# Computing optical meteor flux using Global Meteor Network data

Denis Vida,1,2 Rhiannon C. Blaauw Erskine3, Peter G. Brown1,2, Jonathon
Kambulow1,4, Margaret Campbell-Brown1,2, Michael J. Mazur1
1Department of Physics and Astronomy, University of Western Ontario, London,
Ontario, N6A 3K7, Canada
2Institute for Earth and Space Exploration, University of Western Ontario,
London, Ontario, N6A 5B8, Canada
3Jacobs Space Exploration Group, Marshall Space Flight Center, Huntsville,
Alabama, 35812, USA
4Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario, N2L 3G1, Canada
E-mail<EMAIL_ADDRESS>
(Accepted 2022 June 21. Received 2022 June 21; in original form 2022 June 05.)

###### Abstract

Meteor showers and their outbursts are the dominant source of meteoroid impact
risk to spacecraft on short time scales. Meteor shower prediction models
depend on historical observations to produce accurate forecasts. However, the
current lack of quality and persistent world-wide monitoring at optical
meteoroid sizes has left some recent major outbursts poorly observed. A novel
method of computing meteor shower flux is developed and applied to Global
Meteor Network data. The method is verified against previously published
observations of the Perseids and the Geminids. The complete mathematical and
algorithmic details of computing meteor shower fluxes from video observations
are described. As an example application of our approach, the flux
measurements of the 2021 Perseid outburst, the 2020-2022 Quadrantids, and
2020-2021 Geminids are presented. The flux of the 2021 Perseids reached
similar levels to the 1991-1994 and 2016 outbursts (ZHR $\sim$ 280). The flux
of the Quadrantids shows high year-to-year variability in the core of the
stream while the longer lasting background activity is less variable,
consistent with an age difference between the two components. The Geminids
show a double peak in flux near the time of peak.

###### keywords:

meteors – meteoroids – comets

††pubyear: 2022††pagerange: Computing optical meteor flux using Global Meteor
Network data–5

##  1 Introduction

Modelling and measuring the flux and associated risk of meteoroids to
spacecraft in and beyond Earth’s orbit is essential to all space operations,
especially human spaceflight programs (Moorhead et al., 2020b, 2021). Space
agencies produce meteor shower forecasts (Moorhead et al., 2020a), which
predict the number of meteoroids above a given mass threshold that may impact
a spacecraft over a particular time period.

The sporadic complex is the dominant impact risk to spacecraft over long time
intervals (Moorhead et al., 2019), however the shower meteoroid flux may
dominate the impact risk compared to sporadics in some short periods (hours to
days). Moreover, as shower velocities are typically much higher than those of
sporadic meteoroids, the threshold shower mass for a given kinetic damage is
smaller while the risk from impact plasma is much higher (Moorhead et al.,
2019).

The three strongest annual showers at Earth are the Perseids, Geminids, and
Quadrantids, each regularly displaying a Zenithal Hourly Rate (ZHR) of over
100 (Moorhead et al., 2019). The Perseids and Geminids are comparatively older
($>$1000s of years) streams which have been studied widely (Brown & Jones,
1998; Blaauw et al., 2016; de León et al., 2010). Apart from rare Perseid
outbursts, they display predictable annual behavior (Miskotte & Vandeputte,
2017; Jenniskens & Miskotte, 2021). The Quadrantids, however, are a young,
short-duration shower known to display large year-to-year variations
(Belkovich & Tohktasev, 1974; Rendtel et al., 2016; Abedin et al., 2015, 2018)
making it the least well documented of the major showers.

The Perseids, whose parent body is the comet 109P/Swift–Tuttle, is arguably
the most well known meteor shower. It consists of an old ($\sim 25,000$ yrs)
core component which produces regular returns with a ZHR of $\sim 100$
(Rendtel et al., 2020). In addition, a younger (100s yrs) component (Brown &
Jones, 1998) produces infrequent outbursts with ZHRs as high as 300 (Brown &
Rendtel, 1996). The outbursts are caused by direct gravitational perturbations
from Jupiter and Saturn which shift the nodes of Perseid meteoroids inward,
making them collide with Earth (Brown & Jones, 1998). The outbursts
significantly increase meteoroid impact risk to spacecraft (Beech & Brown,
1993) because the meteoroids are fast, and even $\sim 10^{-5}$ g meteoroids
can cause significant damage (Moorhead et al., 2019). A Perseid meteoroid from
the 1993 outburst is considered the likely cause of the demise of an
operational satellite (Olympus-1; Caswell et al., 1995).

Unlike other meteor showers whose parent bodies are comets, the Geminids
originate from the C-complex asteroid 3200 Phaethon. Their ablation behaviour
in the atmosphere is also distinct and consistent with stronger asteroidal
material (Ceplecha et al., 1998). A major outstanding question related to the
shower is its mode of formation and the ultimate nature of Phaethon - is it an
asteroid or a defunct comet (Ryabova, 2021)? Key to unravelling this mystery
is the structure of the Geminid stream, in particular its current activity
profile.

The mass of the stream is estimated to be as high as $\sim 10\%$ of Phaethon’s
(Hughes & McBride, 1989; Blaauw, 2017), but large uncertainties remain
(Ryabova, 2017). The asteroid currently shows insufficient activity and mass
loss to explain the mass residing in the stream (Jewitt & Li, 2010; Tabeshian
et al., 2019). However, to accurately estimate the stream mass, the structure
of the stream and the way the Earth intersects it must be known. The exact
shape of the activity profile directly informs these, which is why flux
measurements are desirable (Ryabova, 2007).

The Quadrantid meteor shower was first observed in 1835, but modelling
suggests that the broader shower has been active for several thousand years.
The core of the stream where the flux is highest is only two centuries old
(Wiegert & Brown, 2005; Abedin et al., 2015). The proposed parent body of the
core is asteroid 2003 EH1, though other objects have been linked to the
Quadrantids, most notably 96P/Machholz and C/1490 Y1 (Abedin et al., 2015,
2018). The shower displays a sharp flux peak around $283^{\circ}$ degrees
solar longitude (corresponding to approximately January 3) with half width of
only about 6-8 hours. Thus in the visual size ranges, the Quadrantids are only
seen for one night, and mostly in the northern hemisphere which suffers from
poor weather at this time of the year.

Generally, absolute measurement of meteoroid flux has been made from in-situ
impacts (Grün et al., 2002) as well as radar (Kaiser, 1960), visual (Koschack
& Rendtel, 1990a) and video (Molau & Barentsen, 2013; Molau, 2020) meteor
observations. Each technique has inherent biases and significant effort is
required to develop robust flux measurements from raw values. In particular,
though there has been explosive growth in the deployment of video meteor
systems in the last decade (Koten et al., 2019), the video meteor flux
estimation methodology has often been only briefly described in prior
literature (e.g. Brown et al., 2002; Gural, 2002; Campbell-Brown & Jones,
2006; Blaauw et al., 2016).

Here we describe, validate and justify corrections and analysis procedures to
be applied to single-station video meteor measurements to convert raw observed
counts to meteor shower fluxes. The algorithm is generally applicable to any
optical system, but here we focus on fluxes as measured by the Global Meteor
Network (GMN). In our approach, single-station observations from multiple
video systems are aggregated into an equivalent ’global’ instrumental meteor
detector to improve statistics and temporal resolution. We apply this
procedure to the Perseids, Quadrantids, and Geminids to compare with other
measurements and verify our method. In addition, we measure the flux of the
2021 Perseid outburst.

##  2 Methods

###  2.1 Global Meteor Network Hardware and Data Reduction

The Global Meteor Network (GMN) is an amateur-professional collaboration
project with over 600 video meteor cameras in more than 30 countries (Vida et
al., 2021). The GMN uses low-cost consumer-grade security cameras which are a
good match to the sensitivity and field of view of a human observer. In the
most common configuration, a Sony IMX291 CMOS sensor is paired with a 3.6 mm
f/0.95 lens to provide a field of view of $88^{\circ}\times 48^{\circ}$ and a
stellar limiting magnitude of $+6.0\pm 0.5^{\mathrm{M}}$ at 25 frames per
second, though other lens configurations are occasionally used.

