# Etching Plastic Searches for Dark Matter

Amit Bhoonahð, Joseph Bramante†ð, Brian Courtmanð, Ningqiang Song†ð
ð The Arthur B. McDonald Canadian Astroparticle Physics Research Institute and
Department of Physics, Engineering Physics, and Astronomy,
Queen’s University, Kingston, Ontario, K7L 2S8, Canada
†Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5,
Canada

###### Abstract

Large panels of etched plastic, situated aboard the Skylab Space Station and
inside the Ohya quarry near Tokyo, have been used to set limits on fluxes of
cosmogenic particles. These plastic particle track detectors also provide the
best sensitivity for some heavy dark matter that interacts strongly with
nuclei. We revisit prior dark matter bounds from Skylab, and incorporate
geometry-dependent thresholds, a halo velocity distribution, and a complete
accounting of observed through-going particle fluxes. These considerations
reduce the Skylab bound’s mass range by a few orders of magnitude. However, a
new analysis of Ohya data covers a portion of the prior Skylab bound, and
excludes dark matter masses up to the Planck mass. Prospects for future etched
plastic dark matter searches are discussed.

##  1 Introduction

Dark matter (DM) is critical to the formation of the early universe and the
structure of galaxies, and determining dark matter’s mass and couplings to
visible matter is a high priority of modern science. While many searches for
dark matter focus on finding a weakly interacting massive particles (WIMPs),
with a mass adjacent by a few orders of magnitude to Standard Model particles,
it has been known for decades that heavier and more strongly interacting dark
matter could make up the cold material that forms the bulk of matter in our
universe. In particular, it was well appreciated that dark matter could be a
nuclear material, with a large mass and nuclear scattering cross section [1,
2, 3, 4, 5]. Some “nuclearite” models have since been excluded, but
essentially the same considerations apply to models of heavy composite dark
matter [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32]. In this paper we will consider the
detection of strongly interacting massive particle dark matter (SIMPs) using
prior data collected by plastic etch detectors searching for cosmogenic
particles.

Unlike WIMPs, SIMPs often have such a large interaction cross section with
normal matter that their energy is depleted along their path to a detector. As
a consequence, after repeatedly scattering while traveling through the Earth’s
atmosphere and crust, SIMPs are slowed and the kinetic energy they can impart
to electrons and nuclei in underground detectors is reduced. If the SIMP cross
section is large enough, its underground interactions will not overcome the
energy threshold of underground dark matter searches. This makes above-ground
searches for SIMPs beneficial, although in practice this results in additional
particle interaction backgrounds from cosmic rays. On the other hand, because
SIMPs interact very strongly, comparatively simple detectors are capable of
detecting them.

In this work we will study bounds on dark matter interactions with nuclei,
obtained from prior searches for cosmogenic particles in etched plastic track
detectors. We will be particularly focused on plastic track detectors employed
aboard the Skylab space station to search for cosmic rays [33] and underground
in the Ohya quarry to look for monopoles, nuclearites, and exotic particles
[34]. When a particle passes through plastic polymer, its interactions with
nuclei and electrons can break the plastic’s molecular bonds binding them
together. The resulting microscopic damage can be enlarged and observed by
treating the plastic material with acid. After acid treatment, the damage
paths can be observed as holes in the plastic – in the case of many plastic
layers, the orientation of consecutive holes left by a particle’s passage can
be used to validate that particle’s path and even determine its energy as it
traveled through the plastic [35, 36]. However, plastic etch detectors require
large particle energy deposition for detection, around $\sim$ GeV/cm, compared
to underground dark matter experiments, which can require as little as a
single $\sim$eV energy deposition for detection. Compared to sensitive
underground dark matter search experiments, etched plastic polymer detectors
are relatively easy to assemble and maintain, and as a consequence the scale
and mass reach of these detectors achieved in the early 1990s still exceeds
the scale of modern underground dark matter search experiments, albeit for
sensitivity to much larger dark matter cross sections than those being
searched for underground.

Etched plastic track detectors have been used previously to constrain strongly
interacting dark matter. Reference [37] analyzed Skylab’s plastic track data
[33] to bound dark matter’s spin-independent interactions with nucleons. This
bound was re-derived in [38] and later adapted for large composite dark matter
models in [13]. Ohya quarry data [34] was also considered in Reference [38].
After these derivations, the Skylab nucleon cross section bound that has
appeared most in the literature was first presented in [39, 40], before
subsequently being reproduced in many papers that considered heavy strongly
interacting dark matter, $e.g.$ [41, 42, 27, 43, 44, 45, 30, 46, 47, 32]. Our
purpose in this note will be to provide a detailed bound on dark matter’s
interactions using plastic track detectors, by incorporating a realistic dark
matter flux, the geometry and composition of the detector and overburden, and
an accurate energy threshold in the case of [34]. We will find that using a
Maxwell-Boltzmann (MB) dark matter halo velocity distribution (instead of a
fixed DM velocity $\sim 10^{-3}{\rm c}$) and carefully accounting for
background events, appreciably changes plastic etch bounds, lessening the
previously reported Skylab mass reach by orders of magnitude, while somewhat
improving low cross section sensitivity.

The remainder of this note is organized as follows. In Section 2, we detail
the distribution and detection of dark matter at plastic track experiments. In
Section 3 we provide a detailed set of bounds on dark matter’s cross section
with nuclei using data from Skylab and Ohya [33, 34]. In Section 4 we
conclude. Appendix A provides a review of plastic track detectors and
thresholds.

##  2 Dark matter at etched plastic track detectors

As it traverses a polymer, dark matter can scatter against nuclei, breaking
apart molecules in a fashion that can later be detected. In this work, we will
assume two forms for dark matter interactions with nuclei. The first is a
spin-independent nuclear scattering cross section, whereby dark matter
interacts coherently with all nucleons in a nucleus. The nuclear cross section
in this case is customarily defined in terms of a per nucleon cross section
$\sigma_{\chi n}$ as

$\displaystyle\sigma_{\chi A}$ $\displaystyle=A^{2}\frac{\mu^{2}_{\chi
A}}{\mu^{2}_{\chi n}}\sigma_{\chi n},$ $\displaystyle({\rm spin-independent})$
(1)

where $A$ is the number of nucleons in a nucleus, $\mu_{\chi A}$ is the DM-
nucleus reduced mass and $\mu_{\chi n}$ the DM-nucleon reduced mass. Here we
have omitted nuclear form factors [48], as they will have a subdominant effect
on our treatment.

Our second model encompasses the case where that dark matter interacts with
the nucleus through a monotypical contact interaction. This is appropriate if
the dark matter is much larger than the nucleus, and opaque to nuclei [13, 27,
46]. $I.e.$ if dark matter’s radius is much larger than the nuclear radius,
and all incident nuclei scatter, then the cross section is independent of the
size or constitution of the nuclei. In this case for a contact cross section
we will use

$\displaystyle\sigma_{C},$ $\displaystyle({\rm contact})$ (2)

where this also assumes that the DM composite’s structure does not affect
scattering.

In terms of these quantities, the energy imparted to nuclei by dark matter
with mass $m_{\chi}$ passing through a background of nuclei with number
density $n_{A}$ is

$\displaystyle\frac{dE}{dx}=\frac{2E}{m_{\chi}}\sum_{A}\frac{\mu^{2}_{\chi
A}}{m_{A}}n_{A}\sigma_{\chi A},$ (3)

where the summation includes all background nuclei with $A$ nucleons, and
$E=m_{\chi}v^{2}/2$ is the dark matter kinetic energy. The above expression
applies to the spin-independent case. Here and in all that follows, equivalent
expressions for $\sigma_{C}$ are obtained by substituting $\sigma_{\chi
A}\rightarrow\sigma_{C}$.

Note that in Eq. (3) we have averaged over elastic scattering angles. An
averaged energy loss will be appropriate here, since we consider SIMPs that
scatter many times while traversing detector and shielding material.
Integrating this equation, we obtain an energy loss rate for a dark matter
particle traversing a material over distance $x$,

$\displaystyle
E\left(x\right)=E\left(0\right)\,\exp\left[-2\frac{x}{m_{\chi}}\sum_{A}n_{A}\frac{\mu^{2}_{\chi
A}}{m_{A}}\sigma_{\chi A}\right]$ (4)

where $E\left(0\right)=m_{\chi}v^{2}/2$ is the initial energy of the dark
matter particle. In practice, we will want to consider the dark matter’s
energy after traveling fully through an overburden length $x_{O}$ and detector
length $x_{D}$,

$\displaystyle
E\left(x_{O}+x_{D}\right)=E(0)\,\,\exp\left[\frac{-2}{m_{\chi}}\left(x_{O}\sum_{A\subset
O}n_{A}\frac{\mu^{2}_{\chi A}}{m_{A}}\sigma_{\chi A}+x_{D}\sum_{A\subset
D}n_{A}\frac{\mu^{2}_{\chi A}}{m_{A}}\sigma_{\chi A}\right)\right]$ (5)

where the sums are taken over nuclei $A$ in overburden material $O$ and
detector material $D$.

