# Size distributions of bluish and reddish small main-belt asteroids obtained
by Subaru/Hyper Suprime-Cam 111This research is based on data collected at
Subaru Telescope, which is operated by the National Astronomical Observatory
of Japan. We are honored and grateful for the opportunity of observing the
Universe from Maunakea, which has the cultural, historical and natural
significance in Hawaii.

Natsuho Maeda  Department of Planetology, Kobe University, 1-1 Rokkodai-cho,
Nada-ku, Kobe 657-8501, Japan  Tsuyoshi Terai  Subaru Telescope, National
Astronomical Observatory of Japan, National Institutes of Natural Sciences
(NINS), 650 North A‘ohoku Place, Hilo, HI 96720, USA  Keiji Ohtsuki
Department of Planetology, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe
657-8501, Japan  Fumi Yoshida  School of Medicine, University of Occupational
and Environmental Health, 1-1 Iseigaoka, Yahata, Kitakyusyu 807-8555, Japan
Planetary Exploration Research Center, Chiba Institute of Technology, 2-17-1
Tsudanuma, Narashino, Chiba 275-0016, Japan  Kosuke Ishihara  Department of
Planetology, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
Department of Astronomical Science, School of Physical Sciences, Graduate
University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  Takuto Deyama  Department of Planetology, Kobe
University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan  NTT Data MSE Co.,
1-2-33, Miyahara, Yodogawa-ku, Osaka-shi, Osaka 532-0003, Japan

(Received June 1, 2021; Revised September 27, 2021; Accepted September 30,
2021)

###### Abstract

We performed a wide-field survey observation of small asteroids using the
Hyper Suprime-Cam installed on the 8.2 m Subaru Telescope. We detected more
than 3,000 main-belt asteroids with a detection limit of 24.2 mag in the
$r$-band, which were classified into two groups (bluish C-like and reddish
S-like) by the $g-r$ color of each asteroid and obtained size distributions of
each group. We found that the shapes of size distributions of asteroids with
the C-like and S-like colors agree with each other in the size range of
$0.4-5$ km in diameter. Assuming the asteroid population in this size range is
under collision equilibrium, our results indicate that compositional
difference hardly affects the size dependence of impact strength, at least for
the size range between several hundred meters and several kilometers. This
size range corresponds to the size range of “spin-barrier”, an upper limit
observed in the rotation rate distribution. Our results are consistent with
the view that most asteroids in this size range have a rubble-pile structure.

minor planets, asteroids: general

††journal: Astronomical Journal

##  1 INTRODUCTION

The main asteroid belt, located between the orbits of Mars and Jupiter,
consists of asteroids with various spectral types. S-type and C-type are two
of the most major types in the main-belt (DeMeo & Carry, 2014). S-type
asteroids are thought to have a rocky composition, and to be the parent bodies
of ordinary chondrite meteorites because the abundance and spectral features
are consistent with each other (e.g., Binzel et al., 2001; Burbine et al.,
2002). However, the shapes of their spectra do not exactly match and this
problem was previously recognized as a missing link between the asteroids and
the meteorites. Understanding of space weathering was advanced by laboratory
experiments (e.g., Sasaki et al., 2001) as well as spacecraft studies by NEAR
Shoemaker (Prockter et al., 2002) and Hayabusa. The Hayabusa spacecraft
brought a sample of an S-type asteroid (25143) Itokawa back to the Earth, and
analysis of the sample revealed that S-type asteroids are parent bodies of
ordinary chondrites (Nakamura et al., 2011). On the other hand, C-type
asteroids are thought to have a carbon-rich composition with hydrous minerals
(Usui et al., 2019; Potiszil et al., 2020) and are likely to be parent bodies
of carbonaceous chondrites. A sample from a C-type asteroid (162173) Ryugu was
returned by the Hayabusa2 spacecraft and is now under analysis. Another
sample-return mission by OSIRIS-REx for a C-group (a broader definition of
C-type) asteroid (101955) Bennu is also underway. The exact composition of
C-type asteroids will soon be revealed.

S-type and C-type asteroids likely have different compositions, and their
formation regions should also be different. In the current main-belt, S-type
and C-type asteroids dominate in the inner-belt and the middle-/outer-belts,
respectively, and their heliocentric distributions partly overlap (DeMeo &
Carry, 2014; Yoshida & Nakamura, 2007). This indicates that asteroids were
mixed radially by some process (e.g., planetary migration) in the early Solar
System (e.g., Walsh et al., 2011; Nesvorny, 2018; Yoshida et al., 2019, 2020).
Therefore, understanding the evolutions of asteroids with different spectral
types can provide a clue to reveal the Solar System’s history. In the present
work, we focus on the size distribution of asteroids with different spectral
types. Since the current size distributions of asteroids are produced as a
result of continuous collisional evolution, comparing the size distributions
of asteroids with different spectral types is expected to provide important
insights into their collisional evolution.

Many observational studies have been carried out about the size distribution
of main-belt asteroids (MBAs; e.g., Jedicke & Metcalfe, 1998; Ivezić et al.,
2001; Yoshida et al., 2003; Yoshida & Nakamura, 2004, 2007; Wiegert et al.,
2007; Parker et al., 2008; Gladman et al., 2009; Terai et al., 2013; August &
Wiegert, 2013; Ryan et al., 2015; Peña et al., 2020). Some of them studied the
size distributions of MBAs with different spectral types. Ivezić et al. (2001)
analyzed the Sloan Digital Sky Survey (SDSS) data and detected about 13,000
asteroids. They classified their asteroid sample into bluish and reddish
groups, based on the principal component analysis using the $g-r$ and $r-i$
colors. They obtained the size distribution for each group in the range of
absolute magnitude $10<H<19$ ($1\lesssim D\lesssim 40$ km, $D$: diameter).
They approximated their cumulative size distribution with a power-law function

$N(>D)\propto D^{-b},$ (1)

where $N(>D)$ is the number of asteroids with diameters larger than $D$. They
found that the power-law indices of the size distributions of bluish/reddish
groups agree with each other in the size range of $5<D<10$ km ($b=4.00\pm
0.05$) while they are slightly different in the size range of $3<D<5$ km
($b=1.40\pm 0.05$ for the bluish group and $b=1.20\pm 0.05$ for the reddish
group). Yoshida & Nakamura (2007) performed a wide-field survey observation
with the Suprime-Cam installed on the Subaru Telescope and detected about
1,000 MBAs. They classified their asteroid sample into bluish (C-like) and
reddish (S-like) groups based on $B-R$ color, and obtained the size
distributions for each group in the range of $14.6<H<20.2$. They found that
the power-law indices of the bluish and reddish groups agree with each other
within errors at $0.3<D<1$ km ($b=1.33\pm 0.03$ for bluish asteroids and
$b=1.29\pm 0.02$ for reddish asteroids). These works showed slightly different
results, and whether bluish and reddish asteroids have similar size
distributions is still inconclusive.

In this study, we performed a wide-field survey observation with the Hyper
Suprime-Cam installed on the 8.2 m Subaru Telescope. We detected over 3,000
small MBAs and measured the size distributions of bluish and reddish asteroids
using a large homogeneous statistical sample about three times larger than
Yoshida & Nakamura (2007), which allowed us to measure more accurate size
distributions. This can provide a clue to better understanding of the
collisional evolution of asteroids with different spectral types. The
structure of this paper is as follows. In Section 2, we describe our survey
observation and data analysis. In Section 3, we show the results of color and
size distributions. Comparison with previous works and implications of our
results are presented in Section 4. We summarize our conclusions in Section 5.

##  2 OBSERVATION AND DATA ANALYSIS

###  2.1 Observation

We performed the survey observation on January 26, 2015 (UT) using the Hyper
Suprime-Cam (HSC) installed on the 8.2 m Subaru Telescope. HSC is a wide-field
prime focus camera with 104 CCDs for science (2048 pix $\times$ 4096 pix). The
diameter of a field of view is 1.5 degrees with a pixel scale of 0.17
arcseconds (Miyazaki et al., 2018). We used the $g$-band (wavelength
$0.40-0.55\;\mathrm{\mu m}$) and the $r$-band (wavelength
$0.55-0.70\;\mathrm{\mu m}$) filters (Kawanomoto et al., 2018). We obtained
the imaging data of eight fields near the opposition and the ecliptic plane,
covering the sky area of about $14\;\mathrm{deg^{2}}$. The survey area is
shown in Figure 1.

The observation procedure is as follows: First, we visited each field five
times with the $r$-band at intervals of about 40 minutes. Next, we visited
each field in the same way with the $g$-band. We can reduce the effect of
brightness variation due to asteroids’ rotation by averaging five values of an
asteroid’s flux obtained with each filter. The time of single exposure is 200
seconds. We summarize our observation in Table 1.