The camera is controlled via a Raspberry Pi 4 single-board computer (used
currently, circa 2022). This runs the fully automated Raspberry pi Meteor
Station (RMS) software111RMS software:
https://github.com/CroatianMeteorNetwork/RMS which schedules observations,
captures the video stream, detects meteors, and calibrates the data (Vida et
al., 2016). The final data product at each station is a list of time, position
(right ascension, declination) and magnitude measurements for every point on
the meteor’s trajectory. These observations are uploaded to a central server
which correlates them with other stations and computes trajectories and orbits
(Vida et al., 2020a).

To conserve disk space, RMS uses a rolling Four-frame Temporal Pixel video
compression scheme (Gural, 2011). This method assumes that meteors are the
brightest objects visible in a sub-area of the frame for a period on the order
of seconds. The algorithm then compresses blocks of 256 video frames into 4
images: a per-pixel maximum stack, an image where the frame index of pixel
maximum occurrence is encoded as pixel level, a per-pixel average stack, and
per-pixel standard deviation image (Vida et al., 2021). At the normal GMN
frame rate of 25 frames per second each 256-frame block has a duration of
10.24 seconds.

As a first step, a star extraction algorithm identifies stars on the average
pixel stack and fits a Gaussian point spread function on each candidate star
(Vida et al., 2016). The full width at half maximum (FWHM) is determined for
every detected star. If more than 20 stars are detected (which from
examination of many camera images had been determined to indicate that at
least a part of the sky is clear), the meteor detection algorithm is run,
producing measurements of meteor position and brightness.

When a camera is first set up, an initial astrometric and photometric
calibration is manually produced to ensure quality. For the most common 3.6 mm
lens configurations, the average astrometric accuracy is $\sim 1$ arcmin, and
the photometric uncertainty is $\pm 0.2^{\mathrm{M}}$. However, the camera
pointing drifts slightly throughout the night due to thermal expansion of the
camera housing bracket, amounting to as much as 10 arcmin. The sky conditions
such as transparency also change, so an automated recalibration procedure is
run for every detected meteor. In the recalibration step, the lens distortion
and vignetting parameters are kept fixed, and only the pointing and the
photometric offset are fit (Vida et al., 2021).

###  2.2 Shower Association

Single-station shower association is performed by first fitting a great circle
to the observed equatorial coordinates of the meteor (see Section 3.2 in Vida
et al., 2020a). If this great circle at any point is within 3∘ from the
predicted position of the apparent radiant of a meteor shower, the meteor is
tentatively associated with the shower.

In this first association step, the geocentric radiants of all possible meteor
showers occurring at the time of the meteor event are computed by applying
empirically derived radiant drifts based on the shower list from Vida et al.
(2021). The equatorial coordinates of the apparent radiant
($\alpha_{a},\delta_{a}$) are numerically inverted assuming a mean meteor
height of 100 km and the geographical coordinates of the observer (i.e. the
camera). From the great circle fit of the meteor observation, the normal
vector n describing the orientation of the great circle (in equatorial
coordinates) can be simply computed as:

$\displaystyle\hat{\mathrm{\textbf{n}}}=\frac{\mathrm{\textbf{n}}}{|\mathrm{\textbf{n}}|}\,,$
(1)
$\displaystyle\alpha_{n}=\operatorname{arctan2}(\hat{n}_{y},\hat{n}_{x})\,,$
(2) $\displaystyle\delta_{n}=\arcsin\hat{n}_{z}\,.$ (3)

The closest radiant approach distance, which should be $<3^{\circ}$ for a
tentative association, is the angular separation between the normal and the
apparent shower radiant offset by $\pi/2$:

$\displaystyle\theta_{\mathrm{rad}}=|\pi/2-\arccos(\sin\delta_{n}\sin\delta_{a}+\cos\delta_{n}\cos\delta_{a}\cos(\alpha_{n}-\alpha_{a}))|\,.$
(4)

A second check is made ensuring that the meteor is coming from the radiant
instead of going towards it. This is simply achieved by confirming that the
angular separations between the apparent radiant and the meteor begin point is
smaller than the angular separation between the radiant and the end point.

Finally, the meteor is definitively associated with a shower if its observed
angular velocity is consistent with the known shower radiant/speed and the
corresponding begin and end heights are in the expected range of heights
(Duffy et al., 1987). Meteor heights are known to vary with speed (Jenniskens
et al., 2016) and meteoroid type (Vida et al., 2018b); at typical meteoroid
sizes the GMN observes, asteroidal meteoroids penetrate about 5 km deeper than
cometary meteoroids. To derive heights for various showers, we fit an
empirical model to all observed begin and end heights (in kilometers) using
the GMN trajectory database (Vida et al., 2021):

$\displaystyle H_{\mathrm{beg}}=97.84+0.41v_{0}-20919.39v_{0}^{-3}\,,$ (5)
$\displaystyle H_{\mathrm{end}}=59.48+0.37v_{0}-11193.74v_{0}^{-3}\,,$ (6)

where $v_{0}$ is the initial in-atmosphere velocity in km/s. The shape of the
model closely follows the bulk of the observations, but has been shifted such
that 99% of observed meteors are within the empirical height range. The
empirical fits compared to data are shown in Fig. 1.

Figure 1: Comparison between begin (left) and end heights (right) observed for
complete lightcurves from multi-station GMN data. Light grey dots are all
meteors, and dark grey are only meteors from major showers of interest as
specified in Moorhead et al. (2019). The adopted begin height for single-
station shower association is shown with a solid line (top), the reference
height at which collection areas are computed is shown with a dashed line
(middle), and the end for shower association with a dotted line (bottom). The
populations where excursions are seen above the line (near 60 km/s for begin
heights) or below for end heights (at 35 and 60 km/s) correspond to shower
fireballs which are insignificant in flux calculations.

Three pieces of information are available to compute the observed range of
heights assuming a shower membership: the unit vector pointing to the meteor
beginning in horizontal coordinates $\hat{\mathrm{\textbf{v}}}_{b}$, the
vector pointing to the end $\hat{\mathrm{\textbf{v}}}_{e}$, and the total
observed length of the meteor $d$. The total observed length in this construct
is simply the product of the observed duration and the known shower velocity.
Vectors $\hat{\mathrm{\textbf{v}}}$ are computed by taking the observed
azimuths $\Lambda$ and altitudes $\epsilon$ of the beginning and the end of
the meteor and separating them into Cartesian components:

$\displaystyle\mathrm{v}_{x}=\cos\epsilon\cos\Lambda\,,$ (7)
$\displaystyle\mathrm{v}_{y}=\cos\epsilon\sin\Lambda\,,$ (8)
$\displaystyle\mathrm{v}_{z}=\sin\epsilon\,.$ (9)

The same is done using the apparent horizontal coordinates of the radiant,
resulting in a meteor shower radiant vector $\hat{\mathrm{\textbf{v}}}_{r}$.

Taking the curvature of the Earth into account, the observed meteor begin
height $h_{\mathrm{b}}$ and the end height $h_{\mathrm{e}}$ are computed as
follows:

