# Luminosity Function of Faint Sporadic Meteors measured with a Wide-Field
CMOS mosaic camera Tomo-e PM

Ryou Ohsawa<EMAIL_ADDRESS>Shigeyuki Sako  Yuki Sarugaku
Fumihiko Usui  Takafumi Ootsubo  Yasunori Fujiwara  Mikiya Sato  Toshihiro
Kasuga  Ko Arimatsu  Jun-ichi Watanabe  Mamoru Doi  Naoto Kobayashi  Hidenori
Takahashi  Kentaro Motohara  Tomoki Morokuma  Masahiro Konishi  Tsutomu Aoki
Takao Soyano  Ken’ichi Tarusawa  Yuki Mori  Yoshikazu Nakada  Makoto Ichiki
Noriaki Arima  Yuto Kojima  Masahiro Morita  Toshikazu Shigeyama  Yoshifusa
Ita  Mitsuru Kokubo  Kazuma Mitsuda  Hiroyuki Maehara  Nozomu Tominaga  Takuya
Yamashita  Shiro Ikeda  Mikio Morii  Seitaro Urakawa  Shin-ichiro Okumura
Makoto Yoshikawa  Institute of Astronomy, Graduate School of Science,
University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Kiso
Observatory, Institute of Astronomy, Graduate School of Science, The
University of Tokyo, 10762-30 Mitake, Kiso-machi, Kiso-gun, Nagano 397-0101,
Japan  Koyama Astronomical Observatory, Kyoto Sangyo University, Kamigamo
Motoyama, Kita, Kyoto, Kyoto Prefecture 603-8555, Japan  Center for Planetary
Science, Graduate School of Science, Kobe University, 7-1-48, Minatojima-
Minamimachi, Chuo-ku, Kobe 650-0047, Japan  Department of Space Astronomy and
Astrophysics, Institute of Space and Astronautical Science (ISAS), Japan
Aerospace Exploration Agency (JAXA), 3-1-1 Yoshinodai, Chuo-ku, Sagamihara,
Kanagawa 252-5210, Japan  The Nippon Meteor Society, Japan  SOKENDAI (The
Graduate University for Advanced Studies), 10-3 Midoricho, Tachikawa, 190-8518
Tokyo, Japan  National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  Research Center for the Early Universe, School
of Science Bldg. No.4, University of Tokyo, Hongo, 7-3-1, Bunkyo-ku, Tokyo,
113-0033, Japan  Astronomical Institute, Tohoku University, 2 Chome-1-1
Katahira, Aoba Ward, Sendai, Miyagi Prefecture 980-8577, Japan  Department of
Physics, Faculty of Science and Engineering, Konan University, 8 Chome-9-1
Okamoto, Higashinada Ward, Kobe, Hyogo Prefecture 658-0072, Japan
Astronomical Observatory, Graduate School of Science, Kyoto University,
Yoshida-honmachi, Sakyo-ku, 606-8501 Kyoto, Japan  The Institute of
Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
Japan Spaceguard Association, Bisei Spaceguard Center, 1716-3 Ookura, Bisei-
cho, Ibara-shi, Okayama, 714-1411, Japan  Precursory Research for Embryonic
Science and Technology (PRESTO), Japan Science and Technology Agency (JST),
2-21-1 Osawa, Mitaka, Tokyo, 181-0015, Japan

###### Abstract

Imaging observations of faint meteors were carried out on April 11 and 14,
2016 with a wide-field CMOS mosaic camera, Tomo-e PM, mounted on the 105-cm
Schmidt telescope at Kiso Observatory, the University of Tokyo. Tomo-e PM,
which is a prototype model of Tomo-e Gozen, can monitor a sky of
${\sim}1.98\,\mathrm{deg^{2}}$ at 2 Hz. The numbers of detected meteors are
1514 and 706 on April 11 and 14, respectively. The detected meteors are
attributed to sporadic meteors. Their absolute magnitudes range from $+4$ to
$+10\,\mathrm{mag}$ in the $V$-band, corresponding to about
$8.3{\times}10^{-2}$ to $3.3{\times}10^{-4}\,\mathrm{g}$ in mass. The present
magnitude distributions we obtained are well explained by a single power-law
luminosity function with a slope parameter $r=3.1{\pm}0.4$ and a meteor rate
$\log_{10}N_{0}=-5.5{\pm}0.5$. The results demonstrate a high performance of
telescopic observations with a wide-field video camera to constrain the
luminosity function of faint meteors. The performance of Tomo-e Gozen is about
two times higher than that of Tomo-e PM. A survey with Tomo-e Gozen will
provide a more robust measurement of the luminosity function.

###### keywords:

meteors , meteoroids , interplanetary medium

##  1 Introduction

The solar system is composed of a variety of objects with different sizes: the
Sun, the planets and their moons, other small bodies including dwarf planets,
asteroids and comets, and dust grains in interplanetary space. The size of the
small bodies in the solar system covers a wide dynamic range from 1,000 km
(e.g., Ceres) down to about 10 nm ($\beta$-asteroids). The size distribution
of the small bodies provides important information to understand the origin
and the evolution of the solar system. The amount of the interplanetary
materials incoming the Earth is about $10^{5}\,\mathrm{kg\,day^{-1}}$. The
particles in the mass range of $10^{-9}$–$10^{-2}\,$g are the major
contributors of the incoming mass flux [13]. Properties of such small
particles are essential to understand what kind of materials fall into the
Earth. Particles smaller than about $10^{-6}\,$g have been detected in place
with dust counters on satellites and spacecrafts [11, 12, 14, 39], while the
number density of the large particles is so small that their size distribution
is difficult to measure by in-situ observations. The small particles incoming
the Earth are observed as meteors, interacting with the Earth’s atmosphere and
convert part of their kinetic energy into emission lines [1, 7, 16, 28, 37].
Thus, the mass of the incoming particle can be derived from measuring the
brightness of the meteor with an assumption of an energy conversion
efficiency. The Earth works as an extremely wide-area dust detector, providing
an indirect measurement of the size distribution of interplanetary dust around
the Earth. Meteors in the range between $-5$ and $+4\,\mathrm{mag}$. have been
detected by large meteor survey networks, for example, Cameras for Allsky
Meteor Surveillance (CAMS), and SonotaCo network [25]. CAMS obtained more than
135,000 meteor orbits in 2017. In this magnitude range, about 36% of meteors
belong to some meteor showers [26]. The luminosity function of meteors in this
magnitude range was intensively investigated by Hawkins and Upton [20] based
on a large survey with Super-Schmidt cameras. They showed that the meteor
luminosity function was well-approximated by a power-law function. The meteors
detected by these surveys, however, are mostly caused by particles of larger
than ${\sim}1\,\mathrm{g}$. Observations of meteors fainter than
$+5\,\mathrm{mag}$ are required to investigate the size distribution of
interplanetary dust. Cook et al. [9] obtained more than 2,000 sporadic meteors
in four nights using a 10-m optical reflector and a photo multiplier. The
photographic magnitudes of the detected meteors ranged between $+7$ and
$+12\,\mathrm{mag}$. Comparing with literature, they concluded that the meteor
luminosity function was well approximated by a power-law function between
$-2.4$ to $+12.0\,\mathrm{mag}$. They observed meteors as light pulses since
they used the photo multiplier instead of a camera. Thus, the information on
the meteor trajectories was not obtained and the atmospheric correction to
compensate a possibly unstable weather condition was not applied in real time.
Clifton [8] and Hawkes and Jones [17] observed faint meteors down to about $8$
and $6\,\mathrm{mag}$, respectively, in television observations. Imaging
observations of faint meteors using a wide-field video camera are favored to
examine the meteor luminosity function in detail. Radar observations detect
meteors caused by particles in the mass range of about $10^{-6}$–$10^{-2}\,$g.
The Canadian Meteor Orbit Radar has detected more than 3,000,000 meteors down
to $10^{-4}$ g in seven years [4, 5]. Large aperture radars can observe meteor
head echoes. Interplanetary dust grains of about $10^{-6}\,g$ can be detected
by these radars. Kero et al. [29] obtained more than 100,000 meteors using the
Shigaraki Middle and Upper atmosphere radar in Japan. Both the trajectories
and the radar cross-sections of meteors were obtained simultaneously. This is
a great advantage of radar head echo observations. On the other hand, the
radar cross-sections are not easy to convert to the brightness of the meteors,
or the mass of the meteoroids. Still, there is a need for optical observations
of faint meteors. The interplanetary dust grains in the mass range of
$10^{-6}$–$10^{-3}\,\mathrm{g}$ correspond to about $+7$–$14\,\mathrm{mag}$ in
optical observation. To obtain images of such faint meteors, we need a large
photon-collecting mirror, a wide-field optics, and a high-sensitive video
camera [31, 36]. There are only a handful of studies on meteors detected with
large telescopes. Pawlowski and Hebert [35] detected 151 Leonid meteors down
to about $13\,\mathrm{mag}$ with the Liquid Mirror Telescope in Cloudcroft
Observatory, to constrain the mass distribution index of the Leonid shower.
Iye et al. [22] serendipitously detected sporadic meteors with
Subaru/SuprimeCam and constrained the diameter of the collisionally excited
tubes caused by the meteors. Bektešević and Vinković [2] developed a new
method to discover faint meteors obtained in large optical surveys, such as
the Sloan Digital Sky Survey. Currently, no facility conducts regular
observations of faint meteors with a telescope larger than $1\,\mathrm{m}$. We
carried out observations of faint meteors with a wide-field mosaic CMOS camera
Tomo-e PM in 2016. This paper summarizes the results of the observations. This
paper is organized as follows: details of observations are summarized in
Seciton 2; data reduction and observed meteors are presented in Section 3; the
luminosity function of faint meteors is discussed in Section 4; we summarize
the paper in Section 5.