Figure 1: Locations of the surveyed fields of this work. Each circle represents the area covered by the field-of-view of HSC. The solid line shows the ecliptic plane. The cross shows the opposition on the observation day. Table 1: Observation log. Field ID | UT | R.A. | Decl. | Filter | Seeing | Number of
---|---|---|---|---|---|---
|  | (h:m:s) | (d:m:s) |  | (arcsec) | Detection
FIELD01 | 06:51 - 09:25 | 08:20:09 | +18:30:00 | $r$ | 0.59 - 0.99 | 468
| 11:33 - 14:03 |  |  | $g$ | 1.03 - 1.48 |
FIELD02 | 06:55 - 09:29 | 08:25:41 | +19:15:00 | $r$ | 0.60 - 0.99 | 398
| 11:38 - 14:07 |  |  | $g$ | 0.98 - 1.69 |
FIELD03 | 07:00 - 09:33 | 08:25:41 | +17:45:00 | $r$ | 0.54 - 0.90 | 405
| 11:43 - 14:11 |  |  | $g$ | 0.85 - 1.57 |
FIELD04 | 07:05 - 09:37 | 08:36:43 | +17:45:00 | $r$ | 0.58 - 0.78 | 427
| 11:48 - 14:15 |  |  | $g$ | 1.02 - 1.55 |
FIELD05 | 07:16 - 09:45 | 08:36:43 | +16:15:00 | $r$ | 0.55 - 0.69 | 439
| 11:58 - 14:23 |  |  | $g$ | 0.93 - 1.58 |
FIELD06 | 07:21 - 09:49 | 08:31:12 | +17:00:00 | $r$ | 0.54 - 0.76 | 512
| 12:03 - 14:27 |  |  | $g$ | 1.11 - 1.78 |
FIELD07 | 07:26 - 09:53 | 08:25:41 | +16:15:00 | $r$ | 0.59 - 0.95 | 363
| 12:08 - 14:31 |  |  | $g$ | 1.04 - 2.84 |
FIELD08 | 07:31 - 09:56 | 08:20:09 | +17:00:00 | $r$ | 0.57 - 0.97 | 447
| 12:13 - 14:35 |  |  | $g$ | 1.15 - 1.65 |

###  2.2 Image Reduction and Detection

We processed the data into calibrated images and created source catalogs using
the hscPipe (ver.4.0.5), the HSC data reduction/analysis pipeline software
(Bosch et al., 2018). We used the Pan-STARRS 1 (PS1) $3\pi$ catalog (Tonry et
al., 2012; Schlafly et al., 2012; Magnier et al., 2013) for astrometric and
photometric calibrations.

Our developed software extracted candidates of moving objects from the lists
of light sources detected by hscPipe based on the following conditions: (i)
the sky motion is within the range of $\dot{\lambda}>-48$ arcsec hr-1 and
$|\dot{\beta}|<-2.5\dot{\lambda}-55$ arcsec hr-1, where $\dot{\lambda}$ and
$\dot{\beta}$ are the motions along ecliptic longitude and latitude,
respectively (see gray dot-dashed lines in Figure 2). (ii) the objects are
detected from four or more out of five visits in the $r$-band data. Finally,
we checked all of the candidates with eyes to remove false detections. The
resulting data set contains 3,809 MBA candidates.

###  2.3 Measurements

We measured the fluxes of the detected MBAs by aperture photometry using the
same technique as in Yoshida & Terai (2017). (i) We correct the centroid
positions of each object by fitting an object model generated based on its
motion and the seeing size of the image. (ii) We measure the flux of each
object using a moving circular aperture formed based on the sky motion and the
seeing size. If another source such as a field star exists in the aperture, we
subtract another image taken at the same field from the original image using
the Optimal Image Subtraction method (Alard & Lupton, 1998; Bramich, 2008) to
measure the accurate flux of the target object. (iii) We convert the flux to
magnitude using the photometric zero point measured by hscPipe. We defined the
value of each object’s apparent magnitude for each band as the average of all
the visits. Finally, we obtained both $r$\- and $g$-band apparent magnitudes
for 3,472 MBAs.

An apparent magnitude of an asteroid varies with time due to its rotation if
it has a non-axisymmetric shape with respect to its spin axis. We estimated
how our survey cadence reduced this effect using a Monte Carlo simulation.
According to the previous studies for asteroids’ rotation period distribution,
average values of the peak-to-peak amplitude and rotation period are 0.4 mag
and 5.0 hours, respectively, for asteroids with a diameter of several
kilometers (Waszczak et al., 2015; Pravec et al., 2002). Using these values,
we generated 100,000 synthetic objects having a light curve given by a
sinusoidal function with a random initial rotation phase, and performed a
virtual observation of these asteroids with conditions similar to our
observations. We found that the standard deviations of their apparent
magnitudes and colors are 0.034 mag and 0.066 mag, respectively, excluding the
photometric uncertainties.

###  2.4 Orbit Estimation

We estimated the orbital elements for each of the detected asteroids from
their apparent motion. Jedicke (1996) derived the relationship between an
object’s apparent motion velocity in ecliptic coordinates
$(\dot{\lambda},\;\dot{\beta})$ and its orbital elements, assuming that they
are located near the opposition:

$\displaystyle\begin{dcases}\dot{\lambda}=\frac{\sqrt{GM_{\odot}}}{R-1}\left(\frac{\sqrt{a(1-e^{2})}}{R}\cos
I-1\right),\\\
\dot{\beta}=\pm\frac{\sqrt{GM_{\odot}}}{R(R-1)}\sqrt{a(1-e^{2})}\sin
I,\end{dcases}$ (2)

where $a$, $I$, $e$, and $R$ are object’s semi-major axis, inclination,
eccentricity, and heliocentric distance, respectively. $G$ and $M_{\odot}$ are
the gravitational constant and the mass of the Sun, respectively. We cannot
determine the orbital elements of detected asteroids accurately, because we
performed our survey observation only one night. Therefore, we estimated their
semi-major axis $a$ and inclination $I$ under the assumptions that they are on
circular orbits and were observed at the opposition (i.e., $e=0$, $R=a$,
$\lambda^{\prime}=0\arcdeg$, $\beta=0\arcdeg$, where $\lambda^{\prime}$ is
ecliptic longitude from the opposition). These parameters were determined as
the best-fit values for the Equations (2) with the measured sky motion through
a least-square method. If the square root of the sum of squared residuals is
larger than 0.01 arcsec hr-1, we remove the object from our sample (we removed
13 asteroids for this reason). We show the distribution of apparent velocities
of 3,472 asteroids in Figure 2. The removed asteroids are shown with blue
crosses. We obtained a sample of 3,459 MBAs for which $g$\- and $r$-band
magnitudes as well as orbital elements are successfully obtained. We show the
distribution of their orbital elements in Figure 3 (we will explain the
classification into C-like/S-like asteroids in Section 3.3).

The statistical errors in the obtained orbital elements were estimated by a
Monte Carlo simulation as follows (Terai et al., 2013; Nakamura & Yoshida,
2002): (i) We generate 100,000 synthetic asteroids randomly distributed in our
survey field. The orbital elements of these asteroids are given with pseudo-
random numbers that follow the distribution of orbital elements of known MBAs
taken from the ASTORB database (Bowell et al., 1994). (ii) We calculate their
apparent motion velocities under the same condition as our survey, and
estimate their orbital elements from their apparent motion using Equations (2)
under the assumptions of circular orbits and observation near the opposition.
We found that the root mean square errors of estimated $a$, $I$, $R$, and
$\Delta$ are $\sigma_{a}=0.12$ au, $\sigma_{I}=6.8\arcdeg$, $\sigma_{R}=0.35$
au, and $\sigma_{\Delta}=0.35$ au, respectively. These values are roughly
consistent with the previous works that used the same method (Terai et al.,
2013; Nakamura & Yoshida, 2002).

Figure 2: Distribution of apparent velocities of the detected moving objects. The horizontal axis shows the motion along the ecliptic longitude $\dot{\lambda}$, and the vertical axis shows the motion along the ecliptic latitude $\dot{\beta}$. The objects for which photometric measurements with both $r$\- and $g$-bands were successfully performed are plotted. The blue crosses show the objects eliminated from the sample because of large residuals (see in text). The gray dot-dashed lines shows the conditions for MBA detection (Section 2.2), i.e., $\dot{\lambda}>-48\;\mathrm{arcsec\;hr^{-1}}$ and $|\dot{\beta}|<-2.5\dot{\lambda}-55\;\mathrm{arcsec\;hr^{-1}}$. The green dotted lines show $a=1.8$ au (left) and $a=4.0$ au (right), respectively, assuming $R=a$ and $e=0$ in Equations (2). The black dashed lines show $I=40\arcdeg$, $R=a$, and $e=0$ in Equations (2). |
---|---
Figure 3: Semi-major axis vs. inclination of each object in our sample. (a)
C-like asteroids ($g-r<0.50$) and (b) S-like asteroids ($g-r>0.56$). See
Section 3.3 for the classification of these two groups.

##  3 RESULTS

###  3.1 Sample Selection

We derived the absolute magnitude of each asteroid from its apparent magnitude
$m$, heliocentric distance $R$, and geocentric distance $\Delta$ as (Bowell et
al., 1989)

$H=m-5\log(R\Delta)-P(\theta),$ (3)

where $P(\theta)$ is the phase function at a solar phase angle $\theta$
described as

$P(\theta)=-2.5\log[(1-G)\Phi_{1}+G\Phi_{2}].$ (4)

In the above, $G$ is the slope parameter. We use a typical value of MBAs,
$G=0.15$, because there is no way to obtain magnitudes at various solar phase
angles in this survey. $\Phi_{1}$ and $\Phi_{2}$ are functions of $\theta$
given as

$\displaystyle\Phi_{i}=\exp(-A_{i}[\tan(\theta/2)]^{B_{i}})\quad(i=1,2),$
$\displaystyle A_{1}=3.33,\;A_{2}=1.87,\;B_{1}=0.63,\;B_{2}=1.22.$ (5)

We can describe an error of absolute magnitude $\sigma_{H}$ considering error
propagation as follows:

$\sigma_{H}=\sqrt{\sigma_{m}^{2}+\left(-\frac{5}{\ln
10}\frac{\sigma_{R}}{R}\right)^{2}+\left(-\frac{5}{\ln
10}\frac{\sigma_{\Delta}}{\Delta}\right)^{2}},$ (6)

where $\sigma_{m}$ is the photometric error of apparent magnitude. As we
described in Section 2.4, the errors of heliocentric and geocentric distances
are $\sigma_{R}=\sigma_{\Delta}=0.35$ au. Therefore, we can estimate the error
of absolute magnitude to be $\sigma_{H}\simeq 0.4-0.8$ mag.

