# Meteor shower radiant dispersions in Global Meteor Network data

Althea V. Moorhead,1 Tiffany Clements,2 and Denis Vida3
1NASA Meteoroid Environment Office, Marshall Space Flight Center EV44,
Huntsville, Alabama 35812, USA
2Aerodyne Industries, Jacobs Space Exploration Group, Marshall Space Flight
Center EV44, Huntsville, Alabama 35812, USA
3Department of Physics and Astronomy, University of Western Ontario, London,
Ontario, N6A 3K7, Canada E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Meteor showers occur when streams of meteoroids originating from a common
source intersect the Earth. There will be small dissimilarities between the
direction of motion of different meteoroids within a stream, and these small
differences will act to broaden the radiant, or apparent point of origin, of
the shower. This dispersion in meteor radiant can be particularly important
when considering the effect of the Earth’s gravity on the stream, as it limits
the degree of enhancement of the stream’s flux due to gravitational focusing.
In this paper, we present measurements of the radiant dispersion of twelve
showers using observations from the Global Meteor Network. We find that the
median offset of individual meteors from the shower radiant ranges from 0.32∘
for the eta Aquariids to 1.41∘ for the Southern Taurids. We also find that
there is a small but statistically significant drift in Sun-centered ecliptic
radiant and/or geocentric speed over time for most showers. Finally, we
compare radiant dispersion with shower duration and find that, in contrast
with previous results, the two quantities are not correlated in our data.

###### keywords:

meteorites, meteors, meteoroids.

††pubyear: 2021††pagerange: Meteor shower radiant dispersions in Global Meteor
Network data–A.2

##  1 Introduction

Meteor showers are often conceptualized as streams of meteoroids that
encounter the Earth on parallel paths. Meteoroids with identical velocity
vectors can be described by a single radiant (apparent point of origin) and
speed. In actuality, the particles in a meteoroid stream will have some non-
zero dispersion in their radiants and speeds for a variety of reasons:
meteoroids are ejected from their parent comets at varying times and with
varying speeds, directions, and material properties, and these differences are
amplified over time as the meteoroids are exposed to solar radiation and
planetary perturbations (Vaubaillon et al., 2019).

Moorhead et al. (2020) demonstrated that characterizing radiant dispersion is
vital for a realistic computation of the effects of a planet’s gravity on a
meteoroid stream. When zero dispersion is assumed, a singularity in meteoroid
flux and number density appears along the anti-radiant line (that is,
locations in space opposite the planet from the radiant). When the radiant
dispersion is incorporated into the gravitational focusing computation, this
singularity disappears and the flux and number density are finite, with the
magnitude of the flux along the anti-radiant line depending on the radiant
dispersion. Thus, radiant dispersion measurements are necessary to correctly
predict the meteoroid flux onto high-orbiting spacecraft or even the Moon
(Moorhead et al., 2020). It is particularly important to separate measurement
error from the true radiant dispersion, as overestimates in radiant dispersion
produce underestimates in flux.

Meteor shower radiant dispersions probe the degree of orbital dissimilarity
within a meteoroid stream. Another measure of orbital dissimilarity – the
width and shape of a shower’s activity profile – is often used to calibrate or
test stream models (Egal et al., 2019). Thus, the radiant dispersion could be
used as another constraint on the age and/or dynamical history of a meteoroid
stream (Brown & Jones, 1998), particularly if it reflects a non-symmetrical
stream cross-section, as we discuss in Section 3.4.4.

Radiant dispersions are not, however, routinely measured. In many cases this
is because the precision of measured meteor trajectories does not permit a
useful characterization (Koten et al., 2019). For instance, early studies
attempted to measure radiant dispersions using single-station meteor
observations (Jacchia et al., 1950; Whipple & Wright, 1954; Millman &
McKinley, 1963); their methodology involved fitting great circles to the
meteors’ paths, determining the point of closest convergence, and then
measuring the minimum displacement from this point to each meteor arc. This
approach places a lower limit on the individual radiant offsets and thus
likely underestimates the radiant dispersion.

Multi-station meteor observations permit the calculation of radiant points
rather than arcs and, at least in theory, enable better measurements of
radiant dispersion. Kresák & Porubčan (1970) used double-station photographic
observations of meteors to measure the average or median angular deviations of
meteor radiants from the shower average. The authors combined data from
multiple sources: the Harvard (most notably the Super-Schmidt cameras; Hawkins
& Southworth, 1961; McCrosky, 1968), Ondřejov (Ceplecha et al., 1964), and
Soviet meteor observation programs (Babadzhanov & Kramer, 1967). As a result,
the data were not reduced using a consistent method. Furthermore, the data set
did not contain uncertainties on individual radiant measurements and thus no
quality thresholds could be imposed. Although their Perseid radiant dispersion
is based on a large number of meteors, they had to rely on a relatively small
number of meteors for most showers (less than 50 in all but two cases: the
Perseids and the Geminids). No other studies systematically survey meteor
shower radiant dispersions, but the literature contains scattered
characterizations of the dispersion of individual showers (e.g., Jenniskens,
1997; Jenniskens et al., 1998).

This paper presents the first survey of meteor shower radiant dispersions
conducted in the past 50 years. We characterize the Sun-centered geocentric
ecliptic radiant drift and radiant dispersion of twelve meteor showers using
high-quality video meteor trajectories collected by the Global Meteor Network
(Vida et al., 2021). We use a bootstrap approach to compute uncertainties in
our fit parameters and attempt to account for the systematic increase in
radiant dispersion caused by radiant measurement errors. We find that radiant
dispersion tends to shrink as we restrict our solutions to shorter periods of
activity. Finally, we compare our measured dispersions with the shower
durations in an attempt to reproduce the correlation observed by Whipple &
Wright (1954).

##  2 Data

###  2.1 Global Meteor Network

The meteor trajectory data used in this work were collected by the Global
Meteor Network (GMN)111https://globalmeteornetwork.org/ between December 2018
and May 2021; about 300 of the more than 450 GMN cameras contributed (see Vida
et al., 2021, for a map of contributing cameras). GMN video cameras are
operated by professional organizations as well as amateur astronomers and
citizen scientists; all trajectory data are made publicly available in near-
real time.

Because GMN relies on volunteers to build and operate its camera stations, the
specifics of individual cameras can vary. In this work, we use data from
cameras with two different sets of specifications: cameras with a 3.6-mm focal
length, $88^{\circ}\times 48^{\circ}$ field of view, and a limiting stellar
magnitude of about +6; and high-resolution cameras with a 16-mm focal length,
$20^{\circ}\times 10^{\circ}$ field of view, and a limiting stellar magnitude
of about +9 (additional details are provided in Table 1 of Moorhead et al.,
2020). A detailed description of the network, the systems and the methodology
used by GMN is given in Vida et al. (2021).