$\displaystyle\theta_{\mathrm{met}}=\arccos\left(\hat{\mathrm{\textbf{v}}}_{b}\cdot\hat{\mathrm{\textbf{v}}}_{e}\right)\,,$
(10)
$\displaystyle\theta_{\mathrm{b}}=\arccos\left(\hat{\mathrm{\textbf{v}}}_{r}\cdot(-\hat{\mathrm{\textbf{v}}}_{e})\right)\,,$
(11)
$\displaystyle\theta_{\mathrm{e}}=\arccos\left(-\hat{\mathrm{\textbf{v}}}_{r}\cdot(-\hat{\mathrm{\textbf{v}}}_{b})\right)\,,$
(12) $\displaystyle
d_{b}=d\frac{\sin\theta_{\mathrm{b}}}{\sin\theta_{\mathrm{met}}}\,,$ (13)
$\displaystyle
d_{e}=d\frac{\sin\theta_{\mathrm{e}}}{\sin\theta_{\mathrm{met}}}\,,$ (14)
$\displaystyle R_{\mathrm{obs}}=N+h_{\mathrm{obs}}\,,$ (15) $\displaystyle
h_{\mathrm{b}}=\left|-N+\sqrt{d_{b}^{2}+R_{\mathrm{obs}}^{2}-2d_{b}R_{\mathrm{obs}}\cos(\epsilon_{b}+\pi/2)}\right|\,,$
(16) $\displaystyle
h_{\mathrm{e}}=\left|-N+\sqrt{d_{e}^{2}+R_{\mathrm{obs}}^{2}-2d_{e}R_{\mathrm{obs}}\cos(\epsilon_{e}+\pi/2)}\right|\,,$
(17)

where $\theta_{\mathrm{met}}$ is the observed angular extent of the meteor in
the sky, $\theta_{\mathrm{b}}$ the observer-end-begin angle,
$\theta_{\mathrm{e}}$ the observer-begin-end angle, $d_{b}$ and $d_{e}$ are
the linear distances from the observer to the begin and end point,
respectively, $N$ is the distance between the centre of the Earth and surface
at latitude $\varphi$ (see Appendix D in Vida et al., 2020a),
$h_{\mathrm{obs}}$ is the elevation of the observer above the WGS84 ellipsoid,
$R_{\mathrm{obs}}$ is the distance between the centre of the Earth and the
observer, and $\epsilon_{b}$ and $\epsilon_{e}$ are the observed altitude of
the meteor begin and end points above the horizon, respectively. Fig. 2 shows
an illustration explaining the geometry.

Figure 2: Diagram showing how the observed begin and end heights are
estimated. The red arrow is the meteor pointing in reverse, towards the
radiant.

If $H_{\mathrm{beg}}>(h_{\mathrm{b}}+h_{\mathrm{e}})/2>H_{\mathrm{end}}$, the
meteor is considered to be associated with the shower. This method is very
sensitive to the estimated length of the meteor $d$, and may cause short
meteors not to be associated at small distances from the radiant. We perform
additional checks with the duration varied by $\pm 1$ video frame - if any of
these instances satisfy the height criterion, the association is accepted.
Potential issues this method might encounter with meteors that have low entry
angles are avoided by imposing a minimum apparent radiant elevation limit, as
described below in Section 2.4.2.

Fig. 3 is an example of an all-sky map in equatorial coordinates which shows
the paths of observed meteors from one camera on one night, and the associated
shower membership using our criteria.

Figure 3: Top: Individual meteor paths (solid lines ending with cyan circle)
and their partial great circle projections (dotted lines, colored per shower
named following the IAU three letter designation) for single-station shower
association in the night of August 8-9, 2021 from the HR0010 station. All-sky
map (sinusoidal project) is in equatorial coordinates showing mean locations
of apparent shower radiants. The top legend shows the time period of the
observation and the number of single-station shower associations as well as
sporadics (given by “…"). The point of closest approach for each great circle
to the radiant is designated by a cyan cross. Bottom: A list of the annual
showers based on the International Astronomical Union shower catalogue
considered for flux computation and their activity periods (Moorhead et al.,
2019).

###  2.3 Determining the Effective Period of Camera Observation

The first step in measuring the flux for a given camera is choosing observing
periods during which the measurements will be made. We define a favourable
observing period as a time when at least 50% of catalog stars that are
predicted to be observed (based on known pointing and expected sensitivity of
the camera) are registered by the RMS star detector. The method takes into
account lens vignetting, atmospheric extinction, the changing stellar limiting
magnitude, the presence of the Moon, and masked out portions of the field of
view due to the presence of obstructions.

All observations when the Moon was at least 25% illuminated and inside of the
field of view are rejected. The astrometry and photometry recalibration is
performed in 5 minute steps throughout the night to determine the limiting
stellar magnitude. This recalibration procedure produces measurements of the
photometric offset $p_{0}$ which is used to convert instrumental observations
of pixel intensity to magnitudes using the classical relation (Hawkes, 2002):

$\displaystyle M=-2.5\log_{10}(S_{\mathrm{px}})+p_{0}\,,$ (18)

where $M$ is the magnitude and $S_{\mathrm{px}}$ is the sum of pixel
intensities. The effect of vignetting and atmospheric extinction is also
removed, as described in Vida et al. (2021). The variation in the photometric
offset due to changing sky conditions can be significant, up to one magnitude
during the course of the night, as shown in Fig. 4.

Figure 4: Variation of the photometric offset (in units of magnitude) as
observed on August 14, 2021 by a camera in South Dakota, USA. The dip around
9:00 UTC corresponds to a cloudy period, and the error bars indicate $1\sigma$
fit errors.

The stellar limiting magnitude $M_{s}$ is computed using the photometric
offset and the following empirical relation:

$\displaystyle M_{s}=0.83p_{0}-2.59\,.$ (19)

This relation was derived via manually estimated limiting stellar magnitude
measurements covering the whole range of GMN systems, from all sky to narrow
field lenses, as shown in Fig. 5. The manual procedure included increasing the
stellar limiting magnitude of catalog stars in increments of
$0.1^{\mathrm{M}}$ until the stars in the image could not be visually
determined to match the catalog stars. This procedure closely mimics limiting
magnitude estimation during visual meteor observations (Rendtel et al., 2020).
Moreover, it accounts for all end-to-end biases (except those from the RMS
software) such as photometric processing by the camera, noise etc. The linear
empirical fit has a standard deviation of $\pm 0.25^{\mathrm{M}}$.

Figure 5: Manually determined empirical fit of stellar limiting magnitude
measurements as a function of the photometric offset. The focal lengths of the
different GMN systems used for the fit are indicated for every group of
points.

Unlike other source-extraction software, the RMS star detector was designed to
robustly and quickly detect around 50-100 stars during clear skies, even
though more stars can be seen in the image. By comparing the predicted
limiting magnitude from our empirical relation to the limiting magnitude
recovered by the software, we determined that the detection threshold of the
software translates into a $1.2^{\mathrm{M}}$ brighter limiting magnitude than
is given in Eq. 19.

The GAIA DR2 catalog (Gaia Collaboration et al., 2018) is queried for stars up
to the instrumental limiting magnitude and the detected stars are matched to
the catalog. This gives the number of matched stars $N_{\mathrm{matched}}$.
The edge of the image is sampled with 4 vertices on each side to capture the
effects of lens distortion. The list of catalog stars is filtered such that
only the stars inside the polygon in equatorial coordinates described by the
16 vertices are taken. Stars whose apparent magnitude (with vignetting and
atmospheric extinction applied) is fainter than the limiting magnitude are
removed. The celestial coordinates are mapped into image coordinates and stars
in masked regions are rejected. The number of remaining stars is thus the
number of predicted stars $N_{\mathrm{predicted}}$ which a camera should
detect under good conditions.

To determine when conditions are suitable for inclusion in flux estimation,
the ratio $N_{\mathrm{matched}}/N_{\mathrm{predicted}}$ is considered. By
investigating observations from cameras in various conditions, we found
empirically that a ratio of 0.5 accurately discriminates between acceptable
and overcast skies. This method works regardless of observing conditions, sky
quality, and lens used, as long as there is a good initial calibration. It
also identifies poorly calibrated cameras, as the algorithm will not match any
stars and will thus always produce a ratio of 0.

Observing periods are included for flux estimation only for cameras where the
ratio is above the threshold for at least 90 minutes with no more than 15
minutes per 90 minute block below the threshold. Fig. 6 shows an example of
the estimation of good observing periods for one night using this approach.

We tested the method on dozens of nights around the peaks of investigated
showers. The automated results produced cloudy time windows that were within
five minutes in start/stop time compared to manual estimates. Were the
algorithm to produce erroneous results on some nights that were not covered by
the verification procedure, it could only do so conservatively. The
requirement that at least 20 catalog stars are matched to observed stars (all
within 0.5 px for at least 90 minutes) makes it very unlikely for the
algorithm to classify a cloudy night as acceptable.

Figure 6: Observing periods determined for a camera in South Dakota on August
14, 2021. Blue shading indicates the observing periods when the ratio was
above the threshold for at least 90 minutes. The gap around 9:00 UTC
corresponds to a cloudy period which would not be used for flux measurements.

