11institutetext:  Department of Earth Sciences, University of Western Ontario,
London, Ontario, N6A 5B7, Canada 22institutetext: Department of Physics and
Astronomy, University of Western Ontario, London, Ontario, N6A 3K7, Canada
33institutetext: Centre for Planetary Science and Exploration, University of
Western Ontario, London, Ontario, N6A 5B8, Canada

# A new method for measuring the meteor mass index:
application to the 2018 Draconid meteor shower outburst

D. Vida , 11 2 2<EMAIL_ADDRESS>M. Campbell-Brown 22 3 3 P. G. Brown 22 3 3 A.
Egal 22 M. J. Mazur 11 2 2

(Received 13 December 2019 / Accepted 6 February 2020)

###### Abstract

Context. Several authors predicted an outburst of the Draconid meteor shower
in 2018, but with an uncertain level of activity.

Aims. Optical meteor observations were used to derive the population and mass
indices, flux, and radiant positions of Draconid meteors.

Methods. 90 minutes of multi-station observations after the predicted peak of
activity were performed using highly sensitive Electron Multiplying Charge
Coupled Device (EMCCD) cameras. The data calibration is discussed in detail. A
novel maximum likelihood estimation method of computing the population and
mass index with robust error estimation was developed. We apply the method to
observed Draconids and use the values to derive the flux. Meteor trajectories
are computed and compared to predicted radiant positions from meteoroid
ejection models.

Results. We found that the mass index was $1.74\pm 0.18$ in the 30 minute bin
after the predicted peak, and $2.32\pm 0.27$ in the next 60 minutes. The
location and the dispersion of the radiant matches well to modeled values, but
there is an offset of $None$ in solar longitude.

###### Key Words.:

meteors, meteoroids – comets: individual: 21P/Giacobini-Zinner

##  1 Introduction

The Draconids are an annual meteor shower whose parent body is the Jupiter-
family comet 21P/Giacobini-Zinner. The shower usually has a very low activity
with a Zenithal Hourly Rate (ZHR) of $\sim 1$ (Jenniskens, 2006). It produced
large meteor storms in 1933 and 1946 and strong outbursts in a number of other
years (Egal et al., 2019). Many of these outbursts were not predicted
beforehand. The shower’s occasionally high intensity and unpredictability have
made it a focus of research, particularly as it can pose a significant impact
risk to spacecraft in the near-Earth environment (Beech et al., 1995; Cooke &
Moser, 2010; Egal et al., 2018).

In recent years, the shower has produced several notable outbursts. In 2011 an
outburst was predicted in advance (Watanabe & Sato, 2008; Maslov, 2011;
Vaubaillon et al., 2011) and well observed by both radar (Kero et al., 2012;
Ye et al., 2013a) and optical methods (Trigo-Rodríguez et al., 2013; Borovička
et al., 2014; Koten et al., 2014; Šegon et al., 2014; Vaubaillon et al.,
2015). That year the outburst reached a ZHR of 350, and had an average mass
index of $2.0\pm 0.1$, but which varied between 1.84 to 2.30 in one hour
during the peak (Koten et al., 2014).

In contrast, the 2012 outburst was not predicted and was only well observed by
the Canadian Meteor Orbit Radar (CMOR) (Brown & Ye, 2012). The shower produced
a meteor storm at radar sizes (ZHR $\approx 9000\pm 1000$), but visual
observers reported a ZHR almost 2 orders of magnitude lower, ZHR $\sim 200$,
suggesting a high mass distribution index, i.e. that the stream was rich in
small meteoroids. Unfortunately, due to the bad timing of the peak (maximum
over central Asia) and unfavorable weather conditions elsewhere, no optical
orbits associated with the outburst were secured. The 2012 outburst was also
peculiar in that modeling suggested a very high ($>$ 100 m/s) meteoroid
ejection velocity from the parent comet was needed (Ye et al., 2013b).

The 2018 outburst was predicted by various authors (Kresák, 1993; Maslov,
2011; Ye et al., 2013b; Kastinen & Kero, 2017; Egal et al., 2018), but the
predicted activity varied from weak (ZHR 10 - 20) (Maslov, 2011) to possible
meteor storm levels (Kastinen & Kero, 2017). The most recent work by Egal et
al. (2018), which reproduced well most historic Draconids activity, predicted
a peak ZHR of $\sim 80$ at 00:00 UTC on October 9, 2018.

In this work we analyze 1.5 hours of optical observations from Southwestern
Ontario just after the peak111According to visual observations in the IMO
database: https://www.imo.net/members/imo_live_shower?shower=DRA&year=2018 of
the 2018 outburst, from 00:00 UTC to 01:30 UTC. We also develop a novel method
of population and mass index estimation, and compute these indices using our
observations. Finally, we compare model predicted radiants with our multi-
station observations and compute the shower flux.

##  2 Instruments and observations

Optical observations were conducted using low-light level video systems in
Southwestern Ontario, Canada operated by the University of Western Ontario’s
Meteor Physics Group. The observations started shortly after local sunset on
October 8/9 at around 00:00 UTC and ended at 01:30 UTC due to cloudy weather;
thus only a short period after the peak was observed.

The double station observations were made using four Electron Multiplying
Charge Coupled Device (EMCCD) Nüvü HNü 1024
cameras222http://www.nuvucameras.com/fr/files/2019/05/NUVUCAMERAS_HNu1024.pdf
with Nikkor $50\text{\,}\mathrm{mm}$ f/1.2 lenses. The systems were operated
at 32 frames per second, had a limiting stellar magnitude of $+10.0^{M}$, and
a field of view of 15$\times$15 degrees.

Despite clear skies in the region, none of the dozen Southern Ontario Meteor
Network (SOMN) all-sky cameras (Brown et al., 2010) observed a single Draconid
during all of Oct 8-9. These all-sky cameras have an effective limiting meteor
magnitude of $-2^{M}$, qualitatively suggestive of an absence of larger
meteoroids in the falling branch of the outburst. The EMCCD cameras did
observe many Draconid meteors, which we use in the following analysis. In this
section we describe the EMCCD hardware and give details of the data reduction
procedure.

###  2.1 EMCCD systems

The Nüvü HNü 1024 EMCCD cameras are the most recent addition to the Canadian
Automated Meteor Observatory (CAMO) systems (Weryk et al., 2013). These 16-bit
cameras use electron multiplying technology to increase the number of
electrons accumulated in the registers before they are amplified, digitized
and read out, which greatly increases the signal-to-noise ratio. In
combination with a fast $50\text{\,}\mathrm{mm}$ f/1.2 lens, the single frame
limiting magnitude at 32 frames per second for stars is around $+10.0^{M}$,
and $+7.5^{M}$ for meteors. The native resolution of the camera is $1024\times
1024$ pixels, but the frames are binned to $512\times 512$ to make possible
the higher frame rate, which provides for better trajectory precision. The
field of view is $$$\times$$$, and the pixel scale is 1.7 arcmin/pixel. All
camera were run at a gain setting of 200. These systems are more sensitive and
have less noise than the previous generation of systems which relied on Gen
III image intensifiers (Campbell-Brown, 2015). The cameras each have external
GPS timing directly encoded at the frame acquisition stage. Figure 1 shows a
sample meteor captured with the EMCCD camera and Figure 2 shows the light
curve of the same meteor. Note the agreement between the sites in range-
corrected magnitudes (normalized to $100\text{\,}\mathrm{km}$). In Appendix A
we give more examples of multi-station Draconid light curves observed by the
EMCCD systems on Oct 9, 2018.

Two cameras are deployed at the each of the two CAMO sites in Southwestern
Ontario. Cameras 01F and 01G are located at Tavistock ($None$ N, $None$ W,
$329\text{\,}\mathrm{m}$), and 02F and 02G at Elginfield ($None$ N, $None$ W,
$324\text{\,}\mathrm{m}$). The distance between the stations is about
$45\text{\,}\mathrm{km}$. Both pairs are pointed roughly north, one pair (the
G cameras) at the elevation of 45 degrees, the other (F cameras) at 65
degrees. There was no volume overlap between the F and G cameras. One camera
from each site had the Draconid radiant inside the field of view. The EMCCDs
captured a total of 92 double station Draconids ranging in peak magnitudes
from $+1^{M}$ to $+6.5^{M}$, of which only 68 had complete light curves from
at least one camera.

Figure 1: Stack of raw frames showing a $+1^{M}$ Draconid recorded with camera
01G on October 9, 2018 at 00:01:48 UTC. The meteor was first detected at an
apparent magnitude of $+7.0^{M}$. V-band magnitudes of a selection of stars
are shown as well. The vertical gradient of the image background is intrinsic
to this specific camera and is compensated for by flat field correction during
data reduction. The two thinner and fainter lines left of the meteor are
satellites. Figure 2: Absolute GAIA G-band magnitude light curve of the meteor
in Figure 1. The error bars are $\sim 0.1$ magnitude. Observations from camera
01G (Tavistock) are shown in blue, and observations from camera 02G
(Elginfield) are shown in red.

###  2.2 Data reduction and calibration

Using the tools available in the open-source RMS (Raspberry pi Meteor Station)
library333RMS GitHub web page: https://github.com/CroatianMeteorNetwork/RMS,
automated meteor detection (described in detail in Vida et al., 2016) was run
on all collected data and both the astrometric and photometric calibration was
performed manually. The automated detections were manually classified and only
Draconids were extracted for further analysis.

The astrometric and photometric calibrations were done using the Gaia DR2 star
catalog (Gaia Collaboration et al., 2018) as the GAIA G spectral band matches
the spectral response of the EMCCDs well. We applied flat fields and bias
frames to the video data using the method described in Berry & Burnell (2000).
The average residuals of the astrometric calibration were $\sim 1/10$ pixels
or $\sim 10$ arc seconds, and the accuracy of the photometric calibration was
$\pm 0.1$ magnitude. Figure 3 shows the photometric calibration, and Table 1
details the astrometric calibration for the 01F EMCCD camera.