###  2.2 Trajectory solutions

The trajectory solutions for each meteor were calculated using the
observational Monte Carlo (MC) approach detailed in Vida et al. (2020) and
Vida et al. (2021). This technique uses the result provided by the
intersecting planes method (Ceplecha, 1987) as an initial solution for
trajectory minimization using the line-of-sight method developed by Borovička
(1990). This method calculates 3D meteor trajectories by simultaneously using
optical observations from multiple stations. The measurement error is
characterized as angular residuals between the measurements and the fitted
trajectory. Gaussian-distributed noise on the order of the determined errors
is added to the original measurements and the trajectory is recomputed in
every MC run. This gives a purely geometrical trajectory fit uncertainty. The
novelty of the Monte Carlo method is that it utilizes the fact that all
observers must observe the same meteor dynamics (velocity and deceleration) at
the same time, thus a cost function based on residuals of time vs. distance
from the meteor beginning is used to choose the best solution. This results in
a final trajectory solution which is optimal both geometrically and
dynamically, and provides realistic uncertainties for every orbital parameter.
A similar approach was taken by Gural (2012), but their method was forcing an
unphysical model of meteor kinematics to observations, leading to biases in
the computed trajectory solution (Egal et al., 2017). Nevertheless, we note
that without full meteor ablation and fragmentation modeling, measured initial
velocities of meteors can be underestimated by as much as 750 m s-1 (Vida et
al., 2018).

###  2.3 Quality cuts

Our 30-month span of GMN meteor data contains more than 200,000 trajectories,
about 40% of which are tentatively associated with a meteor shower. We
restrict our analysis to the most precise trajectories by applying similar
quality cuts as were used in Moorhead et al. (2020). First, we require that
the maximum convergence angle between camera lines of sight is at least 15∘.
Second, we require that the median angular fit error is no larger than 1
arcmin. Finally, we require that the formal uncertainty in the average in-
atmosphere speed is no greater than 1 km s-1, and that the formal uncertainty
in the radiant is no greater than 0.25∘. These cutoffs allow us to exclude
meteors with the largest errors while retaining a substantial proportion of
the data.

Ideally, we would also restrict our analysis to meteors observed by more than
two stations and/or by high-resolution cameras with 16-mm lenses. However,
this would significantly reduce the number of meteors available to us. We
therefore refrain from completing an analysis using high-resolution or many-
station radiant solutions, but note that Moorhead et al. (2020) found that the
radiant offset distribution of the high-resolution cameras was similar to that
measured using all cameras.

##  3 Methods

###  3.1 Shower member selection and radiant drift

Meteor shower radiants are known to drift in geocentric right ascension and
declination over time. However, this drift is minimized when the radiant is
rotated into ecliptic coordinates and the Sun’s ecliptic longitude subtracted
from the shower longitude (Brown et al., 2008; Jenniskens, 2017). The
resulting Sun-centered geocentric ecliptic coordinates are usually denoted
$\lambda_{g}-\lambda_{\odot}$ and $\beta_{g}$. We will use Sun-centered
ecliptic coordinates to describe the meteor radiants considered in this paper.
The subscript $g$ indicates that these radiants are geocentric (rather than
heliocentric). However, while the Sun-centered ecliptic coordinate system is
known to minimize radiant drift, we do not assume that it eliminates all
drift. Instead, we also test and fit for any drift in Sun-centered ecliptic
radiant and speed.

First, we select meteors that lie near a shower’s established solar longitude,
radiant, and speed. Typically, one would complete this step by selecting
meteors whose radiants lie within a certain great-circle distance of the
shower radiant, or whose orbital elements satisfy some orbital similarity
criterion (e.g., Drummond, 1981). However, in order to minimize any biases in
fitting for linear drift in the individual radiant angles, we instead select
meteors falling within the deliberately large, quasi-rectangular region where

$\displaystyle|\beta_{g}-\beta_{\mathrm{ref}}|$ $\displaystyle<10^{\circ}\,,$
(1)
$\displaystyle|\Delta\lambda_{g}-\Delta\lambda_{\mathrm{ref}}|\cdot\cos\beta_{\mathrm{ref}}$
$\displaystyle<10^{\circ}\,,$ (2)
$\displaystyle|v_{g}-v_{\mathrm{ref}}|/v_{\mathrm{ref}}$
$\displaystyle<0.2\,\textrm{, and}$ (3)
$\displaystyle|\lambda_{\odot}-\lambda_{0}|$
$\displaystyle<\tfrac{1}{2}\mathrm{dur}_{\mathrm{max}}.$ (4)

Here, we use $\Delta\lambda_{g}$ as shorthand for
$\lambda_{g}-\lambda_{\odot}$. We initially centered these windows about the
reference solar longitude, radiant, and speed reported by SonotaCo (2009), but
subsequently shifted them to center around the median solar longitude of the
shower members and the corresponding best-fit radiant and speed at that point
in time (see Table 1).

The variable $\mathrm{dur}_{\mathrm{max}}$ is the estimated duration of the
shower. Ideally, one would choose a very short duration in order to probe the
radiation dispersion of the “core” of the stream. However, for long-lasting,
low-activity showers such as the Taurids, a short duration selects far too few
shower members. For each shower, we ran our shower membership algorithm for
values of $\mathrm{dur}_{\mathrm{max}}$ ranging from 1 day ($\sim
0.986^{\circ}$) to 29 days ($\sim 28.583^{\circ}$) in increments of 2 days. We
then selected as our nominal value of $\mathrm{dur}_{\mathrm{max}}$ the
duration that yielded at least 60% of the estimated true shower members (see
Section 3.2 for our sporadic contamination estimate) yielded by the 29 day
case. This ranged from 1 day for the Quadrantids to 19 days for the Southern
Taurids.

Within our selection window, we use least-squares minimization to derive
linear fits in our parameters. We express these drifts as:

$\displaystyle\Delta\lambda_{\mathrm{ref}}(\lambda_{\odot})$
$\displaystyle=m_{\lambda}\cdot(\lambda_{\odot}-\lambda_{0})+b_{\lambda}$ (5)
$\displaystyle\beta_{\mathrm{ref}}(\lambda_{\odot})$
$\displaystyle=m_{\beta}\cdot(\lambda_{\odot}-\lambda_{0})+b_{\beta}$ (6)
$\displaystyle v_{\mathrm{ref}}$
$\displaystyle=m_{v}\cdot(\lambda_{\odot}-\lambda_{0})+b_{v}$ (7)

As discussed in appendix A.1, we clip our data to exclude the outliers caused
by sporadic contamination. We accomplish this by performing an initial fit
using all data, then removing points that lie more than 2.5∘ from this best-
fit radiant (see equation 8 below), and then perform a second least-squares
fit using the remaining meteors. We bootstrap this process 1000 times and
derive a covariance matrix (and errors) from the bootstrap results.

Figure 1 shows sample linear drift fits for the Orionid meteor shower. In this
case, we find that there is a statistically significant correlation between
solar longitude and $\lambda_{g}-\lambda_{\odot}$, $\beta_{g}$, and $v_{g}$,
but the drift in $\lambda_{g}-\lambda_{\odot}$ is the most pronounced, with a
value of $-0.22^{\circ}$ per degree in solar longitude. It is also apparent
from Fig. 1 that our chosen angular window of $\pm 10^{\circ}$ is much larger
than the typical spread in radiant angles for this shower.

Figure 1: Sun-centered ecliptic longitude (top), latitude (center), and
geocentric speed (bottom) plotted against solar longitude for meteors observed
by GMN in the vicinity of the Orionid meteor shower. A linear fit to the data
is shown as a dashed black line. Plots of the residuals can be found in Fig.
8.