To eliminate biases induced by detection incompleteness, we set the detection
limit magnitude of our survey based on detection efficiencies. We examined the
detection efficiency through the data reduction procedure with implanted
synthetic moving objects (see Yoshida & Terai, 2017, for details). Then, the
detection efficiency for each CCD image obtained as a function of the apparent
magnitude $m$ of implanted objects is fit by the following function

$\eta(m)=\eta_{0}\sum_{k=1,2}\frac{\epsilon_{k}}{2}\left[1-\tanh\left(\frac{m-m_{50}}{w_{k}}\right)\right]$
(7)

with

$\epsilon_{1}(m)=1-\epsilon_{2}(m)=\frac{1}{2}\left[1-\tanh\left(\frac{m-m_{50}}{0.2}\right)\right],$
(8)

where $\eta_{0}$ is the maximum value of the detection efficiency, $m_{50}$ is
the apparent magnitude where detection efficiency is $\eta_{0}/2$, and
$w_{k}\;(k=1,2)$ is the transition width. We calculate the best-fit parameter
values of detection efficiencies for all the exposures for each CCD and set
24.2 mag as the detection limit of apparent magnitude where the detection
efficiency is 50% or more for most ($\sim$90%) of the exposure/CCD images.
Figure 4 shows the plots of $R$ and $H_{r}$ for each of the 3,459 objects in
our sample (we will explain the classification into C-like/S-like asteroids in
Section 3.3). We defined the outer edge of our MBA sample as $R=3.0$ au, where
$m_{r}=24.2$ mag corresponds to $H_{r}=20.3$ mag. Thus, we select objects with
$R\leq 3.0$ au and $H_{r}\leq 20.3$ mag as an unbiased sample. We also
investigated absolute magnitude distribution for each region divided by a
heliocentric distance 2.6 au, i.e., inner region ($R\leq 2.6$ au) and outer
region ($2.6\leq R\leq 3.0$ au), respectively. The detection limit for the
inner region corresponds to $H_{r}=21.1$ mag. Note that the above choice of
the boundaries of the inner and outer regions does not greatly affect our main
results (Section 3.5).

|
---|---
Figure 4: Heliocentric distance vs. $r$-band absolute magnitude of each object
in our sample. (a) C-like asteroids ($g-r<0.50$) and (b) S-like asteroids
($g-r>0.56$). See Section 3.3 for the classification of these two groups. The
dashed line shows apparent magnitude of $m_{r}=24.2$ mag. The solid lines show
the detection limit of the whole region, i.e., heliocentric distance of
$R=3.0$ au and absolute magnitude of $H_{r}=20.3$ mag. The dot-dashed lines
show the detection limit for the inner region, i.e., heliocentric distance of
$R=2.6$ au and absolute magnitude of $H_{r}=21.1$ mag.

###  3.2 Color Distributions

Figure 5 shows the histogram of the $g-r$ color of our unbiased sample
(Section 3.1). In addition to the case of the whole sample, we also plotted
the cases limited by various apparent magnitudes $m_{r}$. When the sample is
limited to those with $m_{r}<21.5$ mag (blue dotted line) or $m_{r}<20.5$ mag
(green dot-dashed line), The peaks are located at $g-r\sim 0.4$ and 0.6,
respectively, which roughly correspond to those of the C-type and S-type
asteroids, respectively, seen in the 4th release of SDSS Moving Object Catalog
(SDSS MOC4, Ivezić et al., 2010; Hasselmann et al., 2011, see Section 3.3,
Figure 6).

Some previous works reported a bimodal color distribution of MBAs, while
others found no such dichotomy. Ivezić et al. (2001) found bimodal
distributions in the color-color diagrams plotted for objects whose
photometric errors are less than 0.05 mag, while they showed that the color
dichotomy becomes unclear for fainter objects as we found in Figure 5. Yoshida
& Nakamura (2007) reported bimodality of $B-R$ color distribution of sub-km to
km-sized MBAs, although it was unclear for the sample for the middle and outer
main-belt. Gladman et al. (2009) found no color dichotomy even though their
sample has a small error of absolute magnitudes (the error values are 0.11 au
and 0.36 mag in heliocentric distance and absolute magnitude, respectively)
due to accurate orbits obtained by multiple-night observations. Peña et al.
(2020) measured $g^{\prime}-r^{\prime}$ colors of about 1,000 MBAs with orbits
determined and found no significant color dichotomy, which might be caused by
uncertainty in absolute magnitudes due to asteroid rotation.

Figure 5: Histogram of the $g-r$ colors of our unbiased sample. Different
lines represent the cases limited by different apparent magnitudes: black for
all the selected sample, red for $m_{r}<22.5$ mag, blue for $m_{r}<21.5$ mag,
and green for $m_{r}<20.5$ mag.

###  3.3 Color Classification

Using the $g-r$ color of each asteroid, we classified our sample into two
groups, i.e., bluish C-like asteroids and reddish S-like asteroids. In order
to define the classification criterion, we use the SDSS MOC4. We corrected the
color of each asteroid in the SDSS MOC4 to that in HSC’s filter system, and
classified these objects into spectral types according to the taxonomy given
by Carvano et al. (2010) and Hasselmann et al. (2011)
($\mathrm{C_{p},\;S_{p}}$, etc.). Figure 6 shows the $g-r$ color histogram of
$\mathrm{C_{p}}$-/$\mathrm{S_{p}}$-type asteroids in the SDSS MOC4. The curves
show the fraction of asteroids belonging to each of the two types for each
color bin; for example, the blue curve shows (the number of
$\mathrm{C_{p}}$-type asteroids)/(the total number of $\mathrm{C_{p}}$-type
and $\mathrm{S_{p}}$-type asteroids) for each bin. We defined the
classification criteria by the $g-r$ color values where the fraction is 0.9;
those with $g-r<0.50$ are defined as C-like asteroids and those with
$g-r>0.56$ as S-like asteroids. In this way, we can get less-contaminated
samples than the previous works, Ivezić et al. (2001) and Yoshida & Nakamura
(2007), which used a single color boundary between bluish and reddish groups.

Table 2: Confusion matrix of the color classification using the SDSS MOC4 asteroids. | Classified
---|---
Actual |  | C-like | S-like | Recall
$\mathrm{C_{p}}$-type | 25472 | 236 | 99.1%
$\mathrm{S_{p}}$-type | 148 | 21602 | 99.3%
Precision | 99.4% | 98.9% |

It should be noted that each of the above two groups likely contains asteroids
with other spectral types because we use only the $g-r$ color. C-like
asteroids likely contain D and X-types as well as their complexes. Also,
S-like asteroids likely contain L, Q, V, D, X, A-types and their complexes.

Figure 6: Histogram of the $g-r$ colors for the SDSS MOC4 (Ivezić et al.,
2010; Hasselmann et al., 2011) $\mathrm{C_{p}}$-type (blue dashed-lines) and
$\mathrm{S_{p}}$-type (red solid-lines) asteroids corrected for the difference
of the filter systems between HSC and SDSS. The classification for each
spectral type is based on the Carvano et al. (2010) taxonomy. The curves show
the fraction of asteroids belonging to each type in each color bin. The
horizontal dotted line shows the fraction of 0.9, and the vertical dotted
lines show the color boundaries used for our classification into C-/S-like
asteroids, corresponding to the $g-r$ colors at the fraction of 0.9 for each
type.

###  3.4 Comparison of Size Distributions of C-like and S-like Asteroids

Based on the photometric measurements and color classification described
above, we obtained absolute magnitude distributions for our MBA sample.
Figures 7(a), (b), and (c) show the absolute magnitude distributions for all,
C-like, and S-like asteroids, respectively. The cumulative numbers shown on
the vertical axis are corrected by the detection efficiency for each asteroid
(Yoshida & Terai, 2017):

$N(<H)=\sum_{j:H_{j}<H}\frac{1}{\eta(m_{j})},$ (9)

where $\eta(m_{j})$ is the detection efficiency of the $j$-th object with
apparent magnitude $m_{j}$ in the $r$-band. As described in Section 2.2, the
sample objects were detected from four or five $r$-band images. In this case,
the detection efficiency of an asteroid in all the four images taken for a
certain field is given by (Yoshida & Terai, 2017):

$\eta(m)=\prod_{i=1}^{4}\eta_{i}(m).$ (10)

where $\eta_{i}(m)$ is the detection efficiency of this object in the $i$-th
image. We estimated $\eta(m)$ as a product of the largest four values of
$\eta_{i}$ for each object.

|
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|
Figure 7: Distributions of the $r$-band absolute magnitudes of the objects in
our unbiased sample for the whole region. (a) all asteroids, (b) C-like
asteroids ($g-r<0.50$), and (c) S-like asteroids ($g-r>0.56$), respectively.
The red dots show cumulative number of the asteroids for each bin. The bin
width is 0.5 mag. The error bars represent Poisson’s statistical error. The
dashed lines show the best-fit models (see Table 3). Diameters shown on the
upper horizontal axis were obtained by converting from absolute magnitudes
under the assumption of a constant albedo of (a) $p=0.12$, (b) $p=0.07$, and
(c) $p=0.21$, respectively (see Section 3.4).

In order to compare the size distributions of C-like and S-like asteroids, we
convert the absolute magnitude of an asteroid to its diameter using the
following equation:

$\log D=0.2r_{\odot}-\log\frac{\sqrt{p}}{2R_{\oplus}}-0.2H_{r},$ (11)

where $r_{\odot}=-26.91$ is the $r$-band magnitude of the Sun (Fukugita et
al., 2011), $p$ is a geometric albedo, and $R_{\oplus}$ is the heliocentric
distance of the Earth (i.e., 1 au). We assumed the constant albedo of $p=0.07$
and $p=0.21$ for C- and S-like asteroids, respectively, based on the mean
values of the C- and S-type MBAs measured by the infrared astronomical
satellite AKARI (Usui et al., 2013).