##  2 Observations

###  2.1 Tomo-e PM

Tomo-e Gozen is a wide-field camera being developed in Kiso observatory, the
University of Tokyo, which will be the world largest video camera for
astronomy. Tomo-e Gozen, mounted on the 105-cm Kiso Schmidt Telescope111Kiso
Observatory is located in $35^{\circ}47^{\prime}38.7^{\prime\prime}$ N and
$137^{\circ}37^{\prime}42.2^{\prime\prime}$ E, and at the altitude of
$1130\,\mathrm{m}$. in Kiso observatory, will continuously monitor a
$20\,\mathrm{deg^{2}}$ area at 2 Hz with 84 CMOS sensors developed by Canon
Inc. The designed pixel scale is about
$1.19{{}^{\prime\prime}}{\times}1.19{{}^{\prime\prime}}$. The limiting
magnitude is estimated to be about $18.5\,\mathrm{mag}$. in the $V$-band in
the 2 Hz observation. The limiting magnitude for meteors will be about
$13\,\mathrm{mag}$. in the $V$-band, on the assumption that meteors moves at
$10^{\circ}\mathrm{s}^{-1}$, which corresponds to the angular velocity of a
meteor at zenith, at 100 km above an observer, at a velocity of
$36\,\mathrm{km\,s^{-1}}$, and at an incident angle of $30\deg$. As a pilot
project, we developed a prototype of Tomo-e Gozen (hereafter, Tomo-e PM),
equipped with the 8 CMOS sensors [38]. Tomo-e PM is mounted on the 105 cm
Schmidt telescope at Kiso Observatory, the University of Tokyo. The pixel
scale is about $1.19{{}^{\prime\prime}}{\times}1.19{{}^{\prime\prime}}$. The
detectors are aligned in a line along the direction of the right ascension
with some gaps (see, Sako et al. [38]). The total area of the field-of-view is
about $1.98\,\mathrm{deg^{2}}$. All the detectors are synchronously operated
by the same control signal. The maximum frame rate is 2 Hz. The overhead time
due to readout is almost negligible thanks to rolling shutter. A NTP
synchronized time is recorded as a time-stamp in a FITS header, although the
time-stamp is not synchronized with a shutter timing. The time-stamp is as
accurate as ${\sim}1\,\mathrm{s}$. The size of the imaging area is
$2000{\times}1128$ pixels. The experimental observations were successfully
completed, to confirm that Tomo-e PM achieved the designed sensitivity.

###  2.2 Observations

Meteor observations with Tomo-e PM were carried out on April 11 and 14, 2016
(UT). Details of the observations are summarized in Table 1. The sky was dark
and clear on April 11, while part of the observations were affected by clouds
on April 14. Meteors are detected as a streak in the 2 Hz observation, while
artificial satellites and space debris are also detected as streaks. To
eliminate these contaminations, the observations were conducted in UT 12–18,
${\pm}3\,$hours around midnight in JST, and the observed regions were set
within the Earth’s shadow. Since the shadow moves during a night, the observed
regions were relocated in every two hours. The captured movie data were
compiled into FITS data cubes in every 3 minutes (hereafter, an observation
unit). Thus, each FITS data cube was composed of 360 frames. In total, 808
observation units (290,880 frames) were obtained on April 11, while 880
observation units (316,800 frames) were obtained on April 14.

Table 1: Observation Logs

Date | 2016-04-11 | 2016-04-14
---|---|---
Weather Condition | Clear Sky | Partly Clouded
Lunar Age | 3.6 | 6.6
Elevation Range |  $13.7^{\circ}$–$37.9^{\circ}$ |  $10.1^{\circ}$–$37.1^{\circ}$
Typical Zero Magnitude | $25.5\,\mathrm{mag}$ | $25.0\,\mathrm{mag}$
Total Observing Time |  ${\sim}5.1\,$hours |  ${\sim}5.5\,$hours
Number of Exposures | 290,880 | 316,800
Number of Detected Meteors | 1,514 | 706
Apparent Magnitudes† |  $4.5$–$12.5\,\mathrm{mag}$ |  $5.5$–$11.0\,\mathrm{mag}$

† Video rate magnitudes defined in Iye et al. [22].

##  3 Results

###  3.1 Data Reduction

Streak-like signals were extracted from the movie data cube with the slant-
and-compress algorithm developed by Ohsawa et al. [34]. First, background
emission was subtracted and stellar signals were masked. A frame was slanted
by a certain angle and compressed into a one-dimensional array with the
normalization such that the noise level should be uniform. A set of one-
dimensional arrays were stacked into a two-dimensional array. The streak-like
signals in the original image were converted as compact sources in the
converted array. The signal-to-noise ratio of faint meteors in the original
image was enhanced by about $\sqrt{n}$, where $n$ was the number of pixels
along the streak, by integrating signals along with the streak. Refer to
Ohsawa et al. [34] for details of the detection algorithm. A number of meteor
candidates were detected by the algorithm. Then, genuine meteor events were
counted by humans. The numbers of the real meteor events were 2002 and 926 on
April 11 and 14, respectively. Some meteors were doubly counted due to the
rolling shutter or passing through multiple detectors. Those duplicated events
were merged. Finally, 1514 and 706 events were extracted as the individual
meteor events detected on April 11 and 14, respectively. Although no prominent
meteor shower activity was predicted at that time, there could be some
contributions from April Lyrids and $\eta$-Aquariids. The separation angles
from the radiant points of these showers were calculated. The fractions of the
meteors which might pass through within 3∘of the radiants were at most 6% and
3% for April Lyrids and $\eta$-Aquariids, respectively. These fractions were
consistent with the number of sporadic meteors randomly coming from the
radiants. We concluded that the contribution from meteor showers was
negligible and the detected meteors were attributed to sporadic meteors. The
time variations in the number of the meteors on April 11 and 14 are
illustrated in the top and bottom panels of Figure 1, respectively. The bar
charts show the numbers of the meteors in the observation unit. The circles
below the bar charts illustrates the time variations in the magnitude zero-
point averaged over the eight detectors, indicating weather conditions in the
observation unit. The small number of the detections on April 14 was simply
attributed to the poor weather conditions.

Figure 1: Time variation in the number of meteors and the magnitude zero-
point. The periods of cloudy weather are indicated by the green hatched
regions. The gray cross-hatched regions mean no data due to pointings or
troubles.

The brightness of the meteors was measured as follows. The projected profile
of the meteor was obtained by averaging the signal along with the streak. The
projected profile was broad and sometimes double-peaked since meteors were
usually defocused [3]. The line-intensity of the meteor, $\tilde{I}$ in units
of $\mathrm{ADU\,pix^{-1}}$, was measured by fitting the projected profile
with a combination of Gaussian profiles. Since the angular velocity of the
meteors were not measured by Tomo-e PM observations, the line-intensity
$\tilde{I}$ was not directly compared with the intensities of nearby stars.
Thus, we adopted the video-rate magnitude defined in Iye et al. [22]. Assuming
that all the meteors moved at an angular velocity of $10^{\circ}s^{-1}$, the
travel distance of the meteors in $0.5\,$s was estimated to be
$1.5{\times}10^{3}$ pixels, where $\theta$ was the pixel scale of Tomo-e PM in
units of arcsecond. The total amount of the signals emitted by the meteor in
$0.5\,$s was obtained by ${I}=1.5{\times}10^{3}\tilde{I}\,\mathrm{ADU}$, which
was transformed into the $V$-band magnitude by comparing with the intensities
of nearby stars. The ranges of the estimated magnitudes are listed in Table 1.
The meteors as faint as about $12.5\,\mathrm{mag}$ in the $V$-band were
detected on April 11. Although the weather condition on April 14 was poorer
than that on April 11, the meteors as faint as about $11.0\,\mathrm{mag}$ were
successfully detected.

###  3.2 System Efficiency

Figure 2: Definition of the meteor detectable volume. The blue dot is the
location of a telescope, which is observing a sky at the elevation $\alpha$.
The thin blue region indicates the meteor detectable layer. The blue cross-
hatched region indicates the meteor detectable volume. The thick blue region
is the test region, where meteors are randomly generated. A generated meteor
with the incident angle $\theta$ is indicated by the red arrow.