###  2.4 Computing Meteoroid Flux

The flux of meteors in the Earth’s atmosphere is defined as the number of
meteors passing through a unit of area in a unit of time, down to a certain
mass limit:

$\displaystyle F(>m_{\mathrm{lim}})=\frac{N_{\mathrm{meteors}}}{A_{e}t}\,,$
(20)

where $N_{\mathrm{meteors}}$ is the number of meteors of mass larger than
$m_{\mathrm{lim}}$ observed over an area $A_{e}$ in time $t$. The $A_{e}t$
term in the denominator is often referred to as the time-area product (TAP)
(Musci et al., 2012). Conventionally, the flux is expressed in the units of
meteoroids km-2 h-1 (Brown et al., 1998). For convenience, the TAP can be
expressed in units of meteoroids per 1000 km2 h, as most major showers have
their peak flux in the range of tens $\times 10^{-3}$ meteoroids km-2 h-1.

For the GMN, the number of meteors is provided as-is by the meteor detector,
the limiting mass is computed using an empirical conversion relating the mass-
peak magnitude and velocity (Verniani, 1973), and the time is simply the
duration of a chosen time bin. The flux is most sensitive to estimation of the
effective collection area $A_{e}$ (also called the “reduced area” in some
literature; Koschack & Rendtel, 1990a) which includes all observational
biases.

####  2.4.1 Raw Atmospheric Collecting Area

The effective collection area is computed by starting with the true area of
the atmosphere observed by the camera at meteor heights (assumed to be a
segment of a spherical shell at a height mid-way between the start and end
height as a function of speed from Fig 1, at around 100 km), and correcting
for observational biases. The effective area is always smaller than the true
area, as observational biases decrease the number of observed meteors compared
to ideal conditions. The bias corrections include terms for the camera
sensitivity, meteor distance, radiant elevation, and meteor angular velocity.
All of these values change across the field of view and over time.

The observing period determined in Section 2.3 is binned into 5 minute
intervals, which was chosen as the smallest unit of time likely to be needed
for flux calculations in most circumstances. For every time bin, the field of
view is divided into $20\times 11$ rectangular blocks (for 16:9 sensor
formats) and the correction variation is applied per block across the field of
view. The outline of the blocks are projected to meteor heights from which raw
atmospheric collecting areas are computed.

To compute the area, the geographical coordinates of the four corners of the
block at the reference meteor height are needed. As a first step, apparent
azimuth $\Lambda$ and altitude $\epsilon$ of the corners are computed using
the astrometric plate, accounting for atmospheric refraction and lens
distortion (Vida et al., 2021). Blocks with all four corners below an altitude
of 20∘ are ignored. This avoids applying large corrections due to high meteor
range and atmospheric extinction for large atmospheric areas. We take this
conservative approach to avoid a major source of uncertainty (Vaubaillon et
al., 2021), namely high corrections which may skew absolute estimates when
debiasing (Galligan & Baggaley, 2004). As part of this procedure we also
remove all meteors observed below an altitude of $\epsilon<20^{\circ}$.

Given the known WGS84 geographical coordinates of the observer (latitude
$\varphi$, longitude $\lambda$, elevation $h_{\mathrm{obs}}$), the range $r$
between the observer and a corner line of sight projected at the meteor height
$h_{\mathrm{met}}$ can be computed with the curvature of the Earth taken into
account as:

$\displaystyle d_{s}=N+h_{\mathrm{obs}}\,,$ (21) $\displaystyle
d_{m}=N+h_{\mathrm{met}}\,,$ (22)
$\displaystyle\beta=\epsilon+\arcsin\frac{d_{s}\cos\epsilon}{d_{m}}\,,$ (23)
$\displaystyle r=d_{m}\frac{\cos\beta}{\cos\epsilon}\,,$ (24)

where $N$ is the distance between the centre of the Earth and surface at
latitude $\varphi$ (see Appendix D in Vida et al., 2020a), $\beta$ is the
angle between the observer, the centre of the Earth, and a point at height
$h_{\mathrm{met}}$ which lies on the corner line.

With the range $r$ known, the Earth-centered Earth-fixed (ECEF) coordinates of
a corner point can be computed as:

$\displaystyle S=-r\cos\epsilon\cos\Lambda\,,$ (25) $\displaystyle
E=r\cos\epsilon\sin\Lambda\,,$ (26) $\displaystyle Z=r\sin\epsilon\,,$ (27)
$\displaystyle
x=x_{\mathrm{obs}}+S\sin\varphi\cos\lambda-E\sin\lambda+Z\cos\varphi\cos\lambda\,,$
(28) $\displaystyle
y=y_{\mathrm{obs}}+S\sin\varphi\sin\lambda+E\cos\lambda+Z\cos\varphi\sin\lambda\,,$
(29) $\displaystyle z=z_{\mathrm{obs}}-S\cos\varphi+Z\sin\varphi\,,$ (30)

where $S,E,Z$ are South, East, and vertical components of the range vector,
$x_{\mathrm{obs}},y_{\mathrm{obs}},z_{\mathrm{obs}}$ are ECEF coordinates of
the observer (see Appendix D1 in Vida et al., 2020a), and $x,y,z$ are ECEF
coordinates of the corner point at the meteor height. The final step is to
convert the resulting ECEF coordinates into geographical coordinates (see
Appendix D2 in Vida et al., 2020a, take apparent sidereal time at Greenwich
$\theta^{\prime}_{0}=0$ to get ECEF instead of Earth-centered inertial
coordinates).

We compute the total area $A$ bound by the four sets of geographical
coordinates ($\varphi_{i}$, $\lambda_{i}$) by performing a line integral on a
sphere (the radius of the sphere is assumed to be $N+h_{\mathrm{met}}$). The
Green’s theorem allows the integration to be split into piece-wise segments
bound by each side of the polygon222The equations were first derived for the
sphericalgeometry library; we found no other published literature on the
subject: https://github.com/anutkk/sphericalgeometry, accessed on 2022/04/05.:

$\displaystyle\chi_{i}=\sin^{2}\left(\frac{\varphi_{i}}{2}\right)+\cos\varphi_{i}\sin^{2}\left(\frac{\lambda_{i}}{2}\right)\,,$
(31)
$\displaystyle\vartheta_{i}=2\operatorname{arctan2}\left(\sqrt{\chi_{i}},\sqrt{1-\chi_{i}}\right)\,,$
(32)
$\displaystyle\bar{\vartheta}_{i}=\vartheta_{i}+(\vartheta_{i}-\vartheta_{i+1})/2\qquad\forall
i\in[1,4]\,,$ (33)
$\displaystyle\gamma^{\prime}_{i}=\operatorname{arctan2}\left(\cos\varphi_{i}\sin\lambda_{i},\sin\varphi_{i}\right)\bmod
2\pi\,,$ (34)
$\displaystyle\Delta\gamma^{\prime}_{i}=\left((\gamma^{\prime}_{i}-\gamma^{\prime}_{i+1}+\pi)\bmod
2\pi\right)-\pi\qquad\forall i\in[1,4]\,,$ (35) $\displaystyle
A^{\prime}=\sum_{i=1}^{4}\Delta\gamma^{\prime}_{i}\left(1-\cos\bar{\vartheta}_{i}\right)\,,$
(36) $\displaystyle
A=\min\left(A^{\prime},1-A^{\prime}\right)(N+h_{\mathrm{met}})^{2}\,.$ (37)

Note that the input vector of latitudes and longitudes has five elements - the
first and the last element are the same to form a closed polygon. To satisfy
the Green’s theorem, the order of the vertices needs to be counter-clockwise.

With the per-block raw area computed, the fraction of the image that is masked
out in the given block (due to the presence of local obstructions near the
camera) is removed from the computed area. This final step yields the total
raw area per block observed by the camera (excluding any observational
biases). This area remains constant over time, as long as the camera pointing
does not change. Fig. 7 shows an example of an image divided into blocks and
the associated projection into geographical coordinates at the height of 100
km.

Figure 7: Top: Blocks plane projected to the Earth’s surface from the station
HR000K at the height of 100 km, with the apparent elevation color coded.
Blocks pointing to lower apparent elevations are significantly larger in area
than those near the zenith, but they also have larger bias corrections.
Bottom: Co-added images including meteors as detected from the station HR000K.
The blocks used for area estimation are overlaid on the image, and the
$20^{\circ}$ elevation limit is plotted as a red line. The colors are
inverted, and the image is upside down so that the orientation matches the top
inset.