It remains to determine the velocity and density distribution of dark matter,
which is an important input to determine the speed and frequency of dark
matter particles incident on plastic track detectors. We will adopt a standard
[48] flux-normalized Maxwell-Boltzmann distribution with $v_{0}$ the Milky
Way’s circular speed and $\vec{v}_{E}$ the earth’s velocity in galactic rest
frame,

$\displaystyle f\left(\vec{v},\vec{v}_{E}\right)\,d^{3}v\
=\frac{2\pi}{\mathcal{N}}\,v^{3}\,\exp\left(-\frac{\tilde{v}^{2}}{v^{2}_{0}}\right)\Theta\left(v_{esc}-\tilde{v}\right)\,\,\sin\,\theta\,d\theta\,dv$
(6)

where

$\displaystyle\tilde{v}^{2}\equiv v^{2}+v_{E}^{2}+vv_{E}\,\cos\,\theta$ (7)

is the velocity of the dark matter particle in the rest frame of the galaxy,
$\mathcal{N}$ is a normalization constant such that $\int
f\left(\vec{v},\vec{v}_{E}\right)\,d^{3}v=1$, the Heaviside theta function
$\Theta$ enforces a cutoff such that dark matter’s speed does not exceed the
galaxy’s escape speed $v_{esc}$ at Earth’s position, and the factor of $2\pi$
indicates the assumption of no azimuthal angular dependence. We use the
following speeds, where in an aim to be conservative in setting bounds we take
the $1\sigma$ lowered value for each of these measured quantities. For the
escape speed from the Milky Way galaxy we take $v_{esc}$ = 503 km s-1 [49],
for the Milky Way circular velocity at the Sun’s position $v_{0}$ = 222 km s-1
[50] (applying their $3\%$ systematic uncertainty), and for the Sun’s relative
motion we take $v_{E}$ = 232 km s-1 [51]. For additional discussion,
especially of the last quantity, see [52, 53, 54, 55].

Now we are ready to specify the flux of dark matter at plastic track
detectors. With a flux-normalized velocity distribution, the average velocity
for incident dark matter is

$\displaystyle
v_{ave}=\int_{0}^{\infty}v\,f\left(\vec{v},\vec{v}_{E}\right)d^{3}v,$ (8)

where it is important to note that $f\left(\vec{v},\vec{v}_{E}\right)$
includes a $\Theta$ function cutting off the escape velocity, meaning the
integral is truncated well before $v=\infty$. The flux (in one direction)
across a planar detector with area $A$ is then

$\displaystyle\Phi_{\chi}=\frac{\rho_{\chi}}{m_{\chi}}v_{ave}\,A\,f_{g}$ (9)

where $\rho_{\chi}=0.3\leavevmode\nobreak\ {\rm GeV/cm^{3}}$ is the dark
matter energy density [50] and $f_{g}=\pi(1-\cos^{2}\theta_{D})/(4\pi)$ is a
geometric flux acceptance factor appropriate for a planar detector that
accepts particles inside a zenith angle $\theta_{D}$, where $\theta_{D}=0$ is
the angle normal to the plane. In order to produce accurate and conservative
bounds, we use Eq. (9) to determine the DM flux. This means we only consider
the DM flux that enters the planar detector from above – we do this because
Skylab and Ohya only provided detailed information about the material above
their detectors.

We are now prepared to discuss dark matter searches at planar plastic track
experiments. Both experiments we consider in the next section will have a
planar design, and each observed $N_{c}>10$ candidate events (most likely
cosmic rays) that we will treat here as indistinguishable from SIMPs to
conservatively set bounds on a potential dark matter flux. To set bounds on
dark matter-nuclear interactions using plastic track detectors, we must
require that more dark matter than $N_{c}$ would have passed through the
material above the detector and the detector itself, with an energy large
enough to leave observable polymer damage during the entire passage, $cf.$
Equation (5). Using the formulae derived in this section, we find it is
possible to set plastic track bounds on dark matter detectors in two steps.

  1. 1.

The maximum dark matter mass an experiment is sensitive to is found by
matching the number of events observed at the plastic detector, $N_{c}$, to
the expected dark matter flux over time $t$, $N_{c}=\Phi_{\chi}t$,

$\displaystyle m_{\chi}^{\rm
max}=\frac{\rho_{\chi}\,v_{ave}\,A\,f_{g}\,t}{N_{c}}.$ (10)

  2. 2.

Plastic track detector damage energy thresholds will favor detection of the
fastest-moving SIMPs in the MB distribution $cf.$ Eq. (6). As mentioned above,
to remain conservative we have selected $1\sigma$ slow values for our velocity
distribution parameters. In addition, for each dark matter mass we determine a
slow initial velocity $v_{i}$, defined by

$\displaystyle
N_{c}=\Phi_{\chi}t\int_{v_{i}}^{\infty}f\left(\vec{v},\vec{v}_{E}\right)d^{3}v$
(11)

For the calculations that follow we use $v_{i}$, which is the slowest speed of
the high velocity portion of dark matter’s MB distribution required to produce
$N_{c}$ events in the detector over time $t$. Then substituting Eq. (5) into
(3), we arrive at

$\displaystyle\frac{dE}{dx}\Big{|}_{th}=\frac{2E_{i}}{m_{\chi}}\left(\sum_{A\subset
D}\frac{\mu^{2}_{\chi A}}{m_{A}}n_{A}\sigma_{\chi
A}\right)\exp\left[\frac{-2}{m_{\chi}}\left(x_{O}\sum_{A\subset
O}n_{A}\frac{\mu^{2}_{\chi A}}{m_{A}}\sigma_{\chi A}+x_{D}\sum_{A\subset
D}n_{A}\frac{\mu^{2}_{\chi A}}{m_{A}}\sigma_{\chi A}\right)\right],$ (12)

where $\frac{dE}{dx}|_{th}$ is the energy threshold for a plastic track
detector and $E_{i}=m_{\chi}v_{i}^{2}/2$. This equation requires the DM energy
to just exceed the detector threshold during its entire transit through the
overburden and detector. Fixing $m_{\chi}$, detector, and overburden
parameters, and solving this equation for $\sigma_{\chi A}$ will yield _two_
real solutions for $\sigma_{\chi A}$, for all validly bounded $m_{\chi}$
masses. Along with $m_{\chi}^{\rm max}$, this defines the region of plastic
track detection sensitivity.

Equation (12) deserves some discussion, since it is a departure from prior
treatments, which separately estimated the threshold and overburden cross
section. As just mentioned, for fixed $m_{\chi}$ there will be two
$\sigma_{\chi A}$ solutions to Eq. (12). The larger cross section solution
corresponds to what has been called the overburden cross section, and
describes a SIMP that scatters a lot with the overburden, but still deposits
sufficient energy to leave etchable plastic tracks because of its relatively
large cross section. The lower cross section corresponds to what has been
called the threshold cross-section, and describes a SIMP that scatters less
with the overburden, and in so doing has enough energy to trigger the detector
during its passage. What is important, is that both of these cross-sections
depend on the dark matter scattering enough with the detector to leave a
visible signature, which is encapsulated by Eq. (12), whether SIMP detection
is limited by excessive overburden scattering or a small cross-section. We
will see that even in the latter case, overburden scattering can still
appreciably affect the cross section bound. In the next section we will use
this formula to set bounds on dark matter using Skylab and Ohya.

##  3 Dark Matter at Skylab and Ohya

We now use the procedure outlined in Section 2 to derive bounds from plastic
track particle searches at Skylab and Ohya. For each experiment we will
describe the number of events $N_{c}$ that cannot be ruled out as SIMPs,
threshold energies $\frac{dE}{dx}|_{th}$, along with detector and overburden
material. Below we also include a brief description of the etching process in
both experiments. Table 1 provides a summary of all essential experimental
parameters. For further details about the etching process, see Appendix A.

Figure 1: Skylab and Ohya bounds are obtained for a spin-independent dark
matter per-nucleon scattering cross section $\sigma_{\chi n}$, using a DM halo
velocity distribution, accounting for background events, and other refinements
detailed in the text. The prior Skylab bound [37], here reproduced from [39],
is shown with a dotted line. The Mica bound is from [56, 22]. The DAMA bound
is from [57]. We also show the CMB distortion [58, 59] and gas cloud heating
[42] bounds from dark matter scattering in the early Universe and in cold gas
clouds. For additional complementary bounds at lower cross-sections from
XENON1T and MAJORANA, see [31]. Figure 2: Skylab and Ohya bounds are obtained
for a fixed contact DM scattering cross section for all nuclei $\sigma_{c}$
(2), for DM mass $m_{\chi}$. The Chicago bound is taken from Ref. [46] and the
Mica bound from Ref. [30]. The limits on contact interactions from DAMA [57]
were derived in [30], using a conversion between spin-independent and contact
interactions for the lower limit, and an overburden analysis for the upper
limit. The CDMS-I and CRESST bound was obtained with a modified version of the
code verne [60, 61] which accounts for the dark matter flux and overburden
[30]; recently reported bounds from XENON1T and MAJORANA are also indicated
[31]. We also show the CMB distortion [58, 59] and gas cloud heating [30]
constraints arising from dark matter nuclear scattering in the early Universe
or in cold gas clouds.