The meteor-collecting power of a observing system is usually evaluated in
units of area. Koschack and Rendtel [30] evaluated the effective meteor-
collecting power from the area of the field-of-view projected onto a meteor
level at an altitude of $100\,\mathrm{km}$ weighted by distance. In
observation with naked eyes (visual observation), a field-of-view was as large
as about $52.5^{\circ}$ [30]. The estimated effective meteor-collecting area
(hereafter, EMCA) in visual observation is $24,400\,\mathrm{km^{2}}$ when
observing at zenith, whereas that area becomes $20,770\,\mathrm{km^{2}}$ when
observing with an elevation angle of $50^{\circ}$, in the case that a meteor
population index222The number of meteors brighter than a magnitude $m$ in
visible wavelengths should follow $N({<}m)\propto r^{m}$. $r$ is 3.5. In
Koschack’s calculation, the region where meteors are observable is
approximated by a thin spherical shell. Hereafter, Koschack’s formalism is
referred as to the thin shell approximation. In meteor observations with
telescopes, the field-of-view is usually smaller than 10∘. In such cases, the
thin shell approximation is not valid. Kresáková and Kresák [31] evaluated the
EMCA by statistically estimating the averaged angular length of meteors. This
method requires a large number of meteors. Since the efficiency in meteor
detection with Tomo-e PM changes with elevation and Tomo-e PM has the multiple
image sensors, their method is not applicable to our observations. Thus, we
evaluated the EMCA of Tomo-e PM by a Monte Carlo simulation. Figure 2
schematically describes the configuration of calculation. When dust grains
penetrate into the upper atmosphere, they are heated by interaction and become
bright in optical wavelengths at a certain altitude, observed as meteors.
Then, when the grain penetrates into a lower atmospheric layer, they cease to
be bright. Here, we define an upper and lower limit of an altitude where
meteors are observable in optical wavelengths by $H_{2}$ and $H_{1}$,
respectively. Thus, the region where meteors are observable in optical
wavelengths is approximated by a spherical shell with a depth of
$H_{2}{-}H_{1}$ (hereafter, meteor shell). In Figure 2, the meteor shell is
indicated by the blue shaded region. The region captured by a camera with a
single rectangular field-of-view is defined by

$\vec{x}\in\left\\{~{}c_{0}\vec{n}_{0}+c_{1}\vec{n}_{1}+c_{2}\vec{n}_{2}+c_{3}\vec{n}_{3}~{}\middle|~{}c_{m}>0~{}\text{for}~{}m=0,1,2,3~{}\right\\},$
(1)

where $\vec{n}_{m}(m=0,1,2,3)$ is a unit vector to point a corner of the
field-of-view. The intersection of the meteor shell and the field-of-view is
defined as a meteor-collecting volume (MCV). The MCV is approximated by a
truncated pyramid as shown by the cross-hatched region in Figure 2. This
approximation does not affect results significantly as long as the dimension
of the field-of-view is smaller than about $1^{\circ}$. The distance to the
MCV is represented by the distance to its geometric center. For a system with
multiple image sensors such as Tomo-e PM, the union of the MCVs for each
sensor is defined as the MCV of the system. An incident meteor is defined by
two properties: an incident point $\vec{p}$ and a pair of incident angles
$(\omega,i)$. The incident point $\vec{p}$ means the location of a dust grain
when they enter the meteor shell. The incident angles correspond to the
velocity of a grain relative to the Earth. The angle $\omega$ is an azimuthal
angle, while the angle $i$ is an incident angle measured from the normal
vector of the meteor shell at $\vec{p}$. Incident meteors are illustrated by
the orange arrows in Figure 2. The evolution of the velocity by interaction
with atmosphere is not taken into account. Thus, we assume that the direction
of a meteor’s velocity is constant. Incident meteors are randomly generated on
the test region, which is a sufficiently large part of the meteor shell, with
a surface density of $\Sigma$ at an altitude of $H_{2}$. An effective meteor-
collecting area (EMCA) at a nominal altitude of $100\,\mathrm{km}$ is defined
by

$A=\frac{N}{\Sigma}\left(\frac{100\,\mathrm{km}}{H_{2}}\right)^{2}=N\frac{A_{0}}{N_{0}}\left(\frac{100\,\mathrm{km}}{H_{2}}\right)^{2},$
(2)

where $N$ is the number of the meteors which enter the MCV, $A_{0}$ is the
total area where the meteors are generated, $N_{0}$ is the total number of the
generated meteor. We developed a code to calculate an EMCA for a given
observing configuration. The shape, the elevation angle $\theta$, and the
position angle $\phi$ of pointing are given. Incident meteors are generated
until $N$ exceeds a threshold $\hat{N}$. Since the detections are given by a
Poisson process, the statistical error of $A$ becomes negligible for
sufficiently large $N_{0}$. When the field-of-view is as large as
${\sim}50^{\circ}$ in diameter, the depth of the meteor shell becomes
negligible. Thus the EMCA calculated here becomes asymptotically identical to
that in the thin shell approximation. As the simplest case, we assumed that
the number density of interplanetary dust around the Earth was uniform and the
velocity of the interplanetary dust grains relative to the Earth was
isotropic. Thus, the distribution of $\vec{p}$ was uniform on the top surface
of the meteor shell, the distribution of $\omega$ distributed uniformly in
$[0,\,2\pi)$, and the distribution of $i$ was set to be proportional to
$\cos{i}$. In calculation, we adopted $H_{1}=80\,\mathrm{km}$ and
$H_{2}=120\,\mathrm{km}$. The EMCA does not significantly depend on the choice
of $H_{1}$ and $H_{2}$ (see, A.3). The threshold $\hat{N}$ was set to
$100,000$ so that the statistical errors become negligible. Note that the
effective meteor-collecting area calculated based on our assumption may
contain a systematic error since the radiant distribution of sporadic meteor
is not uniform [27], and may be incorrect especially for meteor showers.

Figure 3: Time variation in the meteor-collecting areas on April 11 (the red
circles) and April 14 (the blue triangles).

The EMCAs of Tomo-e PM estimated with a Monte-Carlo simulation are shown in
Figure 3. Random errors are as small as the symbols. Figure 3 indicates that
the meteor-collecting area of Tomo-e PM changed from about
$1,000\,\mathrm{km}$ to $10,000\,\mathrm{km}$ with elevation. Note that the
EMCAs calculated here do not include the effect of the distance between the
observer and meteors. When the elevation angle is low, a typical distance to
meteors becomes large and faint meteors are not detected. Thus, the number of
the detected meteors tends to decrease with decreasing elevation angle. Figure
1 shows that the number of the detected meteors peaked around 15:00 UT on
April 11. Detailed discussion is presented in A.4. The meteor travel distances
and a possible bias in the meteor arrival directions are discussed in A.1 and
A.2, respectively.

##  4 Discussion

###  4.1 Absolute Magnitude Distribution

Figure 4: Distributions of the absolute magnitudes of the meteors detected by
Tomo-e PM. The top and bottom panels respectively shows the distributions on
April 11 and 14, 2016.

The observed video-rate magnitudes $m_{V}$ were converted into absolute
magnitudes $M_{V}$, which correspond to $V$-band magnitudes at the distance of
$100\,\mathrm{km}$, by assuming that the distances to meteors were identical
to those to the meteor detectable volume:

$M_{V}=m_{V}+5.0\log_{10}\frac{D(\alpha)}{100\,\mathrm{km}}.$ (3)

Figure 4 shows the distributions of the absolute magnitudes. Meteros of
$M_{V}\lesssim 10$ and $9\,\mathrm{mag}$ were detected on April 11 and 14,
respectively. The histograms increase exponentially with increasing
magnitudes, and then start to decrease at magnitudes of 8th or 9th. The
decrease is basically attributed to sensitivity limits. Detailed discussion is
given in Section 4.2. Jacchia [23] derived an equation to convert the
magnitude of a meteor to the mass of a meteoroid, which were calibrated based
on observations with Super-Schmidt cameras. A modified equation was given by
Jenniskens [24], which derives the mass of a meteoroid from the $V$-band
magnitude of a meteors:

$\log_{10}{M_{g}}=6.31-0.40M_{V}-3.92\log_{10}V_{\infty}-0.41\log_{10}\sin(h_{r}),$
(4)

where $M_{g}$ is the mass of the meteoroid in units of grams, $M_{V}$ is the
$V$-band magnitude of the meteor, $V_{\infty}$ is the incident velocity of the
meteoroid in units of $\mathrm{km\,s^{-1}}$, and $h_{r}$ is the elevation of
the radiant point of the meteor. By assuming that $V_{\infty}$ is
$30\,\mathrm{km\,s^{-1}}$ and $h_{r}$ is $90^{\circ}$, the mass of the meteors
we detected ranges from about $8.3{\times}10^{-2}\,\mathrm{g}$
($4\,\mathrm{mag}$) to about $3.3{\times}10^{-4}\,\mathrm{g}$
($10\,\mathrm{mag}$).

