrm{Pic}(X_{\mathcal{A}})$. If a divisor in
$\textrm{Pic}(X_{\mathcal{A}})$ is effective, then we ask if there is a
$G$-invariant curve in the corresponding linear series. This suggests to we
pay attention to curves invariant under the action of $G$.

######  Observation 4.2.

Consider the following divisor classes

$\displaystyle A=15H$ $\displaystyle-5E_{5}-3E_{3}-2E_{2},$ $\displaystyle
B=\hskip 4.26773pt6H$ $\displaystyle-2E_{2},$ $\displaystyle C=30H$
$\displaystyle-2E_{5}-6E_{3}-6E_{2},$ $\displaystyle D=40H$
$\displaystyle-5E_{5}-7E_{3}-8E_{2}.$

  1. (1)

The line configuration $\mathcal{A}$ is $G$-irreducible.

  2. (2)

A curve $\mathcal{B}$ of divisor class $B$ can be given by the vanishing of
$\phi_{6}$. The curve $\mathcal{B}$ is a union of $6$ lines (Figure 2), all in
a single orbit under the action of $G$. It follows that $\mathcal{B}$ is
$G$-invariant.

  3. (3)

The divisor classes $A,B,C$ have negative self-intersection and $D^{2}=0.$

  4. (4)

The divisor class $D$ may be written as the linear combination $6D=4A+5B+5C$.

Figure 2. The curve $\phi_{6}=0$ is a union of the five solid lines and the
line at infinity.

######  Proposition 4.3.

The divisor class $C$ is effective and there is a $G$-irreducible curve
$\mathcal{C}$ of class $C$.

We dedicate Section 4.3 to give an equation for a $G$-irreducible curve
$\mathcal{C}$ of divisor class $C$. This proposition allows us to prove the
main result.

######  Theorem 4.4.

The divisor class $D=40H-5E_{5}-7E_{3}-8E_{2}$ is nef. Consequently

$\widehat{\alpha}(I_{\mathcal{A}})=\frac{11}{2}.$

###### Proof.

By Observation 4.2 and Proposition 4.3, the divisor classes $A,B,$ and $C$ are
effective and $G$-irreducible. The decomposition $6D=4A+5B+5C$ shows that $D$
is effective. Since an effective divisor will only intersect non-negatively on
irreducible components, it suffices to use the $G$-action on
$\textrm{Pic}(X_{\mathcal{A}})$ to show that $D$ intersects non-negatively on
its $G$-irreducible components. Since $D\cdot A=D\cdot B=D\cdot C=0$, $D$ is
nef. The Waldschmidt constant $\widehat{\alpha}(I_{\mathcal{A}})=11/2$ follows
from Lemma 3.2. ∎

###  4.2. On $G$-invariant curves

To prove Proposition 4.3, we produce an equation for the curve $\mathcal{C}$
and prove it is $G$-irreducible. Since $G$-irreducible curves are also
$G$-invariant, the following lemma says such defining equations are contained
in the subalgebra $T=\mathbb{C}[\phi_{2},\phi_{6},\phi_{10}]\subseteq S$.
Consider the map

$\displaystyle\varphi:\hskip 2.84544pt$
$\displaystyle\mathbb{P}^{2}\rightarrow\mathbb{P}(2,6,10)$ $\displaystyle
p\mapsto[\phi_{2}(p):\phi_{6}(p):\phi_{10}(p)]$

where $\mathbb{P}(2,6,10)$ is a weighted projective space. This space is
isomorphic to the quotient space $\mathbb{P}^{2}/G$ and $\varphi$ is the
quotient map.

######  Lemma 4.5.

Let $\mathcal{C}\subseteq\mathbb{P}^{2}$ be a $G$-invariant curve which does
not contain the line configuration $\mathcal{A}$. Then the defining equation
$f\in S$ of $\mathcal{C}$ is $G$-invariant and is contained in $T$.

###### Proof.

We recall the invariants $\phi_{2}$, $\phi_{6}$ and $\phi_{10}$ from the
preliminaries. The map $\varphi$ is a $60$-uple covering away from the line
configuration and the finitely many points in $(1)-(6)$ of Remark 2.2. In
particular, the map $\varphi$ is a local isomorphism at a general point
$p\in\mathcal{C}$. We have then that $\varphi(\mathcal{C})$ is defined by a
weighted homogeneous equation $g(w_{0},w_{1},w_{2})=0$ where
$w_{0},w_{1},w_{2}$ are the coordinates of $\mathbb{P}(2,6,10)$. Therefore the
pullback $\varphi^{*}g=f$ defines $\mathcal{C}$ and is contained in $T$. ∎

This prompts the following definition: for $m_{5},m_{3},m_{2}\geq 0$, let

$T_{d}(-m_{5}E_{5}-m_{3}E_{3}-m_{2}E_{2})$

be the finite-dimensional subspace of $T$ of homogeneous degree $d$ forms that
are $m_{5}$-uple at the $6$ quintuple points, $m_{3}$-uple at the 10 triple
points and $m_{2}$-uple at the $15$ double points of $\mathcal{A}$. Elements
of this space yield $G$-invariant curves in the linear series $|dH-
m_{5}E_{5}-m_{3}E_{3}-m_{2}E_{2}|$. Consider the following curves. There is a
unique curve given by the vanishing of $\psi_{2}\in T_{2}(0)$. Let
$\psi_{6}\in T_{6}(0)$ be the unique (up to scale) degree $6$ invariant such
that the curve $\psi_{6}=0$ passes through a double point and $\psi_{10}\in
T_{10}(0)$ be the unique (up to scale) invariant such that the curve
$\psi_{10}=0$ passes through a double point and a triple point. Additionally,
there is a unique invariant $\psi_{6}^{\prime}\in T_{6}(0)$ that passes
through a triple point. The invariants $\psi_{6}$,$\psi_{6}^{\prime}$ and
$\psi_{10}$ enjoy incidence relations with respect to $\mathcal{A}$, as the
following lemma shows.

######  Lemma 4.6.

Let $p,p^{\prime}\in\mathcal{A}\subseteq\mathbb{P}^{2}$ be $k$-uple points. If
$C$ is a $G$-invariant curve passing through $p$, then $C$ also passes through
$p^{\prime}$.

###### Proof.

By Remark 2.2, there is an element $g\in G$ such that $g(p)=p^{\prime}$. Since
$p$ vanishes at $C$, then $p^{\prime}=g(p)$ vanishes at $C=g(C)$. ∎

We take advantage of the vanishings at $k$-uple points of $\psi_{d}=0$ to give
an equation for $\mathcal{C}$. An additional exploit is to observe the action
of the stabilizer
$\widetilde{G}_{p}\subseteq\widetilde{G}\subseteq\textrm{PGL}(3,\mathbb{C})$
on the local ring $(\mathcal{O}_{p},\mathfrak{m}_{p})$ at a singularity
$p\in\mathcal{A}$.

######  Lemma 4.7.

Let $p\in\mathcal{A}\subseteq\mathbb{P}^{2}$ be a singularity and
$\widetilde{G}_{p}\subseteq G$ the stabilizer of $p$. Let $w$ be a linear form
not passing through $p$. If $\psi\in T$ is a $\widetilde{G}$-invariant
homogeneous form and vanishes at $p$, then $\psi$ vanishes to order at least
$2$ at $p$ and $\widetilde{\psi}=\psi/w^{d}$ lies in the $1$-dimensional
trivial $\widetilde{G}_{p}$-submodule of
$\mathfrak{m}_{p}^{2}/\mathfrak{m}_{p}^{3}$.

###### Proof.

The proof is analogous to [BDRH19, Lemma 4.8] but we describe the upshot in
the current setting. From Remark 2.1, the stabilizer $\widetilde{G}_{p}$ for a
singularity $p\in\mathcal{A}$ is $D_{2n}$ for $n=2,3$ or $5$. The key
observation used in the proof is that the linear characters of $D_{2n}$ have
order either $1$ or $2$, and therefore must act trivially on a tangent line at
$p$. ∎

Therefore, the previous two lemmas combine to state that if a $G$-invariant
curve passes through a $k$-uple point, it passes through every $k$-uple point
at least twice. This will allow us to restrict the terms of invariant form
vanishing at $p$.

###  4.3. Equation of a curve of divisor class $C$

For constants $\lambda_{i}\in\mathbb{C}$, define the invariant

$\psi_{30}:=\lambda_{1}\psi_{10}^{3}+\lambda_{2}\psi_{2}^{2}\psi_{6}\psi_{10}^{2}+\lambda_{3}\psi_{2}\psi_{6}^{2}\psi_{6}^{\prime}\psi_{10}+\lambda_{4}\psi_{6}^{3}\psi_{6}^{\prime
2}.$

For the invariants $\psi_{10}^{3},\psi_{2}^{2}\psi_{6}\psi_{10}^{2},$
$\psi_{2}\psi_{6}^{2}\psi_{6}^{\prime}\psi_{10}$ and
$\psi_{6}^{3}\psi_{6}^{\prime 2}$, Lemma 4.7 says each invariant vanishes at
order $6$ and $4$ at the double and triple points. To ensure that the curve
$\mathcal{C}$ defined by $\psi_{30}$ is of divisor class $C$, we intend to
select appropriate scalars $\lambda_{i}$ so that $\psi_{30}$ vanishes to two
more orders at the triple points and twice at the quintuple points.