Once we have measured the drift in shower radiant and speed, we re-select
shower member candidates as follows:

$\displaystyle\Theta\left(\Delta\lambda_{g},\,\beta_{g},\,\Delta\lambda_{\mathrm{ref}}(\lambda_{\odot}),\,\beta_{\mathrm{ref}}(\lambda_{\odot})\right)$
$\displaystyle<10^{\circ}$ (8)
$\displaystyle|v_{g}-v_{\mathrm{ref}}(\lambda_{\odot})|/v_{\mathrm{ref}}$
$\displaystyle<0.2\,\textrm{, and}$ (9)
$\displaystyle|\lambda_{\odot}-\lambda_{0}|$
$\displaystyle<\tfrac{1}{2}\mathrm{dur}_{\mathrm{max}}\,.$ (10)

where $\Theta$ is the central angle222computed using the haversine formula
between the two longitude-latitude pairs ($\Delta\lambda_{g}$, $\beta_{g}$)
and ($\Delta\lambda_{\mathrm{ref}}(\lambda_{\odot})$,
$\beta_{\mathrm{ref}}(\lambda_{\odot})$) and our reference radiant and speed
now varies with solar longitude according to equations 5-7.

####  3.1.1 Special handling of the Taurid complex

We wished to include the Northern and Southern Taurid (NTA and STA) meteor
showers in our analysis, but this pair of unusual showers requires special
handling. The Taurids have relatively low peak activity but are very long-
lasting compared to other showers; activity persists for several months rather
than days or weeks. They are embedded in the center of the anti-helion
sporadic source and may arise from the same parent body, 2P/Encke (Whipple,
1940; Wiegert et al., 2009). While they are considered meteor showers, the
Taurids appear to lie somewhere between meteor shower and sporadic source in
their characteristics. Furthermore, because the Nothern and Southern Taurids
are two branches of the same complex, have similar orbits, and are active at
the same time and with similar radiants, most shower identification techniques
struggle to separate the two branches (see, for example, Sugar et al., 2017).

We have found that the simplest and most straightforward way to separate the
two showers is by ecliptic latitude (see Fig. 2). We consider only those
meteors with $\beta_{g}>-1^{\circ}$ when characterizing the NTA radiant
distribution, and only those meteors with $\beta_{g}<-1^{\circ}$ when
characterizing the STAs. This choice is also applied to the sporadic
contamination estimate (see Section 3.2).

Figure 2: Sun-centered ecliptic meteor radiants in the vicinity of the
Northern and Southern Taurid meteor showers. We plot only those meteors that
occur between $\lambda_{\odot}=200$ and $\lambda_{\odot}=250$. The
$\beta_{g}=-1^{\circ}$ dividing line is marked with a dashed line.

###  3.2 Sporadic contamination

It is impossible to tell which of the meteors in our radiant-speed-solar
longitude window are truly shower members and which are false positives; no
shower membership selection method can avoid contamination by sporadic
meteors. We therefore estimate the distribution of sporadic contaminants by
selecting meteors that lie near our reference shower radiant and speed at
times when the shower is _not_ active. We accomplish this by shifting the
solar longitudes of all meteors by 60∘, 63∘, and so on up to 300∘ from their
measured values and rerunning our shower membership algorithm. This allows us
to assess the distribution of sporadic meteoroids around the drifting shower
radiant, averaging over an eight-month period when the shower is not active in
order to minimize the impact of effects such as gaps in observations.

Figure 3 displays the angular distribution of these simulated sporadic meteors
about the drifting shower radiant for the Northern Taurid meteor shower. We
compare these simulated sporadic contaminants with the angular distribution of
unmodified meteors about the same shower radiant. Note that while the number
of shower members per bin is an integer number, the number of sporadic
contaminants is not necessarily an integer; this is because we have divided
the latter by the number of times we have shifted the data in solar longitude.
Based on these results, we expect that approximately 15% of the nominal shower
members are actually sporadic contaminants (we later fit for the level of
sporadic contamination and obtain a value of 12%; see Section 3.3 and Table
2). This fraction ranges from a fraction of a percent for the Geminids (GEM)
to nearly 20% for the alpha Capricornids (CAP).

We consider this sporadic contamination estimation when selecting which
showers to characterize. In this paper, we analyze showers for which the
estimated number of showers members within our nominal solar longitude window
is greater than 100 after subtracting the estimated sporadic contamination
level.

However, it is important to note that these are only rough estimates for the
contamination level. Sporadic activity can vary seasonally (Campbell-Brown &
Jones, 2006; Campbell-Brown & Wiegert, 2009) and therefore we allow the
sporadic contamination level to vary in our fits; see Section 3.3. The true
value of generating simulated sporadic meteors in the manner described here is
that it allows us to characterize the shape of the sporadic radiant
distribution near ecliptic radiants that are similar to a given meteor shower.

###  3.3 Distribution fitting

Because we expect our data to be contaminated by sporadic meteors, we must
incorporate that assumption into our fitting algorithm. We do this by
constructing histograms of the angular offsets between our drifting shower
radiant and both the nominal shower members and our simulated sporadic data.
We then assume that the distribution of true shower members, $f_{i}$, is a
linear combination of the nominal shower members ($h_{i}$) and the simulated
sporadic data ($g_{i}$):

$\displaystyle f_{i}$ $\displaystyle=(1+c)h_{i}-cg_{i}$ (11)

Here, $h_{i}$ and $g_{i}$ are assumed to be the fraction of meteors per bin or
per degree; the choice of coefficients ensures that this is also true for
$f_{i}$. Section 3.4.1 discusses the uncertainties associated with these
quantities in detail. The unitless parameter $c$ is a measure of the degree of
sporadic contamination. Because we include $c$ in our fitting process, we do
not require the sporadic activity level (or observational coverage) during the
shower to be equal to that when the shower is not active; in fact, the
activity of the sporadic sources is known to vary seasonally Campbell-Brown &
Jones (2006); Campbell-Brown & Wiegert (2009). However, our approach does
assume that the distribution of sporadic radiants about the shower’s sun-
centered ecliptic geocentric radiant is constant.

Figure 3: The distribution of radiant offsets (relative to the shower
reference) of GMN meteors that satisfy equations 8-10 for the Northern Taurid
(NTA) meteor shower (“shower members”), and those that satisfy these equations
after being shifted in solar longitude (“simulated sporadic contaminants”).

We first fit a Rayleigh distribution to our data; if the vectorial components
of the meteoroids’ velocity vectors are Gaussian, the radiant dispersion can
be approximated as a Rayleigh distribution (see Appendix A of Moorhead et al.,
2020). We assume that the minimum offset is $0^{\circ}$ and that the only free
parameter is the mode of the distribution, $\mu$. The probability density
function is given by:

$\displaystyle\mathrm{PDF}_{1}(\mu,\Theta)$
$\displaystyle=\frac{\Theta}{\mu^{2}}e^{-\Theta^{2}/(2\mu^{2})}$ (12)

However, our histogram bins have finite width, and we wish to take this into
account in performing our fits. Therefore, we fit the following function to
our data:

$\displaystyle\mathrm{CDF}_{1}(\mu,\Theta_{i+1})-\mathrm{CDF}_{1}(\mu,\Theta_{i})$
(13)

where $\Theta_{i}$ and $\Theta_{i+1}$ mark the edges of a given bin, and CDF
is the cumulative distribution function:

$\displaystyle\mathrm{CDF}_{1}(\mu,\Theta)$
$\displaystyle=1-e^{-\Theta^{2}/(2\mu^{2})}$ (14)

Our nominal bin width is 0.2∘, but we later include bin width as a potential
source of uncertainty (see Section 3.4.2).