###  4.2 Luminosity Function Analysis

Generally, a luminosity function of visible meteors is well approximated by an
exponential function [20]:

$\log_{10}N({<}M)=\log_{10}N_{0}+M\log_{10}r,$ (5)

where $N({<}M)$ and $N_{0}$ are the event rates of meteors brighter than
$M$-th and zero-th magnitude, respectively. The present results are a
composite of the observations in different conditions in terms of the meteor-
collecting area and the limiting magnitudes. Thus, a naive fitting of the
apparent magnitude distribution to derive luminosity functions may bring
biased results. We introduce a statistical model to robustly estimate the
slope parameter $r$ and the meteor rate $\log_{10}N_{0}$. We assume that the
luminosity function of meteors exactly follows Equation (5) and observational
conditions are constant within the observation unit. The number of detectable
meteors per observation unit is expected by

$\left\\{\begin{array}[]{ll}\log_{10}\tilde{n}^{i}&=\log_{10}tN^{i}_{0}A_{i}+\mathcal{M}^{i}\log_{10}r\\\
\log_{10}N^{i}_{0}&=\log_{10}N_{0}+\eta(T^{i}-\mathrm{15:00}_{\mathrm{UTC}})\end{array},\right.$
(6)

where $t$ is the duration of an observation unit, $A_{i}$ is the meteor-
collecting area of the $i$-th observation unit in units of $\mathrm{km^{2}}$,
$\mathcal{M}^{i}$ is a limiting magnitude in the $i$-th observation unit,
$\eta$ is a parameter to describe the diurnal variation [18, 20, 10] in units
of $\mathrm{dex\,hour^{-1}}$, and $T_{i}$ is the observation time of the
$i$-th observation unit in hours in UTC. From Equation (5), expected
magnitudes of detectable meteors in the $i$-th observation unit
$\tilde{M}^{i}$ should follow

$\mathcal{M}^{i}-\tilde{M}^{i}\sim\mathrm{exponential}(\beta),$ (7)

where $\beta$ is a shape parameter of the exponential distribution, given by
$\beta=\log{r}$.

Figure 5: Probability density funciton of the fitting parameters, $r$, $\log_{10}N_{0}$, and $\eta$, are shown in the top, middle, and bottom panels, respectively. The vertical lines with numbers indicate the posterior mean values. Table 2: Statistical Inference with the Simple Model | Parameter | Mean | 2.5% | 50% | 97.5%
---|---|---|---|---|---
2016-04-11 | $r$ | $\phantom{-}3.24$ | $\phantom{-}3.01$ | $\phantom{-}3.23$ | $\phantom{-}3.50$
| $\log_{10}N_{0}$ | $-5.59$ | $-5.91$ | $-5.59$ | $-5.30$
| $10^{2}\eta$ | $\phantom{-}1.16$ | $-0.30$ | $\phantom{-}1.16$ | $\phantom{-}2.62$
2016-04-14 | $r$ | $\phantom{-}3.08$ | $\phantom{-}2.74$ | $\phantom{-}3.06$ | $\phantom{-}3.50$
| $\log_{10}N_{0}$ | $-5.34$ | $-5.82$ | $-5.33$ | $-4.93$
| $10^{2}\eta$ | $\phantom{-}0.50$ | $-1.92$ | $\phantom{-}0.50$ | $\phantom{-}2.91$

The parameters are optimized to match $n^{i}\sim\tilde{n}^{i}$ and
$M^{i}\sim\tilde{M}^{i}$. The optimization is carried out with Stan [6], which
is a software for the Bayesian statistical inference with MCMC sampling [6,
21]. The posterior probability distributions are shown in Figure 5. The
posterior mean values and the 95% confidence intervals of the parameters are
listed in Table 2. No significant differences are detected between the results
on April 11 and 14. The results suggest that the slope parameter is
${\sim}3.1\pm{0.4}$ and the meteor rate is about $-5.5{\pm}0.5$. The data on
April 11 marginally suggest a positive $\eta$ value, while the increase in the
meteor rate is not confirmed on April 14. The present result, $\eta$ of
${\sim}1{\times}10^{-2}\,\mathrm{dex\,hours^{-1}}$, corresponds the increase
by 30% in 12 hours. The increase was much smaller than reported in previous
studies [33, 29]. This could be in part attributable to our assumption in
calculating the EMCA: the radiant distribution of meteors is uniform. Further
observations are required to confirm the non-ditection of the diurnal
variation with Tomo-e PM.

Figure 6: Comparison of luminosity functions obtained in different
observations. The present result is indicated by the blue region.

We compare the present result with luminosity functions in literature in
Figure 6. The purple solid line indicates the luminosity function derived by
Hawkins and Upton [20] based on surveys with Super-Schmidt cameras. Hawkins
and Upton [20] suggested that the $r=3.4{\pm}0.2$ and $\log_{10}N_{0}=-5.2$
using meteors brighter than about $4\,\mathrm{mag}$. The green dashed line
indicates the luminosity function provided by Cook et al. [9] using a 10-m
reflector and phototubes. The slope parameter they derived was about $3.41$.
The blue doted-dashed line and orange double-dotted-dashed line respectively
show the luminosity functions in Clifton [8] and Hawkes and Jones [15]. They
observed meteors using television systems. Since the brightness of meteors
were not directly measured in Clifton [8], we use the luminosity function re-
calibrated by Cook et al. [9]. Hawkes and Jones [17] suggested that $r\sim
2.5$, while Clifton [8] suggested that $r\sim 3.4$. The red triple-dotted-
dashed line shows the luminosity function of fireballs [19], suggesting that
$r\sim 2.5$. Here, we tentatively assume that the conversion factor from
photographic to visual magnitude is ${+}1.0\,\mathrm{mag}$. The present result
is shown by the blue region. The height of the region represents the
uncertainties of the parameters. The present observations constrain the
luminosity function between about $+4$ to $+10\,\mathrm{mag}$, which is the
deepest among the imaging observations. The slope parameter in the present
result is roughly consistent with the other observations except for Hawkins
[19]. The meteor rate $N_{0}$ is by a factor of ${\sim}2.5$ lower than the
value reported in Hawkins and Upton [20]. This can be attributed to a seasonal
variation in the sporadic meteor rate; The number of sporadic meteors in March
and April is about a half of the annual average rate [18, 29, 33]. The meteor
rate of the present result is by a factor of ${\sim}30$ lower than that of
Cook et al. [9]. This difference could be in part attributable to the
difference in the assumed EMCAs. Cook et al. [9] assumed that the EMCA was
$3\,\mathrm{km^{2}}$, which was basically derived with the thin shell
approximation and could be underestimated. Generally, the present result is
consistent with the luminosity functions in literature. Cook et al. [9]
suggested that the luminosity function of meteors was well approximated by a
single slope power-law function from $-2.4$ to $+12\,\mathrm{mag}$. The
present result is in line with the Cook’s suggestion.

###  4.3 System efficiency comparison between Tomo-e PM and Tomo-e Gozen

The operation of Tomo-e PM was completed in 2016. Observations with Tomo-e
Gozen started in February, 2018. Here, we compare the EMCAs between Tomo-e PM
and Tomo-e Gozen, to evaluate the efficiency in the meteor observations with
Tomo-e Gozen. The EMCAs are calculated out for three systems. The shape of
their fields-of-view are illustrated in Figure 7. System 1 has a single field-
of-view with dimensions of $39^{\prime}.7{\times}22^{\prime}.4$, corresponding
to that of one CMOS sensor. System 2 emulates Tomo-e PM, a wide-field camera
equipped with 8 CMOS image sensors mounted on 105 cm Kiso Schmidt telescope
[38]. System 2 has eight image sensors, each of which has a field-of-view with
dimensions of $39^{\prime}.7{\times}22^{\prime}.4$. System 3 is a counterpart
of Tomo-e Gozen. The field-of-view of System 3 is composed of 84 segments. The
size of each segment is the same in Systems 1 and 2. The MCVs with an
elevation angle of $35^{\circ}$, $60^{\circ}$, and $85^{\circ}$ are
illustrated in Figure 8. The position angles of the field-of-view are fixed.
Both the size of the MCV and the distance to the MCV increase with decreasing
elevation angle.

Figure 7: Fields-of-view of the three systems. The violet cross-hatched region
indicates the field-of-view of System 1. The green empty rectangles indicate
the field-of-view of System 2. The blue filled rectangles indicate the field-
of-view of System 3.

Figure 8: Schematic view of meteor-collecting volumes (MCV). The top, middle,
and bottom plots respectively show the MCVs for Systems 1, 2, and 3. The
labels “A”, “B”, and “C” indicate the MCVs for elevation angles of
$85^{\circ}$, $60^{\circ}$, and $35^{\circ}$, respectively. The blue dots
indicate the location of the observing system. The green solid curve in Panels
B indicates the sea level. The gray dashed lines in Panels B respectively show
the top and bottom surface of the meteor sphere.