Figure 8 shows a direct comparison of the cumulative size distributions
between the C- and S-like asteroids. The cumulative number in the vertical
axis is normalized by the value at $D=0.4$ km, which corresponds to the
detection limit of the C-like asteroids. The error bars show Poisson’s
statistical error. We found that the shapes of the cumulative size
distributions of C- and S-like samples agree with each other within
statistical errors for $0.4<D<1.5$ km.

We evaluated the goodness of the agreement by the Kolmogorov-Smirnov test
(hereafter KS test, Press et al., 1992). The statistic $D_{\mathrm{KS}}$ of
the KS test is defined by the maximum deviation between two cumulative
distribution functions $S_{1}(x)$ and $S_{2}(x)$ as

$D_{\mathrm{KS}}=\mathrm{max}|S_{1}(x)-S_{2}(x)|.$ (12)

The significance probability is approximately calculated by the following
function:

$P(D_{\mathrm{KS}}>\lambda)=Q_{\mathrm{KS}}(\sqrt{n_{1}n_{2}/(n_{1}+n_{2})}D_{\mathrm{KS}})$
(13)

with

$Q_{\mathrm{KS}}(\lambda)=2\sum^{\infty}_{j=1}(-1)^{j-1}e^{-2j^{2}\lambda^{2}},$
(14)

where $n_{1}$ and $n_{2}$ are numbers of data points of $S_{1}(x)$ and
$S_{2}(x)$, respectively.

We performed the KS test under the null hypothesis that the two samples are
drawn from the same distribution in the size range of $0.4<D<4.7$ km. We found
that the significance probability is 0.218, i.e., the null hypothesis cannot
be rejected at a 20% significance level. We confirmed that even when we set
the classification criterion of C- and S-like samples more strictly (for
example, $g-r<0.40$ for the C-like asteroids, and $g-r>0.60$ for the S-like
asteroids), the significance probability is no less than 0.2.

|
---|---
|
Figure 8: Cumulative size distributions of bluish C-like asteroids
($g-r<0.50$; shown by blue pluses) and reddish S-like asteroids ($g-r>0.56$;
shown by red crosses). (a), (b), and (c) are for the whole region, the inner
region ($1.8<a<2.6$ au), and the outer region ($2.6<a<3.0$ au), respectively.
The values of the vertical axis are normalized by the values at $D=0.4$ km.
Diameters are obtained by converting from absolute magnitudes under the
assumption of a constant albedo of $p=0.07$ for C-like asteroids, and $p=0.21$
for S-like asteroids (Usui et al., 2013), respectively.

To measure the slopes of obtained absolute magnitude distributions, we
approximated the differential absolute magnitude distributions
$\Sigma(H)=dN(H)/dH$ with the following broken power-law:

$\Sigma(H)=\begin{dcases}10^{\alpha_{1}(H-H_{0})}&(H<H_{\mathrm{break}}),\\\
10^{\alpha_{2}H+(\alpha_{1}-\alpha_{2})H_{\mathrm{break}}-\alpha_{1}H_{0}}&(H\geq
H_{\mathrm{break}}),\end{dcases}$ (15)

where $\alpha_{1}$ and $\alpha_{2}$ are the power-law indices of the absolute
magnitude distributions for the brighter and fainter ends, respectively.
$H_{\mathrm{break}}$ is the absolute magnitude of the breakpoint, and $H_{0}$
is defined so that $\Sigma(H_{0})=1$. We used the maximum-likelihood method
(Bernstein et al., 2004; Yoshida & Terai, 2017) to obtain the best-fit
parameters of Equations (15). For the numerical analysis of the maximum-
likelihood estimation, we used the Markov Chain Monte Carlo method (MCMC) with
a Python package `emcee`222https://emcee.readthedocs.io/en/stable/ (Foreman-
Mackey et al., 2013). Figure 9 shows the post-event distributions of sampling
for each parameter by MCMC for the whole region. We adopted as the best-fit
value for each parameter.

The best-fit parameters are shown in Table 3. In all the cases of the all,
C-like, and S-like asteroids, the slope $\alpha$ changes at $H\sim 15-17$ mag
and the slopes at the fainter end are shallower than those at the brighter
end. The slopes in the fainter end are $\alpha_{2}=0.21^{+0.01}_{-0.01}$ and
$\alpha_{2}=0.28^{+0.01}_{-0.01}$ for C-/S-like asteroids, respectively. On
the other hand, the slopes in the brighter end are
$\alpha_{1}=0.57^{+0.01}_{-0.07}$ and $\alpha_{1}=0.86^{+0.11}_{-0.15}$ for
C-/S-like asteroids, respectively. Note that the slopes in the brighter end
have large uncertainties due to the small sample number, which was partly
caused by CCD saturation for bright objects.

|
---|---
|
Figure 9: Post-event distributions of sampling by MCMC for the whole region. The red dashed lines in the 1D histograms show the best-fit values. Each gray point in the 2D histograms (black contours) shows the result of sampling for each step. The green crosses in the 2D histograms show the error range of the best-fit. Table 3: Numbers of asteroids in our sample and best-fit parameters for the absolute magnitude distributions. Types | number | $\alpha_{1}$ | $\alpha_{2}$ | $H_{0}$ | $H_{\mathrm{break}}$
---|---|---|---|---|---
All | 1814 | $0.55^{+0.04}_{-0.01}$ | $0.23^{+0.01}_{-0.01}$ | $12.58^{+0.17}_{-0.12}$ | $16.69^{+0.04}_{-0.13}$
C-like | 1026 | $0.57^{+0.01}_{-0.07}$ | $0.21^{+0.01}_{-0.01}$ | $13.18^{+0.02}_{-0.33}$ | $16.80^{+0.46}_{-0.01}$
S-like | 479 | $0.86^{+0.11}_{-0.15}$ | $0.28^{+0.01}_{-0.01}$ | $13.80^{+0.20}_{-0.09}$ | $15.14^{+0.29}_{-0.06}$

In Figure 7 (c) for S-like asteroids, we notice a significant deviation of the
model from the data at $H\sim 17$ mag, while they agree well with each other
for the cases of all and C-like asteroids. We confirmed that the absolute
magnitude distribution for S-like asteroids cannot be approximated well with
Equations (15) due to the partial excess at $H\sim 17$ mag, even after fine-
tuning of the parameters.

In Figure 8 (a), the fainter-end slopes of C-like and S-like asteroids agree
with each other, but the best-fit values of $\alpha_{2}$ are slightly
different between them. One reason for this is the discrepancy between the
model and the data described above. Another reason is underestimating the
number of brighter S-like asteroids mainly because of saturation, which makes
the best-fit distribution of S-like asteroids steeper than the actual
distribution. To accurately evaluate the fainter-end slope, we fit a broken
power-law to the S-like asteroid sample again assuming the same values of
$\alpha_{1}$ (0.54) and the diameter at the break corresponding to
$H_{\mathrm{break}}=15.79$ as the best-fit parameters for the C-like asteroid
sample. We found that the best-fit value is $\alpha_{2}=0.234\pm 0.002$, which
is quite similar to that of the C-like asteroids.

###  3.5 Heliocentric Distance Dependence

We also compared the size distributions for the C- and S-like asteroids in
each of the inner and outer regions, separately (Figures 8 (b) and (c)). For
the inner region ($1.8<a<2.6$ au), we found that the shapes of the size
distributions of C- and S-like asteroids agree with each other within the
statistical errors at $0.3<D<1.0$ km. The discrepancy in $D>1.0$ km may be
caused by undetected bright S-like asteroids due to saturation. For the outer
region ($2.6<a<3.0$ au), we found that the shapes of the size distributions of
C- and S-like asteroids agree with each other within the errors at $0.4<D<4.5$
km. We performed the KS test under the null hypothesis that the two samples
are drawn from the same distribution in the size ranges of $0.3<D<2.6$ km and
$0.4<D<4.7$ km for the inner and outer region, respectively. We found that the
significance probabilities are 0.94 for the inner region and 0.36 for the
outer region, respectively, i.e., the null hypothesis cannot be rejected at a
20% significance level for both regions. We also found that their shapes agree
with each other within the errors, at least in the size range of $0.4<D<1.0$
km, even when we change the heliocentric boundary between the two regions by
about 0.1 au.

##  4 DISCUSSION

###  4.1 Comparison with Previous Works

Yoshida & Nakamura (2007) measured size distributions of C-like ($B-R<1.1$)
and S-like ($B-R>1.1$) asteroids obtained by a survey observation with the
Suprime-Cam on the Subaru Telescope. They found that the slopes for C- and
S-like asteroids agree with each other within errors in the fainter end but
disagree in the brighter end. The cumulative size distribution of their C-like
sample was approximated by a single power-law with the slope of $b=1.33\pm
0.03$ for $0.6<D<6.5$ km ($\alpha=0.27\pm 0.01$ for $14.6<H<20.2$ mag), and
that of their S-like sample was approximated by a broken power-law with the
slopes of $b_{1}=2.44\pm 0.09$ for $1.0<D<2.3$ km ($\alpha_{1}=0.49\pm 0.02$
for $15.4<H<17.0$ mag) and $b_{2}=1.29\pm 0.02$ for $0.3<D<1.0$ km
($\alpha_{2}=0.26$ for $17.4<H<20.2$ mag), respectively, where $b_{1}$ and
$b_{2}$ are brighter/fainter end slopes of a cumulative size distribution
represented by a broken power-law (they have the relationship
$b_{1}=5\alpha_{1}$ and $b_{2}=5\alpha_{2}$, respectively, with the
corresponding absolute magnitude distribution slopes $\alpha_{1}$ and
$\alpha_{2}$). Ivezić et al. (2001) measured the size distributions of blue
(C-like) and red (S-like) asteroids, which are obtained by SDSS and classified
based on the principal component analysis using $g-r$ and $r-i$ colors. They
found that the slopes of cumulative size distributions for the brighter ends
are $b_{1}=4.00\pm 0.05$ ($\alpha_{1}=0.80\pm 0.01$) for both the blue and the
red (the size range was $5\lesssim D\lesssim 20$ km for the blue, and
$5\lesssim D\lesssim 40$ km for the red), and that for the fainter ends are
$b_{2}=1.40\pm 0.05$ ($\alpha_{2}=0.28\pm 0.01$) for the blue for $3\lesssim
D\lesssim 5$ km, and $b_{2}=1.20\pm 0.05$ ($\alpha_{2}=0.24\pm 0.01$) for the
red for $0.7\lesssim D\lesssim 5$ km (see Yoshida et al., 2019, for a
discussion of the limiting diameters of their size distributions). Peña et al.
(2020) measured $g^{\prime}$\- and $r^{\prime}$-band absolute magnitude
distributions for $14<H_{g^{\prime}}<18$ mag ($1<D<10$ km) with 1,182 MBAs
detected from HiTS 2015 data. They did not find any color dependence in the
derived absolute magnitude distributions.