In many cases, we found that we could not fully replicate the shape of the
data using a Rayleigh distribution. In these cases, we fit a double Rayleigh
distribution to the data, whose probability and cumulative distribution
functions are:

$\displaystyle\mathrm{PDF}_{2}$
$\displaystyle=(1-\gamma)\,\mathrm{PDF}_{1}(\mu_{a},\Theta)+\gamma\,\mathrm{PDF}_{1}(\mu_{b},\Theta)$
(15) $\displaystyle\mathrm{CDF}_{2}$
$\displaystyle=(1-\gamma)\,\mathrm{CDF}_{1}(\mu_{a},\Theta)+\gamma\,\mathrm{CDF}_{1}(\mu_{b},\Theta)$
(16)

where $\gamma$ is a unitless parameter that measures the contribution of the
second Rayleigh distribution to the overall distribution. Figure 4 compares
both distributions – single and double Rayleigh – with the noise-subtracted
distribution of radiant offsets measured for nine showers. In most cases, the
data have a shallower slope on the right side of the peak than a single
Rayleigh distribution predicts, and thus are better fit by a double Rayleigh
distribution (see Section A.2 for an analysis of the fit residuals). The
Quadrantids (QUA), sigma Hydrids (HYD), and alpha Capricornids (CAP) are
exceptions; use of a double Rayleigh results in no substantial decrease in the
reduced chi-squared statistic (see table 4). However, these three showers have
some of the poorest number statistics out of the set we analyze. Therefore,
throughout this paper we order showers by the number of meteors included in
the nominal fit (see Table 2), so that more reliable fits are located at the
top of figures and tables.

Figure 4: The noise-subtracted distribution of radiant offsets measured by GMN
for twelve meteor showers (black points). For each shower, we also display the
best-fitting Rayleigh distribution (solid blue line) and, if applicable,
double Rayleigh distribution (dashed red line).

We find the best fit Rayleigh or double Rayleigh distribution by minimizing
the chi-squared statistic, an approach that will be very familiar to most
readers. However, we would like to note that one of our fit parameters, $c$,
is not contained within the Rayleigh and double Rayleigh distributions
discussed above. Therefore, we minimize the following quantity:

$\displaystyle\chi^{2}$
$\displaystyle=\sum_{i}{\frac{\left(\mathrm{CDF}(\Theta_{i+1})-\mathrm{CDF}(\Theta_{i})-[(1+c)h_{i}-cg_{i}]\right)^{2}}{(1+c)^{2}\,\sigma_{h_{i}}^{2}+c^{2}\,\sigma_{g_{i}}^{2}}}$
(17)

In the case of a single Rayleigh distribution, we fit two parameters: $\mu$
and $c$. In the case of a double Rayleigh distribution, we fit four
parameters: $\mu_{a}$, $\mu_{b}$, $\gamma$, and $c$.

###  3.4 Uncertainties and sensitivity analysis

####  3.4.1 Uncertainty estimation

We have chosen to fit Rayleigh distributions to a histogram of our data
values. Histogram values are sometimes assigned uncertainties of
$\sigma\approx\sqrt{n_{i}}$, where $n_{i}$ is the number of data points that
fall within bin $i$; this is the root of the Poisson variance $n_{i}$.
However, the probability that a value falls into a given bin is better
described by a binomial distribution. If the probability that a meteor falls
within bin $i$ is $p_{i}$ and the total number of observed meteors is $N$, the
variance is $p_{i}(1-p_{i})/N$ under the normal approximation to a binomial
distribution. When $p_{i}N=n_{i}$ is small, this approximation in turn breaks
down.

We therefore use half the width of a one-$\sigma$ Wilson-Agresti-Coull (WAC;
Agresti & Coull, 1998; Ott & Longnecker, 2016) confidence interval to compute
the uncertainty associated with our histogram bins:

$\displaystyle\sigma_{h_{i}}$
$\displaystyle\simeq\sqrt{\tilde{h}_{i}(1-\tilde{h}_{i})/(N+1)},\,\text{where}$
(18) $\displaystyle\tilde{h}_{i}$
$\displaystyle=\left(n_{i}+\tfrac{1}{2}\right)/(N+1)$ (19)

This approximation allows us to include $n_{i}=h_{i}=0$ bin values in our chi-
squared minimization. (We do not, however, replace $h_{i}$ with
$\tilde{h}_{i}$, as this would artificially inflate the tail of the
distribution.)

As discussed in Section 3.3, we fit Rayleigh distributions to noise-subtracted
data, $f_{i}$ (see equation 11). The uncertainty associated with these data is
therefore:

$\displaystyle\sigma_{f_{i}}$
$\displaystyle=\sqrt{(1+c)^{2}\,\sigma_{h_{i}}^{2}+c^{2}\,\sigma_{g_{i}}^{2}}\,.$
(20)

####  3.4.2 Bootstrap uncertainties

Figure 5: Distribution of simulated meteor radiants (left) following a
Rayleigh distribution with a mode of 0.51∘ and uncertainties in $\lambda_{g}$
and $\beta_{g}$ taken from the observations. The right panel compares the true
underlying spread in radiant (true $\mu_{1}$) with the apparent spread in
radiant (apparent $\mu_{1}$).

Many fitting routines either return parameter uncertainties or a covariance or
inverse Hessian matrix. However, because our algorithm contains multiple steps
(a least-squares drift fit followed by a radiant offset distribution fit), we
estimated our uncertainties using a classic bootstrap approach. From our
initial sample of meteors satisfying equations 1–4, we generated 1000 random
samples of the same size, sampled with replacement. Simultaneously, we
randomly vary the radiant and geocentric velocity drift using the covariance
matrix corresponding to the best fit. Each bootstrap sample was subjected to
our fitting algorithm. We then computed the difference between the 16th and
84th percentiles of each bootstrapped parameter (drift slopes, Rayleigh modes,
noise level, etc.) and assume that this is equivalent to twice the standard
deviation in each case.

The above process allows us to characterize the uncertainty in our fitting
process. However, we also make a number of somewhat arbitrary decisions that
can potentially further alter our best fit. These decisions include the number
and width of bins in the histograms and the number of days of shower activity
analyzed. We conducted a crude sensitivity analysis to determine whether these
sources of error are significant. We found that the total width of our
histogram made a negligible difference in the results, so we keep that fixed
at 10∘. However, the bin width and activity window produced variations in
$\mu_{1}$ that were similar to those produced by the bootstrap alone.

We decided to incorporate all significant sources of error by conducting a
combined random selection and parametric bootstrap in which each fit
iteration:

  1. 1.

uses a shower duration that is randomly selected from the set of duration
values that encompass at least 50 shower members but $\lesssim 90$% of the
estimated total number of shower members,

  2. 2.

uses a histogram bin size selected randomly from the interval [0.1∘, 0.3∘],
and

  3. 3.

is performed on a random sample of the shower members.

We again used the middle 68% of bootstrap fit results to define our parameter
uncertainties; this is equivalent to providing 1-$\sigma$ uncertainties for a
normal distribution.