We now apply Lemma 4.7 to a triple point $p_{3}.$ Choose local affine
coordinates $\widetilde{x},\widetilde{y}$ centered at $p_{3}$ so that
$\mathfrak{m}_{p_{3}}^{k}=(\widetilde{x},\widetilde{y})^{k}$. Let $w$ be a
linear polynomial not vanishing at $p_{3}$ and denote
$\widetilde{\psi}_{d}:=\psi_{d}/w^{d}\in\mathcal{O}_{p_{3}}$. We have the
following decompositions

$\displaystyle\widetilde{\psi}_{10}$ $\displaystyle\equiv
A_{2}+A_{3}\mod\mathfrak{m}_{p_{3}}^{4}$
$\displaystyle\widetilde{\psi^{\prime}}_{6}$ $\displaystyle\equiv
B_{2}+B_{3}\mod\mathfrak{m}_{p_{3}}^{4}$ $\displaystyle\widetilde{\psi}_{6}$
$\displaystyle\equiv C_{0}+C_{1}\mod\mathfrak{m}_{p_{3}}^{2}$
$\displaystyle\widetilde{\psi}_{2}$ $\displaystyle\equiv
D_{0}+D_{1}\mod\mathfrak{m}_{p_{3}}^{2}.$

where $A_{i},B_{i},C_{i},D_{i}$ are homogeneous polynomials of degree $i$ in
$\mathbb{C}[\widetilde{x},\widetilde{y}]$. We therefore have the following
relations

$\displaystyle A_{2}$ $\displaystyle=\mu B_{2}$ $\displaystyle C_{0}$
$\displaystyle=\nu D_{0}$ $\displaystyle C_{1}$ $\displaystyle=3\nu D_{1}$

for some $\mu,\nu\in\mathbb{C}^{*}$, where the third equation is obtained by
computing the image of
$C_{0}^{3}\widetilde{\psi}_{2}^{3}-D_{0}\widetilde{\psi}_{6}$ in
$\mathfrak{m}_{p_{3}}$. Therefore we have

$\displaystyle 0\equiv
C_{0}^{3}\widetilde{\psi}_{2}^{3}-D_{0}\widetilde{\psi}_{6}$
$\displaystyle\equiv
D_{0}^{3}(C_{0}+C_{1})-C_{0}(D_{0}+D_{1})^{3}\mod\mathfrak{m}_{p_{3}}^{2}$
$\displaystyle\equiv D_{0}^{3}(C_{1}-3\nu D_{1})\mod\mathfrak{m}_{p_{3}}^{2}.$

Note that $C_{0}$ and $D_{0}$ are non-zero since $\psi_{2}$ and $\psi_{6}$ do
not vanish at a triple point. The constants $\mu,\nu$ and $D_{0}$ are
dependent on the choice of triple point $p_{3}$ and linear form $w$. Despite
this, we may observe that $\alpha:=\nu/\mu\in\mathbb{C}^{*}$ satisfies

$\alpha\psi_{2}\psi_{10}\equiv\psi_{6}\psi_{6}^{\prime}\mod I_{p_{3}}^{3}.$

This equation is $G$-invariant and thus does not depend on $p_{3}$ or $w$. The
constant of proportionality $\alpha$ is determined by the invariants
$\psi_{d}$.

######  Lemma 4.8.

The curve $\psi_{30}$ is $5$-uple at $p_{3}$ if

$\lambda_{2}+\alpha\lambda_{3}+\alpha^{2}\lambda_{4}=0.$

###### Proof.

We consider $\widetilde{\psi}_{30}$ modulo $\mathfrak{m}_{p_{3}}^{5}$

$\displaystyle\widetilde{\psi}_{30}\equiv\mu^{2}\nu
D_{0}^{3}B_{2}^{2}\lambda_{2}+\mu\nu^{2}D_{0}^{3}B_{2}^{2}\lambda_{3}+\nu^{3}D_{0}^{3}B_{2}^{2}\lambda_{4}\mod\mathfrak{m}_{p_{3}}^{5}\equiv
0.$

The linear condition follows from factoring out by $D_{0}^{3}B_{2}^{2}$ and
dividing by $\mu^{3}$. ∎

######  Lemma 4.9.

The curve $\psi_{30}$ is $6$-uple at $p_{3}$ if it is $5$-uple and

$\displaystyle 5\lambda_{2}+7\alpha\lambda_{3}+9\alpha^{2}\lambda_{4}$
$\displaystyle=0$ $\displaystyle\lambda_{3}+\hskip
4.26773pt2\alpha\lambda_{4}$ $\displaystyle=0$ $\displaystyle
2\lambda_{2}+\hskip 7.11317pt\alpha\lambda_{3}\hskip 39.83368pt$
$\displaystyle=0.$

###### Proof.

Suppose the linear condition of Lemma 4.8 holds. Consider only the degree $5$
terms of $\widetilde{\psi}_{30}$ modulo $\mathfrak{m}_{p_{3}}^{6}$.

$\displaystyle\widetilde{\psi}_{30}\equiv$
$\displaystyle\lambda_{2}(5\mu^{2}\nu D_{0}^{2}D_{1}B_{2}^{2}+2\mu\nu
D_{0}^{3}B_{2}A_{3})+\lambda_{3}(7\mu\nu^{2}D_{0}^{2}D_{1}B_{2}^{2}+\mu\nu^{2}D_{0}^{3}B_{2}B_{3}+\nu^{2}D_{0}^{3}B_{2}A_{3})$
$\displaystyle+\lambda_{4}(9\nu^{3}D_{0}^{2}D_{1}B_{2}^{2}+2\nu^{3}D_{0}^{3}B_{2}B_{3})\mod\mathfrak{m}_{p_{3}}^{5}.$

Observe that we may group these according to the terms
$D_{0}^{2}D_{1}B_{2}^{2},D_{0}^{3}B_{2}A_{3}$ and $D_{0}^{3}B_{2}B_{3}$.

$\displaystyle D_{0}^{2}D_{1}B_{2}^{2}$
$\displaystyle(5\mu\nu\lambda_{2}+7\mu\nu^{2}\lambda_{3}+9\nu^{3}\lambda_{4})=0$
$\displaystyle D_{0}^{3}B_{2}A_{3}$
$\displaystyle(2\mu\nu\lambda_{2}+\nu^{2}\lambda_{3})=0$ $\displaystyle
D_{0}^{3}B_{2}B_{3}$
$\displaystyle(\mu\nu^{2}\lambda_{3}+2\nu^{3}\lambda_{4})=0.\qed$

We now fix multiples for the invariants $\psi_{d}$. The choice of multiples is
due to experimentation that simplifies the polynomial $\psi_{30}$ as much as
possible.

$\displaystyle\psi_{2}$ $\displaystyle=3\phi_{2}$ $\displaystyle\psi_{6}$
$\displaystyle=-3(\omega-1)\phi_{6}$ $\displaystyle\psi_{6}^{\prime}$
$\displaystyle=-25(\phi_{2}^{3}-27(\omega-1)\phi_{6})=-25\left(\frac{1}{27}\psi_{2}^{3}+9\psi_{6}\right)$
$\displaystyle\psi_{10}$
$\displaystyle=\frac{15}{4}\left(25(\omega-1)\phi_{2}^{2}\phi_{6}-(9\omega+3)\phi_{10}\right).$

Note that for the double and triple points $p_{2}=[0:0:1]$ and
$p_{3}=[1:1:1]$, we have

$\displaystyle\varphi(p_{2})$ $\displaystyle=[1:0:0]$
$\displaystyle\varphi(p_{3})$
$\displaystyle=\left[3:\omega:45\omega-60\right]$

from which we see that the $\psi_{d}$’s have the required vanishing
conditions.

######  Lemma 4.10.

The curve $\psi_{30}$ is double at a quintuple point $p_{5}$ if and only if

$32\lambda_{1}=9\lambda_{2}+15\lambda_{3}+25\lambda_{4}$

###### Proof.

Select the quintuple point $p_{5}=[\omega:0:1]$. It is straightforward to
evaluate the terms $\psi_{10}^{3},\psi_{2}^{2}\psi_{6}\psi_{10}^{2},$
$\psi_{2}\psi_{6}^{2}\psi_{6}^{\prime}\psi_{10}$ and
$\psi_{6}^{3}\psi_{6}^{\prime 2}$ at $p_{5}$ to obtain that
$\psi_{30}(p_{5})=0$ if and only if

$32\lambda_{1}-9\lambda_{2}-15\lambda_{3}-25\lambda_{4}=0.\qed$

Observing that the selection of multiples gives $\alpha=-1$, then the linear
system governing these vanishing conditions is the following

$\begin{pmatrix}-32&9&15&25\\\ 0&1&-1&1\\\ 0&5&-7&9\\\ 0&0&1&-2\\\
0&2&-1&0\end{pmatrix}\begin{pmatrix}\lambda_{1}\\\ \lambda_{2}\\\
\lambda_{3}\\\ \lambda_{4}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ 0\\\
0\end{pmatrix}$

######  Theorem 4.11.

The unique solution to the system above is $(2,1,2,1)$. The invariant
$\psi_{30}$ then has the form

$\psi_{30}=2\psi_{10}^{3}+\psi_{2}^{2}\psi_{6}\psi_{10}^{2}+2\psi_{2}\psi_{6}^{2}\psi_{6}^{\prime}\psi_{10}+\psi_{6}^{3}\psi_{6}^{\prime
2}.$

Additionally, the curve $\mathcal{C}$ defined by $\psi_{30}$ is
$G$-irreducible.