Figure 3: Photometric calibration of the 01F camera. The uncalibrated magnitude is simply $-2.5\log_{10}I_{s}$, where $I_{s}$ is the background-subtracted sum of pixel intensities of all star pixels (LSP in the figure). The sensor response was linear (Gamma = 1.0). The magnitudes of stars used for calibration ranged from $+2.3^{M}$ to $9.9^{M}$. Table 1: Example astrometric calibration for the 01F EMCCD camera. The estimated field of view was $None$ $\times$ $None$. The average error was 0.08 px and 0.15 arc minutes. Img X and Y are star image coordinates on the chip, RA and Dec are equatorial coordinates in J2000, Mag is the GAIA G band magnitude, -2.5*LSP is the uncalibrated magnitude (LSP is the logarithm of the sum of pixel intensities), Cat X and Y are predicted catalog star positions in image coordinates, error columns are fit residuals in arc minutes and pixel, and $\theta$ is the angle of a vector pointing from the predicted to the measured star position. The photometric offset was $+16.80\pm 0.11$, but some stars used for photometry were not used for astrometry due to their smaller signal to noise ratio. No | Img X | Img Y | RA | Dec | Mag | -2.5*LSP | Cat X | Cat Y | Error | Error | Error $\theta$
---|---|---|---|---|---|---|---|---|---|---|---
| (px) | (px) | (deg) | (deg) |  |  | (px) | (px) | (arcmin) | (px) | (deg)
1 | 234.34 | 163.91 | 267.193 | +48.836 | +6.91 | -10.02 | 234.50 | 163.97 | 0.29 | 0.17 | +21.8
2 | 370.26 | 15.08 | 272.091 | +44.113 | +6.72 | -9.95 | 370.31 | 15.06 | 0.10 | 0.06 | -21.2
3 | 417.60 | 200.09 | 275.386 | +49.122 | +4.10 | -12.62 | 417.56 | 200.10 | 0.16 | 0.04 | +158.7
4 | 414.03 | 481.25 | 278.144 | +57.046 | +4.56 | -12.37 | 414.02 | 481.28 | 0.02 | 0.03 | +101.4
5 | 187.10 | 62.08 | 264.866 | +46.006 | +3.70 | -13.03 | 187.16 | 61.96 | 0.22 | 0.13 | -64.1
6 | 407.94 | 59.93 | 273.885 | +45.209 | +6.13 | -10.53 | 407.93 | 59.93 | 0.11 | 0.02 | +178.2
7 | 123.98 | 472.02 | 262.682 | +57.877 | +5.72 | -11.05 | 124.05 | 472.06 | 0.12 | 0.08 | +31.2
8 | 100.16 | 294.76 | 261.350 | +52.790 | +6.30 | -10.57 | 100.17 | 294.77 | 0.02 | 0.01 | +26.3
9 | 326.99 | 244.11 | 271.723 | +50.823 | +6.02 | -10.87 | 326.98 | 244.12 | 0.10 | 0.02 | +123.6
10 | 393.84 | 274.09 | 274.983 | +51.348 | +5.99 | -10.82 | 393.82 | 274.09 | 0.12 | 0.02 | +166.3
11 | 130.40 | 107.98 | 262.570 | +47.403 | +7.01 | -9.69 | 130.46 | 108.11 | 0.28 | 0.15 | +64.0
12 | 262.78 | 327.09 | 269.245 | +53.444 | +8.25 | -8.48 | 262.84 | 327.19 | 0.34 | 0.11 | +56.8
13 | 185.98 | 263.89 | 265.341 | +51.818 | +5.70 | -11.20 | 185.95 | 263.97 | 0.09 | 0.08 | +113.6
14 | 344.09 | 278.87 | 272.756 | +51.740 | +6.99 | -9.94 | 344.17 | 278.95 | 0.31 | 0.11 | +45.8
15 | 260.64 | 89.92 | 268.004 | +46.643 | +6.08 | -10.68 | 260.43 | 89.93 | 0.35 | 0.21 | +176.2
16 | 372.69 | 231.22 | 273.677 | +50.244 | +7.85 | -8.89 | 372.67 | 231.23 | 0.12 | 0.03 | +157.8
17 | 92.56 | 381.21 | 260.991 | +55.279 | +7.69 | -9.06 | 92.56 | 381.26 | 0.05 | 0.05 | +80.9
18 | 181.62 | 192.91 | 264.950 | +49.780 | +6.58 | -10.32 | 181.58 | 192.95 | 0.11 | 0.07 | +136.5
19 | 289.11 | 467.85 | 271.500 | +57.360 | +6.58 | -10.23 | 289.12 | 467.85 | 0.14 | 0.00 | +44.8
20 | 89.87 | 441.60 | 260.844 | +57.012 | +6.46 | -10.40 | 89.85 | 441.63 | 0.02 | 0.04 | +118.2
21 | 452.50 | 314.07 | 278.047 | +52.116 | +6.55 | -10.29 | 452.48 | 314.18 | 0.23 | 0.11 | +105.6
22 | 485.15 | 102.90 | 277.345 | +45.982 | +7.04 | -9.62 | 485.25 | 102.86 | 0.19 | 0.10 | -21.0
23 | 491.59 | 327.00 | 279.970 | +52.196 | +5.97 | -10.84 | 491.55 | 327.01 | 0.09 | 0.04 | +167.3
24 | 21.27 | 282.99 | 257.628 | +52.409 | +6.27 | -10.47 | 21.23 | 282.97 | 0.11 | 0.05 | -153.8
25 | 23.80 | 190.04 | 257.918 | +49.746 | +6.14 | -10.69 | 23.79 | 190.07 | 0.21 | 0.03 | +105.6
26 | 73.40 | 134.86 | 260.142 | +48.189 | +6.23 | -10.48 | 73.47 | 134.81 | 0.12 | 0.09 | -36.6
27 | 68.41 | 34.63 | 259.962 | +45.309 | +6.57 | -10.10 | 68.39 | 34.51 | 0.29 | 0.13 | -96.3
28 | 195.19 | 333.37 | 265.997 | +53.802 | +5.72 | -11.28 | 195.18 | 333.44 | 0.18 | 0.07 | +94.7
29 | 232.94 | 443.85 | 268.383 | +56.873 | +3.23 | -13.47 | 232.89 | 443.75 | 0.29 | 0.11 | -115.5
30 | 165.82 | 150.51 | 264.157 | +48.586 | +4.97 | -11.89 | 165.81 | 150.62 | 0.11 | 0.11 | +100.3
31 | 244.07 | 55.53 | 267.197 | +45.700 | +6.11 | -10.51 | 244.16 | 55.46 | 0.20 | 0.12 | -39.3
32 | 327.33 | 365.79 | 272.633 | +54.288 | +5.71 | -11.26 | 327.35 | 365.72 | 0.06 | 0.08 | -74.2
33 | 491.40 | 424.76 | 281.231 | +54.897 | +6.05 | -10.87 | 491.48 | 424.57 | 0.30 | 0.21 | -67.5
34 | 380.38 | 194.53 | 273.734 | +49.159 | +6.45 | -10.33 | 380.49 | 194.43 | 0.22 | 0.15 | -42.7
35 | 71.79 | 66.96 | 260.088 | +46.241 | +4.78 | -11.94 | 71.72 | 66.98 | 0.14 | 0.07 | +161.1
36 | 126.96 | 278.26 | 262.608 | +52.301 | +2.38 | -14.11 | 127.01 | 278.21 | 0.15 | 0.08 | -41.6
37 | 163.55 | 309.90 | 264.399 | +53.172 | +9.01 | -7.92 | 163.53 | 309.75 | 0.27 | 0.15 | -96.3
38 | 280.97 | 220.65 | 269.504 | +50.324 | +8.21 | -8.69 | 280.87 | 220.56 | 0.29 | 0.14 | -140.6
39 | 449.31 | 388.20 | 278.748 | +54.209 | +7.64 | -9.17 | 449.28 | 388.35 | 0.23 | 0.15 | +103.5
40 | 164.16 | 82.80 | 263.945 | +46.642 | +8.17 | -8.75 | 163.97 | 82.90 | 0.37 | 0.21 | +151.7
41 | 323.33 | 411.60 | 272.809 | +55.611 | +9.45 | -7.31 | 323.34 | 411.65 | 0.05 | 0.05 | +74.8
42 | 35.30 | 30.66 | 258.624 | +45.187 | +6.30 | -10.01 | 35.35 | 30.71 | 0.12 | 0.08 | +49.1
43 | 15.62 | 488.13 | 256.793 | +58.269 | +6.90 | -9.83 | 15.60 | 488.13 | 0.26 | 0.03 | +177.5
44 | 13.33 | 400.72 | 256.952 | +55.768 | +6.53 | -10.17 | 13.33 | 400.68 | 0.01 | 0.04 | -85.5
45 | 416.91 | 103.96 | 274.575 | +46.407 | +6.74 | -9.89 | 416.92 | 103.94 | 0.10 | 0.02 | -70.7
46 | 351.89 | 364.77 | 273.815 | +54.140 | +6.56 | -10.24 | 351.84 | 364.75 | 0.09 | 0.05 | -158.5
47 | 494.27 | 388.18 | 280.871 | +53.872 | +6.15 | -10.70 | 494.26 | 388.23 | 0.13 | 0.06 | +104.8
48 | 468.06 | 25.16 | 276.000 | +43.908 | +5.72 | -10.77 | 467.98 | 25.20 | 0.13 | 0.08 | +155.9
49 | 491.94 | 42.70 | 277.079 | +44.257 | +7.12 | -9.36 | 491.91 | 42.71 | 0.05 | 0.03 | +162.6
50 | 400.40 | 344.59 | 275.950 | +53.301 | +6.33 | -10.58 | 400.40 | 344.59 | 0.05 | 0.01 | +45.8
51 | 81.12 | 316.66 | 260.439 | +53.420 | +5.11 | -11.57 | 81.16 | 316.66 | 0.08 | 0.04 | -3.3
52 | 312.67 | 28.33 | 269.867 | +44.718 | +7.79 | -8.89 | 312.73 | 28.39 | 0.16 | 0.08 | +50.5
53 | 168.60 | 438.58 | 264.986 | +56.870 | +6.99 | -9.79 | 168.59 | 438.61 | 0.05 | 0.04 | +101.5
54 | 468.69 | 238.67 | 277.966 | +49.891 | +7.51 | -9.26 | 468.69 | 238.60 | 0.01 | 0.06 | -94.0
55 | 260.70 | 369.63 | 269.380 | +54.665 | +6.80 | -10.01 | 260.67 | 369.60 | 0.07 | 0.04 | -135.3

##  3 Maximum Likelihood Estimation (MLE) method of computing population and
mass indices

Population and mass indices are essential for computing meteor shower fluxes
and investigating their evolutionary history. It is usually assumed that the
cumulative number of meteor peak magnitudes and the cumulative logarithm of
the number of meteoroids above a certain mass threshold follow single power-
law distributions.

Brown & Rendtel (1996) define the population index $r$ as:

$r=\frac{N(M_{v}+1)}{N(M_{v})}$ (1)

where $N(M_{v})$ is the number of meteors with peak magnitude $M\leq M_{v}$.

Assuming that $dN$ is the number of meteoroids with a range of masses between
$m$ and $m+dm$, McKinley (1961) defines the mass index $s$ as:

$dN\propto m^{-s}dm$ (2)

The population and the mass index are the slopes of the best fit lines on a
log-log (magnitude or logarithm of mass vs. logarithm of the number of
meteors) cumulative distribution of magnitudes or masses. These lines can be
simply parameterized as:

$f(x)=10^{ax+b}$ (3)

where $x$ is either the magnitude (in case of the population index) or the
logarithm of the mass (mass index). Note that we assume that the cumulative
distribution is normalized to the $[0,1]$ range.