These studies showed that both of the C-like and S-like asteroids have similar
size distributions with a power-law index of $b\simeq 1.2-1.4$ ($\alpha\simeq
0.24-0.28$) for diameters smaller than several kilometers, but their results
have a slight discrepancy. We confirmed that the size distribution of C- and
S-like asteroids have similar shapes at sub-km, and both of them have a break
point at $D\sim 2$ km. From the results of Ivezić et al. (2001), Peña et al.
(2020), and the present work, the similarity of the shapes of the size
distributions between C-like and S-like asteroids seems to hold up to $\sim$10
km in diameter.

###  4.2 Implications

As we discussed above, the shapes of the size distributions of C- and S-like
asteroids are similar in the size range from several hundred meters up to
about 10 km. This seems to indicate a similarity in the size-dependence of the
impact strength for the C-/S-like asteroids if asteroid population in this
size range is under collision equilibrium. It has been analytically shown that
one of the important factors for determining the shape of a size distribution
in collision equilibrium is the size dependence of the impact strength (the
specific energy needed to disperse half of the target mass $Q_{D}^{*}$) and is
insensitive to the size distributions of fragments produced in individual
impacts (Dohnanyi, 1969; Tanaka et al., 1996; O’Brien & Greenberg, 2003;
Kobayashi & Tanaka, 2010). Therefore, if asteroid population in this size
range is under collision equilibrium, the similarity of their size
distributions seems to indicate the common size-dependence of $Q_{D}^{*}$,
which depends on material, internal structure, and impact velocity (Benz &
Asphaug, 1999; Jutzi et al., 2010). C-type and S-type asteroids are thought to
have different bulk compositions (e.g., Nakamura et al., 2011; Usui et al.,
2019; Potiszil et al., 2020; Kitazato et al., 2021). Considering the above,
our results seem to indicate that bulk compositional difference hardly affects
the size dependence of $Q_{D}^{*}$, at least in size range from several
hundred meters to ten kilometers.

On the other hand, the distribution of asteroids rotation periods shows that
most asteroids at $0.3\lesssim D\lesssim 10$ km have a rotation period longer
than 2.2 hours, the so-called “spin-barrier” (Pravec et al., 2002; Chang et
al., 2015; Carbognani, 2017). This is the centrifugal disruption limit of
asteroids accumulated by only self-gravity. This is often interpreted as
indicating that most asteroids in this size range are rubble-piles. The
similarity of the size distributions between the C-like and S-like asteroids
found in the present work seems to support the view that these asteroids in
this size range have rubble-pile structures.

It should be noted that rotational disruption induced by YORP effect is not
negligible at $D\lesssim 10$ km (Jacobson et al., 2014). Also, removal by
Yarkovsky effect somewhat affects the size distribution, while collisional
disruption primary determine the size distribution of MBAs (O’Brien &
Greenberg, 2005). Similarity of the shapes of the size distributions of C- and
S-like asteroids may indicate that these secondary effect hardly provide
difference in size distributions for different spectral types.

###  4.3 Rough Estimate of the Total Number of MBAs

We can roughly estimate the total number of MBAs by combining the size
distributions of our work and previous works. In order to combine our results
with those obtained in previous works, we converted the $r$-band magnitudes of
our sample into the $V$-band magnitudes, assuming the average $m_{V}-m_{r}$
color of 0.25 mag based on the SDSS MOC4 data. Figure 10 shows the combined
cumulative size distribution of all MBAs. The red and the black lines show our
sample and those derived from the ASTORB database (Bowell et al., 1994),
respectively. For reference, we plotted the slopes obtained by Ivezić et al.
(2001), Gladman et al. (2009), and Peña et al. (2020) with the light-blue
dashed line, the blue dot-dashed line, and green dotted line, respectively
(thin lines represent extrapolations of derived distributions). We combined
the size distribution of our sample with that of ASTORB at $H_{V}=15.4$ mag,
and the distributions obtained by Ivezić et al. (2001), Gladman et al. (2009),
and Peña et al. (2020) are scaled so that they match that of ASTORB at
$H_{V}=13$, 15.5, and $15.5$ mag, respectively, corresponding to the detection
limit of each survey.

We found that the total number of MBAs brighter than $H_{V}=20$ mag is
estimated to be $8.6\times 10^{6}$, i.e., a couple of times as large as that
obtained by extrapolating the results of Ivezić et al. (2001) or Gladman et
al. (2009).

The fact that there are more small MBAs than previous estimate likely affect
the estimation of generation rates of near earth asteroids (NEAs) and
meteorites. Also, our newly-obtained accurate size distribution can be used as
a constrain on theoretical studies of collisional and dynamical evolution of
asteroids (e.g., Bottke et al., 2005a, b; O’Brien & Greenberg, 2005; de Elia &
Brinini, 2007). In particular, the size distribution of small MBAs is
important for comparative studies of asteroids based on analysis of crater
size distribution and estimate of the crater retention ages for those
asteroids visited by spacecraft (e.g., Bottke et al., 2020).

Figure 10: Cumulative size distributions of MBAs obtained from previous
studies and the present work. The red line shows the distribution derived from
our sample (vertical lines show error bars). The black line is derived from
known asteroids based on the ASTORB database (Bowell et al., 1994). The light-
blue dashed line, the blue dot-dashed line, and the green dotted line are
drawn with the model slopes obtained by Ivezić et al. (2001), Gladman et al.
(2009), and Peña et al. (2020), respectively, for reference (thin lines are
their extrapolations). The ASTORB data are connected with our data at
$H_{V}=15.4$ mag (the gray area shows the error range produced by the
uncertainty of the cumulative number at the connection point), with Ivezić et
al. (2001) $H_{V}=13$ mag, and with both Gladman et al. (2009) and Peña et al.
(2020) at $H_{V}=15.5$ mag, respectively. Diameters shown on the upper
horizontal axis are obtained by converting from absolute magnitudes under the
assumption of an albedo $p=0.12$ (Usui et al., 2013).

Figure 11 shows the combined cumulative size distributions of C-type and
S-type asteroids separately. We combined the size distributions of our
C-/S-like sample with the known C-/S-type asteroids (taken from the ASTORB
database, black solid line) by interpolating with the slope of all known
asteroids (dotted lines) and blue/red asteroids of Ivezić et al. (2001)
(dashed lines). Here, we converted the $r$-band magnitudes of our sample into
the $V$-band magnitudes using the average $m_{V}-m_{r}$ colors of 0.20 mag for
the C-type and 0.28 mag for the S-type asteroids, respectively. We can roughly
estimate that the total number of C- and S-type MBAs larger than 400 m in
diameter are about $3.5\times 10^{6}$ and $2.4\times 10^{6}$, respectively.
Namely, the number of C-type asteroids is about 1.5 times as large as that of
S-type asteroids for $D>400$ m.

Yoshida & Nakamura (2007) found that the number ratio of C-like and S-like
asteroids is 2.3 for $0.5<D<10$ km for the whole main-belt. Also, Ivezić et
al. (2001) reported that the number ratio of blue and red asteroids is 2.3 for
$D>1$ km. Therefore, our result also shows a larger number of C-type asteroids
than S-type, and is consistent with these previous works, although our result
shows a smaller ratio. Also, our C-/S-like sample as well as the blue/red
sample of Ivezić et al. (2001) likely contain asteroids with types other than
the C-type and S-type. More spectroscopic or multi-band photometric
observation of asteroids for a wide range of sizes is desirable to more
accurately estimate the total number of asteroids with each spectral type.

|
---|---
Figure 11: Cumulative size distributions of C-type and S-type MBAs obtained
from previous studies and the present work. The size distributions of known
C-/S-type asteroids (black solid line) are connected with those of smaller
sizes using the slopes of all MBAs in the ASTORB database (dotted line) at
$H_{V}=9$ mag for C-type and $H_{V}=7$ mag for S-type, respectively. The size
distribution of all ASTORB MBAs is connected with the blue/red asteroids of
Ivezić et al. (2001) (dashed line) at $H_{V}=13$ mag. The size distributions
of our C-/S-like asteroids are connected with the blue/red asteroids of Ivezić
et al. (2001) at $H_{V}=15.5$ mag, respectively (the gray area shows the error
range produced by the uncertainty of the cumulative number at the connection
point). Error bars represent Poisson’s errors. Diameters on the upper
horizontal axis are obtained by converting from absolute magnitudes under the
assumptions of an albedo (a) $p=0.07$ and (b) $p=0.21$, which are the mean
values for the C-/S-type MBAs, respectively (Usui et al., 2013).

##  5 CONCLUSIONS

We detected a large number of small MBAs in the $g$\- and $r$-band images
taken by the Hyper Suprime-Cam installed on the Subaru Telescope, and obtained
an unbiased sample with a detection limit of $m_{r}=24.2$ mag. We classified
the sample by the $g-r$ color into two groups, bluish C-like and reddish
S-like asteroids.