###### Proof.

We verify the $G$-irreducibility of the curve $\mathcal{C}$ defined by
$\psi_{30}$. It is sufficient to check that the curve $\mathcal{C}^{\prime}$
in $\mathbb{P}(2,6,10)$ defined by

$g:w_{10}^{3}+w_{2}^{2}w_{6}w_{10}^{2}+2w_{10}w_{6}^{2}w_{2}w_{6}^{\prime}+w_{6}^{3}w_{6}^{\prime
2}$

is irreducible, where $w_{2},w_{6},w_{10}$ are the coordinates of
$\mathbb{P}(2,6,10)$ and $w_{6}^{\prime}=\frac{1}{27}\psi_{2}^{3}+9\psi_{6}$.
Consider the quotient map giving an isomorphic quotient space to that in Lemma
4.5

$\displaystyle\widetilde{\varphi}:\hskip 2.84544pt$
$\displaystyle\mathbb{P}^{2}\rightarrow\mathbb{P}(2,6,10)$ $\displaystyle
p\mapsto[\psi_{2}(p):\psi_{6}(p):\psi_{10}(p)].$

The pullback $\widetilde{\varphi}^{*}\mathcal{C}^{\prime}$ is $\mathcal{C}$
and so we prove irreducibility in terms of $w_{i}$. Firstly, assume $g$ has
the factorization

$(F_{2}w_{10}^{2}+F_{1}w_{10}+F_{0})(G_{1}w_{10}+G_{0})$

where $F_{i},G_{i}\in\mathbb{C}[w_{2},w_{6}]$ are homogeneous polynomials of
appropriate degree to make the factors homogeneous. Since $F_{2}G_{1}=1$, then
$G_{1}$ is constant and $G_{0}$ must be also be degree $10$. Since $G_{0}$
divides $w_{6}^{3}w_{6}^{\prime 2}$, $G_{0}$ must necessarily divide
$w_{6}^{\prime}$ or $w_{6}^{\prime 2}$. This contradicts the degree polynomial
$G_{0}$. On the other hand, if the factorization of $g$ were of the form

$(F_{1}w_{10}+F_{0})(G_{1}w_{10}+G_{0})(H_{1}w_{10}+H_{0}).$

Then the $F_{1},G_{1},H_{1}$ must be constant, forcing $F_{0},G_{0},H_{0}$ to
be of degree $10$. A similar reasoning as above concludes that the curve $C$
is $G$-irreducible. ∎

##  5\. Exploring the sub point-configurations

In this section, we consider the sub point-configurations corresponding to the
quintuple, triple and double points of $\mathcal{A}$. For $k=2,3$ and $5$,
denote $I_{k}$ to be the intersection of homogeneous ideals of $k$-uple
points, In contrast to the Waldschmidt constant of $I_{\mathcal{A}}$, the
Waldschmidt constant of $I_{k}$ is easier to obtain due to the smaller number
of considered points.

######  Proposition 5.1.

Let $I_{2}$ be the homogeneous ideal of double points. Then
$\widehat{\alpha}(I_{2})=3$.

###### Proof.

Recall the effective divisor $B=6H-2E_{2}$. Note that this divisor has
negative self-intersection. We know there is a union of $6$ lines that pass
through each double point twice (Figure 2), therefore
$\widehat{\alpha}(I_{2})\leq 3$.

Suppose there is an effective $\mathbb{Z}$-divisor class $F=dH-mE_{2}$ such
that $d/m<3$. This divisor class is $G$-invariant. Observe that

$B\cdot F=(6H-2E_{2})\cdot(dH-mE_{2})=6d-30m<0.$

Considering the curves $\mathcal{B}$ and $\mathcal{F}$ of classes $B$ and $F$
respectively, the curve $\mathcal{B}$ is $G$-irreducible and
$\mathcal{B}\cdot\mathcal{F}<0$. Suppose $\mathcal{B}$ has irreducible
components $\mathcal{B}_{1},\dots,\mathcal{B}_{k}$. Since $\mathcal{B}$ is
$G$-irreducible, for any $i$ we have
$g(\mathcal{B}_{i}\cdot\mathcal{F})=\mathcal{B}_{j}\cdot\mathcal{F}$ for some
$j$. Then $\mathcal{F}$ intersects each $\mathcal{B}_{i}$ negatively and
therefore contains $\mathcal{B}_{i}$. Thus the divisor class $F-B$ is
effective.

Notice that $F-B$ also intersects $B$ negatively. Performing a similar
analysis, we may conclude by induction that $F-kB$ is effective for $k\geq 1$.
This leads to a contradiction as $F-kB$ will eventually have negative degree.
Therefore $\widehat{\alpha}(I_{2})\geq 3.$

∎

We now state the Waldschmidt constants for the remaining point configurations.

######  Proposition 5.2.

For the ideals $I_{3}$ and $I_{5}$, we have $\widehat{\alpha}(I_{3})=3$ and
$\widehat{\alpha}(I_{5})=\frac{12}{5}$.

The values $\widehat{\alpha}(I_{3})$ and $\widehat{\alpha}(I_{5})$ are
obtained similarly. We have from Section 4.2, the divisor class $6H-2E_{3}$ is
effective. For $I_{5}$, there is a degree $12$ invariant vanishing to order
$5$ at the quintuple points thus the divisor $12H-5E_{5}$. Indeed for each
choice of 5 quintuple points, there is a unique conic passing through each
point (Figure 3). The important property these divisors share with $B$ is that
they have negative self-intersection.

Figure 3. A curve of divisor class $12H-5E_{5}$ drawn solid.

## References

  * [AD09] Artebani, Michela, and Igor V. Dolgachev. “The Hesse pencil of plane cubic curves.” L’Enseignement Mathématique 55, no. 3 (2009): 235-273.
  * [BDRH19] Bauer, Thomas, Sandra Di Rocco, Brian Harbourne, Jack Huizenga, Alexandra Seceleanu, and Tomasz Szemberg. “Negative curves on symmetric blowups of the projective plane, resurgences, and Waldschmidt constants.” International Mathematics Research Notices 2019, no. 24 (2019): 7459-7514.
  * [DK15] Derksen, Harm, and Gregor Kemper. Computational invariant theory. Springer, 2015.
  * [DS21] Benjamin Drabkin and Alexandra Seceleanu. “Singular loci of reflection arrangements and the containment problem.” Mathematische Zeitschrift, 299(1):867–895, 2021.
  * [Har18] Harbourne, Brian. “Asymptotics of linear systems, with connections to line arrangements.” arXiv preprint arXiv:1705.09946 (2017).
  * [Kle58] Britton, J. L. “Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree By Felix Klein. Translated by GG Morrice. reprinted. Pp. xvi+ 289. $1.85. 1956.(Dover Publications).” The Mathematical Gazette 42, no. 340 (1958): 139-140.
  * [Nag59] Nagata, Masayoshi. “On the 14-th problem of Hilbert.” American Journal of Mathematics 81, no. 3 (1959): 766-772.
  * [OT13] Peter Orlik and Hiroaki Terao. Arrangements of hyperplanes, volume 300. Springer Science & Business Media, 2013.
  * [ST54] Shephard, Geoffrey C., and John A. Todd. “Finite unitary reflection groups.” Canadian Journal of Mathematics 6 (1954): 274-304.


# Flexible Phased Array Sheets: A Techno-Economic Analysis

Oren S. Mizrahi, Austin Fikes, Ali Hajimiri O.S.M., A.F., and A.H. are with
the Department of Electrical Engineering, California Institute of Technology,
Pasadena, CA, 91125 USA.Manuscript received February 7, 2023

###### Abstract

Phased arrays have enabled advances in communications, sensing, imaging, and
wireless power transfer. In all these applications, large apertures enable
higher power, higher data rates, higher resolution, and complex
functionalities, but are elusive owing to a correspondingly large cost, mass,
and physical size. Flexible phased arrays (FPAs) show potential in breaking
this trade-off. Their thinness and extremely low mass allow FPAs to be folded,
rolled, or otherwise compressed into smaller sizes, thus enabling new regimes
of transport and entirely new applications currently not possible. Though a
number laboratory prototypes of FPAs have been constructed, the economics of
large-scale FPA production has yet to be explored. This paper presents a model
FPA architecture and a cost model for producing it at large-scale. The
estimate of the per-unit-area cost is bounded by a three-tiered approach. The
cost model projects a “middle” estimate for FPA production at $89 per square
meter. Estimates for aerial mass density and startup cost are also discussed.
This cost model demonstrates that an FPA can be produced at an efficient price
point and can potentially replace existing solutions for space,
communications, and vehicular applications that demand lightweight,
portability, and durability in extreme conditions.

##  I Introduction

Phased arrays based on radio frequency integrated circuits (RFICs), introduced
two decades ago, have spurred ongoing inquiry both in academia and industry.
This explosion in interest is owing to RFICs’ incredible capabilities and
diverse applications. By packaging all of the frequency generation, power
amplification, and advanced digital control functionalities into a single,
extremely thin monolithic platform only several square millimeters in area,
integrated circuits enable the large scale and low cost that have allowed
phased arrays to find fertile ground in a plethora of diverse applications.
IC-driven phased arrays have found utility in communications [1, 2], imaging
[3], sensing [4, 5], and wireless power transfer (WPT) [6, 7].

Previously limited by niche technology nodes, RFICs are now commonly
fabricated in CMOS, allowing for incredibly low-cost and highly scalable
technology platforms that can also leverage decades of advanced digital design
legacy. Flexible phased arrays (FPAs) are a natural extension to this
progression, facilitating all the aforementioned applications in novel
physical contexts and potentially at lower cost. FPAs that are thin, low-mass,
durable, low-cost, and driven by integrated electronics have been proposed for
use in communication [8, 9], wireless power transfer [10, 11, 12, 13], rapidly
deployable infrastructure [14, 15], aerodynamic vehicle microwave systems
[16], and more [17].

Collapsible [18], flexible [19, 20], inflatable [21], and foldable [22]
antenna array prototypes have been demonstrated. While these prototypes have
shown they are mechanically and electrically capable of suiting the needs of
the aforementioned applications, large-scale realization cannot come to
fruition unless they prove to be economically efficient relative to the
intended markets. To date, a techno-economic analysis projecting the cost of
FPAs has not been published.