The _Skylab_ plastic track search was conducted on board the Skylab space
station, primarily to study the high $Z$ composition of cosmic rays [33]. The
detector was a 1.17 m2 array of $36$ modules of Lexan plastic track detectors
sheets kept onboard for an observation time of $2.19\times 10^{7}$ s. Each
Lexan module contained thirty-two stacked $250$ $\mu$m thick Lexan sheets.
Assuming uniform stacks laid on top of each other, this amounted to a total of
0.80 cm of Lexan, which has the following composition by weight [62]

$\displaystyle\\{\rm H,C,O\\}$ $\displaystyle=\\{0.0555,0.756,0.189\\}.$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ ({\rm Lexan})$

The overburden for Skylab was a $0.37$ cm aluminum wall to which the Lexan
modules were mounted. We conservatively assume that all Skylab events entered
the aluminum and Lexan stack at the $\theta_{D}=60\degree$ maximum acceptance
angle, and hence took the longest possible path through both the overburden
material and detector. For a particle entering at a zenith angle of
$60\degree$ this corresponds to a distance of $0.74$ cm through the aluminum
and 1.6 cm through the Lexan. After etching, [33] identified 150 events in the
zenith angle range $\theta_{D}\leq 60\degree$ which passed largely undeflected
through the Lexan sheets. While a future analysis might be able to
conclusively attribute these events to high-Z cosmic ray fluxes, here we will
treat them as unreduced background events ($N_{c}$).

The detector threshold at Skylab was determined using fast ion calibration.
Matching the Skylab high-Z ion-scattering threshold using the Bethe formula
yields $\frac{dE}{dx}|_{th}\approx 0.5\ $ GeV/cm.

Skylab Etching. For Skylab, starting from sheet number two, every fourth sheet
in each module (except one which was used for calibration purposes) were
etched for 160 hours at 40$\degree$C in NaOH solution to which 0.057$\%$
Dowfax surfactant had been added. If a track was observed at coincident sites
in multiple sheets, it was assumed that the particle passed through all the
sheets in between. Around 150 events were observed that passed through all
etched sheets.

|  Skylab | Ohya
---|---|---
Area A | 1.17 $m^{2}$ | 2442 $m^{2}$
Duration t | 0.70 yr | 2.1 yr
Zenith cutoff angle | $\theta_{D}=60\degree$ | $\theta_{D}=18.4\degree$
Detector material |  0.25 mm thick Lexan $\times$ 32 sheets |  1.59 mm thick CR-39 $\times$ 4 sheets
Detector density | 1.2 g cm-3 Lexan | 1.3 g cm-3 CR-39
Detector length at $\theta_{D}$ | 1.6 cm | 0.66 cm
Overburden density | 2.7 g cm-3 Aluminum | 2.7 g cm-3 Rock
Overburden length at $\theta_{D}$ | 0.74 cm | 39 m
Table 1: Summary of the two planar etched plastic track experiments considered
in this work, Skylab and Ohya. Along with material elemental compositions
given in the text, the quantities shown are sufficient for setting bounds on
dark matter interactions using Eqs. (10) and (12). $\theta_{D}$ is the maximum
zenith angle of acceptance. For Ohya the value of $\frac{dE}{dx}|_{th}=0.3$
GeV/cm corresponds to the 10$\%$ efficiency threshold for nuclearites in
Figure 2 of [34].

The _Ohya_ experiment was undertaken to search for supermassive relics,
nuclearites, and other exotic particles using a 2442 m2 array of CR-39 plastic
track detectors, consisting of modules of CR-39 placed underground in three
caverns located at the Ohya stone quarries in Japan for 767 days [34]. Each
CR-39 module contained four stacked $1.59$ mm thick CR-39 sheets. This
amounted to a total of 0.64 cm of CR-39 with the chemical formula
$C_{12}H_{18}O_{7}$ indicating the following composition by weight

$\displaystyle\\{\rm H,C,O\\}=\\{0.066,0.526,0.408\\}.$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\text{CR-39})$

The overburden for Ohya was reported as $10^{4}$ g cm-2 of rock. Assuming a
rock density of $\sim$2.7 g cm-3, this corresponds to 37 meters of rock. For
rock’s elemental fractions by weight, we use standard values for Earth’s crust
[63],

$\displaystyle\\{\rm
O,Si,Al,Fe,Ca,Na,K,Mg,Ti,P\\}=\\{0.48,0.28,0.08,0.05,0.03,0.03,0.03,0.02,0.004\\}.\leavevmode\nobreak\
({\rm Rock})$

We conservatively assume that all Ohya events entered the rock and CR-39 stack
at a $\theta_{D}=18.4\degree$ maximum acceptance angle, and hence took the
longest possible path through both the overburden material and the detector. A
particle inbound along this zenith angle traverses 39 m of rock and etches
0.66 cm of CR-39. After etching, 94 coincident holes were identified, of which
74 were ruled out as manufacturing defects due to their irregular shapes. We
therefore consider the remaining 20 holes as candidate dark matter events
($N_{c}$). We note that [34] determined that only a few of these 20 events
could be cosmogenic based on the relative angles of the etched holes. We
nevertheless assume $N_{c}=20$ candidate events in placing dark matter
scattering limits to remain conservative.

The threshold energy deposition at Ohya was explicitly reported for DM
particles that scatter elastically with nuclei. This threshold was given in
terms of isotropic flux acceptance, where less flux was accepted for lower
thresholds, attained when particles impact the detector from a normal angle
(see Appendix A). Here, in order to maximize the accuracy of our overburden
modeling and remain conservative, we will use the Ohya 10$\%$ isotropic flux
detection efficiency, which implies the aforementioned zenith cutoff angle
$\theta_{D}=18.4\degree$. The threshold for detection inside this zenith angle
was reported to be $\frac{dE}{dx}|_{th}=0.3\ {\rm GeV\leavevmode\nobreak\
cm^{-1}}$.

Ohya etching. For Ohya, the etching process involved treating the first two
sheets in each module using a NaOH at a temperature of 90$\degree$ C. For
19$\%$ of these, one or two of the sheets were either over-etched or broken
and replaced by the corresponding third and/or fourth layers for the etching.
The etching process at [34] was slightly different than [33] in that [34]
required the etching process be strong enough to fuse two adjacent cones from
an etch pit into a hole.

The important quantities required to set DM bounds using Skylab and Ohya data
are summarized in Table 1. We note that, as discussed following Eq. (9), we
only consider the downward-going DM flux when setting bounds, since both Ohya
and Skylab only provide information about the material directly above their
detectors. Figures 1 and 2 show bounds obtained for both spin-independent and
contact dark matter scattering on nuclei.

A few features are worth noting in Figures 1 and 2. First, the incorporation
of a realistic MB velocity distribution for dark matter improves the lower DM
cross-section sensitivity of these experiments by an order of magnitude
(compared to assuming a fixed DM velocity $\sim 0.001{\rm c}$). This is
because our analysis has accounted for the high velocity component of the DM
distribution, although we stress that care has been taken not to overstate
this component: see discussion following Eq. (6) and preceding Eq. (12).
Secondly, we note that the mass reach of Skylab is substantially reduced as
compared to prior estimates appearing in the literature. This is partly
because we have accounted for the $\sim 150$ events observed at Skylab as
background events that cannot be ruled out as SIMPs, and partly because we
have properly required more dark matter’s low velocity component to trigger
the detector, as the flux becomes more limited at high DM masses (rather than
assuming that when the bound is becoming flux-limited, all DM particles will
still have $v\sim 0.001{\rm c}$). Finally, we note that requiring the DM
maintain enough energy to trigger Skylab during its entire transit through the
detector, and using a MB distribution when deriving the bound, has lessened
the upper cross section reach by a small amount and altered its scaling with
dark matter mass.

##  4 Conclusions

This work has provided a detailed derivation of constraints on dark matter’s
nuclear interactions in etched plastic track detectors, located on board the
Skylab space station and forty meters underground in the Ohya quarry. While as
compared to previous estimates, the dark matter mass reach of these
experiments has been reduced by a few orders of magnitude, the lower cross
section sensitivity has improved by an order of magnitude for some dark matter
masses.

Both Skylab and Ohya continue to place leading limits on heavy SIMP dark
matter’s interactions more than three decades after they collected data. In
both cases these experiments appear to have been primarily motivated by
searches for non-DM particles (cosmic rays and monopoles). Since there exist
many dark matter models ($e.g.$ [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 32]) that could have large cross
sections and masses, and given the uncharted interaction space evident at high
massses in Figures 1 and 2, it may be prudent to pursue a purpose-built high
mass dark matter search program based on plastic track detectors, located both
above and below ground, like Skylab and Ohya.

## Acknowledgements

We thank Nirmal Raj and Aaron Vincent for useful discussions. The work of AB,
JB, BC, and NS is supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC). Research at Perimeter Institute is supported in
part by the Government of Canada through the Department of Innovation, Science
and Economic Development Canada and by the Province of Ontario through the
Ministry of Colleges and Universities.

##  Appendix A Acid etched plastic track detectors

Plastic track detectors consist of sheets of polymers that can be damaged by
incident fluxes of cosmic rays and other particles, which interact with nuclei
and electrons in the material, breaking the bonds binding the constituent
monomers together, causing localized damage. The damaged plastic is then cured
by immersing it in an acid etching solution. Particle detection is possible,
since the curation rate will not be the same across the entire plastic sheet;
at the damaged sites the track etch rate, $v_{t}$, is different from the bulk
etch rate $v_{b}$, which is the rate at the undamaged sites. The etching
process leaves behind conically shaped damage pits. One can either scan
adjacent plastic sheets for coincident pairs of conical pits, as in [33], or
alternatively, if the acid etching process is rigorous enough, adjacent pits
can merge and form holes that are later observed with a microscope - this was
the procedure adopted in [34]. Neither method of analysis is without
complication. Care must be taken to avoid misidentifying manufacturing
defects, which are often irregularly shaped as compared to signal events,
which have either conical or spherical shapes depending on the etch method.
Coincident etched sites across multiple plastic sheets are needed to establish
a particle event. Because of these difficulties, the identification process
has an efficiency that depends on the size and orientation of the etched hole.