####  3.4.3 The effects of radiant measurement uncertainties

In general, we use a bootstrap approach to estimate the contributions that
various parameters make to our fit uncertainties (Section 3.4.2). However, the
formal uncertainties on the radiant measurements will systematically increase
the measured radiant dispersion, and it is not possible to reverse this
process using a parametric bootstrap.

Therefore, we instead measure the degree of perceived broadening that the
radiant measurement uncertainties add to an assumed Rayleigh distribution. We
determine this by simulating an idealized set of radiants that are evenly
distributed about the nominal shower radiant (that is,
$\Delta\lambda_{\mathrm{ref}}(\lambda_{0})$,
$\beta_{\mathrm{ref}}(\lambda_{0})$). We then vary these simulated radiants by
the measured radiant uncertainties – we generate one simulated radiant per
measured radiant – and measure the difference between the two distributions.
The left panel of Fig. 5 illustrates this process for an assumed Rayleigh
distribution with a mode of 0.51∘. We see that, after applying our parametric
bootstrap, the apparent mode is slightly larger.

We repeated this process for “true” Rayleigh distribution mode values ranging
from 0.05∘ to 1∘. We fit a double Rayleigh distribution to the apparent
distributions, although for values of $\mu_{1}$ greater than $\sim 0.1$, the
second component of the best-fit double Rayleigh was very weak or nonexistent
(that is, $\gamma\lesssim 0.1$). The relationship between the true mode and
the mode of the main component of the best-fit double Rayleigh is shown in the
right panel of Fig. 5. We find that [1] the radiant uncertainties, if
accurate, contribute very little to the measured spread and [2] we can obtain
the true value of $\mu_{1}$ from the apparent value using a simple linear
expression:

$\displaystyle\mu_{1}$
$\displaystyle=1.029\mu_{1,\,\textrm{app}}-0.036^{\circ}~{}~{}(\text{for}~{}\mu_{1}<1)$
(21)

where $\mu_{1,\,\textrm{app}}$ is the apparent or measured value.

For larger values of $\mu_{1}$ ($\mu_{1}\gtrsim 1$), the broadening of the
peak due to radiant measurement uncertainty is undetectable. We fare the two
regimes together as follows:

$\displaystyle\mu_{1}$
$\displaystyle=\begin{cases}~{}1.029\mu_{1,\,\textrm{app}}-0.036^{\circ}&\mu_{1,\,\textrm{app}}<1.2413^{\circ}\\\
~{}\mu_{1,\,\textrm{app}}&\mu_{1,\,\textrm{app}}\geq
1.2413^{\circ}\end{cases}$ (22)

where the boundary value of
$\mu_{1,\,\textrm{app}}=1.2413^{\circ}=0.036^{\circ}/(1.029-1)$ is simply the
value for which the two parts of equation 22 are equal. All values presented
in Section 4 incorporate this correction.

If the radiant uncertainties are underestimated, this effect could be larger.
For instance, Vida et al. (2021) suggests that the uncertainties might be
underestimated by as much as a factor of 2; if we multiply all uncertainties
by 2, the slope of equation 21 increases to 1.067. However, this would still
be a fairly minor source of error in our fit parameters.

####  3.4.4 The correlation between radiant dispersion and shower duration

Meteor showers contain particles of varying sizes and of varying ages. Small
particles will typically be ejected from their parent comets at greater speeds
and will be more affected by radiation pressure and Poynting-Robertson drag.
Over time, the orbits of particles of all sizes will tend to diverge. If a
meteoroid stream contains a component that is more evolved, we would expect
this to be reflected in both the radiant dispersion and the activity duration.
For instance, Jenniskens et al. (1998) found that the radiant dispersion of
Perseid meteors was smaller during an outburst than it is in a typical year. A
similar effect has been reported for the mass distribution within a shower;
Arlt (2000) report a larger proportion of bright meteors near the peak of
Taurid activity, particularly during outburst years.

Figure 6: Bootstrap fit results (blue dots) for the median angular dispersion
as a function of the width of the activity interval used for the fit. An
exponential fit to the data is shown as a dashed line, and the extrapolated
value corresponding to one night’s worth of data is marked with a black star.

The number statistics for most of our showers do not permit us to control for
this effect by restricting our analysis to one night of observations. However,
our bootstrap uncertainty analysis does include several variations per shower
of the number of days of data analyzed. We can use these bootstrap results to
test whether there is a correlation between the duration of the analysis
window and the derived radiant dispersion. Figure 6 shows such an analysis for
all twelve meteor showers.

Nine of our showers show a correlation between the duration of the activity
window and the median dispersion. We see that, for these showers, the
dispersion is lowest when the analysis window is shortest, and plateaus to a
larger value for longer analysis windows. A simple linear fit does not capture
this behavior; instead, we fit the following equation to the results:

median $\displaystyle=A+B\cdot C^{\text{dur}_{\text{max}}/2}$ (23)

where dur${}_{\text{max}}$ is the width of the analysis window, $A$ and $B$
are positive constants, and $C$ is a constant between 0 and 1. This formula is
a reasonable description of the relationship between activity window duration
and median dispersion for the Perseids (PER), Geminids (GEM), Orionids (ORI),
eta Aquariids (ETA), Northern Taurids (NTA), April Lyrids (LYR), Southern
delta Aquariids (SDA), Quadrantids (QUA), and Leonids (LEO): data and trends
are shown in Fig. 6. The Southern Taurids (STA), sigma Hydrids (HYD), and
alpha Capricornids (CAP) did not show clear signs of such a trend, but we also
note that these three showers have the largest spread in bootstrap solutions.

##  4 Results

This section summarizes our radiant dispersion measurements for the twelve
showers we considered. Many of these results were depicted graphically in
Section 3, but here we provide additional details, compare our results with
previous measurements, and discuss the implications for stream shape.

###  4.1 Sun-centered ecliptic radiant drift

Most showers displayed a statistically significant drift in Sun-centered
ecliptic radiant when we set our significance level at 0.05. Table 1 gives our
best fit and corresponding uncertainties for the constants listed in equations
5–7. In those cases where the null hypothesis (i.e., that there is no
correlation between the parameter and solar longitude) could not be ruled out
at the 95% confidence level, the best-fit slope appears in gray. We have still
chosen to present these fit values, however, because they are a key component
of our bootstrap uncertainty analysis.