Because all observations systems have some limiting sensitivity, the number of
observed meteors is finite and detection efficiency drops off rapidly at
magnitudes close to the limiting sensitivity. Thus there is only one part of
the parameter space in which the power law assumption is valid. As a result,
operationally a power law is only fit up to some limiting sensitivity
$x_{min}$. A similar approach is used in other fields as well, although ways
of estimating the value $x_{min}$ differ (Corral & González, 2019). Pokornỳ &
Brown (2016) suggested fitting two parameters $x_{min},x_{max}$ which describe
the range of magnitudes, radar echo amplitudes (or equivalently masses) for
which the linear approximation is valid. They argue that the addition of the
upper boundary $x_{max}$ is necessary as small number statistics may skew the
power law at larger amplitudes/masses. They used MultiNest (Feroz et al.,
2009) to fit their model, an advanced Bayesian regression method which can
also perform robust error estimation.

In many studies (e.g. Brown et al., 2002), the meteoroid mass or population
index is found by constructing a cumulative histogram and performing linear
regression on binned data. This approach has many serious problems which are
detailed in Clauset et al. (2009), the most important of which is that the
result may change with histogram binning, especially when data sets with a
small number of meteors are used. Molau et al. (2014) proposed a more advanced
method which is capable of estimating the population index and the meteor flux
at the same time. The drawback of this method is that it requires a running
estimation of the meteor limiting magnitude from observations. It also is
constructed such that estimating the exact value of the population index is
rather subjective, and no details of the error analysis are provided.

###  3.1 Description of the new method

To estimate the population and the mass index robustly, in a statistical
manner and with a reliable error estimate, we use the maximum likelihood
estimation (MLE) method to fit a gamma distribution to the observed
distribution of magnitudes and/or masses. This approach was inspired by the
work of Clauset et al. (2009) where they question the validity of the power
law assumption in many real-world applications.

First, we tried to fit a Gumbel distribution to the mass distribution
following Blaauw et al. (2016), but we found that it always produced fits with
smaller p-values than the gamma distribution, under the null hypothesis the
data and model distributions are identical.

Our choice of the gamma distribution for our probability distribution has a
theoretical basis. If we assume that meteors appear at a rate $\lambda$ per
unit time, then the time between meteors follows an exponential distribution
with rate $\lambda$ while the total number of meteors in a given time span $t$
follows a Poisson distribution $P(\lambda t)$. A gamma distribution
$Gamma(n,\lambda)$ is the sum of independent exponentially distributed random
variables, and describes the length of time needed to observe a number of
meteors $n$ (Akkouchi, 2008; Leemis & McQueston, 2008).

By replacing the time in this analogy with magnitude or mass, the theory still
applies. Given a meteor rate per unit magnitude or mass (i.e. a population or
mass index), a gamma distribution describes the distribution of magnitudes or
masses one expects given a certain total number of meteors. In the special
case where the rate is a constant in our chosen time interval, the sum can be
simplified to an Erlang distribution. Because of historical observations that
show that the mass index is not constant even on short time scales for certain
showers (Koten et al., 2014), we use the more general gamma distribution.

We can show the above argument is correct quantitatively. Note that in the
paragraph above we implicitly assume an exponential distribution, not a power
law. The power law assumption for the distribution of meteoroid masses used in
meteor astronomy comes from the foundational theoretical paper by Dohnanyi
(1969) who assumed that when asteroids/meteoroids collide, the resulting
fragment mass distribution follows a power law (Clauset et al., 2009):

$p(x)=\frac{s-1}{x_{min}}\left(\frac{x}{x_{min}}\right)^{-s}$ (4)

where $s$ is the scaling parameter (differential mass index in our example)
and $x_{min}$ is the lower bound of the power law behavior. The power law
assumption continues to propagate through the analytical derivation of
Dohnanyi (1969), resulting in a population-wide differential mass distribution
index of $s=1.834$, assuming collisional steady-state conditions. Dohnanyi
(1969) justified the use of the power law by appealing to the experimental
work of Gault et al. (1963), who showed that the cumulative mass-size
distribution of fragmented rock from craters formed by hyper-velocity impacts
in the lab follows a power-law, a result confirmed in more recent experiments
and simulations (e.g. Kun & Herrmann, 1996).

In reality, the number of meteors naturally tapers off as the limiting system
magnitude is approached. This can be modeled as a power law distribution with
an exponential cutoff. Its probability density function $p(x)$ is given by
Clauset et al. (2009):

$\displaystyle p(x)=Cf(x)$ (5) $\displaystyle f(x)=x^{-s}e^{-\lambda x}$ (6)
$\displaystyle C=\frac{\lambda^{1-s}}{\Gamma(1-s)}$ (7)

where $\lambda$ is the decay rate, $f(x)$ the basic functional form, $C$ the
normalization constant such that $\int^{\infty}_{0}Cf(x)dx=1$, and $\Gamma$ is
the gamma function. If we define a shape parameter as $k=1-s$, it can easily
be shown that the resulting probability distribution is the gamma
distribution:

$p(x)=\frac{\lambda^{k}x^{k-1}e^{-\lambda x}}{\Gamma(k)}$ (8)

Figure 4 shows comparison of the cumulative density functions (CDF) of a
power-law and of a gamma distribution. Both distributions follow the same
trend at smaller values, but the exponential tapering overwhelms the power law
component in the gamma distribution at around $x\sim 10^{-1}$ in this example.
To estimate the power law scaling parameter from the gamma distribution, one
can compute the slope of the gamma CDF before the tapering becomes
significant. In theory, one may derive the slope directly as $s=1-k$, but in
practice (especially when one works on a small data set) we find that this
does not give robust estimates of population and mass indices. It appears that
the fit sometimes prefers larger values of $s$ which it compensates for by
using larger values of $\lambda$. For that reason, we develop a more robust
method described below.

Figure 4: Comparison of the cumulative density function for a power law and a
gamma function.

We use the gamma distribution MLE fit from the functions in the Python scipy
library444Gamma function, scipy library:
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gamma.html.We
refer readers to the documentation of the scipy library for details. The
Python source code of our new method can be found in the Western Meteor Python
Library555WMPL source code: https://github.com/wmpg/WesternMeteorPyLib under
wmpl/Analysis/FitPopulationAndMassIndex.py.

Figure 5 shows an example of the gamma distribution fit to meteor magnitude
data. After the fit is performed, the inflection point in the probability
density function (PDF) is found (the minimum of the first derivative of the
PDF). This inflection point is taken as the limiting sensitivity at which data
is complete; this means that the number of meteors has started decreasing at
this point purely due to sensitivity losses. In our approach, the inflection
point is used as an anchor; it indicates that the optimal position where the
slope of the population or mass index should be estimated is at brighter
magnitudes or larger masses from that point. We operationally define the
optimal reference point $x_{ref}$ for:

  * •

The population index, as one magnitude brighter than the inflection point.

  * •

The mass index, as 0.4 dex larger in mass than the inflection point. A 0.4 dex
in mass roughly corresponds to the difference of one meteor magnitude (Vida et
al., 2018).

Assuming an 8-bit camera is used to capture meteor data, it gives
$\log_{2.512}256\approx 6$ magnitudes of dynamic range. As examples below
show, the data is often not complete for the faintest three magnitudes,
leaving only three magnitudes before reaching the point of saturation. One
magnitude brighter than the completeness magnitude is a rough midpoint between
the point of completeness and sensor saturation, allowing for some leeway in
case the saturation point is shifted due different meteor angular speeds.

The mass index $s$ or the population index $r$ can then be computed from the
slope $a$ of the log survival function $S$ (i.e. complementary cumulative
distribution function) of the gamma distribution at the reference point
$x_{ref}$ as:

$\displaystyle a_{s}=10^{-\frac{d}{dx}\log S_{s}(x_{s,ref})}$ (9)
$\displaystyle s=1+a_{s}$ (10) $\displaystyle a_{r}=10^{-\frac{d}{dx}\log
S_{r}(x_{r,ref})}$ (11) $\displaystyle r=10^{a_{r}}$ (12)

where $S_{s}$ and $x_{s,ref}$ are fit to logarithms of mass, and $S_{r}$ and
$x_{r,ref}$ are fit to magnitudes. Given a generic cumulative density function
$F(x)$, the survival function $S(x)$ is simply:

$S(x)=1-F(x)$ (13)

and we compute the derivative in $a$ numerically. The intercept $b$ in
equation 3 can be computed as:

$b=\log S(x_{ref})+ax_{ref}$ (14)

The Kolmogorov-Smirnov test is used to compute the p-value which we use as a
measure of goodness of fit.

Next, we compute the effective meteor limiting magnitude/mass $l_{m}$ used in
flux estimation as a point where the line $f(x)=10^{ax+b}$ is equal to the
normalized cumulative value of 1, i.e. $F(x)=1$:

$l_{m}=-b/a$ (15)

We find that this approach is equivalent to the method of Blaauw et al. (2016)
who take the mode of the fitted Gumbel distribution as the effective limiting
magnitude. We suggest that estimating the limiting meteor magnitude directly
from meteor data is preferable to estimating it using stellar limiting
magnitude and then applying angular velocity corrections, as our approach
naturally includes all observational/instrumental biases.

Finally, the uncertainty of measured slopes is estimated using the bootstrap
method (Ivezić et al., 2014, page 140). The input data set is resampled with
replacement, with the sample size equal to the size of the input data set, and
the fit is repeated on the sample. The procedure is repeated 1000 times and
population or mass indices are computed for each run. Slopes that are outside
3 sigma of the mean are rejected, and the resulting standard deviation is
reported as the uncertainty.

###  3.2 Testing the method

Figure 5 shows the distribution of peak magnitudes of 2604 manually reduced
double station sporadic meteors captured by the Canadian Automated Meteor
Observatory’s (CAMO) influx camera (Weryk et al., 2013). The data set covers a
period between June 2009 and August 2014, and only those meteors which had
complete light curves (both the beginning and the end was visible inside the
field of view of at least one camera) were used in the analysis. We note that
imposing the visibility criterion was essential, as including meteors with
partially observed tracks produced a bimodal magnitude distribution which
greatly skewed the results. Although the CAMO influx camera is not used for
the analysis of the Draconids, we use this high quality data set used in
previous work (e.g. Campbell-Brown, 2015) to demonstrate the robustness of the
method. The system has a meteor limiting magnitude of $+5.5^{M}$, but using
our approach we find that the data set is complete only up to magnitude
$+3.2^{M}$, indicated by the inflection point, while the population index is
estimated at magnitude $+2.2^{M}$.

Figure 5: Distribution of peak R-magnitudes of sporadic meteors from the CAMO
influx camera (histogram) and the gamma distribution fit (blue line is the
main fit, black lines are all Monte Carlo runs). The jaggedness of the
histogram is due to the limited precision of the magnitudes in the data file.
The yellow shaded area around the inflection point represents the uncertainty
in the inflection point location, and the green shaded area the uncertainty
around the reference point. Figure 6: Cumulative distribution of peak
magnitudes of sporadic meteors captured by the CAMO influx camera. The red
line indicates the slope of the fitted population index.