We found that the shapes of the size distributions of C-like and S-like
asteroids agree well with each other for $0.4<D<5$ km. The similarity of their
size distributions represents a similarity in the size dependence of impact
strength of C-like and S-like asteroids for the size range from several
hundred meters to several kilometers. Assuming the asteroid population is
under collision equilibrium, our results indicate that compositional
difference hardly affects the size dependence of impact strength, at least for
the above size range. This size range is consistent with the size range where
the observed distribution of asteroid rotation rates has an upper limit,
called the spin-barrier. The existence of the spin-barrier can be interpreted
as showing that most asteroids in this size range have rubble-pile structures,
and our results are consistent with such a view.

By combining our results with previously known size distributions, we found
that the total number of MBAs brighter than $H_{V}=20$ mag is a few times as
large as that estimated by extrapolation of previous results. This would
affect the estimation of the generation rates of NEAs and meteorites.

We are grateful to Takafumi Ootsubo, Eri Tatsumi, Tomokatsu Morota, and
Hiroshi Kobayashi for constructive discussions. We also thank the anonymous
referee for providing helpful comments and suggestions. This work was
supported by JSPS KAKENHI grant Nos. JP15H03716, JP16H04041, JP18K13607,
JP20H04617, and JP21H00043.

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# No-Three-in-a-$\Theta:$ Variations on the No-Three-in-a-Line Problem

Natalie Dodson  Middlebury College  Anant Godbole  East Tennessee State
University  Dashleen Gonzalez  University of Puerto Rico Mayagüez  Ryan Lynch
University of Minnesota, Twin Cities  Lani Southern  Willamette University

###### Abstract

We pose a natural generalization to the well-studied and difficult no-three-
in-a-line problem: How many points can be chosen on an $n\times n$ grid such
that no three of them form an angle of $\theta$? In this paper, we classify
which angles yield nontrivial problems, noting that some angles appear in
surprising configurations on the grid. We prove a lower bound of $2n$ points
for angles $\theta$ such that $135\degree\leq\theta<180\degree$, and further
explore the case $\theta=135\degree$, utilizing geometric properties of the
grid to prove an upper bound of $3n-2$ points. Lastly, we generalize the proof
strategy used in proving the upper bound for $\theta=135\degree$ to provide a
general upper bound for all angles.

##  1 Introduction

The no-three-in-a-line problem is an open problem first proposed by Henry
Dudeney in the early 20th century. It asks, what is the maximum number of
points that can be placed on an $n\times n$ grid such that no three of them
lie in a line? [4] This problem is equivalent to the $\theta=180\degree$ case.
Directly applying the pigeonhole principle on the rows of the grid gives an
upper bound of $2n$ for the no-three-in-a-line-problem. However, it is unknown
whether this upper bound is tight. This uncertainty is not due to lack of
effort from many mathematicians. Hall, Jackson, Sudbery, and Wild proved a
lower bound of $(\frac{3}{2}-\epsilon)n$ for all $n$ [4]. Guy and Kelly also
provided a probabilistic conjecture for an upper bound of about $1.874n$,
which was later corrected to $1.814n$ by Gabor Ellmann [3].

Gossell and Johnson solved a related problem, determining that the maximum
number of points that lie on an $n\times n$ grid such that no three of them
lie in an “L,” or an angle of $90\degree$ is $2n-2$ [2]. An alternate proof
for the “L” case was also presented by Godbole and LaFollette [1].

We ask and begin to answer a generalization of these problems: How many points
can be placed on an $n\times n$ grid such that no three of them form an angle
of $\theta$ ? In Section 2, we classify which choices of $\theta$ yield
interesting problems. In Section 3, we prove a lower bound for interesting
angles between $135\degree$ and $180\degree$. In Section 4, we find an upper
bound for the $135\degree$ case, relying on geometric properties specific to
$\theta=135\degree$. We generalize to all interesting angles in Section 5 and
obtain upper bounds for all angles that can appear on the grid.

##  2 Interesting Angles

In this section, we determine which choices of $\theta$ yield interesting
problems. We begin with the following definitions:

######  Definition 1.

A _grid_ $G_{n}$ is a set of $n^{2}$ points arranged with $n$ points in each
row and column for some natural number $n$. We denote points in $G_{n}$ by
their corresponding integer coordinates in $\mathbb{R}^{2},$ starting with
$(1,1)$ for the bottom left hand corner.

######  Definition 2.

A _construction_ $C_{n}$ is a subset of points of $G_{n}$.

######  Definition 3.

Every point $P\in C_{n}$ is called a _chosen point_.

######  Definition 4.

An angle $\theta$ is _interesting_ if there exists a construction $C_{n}$ for
some $n\in\mathbb{N}$ such that for some $A,B,$ and $C\in C_{n}$, one of the
interior angles of triangle $ABC$ is equal to $\theta.$

######  Theorem 1.

An angle $\theta$ is interesting if and only if $\tan(\theta)$ is rational.

###### Proof.

Figure 1: A construction of $\theta$ for $\theta=\arctan(\frac{-3}{2}).$

Let $\theta$ be an arbitrary angle such that $\tan(\theta)=\frac{p}{q}$ for
some integers $p$ and $q.$ If $\frac{p}{q}<0,$ choose $q$ to be negative. Let
$n=2(\text{max}\\{\lvert P\rvert,\lvert q\rvert\\})+2.$ Take the points
$(\frac{n}{2}+1,1),$ $(\frac{n}{2}+2,1),$ and $(\frac{n}{2}+1+q,1+p)$ in
$C_{n}.$ These points form an angle of $\theta,$ as shown in Figure 1, so
$\theta$ is interesting.

Let $\theta$ be an arbitrary interesting angle. Then for some $C_{n}$, there
exists $A=(A_{x},A_{y}),B=(B_{x},B_{y}),C=(C_{x},C_{y})\in C_{n}$ such that
$\angle ABC=\theta$.

The magnitude of the cross product is given by
$\|\overrightarrow{BA}\times\overrightarrow{BC}\|=\|\overrightarrow{BA}\|\|\overrightarrow{BC}\|\sin{\theta}.$
The cosine of the angle is given by
$\cos{\theta}=\frac{\overrightarrow{BA}\cdot\overrightarrow{BC}}{\|\overrightarrow{BA}\|\|\overrightarrow{BC}\|}.$
Thus

$\displaystyle\tan{\theta}$ $\displaystyle=\frac{\sin{\theta}}{\cos{\theta}}$
$\displaystyle=\left(\frac{\|\overrightarrow{BA}\times\overrightarrow{BC}\|}{\|\overrightarrow{BA}\|\|\overrightarrow{BC}\|}\right)\left(\frac{\|\overrightarrow{BA}\|\|\overrightarrow{BC}\|}{\overrightarrow{BA}\cdot\overrightarrow{BC}}\right)$
$\displaystyle=\frac{\|\overrightarrow{BA}\times\overrightarrow{BC}\|}{\overrightarrow{BA}\cdot\overrightarrow{BC}}$
$\displaystyle=\frac{(A_{x}-B_{x})(C_{y}-B_{y})-(C_{x}-B_{x})(A_{y}-B_{y})}{(A_{x}-B_{x})(C_{x}-B_{x})+(A_{y}-B_{y})(C_{y}-B_{y})},$

which is rational since $A,$ $B,$ and $C$ have integer coordinates. ∎

##  3 Lower Bound for $135\degree\leq\theta<180\degree$

######  Definition 5.

A construction $C_{n}$ is _peaceful_ for $\theta$ if no three points in
$C_{n}$ form an angle of $\theta$.

######  Theorem 2.

For $135\degree\leq\theta<180\degree,$ there exists a peaceful $C_{n}$ with
$\lvert C_{n}\rvert=2n$.

###### Proof.

Figure 2: Lower Bound Construction $135\degree\leq\theta<180\degree$ for
$G_{6}$

Figure 3: Case 1 of Theorem 2

Let $\theta$ be an arbitrary angle such that
$135\degree\leq\theta<180\degree.$ Let $n\in\mathbb{N}$ be arbitrary. Consider
the construction $C_{n}=\\{(x,y)\mid(x,y)\in G_{n},y=1\text{ or }n\\}$, as
shown for $G_{6}$ in Figure 2. Let $A=(A_{x},A_{y}),B=(B_{x},B_{y}),$ and
$C=(C_{x},C_{y})\in C_{n}$. There are three cases for $\angle{ABC}$:

  1. 1.

$A_{y}=B_{y}=C_{y}$, as shown for $A_{y}=n$ in Figure 3. In this case
$\angle{ABC}=180\degree$, so $\angle{ABC}\neq\theta.$

  2. 2.

Without loss of generality, $A_{y}=C_{y}=n$ and $B_{y}=1$. An example is shown
in Figure 4. The largest angle is constructed if $A=(1,n)$, $B=(B_{x},1)$ and
$C=(n,n)$. By way of contradiction, suppose that $\angle ABC>90\degree.$ Then
$ABC$ is an obtuse triangle and the side opposite the angle, $\overline{AC}$
must be the longest side of the the triangle. $\overline{AC}$ has length
$n-1.$ However, by the Pythagorean theorem, $\overline{AB}$ has length
$\sqrt{(n-1)^{2}+(B_{x}-1)^{2}}\geq n-1$ and $\overline{BC}$ has length
$\sqrt{(n-1)^{2}+(n-B_{x})^{2}}\geq n-1$, which is a contradiction. Therefore,
$\angle ABC\not>90\degree$ and $\angle ABC\neq\theta.$

  3. 3.