|  | $\lambda_{g}-\lambda_{\odot}$ | $\beta_{g}$ | $v_{g}$
---|---|---|---|---
shower | $\lambda_{0}$ | $b_{\lambda}$ | $m_{\lambda}$ | $b_{\beta}$ | $m_{\beta}$ | $b_{v}$ | $m_{v}$
code | (∘) | (∘) |  | (∘) |  | (km s-1) | (km s-1 deg-1)
PER | 140.10 | 283.531(41) | 0.068(14) | 38.362(31) | -0.059(10) | 58.680(36) | -0.011(11)
GEM | 262.20 | 207.995(12) | -0.092(17) | 10.466(12) | -0.103(15) | 33.881(22) | 0.088(29)
ORI | 209.20 | 246.670(25) | -0.242(09) | -7.557(17) | 0.063(06) | 65.548(48) | -0.087(16)
ETA | 45.12 | 293.813(19) | -0.254(06) | 7.696(21) | 0.063(07) | 65.508(48) | 0.058(13)
NTA | 235.80 | 189.780(66) | -0.260(10) | 2.613(45) | 0.014(06) | 27.064(51) | -0.138(08)
STA | 218.00 | 191.821(98) | -0.213(11) | -4.911(47) | -0.021(06) | 27.192(96) | -0.066(10)
LYR | 32.50 | 240.927(86) | 0.64(16)0 | 56.639(42) | -0.22(10)0 | 46.657(54) | 0.233(72)
SDA | 126.70 | 208.629(48) | -0.237(29) | -7.488(35) | -0.123(18) | 40.419(66) | -0.196(29)
QUA | 283.00 | 275.67(24) |  0.22(59)0 | 63.458(56) |  0.17(13)0 | 40.851(68) |  -0.01(17)0
HYD | 254.80 | 231.099(83) | -0.098(18) | -16.421(53) | -0.001(14) | 58.766(94) | -0.059(22)
LEO | 235.50 | 272.525(38) | -0.295(21) | 10.225(43) | -0.169(21) | 69.839(85) | 0.056(27)
CAP | 125.48 | 179.27(14) | -0.359(56) | 9.743(64) | 0.104(19) | 22.364(69) | -0.158(23)
Table 1: Solar longitude of peak activity (as observed by GMN cameras;
$\lambda_{0}$) and linear fits for drift in Sun-centered ecliptic radiant
($\lambda_{g}-\lambda_{\odot}$, $\beta_{g}$) and geocentric speed ($v_{g}$)
around this time. The fit parameters correspond to equations 5–7, where $m$ is
used for slope and $b$ is used for the “intercept” (value when
$\lambda_{\odot}=\lambda_{0}$). Slopes that lie within two $\sigma$ of zero
appear in gray.

Eleven out of twelve showers exhibit a statistically significant drift in
either $\lambda_{g}-\lambda_{\odot}$ or $\beta_{g}$, with one exception: the
Quadrantids (QUA). However, the Quadrantids are also very brief in duration,
which makes the detection of a drift difficult and also less critical for
measuring radiant offsets.

###  4.2 Nominal radiant dispersions

Table 2 lists our nominal radiant dispersion fit parameters for each meteor
shower. We include the best-fit parameters for equation 12 or equation 15,
depending on whether a Rayleigh or double Rayleigh provides a better fit to
the data, and the corresponding mode and median dispersion. The last two
columns provide the median dispersion from Kresák & Porubčan (1970) and an
estimate of the median dispersion derived from Whipple & Wright (1954).

shower |  $N$ | $\mu_{1}$ | $\mu_{2}$ | $\gamma$ |  $c$ | mode | median | Kresák & Porubčan | Whipple & Wright
---|---|---|---|---|---|---|---|---|---
|  | (∘) | (∘) |  |  | (∘) | (∘) | median (∘) | adjusted (∘)
PER | 1445 | 0.77(10) | 1.53(31) | 0.39(14) | 0.009(12) | 0.84(08) | 1.14(11) |  01.26 ($N=254$) |  01.02 ($N=109$)
GEM | 1279 | 0.26(03) | 0.53(06) | 0.35(12) | 0.001(01) | 0.28(02) | 0.38(02) | 0.49 ($N=81$) | 0.33 ($N=46$)
ORI | 777 | 0.38(03) | 0.88(16) | 0.32(11) | 0.018(09) | 0.40(03) | 0.55(02) | 0.84 ($N=30$) | 0.57 ($N=28$)
ETA | 441 | 0.21(03) | 0.49(07) | 0.37(11) | 0.018(05) | 0.22(02) | 0.32(02) | – | –
NTA | 410 | 0.61(12) | 1.32(20) | 0.45(19) | 0.120(24) | 0.67(12) | 0.97(11) | 1.94 ($N=25$) | 1.58 ($N=25$)
STA | 345 | 0.64(21) | 1.44(18) | 0.78(17) | 0.064(45) | 0.89(23) | 1.41(12) | 2.20 ($N=46$) | 0.86 ($N=47$)
LYR | 333 | 0.41(06) | 1.09(18) | 0.38(14) | 0.020(08) | 0.43(06) | 0.64(05) | 0.25 ($N=7$)0 | –
SDA | 285 | 0.52(16) | 1.28(48) | 0.20(30) | 0.043(12) | 0.54(12) | 0.70(06) | 1.41 ($N=27$) | 1.20 ($N=40$)
QUA | 253 | 1.05(08) | – | – | 0.016(35) | 1.05(08) | 1.23(09) | 1.13 ($N=16$) | –
HYD | 235 | 0.74(07) | – | – | 0.10(11)0 | 0.74(07) | 0.87(08) | – | –
LEO | 170 | 0.16(06) | 0.49(09) | 0.65(21) | 0.036(27) | 0.18(07) | 0.38(05) | 0.32 ($N=9$) | 0.33 ($N=32$)
CAP | 115 | 0.59(10) | – | – | 0.187(61) | 0.59(10) | 0.69(12) | – | –
Table 2: Radiant distribution fit parameters and their corresponding median
and mode for nine showers observed by GMN. Bootstrap uncertainties are
provided in parentheses, and showers are sorted by decreasing number of
observed meteors. The Quadrantids (QUA), sigma Hydrids (HYD), and alpha
Capricornids (CAP) are best described by a single Rayleigh distribution and so
have no values quoted for $\mu_{2}$ and $\gamma$. Kresák & Porubčan (1970)
median angular offset values are provided in the second-to-last column, and
Whipple & Wright (1954) values, corrected using the factor in equation 24 in
order to obtain a value equivalent to the median dispersion, are provided in
the last column.

We found that the Rayleigh and double Rayleigh fits for most showers had very
similar values for the median offset angle, so this is the most stable measure
of the dispersion. However, Moorhead et al. (2020) found that the mode was a
more appropriate choice for computing the effects of gravitational focusing,
and therefore we provide both statistics.

We have included prior dispersion measurements from Kresák & Porubčan (1970)
for several showers. However, this earlier work was based on many fewer
meteors in most cases; for instance, we have almost 50 times as many Lyrid
radiants on which to base our analysis. Of the showers considered by this
earlier work, only the Perseids would have satisfied our threshold for
analysis ($N>100$). We note that in this case, our measurement agrees with
that of Kresák & Porubčan (1970) in that they differ by about 1 $\sigma$.

We have also included radiant dispersion measurements from Whipple & Wright
(1954), adjusted to account for differences in methodology. Whipple & Wright
cite mean rather than median deviations; we multiply by a factor of
$\sqrt{\pi/4\ln 2}$ to account for the difference between the median and the
mean of a Rayleigh distribution. We multiply by an additional factor of
$\sqrt{2}$ to account for the fact that Whipple & Wright (1954) measures only
one of two radiant offset dimensions. Thus, the total correction factor is:

$\displaystyle\sqrt{\frac{\pi}{2\ln 2}}$ $\displaystyle\simeq 1.5$ (24)

This factor has been applied to obtain the values in the last column of Table
2. After applying this correction, the Whipple & Wright (1954) results for the
Perseids, Geminids, Orionids, and Leonids are within one-to-two $\sigma$ of
our measurements; however, the Taurid and Southern delta Aquariid values are
substantially different. Discrepancies between our values, those of Kresák &
Porubčan (1970), and those of Whipple & Wright (1954) indicate that the
Taurids in particular may be difficult to characterize. Additionally, all
three studies measure disparate dispersions for the Northern and Southern
Taurids, but do not all measure a larger dispersion for the same branch. Both
Kresák & Porubčan (1970) and this work measure larger dispersions for the
Southern Taurids, while Whipple & Wright (1954) measures a larger dispersion
for the Northern Taurids.