Figure 6 shows the cumulative distribution of peak magnitudes and the line
representing the tangent to the log cumulative distribution function of the
fitted gamma function. The estimated slope of the line, i.e. the population
index, is $r=2.55\pm 0.06$, which is consistent with other studies which have
measured the expected yearly average sporadic population index from video
data, e.g. (Molau, 2015). Note that the estimated slope follows the data
histogram well. If the slope was to be estimated at the inflection point, the
population index would be $r=2.20$ and the slope would significantly deviate
from the data histogram.

Figures 7 and 8 show the distribution of masses and the cumulative histogram
of masses for the same CAMO data set. The computed mass index of $s=2.18\pm
0.05$ is very consistent with expected values from other authors (e.g. Blaauw
et al., 2011a). We note that Pokornỳ & Brown (2016) used the same data set,
but their algorithm fitted a line to the histogram on the part after the
inflection point, slightly reducing the value of the mass index to $s=2.09$.

Figure 7: Distribution of masses of sporadic meteors from the CAMO influx
camera (histogram) and the gamma distribution fit (blue line). Masses were
computed using a fixed dimensionless luminous efficiency $\tau=0.7\%$. Figure
8: Cumulative distribution of masses of sporadic meteors from the CAMO influx
camera. The red line indicates the slope of the fitted mass index, while the
shaded black lines represent the Monte Carlo fits of the resampled set as
described in the text.

Due to the large number of data points, the estimate of the mass index of
sporadic meteors can be accurately estimated, but often in practice, only tens
of shower meteors are available. To demonstrate the robustness of the method,
we estimate the mass index from 38 Geminids (all observed in 2012 within
$None$ of solar longitude around the peak) from the CAMO influx data set
(Figures 9 and 10). The fit is much more uncertain, but the estimated value of
the mass index for the Geminids is $s=1.70\pm 0.14$, which is consistent with
previous work (Jones & Morton, 1982; Arlt & Rendtel, 2006; Blaauw et al.,
2011b).

Figure 9: Distribution of masses of 38 Geminid meteors from the CAMO influx
camera and the gamma distribution fit. See Fig 5 caption for the explanation
of plot details. Figure 10: Cumulative distribution of masses of 38 Geminid
meteors from the CAMO influx camera. The line indicates the slope of the
fitted mass index. See Fig 5 caption for the explanation of plot details.

In the examples above, the p-values derived using the Kolmogorov-Smirnov test
are larger than 0.05, which means that the null hypothesis that the model fits
well to the data cannot be rejected with a $95\%$ confidence level in any of
the cases, even for the Geminids with a small number of meteors. This suggests
that our theoretical assumptions are valid.

We investigated how the population and the mass index of sporadic meteors
change throughout the year using the new method. The results are shown in
Figures 11 and 12. The data was divided into 10 bins of equal number of
meteors (around 260 in each bin) but of variable ranges of solar longitudes.
Large variations in both can be seen at these optical meteoroid sizes, larger
than the fairly stable values in radar sizes (Blaauw et al., 2011a; Pokornỳ &
Brown, 2016) but following the same trend. The variations and features in the
population index are comparable to those derived from visual data by Rendtel
(2004), except after $\lambda_{\astrosun}>$$$ where the number of CAMO
observations is very low due to bad weather conditions in winter and large
bins of solar longitude had to be used. In Figure 12, the mass index estimated
using our novel method is plotted as a solid line, and the mass index computed
from the population index using the classical relation $s=1+2.3\log r$
(McKinley, 1961) is shown as a dashed line. Comparing the two, it can be seen
that the peak magnitude alone is not a good proxy of the total meteoroid mass
nor the mass distribution for sporadics.

Figure 11: Annual variation in the population index for sporadic meteors using
data from the CAMO influx camera. Figure 12: Annual variation in the sporadic
mass index (solid line) as derived using the CAMO influx camera. The dashed
line is the mass index derived from the population index using the classical
relation $s=1+2.3\log r$.

Finally, we apply the method to EMCCD data collected on the night of August
13, 2018 ($\lambda_{\astrosun}=$$$). The meteors were detected using a hybrid
cluster - template-matching matched filter algorithm (Gural, 2007, 2016),
which detects even very faint meteors close to the noise floor. The method
will be described in a future paper. In total, 134 double-station Perseids
were observed with the two pairs of cameras during 6.5 hours of observation.
The meteor limiting magnitude was +5.0 for these fast Perseid meteors, and
only those meteors which had complete light curves by at least one camera were
used in the analysis. We measured the population index for the Perseids to be
$r=2.43\pm 0.29$, and the mass index $s=1.90\pm 0.12$, comparable to long-term
visual measurements at the same solar longitude (Rendtel, 2014). In the same
night, there were 283 sporadics whose population index was $r=3.56\pm 0.50$
and mass index $s=2.30\pm 0.14$, significantly higher than that of the
Persieds, as expected.

##  4 Mass and population index of the 2018 Draconids

We apply the new population and mass index estimation method to the Draconids
observed on Oct 8/9, 2018 using the EMCCD cameras. Meteor detection and
calibration was done using RMS software (Vida et al., 2016). Instead of
computing photometric masses, we computed the integrated luminous intensity
and used it as a proxy for mass.

Following Vida et al. (2018), the meteoroid mass $m$ can be computed as:

$\displaystyle I(t)=P_{0m}10^{-0.4M_{A}(t)}$ (16) $\displaystyle
m=\frac{2}{v^{2}\tau}\int_{0}^{t}I(t)$ (17)

where $I$ is the luminous intensity, $P_{0M}$ is the power of a zero magnitude
meteor, $M_{A}$ the range-corrected GAIA G-magnitude (normalized to
$100\text{\,}\mathrm{km}$), $\tau$ the luminous efficiency, and $v$ the
velocity of the meteor. $P_{0M}$ and the term before the integral simply scale
the mass, and as we are only interested in the slope of cumulative logarithms
of mass, the scaling has no influence on the slope if we assume that all
meteors have similar velocities. This is exactly the case, as all meteors are
members of the same shower. Thus, we drop these terms and compute the
dimensionless integrated luminous intensity $I^{*}$ as a proxy for mass:

$I^{*}=\int_{0}^{t}10^{-0.4M_{A}(t)}$ (18)

The two pairs of EMCCDs recorded a total of 68 double-station Draconids with
complete light curves, ranging in peak magnitudes from $+1^{M}$ to $+6.6^{M}$.
We divided the data into two time bins:

  1. 1.

Bin 1 - Is a $\sim 30$ minute bin from 00:03:33 UTC (beginning of observation)
to 00:32:01 UTC. This bin had a total of 30 Draconids and captures a part of
the predicted peak.

  2. 2.

Bin 2 - Is a $\sim 60$ minute bin from 00:32:01 UTC to 01:28:58 UTC (end of
observation). This bin had a total of 38 Draconids and captures the post-peak
declining activity.

###  4.1 Bin 1

The first bin close to the peak, produced a population index of $1.98\pm 0.50$
with an estimated completeness magnitude of $+4.30^{M}\pm 0.97$. Figure 13
shows that the observations roughly follow a power-law until the reference
point at $+3.30^{M}\pm 0.97$, after which there is a rapid drop-off in the
number of observed meteors. The model fit in this case does not seem to be
robust, as indicated by the uncertainty, despite the large p-value of 0.843.
The population index might have been underestimated due to the overabundance
of meteors with peak magnitudes of around $4^{M}$, possibly because of small
number statistics. Alternatively, the population index might have been
increasing rapidly, as evidenced by the measured value in the second bin; thus
the measured value is only the average of the population index function.
Because this would break the underlying assumption of a fixed power-law, this
population index, the associated effective meteor limiting magnitude of
$+5.42^{M}\pm 0.64$ and the flux derived from it are suspect.

Figure 13: Cumulative histogram of magnitudes using the Draconid data in the
first time bin.

Figure 14 shows the cumulative histogram of dimensionless integrated luminous
intensities and the mass index estimate of $1.74\pm 0.18$ with the value of
integrated intensity completeness being $-2.40\pm 0.39$. The fit appears to be
robust, and we note that the mass index estimate has a much lower uncertainty
than the population index estimate. This value of the mass compares well to
the value for the 2011 outburst at the peak (Koten et al., 2014).

Figure 14: Cumulative histogram of integrated luminous intensities using the
Draconid data in the first time bin.

###  4.2 Bin 2

For the data in the second bin, we measure a much larger and more uncertain
population index of $3.36\pm 1.31$ (Figure 15), and a mass index of $2.32\pm
0.26$ (Figure 16). The large difference from the values estimated for bin 1
indicate a fast change in the particle size distribution as the Earth moves
through the stream. This rapid change is the probable cause of the increased
uncertainty, and this 60 minute bin likely encompasses a range of meteoroid
size distributions. We note that the speed of the mass index increase is
similar to the 2011 Draconids, when the mass index changed from 1.84 to 2.30
in one hour (Koten et al., 2014).

Figure 15: Cumulative histogram of magnitudes using the Draconid data in the
second time bin. Figure 16: Cumulative histogram of integrated luminous
intensities using the data in the second time bin.

##  5 Flux

Collecting areas for the two pairs of EMCCD cameras were calculated using the
method of Campbell-Brown et al. (2016). A grid of points, spaced by 4 km, was
placed at heights between 92 and 98 km, to capture the typical heights of
Draconid meteors. At each height, the bias for each point in the grid was
calculated, taking into account the radial speed of Draconids at the time of
the observation, the sensitivities of the two cameras, and the range to each
camera. The area of the 4$\times$4 km region was scaled to account for meteors
not seen due to lower sensitivity, using the measured mass index of the
stream. An average of the areas at the heights of interest was taken, though
the collecting area of both systems was relatively constant at those heights.

The flux was calculated by dividing the number of two-station Draconids by the
overlapping collecting area of each camera pair. The same two time intervals
used in the mass index calculation were used for this calculation, and the
flux of the first (half-hour) bin multiplied by two to get the number of
meteoroids per hour. This calculation was performed for each pair of cameras
separately, and then the numbers and collecting areas were added and the flux
from both sets of cameras was calculated. The flux was then corrected, using
the $s$ value, from the system limiting magnitudes of +5.4 and +4.5 to a
limiting magnitude of +6.5, which is standard when, for example, calculating
Zenithal Hourly Rates. All of these numbers, including the ZHR, are given in
Table 2.