The EMCAs for the three systems are calculated for elevation angles from
$10^{\circ}$ to $90^{\circ}$ at $5^{\circ}$ intervals. Figure 9 shows the
dependence of the EMCA on the elevation angle $\alpha$. The EMCAs at zenith
are about $43$, $281$, and $630\,\mathrm{km^{2}}$ for System 1, 2, and 3,
respectively. At $\alpha=10^{\circ}$, the EMCAs increase to about
$1.1{\times}10^{3}$, $7.8{\times}10^{3}$, and
$25.8{\times}10^{3}\,\mathrm{km^{2}}$ for System 1, 2, and 3, respectively.
The EMCAs for Systems 1, 2, and 3 similarly increase with decreasing elevation
angle. The dependence of the EMCA on $\alpha$ is well approximated by
$\sin^{-2}\alpha$. Note that the EMCAs, which do not include the effect of the
distance to meteors, do not reflect the expected number of meteors detected.
Refer to A.4 for detailed discussion.

Figure 9: Effective meteor-collecting areas (EMCAs) in units of
$\mathrm{km^{2}}$ against an elevation angle $\alpha$. The error bars are as
small as the line width.

The EMCAs for Systems 2 is about 7 times larger than that of System 1. This
increase is almost proportional to the number of the image sensors. The
efficiency of System 3 is about two times larger than that of System 2. Since
the detectors of Tomo-e Gozen are tiled in the image circle of the telescope,
a large fraction of meteors will be detected first in peripheral detectors;
the detectors around the center of the field-of-view have little contribution
to the meteor-collecting efficiency.

Table 3: EMCA for Square Fields-of-View

| $10^{\prime}{\times}10^{\prime}$ | $20^{\prime}{\times}20^{\prime}$ | $30^{\prime}{\times}30^{\prime}$ | $40^{\prime}{\times}40^{\prime}$ | $50^{\prime}{\times}50^{\prime}$ | $60^{\prime}{\times}60^{\prime}$
---|---|---|---|---|---|---
Side Length (′) | $10$ | $20$ | $30$ | $40$ | $50$ | $60$
FOV area ($\mathrm{deg}^{2}$) | $0.03$ | $0.11$ | $0.25$ | $0.44$ | $0.69$ | $1.00$
EMCA ($\mathrm{km^{2}}$) | $14.6$ | $27.2$ | $41.4$ | $55.1$ | $69.2$ | $83.2$
Projected Areaa ($\mathrm{km^{2}}$) | $0.09$ | $0.34$ | $0.76$ | $1.35$ | $2.12$ | $3.05$
Event Rateb ($\mathrm{hour^{-1}}$) | $22$ | $45$ | $68$ | $90$ | $114$ | $136$

a The area of the field-of-view projected on the sphere at an altitude of
$100\,\mathrm{km}$.

b Estimated meteor event rates for meteors brighter than $10$ magnitude.

To investigate the dependence of the EMCA on the field-of-view, we calculate
the EMCA for simple systems which have a single square field-of-view pointing
at the zenith. The side lengths of the fields-of-view are $10^{\prime}$,
$20^{\prime}$, $30^{\prime}$, $40^{\prime}$, $50^{\prime}$, and $60^{\prime}$.
Calculated EMCAs are listed in Table 3 as well as the areas of the fields-of-
view projected on the surface at an altitude of $100\,\mathrm{km}$ above the
sea level (hereafter, referred as to projected fields-of-view). The EMCA is
larger than the area of the projected field-of-view. In the case of the
$60^{\prime}{\times}60^{\prime}$ system, the EMCA is larger by a factor of
about 30. This simply illustrates the thin shell approximation is not
appropriate for systems with small fields-of-view. The EMCAs are almost
proportional to the side lengths of the fields-of-view, rather than the areas
of the fields-of-view. The meteor-collecting efficiency, in general, depends
on the surface area of the MCV. For a narrow field-of-view, the surface area
of the MCV is approximately proportional to the side length of the field-of-
view. This simply explains the dependence of the EMCA on the side length; The
EMCA of narrower field-of-view systems does not decrease as with the area of
the field-of-view. This is generally consistent with the fact that the EMCA of
System 3 is about two times larger than that of System 2; The summation of the
most outer circumference sides of System 2 is about $11.3^{\circ}$, while that
of System 3 is about $19.5^{\circ}$. The expected number of detected meteors
per hour is estimated by $AN_{0}r^{\mathcal{M}}$, where $\mathcal{M}$ is a
limiting magnitude, $r$ is a meteor index, $A$ is an EMCA, and $N_{0}$ is a
rate of meteors brighter than zeroth magnitude. Here, we adopt $r=3.4$ and
$\log_{10}N_{0}=-5.1$ [20]. Table 3 lists the event rate calculated for
$\mathcal{M}=10\,\mathrm{mag}$ in optical wavelengths. A system with a
$1^{\circ}{\times}1^{\circ}$ field-of-view and a limiting magnitude of
${\sim}10$ will detect more than 100 meteors per hour. This result encourages
meteor survey observations with wide-field cameras mounted on large aperture
telescopes (e.g. Subaru/HSC [32]). Tomo-e Gozen is expected to detect more
than about 2,000 faint meteors a night. Since meteors will run across multiple
detectors of Tomo-e Gozen, faint meteors are more robustly detected with
Tomo-e Gozen than Tomo-e PM. Tomo-e Gozen will be an ideal instrument to
investigate variations or structures in the luminosity function of meteors
brighter than about ${+}10\,\mathrm{mag}$.

##  5 Conclusion

Tomo-e Gozen is a mosaic CMOS camera developed in Kiso Observatory, the
University of Tokyo. Tomo-e Gozen, equipped with 84 CMOS sensors, will
continuously obtain images of $20\,\mathrm{deg^{2}}$ at $2\,\mathrm{Hz}$. In
this sense, Tomo-e Gozen will be the world largest video camera for astronomy.
As a pilot project, we developed a prototype model of Tomo-e Gozen, Tomo-e PM,
which has 8 CMOS sensors. Tomo-e PM can monitor a sky of about
$2\,\mathrm{deg^{2}}$ at $2\,\mathrm{Hz}$. We carried out imaging observations
of faint meteors with Tomo-e PM on April 11 and 14, 2016. We set the
observation field within the Earth’s shadow to eliminate contamination from
debris and satellites in the low Earth orbit. The total observing time was
about 10 hours, and 2,220 meteor events were detected in total. Our
observations provide a new measurement of the meteor luminosity function. The
video rate magnitudes of the meteors we detected are typically from $+4$ to
$+12\,\mathrm{mag}$. The corresponding mass of the meteors ranges from
$8.3{\times}10^{-2}$ to $3.3{\times}10^{-4}\,\mathrm{g}$. The present results
are consistent with a single power-law luminosity function. A statistical
model suggest that the slope parameter $r=3.1{\pm}0.4$ and the meteor rate
$\log_{10}N_{0}=-5.5{\pm}0.5$. The diurnal variation in the sporadic meteor
rate is marginally confirmed only on April 11. The slope is roughly consistent
with those of luminosity functions in literature. The meteor rate is lower
than those in literature. The discrepancy is, in part, attributed to the
seasonal variation and different assumptions on the EMCAs. The operation of
Tomo-e PM was completed in 2016. Observations with Tomo-e Gozen started in
February, 2018. We compare the effective meteor-collecting areas between
Tomo-e PM and Tomo-e Gozen, suggesting that Tomo-e Gozen is twice as efficient
as Tomo-e PM in detecting meteors. More than 2,000 meteors will be detected
every night with Tomo-e Gozen. Kiso Observatory has the capability to make it
a leading place in time-domain astronomy including observations of faint
meteors.

## Acknowledgments

This research is supported in part by JSPS Grants-in-Aid for Scientific
Research (KAKENHI): Grant Numbers JP16H02158, JP16H06341, JP18H01261, and
JP18K13599. This research is in part supported by JSPS Grant-in-Aid for
Scientific Research on Innovative Areas: Grant Number JP2905. This is also
supported in part by PRESTO, Japan Science and Technology Agency (JST). This
work is also supported by the Optical and Near-infrared Astronomy Inter-
University Cooperation Program by the MEXT of Japan. This work is achieved
using the grant of Joint Development Research by the Research Coordination
Committee, National Astronomical Observatory of Japan (NAOJ). The fabrication
of Tomo-e PM is conducted in collaboration with the Advanced Technology Center
of NAOJ. This research is supported by Canon Inc. for developing the full-HD
high-sensitivity CMOS sensors.

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##  Appendix A Validation of the EMCA Calculation

###  A.1 Travel Distances

In the present formalism, we assume that an incident meteor never fades away
inside the meteor shell. This assumption may bring an overestimate of a rate
of meteors with a long travel distance, and an EMCA could be overestimated
accordingly. Figure 10 shows the distribution of the travel distances of the
meteors for System 1 with $\alpha=60^{\circ}$. The distribution neither
depends on the systems nor elevation angles. The fraction of the meteors with
a travel distance longer than $400\,\mathrm{km}$ was $\lesssim 10\%$. Thus, we
concluded that the assumption had virtually no effect on the present results.