As one might anticipate, the rate of acid etching where particle damage has
occurred ($v_{t}$) depends on the energy deposited by the cosmic ray at a
damage site. This relationship can be obtained through calibration with a beam
of ions, typically iron [33]. The process involves firstly relating $v_{t}$,
obtained by dividing the measured size of etch pits by the total etch time
[33], to the effective charge and velocity of the incident beam of charged
particles through a power law phenomenological formula. This can then be
related to the energy transfer per unit length, $\frac{dE}{dx}$, using $e.g.$
the Bethe-Bloch formula.

The plastic etch method has a threshold, since the process will not identify
etch pits below a certain size, implying that the damage done at a site needs
to exceed some amount. This sets the minimum amount of energy transfer for
detection. For etch holes to form after curation, $v_{t}$ must be greater than
$v_{b}$. The exact criterion is given by an expression that depends on the
zenith angle $\theta_{c}$ along which the particle passes through the sheet,
since the acid will not etch oblique damage paths as efficiently as for damage
paths that are normal to the plastic sheet [34],

$\displaystyle\theta_{c}=\cos^{-1}\left(\frac{v_{b}}{v_{t}}\frac{D}{D-d}\right).$
(13)

Here $D$ and $d$ are the thickness of the plastic track before and after the
etching process respectively and $\theta_{c}$ defines a minimum zenith angle
below which damaged pits cannot be observed. We note in passing that in the
limit $d\ll D$ this is the same requirement set by Shirk and Price in [33]:
that the horizontal component of the track etch rate equals the bulk etch
rate.

## References

  * [1] E. Witten, _Cosmic Separation of Phases_ , _Phys. Rev. D_ 30 (1984) 272.
  * [2] E. Farhi and R. Jaffe, _Strange Matter_ , _Phys. Rev. D_ 30 (1984) 2379.
  * [3] A. De Rujula and S. Glashow, _Nuclearites: A Novel Form of Cosmic Radiation_ , _Nature_ 312 (1984) 734.
  * [4] M.W. Goodman and E. Witten, _Detectability of Certain Dark Matter Candidates_ , _Phys. Rev. D_ 31 (1985) 3059.
  * [5] A. Drukier, K. Freese and D. Spergel, _Detecting Cold Dark Matter Candidates_ , _Phys. Rev. D_ 33 (1986) 3495.
  * [6] S. Nussinov, _Technocosmology: could a technibaryon excess provide a ’natural’ missing mass candidate?_ , _Phys. Lett. B_ 165 (1985) 55.
  * [7] J. Bagnasco, M. Dine and S.D. Thomas, _Detecting technibaryon dark matter_ , _Phys. Lett. B_ 320 (1994) 99 [hep-ph/9310290].
  * [8] D.S.M. Alves, S.R. Behbahani, P. Schuster and J.G. Wacker, _Composite Inelastic Dark Matter_ , _Phys. Lett._ B692 (2010) 323 [0903.3945].
  * [9] G.D. Kribs, T.S. Roy, J. Terning and K.M. Zurek, _Quirky Composite Dark Matter_ , _Phys. Rev. D_ 81 (2010) 095001 [0909.2034].
  * [10] H.M. Lee, M. Park and V. Sanz, _Gravity-mediated (or Composite) Dark Matter_ , _Eur. Phys. J. C_ 74 (2014) 2715 [1306.4107].
  * [11] G. Krnjaic and K. Sigurdson, _Big Bang Darkleosynthesis_ , _Phys. Lett._ B751 (2015) 464 [1406.1171].
  * [12] W. Detmold, M. McCullough and A. Pochinsky, _Dark Nuclei I: Cosmology and Indirect Detection_ , _Phys. Rev._ D90 (2014) 115013 [1406.2276].
  * [13] D.M. Jacobs, G.D. Starkman and B.W. Lynn, _Macro Dark Matter_ , _Mon. Not. Roy. Astron. Soc._ 450 (2015) 3418 [1410.2236].
  * [14] M.B. Wise and Y. Zhang, _Stable Bound States of Asymmetric Dark Matter_ , _Phys. Rev. D_ 90 (2014) 055030 [1407.4121].
  * [15] M.B. Wise and Y. Zhang, _Yukawa Bound States of a Large Number of Fermions_ , _JHEP_ 02 (2015) 023 [1411.1772].
  * [16] E. Hardy, R. Lasenby, J. March-Russell and S.M. West, _Big Bang Synthesis of Nuclear Dark Matter_ , _JHEP_ 06 (2015) 011 [1411.3739].
  * [17] E. Hardy, R. Lasenby, J. March-Russell and S.M. West, _Signatures of Large Composite Dark Matter States_ , _JHEP_ 07 (2015) 133 [1504.05419].
  * [18] M.I. Gresham, H.K. Lou and K.M. Zurek, _Nuclear Structure of Bound States of Asymmetric Dark Matter_ , _Phys. Rev._ D96 (2017) 096012 [1707.02313].
  * [19] M.I. Gresham, H.K. Lou and K.M. Zurek, _Early Universe synthesis of asymmetric dark matter nuggets_ , _Phys. Rev._ D97 (2018) 036003 [1707.02316].
  * [20] J. Bramante, B. Broerman, R.F. Lang and N. Raj, _Saturated Overburden Scattering and the Multiscatter Frontier: Discovering Dark Matter at the Planck Mass and Beyond_ , _Phys. Rev. D_ 98 (2018) 083516 [1803.08044].
  * [21] M.I. Gresham, H.K. Lou and K.M. Zurek, _Astrophysical Signatures of Asymmetric Dark Matter Bound States_ , _Phys. Rev._ D98 (2018) 096001 [1805.04512].
  * [22] J. Bramante, B. Broerman, J. Kumar, R.F. Lang, M. Pospelov and N. Raj, _Foraging for dark matter in large volume liquid scintillator neutrino detectors with multiscatter events_ , _Phys. Rev._ D99 (2019) 083010 [1812.09325].
  * [23] M. Ibe, A. Kamada, S. Kobayashi and W. Nakano, _Composite Asymmetric Dark Matter with a Dark Photon Portal_ , _JHEP_ 11 (2018) 203 [1805.06876].
  * [24] D.M. Grabowska, T. Melia and S. Rajendran, _Detecting Dark Blobs_ , _Phys. Rev. D_ 98 (2018) 115020 [1807.03788].
  * [25] A. Coskuner, D.M. Grabowska, S. Knapen and K.M. Zurek, _Direct Detection of Bound States of Asymmetric Dark Matter_ , _Phys. Rev._ D100 (2019) 035025 [1812.07573].
  * [26] Y. Bai, A.J. Long and S. Lu, _Dark Quark Nuggets_ , _Phys. Rev. D_ 99 (2019) 055047 [1810.04360].
  * [27] M.C. Digman, C.V. Cappiello, J.F. Beacom, C.M. Hirata and A.H. Peter, _Not as big as a barn: Upper bounds on dark matter-nucleus cross sections_ , _Phys. Rev. D_ 100 (2019) 063013 [1907.10618].
  * [28] Y. Bai and J. Berger, _Nucleus Capture by Macroscopic Dark Matter_ , _JHEP_ 05 (2020) 160 [1912.02813].
  * [29] J. Bramante, J. Kumar and N. Raj, _Dark matter astrometry at underground detectors with multiscatter events_ , _Phys. Rev. D_ 100 (2019) 123016 [1910.05380].
  * [30] A. Bhoonah, J. Bramante, S. Schon and N. Song, _Detecting Composite Dark Matter with Long Range and Contact Interactions in Gas Clouds_ , 2010.07240.
  * [31] M. Clark, A. Depoian, B. Elshimy, A. Kopec, R.F. Lang and J. Qin, _Direct Detection Limits on Heavy Dark Matter_ , 2009.07909.
  * [32] J.F. Acevedo, J. Bramante and A. Goodman, _Nuclear Fusion Inside Dark Matter_ , 2012.10998.
  * [33] E.K. Shirk and P.B. Price, _Charge and energy spectra of cosmic rays with Z $\gtrsim$ 60: the Skylab experiment._, _The Astrophysical Journal_ 220 (1978) 719.
  * [34] S. Orito et al., _Search for supermassive relics with 2000-m 2 array of plastic track detector_, _Phys. Rev. Lett._ 66 (1991) 1951.
  * [35] S.A. Durrani and R.K. Bull, _Solid State Nuclear Track Detection. Principles, Methods and Applications_ (1987).
  * [36] R. Fleischer, P. Price, R. Walker and R. Walker, _Nuclear Tracks in Solids: Principles and Applications_ , University of California Press (1975).
  * [37] G.D. Starkman, A. Gould, R. Esmailzadeh and S. Dimopoulos, _Opening the Window on Strongly Interacting Dark Matter_ , _Phys. Rev. D_ 41 (1990) 3594.
  * [38] P.C. McGuire, _Low background balloon borne direct search for ionizing massive particles as a component of the dark galactic halo matter_ , Ph.D. thesis, Arizona U., 1994.
  * [39] B.D. Wandelt, R. Dave, G.R. Farrar, P.C. McGuire, D.N. Spergel and P.J. Steinhardt, _Selfinteracting dark matter_ , in _4th International Symposium on Sources and Detection of Dark Matter in the Universe (DM 2000)_ , pp. 263–274, 6, 2000 [astro-ph/0006344].
  * [40] A.L. Erickcek, P.J. Steinhardt, D. McCammon and P.C. McGuire, _Constraints on the Interactions between Dark Matter and Baryons from the X-ray Quantum Calorimetry Experiment_ , _Phys. Rev. D_ 76 (2007) 042007 [0704.0794].
  * [41] G.D. Mack, J.F. Beacom and G. Bertone, _Towards Closing the Window on Strongly Interacting Dark Matter: Far-Reaching Constraints from Earth’s Heat Flow_ , _Phys. Rev. D_ 76 (2007) 043523 [0705.4298].
  * [42] A. Bhoonah, J. Bramante, F. Elahi and S. Schon, _Galactic Center gas clouds and novel bounds on ultralight dark photon, vector portal, strongly interacting, composite, and super-heavy dark matter_ , _Phys. Rev. D_ 100 (2019) 023001 [1812.10919].
  * [43] J. Bramante, A. Buchanan, A. Goodman and E. Lodhi, _Terrestrial and Martian Heat Flow Limits on Dark Matter_ , _Phys. Rev. D_ 101 (2020) 043001 [1909.11683].
  * [44] R. Garani and P. Tinyakov, _Constraints on Dark Matter from the Moon_ , _Phys. Lett. B_ 804 (2020) 135403 [1912.00443].
  * [45] Y. Bai, M. Korwar and N. Orlofsky, _Electroweak-Symmetric Dark Monopoles from Preheating_ , _JHEP_ 07 (2020) 167 [2005.00503].
  * [46] C.V. Cappiello, J. Collar and J.F. Beacom, _New Experimental Constraints in a New Landscape for Composite Dark Matter_ , 2008.10646.
  * [47] J.F. Acevedo, J. Bramante, A. Goodman, J. Kopp and T. Opferkuch, _Dark Matter, Destroyer of Worlds: Neutrino, Thermal, and Existential Signatures from Black Holes in the Sun and Earth_ , 2012.09176.
  * [48] J. Lewin and P. Smith, _Review of mathematics, numerical factors, and corrections for dark matter experiments based on elastic nuclear recoil_ , _Astropart. Phys._ 6 (1996) 87.
  * [49] A.J. Deason, A. Fattahi, V. Belokurov, N.W. Evans, R.J.J. Grand, F. Marinacci et al., _The local high-velocity tail and the Galactic escape speed_ , _Monthly Notices of the Royal Astronomical Society_ 485 (2019) 3514 [1901.02016].
  * [50] A.-C. Eilers, D.W. Hogg, H.-W. Rix and M.K. Ness, _The Circular Velocity Curve of the Milky Way from 5 to 25 kpc_ , _The Astrophysical Journal_ 871 (2019) 120 [1810.09466].
  * [51] R. Schönrich, J. Binney and W. Dehnen, _Local kinematics and the local standard of rest_ , _Monthly Notices of the Royal Astronomical Society_ 403 (2010) 1829 [0912.3693].
  * [52] J. Bovy, _A purely acceleration-based measurement of the fundamental Galactic parameters using Gaia EDR3_ , _arXiv e-prints_ (2020) arXiv:2012.02169 [2012.02169].
  * [53] J. Bovy, C. Allende Prieto, T.C. Beers, D. Bizyaev, L.N. da Costa, K. Cunha et al., _The Milky Way’s Circular-velocity Curve between 4 and 14 kpc from APOGEE data_ , _The Astrophysical Journal_ 759 (2012) 131 [1209.0759].
  * [54] C. Wegg, O. Gerhard and M. Bieth, _The gravitational force field of the Galaxy measured from the kinematics of RR Lyrae in Gaia_ , _Monthly Notices of the Royal Astronomical Society_ 485 (2019) 3296 [1806.09635].
  * [55] S. Chakrabarti, P. Chang, M.T. Lam, S.J. Vigeland and A.C. Quillen, _A measurement of the Galactic plane mass density from binary pulsar accelerations_ , _arXiv e-prints_ (2020) arXiv:2010.04018 [2010.04018].
  * [56] P. Price and M. Salamon, _Search for Supermassive Magnetic Monopoles Using Mica Crystals_ , _Phys. Rev. Lett._ 56 (1986) 1226.
  * [57] R. Bernabei et al., _Extended limits on neutral strongly interacting massive particles and nuclearites from NaI(Tl) scintillators_ , _Phys. Rev. Lett._ 83 (1999) 4918.
  * [58] C. Dvorkin, K. Blum and M. Kamionkowski, _Constraining Dark Matter-Baryon Scattering with Linear Cosmology_ , _Phys. Rev. D_ 89 (2014) 023519 [1311.2937].
  * [59] V. Gluscevic and K.K. Boddy, _Constraints on Scattering of keV-TeV Dark Matter with Protons in the Early Universe_ , _Phys. Rev. Lett._ 121 (2018) 081301 [1712.07133].
  * [60] B.J. Kavanagh. Verne, Astrophysics Source Code Library, record ascl:1802.005, available at https://github.com/bradkav/verne, 2016.
  * [61] B.J. Kavanagh, _Earth scattering of superheavy dark matter: Updated constraints from detectors old and new_ , _Phys. Rev. D_ 97 (2018) 123013 [1712.04901].
  * [62] R.G. Williams, C.J. Gesh and R.T. Pagh, _Compendium of Material Composition Data for Radiation Transport Modeling_ (2006), 10.2172/902408.
  * [63] B. Mason and C. Moore, _Principles of Geochemistry_ , Wiley (1982).