$A_{y}=1=B_{y}$ and $C_{y}=n.$ An example is shown in Figure 5. Then the
largest angle that can occur comes from taking $A=(1,1),$ $B=(B_{x},1)$ and
$C=(n,n)$ for some natural number $B_{x}$ such that $1<B_{x}\leq n.$ If
$B_{x}=n,$ $\angle ABC=90\degree\neq\theta.$ If $B_{x}<n,$ consider the right
triangle formed by the points $(B_{x},1),$ $(n,1),$ and $(n,n).$ Let
$\theta_{1}$ be the angle inside the triangle with $(B_{x},1)$ at its vertex,
as shown in Figure 5. Note that $\angle ABC=180\degree-\theta_{1}.$ Since,
$1<B_{x}<n,$ $\tan(\theta_{1})=\frac{n-1}{n-B_{x}}>1$ so
$\theta_{1}>45\degree.$ Thus $\angle ABC<180\degree-45\degree=135\degree$ and
$\angle ABC\neq\theta.$

Figure 4: Case 2 of Theorem 2

Figure 5: Case 3 of Theorem 2

∎

##  4 Upper Bound for $\theta=135\degree$

######  Definition 6.

A _$\frac{p}{q}$ -slope-bucket_ is a subset of points of $G_{n}$ that lie on a
line of slope $\frac{p}{q}$ for some fixed $\frac{p}{q}.$ When the choice of
$\frac{p}{q}$ is clear, we may refer to it simply as a slope bucket.

In this section, “slope buckets” refer to $-1$-slope-buckets. The $-1$ slope
buckets of $G_{4}$ are shown in Figure 6.

Figure 6: The $-1$-slope-buckets $G_{4}$

######  Definition 7.

An _interior point_ $A=(A_{x},A_{y})$ is a point in a construction $C_{n}$
such that there exists distinct $B=(B_{x},B_{y})$ and $C=(C_{x},C_{y})\in
C_{n}$ where either

  * •

$A_{x}=B_{x}=C_{x}$ and $B_{y}<A_{y}<C_{y}$, or

  * •

$A_{y}=B_{y}=C_{y}$ and $B_{x}<A_{x}<C_{x}$, or

  * •

$A,B,$ and $C$ are in the same slope bucket and $B_{x}<A_{x}<C_{x}$.

Figure 7: $A,B$ and $C$ are interior points with respect to its $-1$-slope
bucket, its column, and its row, respectively.

######  Lemma 3.

Suppose there exist two chosen points $A,B$ with $A_{y}>B_{y}$ in the same
column, $k$. The two cases enumerated below are shown in Figure 8.

  1. 1.

If another point, $C,$ is chosen in the slope bucket of $A$ with $C_{x}<k,$
$\angle CAB=135\degree.$

  2. 2.

If another point, $D$, is chosen in the slope bucket of $B$ with $D_{x}>k$,
$\angle ABD=135\degree$.

Figure 8: Lemma 3 $\angle CAB$ and $\angle ABD$

###### Proof.

  1. 1.

Since $A$ and $C$ are in the same $-1$-slope-bucket,
$C_{x}-A_{x}=-(C_{y}-A_{y})$. The vectors between points are given by
$\vec{AC}=\langle C_{x}-A_{x},C_{y}-A_{y}\rangle$ and $\vec{AB}=\langle
B_{x}-A_{x},B_{y}-A_{y}\rangle=\langle 0,B_{y}-A_{y}\rangle$. Then
$\angle{CAB}$ is given by

$\displaystyle\angle{CAB}$
$\displaystyle=\arccos{\frac{\vec{AC}\cdot\vec{AB}}{\|\vec{AC}\|\|\vec{AB}\|}}$
$\displaystyle=\arccos{\frac{-1}{\sqrt{2}}}=135\degree.$

  2. 2.

Since $B$ and $D$ are in the same $-1$-slope-bucket,
$D_{x}-B_{x}=-(D_{y}-B_{y})$. The vectors between the points are given by
$\vec{BD}=\langle D_{x}-B_{x},D_{y}-B_{y}\rangle$ and $\vec{BA}=\langle
A_{x}-B_{x},A_{y}-B_{y}\rangle=\langle 0,A_{y}-B_{y}\rangle.$ Then
$\angle{ABD}$ is given by

$\displaystyle\angle{ABD}$
$\displaystyle=\arccos{\frac{\vec{BD}\cdot\vec{BA}}{\|\vec{BD}\|\|\vec{BA}\|}}$
$\displaystyle=\arccos{\frac{-1}{\sqrt{2}}}=135\degree.$

∎

######  Lemma 4.

Suppose there exist two chosen points $A,B$ with $A_{x}>B_{x}$ in the same
row, $k$. The two cases enumerated below are shown in Figure 9.

  1. 1.

If another point, $C,$ is chosen in the slope bucket of $A$ with $C_{y}<k,$
$\angle CAB=135\degree.$

  2. 2.

If another point, $D,$ is chosen in the slope bucket of $B$ with $D_{y}>k,$
$\angle DBA=135\degree.$

Figure 9: Lemma 4 $\angle CAB$ and $\angle DBA$

###### Proof.

These cases are analogous to the cases Lemma 3. ∎

######  Lemma 5.

Suppose there exist two chosen points $A,B$ with $A_{y}>B_{y}$ in the same
slope bucket, $k$. There are four cases:

  1. 1.

If another point, $C,$ is chosen in the column of $A$ with $A_{y}<C_{y},$
$\angle CAB=135\degree.$

  2. 2.

If another point, $D,$ is chosen in the column of $B$ with $B_{y}>D_{y},$
$\angle ABD=135\degree.$

  3. 3.

If another point, $E,$ is chosen in the row of $A$ with $A_{x}>E_{x},$ $\angle
EAB=135\degree$.

  4. 4.

If another point, $F,$ is chosen in the row of $B$ with $B_{x}<F_{x},$ $\angle
ABF=135\degree.$

The four cases below are shown in Figure 10.

Figure 10: Lemma 5 $\angle CAB,\angle DBA,\angle EAB,$ and $\angle ABF$

###### Proof.

  1. 1.

Consider a function from Lemma 5 to Lemma 3 defined by

$\displaystyle C$ $\displaystyle\rightarrow A$ $\displaystyle A$
$\displaystyle\rightarrow B$ $\displaystyle B$ $\displaystyle\rightarrow D.$

By Part 2 of Lemma 3, $\angle CAB=135\degree.$

  2. 2.

From Lemma 5 to Lemma 3, map

$\displaystyle A$ $\displaystyle\rightarrow C$ $\displaystyle B$
$\displaystyle\rightarrow A$ $\displaystyle D$ $\displaystyle\rightarrow B.$

By Part 1 of Lemma 3, $\angle ABD=135\degree$.

  3. 3.

From Lemma 5 to Lemma 4, map

$\displaystyle E$ $\displaystyle\rightarrow B$ $\displaystyle A$
$\displaystyle\rightarrow A$ $\displaystyle B$ $\displaystyle\rightarrow C$

From Part $1$ of Lemma 4, $\angle EAB=135\degree$.

  4. 4.

From Lemma 5 to Lemma 4, map

$\displaystyle A$ $\displaystyle\rightarrow D$ $\displaystyle B$
$\displaystyle\rightarrow B$ $\displaystyle F$ $\displaystyle\rightarrow A$

By Part $2$ of Lemma 4, $\angle ABF=135\degree$.

∎

######  Lemma 6.

Let $n\in\mathbb{N}$ be arbitrary. Let $A$ and $B$ be distinct chosen points
in an outside column, $k$, of $G_{n}$ with $A_{y}>B_{y}$.

  1. 1.

$k=1$: If $B$ has a point, $D$, in the same slope bucket,
$\angle{ABD}=135\degree$.

  2. 2.

$k=n$: If $A$ has a point, $C$ in the same slope bucket,
$\angle{CAB}=135\degree$.

Figure 11: Lemma 6 $\angle ABD$ and $\angle CAB$

###### Proof.

Part 1 follows from Lemma 3 Case 2 and Part 2 follows from Lemma 3 Case 1. The
angles are illustrated in Figure 11. ∎

######  Lemma 7.

A point that is an interior point with respect to its slope bucket and has
another chosen point in its row or column is part of a $135\degree$ angle.

Figure 12: Lemma 7 \- Row

Figure 13: Lemma 7 \- Column

###### Proof.

This lemma follows from Lemma 5. The two possible cases are illustrated in
Figure 12 and Figure 13. ∎

######  Lemma 8.

A point that is an interior point with respect to its row or column and has
another chosen point in its slope bucket is part of a $135\degree$ angle.

Figure 14: Lemma 8 \- Column

Figure 15: Lemma 8 \- Row

###### Proof.

This lemma follows from Lemma 3 and Lemma 4. The possible cases are shown in
Figure 14 and Figure 15. ∎

We have now arrived at the two most critical lemmas for proving our upper
bound for angles of $135\degree$.

######  Lemma 9.

Let $n\in\mathbb{N}$. Let $C_{n}$ be arbitrary such that $C_{n}$ contains $2n$
points and is peaceful. If $k$ is the number of slope buckets occupied by the
points in $C_{n}$, at most $2n-1-k$ additional points can be added to $C_{n}$
without creating a $135\degree$ angle.

###### Proof.

Let $C_{n}$ be an arbitrary peaceful construction containing $2n$ points for
$\theta=135\degree$. Let $k$ be the number of slope buckets occupied by
$C_{n}$. By the pigeonhole principle, for every additional point added to
$C_{n}$, there exists some point in $C_{n}$ which is an interior point in its
column. By Lemma 8, an interior point in its column must be the only point in
its slope bucket. Since $G_{n}$ has $2n-1$ slope buckets and $k$ of them are
occupied by $C_{n}$, at most $2n-1-k$ additional points can be added to
$C_{n}$. ∎

######  Lemma 10.

Let $C_{n}$ be an arbitrary peaceful construction containing $2n$ points for
$\theta=135\degree$. Let $k$ be the number of slope buckets occupied by
$C_{n}$ points. Then $k\geq n+1$.