Figure 10: Distribution of the travel distances of observable meteors. The
filled bars show the histogram the travel distances for System 1 with an
elevation angle $\alpha=60^{\circ}$. The cumulative distribution is described
by the solid step line.

###  A.2 Position Angles

Artificial patterns are found in the distribution of position angles on the
focal plane. Position angles (PA) of meteors for System 2 are summarized in
Figure 11. Dips around $\mathrm{PA}=-90^{\circ}$ and $90^{\circ}$ in the top
panel of Figure 11 are attributed to the alignment of the detectors. A
fraction of meteors with $\mathrm{PA}\sim{{\pm}180^{\circ}}$ increased with
decreasing elevation angle. Thus, a large fraction of meteors run from the
zenith to the nadir in a field-of-view. The observations on April 11 and 14
were carried out with elevation lower than 38∘. Similar distributions of the
PAs were confirmed in the observations. Our calculation is qualitatively
consistent with the observations. The degree of the concentration depends on
the distribution of the incident angle $i$. Since it was difficult to
constrain the distribution of $i$ from the observations, we did not discuss
the PA distribution of the meteors in the paper.

Figure 11: Distributions of the position angles of mock meteors. A position
angle of zero corresponds to a meteor moving toward the zenith, or upward in
the field-of-view.

###  A.3 Depth of the Meteor Shell

We assume that the top and bottom boundaries of the meteor shell were at $120$
and $80\,\mathrm{km}$, respectively. They are chosen as nominal values [24].
An appearing and disappearing altitude of a meteor, however, depend on the
size or composition of the meteoroid. We calculate the EMCAs for
$H_{1}=60\,\mathrm{km}$ and $H_{2}=140\,\mathrm{km}$, the results are almost
the same. Thus, we conclude that the choice of $H_{1}$ and $H_{2}$ has little
impact on the present results.

###  A.4 Reduced Efficient Meteor-Collecting Area

The EMCA defined in Equation (2) does not take into account the apparent
brightness of incident meteors and atmospheric extinction. A practical meteor-
collecting area is given by reducing $A$ as

$A^{\mathrm{red}}=\frac{A_{0}}{N_{0}}\left(\frac{100\,\mathrm{km}}{H_{2}}\right)^{2}\sum^{N}_{i}r^{-5\log_{10}\frac{d_{i}}{100\,\mathrm{km}}-\epsilon_{i}},$
(8)

where $r$ is the population index, $d_{i}$ is the distance to the $i$-th
meteor, and $\epsilon_{i}$ is the amount of extinction of the $i$-th
observable meteor. The reduced effective meteor-collecting area (hereafter,
referred as to RMCA) corresponds to the area defined in Equation (1) of
Koschack and Rendtel [30]. For the sake of simplicity, we adopt
$\epsilon_{i}=0$ in the rest of the paper. We calculate the RMCAs for Systems
2. The position angle $\phi$ is fixed at zero. Figure 12 shows the RMCA for
$r=2.0$, $2.5$, $3.0$, $3.5$, and $4.0$ against an elevation angle. While the
RMCA is almost the same as the EMCA when the elevation angle $\alpha$ is
larger than about $80^{\circ}$, the differences between the EMCA and RMCA
become significant at low elevation angles. The differences increase as $r$
becomes large. Hawkins and Upton [20] and Cook et al. [9] independently
obtained $r\simeq 3.4$ for sporadic meteors with $M_{\mathrm{pg}}<{+}5$ and
${+}7<M_{\mathrm{pg}}<{+}12$, respectively. Thus, we safely assume $r>3.0$ for
visible sporadic meteors. In such cases, the RMCA is largest at the zenith.
This does not depend on systems. In observations of sporadic meteors, a
monitoring observation close to the zenith is favored. When $r$ is larger than
about $3.0$, the RMCA monotonically decreased with decreasing elevation angle
in contrast to the EMCA. The RMCA is well-approximated by

$\tilde{A}^{\mathrm{red}}=Ar^{-5\log_{10}\frac{\langle
d\rangle}{100\,\mathrm{km}}},$ (9)

where $\langle d\rangle$ is the averaged distance to the MCVs. Since the
field-of-view is sufficiently narrow, the distances to the meteors are
approximately identical. This justifies Equations (6) and (9).

Figure 12: Reduced effective meteor-collecting areas (RMCAs) in units of
$\mathrm{km^{2}}$ against an elevation angle $\alpha$. The symbols indicate an
assumed meteor index value $r$. The gray dashed line indicates the EMCA for
System 3 as a reference.

Equation (8) indicates that an expected number of meteors can be given by
$\tilde{n}=A^{\mathrm{red}}N_{0}r^{\mathcal{M}}$, which is a function of the
meteor index $r$, the elevation angle $\alpha$, and the limiting magnitude
$\mathcal{M}$. Figure 12 illustrates the dependence of $A^{\mathrm{red}}$ on
the meteor index $r$; the RMCA ratio at $\alpha=30^{\circ}$ to
$\alpha=90^{\circ}$ decreases from 1.74 to 0.63 when the meteor index $r$
increases from 2.0 to 4.0. Thus, comparing the number of meteors detected in
different elevation angles provides an estimate of the meteor index $r$
without measuring the brightness of the meteors. Figure 13 shows an
application of this method. The orange crosses show the numbers of meteor
events in 3 minute on April 11, 2016333The data on April 14, 2016 are not
shown, since large part of the data with $\alpha>30^{\circ}$ were obtained in
bad conditions.. The number of detections decreases with decreasing elevation
angle. The lines indicate the expected number of meteor detections for
different $r$ values. The limiting magnitudes are estimated from neighboring
stars. The observations are well explained by the expectations with
$r=3.0$–$4.0$, consistent with previous studies [9, 17, 20]. This confirms
that the present results are self-consistent.

Figure 13: Deriving the meteor index $r$ by counting the number of meteors.
The orange crosses shows the number of the meteors in 3 minute against the
elevation angle in observation. The lines show the expected number of
detections for different $r$ values.


# Predicting Twitter User Socioeconomic Attributes with Network and Language
Information

Nikolaos Aletras  University of Sheffield<EMAIL_ADDRESS>and
Benjamin Paul Chamberlain  Imperial College London Asos.com
<EMAIL_ADDRESS>
(2018)

###### Abstract.

Inferring socioeconomic attributes of social media users such as occupation
and income is an important problem in computational social science. Automated
inference of such characteristics has applications in personalised recommender
systems, targeted computational advertising and online political campaigning.
While previous work has shown that language features can reliably predict
socioeconomic attributes on Twitter, employing information coming from users’
social networks has not yet been explored for such complex user
characteristics. In this paper, we describe a method for predicting the
occupational class and the income of Twitter users given information extracted
from their extended networks by learning a low-dimensional vector
representation of users, i.e. graph embeddings. We use this representation to
train predictive models for occupational class and income. Results on two
publicly available datasets show that our method consistently outperforms the
state-of-the-art methods in both tasks. We also obtain further significant
improvements when we combine graph embeddings with textual features,
demonstrating that social network and language information are complementary.

social media, graph embeddings, user profiling

††copyright: rightsretained††conference: ACM HT; July 2018; Baltimore,
Maryland††journalyear: 2018

##  1\. Introduction

The daily interaction of billions of users with online social platforms such
as Facebook and Twitter has made available enormous amounts of user generated
content. The plethora and diversity of this data (e.g. text, images or
interactions with other users such as ‘retweets’ or ‘likes’) enables studies
in computational social science (Lazer et al., 2009; Conte et al., 2012) to
analyse human behaviour on a large scale and automatically infer user latent
attributes.

Automatic inference of user characteristics includes studies on inferring age
and gender (Rao et al., 2010; Burger et al., 2011; Schwartz et al., 2013),
location (Cheng et al., 2010; Han et al., 2014; Dredze et al., 2016),
personality traits (Quercia et al., 2011; Kosinski et al., 2013; Volkova and
Bachrach, 2015) and political orientation (Tumasjan et al., 2010;
Pennacchiotti and Popescu, 2011; Cohen and Ruths, 2013) inter alia. More
recently, there has been a particular focus on inferring complex user
socioeconomic characteristics such as occupational class (Li et al., 2014;
Huang et al., 2015; Preoţiuc-Pietro et al., 2015a), income (Preoţiuc-Pietro et
al., 2015b; Volkova and Bachrach, 2015) and socioeconomic class (Miranda Filho
et al., 2014; Lampos et al., 2016). Apart from their importance in
computational social science, such methods are also useful in downstream
applications such as targeted advertising and online political campaigning.