Table 2: Fluxes of Draconids, in meteoroids per square km, per hour per EMCCD camera pair (F and G). The first two rows show the measured mass index per time bin; the last two rows show the upper and lower bounds to within 1$\sigma$ of the mass index for bin 1, demonstrating the large sensitivity in flux and ZHR depending on the adopted mass index. The number of Draconids per camera pair are shown together with collecting areas per camera pair and flux. Bin | s | Num F | Num G | F collect. | G collect. | Flux F | Flux G | Flux to LM | Flux +6.5M | ZHR
---|---|---|---|---|---|---|---|---|---|---
|  |  |  | area (km2) | area (km2) |  |  |  |  |
Bin 1 | 1.74 | 14 | 13 | 328 | 687 | 0.085 | 0.038 | 0.053 | 0.11 | 586
Bin 2 | 2.32 | 22 | 29 | 214 | 420 | 0.10 | 0.071 | 0.082 | 0.99 | 681
Bin 1 | 1.56 | 14 | 13 | 373 | 784 | 0.075 | 0.033 | 0.047 | 0.081 | 1150
Bin 1 | 1.92 | 14 | 13 | 288 | 603 | 0.097 | 0.043 | 0.061 | 0.15 | 389

It is obvious that the collecting area of these systems is a strong function
of the mass index: if the population index is higher (more faint meteors) the
effective collecting area is smaller because many more meteors are expected to
be missed in the less sensitive parts of the collecting area. We expect the
flux in the second time bin to be lower than in the first, because the peak
occurred just before observations began, but the flux is nearly an order of
magnitude larger, and the Zenithal Hourly Rate slightly higher. We note that
both camera pairs were pointed close to the apparent radiant, which may skew
flux calculations.

To get a sense of the uncertainty in the flux which comes from the uncertainty
in the mass index, for the first time bin we recomputed the collecting area
with the mass index varying from 1.56 to 1.92, the uncertainty limits. At the
system’s limiting magnitude, this changed the flux from 0.047 meteoroids km-2
hr-1 (at the lowest mass index) to 0.061 (at the highest); extrapolating to a
limiting magnitude of +6.5 produced values from 0.081 to 0.15 meteoroids km-2
hr-1. The uncertainty in the ZHR is significantly higher, varying from 389 to
1150 for the two different mass indices. There is some uncertainty in the
limiting magnitude as well, which will contribute further uncertainty to these
values. All of the ZHR values are higher than those observed visually by the
IMO666IMO web site: www.imo.net/draconids-outburst-on-oct-8-9/, which were of
order 100. Because of the rapidly changing mass index, the flux values here
should be treated with caution.

##  6 Radiant distribution

Trajectories and orbits were computed with the Monte Carlo method of Vida et
al. (2020b), and the initial velocities were estimated using the sliding fit
proposed in the that paper using 40% or more points from the beginning of the
meteor. We did not apply the deceleration correction proposed by Vida et al.
(2018) as accurate velocities were not the focus of this work, and this should
not influence the radiant dispersion.

Figure 17 shows a probabilistic radiant map in Sun-centered geocentric
ecliptic coordinates generated by combining trajectory solutions of all Monte
Carlo runs (200 per meteor) using kernel density estimation and a Gaussian
kernel. Figure 18 shows the nominal radiants with error bars in equatorial
coordinates with the solar longitude color coded. Both plots include
comparison between the observed and simulated radiants as produced by Egal et
al. (2018). In the simulation, the meteors were produced by meteoroids ejected
in 1953, and we only selected simulations which hit the Earth in the range of
solar longitude between $None$ and $None$.

The observed radiant is quite compact and the bulk of the shower falls within
a circle of $None$ in diameter (dashed circle in the plot), centered around a
median shower radiant $\alpha_{g}=$$$ and $\delta_{g}=$$$, or
$\lambda_{g}-\lambda_{\astrosun}=$$$ and $\beta_{g}=$$$. Only radiants with
high measurement uncertainties lie outside of this circle. The probabilistic
radiant map shows an area of higher density at $\alpha_{g}=$$$ and
$\delta_{g}=$$$, or $\lambda_{g}-\lambda_{\astrosun}=$$$ and $\beta_{g}=$$$,
which is marked with a smaller circle in Figure 17, but we cannot exclude this
being an artifact due to small number statistics (only 5 concentrated radiants
with small uncertainty). The observed radiants match the width of simulated
radiants well, but there is an offset in solar longitude.

Following Kresák & Porubčan (1970), we compute the dispersion as the median
angular offset from the mean radiant. Because the observation period was
short, we did not perform the radiant drift correction. Figure 19 shows a
histogram of angular offsets from the mean radiant. The median angular offset
was $None$.

In Figure 18, the Egal et al. (2018) simulations show an obvious correlation
of the radiant position with the solar longitude. Our observing period spanned
the solar longitudes from $None$ to $None$, but the simulated radiant
positions matched the observations best for $\lambda_{\astrosun}$$$,
indicating that the position of simulated particles is shifted along the orbit
when compared to observations. Due to the high precision of these
observations, it may possible to optimize the meteoroid ejection and
integration parameters until a good match in both the position and time is
achieved, improving future predictions.

Figure 17: Sun-centered geocentric ecliptic radiant distribution of the 2018
Draconids observed with the EMCCD systems. The normalized density is color
coded. Figure 18: Geocentric radiant distribution of the 2018 Draconids
observed with the EMCCD systems. The solar longitude is color coded, and the
error bars show the one standard deviation uncertainty. The squares show the
simulated radiants by (Egal et al., 2018). Figure 19: Distribution of angular
offsets from the mean radiant of the 2018 Draconids. The vertical line marks
the median angular offset of $None$.

Figure 20 shows histograms of simulated and measured geocentric velocities for
the 2018 Draconids. The histograms were normalized so that their area is equal
to 1. The mean and standard deviation were 20.05 $\pm
0.93\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and 20.94 $\pm
0.08\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ for observed and simulated
data, respectively. The observed geocentric velocities have a larger scatter
than the simulations due to the limited velocity measurement precision.
Furthermore, there is a systematic $\sim
0.9\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ underestimation of the
geocentric velocity, a combined effect of meteor deceleration prior to
detection (Vida et al., 2018) and using the average velocity of the first 40%
(or more) of the meteor as the initial velocity. In Table
LABEL:tab:meteor_list we give a list of all meteors and their geocentric
radiants, orbits, magnitudes, and mass proxies. We note that due to the
underestimation of the geocentric velocity from lack of deceleration
correction, the semi-major axes listed in the table are also underestimated.

Figure 20: Comparison of simulated (blue) and observed (red) geocentric
velocities.

##  7 Conclusions

We analyzed 90 minutes of optical meteor data of the 2018 Draconid meteor
shower outburst. The observations started just after the predicted peak at
00:00 UTC on October 9, 2018. We performed multi-station observations with the
SOMN all-sky network and four sensitive EMCCD cameras. The all-sky system did
not record any bright Draconid meteors, but 68 multi-station Draconids with
complete light curves were recorded with the EMCCD systems. The astrometric
and photometric calibration of the EMCCD systems has been discussed in detail,
and we show that the EMCCD systems have a limiting stellar magnitude of
$+10.0^{M}$ and an astrometric precision of 10 arc seconds.

We developed a novel method of population and mass index computation which is
based on maximum likelihood estimation. The method fits a gamma function to
the distribution of magnitudes or logarithms of mass to estimate the slope of
the cumulative distribution (i.e. population or mass index) before the
sensitivity effects become important. We validated this approach on past data
obtained with the CAMO influx system. The method also enables robust error
estimation.

We divided EMCCD data of the 2018 Draconid outburst into two time bins of
approximately equal number of meteors: the first bin spanned the time from
00:00 UTC to 00:30 UTC, and the second bin spanned from 00:30 UTC to 01:30
UTC, Oct 9, 2018. The mass index in the first bin was $1.74\pm 0.18$, and in
the second bin $2.32\pm 0.27$, an increase of similar magnitude and rate of
change to the 2011 Draconids (Koten et al., 2014).

Because of the rapid rate of change of the mass index of the shower and small
number statistics, the values derived for the mass index and flux are very
uncertain, particularly for the first time bin. The fluxes in the two bins
were 0.11 and 0.99 meteoroids km-2 hr-1, respectively, corrected to a limiting
meteor magnitude of +6.5. From visual observations, we expect the flux was
actually falling from the first to the second time interval; the reversed
trend in this data is most likely due to the assumption of a single power law
in each time bin, which does not adequately describe the data. For this
reason, we do not attempt to describe the flux profile of the shower.

The measured shower radiant diameter was $None$ which matches simulations by
Egal et al. (2018) well, but there is an offset in solar longitude of $None$.
The measured radiant dispersion is a good match to the 2011 outburst, both the
high-precision radiant measurements by Borovička et al. (2014), and the
meteoroid ejection simulations by Vida et al. (2020a).

###  7.1 Complete input data and trajectory solutions for EMCCD meteors

We provide the input data and detailed trajectory solutions for the Draconids
observed with the EMCCD systems, in electronic form at the CDS via anonymous
ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.

###### Acknowledgements.

We thank Dr. Gunther Stober for helping with the equipment during the
observing campaign. Furthermore, we thank Dario Zubović, Damir Šegon,
Aleksandar Merlak, Damjan Nemarnik, Vladimir Jovanović, Peter Eschman, Dr. Dan
Kinglesmith, Przemek Naganski, Dmitrii Rychkov, Stanislav Korotkiy, and Milan
Kalina for supporting the RMS project. This work was supported in part by the
NASA Meteoroid Environment Office under cooperative agreement 80NSSC18M0046.
Funding from the Natural Sciences and Engineering Research Council of Canada
(Grant no. RGPIN-2016-04433) and the Canada Research Chairs Program is
gratefully acknowledged.