# Poncelet and the arquimedean twins

Liliana Gabriela Gheorghe

Abstract. We give a sharp construction for twins in arbelos, based on polar
reciprocity. In the process, new circles displaying arquimedean afinities came
into scene.

Figure 1. Twins in arbelos are solutions of two distinct Apollonius’ problems:
but why are they arquimedean?

##  1\. Introduction

Arbelos is the greek for ’shoemaker’s knife’, the shape bounded by three
pairwise tangent semicircles with diameters lying on the same line. As a
geometric object, it was first studied by Arquimedes in his Book of Lemmas,
hence dates back more than 2200 years ago. If we draw the common (internal)
tangent to the arbelos circles then the two circles that tangent this linee,
the arbelos outer circle and one of the arbelos’ circles are called "the
twins"; see figure 5. Arquimedes had already spotted them and proved that
their radius is half the harmonic mean of the two arbelos i-circles. As a
tribute, circles in arbelos, congruent with the twins are called arquimedean;
chasing arquimedean circles in arbelos was is a constant theme ever-since.

Perhaps the most humble arquimedean circle is those whose diameter is the
parallel through $M$ at the bases of the (rectangular) trapeze of basis
$R_{1},R_{2}$ and altitude $R_{1}+R_{2}$; see figure 2; in fact the proof of
this fact uses elementary proprieties in trapezius. Longer, yet
straightforward computation confirms that the tangents from $O_{1}$ and
$O_{2}$ to this circle meet precisely at the center of the arbelos i-circle.
This circle was spotted in [B] (somehow backwardly that presented here), but
we presume Arquimedes already knew about it.

### Main results

Rather than chasing arquimedean circles, here we are interested in explain why
the twins are identical and how to draw them. We foreseen the (classic) twins
as solutions of two degenerated Apollonius’ problem, and we find their centers
as intersection of special conics.

Keywords: arbelos, Apollonius’s problem, circle inversion, poles, polar duals.

2020 Mathematics Subject Classification: 51A05, 51A30, 51M15.

To perform their intersection, we use polar reciprocity, the method tailored
by Poncelet ([P]) in order to proof his Porism. The proof itself led to a
sharp geometric construction for their centers, which is our main result; see
theorem 1. The method also explains why the radii of the twins is the same. We
then attach another pair of circles, which has twice the radius of the initial
one, and study a new tern of related circles. One of the circles of the tern
is arquimedean, and was first spotted by Scotch in [S]; the other two are new
and verify an arquimedean-type relation which we prove in theorem 4.

### Related work

The literature on arbelos is so rich and the references are so abundant, that
we limit ourselves to a small sample that show that interest for this problem
is still vivid. In [B] a famous construction for the i-circle in arbelos, much
simpler than the one from Archimedes’ proof, due to the existence of a third
arquimedean circle; [W] also provides a simple construction of the i-circle in
(classic) arbelos; in [D-L] the authors study inversions mapping that switch
two given circles and apply it to arbelos; [O] studies a generalization of
arbelos; [O-W] study twin circles in skewed arbelos. [We] is a handy source
that collect many of these facts and related bibliography.

Figure 2. An arquimedean circle (bordeaux) canonically attached to an arbelos;
two other circles of same radius (solid yellow) can be drawn tangent to the
arbelos’ circles (grey) and to their common tangent; but in this case, why do
they also the arbelos’ external circle?

### Notations

We shall note by $(O)$ a circle centered in $O$. By reflection in $(O)$ we
mean the symmetry (or inversion) with respect to circle $(O).$ We freely use
"dual curve", "polar dual" or "reciprocal curve" as synonymous.