Presented in this paper is a cost model that seeks to fill this gap.
Envisioned is a model FPA that is light, durable, low-cost, thin, and boasts
functionalities packaged into an RFIC. The manufacturing process for such a
FPA is discussed with respect to four “layers:” the IC, the flexible PCB, the
radiators, and assembly. At each phase, three tiers of cost projection are
discussed: a “current tier” which assumes no changes to the current
manufacturing process, a “middle tier” which assumes large scale manufacturing
and conservative projections for the future, and an “asymptotic” tier which
assumes large scale manufacturing and great advancements in technology. The
goal is to arrive at a projection for the per area cost of producing a FPA as
herein envisioned, with a low and high bound coming from the lowest and
highest projection tiers. We also discuss a mass projection and use capital
expense estimates to project a startup costs.

While the envisioned FPA is well-suited for terrestrial and aquatic
communications, wireless power, and sensing, our analysis is particularly
targeted for applications in space, such as satellite communications and space
solar power, where large aperture, low mass, and deployability are critical
and where current solutions are very limited.

In Section II, the model FPA is discussed. In Section III, the manufacturing
process is discussed and relevant cost details are presented. In Section IV,
the total FPA sheet cost is presented. In In Section V, a total startup cost
is presented. In Section VI, an aerial mass density projection is presented.
Section VII, a cost comparison to various competitors in different markets is
presented. Listed in the Appendix are cost model variables, and a discussion
on price sensitivity to design changes.

##  II A Model Flexible Phased Array Sheet

Figure 1: Center: Envisioning the ideal FPA: thin, low mass, durable, low
cost, with integrated electronics. Outside, clockwise from the top right: (a)
A wireless power transfer tile with photovoltaic concentrators on one side, a
FPA in the middle, and 10GHz flexible patch antennas on the other side. (b) A
256-element FPA with integrated power generation and 10GHz patch antennas. (c)
An 8-element FPA tile with 10GHz dipole radiators and integrated power
control. (d) A 256-element FPA with integrated power generation and a flexible
sheet of 10GHz patch antennas.

A model FPA can be stripped down to three fundamental components:

  1. 1.

A RFIC which is powered by a DC power supply and performs all the necessary
frequency locking, power amplification, phase shifting, signal modulation, and
digital control functionalities necessary for the desired application.

  2. 2.

A thin and flexible PCB with radiator ground plane, microstrip routing between
the IC and the radiators, and microstrip lines for power and digital control.

  3. 3.

Thin and flexible radiators which are suited for the relevant physical context
and application111Different radiators offer tradeoffs vis-a-vis radiation
polarization, bandwidth, form-factor, etc. Almost all designs can be made
flexible, as shown in various publications [7, 18, 22]..

These descriptions are intentionally general: they are common to almost any
imaginable embodiment of a FPA and variations on the details do not greatly
affect the cost projection presented below. For example, because ICs are paid
for by the wafer and not by the transistor, our model accounts for many
applications that might use different IC architectures and building blocks
(single frequency wireless power transfer, wideband communications, etc.),
assuming area is not greatly affected.

However, to obtain some of the specific details presented herein, we make
assumptions about the exact nature of the FPA:

  1. 1.

The RFIC is manufactured in 65nm CMOS, with an integrated micro-controller,
bumping, and potentially an interposer.

  2. 2.

The PCB is etched out of 2-layer, rolled polyimide and copper clad.

  3. 3.

The radiators are 10GHz co-cured popup dipoles as presented in [18]. The same
composite-electromagnetic co-design can be applied to different radiator
architectures with a similar cost.

Other system variables used in the model are listed in the Appendix below:

##  III Cost Model

###  III-A RFIC

Fundamentally, the RFIC cost is a function of two numbers: The cost per wafer
and the number of chips per wafer (wafer yield).

####  III-A1 Wafer Cost

Assuming a 65 nm process, a 300mm wafer costs $2000. Considering the
additional $60 for bumping the wafer, total wafer cost at scale,
$C_{\text{wafer}}$, is 2060 $/wafer.

####  III-A2 Wafer Yield

Currently, the model 16-channel WPT RFIC [12] is 3mm x 3mm = 9mm2. Adding an
additional 0.1mm/side to account for the width of the dicing lane and an
additional $\approx 4\text{mm}^{2}$ to integrated a microcontroller, total
area is 13.61mm2.

Wafer yield is not simply the quotient of the wafer area and the chip area;
geometric incompatibility of cutting squares out of a circle reduces yield.
True yield can be estimated using the following formula:

$\displaystyle N_{\text{chips}}$ $\displaystyle=\left\lfloor\frac{\pi
R_{\text{wafer}}^{2}}{A_{\text{chip}}}-\frac{2\pi
R_{\text{wafer}}}{\sqrt{2A_{\text{chip}}}}\right\rfloor$ (1)
$\displaystyle=\left\lfloor\frac{\pi\cdot(150\text{mm})^{2}}{13.61\text{mm}^{2}}-\frac{2\pi\cdot
150\text{mm}}{\sqrt{2\cdot 13.61\text{mm}^{2}}}\right\rfloor$ (2)
$\displaystyle=\left\lfloor 5013.023\right\rfloor$ (3) $\displaystyle=5013$
(4)

Defect yield (the fraction of chips that pass inspection) is highly variable
and subject to changes based on technology node, devices used, desired
accuracy, etc. Assuming 5% of chips are discarded because they contain
defective devices, chip yield becomes 4762 chips per wafer. Thus, our chip
cost ($C_{\text{chip}}$) is:

$\displaystyle C_{\text{chip}}$
$\displaystyle=\frac{C_{\text{wafer}}}{N_{\text{chips}}}$ (5)
$\displaystyle=\frac{\$2060}{4762}$ (6) $\displaystyle=\$0.4326/\text{chip}$
(7)

Finally, price per unit area ($C_{\text{chip}}^{A}$) can be computed by
dividing chip cost by the total area driven by a single chip (herein called a
tile) ($A_{\text{tile}}$):

$\displaystyle C_{\text{chip}}^{A}$
$\displaystyle=\frac{C_{\text{chip}}}{A_{\text{tile}}}$ (8)
$\displaystyle=\frac{C_{\text{chip}}}{Kd^{2}}$ (9)
$\displaystyle=\frac{\$0.4326}{16\cdot(18\times 10^{-2}\text{m})^{2}}$ (10)
$\displaystyle=83.45\frac{\$}{\text{m}^{2}}$ (11)

where $K$ is the number of radiators per chip and $d=0.6\lambda$ is the
radiator spacing.

Additional costs include the cost to dice the wafer (assumed to be 3% of the
chip cost), the cost to produce the mask set ($ 500,000 for 65nm, amortized
over lifetime production), and the cost to make an interposer, which we assume
is made of the same PCB (and carrying the same cost per unit area) as that
discussed in the next section). If the interposer is 3x the area of the chip,
our final cost per unit area for the RFIC is $88.39\frac{\$}{\text{m}^{2}}$ \-
a marginal increase over the cost before these additional costs.

This cost projection is based on our current process, but if we account for an
improved process and lower prices with further advancements in technology
nodes, we can project both a “middle” projection and an “asymptotic”
projection:

Variable | Unit | Current | Middle | Asymptotic
---|---|---|---|---
$C_{\text{wafer}}$ (65nm, 300mm) | [$/wafer] | 2000 | 1500 | 1200
Bumping | [$/wafer] | 60 | 40 | 20
Mask set | [$] | 500,000 | 350,000 | 250,000
RF chip area | [mm2] | 9 | 6 | 4
$\mu$C chip area | [mm2] | 4 | 2 | 1
Dicing Lane width | [$\mu$m] | 100 | 70 | 50
Interposer | [ ] | YES | YES | NO
Total Price | [$/m2] | 88.39 | 39.67 | 19.51
TABLE I: RFIC Cost Projections

###  III-B Thin, Flexible PCB

Flexible PCBs are new and uncommon and, thus, their manufacturing process is
less known - especially at high scale.

####  III-B1 Clad Manufacturing

The process begins with the construction of the PCB clad: a stack of layers of
interleaved metal and insulator. Assumed herein are thin layers of copper and
polyimide (Kapton). Clad comes either rolled or autoclaved; rolled clad is
about 6x cheaper (owing to higher throughput) but has higher surface roughness
222The inner surface of the metal has a high density of dendrites which allow
it to adhere to the substrate without baking the stackup. The dendrites
degrade the performance of microstrip lines and other electromagnetic
structures.. Assumed for the purposes of this model is the use of rolled clad,
due to the importance of low-cost and scalability and the limited added
losses.

The rolled clad is drilled to produce via holes which are then plated. The
board is patterned and etched on both sides to generate RF and DC traces.
Next, the board is conformal coated to protect parts from the environment and
from potential shorts. Higher tier projections assume the use of less
conformal coating which reduces mass and cost.

The following table lists expenses tracked along this manufacturing process:

Variable | Unit | Current | Middle | Asymptotic
---|---|---|---|---
Materials | [$/m2] | 98.79 | 4.94 | 2.66
Labor | [$/m2] | 14.25 | 1.78 | 1.78
Energy | [$/m2] | 0.16 | 0.16 | 0.16
Capex | [$/m2] | 1.11 | 1.11 | 1.11
Total Price | [$/m2] | 114.31 | 7.99 | 5.71
TABLE II: Flexible PCB Cost Projections

The “current” number is dominated by materials and labor costs because it
assumes clad is purchased already manufactured instead of rolled in-house; the
other two categories add the cost of manufacturing clad but assume materials
purchased are raw copper and polyimide sheets instead. The great disparity in
materials cost is owing to a 40x markup of off-the-shelf clad relative to the
raw materials.

Figure 2: FPA cost [$/m2] at different projection tiers: (a) Current
($566.21/m2), (b) Middle ($88.75/m2), and (c) Asymptotic ($45.37/m2).

###  III-C Flexible Radiators

Of critical importance in large, scalable, flexible, lightweight, and durable
phased array systems are radiators which themselves have these
characteristics. The most compelling platform for designing and manufacturing
flexible radiators, presented in [18], is one in which an electronic antenna
layer is “co-cured” to a flexible substrate. This allows for complete control
over the physical form, mechanical strength, and thermal properties of the
radiator. Moreover, any incarnation of a flexible radiator using this platform
would go through a similar manufacturing process, allowing for the projection
of a wide range of radiators using a single model.