####  2.4.2 Correcting for Observational Biases

Prior to introducing any corrections, area blocks which are within 15∘ from
the radiant are excluded, as are observation periods when the radiant was
within 15∘ of the middle of the field of view. This is done to avoid the
radiant singularity and unphysically large corrections as the apparent meteor
angular velocity is zero at the radiant. Additionally, all observation periods
when the apparent radiant elevation was below 15∘ are rejected. These limits
were chosen such that no corrections exceed a factor of 10, minimizing
potential systematic errors due to modelling assumptions, as adopted in
earlier meteor debiasing work (Galligan & Baggaley, 2004).

For every time bin, a reference meteor limiting magnitude $M_{m}$ in the
middle of the focal plane is computed following Blaauw et al. (2016):

$\displaystyle
M_{m}=M_{s}-5\log_{10}\left(\frac{r_{\mathrm{mid}}}{100}\right)-2.5\log_{10}\left(\frac{v_{0}F\sin\theta}{\mathrm{{FPS}~{}r_{\mathrm{mid}}~{}\mathrm{FWHM}}}\right)\,,$
(38)

where $r_{mid}$ is the range in kilometers from the centre of the focal plane
to a point at the meteor height (computed using Eq. 24), $v_{0}$ is the in-
atmosphere meteor velocity in km/s, $F$ is the plate scale in px/deg, FPS is
the frame rate, and FWHM is the full width at half maximum of the point spread
function in px. The FWHM is determined by fitting a Gaussian point spread
function on individual detected stars every 5 minutes and taking a median
value. The meteor limiting magnitude $M_{m}$ varies across the field of view;
however the per-block corrections given below even out this variation so that
only one reference $M_{m}$ need be used at any time.

The collection area of a meteor shower is estimated at the reference height at
which meteors are observed (see Fig. 1). The reference height is simply the
mean of the model begin and end height, given in Eq. 5 and 6. The reference
height can also be manually specified. Because meteor showers contain
meteoroids of different masses, the heights can vary by a few kilometers. From
GMN meteor observations, we find that a standard deviation of $\sigma_{h}=2$
km represents this variation well. To take this effect into account, we use a
Gaussian weighting scheme. The distribution is centered on the estimated
reference height and probabilities are computed with a height delta of 2 km
between $\pm 3\sigma_{h}$. The weights are normalized such that
$\sum_{h=60}^{130}w(h)=1$, where the limits corresponds to the model limits.

The total effective collection area $A_{e}$ at the given time is computed as:

$\displaystyle
A_{e}=\sum_{h=60}^{130}\sum_{x,y}w(h)A(x,y,h)\left(c_{\mathrm{sensitivity}}c_{\mathrm{range}}c_{\epsilon}c_{\theta}\right)^{s-1}\,,$
(39)

where $w(h)$ is the weight at each height, $A(x,y,h)$ is the raw area of each
block at the height $h$, the $c$ parameters are individual per-block
corrections, and $s$ is the differential mass index of the shower. Note that
this is physically the same basic approach as is used to compute meteor radar
collecting areas (Kaiser, 1960).

The sensitivity correction takes into account optical vignetting and
atmospheric extinction. The GMN photometric calibration models both of these
effects (Vida et al., 2021). A no-atmosphere magnitude $M_{\mathrm{na}}$ is
computed using the full model and an arbitrary pixel level sum value
$S_{\mathrm{px}}=400$; this magnitude includes the vignetting and extinction
corrections. The magnitude is then converted back into a pixel value without
these corrections (simply
$S_{\mathrm{raw}}=10^{(M_{\mathrm{na}}-p_{0})/(-2.5)}$ recalling that p0 is
the photometric offset). The total sensitivity correction is thus:

$\displaystyle
c_{\mathrm{sensitivity}}=\frac{S_{\mathrm{px}}}{S_{\mathrm{raw}}}\,.$ (40)

The range correction is:

$\displaystyle c_{\mathrm{range}}=\left(\frac{100}{r}\right)^{2}\,,$ (41)

where $r$ is the range in km along the middle of the block to the meteor
height, as the meteor absolute magnitude is defined as the magnitude of a
meteor seen from a range of 100 km.

The radiant elevation correction is:

$\displaystyle c_{\epsilon}=\sin^{\gamma}\epsilon$ (42)

where $\epsilon$ is the radiant elevation at the considered time, and $\gamma$
is the zenith exponent (Zvolankova, 1983) which we keep fixed at 1.5 for all
showers. $\gamma>1$ indicates that there is a non-geometrical dependence on
the observation efficiency with the radiant elevation, possibly due to
atmosphere or ablation effects, and can range between 1 and 2. Molau &
Barentsen (2013) found that $\gamma$ varies from shower to shower; however
they also found it very difficult to measure.

The angular velocity correction accounts for sensitivity loss due to the
shortened effective exposure per pixel during a frame integration caused by
the meteor angular velocity. Because of this, the meteor limiting magnitude
varies across of the field of view. The correction normalizes the angular
velocity to the limiting magnitude in the middle of the field of view. The
per-block correction is:

$\displaystyle c_{\theta}=\frac{\theta_{\mathrm{block}}}{\theta_{0}}\,,$ (43)

where $\theta_{\mathrm{block}}$ is the angular velocity in the middle of the
considered block, and $\theta_{0}$ is the angular velocity in the middle of
the focal plane. The angular velocity is computed as:

$\displaystyle\theta=v_{0}\frac{\sin\eta}{r}\,,$ (44)

where $\eta$ is the angular distance from the radiant to the considered
direction.

We ignore any correction for partial trails in the field of view as the GMN
fields are large compared to typical apparent meteor path lengths that almost
all meteors occur entirely inside the FOV. For narrower field of view this
correction may need to be added (Blaauw et al., 2016).

###  2.5 Scaling the Flux to a Target Limiting Magnitude or Mass

Optical observations of meteor showers are by definition brightness limited
surveys. In this work, we use as reference sensitivity the meteor limiting
magnitude in the middle of the field of view. However, as the limiting
magnitude can change over time, an approach is required to scale observations
to a standardized reference, which is traditionally taken as a meteor limiting
magnitude of $+6.5^{\mathrm{M}}$ (Blaauw et al., 2016). From an operational
standpoint, however, it is helpful to have the sensitivity expressed as a
limiting mass, as it will not depend on the spectral response of the system
nor the velocity, so fluxes can be compared between different showers on the
same mass scale.

From atmospheric observations (Vojáček et al., 2019; Vida et al., 2020b) and
in-situ measurements of dust at parent comets (Fulle et al., 2016), it is
known that masses of shower meteors are usually distributed according to a
power-law distribution:

$\displaystyle dN\propto m^{-s}dm\,,$ (45)

where $dN$ is the number of meteoroids with mass in the interval $dm$ and $s$
is the differential mass index. The mass indices of showers can be measured
using various methods (Koschack & Rendtel, 1990a; Blaauw et al., 2011; Pokorný
& Brown, 2016; Ehlert & Erskine, 2020; Vida et al., 2020b). Although it is
possible to measure the mass index from the actual data set used to compute
the flux (Molau et al., 2014), in this work we use accepted values from the
literature.

Traditionally, the meteor peak magnitude is taken as a proxy for mass (Jacchia
et al., 1967). From the magnitude-mass relation (Vida et al., 2020b), it can
be shown that the number of meteors of a given peak magnitude should follow:

$\displaystyle r=\frac{N(M_{m}+1)}{N(M_{m})}\,,$ (46)

where $r$ is the population index (Brown & Rendtel, 1996) and $N(M_{m})$ is
the number of meteors in the integer magnitude bin $M_{m}$. Thus, to scale the
flux from the observed meteor limiting magnitude $M_{m}$ to a reference
limiting magnitude of $+6.5^{\mathrm{M}}$, the following relation can be used:

$\displaystyle F(6.5^{\mathrm{M}})=F(M_{m})r^{6.5-M_{m}}\,.$ (47)

To minimize the effect of the assumed mass index on the final result, a mean
observed limiting magnitude can also be chosen as reference, in which case the
factor of $6.5$ in the equation would be replaced with the limiting magnitude
of choice.