## References

  * Akkouchi (2008) Akkouchi, M. 2008, J. Chungcheong Math. Soc, 21, 501
  * Arlt & Rendtel (2006) Arlt, R. & Rendtel, J. 2006, Monthly Notices of the Royal Astronomical Society, 367, 1721
  * Beech et al. (1995) Beech, M., Brown, P. G., & Jones, J. 1995, Quarterly Journal of the Royal, 36, 127
  * Berry & Burnell (2000) Berry, R. & Burnell, J. 2000, Astronomical Image Processing (Willmann-Bell)
  * Blaauw et al. (2011a) Blaauw, R., Campbell-Brown, M., & Weryk, R. 2011a, Monthly Notices of the Royal Astronomical Society, 412, 2033
  * Blaauw et al. (2011b) Blaauw, R., Campbell-Brown, M., & Weryk, R. 2011b, Monthly Notices of the Royal Astronomical Society, 414, 3322
  * Blaauw et al. (2016) Blaauw, R. C., Campbell-Brown, M., & Kingery, A. 2016, Monthly Notices of the Royal Astronomical Society, 463, 441
  * Borovička et al. (2014) Borovička, J., Koten, P., Shrbený, L., Štork, R., & Hornoch, K. 2014, Earth, Moon, and Planets, 113, 15
  * Brown & Rendtel (1996) Brown, P. & Rendtel, J. 1996, Icarus, 124, 414
  * Brown et al. (2010) Brown, P., Weryk, R., Kohut, S., Edwards, W., & Krzeminski, Z. 2010, WGN, Journal of the International Meteor Organization, 38, 25
  * Brown & Ye (2012) Brown, P. & Ye, Q. 2012, Central Bureau Electronic Telegrams, 3249, 1
  * Brown et al. (2002) Brown, P. G., Campbell-Brown, M. D., Hawkes, R., Theijsmeijer, C., & Jones, J. 2002, Planetary and Space Science, 50, 45
  * Campbell-Brown (2015) Campbell-Brown, M. 2015, Planetary and Space Science, 118, 8
  * Campbell-Brown et al. (2016) Campbell-Brown, M. D., Blaauw, R., & Kingery, A. 2016, Icarus, 277, 141
  * Clauset et al. (2009) Clauset, A., Shalizi, C. R., & Newman, M. E. 2009, SIAM review, 51, 661
  * Cooke & Moser (2010) Cooke, W. J. & Moser, D. E. 2010, Meteoroids 2010: An International Conference on the Minor Bodies in the Solar System; 24-28 May 2010; Breckenridge, CO; United States
  * Corral & González (2019) Corral, Á. & González, Á. 2019, Earth and Space Science, 6, 673
  * Dohnanyi (1969) Dohnanyi, J. 1969, Journal of Geophysical Research, 74, 2531
  * Egal et al. (2019) Egal, A., Wiegert, P., Brown, P. G., et al. 2019, Icarus, 330, 123
  * Egal et al. (2018) Egal, A., Wiegert, P., Brown, P. G., et al. 2018, The Astrophysical Journal Letters, 866, L8
  * Feroz et al. (2009) Feroz, F., Hobson, M., & Bridges, M. 2009, Monthly Notices of the Royal Astronomical Society, 398, 1601
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Gault et al. (1963) Gault, D. E., Shoemaker, E. M., & Moore, H. J. 1963, Spray ejected from the lunar surface by meteoroid impact (National Aeronautics and Space Administration)
  * Gural (2016) Gural, P. 2016, in International Meteor Conference Egmond, the Netherlands, 2-5 June 2016, 96–104
  * Gural (2007) Gural, P. S. 2007, in Advances in Meteoroid and Meteor Science (Springer), 269–275
  * Ivezić et al. (2014) Ivezić, Ž., Connolly, A. J., VanderPlas, J. T., & Gray, A. 2014, Statistics, Data Mining, and Machine Learning in Astronomy: A Practical Python Guide for the Analysis of Survey Data (Princeton University Press)
  * Jenniskens (2006) Jenniskens, P. 2006, Meteor showers and their parent comets (Cambridge University Press)
  * Jones & Morton (1982) Jones, J. & Morton, J. 1982, Monthly Notices of the Royal Astronomical Society, 200, 281
  * Kastinen & Kero (2017) Kastinen, D. & Kero, J. 2017, Planetary and Space Science, 143, 53
  * Kero et al. (2012) Kero, J., Fujiwara, Y., Abo, M., Szasz, C., & Nakamura, T. 2012, Monthly Notices of the Royal Astronomical Society, 424, 1799
  * Koten et al. (2014) Koten, P., Vaubaillon, J., Tóth, J., Margonis, A., & Ďuriš, F. 2014, Earth, Moon, and Planets, 112, 15
  * Kresák (1993) Kresák, L. 1993, in Meteoroids and their parent bodies, Proceedings of the International Astronomical Symposium held at Smolenice, Slovakia, July 6-12, 1992, Bratislava, edited by J. Stohl and I.P. Williams (Astronomical Institute, Slovak Academy of Sciences), 147
  * Kresák & Porubčan (1970) Kresák, L. & Porubčan, V. 1970, Bulletin of the Astronomical Institutes of Czechoslovakia, 21, 153
  * Kun & Herrmann (1996) Kun, F. & Herrmann, H. J. 1996, Computer Methods in Applied Mechanics and Engineering, 138, 3
  * Leemis & McQueston (2008) Leemis, L. M. & McQueston, J. T. 2008, The American Statistician, 62, 45
  * Maslov (2011) Maslov, M. 2011, WGN, Journal of the IMO, 39, 3
  * McKinley (1961) McKinley, D. W. R. 1961, New York, McGraw-Hill, 1961.
  * Molau (2015) Molau, S. 2015, in International Meteor Conference Mistelbach, Austria, 11–15
  * Molau et al. (2014) Molau, S., Barentsen, G., & Crivello, S. 2014, in Proceedings of the International Meteor Conference, Giron, France, 18-21 September 2014, 74–80
  * Pokornỳ & Brown (2016) Pokornỳ, P. & Brown, P. G. 2016, Astronomy & Astrophysics, 592, A150
  * Rendtel (2004) Rendtel, J. 2004, in Proceedings of the International Meteor Conference, 22nd IMC, Bollmannsruh, Germany, 2003, 114–122
  * Rendtel (2014) Rendtel, J. 2014, Meteor Shower Workbook 2014 (International Meteor Organization), 88
  * Šegon et al. (2014) Šegon, D., Andreić, Ž., Gural, P. S., et al. 2014, Earth, Moon, and Planets, 112, 33
  * Trigo-Rodríguez et al. (2013) Trigo-Rodríguez, J. M., Madiedo, J. M., Williams, I. P., et al. 2013, Monthly Notices of the Royal Astronomical Society, 433, 560
  * Vaubaillon et al. (2015) Vaubaillon, J., Koten, P., Margonis, A., et al. 2015, Earth, Moon, and Planets, 114, 137
  * Vaubaillon et al. (2011) Vaubaillon, J., Watanabe, J., Sato, M., Horii, S., & Koten, P. 2011, WGN, Journal of the IMO, 39, 3
  * Vida et al. (2018) Vida, D., Brown, P. G., & Campbell-Brown, M. 2018, Monthly Notices of the Royal Astronomical Society, 479, 4307
  * Vida et al. (2020a) Vida, D., Brown, P. G., Campbell-Brown, M., Wiegert, P., & Gural, P. S. 2020a, Monthly Notices of the Royal Astronomical Society, 491, 3996
  * Vida et al. (2020b) Vida, D., Gural, P. S., Brown, P. G., Campbell-Brown, M., & Wiegert, P. 2020b, Monthly Notices of the Royal Astronomical Society, 491, 2688
  * Vida et al. (2016) Vida, D., Zubović, D., Šegon, D., Gural, P., & Cupec, R. 2016, in Proceedings of the International Meteor Conference (IMC2016), Egmond, The Netherlands, 2–5
  * Watanabe & Sato (2008) Watanabe, J.-I. & Sato, M. 2008, in Advances in Meteoroid and Meteor Science (Springer), 111–116
  * Weryk et al. (2013) Weryk, R., Campbell-Brown, M., Wiegert, P., et al. 2013, Icarus, 225, 614
  * Ye et al. (2013a) Ye, Q., Brown, P. G., Campbell-Brown, M. D., & Weryk, R. J. 2013a, Monthly Notices of the Royal Astronomical Society, 436, 675
  * Ye et al. (2013b) Ye, Q., Wiegert, P. A., Brown, P. G., Campbell-Brown, M. D., & Weryk, R. J. 2013b, Monthly Notices of the Royal Astronomical Society, 437, 3812

##  Appendix A Light curves of selected Draconids

Figure 21: Example light curves of EMCCD Draconids, covering the range of
observed peak magnitudes.

##  Appendix B Table of EMCCD Draconids

1]