####  III-C1 Composite Substrate Preparation

The manufacturing process begins with a single-ply composite mesh. The mesh
can either be purchased pre-impregnated with an epoxy matrix or raw and then
impregnated in house. Owing to the extremely high-cost of current off-the-
shelf options, this model assumes in-house impregnation for the middle and
asymptotic tiers.

The impregnated fiber plies are stacked with respect to a particular
weave333Assumed in this case is the 3-ply 45∘/90∘/45∘ stack presented in [18].
The plies are laminated under light heat and cut into the final fiber shape.

####  III-C2 Electronic Antenna Preparation

The electronic layer of the antenna - envisioned as the etched, 2-layer
polyimide flexible PCB used in [18] \- is prepared in parallel. These antennas
are processed in a manner similar to the PCB but without any conformal coating
and utilizing a much thinner substrate. Furthermore, empty area on antenna
sheets not containing metal traces are cut out to reduce mass and dielectric
that might interfere electromagnetically.

####  III-C3 Alignment and Curing

Finally, the fiber backing and the antenna layers are aligned and laminated to
produce the final radiator stack. This stack is then fit into a custom
silicone mold which is vacuum-sealed and cured under temperature and pressure
for 2 hours. The combined curing of both layers - called “co-curing” - is a
method of producing flexible radiators with great potential in scalability
without compromising on consistency.

Listed below are estimates of costs for the various stages of manufacturing:

Variable | Unit | Current | Middle | Asymptotic
---|---|---|---|---
Impregnated Fiber | [$/m2] | 196.85 | 16.79 | 1.68
Polyimide clad | [$/m2] | 86.00 | 0.53 | 0.53
Fiber/Curing Processing | [$/m2] | 5.02 | 5.02 | 5.02
Antenna Processing | [$/m2] | 21.94 | 4.47 | 4.47
Labor | [$/m2] | 8.75 | 0.88 | 0.88
Capex | [$/m2] | 14.68 | 7.41 | 1.59
Total Price | [$/m2] | 333.24 | 35.09 | 14.16
TABLE III: Co-Cured Popup Antenna Cost Projections

The current projection is dominated by materials costs owing to the extremely
high price of pre-impregnated fiber and off-the-shelf polyimide clad. When
produced in house, these materials costs reduce by an order of magnitude.
Further savings are achieved through increased automation and reduced capex by
assuming more durable silicone molds in the latter tiers.

###  III-D Assembly

Finally, the three components are assembled using highly scalable
manufacturing steps that are extremely standard in industry and easy to model:
pick-and-place assembly and flip-chip bonding for IC. In addition, the cost of
several resistors, capacitors (“passives”), and a high emissivity metal
“thermal spreader” mounted on top of the RFIC is accounted for.

Variable | Unit | Current | Middle | Asymptotic
---|---|---|---|---
Passives | [$/m2] | 5.15 | 2.58 | 2.58
Processing | [$/m2] | 0.25 | 0.21 | 0.21
Labor | [$/m2] | 23.31 | 1.93 | 1.93
Capex | [$/m2] | 1.55 | 1.29 | 1.29
Total Price | [$/m2] | 30.26 | 6.00 | 6.00
TABLE IV: WPT Layer Assembly Cost Projections

In this case, cost reductions come mostly from high automation and redesigning
the ICs to use fewer passive components.

##  IV Flexible Phased Array Sheet Cost

The full, per-area cost for the envisioned FPA sheet can be projected by
summing the costs from the four manufacturing steps discussed above:

Variable | Unit | Current | Middle | Asymptotic
---|---|---|---|---
RFIC | [$/m2] | 89.23 | 39.67 | 19.51
Flexible PCB | [$/m2] | 114.31 | 7.99 | 5.71
Radiators | [$/m2] | 333.24 | 35.09 | 14.16
Assembly | [$/m2] | 30.26 | 6.00 | 6.00
Total Price | [$/m2] | 566.21 | 88.75 | 45.37
TABLE V: FPA Sheet Cost Projection

As we can see in the “Middle” column, scaling up and automating current
processes is responsible for a 6x reduction in cost. Moreover, the entire
bound between current and asymptotic spans only a single order of magnitude,
providing us with a narrow estimate for FPA sheet cost.

Figure 3: Tile mass [mg] at different projection tiers: (a) Current (1012mg),
(b) Middle (433mg), and (c) Asymptotic (403mg).

##  V Startup Cost

Relevant to large-scale production of an device with a novel manufacturing
chain is the cost to outfit a facility for production. Startup cost, measured
in $\frac{\$\cdot\text{yr}}{\text{m}^{2}}$, was projected by tracking only the
capital expenses and normalizing them by their associated annual throughput,
measured in m${}^{2}/\text{yr}$. This number is subsequently scaled by the
desired facility capacity, measured in m${}^{2}/\text{yr}$, to calculate the
cost of the facility.

This cost does not include the price of land or taxes, which are subject to
changes based on location, market conditions, local taxes, and potential
subsidies obtained from local or federal government(s). Moreover, these
startup costs are not in addition to the total costs per unit array area
presented earlier; per-unit-array-area costs include capital expenditures that
overlap with those included in the startup cost, though many of the capital
expenses will be recurring based on machine lifetime. Startup costs can be
seen simply as representing the initial investment necessary before any arrays
can be produced or sold.

Variable | Unit | Current | Middle | Asymptotic
---|---|---|---|---
Flexible PCB | [$\$\cdot\text{yr}/\$\text{m}^{2}$] | 31.68 | 31.68 | 31.68
Radiators | [$\$\cdot\text{yr}/\$\text{m}^{2}$] | 37.18 | 37.18 | 37.18
Assembly | [$\$\cdot\text{yr}/\$\text{m}^{2}$] | 31.13 | 25.76 | 25.76
Total Price | [$\$\cdot\text{yr}/\$\text{m}^{2}$] | 99.98 | 94.61 | 94.61
TABLE VI: Flexible Phased Array Sheets Startup Cost

Projections are similar for all tiers owing to equal machine pricing and
roughly constant throughput. For a facility capable of producing 10,000 m2 of
FPA sheets in a year, an initial investment of between $946,100-$999,800 would
be required.

##  VI Aerial Mass Density

A mass model is populated by investigating the same three manufacturing layers
discussed above: each component contributes predictable mass. These masses are
tabulated below to generate a projection of the aerial mass density of a
single FPA sheet tile:

Component | Unit | Current | Middle | Asymptotic
---|---|---|---|---
RFIC | [mg] | 2.55 | 1.70 | 1.13
Flexible PCB | [mg] | 199.58 | 171.59 | 143.60
RF signal path | [mg] | 5.60 | 5.60 | 5.60
Radiator | [mg] | 752.00 | 203.04 | 203.04
DC power lines | [mg] | 47.24 | 47.24 | 47.24
Thermal spreader | [mg] | 5.04 | 3.36 | 2.24
Total mass | [mg] | 1012.01 | 432.53 | 402.85
Total mass density | [g/m2] | 195.22 | 83.43 | 77.71
TABLE VII: FPA Tile Mass Projection

Mass reductions in the tiers can be attributed to design decisions that were
also responsible for cost reductions. Higher tiers assume smaller RFICs. This
also reduces the size (and thus mass) of the thermal spreader which is sized
in proportion to the IC. Some mass reductions, like that for the flexible PCB
and radiators, are owing to design decisions made specifically to reduce mass.
The middle and asymptotic tiers assume “cheesing”444“Cheesing” refers to the
etching of a grid pattern in an otherwise solid ground plane at a pitch much
lower than the wavelength for the purpose of minimizing mass. in the PCB
ground plane and less glass fiber substrate in the radiator. Some components,
such as the DC power lines and RF signal path, are sized based on electrical
considerations and won’t reduce in mass with higher tiers.

##  VII Market Competitiveness

Absolute cost is useful but far more relevant is cost efficiency relative to
market competitors. As there are no FPAs on the market, the closest
competitors are other phased array solutions for the relevant applications.
Currently, there are market offerings for typical phased array transceivers,
but more relevant are the offerings advertising mobile connectivity (portable
transceivers and arrays that are designed to be mounted on cars, boats, and
planes).

The cost comparison below was enabled by the relentless gains in the number of
offerings and in the market size. There are now consumer offerings (like
Starlink’s “Dishy McFlatface”) which have spurred detailed breakdowns and in-
depth device specifications. Elusive details were filled in using assumptions
listed below.

|  | Starlink | Starlink | Kymeta | KVH
---|---|---|---|---|---
Variable | Unit | (Standard) | (High Perf) | (u8) | (TracVision A9)
Electronics Cost555Assumed to be 80% of market cost. | [$] | 479 | 2,000 | 15,160 | 46,800
Electronics Mass666Assumed to be 25% of total mass if number wasn’t available. 25% is based on the measured ratio for Starlink’s Dishy McFlatface V2. | [g] | 725 | 1,725 | 8,000 | 5,625
Area | [m2] | 0.155 | 0.294 | 0.810 | 0.656
Cost density | [$/m2] | 3,083 | 6,807 | 18,716 | 7,316
Mass density | [g/m2] | 4,664 | 5,870 | 9,877 | 8,573
TABLE VIII: Phased Array Cost and Mass Comparison Table

The best comparison on the market is Starlink’s “Standard” transceiver,
because it is being produced at scale with among the most modern hardware on
the market, which has aerial mass density of 4,664 g/m2 and a cost density of
3,083 $/m2. Even adjusting the presented cost model to a frequency of 40GHz
(in the middle of Starlink’s uplink and downlink bands), this model projects
an aerial mass density of 2,396 g/m2 and a cost density of 761 $/m2 (for the
“middle” projection).