Formally, from the definition of the magnitude, the conversion between the
mass and the population index is (Ceplecha et al., 1998):

$\displaystyle s=1+2.5\log_{10}r\,.$ (48)

Verniani (1973) found that a factor of 2.3 better describes the conversion
based on empirical observations of the meteor radar magnitude and mass.
However, Vida et al. (2020b) compared the directly measured population and
mass indices using a robust statistical method on a high-quality manually
reduced data set of optical meteors. They found that the conversion with the
factor of 2.3 ($s=1+2.3\log_{10}r$) is not a good fit to the true mass index
of sporadics. Of importance is that the mass index computed in that work was
found from full integration of the complete light curves and conversion to
total mass using a fixed luminous efficiency value, not just peak magnitudes.
Here we repeat the analysis but also include the conversion using a factor of
2.5, which is shown in Fig. 8. A significantly better (though still not
perfect) correspondence is found using 2.5 as compared to 2.3.

We adopt a factor of 2.5 for all fluxes in this work. The remaining disparity
may be due to systematic differences in the ablation behaviour of different
populations of meteoroids. Another potential source of error might be the
power law assumption. For example, using a numerical model of meteoroid
ejection from comet 1P/Halley, Egal et al. (2020) predicted that Orionid
meteoroids arriving at the Earth should have a mass distribution which follows
a broken power law, notably with several sharp slope changes around 1 mm
sizes.

Figure 8: Annual variation in the measured sporadic mass index (solid line,
$1\sigma$ errors shown) as derived using the Canadian Automated Meteor
Observatory’s influx camera using complete light curves to estimate mass (Vida
et al., 2018a). Every bin has a total of 250 meteors. The dashed line is the
mass index derived from the population index using the factor of 2.5, and the
dotted line is using the factor 2.3, all based on meteor peak magnitude only.

After the fluxes in every time bin have been normalized to a single limiting
magnitude of choice, an equivalent limiting mass can be computed. The
magnitude to mass conversion used here is found in Peterson (1999). This
relationship employs a speed-dependent luminous efficiency from Verniani
(1965), which Jacchia et al. (1967) applied to Super-Schmidt meteors:

$\displaystyle
2.25\log_{10}(m_{\mathrm{lim}})=-8.75\log_{10}\left({v_{0}}\right)-M_{\mathrm{lim}}+11.59\,,$
(49)

where $m_{\mathrm{lim}}$ is the limiting mass in grams, $v_{0}$ is the initial
in-atmosphere velocity in km/s, and $M_{\mathrm{lim}}$ is the peak meteor
limiting magnitude in the panchromatic bandpass. As an example, for the
Quadrantids with a velocity of 41 km/s the limiting mass at +6.5 magnitude is
$9\times 10^{-5}$ grams.

We estimate the errors in the computed flux using a classical assumption that
the meteor events are distributed according to a Poisson distribution (Vida et
al., 2020b). A confidence interval for the flux based on the number of
observed events can be computed using the “exact” method (Ulm, 1990):
$[\mathrm{Pr}(\alpha/2,2N),\mathrm{Pr}(1-\alpha/2,2(N+1))]$, where
$\mathrm{Pr}$ is the percent point function of the $\chi^{2}$ distribution,
$N$ is the number of meteors, and $\alpha$ is the confidence interval
($\alpha=0.05$ for the 95% confidence interval). The resulting range in the
number of meteors expected based on Poisson statistics yields a flux
confidence interval via Eq. 20, keeping the effective collection area fixed.

The zenithal hourly rate (ZHR) can be computed from the flux following
Koschack & Rendtel (1990b):

$\displaystyle\mathrm{ZHR}=\frac{37200F(6.5^{\mathrm{M}})}{(13.1r-16.5)(r-1.3)^{0.748}}$
(50)

where $F(6.5^{\mathrm{M}})$ is the flux to the limiting meteor magnitude of
$+6.5^{\mathrm{M}}$ in meteoroids km-2 h-1.

Fig. 9 is an example of the single-station flux output the software generated
for the Geminids during the peak night of the shower. These reports are
automatically generated for all showers relevant for spacecraft meteoroid
impact risk assessment (Moorhead et al., 2019).

Figure 9: An example showing the single-station flux of the 2020 Gemind peak
observed by station HR000K. Individual computation steps and corrections are
shown. The corrections shown are the median across the field of view. A fixed
number of 100 meteors per bin is used, and the flux is computed to a limiting
mass of $2\times 10^{-4}$ grams (+6.5M).

####  2.5.1 Combining Single-station Measurement

Figure 10: Flux measurements of the 2020 Perseids using all cameras reporting
data for the shower in the GMN (a total of 130 cameras in this example). A
minimum TAP was set to 100,000 km2 h to dynamically define the bin lengths.
Error bars for some points in the wings are smaller than plot markers.
Individual insets show mean TAP-weighted shower properties and limiting
magnitude. Note that for this bin size and number of cameras each equivalent
”global” measurement has between hundreds to thousands of meteors per data
point. The top inset shows the mid-bin flux and confidence interval for a
reference mass of $2\times 10^{-5}$ g (black dots) while the grey symbols
represent the flux to a limiting mass of $3\times 10^{-4}$ g, which was
derived from the average effective meteor limiting sensitivity of the
aggregated network data ($+3.95^{\mathrm{M}}$). The former is scaled from the
latter using the mass index of 1.8 as shown in the figure title. The second
inset from top shows the time-area product (TAP) for the network as a function
of solar longitude with the vertical grey lines denoting the time bins needed
to achieve the constant TAP of 100,000 km2 h. The black dots represent the
total number of raw meteor detections by all cameras included in the flux
calculation for a given bin. The third inset from the top shows the average
radiant elevation across all cameras as well as the radiant distance from the
center of the camera FOVs weighted by the TAP per bin. The phase of the moon
is also shown for ease of interpretation, with 100 being full moon. Finally,
the bottom inset shows the average limiting magnitude and average meteor
shower angular velocity in the center of the field of view per bin, TAP
weighted.

To improve number statistics and produce robust and continuous flux
measurements, we create a “virtual" global instrument by combining single-
station meteor counts and time-area product (TAPs) observations across the
entire meteor network. The single-station flux code is run automatically on
data from all available stations. To ensure that possible false positives (not
real meteors) do not pollute the data set, all nights when more than 20
sporadic meteors per hour were detected are ignored. This value was found to
be much higher than the number of real sporadic meteors ever observed in
practice by any of the GMN cameras.

All observing periods are divided into fixed $\sim 5$ minute bins (by solar
longitude) which do not change across the years (if more than one year is
being analysed). Meteor counts and TAPs are summed in each bin across all
cameras. The implicit assumption is that all measurements have been scaled to
the same limiting magnitude, which is achieved by scaling the TAP using Eq.
47. The meteor count remains as-is because it is used to compute the Poisson
confidence interval.

Next, the 5 minute bins are combined to achieve a target minimum number of
meteors and/or a minimum TAP per bin (Molau & Barentsen, 2014; Molau, 2020).
This allows a user to choose between precision and time resolution. Fig. 10
shows the global instrument equivalent flux of the 2020 Perseids obtained by
combining measurements across the whole network.

####  2.5.2 Method Verification

To verify the code implementation, we compare the fluxes derived for two major
showers with consistent year-to-year activity: the Perseids and the Geminids.
A peak flux of $30.5^{+1.1}_{-1.1}\times 10^{-3}$ meteoroids km-2 h-1 to a
limiting mass and magnitude of $2.2\times 10^{-5}$ g and $+6.5^{\mathrm{M}}$
($4.68^{+.0.17}_{-.0.16}\times 10^{-3}$ meteoroids km-2 h-1 at $3\times
10^{-4}$ g, $+3.95^{\mathrm{M}}$) was measured for the Perseids in 2020 (ZHR =
$124.7^{+4.5}_{-4.3}$, $s=1.8$, $r=2.1$) at solar longitude $140.4^{\circ}\pm
0.2^{\circ}$, as shown in Fig. 10. This value compares favourably to the
predicted flux at the given mass limit by Ehlert & Erskine (2020), who based
their model on independent measurements across 5 orders of magnitude in mass,
as shown in Fig. 11. This value is also consistent with both past visual and
video flux measurements of the Perseids (Brown & Rendtel, 1996; Molau et al.,
2019), as given in Table 2.5.2. The measured flux as a function of solar
longitude is available in Supplementary Materials.