Finally, we note that in Moorhead et al. (2020), we obtained a crude
measurement of the radiant dispersion of Orionid and Perseid meteors using the
first 18 months out of the 30 months of data used in this work. This earlier
attempt does not subtract sporadic contamination or account for radiant
measurement uncertainties, and fits a single Rayleigh distribution to a
cropped portion of the radiant dispersion data using a maximum likelihood
estimation algorithm. The inclusion of a larger data set and refinement of our
fitting algorithms resulted in a measured mode that is smaller for both
showers: we measure the mode of the Perseid offsets as $0.84\pm 0.08^{\circ}$
instead of 1.02∘, and for the Orionids we now obtain $0.40\pm 0.03^{\circ}$
instead of 0.52∘

###  4.3 Radiant dispersions on the most active night

The bootstrap fits for nine of our showers exhibited a correlation between the
number of days of shower activity analyzed and the median radiant dispersion
(see Section 3.4.4 and Fig. 6). We use these fits to estimate the median
radiant dispersion on the night that yielded the largest number of GMN
observations; these values are listed in Table 3.

We note that equation 23 was concocted to resemble the behavior seen in Fig. 6
and has no physical motivation. Therefore, the numbers in Table 3 should be
considered crude estimates of how tight the dispersion may be on the most
active night, and are presented without formal uncertainties.

###  4.4 Shower duration and radiant dispersion

One might expect that both radiant dispersion and shower duration probe the
spread of orbital elements within a meteor shower and thus could be
correlated. Whipple & Wright (1954) reported such a correlation for
measurements of the Draconids, Leonids, Geminids, Orionids, Perseids, delta
Aquariids, and Northern and Southern Taurids. We test for this correlation in
our data by comparing our median radiant dispersions with the duration of each
shower.

We do not measure shower duration from this data, as we have not corrected for
issues such as gaps in data collection due to cloudy weather. Instead, we will
rely on existing measures of the showers’ activity profiles, which are often
modeled as a double exponential function: (Jenniskens, 1994; Moorhead et al.,
2017):

ZHR
$\displaystyle=\text{ZHR}_{0}\times\begin{cases}10^{+B_{p}(\lambda_{\odot}-\lambda_{0})}&\lambda_{\odot}<\lambda_{0}\\\
10^{-B_{m}(\lambda_{\odot}-\lambda_{0})}&\lambda_{\odot}\geq\lambda_{0}\end{cases}$
(25)

Here, ZHR refers to the zenithal hourly rate, ZHR0 is the rate at the shower’s
peak, $\lambda_{0}$ is the solar longitude corresponding to the peak, and
$B_{p}$ and $B_{m}$ are parameters that govern how steeply activity rises in
advance of the peak or drops after it.

shower | nom. window | nom. median | median1
---|---|---|---
code | (days) | (∘) | (∘)
PER | 9 | 1.14(11) | 0.93
GEM | 3 | 0.38(02) | 0.34
ORI | 7 | 0.55(02) | 0.53
ETA | 9 | 0.32(02) | 0.28
NTA | 15 | 0.97(11) | 0.55
STA | 19 | 1.41(12) | –
LYR | 3 | 0.64(05) | 0.58
SDA | 5 | 0.70(06) | 0.64
QUA | 1 | 1.23(09) | 1.19
HYD | 15 | 0.87(08) | –
LEO | 7 | 0.38(05) | 0.18
CAP | 7 | 0.69(12) | –
Table 3: Estimated median meteor shower radiant dispersion on the night when
the largest number of meteors were detected by GMN (median1). We compare these
values with the median angular offset of our nominal fit (taken from Table 2)
and provide the duration of the activity window used in our nominal fit.

The full duration at half maximum (FDHM) of such an activity curve has the
following analytic solution:

FDHM
$\displaystyle=\left(\frac{1}{B_{p}}+\frac{1}{B_{m}}\right)\cdot\log_{10}2$
(26)

Some showers, such as the Geminids and Perseids, are better described by a
superposition of two double-exponential curves; in these cases, we solve for
the FDHM numerically. In general, we take the $B_{p}$ and $B_{m}$ parameters
from Moorhead et al. (2017), except in the case of the Geminids. We conducted
spot-checks of the radar profiles against publicly available optical meteor
fluxes333https://www.imo.net/members/imo_live_shower444https://meteorflux.org
and find that, except for the Geminids, the Moorhead et al. (2017) profiles
are also a good fit to optical data. Because optical Geminid activity tends to
be briefer than radar Geminid activity, we use the activity parameters
computed by Jenniskens (1994) using visual Geminid observations.

We use equation 24 of Kresák & Porubčan (1970) to compute the “width” of a
meteoroid stream from its duration:

$\displaystyle S$
$\displaystyle=r_{h}\tan\left(\text{FDHM}/2\right)\sin\epsilon_{h}$ (27)

where $r_{h}\simeq 1$ au is the heliocentric distance at the time of
observation and $\epsilon_{h}$ is the angle between the apex direction and the
heliocentric radiant. In Fig. 7, we compare this quantity with both the median
geocentric radiant dispersion and the equivalent heliocentric radiant
dispersion, which is given by

$\displaystyle\Theta_{\text{med},\,h}$
$\displaystyle=\tan^{-1}\left(\frac{v_{g}}{v_{h}}\tan\Theta_{\text{med},\,g}\right)$
(28)

where $\Theta_{\text{med}}$ is used to indicate the median angular offset and
$v_{h}$ is the heliocentric speed of the stream. Note that we plot the median
dispersion within our nominal window for each shower (see Table 3 for these
durations). The radiant dispersion for the entire stream may be a little
broader than that for our nominal windows, as indicated by Fig. 6, but the
difference is small for most showers.

Figure 7: A comparison of the median geocentric radiant dispersion (top) and
heliocentric radiant dispersion (bottom) with a measure of the stream’s width
derived from its duration.

We do not observe a correlation between either geocentric or heliocentric
radiant dispersion and stream width/duration. A Spearman rank-order
correlation test fails in both cases. Thus, we do not reproduce the result of
Whipple & Wright (1954). This is likely due to the difference in methodology
in determining the shower duration. Whipple & Wright (1954) note that their
durations are highly uncertain and are based on the elapsed time between the
first observed meteor attributed to a shower and the last; this quantity will
depend on the level of activity as well as the shape of the time profile. For
instance, their Perseid duration is approximately 27 days, while ours is about
2 days. Furthermore, they lack dispersion measurements for the Quadrantids and
eta Aquariids, whose placements in Fig. 7 tend to inhibit a correlation.

This has several implications: first, shower duration cannot be used to
predict the radiant dispersion of a shower. Second, variations in the ratio of
shower duration to radiant dispersion may indicate that the cross section of
some streams are elliptical or asymmetrical (Vaubaillon et al., 2005). These
two quantities could also potentially be used as an independent constraint on
dynamical models of meteoroid streams.