##  VIII Conclusion

Presented here is an envisioned model FPA and a corresponding cost model which
projects the price per unit area of producing it at large scale. The FPA is
constructed of an RFIC and flexible radiators mounted to a thin, flexible PCB.
This platform is equipped for numerous applications in communications,
sensing, and wireless power transfer; small changes to design details do not
greatly affect the projected cost.

The cost model projects prices at three tiers, with each tier making
progressively more assumptions about scaling and future technology pricing.
Using the “current” projection which assumes current pricing and no scaling,
FPAs can be produced at $566.21 per square meter, a number strongly inflated
by off-the-shelf materials costs. More accurate for a high-scale production is
the “middle” projection, which estimates production at $88.75 per square
meter. With further advances in technology, the asymptotic price limit is
projected at $45.37 per square meter.

Presented alongside the cost model is a mass model which projects an aerial
mass density of between 195.22 g/m2 and 77.71 g/m2 for the current and
asymptotic tiers, respectively. Additionally, capex costs throughout the model
were collected and normalized to annual throughput to generate a startup cost
rate for the desired facility capacity. A facility producing 10,000 m2 of FPA
sheets would require an up-front investment of about $1 million.

Projected prices are highly competitive relative to current offerings in the
market and are approximately $1/4$ the cost of the leading rigid phased array
transceiver. Projected aerial mass density is also between one-half and one-
tenth as large as current rigid offerings. The envisioned FPA can enable a
factor reduction in launch costs for satellites and can be integrated into
air-borne designs in ways current market offerings are simply incapable of.

This analysis lends credence to the notion that FPAs are not only mechanically
and electrically qualified to be implemented in the desired applications, but
also economically efficient relative to the corresponding markets. FPAs are
relatively cheap to produce, extremely light, and require a small initial
capital investment to produce. Low-cost FPAs are uniquely positioned to enable
the deployment of ultra-large apertures in novel contexts - space, aquatic and
aerodynamic vehicles, extreme conditions, and others - facilitating greater
power, higher bandwidth, higher power levels, and new applications previously
prohibited or not yet envisioned.

##  IX Appendix

###  IX-A System Variables

Listed below are variables assumed for this model.

Symbol | Description | Unit | Value
---|---|---|---
$f$ | Frequency | [GHz] | 10
$d$ | Radiator spacing | [mm] | 18 ($0.6\lambda$)
$K$ | Output channels per RFIC | [ ] | 16
$L$ | Flexible PCB metal layers | [ ] | 2
$t_{m}$ | Metal thickness | [$\mu$m] | 10
$t_{p}$ | Polyimide thickness | [$\mu$m] | 12.5
TABLE IX: FPA Variables and Assumed Values

###  IX-B Economic Variables

A few market variables factored into our calculations. These are middle values
that change as a function of time and place but were chosen to represent
realistic market conditions:

Variable Description | Unit | Value
---|---|---
Cost of labor, adjusted777Adjustment is based on a cost of wage of $25/hour adjusted by 2x to include the cost of benefits, workplace insurance, labor expenses (cleaning, bathroom, facilities, etc.), and other expenses. | [$/hour] | 50
Cost of electricity888In Texas, USA. 2022 | [¢/kWh] | 8.05
Inflation rate | [%/year] | 3.8
TABLE X: Economic Variables and Assumed Values

###  IX-C Cost Variability With Different Design Variables

Though some design decisions were necessary to identify certain prices, the
model is fairly amenable to changes in these details. We used this flexibility
to demonstrate how different design variables would affect the final cost. All
percentages are relative to the “middle” projection of 88.75 $/m2.

  * •

Changing from a Si CMOS process to a GaAs process increases the IC cost by a
factor of 4.8x. This increases total cost to 239.50 $/m2 (a 170% increase).

  * •

Increasing operating frequency increases the density of RFICs, increasing
overall cost. At $f=20$GHz, total cost is 220.62 $/m2 (a 149% increase); at
$f=40$GHz, total cost is 761.35 $/m2 (a 758% increase).

  * •

Increasing the number of PCB layers from 2 to 4, for the purpose of more real-
estate for routing or advanced functionalities, increases cost slightly to
98.40 $/m2 (a 10% increase).

  * •

Autoclaved clad, necessary for applications requiring extremely low surface
roughness for low attenuation, increases total cost to 113.00 $/m2 (a 28%
increase).

  * •

Adding a conformal coating to protect the electronics in harsh environments
(outdoors, high radiation conditions, space, etc.) increases total cost to
91.60 $/m2 (a 3% increase).

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11institutetext: School of Physics, Free University of Tbilisi, 0183, Tbilisi,
Georgia 11email<EMAIL_ADDRESS>22institutetext: E. Kharadze
Georgian National Astrophysical Observatory, Abastumani, 0301, Georgia
33institutetext: Institute for Fusion Studies, The University of Texas at
Austin, Austin, TX 78712, USA 33email<EMAIL_ADDRESS>
# Radio pulsars resonantly accelerating electrons

Zaza N. Osmanov 1122 Swadesh M. Mahajan 33

(Received ; accepted )

###### Abstract

Context.

Aims.

Methods. Based on the recently demonstrated resonant wave-wave process, it is
shown that electrons can be accelerated to ultra-relativistic energies in the
magnetospheres of radio pulsars. The energization occurs via the resonant
interaction of the electron wave (described by a Klein-Gordon (KG) equation)
moving in unison with an intense electromagnetic (EM) wave; the KG
wave/particle continuously draws energy from EM. In a brief recapitulation of
the general theory, the high energy (resonantly enhanced) electron states are
investigated by solving the KG equation, minimally coupled to the EM field.

Results. The restricted class of solutions, that propagate in phase with EM
radiation (functions only of $\zeta=\omega t-kz$), are explored to serve as a
possible basis for the proposed electron energization in the radio pulsars.

Conclusions. We show that the wave-wave resonant energization mechanism could
be operative in a broad class of radio pulsars with periods ranging from
milliseconds to the normal values ($\sim 1$ sec); it could drive the
magnetospheric electrons to acquire energies from $100$s of TeVs (millisecond
pulsars) to $10$ ZeVs (normal pulsars).

###### Key Words.:

acceleration of particles; pulsars: general; plasmass

##  1 Introduction

In a recent work a new mechanism of wave-particle resonant interaction has
been discovered (Mahajan & Asenjo, 2016), that might be a significant process
for acceleration of particles. The study of acceleration mechanisms is crucial
for understanding the nature and origin of cosmic rays. Earlier research
suggested that very high energy (VHE) particles (Baade & Zwicky, 1934) could
be emitted in supernovae events. The Crab Nebula, for instance, was identified
as the source of the cosmic rays by Sekido et al. (1951). The discovery of
pulsars in 1967 brought another possible player into the game. In the
pioneering work of Gunn & Ostriker (1969), the role of pulsars in accelerating
protons to $10^{18}$eV was established. Since then, a number of authors have
investigated pulsars as an energizing source (Bednarek & Protheroe, 2002;
Lemoine et al., 2015; Osmanov & Rieger, 2009, 2017; Mahajan et al., 2013;
Osmanov et al., 2015). It is also strongly believed that extremely high energy
cosmic rays may emerge from active galactic nuclei (AGN) (e.g. Protheroe &
Szabo, 1992; Asaf et al., 2009) and gamma ray bursts (e.g. Schady, 2017; Zhang
et al., 2021). In this context it is worth noting that termination shocks of
galactic outflows and star-forming regions are also considered as possible
sources for ultra-high energy particles (Marcowith et al., 2021; Globus &
Blandford, 2023; Owen et al., 2023).

It is, perhaps, useful to list the generally invoked mechanisms that might,
potentially, energize the cosmic particles to VHE. The Fermi-type acceleration
mechanisms (Fermi, 1949; Bell, 1978; Catanese & Weeks, 1999) can explain the
generation of ultra-high energies. But the Fermi process is very efficient
only if the particles have already been pre-accelerated (Rieger & Mannheim,
2000). In contrast, magneto-centrifugal acceleration (MA) does not require
pre-acceleration; it can operate on particles starting with lower energies.
The high efficiency of the mechanism has been demonstrated in the context of
pulsars (Osmanov & Rieger, 2009, 2017; Mahajan et al., 2013; Osmanov et al.,
2015) and AGN (Osmanov et al., 2014). The MA, however, works only if magnetic
fields are strong enough to ensure that the plasma obeys the ”frozen-in
condition” forcing the particles follow the co-rotating field lines (Osmanov &
Rieger, 2009, 2017; Mahajan et al., 2013; Osmanov et al., 2015).

In our recent work (Mahajan & Osmanov, 2022) we explored the resonant wave-
wave mechanism (just the one we will be invoking in this paper) of particle
energization in the context of the radio-loud AGN.

Taking into account the factors limiting the maximum energy (inverse Compton
mechanism, synchrotron radiation and curvature emission), we have found that
the electrons and protons (in the magnetosphere of a radio-loud AGN) might
reach energies of the order of $10^{21}$ eV. It has been shown that the most
efficient factor constraining the maximum energies is the radiation field
energy density, and consequently inverse Compton scattering becomes
significant. In light of the Greisen-Zatsepin-Kuzmin (GZK) limit (a
theoretical upper limit of extragalactic cosmic ray protons: $5\times 10^{19}$
eV (Greisen, 1966; Zatsepin & Kuzmin, 1966)), a further examination of the
efficiency of this new mechanism may be warranted.

In this paper, however, we turn our attention to pulsars whose magnetospheres
are populated with electron-positron pairs. Since many pulsars emit in the
radio spectral band, it is interesting to explore whether the resonant
energization will, efficiently, operate in this setting.

The paper is organized as follows: in Sec. 2 we summarize the general theory
of resonant acceleration(Mahajan & Asenjo, 2016, 2022). In Sec. 3, we apply
the mechanism to pulsars, derive and discuss the principal results. In Sec. 4,
we put our results into perspective.