##  2\. Twins’ centers and the Apolonius’ problem

Twins in arbelos are the two circles that tangent the arbelos’ nested circles
and their common (internal) tangent line; thus, each twin is a solution of an
Apollonius problem; hence, as in [G], their centers can be obtained by
intersecting some special conics.

Let a circle and a tangent line to it be given, as in figure 3.

######  Lemma 1.

The locus of the centers of the circles that tangent both the circle and the
line is a parabola focused at circle’s center and whose vertex is at the
tangency point of the circle and the line.

Let $(O)$ and $(O_{1})$ be two internally tangent circles, as in figure 3.

######  Lemma 2.

The locus of the centers of the circles that tangent two nested (internally)
tangent circles is an ellipse focused at the centers of the two nested circles
and passing through their common tangency point.

Both lemma 1 and 2 have elementary proofs that we omit. When we specialize to
arbelos, we get a sharper result. Refer to figure 3.

######  Proposition 1.

The center of each twin is the intersection between an ellipse and a parabola.

Each ellipse has one vertex at the common tangency point of each internal
arbelos circle with the external one, (points $A_{1}$ and $A_{2}$
respectively), has one focus in $O$ and the other focus in $O_{1}$ and
$O_{2},$ respectively. The vertex of the parabolas is the common tangency
point of the internal arbelos’ circles $(O_{1})$, $(O_{2})$ (point $M$) and
their focus is in $O_{1}$ and $O_{2},$ respectively.

At this point, any drawing software is able to perform the intersection of
these two conics. Nevertheless, we are "not done", since the task is to
perform a geometric construction: a construction with a straight-line and a
compass only. In a geometric construction, one cannot intercept "continuous
curves", other then circles and lines.

In fact, while two arbitrary conics cannot be (geometrically) intersected, the
conics that aroused here, have a special feature: a common focus. And here is
where polar reciprocity comes into scene. The reader not acquainted with this
topic, may see Appendix.

######  Lemma 3.

The intersection of two conics that have a common focus are the poles of the
common tangents of their duals, w.r. to an inversion circle centered at their
(common) focus.

For more details on poles, polars and polar reciprocity, see [A], [S], [GSO].

Figure 3. I) (left) The locus of the centers of the circles that tangents two
internally tangent circles is an ellipse (orange) which has the foci at the
centers of the two circles and one vertice at their common tangency point. II)
(right) The locus of the centers of the circles that tangents externally one
circle (solid grey) and a line (which tangent the circle) is a parabola
(purple), which has the focus into the center of the circle and one vertice at
the common tangency point of the circle and the line.  Figure 4. I) The polar
dual w.r. to $(O_{1})$ of the ellipse focused in $O_{1}$ and $O$ and passing
through $A_{1}$ is a circle (dotted orange) tangent at $A_{1}$ to $(O)$. Its
diameter is $[A_{1}O_{2}^{\prime}]$ where $O_{2}^{\prime}$ is the reflection
of $O_{2}$ in $(O).$ II) The polar dual w.r. to $(O_{2})$ of the parabola
focused in $O_{1}$ and vertex $M$ is the circle of diameter $[O_{2}M].$ III)
The similitude center of these two circles is $O_{2},$ the center of the
second arbelos’ circle. IV) The two (real) intersection points of the parabola
and the ellipse are the poles (w.r. to $(O_{2})$) of their common tangents :
shown is point $D_{1},$ the center of one of the arbelos’ twins  Figure 5.
Twins in arbelos: a solution via siblings, tangents, and poles.

Now we may specify who these duals are. Refer to figure 4 and choose as the
inversion circle $(O_{1})$.

######  Lemma 4.

The polar duals (w.r. to $(O_{1})$) of the ellipses $\mathcal{E}_{1}$ and the
parabola $\mathcal{P}_{1}$ in figure 4, are two circles that:

  1. (i)

are tangent to $(O_{1})$ at $A_{1}$ and $M$ respectively;

  2. (ii)

the diameter of the parabola’s dual is $[O_{1}M]=R_{1}$ and the diameter of
the ellipse’s dual is
$[A_{1}O_{2}^{\prime}]=\frac{2R_{1}^{2}+R_{1}R_{2}}{R_{1}+R_{2}}$ where
$O_{2}^{\prime}$ is the reflection of $O_{2}$ in $(O_{1});$

  3. (iii)

the similitude centers of these two circles is $O_{2}.$

###### Proof.

The proof uses known facts on the polar of a conic, that we collect in
Appendix. Both the parabola and the ellipse in figure 4 have one common focus
in $O_{1};$ therefore, their duals w.r. to $(O_{1})$ are circles. The dual of
a parabola w.r. to an inversion circle centered at its focus is a circle,
whose diameter is $[O_{1}M]$, where $M$ is the reflection of the parabola’s
vertice and $O_{1}$ is the center of inversion. Since the vertice $M$ is
located on the inversion circle, its is invariant by reflection; thus, the
dual of the parabola is a circle with diameter $[O_{1}M]=R_{1}.$

The ellipse focused in $O_{1}$ and $O$, and passing through $A_{1}$ has its
second vertice at $O_{2}.$ Therefore, its polar dual w.r. to $(O_{1})$ is a
circle whose diameter is $[A_{1}O_{2}^{\prime}]$, where $O_{2}^{\prime}$ is
the reflection of $O_{2}$ in $(O_{1})$; the vertice $A_{1}$ is invariant,
since it is a point of the inversion circle. Thus, the diameter of the dual is

(1)
$A_{1}O_{2}^{\prime}=A_{1}O_{1}+O_{1}O_{2}^{\prime}=R_{1}+\frac{R_{1}^{2}}{R_{1}+R_{2}}=\frac{2R_{1}^{2}+R_{1}R_{2}}{R_{1}+R_{2}}.$

Let $o_{1}$ and $S_{1}$ be the centers of these dual circles; their radius
are, respectively

(2)
$r_{1}=\frac{A_{1}M}{2}=\frac{2R_{1}^{2}+R_{1}R_{2}}{2(R_{1}+R_{2})}\;\;\mbox{
and}\;\;r_{m}=\frac{R_{1}}{2}.$

In order to prove that the similitude center is $O_{2},$ we prove that

(3) $\frac{o_{1}O_{2}}{S_{1}O_{2}}=\frac{r_{1}}{r_{m}}.$

In fact,

$S_{1}O_{2}=\frac{R_{1}}{2}+{R_{2}}=\frac{R_{1}+2R_{2}}{2}$

and

$o_{1}O_{2}=A_{1}O_{2}-A_{1}o_{1}=2R_{1}-\frac{R_{1}(2R_{1}+R_{2})}{2(R_{1}+R_{2})}=\frac{(2R_{1}+R_{2})(R_{1}+2R_{2})}{2(R_{1}+R_{2})};$

hence

$\frac{o_{1}O_{2}}{S_{1}O_{2}}=\frac{(2R_{1}+R_{2})}{(R_{1}+R_{2})};$
$\frac{r_{1}}{r_{m}}=\frac{R_{1}(2R_{1}+R_{2})}{2(R_{1}+R_{2})}\cdot\frac{2}{R_{1}}=\frac{(2R_{1}+R_{2})}{(R_{1}+R_{2})},$

hence equation 3 is verified.

∎

We may attach to any arbelos a pair of tangent circles, "the siblings:" these
are two circles, mutually tangent at the common tangency points of the arbelos
circles, and of half radius each.

The results proved above justifies the following new and sharp construction of
the centers of the twins; refer to figure 5.

######  Theorem 1.

The centers of the twins are the poles of the tangents drawn from the center
of a arbelos’ circle, to the opposite sibling.

##  3\. Twins’ radii

At this point, we draw the twins’ centers using polar reciprocity. This does
not guarantees that the circles are arquimedean. Fortunately, the method is
proper for computation purposes, as well. Refer to figure 5.

######  Lemma 5.

Let $(O_{1})$ and $(O_{2})$ the (internal) arbelos circles and let $(S_{1})$
and $(S_{2})$, be their siblings: two circles whose diameters are $[O_{1}M]$
and $[O_{2}M]$. Let $O_{1}T_{2}$ the tangent from $O_{1}$ to circle $(S_{2});$
let $D_{2}$ be its pole w.r. to $(O_{2})$; construct similarly $D_{1}.$ Then

$O_{2}D_{2}-R_{2}=O_{1}D_{1}-R_{1}.$

###### Proof.

Let $r_{1},r_{2}$ be half of the radius $R_{1},R_{2}$. By construction,

$\triangle{O_{1}O_{2}P_{2}}\sim\triangle{O_{1}S_{2}T_{2}},$

hence

$\frac{r_{2}}{O_{2}P_{2}}=\frac{2r_{1}+r_{2}}{2r_{1}+2r_{2}},\;\;\;\mbox{or}\;\;\;O_{2}P_{2}=\frac{2r_{2}(r_{1}+r_{2})}{2r_{1}+r_{2}}.$

Since $D_{2},$ the pole of the line $O_{1}T_{2}$ is the reflection of $P_{2},$
the projection of $O_{2}$ to the line $O_{1}T_{2}$, with respect to $(O_{2}).$

$O_{2}P_{2}\cdot O_{2}D_{2}=4r_{2}^{2},$

hence

$O_{2}D_{2}=\frac{4r_{2}^{2}}{O_{2}P_{2}}=\frac{4r_{2}^{2}(2r_{1}+r_{2})}{2r_{2}(r_{1}+r_{2})}=\frac{2r_{2}(2r_{1}+r_{2})}{r_{1}+r_{2}}.$

This gives

$O_{2}D_{2}-R_{2}=O_{2}D_{2}-2r_{2}=\frac{2r_{2}(2r_{1}+r_{2})}{r_{1}+r_{2}}-2r_{2}=2r_{2}\bigg{[}\frac{(2r_{1}+r_{2})}{r_{1}+r_{2}}-1\bigg{]},$

hence

(4) $r=O_{2}D_{2}-R_{2}=\frac{2r_{1}\cdot r_{2}}{r_{1}+r_{2}}.$

Formula 4 is symmetric with respect to $r_{1}$ and $r_{2};$ an identical
computation, obtained by switching the indices $1$ and $2,$ ends the proof.