###### Proof.

Let $C_{n}$ be an arbitrary peaceful construction containing $2n$ points for
$\theta=135\degree$. Let $k$ be the number of slope buckets that have at least
one chosen point. Let $\mathcal{B}$ be the set of slope buckets containing at
least $2$ points in $C_{n}$. Let $P\subseteq C_{n}$ be the points in $C_{n}$
belonging to a slope bucket of $\mathcal{B}$. Since $2n-\lvert P\rvert$ is the
number of slope buckets containing one point in $C_{n}$ and
$\lvert\mathcal{B}\rvert$ is the number of slope buckets containing $\geq 2$
points in $C_{n}$, by substitution,

$k=2n-\lvert P\rvert+\lvert\mathcal{B}\rvert.$ (1)

Let $M$ be the set of interior points in slope buckets of $\mathcal{B}.$ Since
each slope bucket contains a finite number of points in $G_{n}$, there are
$2\lvert\mathcal{B}\rvert$ non-interior points in the slope buckets of
$\mathcal{B}$. Therefore,

$\lvert M\rvert=\lvert P\rvert-2\lvert\mathcal{B}\rvert$ (2)

By Lemma 8, each column of the grid can contain at most two points of $P.$
Since there are $n$ columns on the grid,

$\lvert P\rvert\leq 2n.$ (3)

By Lemma 6, the two exterior columns can contain at most $1$ point in a slope
bucket of $\mathcal{B}$. Thus,

$\lvert P\rvert\leq 2n-2.$ (4)

Additionally, by Lemma 7, any point in $M$ is the only point in its column.
Therefore,

$\begin{split}\lvert P\rvert&\leq 2n-\lvert M\rvert-2.\text{ Equivalently,}\\\
-\lvert P\rvert&\geq-(2n-\lvert M\rvert-2).\text{ Adding 2n to both sides,}\\\
2n-\lvert P\rvert&\geq 2n-(2n-\lvert M\rvert-2)\\\ &=\lvert
M\rvert+2.\end{split}$ (5)

By substitution for $\lvert M\rvert$ of Equation 2 into Inequality 5,

$\begin{split}2n-\lvert P\rvert&\geq\lvert
P\rvert-2\lvert\mathcal{B}\rvert+2\\\ 2\lvert\mathcal{B}\rvert&\geq 2\lvert
P\rvert-2n+2\\\ \lvert\mathcal{B}\rvert&\geq\lvert P\rvert-n+1.\end{split}$
(6)

By substitution of Inequality 6 into Equation 1,

$\begin{split}k&\geq 2n-\lvert P\rvert+(\lvert P\rvert-n+1)\\\
&=n+1\end{split}$ (7)

∎

######  Theorem 11.

A peaceful construction $C_{n}$ can have at most $3n-2$ points for
$\theta=135\degree$.

###### Proof.

Let $C_{n}$ be a peaceful construction containing $2n$ points. By Lemma 9, we
can add at most $2n-1-k$ additional points to $C_{n}$ for a total of
$2n+2n-1-k=4n-1-k$ points. By Lemma 10, $k\geq n+1.$ Thus $C_{n}$ can contain
at most $4n-1-(n+1)=3n-2$ points. ∎

##  5 Upper Bound for all $\theta$

We now generalize Theorem 11 to prove an upper bound for all interesting
angles $\theta$ such that $0\degree<\theta<180\degree.$ If $p,q<n$, let
$f(p,q)$ be the number of $\frac{p}{q}$ slope buckets on $G_{n}$.

######  Lemma 12.

We have the equality $f(p,q)=pn+qn-pq.$

###### Proof.

Let $n,p,q\in\mathbb{N}$ such that $p,q<n$. Let $\mathcal{S}$ be the set of
points that are either in the first $p$ rows or the first $q$ columns of
$G_{n}$. We now show that every point in $G_{n}$ is in the same slope bucket
as one point in $\mathcal{S}.$ Let $B=(B_{x},B_{y})\in G_{n}$ be arbitrary. By
the division algorithm,

$B_{x}=qs+B_{x}\text{ mod }q\text{ and }B_{y}=pt+B_{y}\text{ mod }p$

for some $s\text{ and }t\in\mathbb{N}\cup\\{0\\}$. Let

$A=(A_{x},A_{y})=\begin{cases}(B_{x}-qt\text{, }B_{y}\text{ mod }p)&\text{if
}t\leq s\\\ (B_{x}\text{ mod }q\text{, }B_{y}-ps)&\text{if }t>s.\end{cases}$

It suffices to prove that the slope between $A$ and $B$ is $\frac{p}{q}$ and
that $A_{x}\leq q$ or $A_{y}\leq p.$ Let $z_{A,B}$ be the slope between $A$
and $B$. If $t\leq s,$ $z_{A,B}$ is given by

$\displaystyle z_{A,B}$ $\displaystyle=\frac{B_{y}-A_{y}}{B_{x}-A_{x}}$
$\displaystyle=\frac{(pt+B_{y}\text{ mod }p)-(B_{y}\text{ mod
}p)}{(qs+B_{x}\text{ mod }q)-(B_{x}-qt)}.$ Substituting $B_{x}-qs$ for $B_{x}$
mod $q,$ $\displaystyle=\frac{(pt+B_{y}\text{ mod }p)-(B_{y}\text{ mod
}p)}{(qs+B_{x}-qs)-(B_{x}-qt)}=\frac{pt}{qt}=\frac{p}{q}.$

If $t>s,$ $z_{A,B}$ is given by

$\displaystyle z_{A,B}$ $\displaystyle=\frac{B_{y}-A_{y}}{B_{x}-A_{x}}$
$\displaystyle=\frac{(pt+B_{y}\text{ mod }p)-(B_{y}-ps)}{(qs+B_{x}\text{ mod
}q)-(B_{x}\text{ mod }q)}.$ Substituting $B_{y}-pt$ for $B_{y}$ mod $p,$
$\displaystyle=\frac{(pt+B_{y}-pt)-(B_{y}-ps)}{qs+B_{x}\text{ mod
}q-(B_{x}\text{ mod }q)}=\frac{ps}{qs}=\frac{p}{q}.$

In either case, $B$ falls into the $\frac{p}{q}$ slope bucket of $A.$ When
$t\leq s,$ $A_{y}=B_{y}\text{ mod }p$ so $B_{y}<p.$ Thus, $A$ is in the first
$p$ rows of the grid. When $t>s,$ $A_{x}=B_{x}\text{ mod }q$ so $A_{x}<q.$
Thus $A$ is in the first $q$ columns of the grid. Since
$\lvert\mathcal{S}\rvert=pn+qn-pq$ and no points in $\mathcal{S}$ are in the
same slope bucket, $f(p,q)=pn+qn-pq.$ ∎

######  Theorem 13.

A construction $C_{n}$ that is peaceful for $\theta=\arctan(\frac{p}{q})$ has
at most $2n+f(p,q)-2(\max\left\\{\lvert P\rvert,\lvert q\rvert\right\\})$
points.

###### Proof.

Let $G_{n}$ be an arbitrary grid and $\theta$ an arbitrary interesting angle.
By Theorem 1, $\tan(\theta)=\frac{p}{q}$ for some integers $p$ and $q.$
Suppose that $C_{n}$ has $2n$ points and is peaceful for $\theta.$ The number
of grid points in a $\frac{p}{q}-$slope-bucket is at most
$\frac{n}{\text{max}\\{\lvert P\rvert,\lvert q\rvert\\}}.$ Then the $2n$
points must occupy at least $\frac{2n}{\left(\frac{n}{\text{max}\\{\lvert
p\rvert,\lvert q\rvert\\}}\right)}=2(\text{max}\\{\lvert P\rvert,\lvert
q\rvert\\}).$

By the pigeonhole principle, for every additional point chosen on the grid,
there exists some point which is an interior point in its column. A point
which is an interior point in its column will be part of an angle of $\theta$
if there are any other points chosen in its slope bucket. Thus any additional
points added to $C_{n}$ must be the only chosen points in their slope buckets
for $C_{n}$ to remain peaceful for $\theta.$

Since there are $f(p,q)$ slope buckets on the grid and at least
$2(\text{max}\\{\lvert P\rvert,\lvert q\rvert\\})$ have been used to place the
first $2n$ points, at most $f(p,q)-2(\text{max}\\{\lvert P\rvert,\lvert
q\rvert\\})$ more points can be placed on the grid without forming an angle of
$\theta.$ Thus at most $2n+f(p,q)-2(\text{max}\\{\lvert P\rvert,\lvert
q\rvert\\})$ points can be placed on the grid without forming an angle of
$\theta=\tan^{-1}(\frac{p}{q}).$ ∎

##  6 Conclusion

When proving the upper bound for $135\degree$ angles, we considered only
angles that had one side in a row or column, which we called _grid fitted_.
However, other, _sneaky_ $135\degree$ angles can be formed by grid points. An
example is shown in Figure 16. The prevalence of sneaky angles makes finding
configurations peaceful for $135\degree$ a difficult task. For this reason, we
believe the upper bound is not tight and more work remains to be done on this
problem and many related problems. We end with a list of further questions
about the combinatorial properties of objects on a grid.

Figure 16: A grid fitted and a sneaky $135\degree$ angle on $G_{6}.$

  * •

What other $135\degree$ free constructions exist and what do they have in
common?

  * •

How does the number of possible sneaky angles relate to grid size?

  * •

How may considering the specific geometry of angles improve their upper and
lower bounds?

  * •

What other shapes are interesting to consider? How many points can be placed
on a grid such that no four make a square of any size (and rotation)?

##  7 Statements and Declarations

The work of each of the five authors was supported by National Science
Foundation Grant 1852171. The authors have no relevant financial or non-
financial interests to disclose.

## References

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  * [3] Richard K. Guy and Patrick A. Kelly, _The no-three-in-line problem_ , Canadian Mathematical Bulletin 11 (1968), no. 4, 527–531.
  * [4] R.R Hall, T.H Jackson, A Sudbery, and K Wild, _Some advances in the no-three-in-line problem_ , Journal of Combinatorial Theory, Series A 18 (1975), no. 3, 336–341.