##  2 Brief outline of theory

For a particle with rest mass $m$, momentum, $P=\gamma m\upsilon$, and energy
$E=(P^{2}+m^{2})^{1/2}$, the group velocity of the associated KG wave
($\hbar=c=1$),

$\upsilon_{g}=\frac{\partial E}{\partial
P}=\frac{P}{\left(P^{2}+m^{2}\right)^{1/2}},$ (1)

tends to unity for $P>>m$. One then, expects, that such a matter wave, could
strongly resonate with an EM wave. Moving almost in phase the particle wave
could continuously draw energy from the EM wave.

The analysis of Mahajan & Asenjo (2016, 2022) begins with ($\omega=2\pi f$ is
the emission cyclic frequency),

$\left(\partial^{2}_{t}-\partial_{z}^{2}+2qAK_{\perp}\cos\left(\omega
t-kz\right)+K_{\perp}^{2}+m^{2}+q^{2}A^{2}\right)\Psi=0,$ (2)

describing a KG particle wave $(\Psi)$ in the presence of a circularly
polarized EM wave $(A^{0}=A^{z}=0,\;A^{x}=A\cos\left(\omega
t-kz\right),\;A^{y}=-A\cos\left(\omega t-kz\right)),$ propagating in the z
direction. In Eq. (2), $q$ denotes the charge of the particle and $K_{\perp}$
represents the perpendicular momentum of the KG wave. For the resonant KG-EM
system, we seek solutions where $\Psi=\Psi(\omega t-kz)=\Psi(\xi)$. Equation
(2), then, converts to an ordinary differential equation

$\left(\omega^{2}-k^{2}\right)\frac{d^{2}\psi}{d\xi^{2}}+\left(\mu+\lambda\cos\xi\right)\psi=0,$
(3)

a generic Mathew equation with $\mu=K_{\perp}^{2}+m^{2}+q^{2}A^{2}$ and
$\lambda=2qAK_{\perp}$.

Since $\omega$ and $k$ are very close (in vacuum, they are exactly the same),
Eq. (3) is a singular equation. This singular feature ($d^{2}\Psi/d\xi^{2}$
must become large to satisfy the equation) is the reason why the KG particle
obeying Eq. (3) must have solutions with high energy $(E)$ and momentum
($K_{z}$). The novelty of our mechanism is that the enhancement of electron
energy (and the rate of energy gain) depends crucially on the resonance
between the two hyperbolic waves, i.e, $E/K_{z}$. of the electron wave
$=\omega/k$ of the EM wave. Both of these are greater than unity (the latter
only in a plasma).

It is worth noting that a classical electron is not a hyperbolic wave, but the
physics that this theory is based on depends on the resonance of two
hyperbolic waves, therefore, the mechanism of particle energization strongly
depends on the quantum nature of a particle (Mahajan & Asenjo, 2022). bf In
fact, we will now show explicitly, that quantum effects are essential for the
acceleration process.

That these high energy solutions are actually accessible, requires further
enquiry. This was done by Mahajan & Asenjo (2016, 2022) where an expression
for the rate of energy gain was derived as

$\frac{dE}{dt}=\frac{\sqrt{2}\omega^{2}}{\left(\omega^{2}-k^{2}\right)^{1/2}}\frac{\hbar
qAK_{\perp}}{\left(m^{2}+K_{\perp}^{2}+q^{2}A^{2}\right)^{1/2}}\frac{\sin
kl}{kl}.$ (4)

(where we have restored the $\hbar$ in the numerator) assuming that the
particle acceleration takes place in the region $z=\\{-l,l\\}$. Eq. (4) fully
illustrates the resonant character of the acceleration process. Indeed, for
small values of $\omega^{2}-k^{2}$ the rate of the energy gain becomes very
large. One must note :
1) the rate of energy gain stems from a quantum effect
2) it requires the perpendicular momentum $K_{\perp}$ to be non zero,
3) it is resonantly enhanced.
To make an estimate for the resonant factor, after restoring dimensions (In
the pulsar context, $qA>>mc^{2},K_{\perp}c$), and using the dispersion
relation (see Mahajan & Asenjo (2022)),
$\omega^{2}-k^{2}c^{2}=\omega_{p}^{2}\left(1+q^{2}A^{2}/(m^{2}c^{4})\right)^{-1/2}\approx
mc^{2}\omega_{p}^{2}/qA$, one obtains

$\frac{dE}{dt}\simeq\frac{\sqrt{2}\hbar\omega^{2}}{\omega_{p}}\left(\frac{qA}{mc^{2}}\right)^{1/2}K_{\perp}c.$
(5)

Here, $\omega_{p}=\sqrt{4\pi nq^{2}/m}$ is the Langmuir frequency, $n$ denotes
the plasma number density. A good estimate for the amplitude of the EM wave is
$A=c\sqrt{4\pi cF_{r}}/\omega$, where $F_{r}\simeq L_{r}/(2\pi
r^{2}\theta^{2})$ is the radio flux of a pulsar, $\theta\simeq\frac{\Omega
r_{s}}{2c}\left(\frac{R_{lc}}{r_{s}}\right)^{1/2}$ is the half opening angle
of the pulsar’s emission beam (Machabeli & Usov, 1979), $L_{r}$ represents the
radio luminosity and $r_{s}\simeq 10$ km is the neutron star’s radius.

What we have shown in the paper is a promise- that resonant high energy
electron (wave) states exist, and one can qualitatively see an energy-gain
pathway through which the electron could be driven to such states.

##  3 Theory in the Pulsar Context

In this section, we apply the aforementioned mechanism to radio pulsars,
studying efficiency of the process over a range of physical parameters.

How long can the accelerating process last?f Until some energy loss
mechanisms, that become more efficient at higher energies, can balance the
growth. We will discuss three such loss mechanisms and assess their role in
limiting the maximum particle energy.

Synchrotron Radiation: Though the proposed acceleration mechanism was
independent of the magnetic field, there does exist a strong magnetic field in
the pulsar; the relativistic electrons, moving in such a magnetic field, will
experience extreme synchrotron losses, characterized by the emission power
(Rybicki & Lightman, 1979)

$P_{syn}\simeq\frac{2e^{4}B^{2}\gamma^{2}}{3m^{2}c^{3}},$ (6)

where $\gamma$ is the particle Lorentz factor, $B\simeq
B_{st}\times(r_{s}/r)^{3}$ is magnetic field, $B_{st}\simeq 3.2\times
10^{19}\sqrt{P\dot{P}}$ Gauss is the magnetic field close to the surface of
the neutron star (Shapiro & Teukolsky, 2004). Also $P$ ( $\dot{P}$) is the
period of rotation (its time derivative), $r_{s}\simeq 10^{6}$ cm denotes the
star radius, and $r$ is the radial coordinate. Throughout the paper we use the
$P-\dot{P}$ relation corresponding to the death line (traditionally the radio
pulsars are assumed to locate above the so-called death line in $P-\dot{P}$
diagram) $\dot{P}\simeq 10^{-15}\times P^{11/4}$ (Ruderman & Sutherland,
1979). Therefore, $\dot{P}$ might be even higher, giving a more efficient
picture of acceleration. Since the synchrotron induced cooling rate goes up as
$\gamma^{2}$ (Eq. (6)), it will eventually balance the rate of energy gain
putting a limit on the achieved energy. In the theory of pulsars, it is
generally believed that acceleration of particles is very efficient on the
length-scales of the light cylinder (LC - a hypothetical area where the linear
velocity of rotation equals the speed of light). Applying the synchrotron
limiting condition ($dE/dt\simeq P_{syn}$) to electrons in the LC zone, one
can obtain an expression for the maximum attainable energy:

$\gamma_{syn}^{max}\simeq\left(\frac{dE}{dt}\times\frac{3m^{2}c^{3}}{2e^{4}B^{2}}\right)^{1/2}\simeq$
$\simeq 1.1\times 10^{15}\times\left(\frac{f}{10^{8}\;Hz}\right)^{5/8}\times
P^{(2-3\alpha)/16}\times$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\;\left(\frac{\dot{P}}{10^{-15}s\;s^{-1}}\right)^{(3\alpha-12)/16}\times\left(\frac{r}{R_{lc}}\right)^{15/4},$
(7)

where $\dot{P}$ is normalized by a value typical for normal pulsars ($P\sim 1$
sec), $R_{lc}=cP/2\pi$ represents the LC radius and we have taken into account
that the radio spectral luminosity of pulsars, can be approximated as
$L_{r,\nu}\simeq 8.5\times 10^{3}\times\left(B_{st}/P^{2}\right)^{\alpha}$ erg
s-1Hz-1 with $\alpha\simeq 0.98$ (Stollman, 1987). Therefore, it is clear that
the resonant mechanism of particle energization is extremely efficient
providing relativistic factors of the order of $10^{16}$ on the LC area for
the emission frequency $100$ MHz.

It is also worth noting that any calculation concerning pulsars must take
cognizance of the large magnetic fields. Such a calculation will necessarily
add effects of order $e\hbar B/(mcE)$ to the energy spectrum, and also cause
asymmetry in the wave function. However, given that even for the pulsar
fields, the Landau level correction will be, at best, in 10-100 eV range, it
is not relevant for the present study.

Curvature Emission could be another mechanism that, in principle, might limit
the maximum Lorentz factor. However, the particles, following the curved
trajectories, will radiate (Ochelkov & Usov, 1980) if only if co-rotation is
maintained. The maximum possible $\gamma$ for which the particles co-rotate is
given by $\gamma_{co-rot}\simeq B_{lc}^{2}/(8\pi mc^{2}n_{{}_{GJ}})$.For
normal pulsars ($P\simeq 1$ s, $\dot{P}\simeq 10^{-15}s\;s^{-1}$), it reaches
the value $6.5\times 10^{6}$ on the LC zone. Here $n_{{}_{GJ}}={\bf\Omega\cdot
B}/(2\pi ec)$ is the so-called Goldreich-Julian number density (or the number
density of pulsar’s magentospheric particles) (Goldreich & Julian, 1969), and
$\Omega=2\pi/P$ represents the angular velocity of the pulsar. Thus for very
high energy particles ($\gamma>>\gamma_{co-rot}$), curvature losses will not
be relevant.