∎

Lemma 3 and Lemma 5, prove the result on the twins; refer to figure 5.

######  Theorem 2.

The circles centered in $D_{1},D_{2}$ and of radius $r$ are arquimedean twins
in arbelos: they are tangent to both $(O_{1})$ $(O_{2})$ and $(O)$, and also
are tangents to the line $l$ and their (common) radius $r$ is

$\frac{1}{r}=\Big{[}\frac{1}{R_{1}}+\frac{1}{R_{2}}\Big{]},$

where $R_{1},R_{2}$ denotes the radii of the arbelos’ circles.

###### Proof.

The fact that the points $D_{1}$ and $D_{2}$ are the centers of the twins is
guaranteed by Lemma 3. Thus, the circle centered in $D_{2}$ and whose radius
is $D_{2}O_{2}-R_{2}$ tangents the line $l,$ as well, hence these are the
twins. The relation 4 ends the proof. ∎

Figure 6.  The center of the arbelos’ i-circle as intersection of two
ellipses: one focused in $O$ and $O_{1}$ and passing through $O_{2}$, (green)
and the other focused in $O$ and $O_{2}$ and passing through $O_{1}.$ Their
intersection is the pole w.r. to $(O)$ of the common tangent (blue) to their
reciprocal circles (orange and green circles).

### Arbelos i-circle

For the sake of completeness, and because is effortless, we show how may draw
the arbelos’ i-circle $\mathcal{I}$, the circle that tangents the three
arbelos’ circles.

######  Theorem 3.

Refer to figure 6.

  1. (i)

The center of $\mathcal{I}$ is obtainable as the intersection of two ellipses:
one focused in $O$, and $O_{1}$, and axis $[O_{2}A_{1}]$ and the other focused
in $O$ and $O_{2}$ and axis $[O_{1}A_{2}]$.

  2. (ii)

The intersection points of these ellipses are the poles of the common tangents
to their reciprocal circles, w.r. to $(O)$.

  3. (iii)

The radius of the arbelos i-circle is

$R=\frac{R_{1}R_{2}(R_{1}+R_{2})}{R_{1}^{2}+R_{1}R_{2}+R_{2}^{2}}$

.

###### Proof.

The two ellipses have a common focus in $O$ and tangent the inversion circle
$(O)$ at their vertices $A_{1}$ and $A_{2},$ respectively. Hence their duals
are two circles of diameters $[A_{1}O_{2}^{\prime}]$ and
$[A_{2}O_{1}^{\prime}]$ respectively, where $O_{1}^{\prime}$ is the reflection
of $O_{1}$ in $(O)$ and $O_{2}^{\prime}$ is the reflection of $O_{2}$ in
$(O)$.

Let $H$ be the similitude center of the reciprocal circles.

Then

$\triangle{HC_{1}P_{1}}\sim\triangle{HOP}\sim\triangle{HC_{2}P_{2}}.$

Hence

$OP=\frac{(R_{1}+R_{2})(R_{1}^{2}+R_{1}R_{2}+R_{2}^{2})}{R_{1}^{2}+R_{2}^{2}}.$

Since $D$ is the pole of $HP,$

$OD\cdot OP=(R_{1}+R_{2})^{2}.$

The radius of the i-circle is

$R=(R_{1}+R_{2})-OD=(R_{1}+R_{2})-\frac{(R_{1}+R_{2})^{2}}{OP}$

which gives

$R=\frac{R_{1}R_{2}(R_{1}+R_{2})}{R_{1}^{2}+R_{1}R_{2}+R_{2}^{2}}.$

∎

Figure 7.  The cousin i-circle (golden) tangents internally $(O)$ and
externally $(A_{1})$ and $(A_{2})$. Its center obtains as intersection of an
ellipse (violet) and a hyperbola, focused in $A_{2},O$ and $A_{1},A_{2},$
respectively. $S$ is the pole of the common tangent at their reciprocal
circles, w.r. to $(A_{2});$ its radius is the same as those of the classic
twins: $\frac{1}{s}=\big{[}\frac{1}{R_{1}}+\frac{1}{R_{2}}\big{]}.$

##  4\.  Arquimedean circles in doubling arbelos

We now associate to a classic arbelos, two other circles, passing through the
common tangency points $M$ and centered at each of the end-point of arbelo’s
diameter; we call it a doubling arbelos; see figure 7 and 9.

By i-circle in a doubling arbelos, we mean the circle that tangents externally
the two new-added circles and internally the arbelos diametral circle. Such
circle was first spoted by Scotch, in [S], who proved that it is arquimedean;
above, we indicate another proof for this fact, and also a construction for
the center of this circle, both based on polar reciprocity.

######  Theorem 4.

The i-circle in a doubling arbelos is arquimedean:
$\frac{1}{s}=\big{[}\frac{1}{R_{1}}+\frac{1}{R_{2}}\big{]}.$

###### Proof.

Refer to figure 7. We foresee the i-circle associated to a doubling arbelos’
as a solution of an Apolonius’ problem; therefore, its center $S,$ can be
obtained as an intersection between an ellipses focused in $O$ and $A_{2},$
passing through $O_{1}$ and a hyperbola focused in $A_{1}$ and $A_{2},$ and
passing through $M.$ Since these two conics have a common focus in $A_{2},$ a
polar dual w.r. to $(A_{2})$ maps them into a pair of circles, whose diameters
are the the reflection of their (ellipses and hyperbola’s) vertices. $S,$ the
center of the i-circle in a doubling arbelos is the pole of the common tangent
to their reciprocal circles. Straightforward computations, similar to those in
the proof of theorem 2 provide an give proof to the Scotch’s result.

These method embeds the geometric construction of $S$, as a pole of the common
tangents to these reciprocal circles.

∎

Now we consider two new circles associated to a doubling arbelos: the twin-
cousins are the circles that tangents externally the two arbelo’s circles of
diameters $[A_{1}A_{2}]$ and $[A_{1}M]$, as well as internally the arbelos
external circle and one of the (new) circles centered at $A_{1}$ and of radii
$A_{1}M$ or $A_{2}$ and of radii $A_{2}M;$ see figure 9 (bottom side).

The twin-cousins (solid blue circles) are not congruent; nevertheless they
verify an arquimedean-type metric relation.

######  Theorem 5.

Let $s_{1},s_{2}$ be the radius of the twin-cousins; then

$\frac{1}{s_{1}}+\frac{1}{s_{2}}=3\big{[}\frac{1}{R_{1}}+\frac{1}{R_{2}}\big{]}$

Figure 8. $S_{1},$ the center of a twin-cousin, obtained as intersection of
two ellipses, as well as a pole of the common tangent at their dual circles
(dotted, centered at $o_{1}^{\prime}$ and $o_{2}^{\prime}$, respectively.
Figure 9.  Metric coincidences in arbelos: I) The (classic) twins (solid
golden) and the cousin i-circle (blue) are congruent and arquimedean:
$s=r_{1}=r_{2}$ and $\frac{1}{s}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$. II) The two
cousin-circles (solid blue) verifies
$\frac{3}{s}=\big{[}\frac{1}{s_{1}}+\frac{1}{s_{2}}\big{]}$ hence
$\frac{1}{s_{1}}+\frac{1}{s_{2}}=3\big{[}\frac{1}{R_{1}}+\frac{1}{R_{2}}\big{]}$

###### Proof.

Refer to figure 8. The center of twin cousin $S_{1}$, obtains as intersection
of two ellipses: $\mathcal{E}_{1}$ the ellipses focused in $O$ and $O_{1}$,
and passing through $A_{1},$ and another ellipses, $\mathcal{E}^{\prime}$,
focused in $O_{1}$ and $A_{1}$, and passing through $M.$ Perform a dual
transform w.r. to $(O_{1})$; since $O_{1}$ is the common focus of the ellipses
$\mathcal{E}_{1}$ and $\mathcal{E}^{\prime},$ their duals are two circles. The
dual of $\mathcal{E}_{1}$ is the circle $\mathcal{C}^{\prime}_{2}$ of diameter
$[A_{1}O_{2}^{\prime}],$ where $O_{2}^{\prime}$ is the reflection of $O_{2}$
in $(O_{1})$; denote by $o_{2}^{\prime}$ and $R_{2}^{\prime}$ be its center
and radius. Similarly, the dual of $\mathcal{E}^{\prime}$ is the circle
$\mathcal{C}^{\prime}_{2}$ of diameter $[B_{1}^{\prime}M],$ where
$B_{1}^{\prime}$ is the reflection of the ellipse’s vertice $B_{1}$ in
$(O_{1})$; denote by let $o_{1}^{\prime}$ and $R_{1}^{\prime}$ be its center
and radius. The common tangent, $p$ of these dual circles intercept the line
of centers in $\Omega_{1},$ their similitude center. Let
$P_{1},P_{1}^{\prime},P_{2}^{\prime}$ denote respectively the projections of
$O_{1},o_{1}^{\prime},o_{2}^{\prime}$ on the tangent $p.$