Table 3: Radiants, orbits, magnitudes and mass proxies for the observed EMCCD Draconids. Rows below every entry list $1\sigma$ uncertainties. Date and time (UTC) | $\lambda_{\astrosun}$ | $\alpha_{g}$ | $\delta_{g}$ | $v_{g}$ | $a$ | $e$ | $q$ | $\omega$ | $i$ | $\pi$ | Mag | $\log_{10}I^{*}$
---|---|---|---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (deg) | ($\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$) | (AU) |  | (AU) | (deg) | (deg) | (deg) |  |
2018-10-09 00:06:37 | 195.4007 | 261.148 | 55.909 | 18.733 | 2.393 | 0.5840 | 0.9954 | 171.91 | 29.37 | 7.31 | 4.19 | -2.20
|  | 0.043 | 0.025 | 0.064 | 0.020 | 0.0035 | 0.00004 | 0.04 | 0.07 | 0.04 |  |
2018-10-09 00:07:52 | 195.4016 | 262.200 | 56.273 | 20.328 | 2.993 | 0.6673 | 0.9960 | 172.85 | 31.30 | 8.26 | 4.05 | -2.15
|  | 0.025 | 0.036 | 0.044 | 0.016 | 0.0020 | 0.00002 | 0.03 | 0.06 | 0.03 |  |
2018-10-09 00:08:31 | 195.4020 | 262.814 | 55.988 | 21.698 | 4.069 | 0.7552 | 0.9961 | 173.24 | 32.69 | 8.65 | 4.73 | -2.31
|  | 0.152 | 0.034 | 0.667 | 0.444 | 0.0394 | 0.00001 | 0.12 | 0.73 | 0.12 |  |
2018-10-09 00:10:08 | 195.4031 | 262.062 | 56.168 | 19.090 | 2.501 | 0.6018 | 0.9960 | 172.68 | 29.84 | 8.08 | 3.32 | -1.86
|  | 0.018 | 0.016 | 0.043 | 0.014 | 0.0022 | 0.00001 | 0.01 | 0.05 | 0.01 |  |
2018-10-09 00:10:20 | 195.4033 | 260.831 | 55.913 | 20.383 | 3.024 | 0.6710 | 0.9949 | 171.75 | 31.31 | 7.16 | 3.84 | -1.96
|  | 0.107 | 0.010 | 0.046 | 0.029 | 0.0030 | 0.00007 | 0.08 | 0.05 | 0.08 |  |
2018-10-09 00:10:47 | 195.4036 | 262.436 | 55.955 | 20.823 | 3.356 | 0.7032 | 0.9959 | 172.94 | 31.74 | 8.35 | 0.94 | -0.82
|  | 0.015 | 0.012 | 0.008 | 0.003 | 0.0002 | 0.00001 | 0.01 | 0.01 | 0.01 |  |
2018-10-09 00:12:51 | 195.4050 | 261.350 | 55.779 | 19.598 | 2.715 | 0.6334 | 0.9953 | 172.06 | 30.34 | 7.46 | 5.04 | -2.36
|  | 0.018 | 0.022 | 0.032 | 0.011 | 0.0014 | 0.00001 | 0.02 | 0.04 | 0.02 |  |
2018-10-09 00:14:00 | 195.4058 | 261.414 | 56.034 | 19.739 | 2.735 | 0.6361 | 0.9955 | 172.19 | 30.59 | 7.59 | 5.41 | -2.51
|  | 0.025 | 0.008 | 0.021 | 0.010 | 0.0013 | 0.00002 | 0.02 | 0.02 | 0.02 |  |
2018-10-09 00:14:36 | 195.4062 | 262.067 | 55.961 | 19.702 | 2.758 | 0.6389 | 0.9959 | 172.64 | 30.49 | 8.05 | 5.56 | -2.72
|  | 0.045 | 0.023 | 0.077 | 0.034 | 0.0048 | 0.00002 | 0.03 | 0.08 | 0.03 |  |
2018-10-09 00:14:54 | 195.4064 | 262.214 | 55.958 | 21.019 | 3.472 | 0.7132 | 0.9958 | 172.79 | 31.97 | 8.20 | 2.77 | -1.62
|  | 0.015 | 0.024 | 0.023 | 0.010 | 0.0008 | 0.00001 | 0.02 | 0.03 | 0.02 |  |
2018-10-09 00:17:38 | 195.4083 | 262.062 | 56.143 | 20.165 | 2.932 | 0.6604 | 0.9958 | 172.71 | 31.08 | 8.12 | 3.24 | -1.86
|  | 0.026 | 0.015 | 0.112 | 0.062 | 0.0067 | 0.00001 | 0.02 | 0.12 | 0.02 |  |
2018-10-09 00:17:52 | 195.4084 | 262.133 | 55.744 | 21.280 | 3.724 | 0.7326 | 0.9956 | 172.68 | 32.19 | 8.09 | 5.98 | -2.83
|  | 0.139 | 0.047 | 0.398 | 0.250 | 0.0238 | 0.00003 | 0.10 | 0.43 | 0.10 |  |
2018-10-09 00:17:54 | 195.4084 | 261.603 | 56.081 | 19.933 | 2.817 | 0.6466 | 0.9956 | 172.35 | 30.82 | 7.76 | 5.12 | -2.36
|  | 0.042 | 0.007 | 0.031 | 0.015 | 0.0019 | 0.00002 | 0.03 | 0.04 | 0.03 |  |
2018-10-09 00:18:22 | 195.4088 | 262.639 | 55.737 | 19.682 | 2.808 | 0.6453 | 0.9961 | 172.99 | 30.37 | 8.40 | 1.99 | -1.23
|  | 0.348 | 0.130 | 0.316 | 0.168 | 0.0191 | 0.00025 | 0.28 | 0.34 | 0.28 |  |
2018-10-09 00:18:25 | 195.4088 | 261.974 | 56.042 | 19.818 | 2.789 | 0.6430 | 0.9958 | 172.60 | 30.65 | 8.01 | 5.21 | -2.47
|  | 0.013 | 0.008 | 0.026 | 0.013 | 0.0016 | 0.00001 | 0.01 | 0.03 | 0.01 |  |
2018-10-09 00:18:44 | 195.4090 | 262.370 | 55.669 | 19.780 | 2.851 | 0.6507 | 0.9959 | 172.78 | 30.47 | 8.19 | 3.58 | -1.95
|  | 0.013 | 0.026 | 0.029 | 0.009 | 0.0011 | 0.00001 | 0.02 | 0.04 | 0.02 |  |
2018-10-09 00:19:46 | 195.4097 | 262.432 | 55.913 | 21.927 | 4.300 | 0.7684 | 0.9958 | 172.96 | 32.93 | 8.37 | 1.56 | -1.05
|  | 0.128 | 0.066 | 0.040 | 0.025 | 0.0013 | 0.00010 | 0.10 | 0.06 | 0.10 |  |
2018-10-09 00:19:54 | 195.4098 | 260.962 | 56.355 | 19.669 | 2.642 | 0.6233 | 0.9953 | 171.95 | 30.63 | 7.36 | 5.49 | -2.76
|  | 0.105 | 0.064 | 0.184 | 0.062 | 0.0093 | 0.00009 | 0.10 | 0.23 | 0.10 |  |
2018-10-09 00:19:56 | 195.4098 | 260.852 | 55.758 | 19.673 | 2.729 | 0.6354 | 0.9950 | 171.69 | 30.44 | 7.10 | 5.64 | -2.57
|  | 0.388 | 0.141 | 0.040 | 0.013 | 0.0017 | 0.00030 | 0.33 | 0.07 | 0.33 |  |
2018-10-09 00:20:13 | 195.4100 | 261.106 | 55.483 | 21.299 | 3.739 | 0.7339 | 0.9948 | 171.87 | 32.17 | 7.28 | 3.59 | -1.91
|  | 0.093 | 0.034 | 0.022 | 0.019 | 0.0013 | 0.00007 | 0.08 | 0.02 | 0.08 |  |
2018-10-09 00:24:15 | 195.4128 | 261.709 | 56.185 | 19.394 | 2.592 | 0.6159 | 0.9957 | 172.43 | 30.22 | 7.85 | 5.50 | -2.72
|  | 0.041 | 0.041 | 0.134 | 0.058 | 0.0081 | 0.00003 | 0.03 | 0.14 | 0.03 |  |
2018-10-09 00:24:52 | 195.4132 | 261.638 | 55.836 | 20.452 | 3.118 | 0.6807 | 0.9954 | 172.32 | 31.32 | 7.73 | 5.54 | -2.55
|  | 0.018 | 0.024 | 0.045 | 0.022 | 0.0023 | 0.00001 | 0.02 | 0.05 | 0.02 |  |
2018-10-09 00:25:23 | 195.4136 | 262.628 | 56.100 | 21.198 | 3.589 | 0.7225 | 0.9961 | 173.13 | 32.20 | 8.55 | 3.09 | -1.68
|  | 0.034 | 0.004 | 0.023 | 0.019 | 0.0015 | 0.00002 | 0.03 | 0.02 | 0.03 |  |
2018-10-09 00:25:59 | 195.4140 | 262.163 | 56.083 | 20.088 | 2.911 | 0.6579 | 0.9959 | 172.76 | 30.97 | 8.17 | 3.32 | -1.83
|  | 0.009 | 0.014 | 0.017 | 0.006 | 0.0007 | 0.00001 | 0.01 | 0.02 | 0.01 |  |
2018-10-09 00:26:48 | 195.4145 | 262.170 | 56.165 | 19.224 | 2.550 | 0.6095 | 0.9960 | 172.76 | 29.99 | 8.18 | 5.53 | -2.67
|  | 0.017 | 0.019 | 0.038 | 0.010 | 0.0016 | 0.00001 | 0.02 | 0.05 | 0.02 |  |
2018-10-09 00:29:09 | 195.4162 | 262.108 | 56.096 | 19.895 | 2.819 | 0.6467 | 0.9959 | 172.72 | 30.76 | 8.13 | 2.31 | -1.37
|  | 0.012 | 0.010 | 0.009 | 0.003 | 0.0003 | 0.00001 | 0.01 | 0.01 | 0.01 |  |
2018-10-09 00:30:10 | 195.4169 | 262.123 | 55.988 | 20.920 | 3.391 | 0.7064 | 0.9957 | 172.73 | 31.88 | 8.15 | 2.50 | -1.26
|  | 0.014 | 0.007 | 0.051 | 0.035 | 0.0031 | 0.00001 | 0.01 | 0.05 | 0.01 |  |
2018-10-09 00:30:24 | 195.4170 | 262.482 | 55.895 | 20.043 | 2.938 | 0.6610 | 0.9960 | 172.93 | 30.84 | 8.35 | 4.51 | -2.34
|  | 0.125 | 0.086 | 0.096 | 0.034 | 0.0038 | 0.00010 | 0.11 | 0.13 | 0.11 |  |
2018-10-09 00:30:45 | 195.4173 | 262.300 | 56.127 | 19.325 | 2.596 | 0.6162 | 0.9961 | 172.85 | 30.10 | 8.27 | 4.33 | -2.33
|  | 0.015 | 0.032 | 0.029 | 0.006 | 0.0009 | 0.00001 | 0.02 | 0.04 | 0.02 |  |
2018-10-09 00:30:57 | 195.4174 | 263.270 | 55.346 | 19.881 | 2.997 | 0.6676 | 0.9964 | 173.34 | 30.45 | 8.76 | 4.93 | -2.47
|  | 0.232 | 0.136 | 0.109 | 0.035 | 0.0034 | 0.00017 | 0.18 | 0.16 | 0.18 |  |
2018-10-09 00:33:05 | 195.4189 | 262.340 | 55.534 | 20.073 | 3.015 | 0.6697 | 0.9958 | 172.72 | 30.77 | 8.14 | 3.04 | -1.46
|  | 0.014 | 0.053 | 0.065 | 0.026 | 0.0028 | 0.00001 | 0.02 | 0.09 | 0.02 |  |
2018-10-09 00:33:54 | 195.4194 | 261.699 | 56.214 | 19.620 | 2.672 | 0.6273 | 0.9957 | 172.44 | 30.49 | 7.86 | 3.68 | -1.94
|  | 0.014 | 0.018 | 0.033 | 0.011 | 0.0016 | 0.00001 | 0.01 | 0.04 | 0.01 |  |
2018-10-09 00:34:12 | 195.4196 | 261.985 | 55.767 | 20.477 | 3.165 | 0.6854 | 0.9956 | 172.55 | 31.31 | 7.97 | 2.88 | -1.61
|  | 0.021 | 0.029 | 0.036 | 0.017 | 0.0017 | 0.00001 | 0.01 | 0.05 | 0.01 |  |