Inverse Compton Scattering: The magnetosphere of pulsars is full of thermal
photons originating from the star surface. The temperature of the photon gas
depends on the pulsar’s age $\tau=P/(2\dot{P})$ (Zhang & Harding, 2000)

$T\simeq\cases{5}.9\times
10^{5}\times\left(\frac{10^{6}\;yrs}{\tau}\right)^{0.1}K&\tau>10^{5.2}\;yrs\\\
2.8\times
10^{5}\times\left(\frac{10^{6}\;yrs}{\tau}\right)^{0.5}K\tau>10^{5.2}\;yrs{}$
(8)

that has been obtained by a rough fit to a numerical model presented in
(Schaab et al., 1999).

For $P=1$ sec, $\dot{P}=10^{-15}$ s s-1, $T\simeq 7\times 10^{4}K\equiv
T_{1}$. The accelerated electrons will, inevitably, encounter the thermal
photons and lose energy via inverse Compton (IC) scattering. We will consider
the IC cooling rates in two different domains, the Compton and the Klein-
Nishina regimes. The maximum $\gamma$ acquired by the particle is, as before,
attained when the energization rate is balanced by the cooling rate. In the
Compton regime, the cooling rate is given by (Rybicki & Lightman, 1979)

$P_{{}_{IC}}\simeq\sigma_{{}_{T}}c\gamma^{2}U,$ (9)

(where $\sigma_{{}_{T}}$ represents the Thomson cross section, $U\simeq
L/(4\pi r^{2}c)$ is the thermal energy density and $L\simeq 4\pi
r_{s}^{2}\sigma_{{}_{T}}T^{4}$ \- the corresponding luminosity) and would
yield the maximum relativistic factor

$\gamma_{{}_{IC}}^{max}\simeq\left(\frac{\bar{d}E}{dt}\times\frac{4\pi
r^{2}}{\sigma_{{}_{T}}L}\right)^{1/2}\simeq$ $\simeq 6.9\times
10^{6}\times\left(\frac{f}{10^{8}\;Hz}\right)^{5/8}\times
P^{(10-3\alpha)/16}\times$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\left(\frac{\dot{P}}{10^{-15}\;s\;s^{-1}}\right)^{(3\alpha-4)/16}\times\left(\frac{r}{R_{lc}}\right)^{7/4}\times\left(\frac{T_{1}}{T}\right)^{2},$
(10)

which comes out to be so high that the condition-
$\gamma\epsilon_{ph}/(mc^{2})\ll 1$ \- necessary for using the formula in the
Compton regime is violated ($\epsilon_{ph}\sim kT$ for thermal photons).

Figure 1: Here we show the plots of the maximum energy E(P) for different
emission frequencies, $f=(1;10;100)\times 10^{8}$ Hz on the LC zone
$r=R_{lc}$. Figure 2: For $P=1$ sec we show the maximum energy E(r) versus the
radial coordinate for the same set of frequencies, $f=(1;10;100)\times 10^{9}$
Hz.

In the Klein-Nishina regime, characterized by the cooling power (Blumenthal &
Gould, 1970)

$P_{{}_{KN}}\simeq\frac{\sigma_{{}_{T}}\left(mckT\right)^{2}}{16\hbar^{3}}\left(\ln\frac{4\gamma
kT}{mc^{2}}-1.981\right)\left(\frac{r_{s}}{r}\right)^{2},$ (11)

a similar prescription (equating $dE/dt\simeq P_{{}_{KN}}$ ) limits the
maximum Lorentz factor to

$\gamma_{{}_{KN}}^{max}\simeq 1.5\times
10^{5}\times\frac{T_{1}}{T}\times\exp\left(1.8\times 10^{4}\times
g\left(r,P,\dot{P}\right)\times\left(\frac{f}{10^{8}\;Hz}\right)^{5/4}\times\left(\frac{T_{1}}{T}\right)^{2}\right),$
(12)

where

$g\left(r,P,\dot{P}\right)=P^{(10-3\alpha)/8}\times\left(\frac{\dot{P}}{10^{-15}\;s\;s^{-1}}\right)^{(3\alpha-4)/8}\times\left(\frac{r}{R_{lc}}\right)^{3/2}.$
(13)

High values of the factor in the exponent means that the IC process in the KN
regime becomes significant only for unrealistically high Lorentz factors. It
is worth noting that the so-called Photo-pion reactions, when photons
interacting with protons can produce pions, might become significant, but by
taking the corresponding time-scale, $t_{p\gamma}\simeq 10^{6}\times
10^{19}eV/E$ sec (Aharonian, 2004) into account ($E$ denotes the proton’s
energy), one can straightforwardly show that the maximum attainable energy is
of the order of $10^{28}$ eV, that exceeds by many orders of magnitude the
limit corresponding to the synchrotron process. Therefore, inverse Compton
scattering and the photo-pion processes, like the curvature radiation, do not
impose any stringent limits on the maximum allowable energy; the synchrotron
cooling process (see Eq. (7)) is the only relevant one.

In Fig. 1, we plot the behavior of electron energy, $E=\gamma mc^{2}$
($\gamma\equiv\gamma_{syn}^{max}$), versus the pulsar period of rotation for
three different radio frequencies $f=(1;10;100)\times 10^{8}$ Hz on the LC
surface, $r=R_{lc}$. The maximum attainable energy is a continuously
increasing function of $P$; resonant energization can catapult electrons to
energies of the order of $10^{18-22}$ eV. In the following figure (Fig. 2),
for the same set of frequencies and a fixed period of rotation( $P=1$ sec), we
plot $E$ versus the non-dimensional radial coordinate (normalized by
$R_{lc})$); the maximum achievable energy falls in the range $10^{15-22}$ eV.

It is natural to assume that not all of the magnetospheric particles are
involved in the energization process. One can estimate a theoretical upper
limit on the number density, $n_{m}$ of such electrons. The total kinetic
power of accelerated particles $dK/dt\simeq 2\gamma mn_{m}c^{3}\pi
r^{2}\theta^{2}$ must remain below the slow-down luminosity,
$L_{sd}=I\Omega|\dot{\Omega}|=4I\pi^{2}\dot{P}/P^{3}$, of the pulsar. In the
expression for $L_{sd}$, $I=2Mr_{s}^{2}/5$ is the moment of inertia and
$M\approx 1.5\times M_{\odot}$ is the pulsar mass ($M_{\odot}\approx 2\times
10^{33}$g is the solar mass). The constraint, $dK/dt<L_{sd}$, puts an absolute
upper limit on $n_{m}$ that varies from $4.4\times 10^{-3}$ cm-3 (for
$P=0.001$ sec) to $4.4\times 10^{-4}$ cm-3 (for $P=1$ sec). Realistic number
densities will be, surely, much less because pulsars slow down via the
magneto-dipole radiation. Therefore, the flux of ultra-high energy particles
will be much less than the theoretical limit, $F\simeq L_{sd}/(4\pi r^{2})\sim
4\times 10^{-13}$ erg cm-2 s-1.

Before summarizing our results, we point a longstanding problem concerning the
fate of the low frequency magneto-dipole radiation. Gunn and Ostriker Gunn &
Ostriker (1969, 1969) have proposed that the magnetic dipole radiation could
accelerate the particles to extremely high energies. This work has been
generalized by introducing a refractive index different from vacuum (Kegel et
al., 1995), but the details on the working and efficiency of such a mechanism
are far from clear (Pétri, 2021). It may be interesting to explore the
interaction of such EM waves with the KG waves.

##  4 Summary

Resonant energization of relativistic electrons via the immense EM fields of a
radio pulsar was explored as a mechanism for producing very high energy cosmic
rays.

By considering several ”cooling” mechanisms that will eventually compete with
resonant energization, we calculated the limiting energies to which the
particles could be accelerated. For typical radio pulsars, these limits are
set by the synchrotron radiation emission.

Alternative cooling mechanisms like curvature radiation or Inverse Compton
scattering (Thomson or KN regime) on thermal photons, were found not to impose
any significant restrictions on energy gain

The situation could drastically change for the radio-loud pulsars that are
gamma-ray sources in the GeV-TeV regime. For such highly energetic photons, IC
scattering will become so strong that the particles will not accelerate at
all.

When limited, primarily, by synchrotron radiation, the electrons could attain
extremely high relativistic factors, $\sim 10^{15}$. This will translate to an
energy range $10^{15-22}$ eV for the large range of radio pulsars with periods
varying from millisecond to normal values($P=1$ sec).

A similar mechanism of acceleration could also pertain for protons, or
hadrons; the roles of limiting factors, however, will need to be reassessed.
For hadrons, for example, the IC scattering does not impose any constraints on
maximum energies (Aharonian, 2004).

By taking the pulsar’s slow-down luminosity into account we have also
estimated the upper limit on the number density of electrons that undergo
resonant energization; the fraction of particles that could be so energized,
will constitute only an insignificant fraction of the Goldreich-Julian number
density.

## Acknowledgments

The work of Z.O. was partially supported by the EU fellowships for Georgian
researchers, 2023 (57655523). Z.O. also would like to thank Torino
Astrophysical Observatory and Universitá degli Studi di Torino for hospitality
during working on this project. SMM acknowledges the support of US DOE grants:
DE-FG02-04ER54742 and DE-AC02- 09CH11466.

## Data Availability

Data are available in the article and can be accessed via a DOI link.

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