Since $\mathcal{E}^{\prime}$ the ellipses focused in $O_{1}$ and $A_{1}$, and
passing through $M,$ and since $O_{1}B_{1}=2\cdot R_{1}$,

$O_{1}B_{1}\cdot
O_{1}B_{1}^{\prime}=R_{1}^{2},\;\;\;O_{1}B_{1}^{\prime}=\frac{R_{1}}{2};$

therefore,

(5)
$R_{1}^{\prime}=\frac{3R_{1}}{4},\;\;\;O_{1}O_{1}^{\prime}=\frac{R_{1}}{4}.$

Now we find the segment $O_{1}O_{2}^{\prime}$. The diameter
$A_{1}A_{1}^{\prime}$ of the dual circle of the ellipses
$\mathcal{E}^{\prime}$ is

$A_{1}A_{1}^{\prime}=O_{1}A_{1}+O_{1}A_{1}^{\prime}=R_{1}+\frac{R_{1}^{2}}{R_{1}+R_{2}}=\frac{2R_{1}^{2}+R_{1}R_{2}}{R_{1}+R_{2}};$

hence

(6) $R_{1}^{\prime}=O_{1}A_{1}^{\prime}=\frac{R_{1}^{2}}{R_{1}+R_{2}}$

Finally,

$O_{1}O_{2}^{\prime}=\frac{A_{1}A_{1}^{\prime}}{2}-O_{1}A_{1}^{\prime}=\frac{2R_{1}^{2}+R_{1}R_{2}}{2(R_{1}+R_{2})}-\frac{R_{1}^{2}}{(R_{1}+R_{2})}=\frac{R_{2}^{2}}{2(R_{1}+R_{2})}.$

Thus,

$O_{1}O_{2}^{\prime}=O_{1}A_{1}-O_{2}^{\prime}A_{1}=R_{1}-\frac{A_{1}A_{1}^{\prime}}{2}=R_{1}-\frac{2R_{1}^{2}+R_{1}R_{2}}{2(R_{1}+R_{2})}.$

In other words,

$O_{1}O_{1}^{\prime}=O_{1}M-O_{1}^{\prime}M=R_{1}-R_{1}^{\prime}=\frac{R_{1}R_{2}}{2(R_{1}+R_{2})}.$

and

(7)
$O_{1}^{\prime}O_{2}^{\prime}=O_{1}O_{2}^{\prime}+O_{1}O_{1}^{\prime}=\frac{R_{1}R_{2}}{2(R_{1}+R_{2})}+\frac{R_{1}}{4}=\frac{R_{1}^{2}+3R_{1}R_{2}}{4(R_{1}+R_{2})}$

Now let $x=\Omega_{1}O_{1}^{\prime}$ and $p=O_{1}P_{1};$ by hypothesis,

$\triangle{\Omega O_{1}^{\prime}P_{1}^{\prime}}\sim\triangle{\Omega
O_{1}P_{1}}\sim\triangle{\Omega O_{2}^{\prime}P_{2}^{\prime}}.$

Using the relations above, we obtain

$x=\frac{3R_{1}(R_{1}+3R_{2})}{4(R_{1}-R_{2})},\;\;\;p=\frac{R_{1}(R_{1}+2R_{2})}{(R_{1}+3R_{2})}.$

Hence

$O_{1}^{\prime}O_{2}^{\prime}=\frac{R_{1}^{2}+3R_{1}R_{2}}{4(R_{1}+R_{2})},\;\;R_{1}^{\prime}=\frac{3R_{1}}{4},\;\;R_{2}^{\prime}=\frac{2R_{1}^{2}+R_{1}R_{2}}{2(R_{1}+R_{2})}.$

With these ingredients in place, a straightforward computations led to the
radius of the cousin-twin centered at $S_{1}$

(8) $s_{1}=\frac{R_{1}R_{2}}{R_{1}+2R_{2}}.$

If we interchange the indices $1$ and $2$, we obtain

(9) $s_{2}=\frac{R_{1}R_{2}}{R_{2}+2R_{1}}.$

Thus

$\frac{1}{s_{1}}+\frac{1}{s_{2}}=3\big{[}\frac{1}{R_{1}}+\frac{1}{R_{2}}\big{]}.$

∎

## Appendix. Brief recall on polar reciprocity

We include here a brief recall on polar reciprocity.

Let us fix $C(\Omega,R)$ a circle centered in $\Omega$ and of radius $R,$
which we shall call inversion circle.

######  Definition 1.

If $p_{0}$ is a line that does not pass through $\Omega,$ its pole is the
inverse of the projection of the center $\Omega,$ on the line $p_{0}.$

######  Definition 2.

If $P_{0}$ is a point $(P_{0}\neq\Omega),$ the polar of $P_{0}$ is the
perpendicular line on $\Omega P_{0},$ that pass through $P_{1},$ the inverse
of $P_{0}.$

######  Definition 3.

The polar dual (or a reciprocal curve) of a regular curve (w.r. to an
inversion circle) defines as the curve whose points are the poles of the
tangents of the original curve.

When we perform the dual of a circle, w.r. to an inversion circle, we obtain
conics.

######  A 1.

(see e.g. [S], art. 306 and 309) The dual of a circle $\gamma=C(O,r),$ w.r. to
an inversion circle ${C(\Omega,R)},$ is a conic, ${\Gamma};$ if $d$ denotes
the distance between the centres of the reciprocated and inversion circles,
$d=\Omega O,$ then:

i) $\Gamma$ is an ellipse, if $r<d;$

ii) $\Gamma$ is a parabola, if $r=d;$

iii) $\Gamma$ is a hyperbola, if $r>d.$

Moreover, (one of) the the focus of the dual conic $\Gamma$ is precisely
$\Omega,$ the center of the inversion circle; its directrix is the polar of
$O,$ the center of the reciprocated circle and the eccentricity is
$e=\frac{r}{d}.$

This theorem has a very useful counterpart.

######  A 2.

The dual of a conic $\Gamma,$ w.r. to an inversion circle centered into its
focus, is a circle, $\gamma.$ The symmetric of the vertices of the conic
$\Gamma,$ are a pair of diametrically opposite points of the dual circle,
$\gamma.$ The pole of the directrix of $\Gamma,$ is the center of the circle
$\gamma.$

All the geometrical elements of the dual conic, can be drawn with straight-
line and compass, since all the steps involves drawing the symmetric of a
point and the pole of a line.

A final useful fact regards intersections of (regular) curves.

######  A 3.

The intersection of two curves are the poles of their common tangents.

## Bibliography

[AZ] A. V. Akopyan and A. A. Zaslavsky, Geometry of Conics, Amer. Math. Soc.,
Providence, RI, 2007 [Ba] Bankoff,L.,Are the twin circles of Archimedes really
twins?, Mathematics Magazine, 47: 214–218. [B] Bergsten, C.,Magic Circles in
the arbelos TMME, vol7, n.2-3, 209-222, ISSN 1551-3440

[D-L] Danneels, E., van Lamoen, F. : Midcircles and the arbelos. Forum
Geometricorum 7 (2007), 53-65.

[G] Gheorghe, L.G. _Apollonius’ problem: in pursuit of a natural solution_ ,
Int.J.Geom.,9 (2) (2020), 39–51.

[G-S-O] Glaeser,G., Stachel, H., Odehnal, B., _The universe of Conics_ ,
Springer Specktrum, Springer-Verlang Berlin Heidelberg, 2016.

[L-W] van Lamoen, F. and Weisstein, E. W. : "Pappus Chain." From MathWorld - A
Wolfram Web Resource. [ http://mathworld.wolfram.com]

[O] Oller-Marcen´,A.,The f-belos Forum Geometricorum Volume 13 (2013)
103–-111.

[O-W] H. Okumara and M. Watanabe, The twin circles of Archimedes in a skewed
arbelos, Forum Geom., 4 (2004) 229–251. [O-W] H. Okumura and M. Watanabe, A
generalization of Power’s Archimedean circle, Forum Geom., 6 (2006), 103–105.

[P] Poncelet, J.V., _Traité de propriétés projectives des figures_ , Gauthier-
Villars, Paris, 1866

[S] Salmon, G.,_A treatise on conic sections_ , Longman, Green, Reader and
Dyer, London, 1869.

[S] Schoch, T.: “arbelos: The Woo circles”. [http://www.retas.de/thomas

/arbelos/woo.html ]

[We] Weisstein, E. W.: "arbelos." From MathWorld - A Wolfram Web Resource. [
http://mathworld.wolfram.com/arbelos.html ]

[W] Woo, P., Simple Constructions of the Incircle of an arbelos, Forum
Geometricorum Volume 1 (2001) 133–-136.

DEPARTAMENTO DE MATEMÁTICA

UNIVERSIDADE FEDERAL DE PERNAMBUCO

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