Figure 11: Perseid fluxes from Ehlert & Erskine (2020) using measurements by
the Canadian Orbit Meteor Radar (CMOR), NASA’s Wide-Field meteor cameras and
NASA’s All-Sky meteor cameras. The GMN camera fluxes are also shown from this
work. From these relative fluxes, a mass index of 1.54 $\pm$ 0.09 was
measured. The GMN uncertainties are smaller than the symbol size.

Study | Years | Component | Method | s | r |  Flux (met km-2 hr-1) |  |
---|---|---|---|---|---|---|---|---
| ZHR |  |  |  | at $2.2\times 10^{-5}$g ($+6.5^{\mathrm{M}}$) |  |
Brown & Rendtel (1996) | 1991 | Annual | Visual | 1.80 | 2.10 | 0.024 $\pm$ 0.001† |  $97$ $\pm$ | $2$
Brown & Rendtel (1996) | 1992 | Annual | Visual | 1.80 | 2.10 | 0.021 $\pm$ 0.009† |  $84$ $\pm$ | $34$
Brown & Rendtel (1996) | 1993 | Annual | Visual | 1.80 | 2.10 | 0.022 $\pm$ 0.001† |  $86$ $\pm$ | $2$
Brown & Rendtel (1996) | 1994 | Annual | Visual | 1.80 | 2.10 | 0.022 $\pm$ 0.001† |  $86$ $\pm$ | $2$
Molau et al. (2019) | 2015 | Annual | Video | 1.80 | 2.20 | 0.038 $\pm$ 0.002* |  $124$ $\pm$ |  $5$*
Molau et al. (2019) | 2018 | Annual | Video | 1.80 | 2.20 | 0.028 $\pm$ 0.001* |  $92$ $\pm$ |  $3$*
meteorflux.org | 2020 | Annual | Video | 1.80 | 2.20 | 0.027 $\pm$ 0.001 |  $86$ $\pm$ | $5$
VMDB | 2020 | Annual | Visual | 2.00 | 2.50 | 0.039 $\pm$ 0.004† |  $77$ $\pm$ | $7$
Our study | 2020 | Annual | Video | 1.80 | 2.09 | 0.031 $\pm$ 0.001 |  $125$ $\pm$ | $5$
Brown & Rendtel (1996) | 1991 | Outburst | Visual | 1.80 | 2.10 | 0.071 $\pm$ 0.016† |  $284$ $\pm$ | $63$
Jenniskens et al. (1998) | 1991 | Outburst | Photo | 1.73 | 1.96 | 0.063 $\pm$ 0.014† |  $350$ $\pm$ | $75$
Brown & Rendtel (1996) | 1992 | Outburst | Visual | 1.80 | 2.10 | 0.055 $\pm$ 0.006† |  $220$ $\pm$ | $22$
Jenniskens et al. (1998) | 1992 | Outburst | Photo | 1.73 | 1.96 | 0.099 $\pm$ 0.027† |  $550$ $\pm$ | $150$
Brown & Rendtel (1996) | 1993 | Outburst | Visual | 1.80 | 2.10 | 0.066 $\pm$ 0.004† |  $264$ $\pm$ | $17$
Jenniskens et al. (1998) | 1993 | Outburst | Photo | 1.73 | 1.96 | 0.045 $\pm$ 0.006† |  $250$ $\pm$ | $35$
Brown & Rendtel (1996) | 1994 | Outburst | Visual | 1.80 | 2.10 | 0.060 $\pm$ 0.004† |  $238$ $\pm$ | $17$
Jenniskens et al. (1998) | 1994 | Outburst | Photo | 1.73 | 1.96 | 0.034 $\pm$ 0.005† |  $190$ $\pm$ | $35$
Ehlert & Erskine (2020) | 2016 | Outburst | Video | 1.54 | 1.72 | 0.011 $\pm$ 0.003† |  $134$ $\pm$ | $39$
Miskotte & Vandeputte (2017) | 2016 | Outburst | Visual | 1.82 | 2.12 | 0.084 $\pm$ 0.004† |  $320$ $\pm$ |  $15$*
Miskotte et al. (2021) | 2021 | Outburst | Radio | 1.95 | 2.39 | 0.114 $\pm$ 0.003† |  $269$ $\pm$ |  $6$*
Miskotte et al. (2021) | 2021 | Outburst | Visual | 2.10 | 2.76 | 0.136 $\pm$ 0.011† |  $195$ $\pm$ |  $16$*
Jenniskens & Miskotte (2021) | 2021 | Outburst | Video | 2.39 | 3.59 | 0.259 $\pm$ 0.030† |  $170$ $\pm$ |  $20$*
VMDB | 2021 | Outburst | Visual | 1.86 | 2.20 | 0.044 $\pm$ 0.003† |  $144$ $\pm$ | $9$
Our study | 2021 | Outburst | Video | 1.80 | 2.09 | 0.068 $\pm$ 0.004 |  $277$ $\pm$ | $18$

Table 1: Perseid flux measurements for the annual and the outburst component
compared from this study to other previously published values. * - numbers
taken from a plot, not given explicitly in text, † \- computed either from the
flux or ZHR, whichever one was originally given in the paper.

A combined peak flux of $20.04^{+.51}_{-.50}\times 10^{-3}$ meteoroids km-2
h-1 at a limiting mass and magnitude of $2\times 10^{-4}$ g and
$+6.5^{\mathrm{M}}$ ($6.40^{+.16}_{-.16}\times 10^{-3}$ meteoroids km-2 h-1 at
$1\times 10^{-3}$ g, $+4.73^{\mathrm{M}}$) was measured for the Geminids in
2020 and 2021 (ZHR = $128.2^{+3.3}_{-3.2}$, $s=1.7$, $r=1.9$) at solar
longitude $261.98^{\circ}\pm 0.08^{\circ}$. The full flux profile is shown in
Fig. LABEL:fig:GEM_flux, and the data are given in Supplementary Materials.
These values are in accordance with recent visual and video flux measurements
(Molau et al., 2015; Molau et al., 2016; Miskotte, 2019), as given in Table
LABEL:tab:geminid_flux. Ehlert & Erskine (2020) measured their fluxes in 2015
the night after the peak, at solar longitude $\sim 263.0$, and our measurement
can be seen in Fig. LABEL:fig:GEMmassIndex. Our measurements lie below their
curve by a factor of two, however the other data points across the large mass
range show a similar scatter.

https://github.com/CroatianMeteorNetwork/RMS

WesternMeteorPyLib: https://github.com/wmpg/WesternMeteorPyLib

Readers are encouraged to contact the lead author (DV) in the event they are
not able to obtain the code on-line.

###  4.2 Data Availability

Per-station and summary flux measurements are available upon request. All flux
data of the Perseids, Geminids, and the Quadrantids presented in this paper
are given in supplementary materials.

##  5 Acknowledgements

Funding for this work was provided by the NASA Meteoroid Environment Office
under cooperative agreement 80NSSC21M0073. The authors would like to thank Dr.
Bill Cooke for providing insight and expertise that assisted the research and
spurred good discussion.