Following the hypothesis that language is indicative of the social status of a
person (Bernstein, 1960, 2003; Labov, 2006), previous research analysed user
generated written content to derive text based features such as bag-of-words
or clusters of words. These features are used to train predictive models for
inferring socioeconomic attributes (Li et al., 2014; Preoţiuc-Pietro et al.,
2015a). Despite the fact that these methods have proved to perform well, they
have not considered any relations and interactions between users. Moreover,
there is a large proportion of inactive users that do not produce any content.
For example, previous studies have shown that only around two thirds of the
users are active (i.e. posted at least twice) on Twitter (Huberman et al.,
2008; Liu et al., 2014). This makes it impossible to solely utilise language
based models to infer socioeconomic or other characteristics of inactive
users.

A different approach to the problem is to include information from the social
network structure. Socioeconomic status can be indicated by looking into the
range and the composition of the social network of a person (Campbell et al.,
1986). That is because people who belong to the same social circles often
share common characteristics. This is known as social network homophily, i.e.
the inclination of people towards developing social ties with similar others
(Lazarsfeld et al., 1954; McPherson et al., 2001). Despite expected
differences to real life social networks, it has been shown that online social
networks, e.g. Facebook and Twitter, exhibit some levels of homophily
(Kosinski et al., 2013; Al Zamal et al., 2012). People that follow each other
on Twitter usually share common topical interests (Kwak et al., 2010; Weng et
al., 2010). Previous work utilised the social network structure to infer user
attributes such as gender and age, personality traits and sentiment (Tan et
al., 2011; Al Zamal et al., 2012; Staiano et al., 2012; Kosinski et al., 2013;
Perozzi and Skiena, 2015), but not any socioeconomic attributes.

In this paper, we focus on using social network information to infer user’s
occupational class and income. Following that direction, we explore two
hypotheses using data from Twitter: (1) a user’s social network is indicative
of their income and occupational class; and (2) the information from the
social network structure and textual information are complementary. To answer
these hypotheses, we extract information from a user’s social network and
encapsulate it in user graph embeddings (Perozzi et al., 2014; Tang et al.,
2015). Graph embeddings place Twitter users in a vector space where similar
users are likely to be close to each other. The user graph embeddings are
treated as features to train linear and non-linear supervised models for
predicting income and occupational class.

The major contributions of our paper are:

  * •

To the best of our knowledge, this work is the first to introduce neural graph
embeddings to predict income and occupational class on Twitter.

  * •

Our model can be used to infer complex socioeconomic characteristics of
inactive users, exploiting the fact that user graph embeddings do not rely on
any textual information.

  * •

We show that a user’s social network and written content, i.e. tweets, contain
complementary information. This is demonstrated by training models that
combine both feature sets. Our evaluation on two standard, publicly available
datasets of Twitter users that are labelled with occupational class and income
shows that they outperform models using solely language or solely network
information.

  * •

Our proposed model achieves state-of-the-art performance in these two datasets
of income and occupational class significantly outperforming models introduced
by Preoţiuc-Pietro et al. (2015a, b).

##  2\. User Neural Graph Embeddings

User neural graph embeddings are dense vector representations that position
similar users close together in a high-dimensional Euclidean space. Neural
embeddings are popular in natural language processing for learning vector
representations of words (Mikolov et al., 2013; Bengio et al., 2003).

The only inputs required to learn word embedding models are sequences of words
in documents and so the concept can be extended to network structured data
using random walks to create sequences of vertices. In our case, vertices
represent Twitter users and edges represent a follower/followee relationship,
and we treat edges as if they were undirected. This is justified because
Twitter is predominantly an interest graph (Gupta et al., 2013) and a large
body of research has shown that the homophily principle applies to users who
express similar interests in social networks (Chamberlain et al., 2017;
Kosinski et al., 2013; Tan et al., 2011). By treating the graph as undirected
we ensure that all users that follow a common account (indicative of an
interest) have a maximum path distance of two. Vertices are embedded by
treating them exactly analogously to words in the text formulation of the
model (Perozzi et al., 2014). Extensions varying the nature of the random
walks have been explored in LINE (Tang et al., 2015) and Node2vec (Grover and
Leskovec, 2016). The main justification for this idea is that social networks
are a form of noisy measurement of a true underlying network. Random walks
have been shown extensively to mitigate for false edges and infer the presence
of missing ones (Page et al., 1999).

###  2.1. Generating User Sequences and Contexts

Figure 1. The Skipgram model uses two vector representations $\mathbf{W}$ and
$\mathbf{W^{\prime}}$ to predict the context vertices from a single input
vertex.

Given a network of users connected with unweighted edges, random walks are
generated by repeatedly sampling an integer uniformly from {1,2,…, $D_{v}$}
where $D_{v}$ is the vertex degree and moving to a new vertex. Concretely, for
a random walk starting at vertex $v_{0}$ we would sample $x\sim
U(\\{1,2,\dots,D_{v_{0}}\\})$ where U is a uniform distribution and
$D_{v_{0}}$ is the degree of $v_{0}$. If $x=1$ we move to the lowest indexed
neighbour of $v_{0}$, append that vertex to the random walk and repeat the
process at vertex $v_{1}$.

###  2.2. The Skipgram Model on User Sequences

After we have sampled user sequences with random walks, we can use them to
train user embeddings. There are several related embedding models, i.e.
SkipGram and Continuous Bag of Words (Mikolov et al., 2013). Here we adopt the
SkipGram with Negative Sampling (SGNS) model that is depicted in Figure 1. The
figure shows a shallow neural network with a single hidden layer and two
separate vector representations labelled as W and W’. The input to SGNS is a
sequence of users, which are mapped to (input, context) pairs by sliding a
context window over the input sequences. The input user representation is in W
and the neighbouring (i.e. context) users share a representation in W’. Users
are initially randomly allocated within the two vector spaces and then the
model is trained using Stochastic Gradient Descent (SGD). The objective
function gives the probability of the context users given the input user,
which is modelled by a softmax. We optimise the negative log likelihood given
by

(1) $\displaystyle L$ $\displaystyle=-\log
p(w_{o,1},w_{o,2},w_{o,3},...,w_{o,C}|w_{I})$ (2)
$\displaystyle=-\log\prod_{c=1}^{C}\frac{\exp{\mathbf{v}^{{}^{\prime}T}_{c}}\mathbf{v}_{I}}{\sum_{j=1}^{V}\exp{\mathbf{v}^{{}^{\prime}T}_{j}\mathbf{v}_{I}}}$

where $w_{(.)}$ is a user and $\mathbf{v}_{(.)}$ and
$\mathbf{v}_{(.)}^{{}^{\prime}}$ are the input and output vector
representations of that user and $C$ is the context size, typically ten. In
practice, it is expensive to evaluate Equation (2) as the sum in the
denominator is over all of the users in the network. Instead we use negative
sampling, which is a form of Noise Contrastive Estimation (NCE) (Gutmann and
Hyvärinen, 2012), to estimate the function by only evaluating a small number
of negative samples in addition to the observed positive example. The gradient
descent update rules for a user pair $(w_{I},w_{O})$ with vector
representations $(\mathbf{v}_{I},\mathbf{v}^{\prime}_{O})$ are found by
applying the chain rule to Equation (2) and are given by

(3) $\displaystyle\mathbf{v}_{j}^{{}^{\prime}new}$
$\displaystyle=\begin{cases}\mathbf{v}_{j}^{{}^{\prime}old}-\eta(\sigma(\mathbf{v}_{j}^{{}^{\prime}T}\mathbf{v}_{I})-t_{j})\mathbf{v}_{i},&w_{j}\in\chi\\\
\mathbf{v}_{j}^{{}^{\prime}old},&\text{otherwise}\end{cases}$

where $\chi=\\{w_{O}\\}\cup W_{neg}$. For the output representation and

(4) $\displaystyle\mathbf{v}_{I}^{new}$
$\displaystyle=\mathbf{v}_{I}^{old}-\eta\sum_{j:w_{j}\in\chi}(\sigma(\mathbf{v}^{{}^{\prime}T}_{j}\mathbf{v}_{I})-t_{j})\mathbf{v}^{\prime}_{j}$

for the input representation. In these equations $t_{j}$ is an indicator
variable that is one if and only if $w_{j}=w_{O}$ and zero otherwise, $\eta$
is the SGD learning rate and $W_{\rm neg}$ is the set of negatively sampled
users. We follow (Mikolov et al., 2013) and draw $W_{\rm neg}$ from the
distribution of users in the random walks raised to the power of
$\frac{3}{4}$.

##  3\. Experimental Setup

C | Title | U
---|---|---
1 | Managers, Directors and Senior Officials | 461
2 | Professional Occupations | 1615
3 | Associate Profess. and Technical Occupations | 950
4 | Administrative and Secretarial Occupations | 168
5 | Skilled Trades Occupations | 782
6 | Caring, Leisure and Other Service Occup. | 270
7 | Sales and Customer Service Occupations | 56
8 | Process, Plant and Machine Operatives | 192
9 | Elementary Occupations | 131
Total | 4625
Table 1. Distribution of users (U) across occupational classes (C). Figure 2.
Distribution of users and income. Income is calculated in British pounds (£).