2018-10-09 00:34:35 | 195.4199 | 261.820 | 55.782 | 20.460 | 3.143 | 0.6832 | 0.9955 | 172.43 | 31.31 | 7.85 | 4.71 | -2.49
|  | 0.025 | 0.091 | 0.080 | 0.025 | 0.0025 | 0.00002 | 0.03 | 0.12 | 0.03 |  |
2018-10-09 00:34:57 | 195.4201 | 261.592 | 56.010 | 18.759 | 2.401 | 0.5854 | 0.9957 | 172.27 | 29.42 | 7.69 | 5.19 | -2.52
|  | 0.018 | 0.034 | 0.036 | 0.008 | 0.0014 | 0.00002 | 0.02 | 0.05 | 0.02 |  |
2018-10-09 00:40:00 | 195.4236 | 262.710 | 54.921 | 17.099 | 2.105 | 0.5266 | 0.9963 | 172.76 | 27.04 | 8.18 | 5.63 | -2.83
|  | 0.193 | 0.127 | 0.370 | 0.102 | 0.0183 | 0.00012 | 0.15 | 0.46 | 0.15 |  |
2018-10-09 00:40:02 | 195.4236 | 262.162 | 56.212 | 20.711 | 3.210 | 0.6897 | 0.9959 | 172.82 | 31.72 | 8.24 | 4.69 | -2.35
|  | 0.041 | 0.011 | 0.027 | 0.017 | 0.0016 | 0.00002 | 0.03 | 0.03 | 0.03 |  |
2018-10-09 00:42:09 | 195.4251 | 261.976 | 55.944 | 19.376 | 2.629 | 0.6211 | 0.9958 | 172.55 | 30.11 | 7.98 | 4.78 | -2.45
|  | 0.262 | 0.136 | 0.059 | 0.010 | 0.0016 | 0.00020 | 0.22 | 0.10 | 0.22 |  |
2018-10-09 00:42:24 | 195.4252 | 261.600 | 55.887 | 19.625 | 2.719 | 0.6339 | 0.9955 | 172.27 | 30.40 | 7.70 | 4.43 | -2.30
|  | 0.010 | 0.013 | 0.022 | 0.007 | 0.0010 | 0.00001 | 0.01 | 0.03 | 0.01 |  |
2018-10-09 00:43:13 | 195.4258 | 261.918 | 56.138 | 20.173 | 2.930 | 0.6601 | 0.9957 | 172.60 | 31.10 | 8.03 | 4.36 | -2.05
|  | 0.014 | 0.019 | 0.020 | 0.006 | 0.0006 | 0.00001 | 0.01 | 0.03 | 0.01 |  |
2018-10-09 00:44:30 | 195.4267 | 262.635 | 56.235 | 21.228 | 3.572 | 0.7211 | 0.9961 | 173.18 | 32.28 | 8.60 | 2.06 | -1.33
|  | 0.281 | 0.125 | 0.017 | 0.023 | 0.0018 | 0.00022 | 0.24 | 0.04 | 0.24 |  |
2018-10-09 00:45:10 | 195.4272 | 261.695 | 56.232 | 21.624 | 3.816 | 0.7391 | 0.9954 | 172.52 | 32.76 | 7.95 | 3.65 | -1.80
|  | 0.032 | 0.043 | 0.058 | 0.035 | 0.0026 | 0.00003 | 0.04 | 0.08 | 0.04 |  |
2018-10-09 00:46:20 | 195.4280 | 262.032 | 56.029 | 20.632 | 3.198 | 0.6886 | 0.9957 | 172.66 | 31.57 | 8.09 | 4.19 | -2.12
|  | 0.030 | 0.030 | 0.038 | 0.015 | 0.0014 | 0.00002 | 0.02 | 0.05 | 0.02 |  |
2018-10-09 00:48:59 | 195.4298 | 262.503 | 56.053 | 20.324 | 3.048 | 0.6733 | 0.9961 | 173.00 | 31.21 | 8.43 | 4.49 | -2.09
|  | 0.143 | 0.068 | 0.016 | 0.009 | 0.0010 | 0.00011 | 0.12 | 0.03 | 0.12 |  |
2018-10-09 00:49:03 | 195.4298 | 262.424 | 56.205 | 22.928 | 5.589 | 0.8218 | 0.9958 | 173.07 | 34.10 | 8.50 | 3.45 | -1.57
|  | 0.034 | 0.008 | 0.105 | 0.173 | 0.0067 | 0.00001 | 0.03 | 0.11 | 0.03 |  |
2018-10-09 00:50:26 | 195.4308 | 260.164 | 56.377 | 18.070 | 2.146 | 0.5364 | 0.9950 | 171.25 | 28.75 | 6.69 | 6.31 | -3.11
|  | 0.089 | 0.040 | 0.102 | 0.031 | 0.0062 | 0.00005 | 0.07 | 0.11 | 0.07 |  |
2018-10-09 00:50:30 | 195.4308 | 261.233 | 56.480 | 18.889 | 2.376 | 0.5810 | 0.9956 | 172.14 | 29.73 | 7.57 | 4.02 | -2.12
|  | 0.016 | 0.013 | 0.013 | 0.003 | 0.0005 | 0.00001 | 0.01 | 0.02 | 0.01 |  |
2018-10-09 00:53:49 | 195.4331 | 262.207 | 56.102 | 20.183 | 2.954 | 0.6629 | 0.9959 | 172.80 | 31.08 | 8.23 | 4.53 | -2.20
|  | 0.026 | 0.007 | 0.017 | 0.010 | 0.0011 | 0.00001 | 0.02 | 0.02 | 0.02 |  |
2018-10-09 00:54:02 | 195.4332 | 262.175 | 56.064 | 19.528 | 2.676 | 0.6278 | 0.9960 | 172.74 | 30.32 | 8.17 | 6.61 | -3.38
|  | 0.010 | 0.020 | 0.020 | 0.005 | 0.0006 | 0.00001 | 0.01 | 0.03 | 0.01 |  |
2018-10-09 00:54:33 | 195.4336 | 262.130 | 55.991 | 18.750 | 2.416 | 0.5878 | 0.9960 | 172.66 | 29.38 | 8.09 | 3.32 | -1.72
|  | 0.010 | 0.019 | 0.015 | 0.002 | 0.0004 | 0.00001 | 0.01 | 0.02 | 0.01 |  |
2018-10-09 00:59:05 | 195.4367 | 262.004 | 56.005 | 21.422 | 3.746 | 0.7342 | 0.9956 | 172.66 | 32.44 | 8.10 | 1.05 | -0.76
|  | 0.029 | 0.008 | 0.041 | 0.036 | 0.0024 | 0.00002 | 0.02 | 0.04 | 0.02 |  |
2018-10-09 00:59:12 | 195.4368 | 261.853 | 56.040 | 19.636 | 2.710 | 0.6325 | 0.9957 | 172.50 | 30.45 | 7.94 | 5.94 | -2.86
|  | 0.054 | 0.068 | 0.070 | 0.019 | 0.0026 | 0.00005 | 0.06 | 0.10 | 0.06 |  |
2018-10-09 00:59:30 | 195.4370 | 260.516 | 55.773 | 18.862 | 2.428 | 0.5902 | 0.9949 | 171.39 | 29.51 | 6.83 | 6.04 | -2.94
|  | 0.069 | 0.014 | 0.024 | 0.009 | 0.0015 | 0.00005 | 0.06 | 0.03 | 0.06 |  |
2018-10-09 00:59:35 | 195.4370 | 261.633 | 56.017 | 19.653 | 2.712 | 0.6328 | 0.9956 | 172.33 | 30.47 | 7.77 | 3.46 | -1.62
|  | 0.009 | 0.010 | 0.013 | 0.004 | 0.0005 | 0.00001 | 0.01 | 0.02 | 0.01 |  |
2018-10-09 00:59:50 | 195.4372 | 261.715 | 55.772 | 19.379 | 2.644 | 0.6235 | 0.9956 | 172.31 | 30.07 | 7.75 | 2.37 | -1.28
|  | 0.014 | 0.019 | 0.015 | 0.003 | 0.0005 | 0.00001 | 0.01 | 0.02 | 0.01 |  |
2018-10-09 01:01:43 | 195.4385 | 261.983 | 56.221 | 20.594 | 3.132 | 0.6820 | 0.9958 | 172.68 | 31.60 | 8.12 | 6.04 | -2.97
|  | 0.117 | 0.047 | 0.077 | 0.039 | 0.0049 | 0.00008 | 0.10 | 0.08 | 0.10 |  |
2018-10-09 01:08:52 | 195.4434 | 262.068 | 55.950 | 20.164 | 2.967 | 0.6643 | 0.9958 | 172.65 | 31.02 | 8.09 | 5.31 | -2.59
|  | 0.174 | 0.042 | 0.501 | 0.331 | 0.0298 | 0.00004 | 0.13 | 0.54 | 0.13 |  |
2018-10-09 01:10:16 | 195.4444 | 263.413 | 56.213 | 21.256 | 3.654 | 0.7273 | 0.9965 | 173.71 | 32.27 | 9.16 | 4.22 | -2.13
|  | 0.088 | 0.017 | 0.047 | 0.033 | 0.0028 | 0.00005 | 0.07 | 0.05 | 0.07 |  |
2018-10-09 01:12:26 | 195.4458 | 261.633 | 55.913 | 19.390 | 2.625 | 0.6208 | 0.9956 | 172.29 | 30.13 | 7.74 | 4.80 | -2.38
|  | 0.043 | 0.012 | 0.022 | 0.009 | 0.0013 | 0.00003 | 0.04 | 0.03 | 0.04 |  |
2018-10-09 01:13:18 | 195.4464 | 261.579 | 55.956 | 19.086 | 2.511 | 0.6035 | 0.9956 | 172.25 | 29.79 | 7.70 | 5.14 | -2.54
|  | 0.261 | 0.155 | 0.102 | 0.040 | 0.0062 | 0.00023 | 0.24 | 0.12 | 0.24 |  |
2018-10-09 01:14:12 | 195.4471 | 261.196 | 56.256 | 19.152 | 2.483 | 0.5990 | 0.9955 | 172.06 | 29.98 | 7.50 | 4.57 | -2.28
|  | 0.019 | 0.021 | 0.013 | 0.002 | 0.0004 | 0.00001 | 0.02 | 0.02 | 0.02 |  |
2018-10-09 01:14:40 | 195.4474 | 261.443 | 56.452 | 20.109 | 2.823 | 0.6474 | 0.9956 | 172.34 | 31.15 | 7.79 | 5.30 | -2.63
|  | 0.080 | 0.060 | 0.039 | 0.011 | 0.0014 | 0.00007 | 0.07 | 0.06 | 0.07 |  |
2018-10-09 01:15:32 | 195.4480 | 261.452 | 55.902 | 20.320 | 3.024 | 0.6709 | 0.9953 | 172.19 | 31.21 | 7.64 | 3.44 | -1.92
|  | 0.139 | 0.086 | 0.009 | 0.012 | 0.0013 | 0.00013 | 0.13 | 0.02 | 0.13 |  |
2018-10-09 01:17:31 | 195.4493 | 261.989 | 56.232 | 20.535 | 3.097 | 0.6785 | 0.9958 | 172.69 | 31.54 | 8.14 | 4.19 | -1.99
|  | 0.106 | 0.062 | 0.018 | 0.013 | 0.0013 | 0.00009 | 0.10 | 0.03 | 0.10 |  |
2018-10-09 01:21:22 | 195.4520 | 261.745 | 55.998 | 19.711 | 2.742 | 0.6369 | 0.9956 | 172.41 | 30.53 | 7.86 | 4.64 | -2.35
|  | 0.101 | 0.062 | 0.017 | 0.010 | 0.0013 | 0.00009 | 0.09 | 0.02 | 0.09 |  |
2018-10-09 01:21:30 | 195.4521 | 262.764 | 56.139 | 20.150 | 2.956 | 0.6630 | 0.9963 | 173.20 | 31.03 | 8.66 | 5.20 | -2.69
|  | 0.145 | 0.021 | 0.168 | 0.066 | 0.0094 | 0.00008 | 0.12 | 0.19 | 0.12 |  |
2018-10-09 01:23:21 | 195.4533 | 262.104 | 55.876 | 20.780 | 3.327 | 0.7007 | 0.9957 | 172.67 | 31.69 | 8.12 | 1.82 | -1.19
|  | 0.026 | 0.004 | 0.014 | 0.009 | 0.0008 | 0.00002 | 0.02 | 0.02 | 0.02 |  |
2018-10-09 01:27:34 | 195.4562 | 261.840 | 55.849 | 20.001 | 2.897 | 0.6563 | 0.9956 | 172.44 | 30.81 | 7.90 | 4.11 | -2.25
|  | 0.057 | 0.083 | 0.121 | 0.061 | 0.0077 | 0.00003 | 0.04 | 0.13 | 0.04 |  |
Table 3: continued.