We would also like to thank the following GMN station operators whose cameras
provided the data used in this work and contributors who made important code
contributions (in alphabetical order): Richard Abraham, Victor Acciari, Rob
Agar, David Akerman, Daknam Al-Ahmadi, Jamie Allen, Edison Álvarez José Felipe
Pérezgómez, Alexandre Alves, Željko Andreić, Martyn Andrews, Miguel Angel
Diaz, Enrique Arce, Georges Attard, Chris Baddiley, David Bailey, Erwin van
Ballegoij, Roger Banks, Hamish Barker, Jean-Philippe Barrilliot, Richard
Bassom, Ricky Bassom, Ehud Behar, Josip Belas, Alex Bell, Serge Bergeron,
Denis Bergeron, Jorge Bermúdez Augusto Acosta, Steve Berry, Adrian Bigland,
Chris Blake, Arie Blumenzweig, Ventsislav Bodakov, Claude Boivin, Robin
Boivin, Bruno Bonicontro, Fabricio Borges, Ubiratan Borges, Dorian Božičević,
David Brash, Ed Breuer, Martin Breukers, John W. Briggs, Peter G. Brown,
Laurent Brunetto, Tim Burgess, Ludger Börgerding, Sylvain Cadieux, Peter
Campbell-Burns, Andrew Campbell-Laing, Pablo Canedo, Seppe Canonaco, Jose
Carballada, Steve Carter, David Castledine, Gilton Cavallini, Brian Chapman,
Jason Charles, Matt Cheselka, Tim Claydon, Trevor Clifton, Manel
Colldecarrera, Michael Cook, Bill Cooke, Christopher Coomber, Brendan Cooney,
Edward Cooper, Jamie Cooper, Andrew Cooper, Marc Corretgé Gilart, Paul Cox,
Llewellyn Cupido, Christopher Curtis, Ivica Ćiković, Dino Čaljkušić, Chris
Dakin, Alfredo Dal’Ava Júnior, James Davenport, Richard Davis, Steve Dearden,
Christophe Demeautis, Bart Dessoy, Paul Dickinson, Ivo Dijan, Pieter Dijkema,
Tammo Jan Dijkema, Marcelo Domingues, Stacey Downton, Zoran Dragić, Iain Drea,
Igor Duchaj, Jean-Paul Dumoulin, Garry Dymond, Jürgen Dörr, Robin Earl, Howard
Edin, Ollie Eisman, Carl Elkins, Ian Graham Enting, Peter Eschman, Bob Evans,
Andres Fernandez, Barry Findley, Rick Fischer, Richard Fleet, Jim Fordice,
Jean Francois Larouche, Kyle Francis, Patrick Franks, Gustav Frisholm, Enrique
Garcilazo Chávez, José García María, Mark Gatehouse, Ivan Gašparić, Chris
George, Megan Gialluca, Kevin Gibbs-Wragge, Jason Gill, Philip Gladstone, Uwe
Glässner, Hugo González, Nikola Gotovac, Neil Graham, Pete Graham, Colin
Graham, Sam Green, Bob Greschke, Daniel J. Grinkevich, Larry Groom, Dominique
Guiot, Tioga Gulon, Margareta Gumilar, Peter Gural, Nikolay Gusev, Kees
Habraken, Alex Haislip, John Hale, Peter Hallett, Graeme Hanigan, Erwin
Harkink, Ed Harman, Marián Harnádek, Ryan Harper, David Hatton, Tim Havens,
Mark Haworth, Paul Haworth, Richard Hayler, Sam Hemmelgarn, Rick Hewett, Don
Hladiuk, Alex Hodge, Simon Holbeche, Jeff Holmes, Nick Howarth, Matthew
Howarth, Jeff Huddle, Bob Hufnagel, Roslina Hussain, Russell Jackson, Jean-
Marie Jacquart, Jost Jahn, Phil James, Ron James Jr, Nick James, Rick James,
Ilya Jankowsky, Alex Jeffery, Klaas Jobse, Richard Johnston, Dave Jones,
Fernando Jordan, Vladimir Jovanović, Jocimar Justino de Souza, Javor Kac,
Richard Kacerek, Milan Kalina, Jonathon Kambulow, Steve Kaufman, Paul
Kavanagh, Alex Kichev, Harri Kiiskinen, Jean-Baptiste Kikwaya, Sebastian
Klier, Dan Klinglesmith, John Kmetz, Zoran Knez, Korado Korlević, Stanislav
Korotkiy, Danko Kočiš, Josip Krpan, Zbigniew Krzeminski, Patrik Kukić,
Reinhard Kühn, Remi Lacasse, Gaétan Laflamme, Steve Lamb, Hervé Lamy, Jean
Larouche Francois, Ian Lauwerys, David Leurquin, Gareth Lloyd, Rob Long De
Corday, Eric Lopez, Jose Lopez Galindo, José Luis Martin, Juan Luis Muñoz,
Pete Lynch, Frank Lyter, Anton Macan, Jonathan Mackey, John Maclean, Igor
Macuka, Nawaz Mahomed, Simon Maidment, Mirjana Malarić, Nedeljko Mandić, Alain
Marin, Bob Marshall, Colin Marshall, Andrei Marukhno, Keith Maslin, Nicola
Masseroni, Bob Massey, Jacques Masson, Damir Matković, Filip Matković, Dougal
Matthews, Michael Mazur, Sergio Mazzi, Stuart McAndrew, Alex McConahay, Robert
McCoy, Charlie McCormack, Mark McIntyre, Peter Meadows, Edgar Merizio Mendes,
Aleksandar Merlak, Filip Mezak, Pierre-Michael Micaletti, Greg Michael, Matej
Mihelčić, Simon Minnican, Wullie Mitchell, Edson Morales Valencia, Nick
Moskovitz, Dave Mowbray, Andrew Moyle, Gene Mroz, Muhammad Luqmanul Hakim
Muharam, Brian Murphy, Carl Mustoe, Przemek Nagański, Jean-Louis Naudin,
Damjan Nemarnik, Attila Nemes, Dave Newbury, Colin Nichols, Nick Norman,
Philip Norton, Zoran Novak, Gareth Oakey, Washington Oliveira, Angélica Olmos
López, Jamie Olver, Christine Ord, Nigel Owen, Michael O’Connell, Dylan
O’Donnell, Thiago Paes, Carl Panter, Neil Papworth, Filip Parag, Gary Parker,
Simon Parsons, Ian Pass, Igor Pavletić, Lovro Pavletić, Richard Payne, Pierre-
Yves Pechart, William Perkin, Enrico Pettarin, Alan Pevec, Mark Phillips,
Patrick Poitevin, Tim Polfliet, Pierre de Ponthière, Derek Poulton, Janusz
Powazki, Aled Powell, Alex Pratt, Miguel Preciado, Chuck Pullen, Terry
Pundiak, Lev Pustil’Nik, Dan Pye, Chris Ramsay, David Rankin, Steve Rau,
Dustin Rego, Chris Reichelt, Danijel Reponj, Fernando Requena, Maciej
Reszelsk, Ewan Richardson, Martin Richmond-Hardy, Mark Robbins, David
Robinson, Martin Robinson, Heriton Rocha, Herve Roche, Adriana Roggemans, Paul
Roggemans, Alex Roig, David Rollinson, Jim Rowe, Dmitrii Rychkov, Michel
Saint-Laurent, Clive Sanders, Jason Sanders, Ivan Sardelić, Rob Saunders, John
Savage, Lawrence Saville, Vasilii Savtchenko, William Schauff, Ansgar Schmidt,
Jim Seargeant, Jay Shaffer, Steven Shanks, Mike Shaw, Angel Sierra, Ivo
Silvestri, Ivica Skokić, Dave Smith, Tracey Snelus, Warley Souza, Mark Spink,
Denis St-Gelais, James Stanley, Radim Stano, Laurie Stanton, Robert D. Steele,
Yuri Stepanychev, Graham Stevens, Peter Stewart, William Stewart, Con
Stoitsis, Andrea Storani, Andy Stott, David Strawford, Rajko Sušanj, Bela
Szomi Kralj, Marko Šegon, Damir Šegon, Jeremy Taylor, Yakov Tchenak, Eric
Toops, Torcuill Torrance, Steve Trone, Wenceslao Trujillo, John Tuckett, Myron
Valenta, Jean Vallieres, Paraksh Vankawala, Neville Vann, Marco Verstraaten,
Arie Verveer, Predrag Vukovic, Martin Walker, Aden Walker, Bill Wallace, John
Waller, Jacques Walliang, Didier Walliang, Jacques Walliang, Christian Wanlin,
Tom Warner, Neil Waters, Steve Welch, Alexander Wiedekind-Klein, John
Wildridge, Ian Williams, Guy Williamson, Urs Wirthmueller, Bill Witte, Martin
Woodward, Jonathan Wyatt, Anton Yanishevskiy, Penko Yordanov, Stephane Zanoni,
and Dario Zubović.

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