###  3.1. Data

We experiment using two publicly available datasets that contain Twitter users
mapped to their occupational class and income (Preoţiuc-Pietro et al., 2015b,
a). The datasets contain the same group of 5,191 users in total. However, some
of the accounts are not considered in our experiments since we were not able
to extract their social network information. These accounts may have been
deleted or become private since the release of the datasets. Therefore, we
report results on a subset of the original set of users, i.e. 4,625 users,
that are still publicly available.

#### Occupational Class

Users are mapped with an occupation using the Standard Occupation
Classification (SOC) taxonomy devised by the Office of National Statistics in
the UK, based on skill requirements. The SOC taxonomy has a hierarchical
structure with 9 major groups (e.g managers or elementary occupations). Users
in the dataset have been mapped to one of these major groups. Table 1 shows
the distribution of users across the nine occupational classes. The Pearson’s
correlation between the original distribution of users and our subset
distribution is $0.93$.

#### Income

The occupational class of users has further been used as a proxy to infer
their income from the Annual Survey of Hours and Earnings. The income
represents the mean yearly earnings for 2013 in British Pounds (GBP) for each
occupational class. Figure 2 shows the distributions of users and income in
the dataset. The mean user income in the original dataset is $32,509.74$,
while the mean of the subset we use in our experiments is $32,727.92$.

###  3.2. Implementation of the Graph Embeddings

To construct graph embeddings we downloaded the Twitter IDs of everyone
followed by the $4,625$ accounts. This produced a set of $3,925,702$ users in
total. We considered only accounts followed by at least 10 users, which
reduced the number of the unique accounts to $53,199$. To produce sequences of
users we treat the edges of the Twitter graph as undirected and take 80 step
random walks initiated at each vertex in this network.

The dimensionality of the embedding affects the performance in predictive
tasks. We experimented with dimensionalities of $16,32,64$ and $128$ and chose
the optimal value following a nested 10-fold cross-validation approach as in
Preoţiuc-Pietro et al. (2015a, b). We found that the best performing
embedding111During initial experimentation, we noticed that varying the length
of the random walk between 40 and 100 did not substantially affect the quality
of the embeddings. dimensionality is 32. The user embeddings and the code to
generate them are available to download from
https://github.com/melifluos/income-prediction.

###  3.3. Predictive Models

#### Occupational Class

Predicting the occupational class of a user is defined as a 9-way
classification task. Given a user feature representation, our goal is to
assign the most probable class label. For that purpose, we use the graph
embeddings as features and a concatenation of the graph embeddings with the
topics introduced in (Preoţiuc-Pietro et al., 2015a) to train Logistic
Regression (LR) (Zou and Hastie, 2005), Support Vector Machines (SVM)
(Joachims, 1998) and Gaussian Process Classifiers (GPC) (Williams and
Rasmussen, 1996). All of the classifiers222The Gaussian Process models are
trained using GPy (http://github.com/SheffieldML/GPy). All the other models
are trained using Scikit-learn (http://scikit-learn.org/). are trained
following the one-vs-all approach333All the hyperparameters of the baseline
predictive models using _Topics_ as features are identical to the models
presented in (Preoţiuc-Pietro et al., 2015a, b). We tune the hyperparameters
of our proposed models (_Graph_ and _Topics+Graph_) performing a nested
10-fold cross-validation, identical to the data splits used in previous work..

#### Income

Inferring income is defined as a regression task. Given the user feature
representation as input, we try to predict a real value representing the
user’s income. The goal is to minimise the absolute error between the actual
and inferred income. We also compare three popular models: (1) linear
regression (LR), (2) Support Vector Regression (SVR) (Drucker et al., 1997),
and (3) Gaussian Process Regression (GPR) (Williams and Rasmussen, 1996).

##  4\. Results and Discussion

Tables 2 and 3 show the results obtained by the proposed models using the
graph embeddings (_Graph_) and their combination (_Graph+Topics_) as feature
representations for users. Note that models using _Topics_ (i.e. word
frequency of user’s tweets in a set of 200 precomputed word clusters) and
_Temporal Orientation_ as features are the baseline methods presented in
Preoţiuc-Pietro et al. (2015a, b) and Hasanuzzaman et al.
(2017).444Replicating the method of Hasanuzzaman et al. (2017) was not
possible, hence we report results only for income from their paper. To compare
against Preoţiuc-Pietro et al. (2015b, a), we retrain these models using the
user accounts in the dataset that are publicly available (see Subsection 3.1).

Occupational Class
---
Method | Accuracy (%)
Majority Class | 35.00
Preoţiuc-Pietro et al. (2015a)
LR-Topics | 46.57
SVM-Topics | 49.47
GP-Topics | 49.64
Ours
LR-Graph | 46.24
SVM-Graph | 50.14
GP-Graph | 50.44
LR-Graph+Topics | 48.84
SVM-Graph+Topics | 52.00†
GP-Graph+Topics | 51.46†
Table 2. Accuracy of models in predicting user occupational class.$\dagger$
denotes statistical significant different (t-test, $p<0.01$) method to _GP-
Topics_.

Our best performing model using graph based features (_Graph_) achieves an
accuracy of $50.44\%$ in the occupational classification task. In income
prediction, the MAE is $9,048$ and Pearson’s correlation is $0.63$. This
implies that graph embeddings carry meaningful information about user’s
socioeconomic attributes making them an effective user representation. The
graph embedding features perform consistently better than the textual features
(_Topics_) for the majority of the predictive models on both tasks except for
the LR model. This confirms our first hypothesis that information from the
network structure of a user is indicative of socioeconomic attributes. Figure
3 shows a 2-d t-SNE plot of the best performing user embedding, where we
observe many distinct “communities” of low and high income users that appear
together. This confirms our assumption about the homophilic nature of the
Twitter network.

The combination of user embeddings and topics (_Graph+Topics_) outperforms
either feature set used individually. More specifically, our _GPC-
Graph+Topics_ model significantly outperforms (t-test, $p<0.01$) the previous
state-of-the-art method, _GPC-Topics_ introduced in Preoţiuc-Pietro et al.
(2015a) on occupational classification. Moreover, our _SVR-Graph+Topics_ model
significantly outperforms ($p<0.001$) the best baseline method, i.e.
_SVM+Topics_. This confirms our second hypothesis that network structure and
linguistic information are complementary.

The above findings shed light on the homophilic behaviour of users on Twitter.
That might have further implications on user behaviour when selecting friends
and forming social networks online. Our results suggest that a stronger bias
might exist towards selecting friends with common socioeconomic backgrounds in
contrast to common topics of interest and that needs to be explored further.

Non-linear models (i.e. SVM, SVR, GPC, GPR) achieve better results in
inferring user socioeconomic attributes than linear (LR) models. While in the
occupational classification task our best performing model is GPC, the best
model in income inference is the SVR instead of GPR. This implies that model
selection is important in these tasks.

An analysis of the errors in the occupational classification task shows that
most misclassifications come from adjacent classes. For example, users in
classes 1, 3 and 4 are mistakenly classified as class 2. This happens because
adjacent classes contain related occupations. However, we notice less
dispersion of errors caused by other classes misclassified as class 2 when we
use graph embeddings and the combination of graph embeddings and topics. This
might be explained by the homophily of users’ networks captured by graph
embeddings.

Income
---
Method | MAE (£) | $\mathbf{\rho}$
Preoţiuc-Pietro et al. (2015b)
LR-Topics | 10573 | .50
SVR-Topics | 9528 | .59
GPR-Topics | 9883 | .60
Hasanuzzaman et al. (2017)
LR-Temporal Or. | 10850 | .45
GP-Temporal Or. | 10235 | .51
Ours
LR-Graph | 10811 | .50
SVM-Graph | 9048‡ | .62
GP-Graph | 9532 | .63
LR-Graph+Topics | 10326 | .54
SVM-Graph+Topics | 9072‡ | .64
GP-Graph+Topics | 9488 | .64
Table 3. Mean Absolute Error (MAE) and Pearson’s correlation coefficient
($\rho$) between actual and predicted income. ${\ddagger}$ denotes statistical
significant different (t-test, $p<0.001$) method to _SVM-Topics_. Figure 3. A
2-d t-SNE plot of the best performing user embedding (32D). Black and white
represent users with above and below median income respectively.

##  5\. Conclusions

We presented a method to reliably predict user occupational class and income
on Twitter. Information from a user’s social network is represented by graph
embeddings (Perozzi et al., 2014; Tang et al., 2015) and is used to train
predictive models. To the best of our knowledge, this work is the first to
introduce graph embeddings for automatically inferring socioeconomic
characteristics. We also demonstrated that the information extracted from the
user’s social network and their language use are complementary. That
combination significantly improves predictive performance. Finally, our
proposed models achieve state-of-the-art results in two standard datasets of
income and occupational class, significantly outperforming previous methods.

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