##  5 Conclusions

We have presented measurements of Sun-centered ecliptic radiant drift and
radiant dispersions of meteors belonging to twelve major showers: the
Perseids, Geminids, Orionids, eta Aquariids, Northern Taurids, Southern
Taurids, April Lyrids, Southern delta Aquariids, Quadrantids, sigma Hydrids,
Leonids, and alpha Capricornids. These are the first radiant dispersion
measurements for the eta Aquariids, sigma Hydrids, and alpha Capricornids. Our
measurements are based on high quality meteor trajectories from the Global
Meteor Network which, critically, include formal measurement uncertainties.

Meteor radiants are known to drift in right ascension and declination over
time, and linear approximations of these drifts are published in observer
tables. Meteor astronomers therefore often use Sun-centered ecliptic longitude
and latitude to describe the position of a meteor shower, as shower radiants
tend to be more stationary in this non-inertial coordinate system. However, we
have measured small but statistically significant drifts in Sun-centered
ecliptic radiant for eleven of the twelve showers considered here. Ten showers
also show signs of a drift in geocentric velocity.

While the distribution of radiant offsets is similar to a Rayleigh
distribution (as established by Moorhead et al., 2020), most showers show a
slight-to-moderate excess of meteors towards the tail-end of the distribution.
We have found that a double Rayleigh distribution is better able to replicate
this shape. We have not offered a physical justification for the double
Rayleigh distribution, but possible explanations are that streams consist of
meteoroid populations of different sizes or ages that have varying
dispersions, or that the stream cross-section is not axisymmetric.

We have placed particular emphasis on determining the smallest plausible
dispersion; our motivation in doing so is to enable conservativism in applying
these results to spacecraft risk predictions (Moorhead et al., 2020). While
radiant measurement errors have a modest effect on the radiant dispersion (see
Section 3.4.3), the duration of the analysis window appears to have a
significant effect. We have estimated the one-night radiant dispersion for our
showers and find that the Northern Taurids and Leonids could have a
significantly more compact radiant near the time of peak activity.

Our results suggest that radiant dispersion measurements could offer new
constraints on meteoroid stream structure. Unlike Whipple & Wright (1954), we
do not observe a simple linear correlation between radiant dispersion and
stream width (as determined from shower duration). This opens up the
possibility that some of these meteoroid streams may be elliptical in cross-
section. This is supported by other studies: for instance, both observations
(Vida, 2020) and models (Egal et al., 2020) of the Orionid meteor shower
indicate a complex radiant structure. An apparently asymmetric Orionid radiant
distribution is also presented in Fig. 1 of Moorhead et al. (2020). Thus, we
encourage additional characterizations of meteor shower radiant distributions.
Future improvements could include investigations of whether this dispersion
varies with limiting meteor magnitude, and the degree of asymmetry present in
the stream cross section (a somewhat tricky measurement for a group of points
that slowly drifts across a sphere). We also suggest that measured radiant
distributions be compared with those predicted by meteoroid stream models.

## Acknowledgments

This work was supported in part by NASA Cooperative Agreement 80NSSC18M0046,
by the Natural Sciences and Engineering Research Council of Canada, and by
Jacobs contract 80MSFC18C0011. Initial funding for the 16 mm GMN stations was
provided by the Istria County Association of Technical Culture.

The authors would like to thank the following GMN station operators and
contributors whose stations provided the data used in this work (in
alphabetical order): Victor Acciari, Alexandre Alves, Željko Andreić, Georges
Attard, Roger Banks, Hamish Barker, Ricky Bassom, Richard Bassom, Jean-
Philippe Barrilliot, Dr. Ehud Behar, Josip Belas, Alex Bell, Serge Bergeron,
Arie Blumenzweig, Ventsislav Bodakov, Ludger Börgerding, Claude Boivin, Bruno
Bonicontro, Fabricio Borges, Dorian Božičević, Martin Breukers, John Briggs,
Laurent Brunetto, Tim Burgess, Peter Campbell-Burns, Pablo Canedo, Seppe
Canonaco, Jose Carballada, Gilton Cavallini, Brendan Cooney, Edward Cooper,
Dino Čaljkušić, Tim Claydon, Manel Colldecarrera, Christopher Curtis, Ivica
Ćiković, Jürgen Dörr, Chris Dakin, Alfredo Dal’Ava Júnior, Steve Dearden,
Christophe Demeautis, Paul Dickinson, Ivo Dijan, Tammo Jan Dijkema, Pieter
Dijkema, Stacey Downton, Zoran Dragić, Jean-Paul Dumoulin, Garry Dymond, Robin
Earl, Ollie Eisman, Carl Elkins, Peter Eschman, Rick Fischer, Richard Fleet,
Jim Fordice, Mark Gatehouse, Megan Gialluca, Jason Gill, Hugo González, Philip
Gladstone, Uwe Glässner, Nikola Gotovac, Neil Graham, Bob Greschke, Sam Green,
Daniel J. Grinkevich, Larry Groom, Tioga Gulon, Dominique Guiot, Margareta
Gumilar, Pete Gural, Kees Habraken, Erwin Harkink, Ed Harman, Tim Havens,
Richard Hayler, Alex Hodge, Bob Hufnagel, Russell Jackson, Jean-Marie
Jacquart, Ron James Jr., Ilya Jankowsky, Klaas Jobse, Dave Jones, Vladimir
Jovanović, Milan Kalina, Jonathon Kambulow, Richard Kacerek, Steve Kaufman,
Alex Kichev, Jean-Baptiste Kikwaya, Zoran Knez, Dan Klinglesmith, Danko Kočiš,
Korado Korlević, Stanislav Korotkiy, Josip Krpan, Zbigniew Krzeminski, Patrik
Kukić, Reinhard Kühn, Gaétan Laflamme, David Leurquin, Anton Macan, John
Maclean, Igor Macuka, Mirjana Malarić, Nedeljko Mandić, Bob Marshall, José
Luis Martin, Colin Marshall, Andrei Marukhno, Keith Maslin, Bob Massey, Damir
Matković, Michael J. Mazur, Sergio Mazzi, Alex McConahay, Robert McCoy,
Charlie McCromack, Mark J.M. McIntyre, Aleksandar Merlak, Filip Mezak, Pierre-
Michael Micaletti, Matej Mihelčić, Simon Minnican, Wullie Mitchell, Nick
Moskovitz, Gene Mroz, Brian Murphy, Carl Mustoe, Przemek Nagański, Jean-Louis
Naudin, Damjan Nemarnik, Colin Nichols, Zoran Novak, Michael O’Connell,
Washington Oliveira, Thiago Paes, Carl Panter, Lovro Pavletić, Filip Parag,
Igor Pavletić, Richard Payne, Pierre-Yves Pechart, Enrico Pettarin, Alan
Pevec, Patrick Poitevin, Pierre de Ponthière, Alex Pratt, Miguel Preciado,
Chuck Pullen, Dr. Lev Pustil’nik, Chris Ramsay, Danijel Reponj, David
Robinson, Martin Robinson, Heriton Rocha, Herve Roche, Adriana Roggemans, Paul
Roggemans, Alex Roig, James Rowe, Dmitrii Rychkov, Michel Saint-Laurent, Jason
Sanders, Rob Saunders, William Schauff, Ansgar Schmidt, Jay Shaffer, Jim
Seargeant, Ivica Skokić, Dave Smith, Tracey Snelus, James Stanley, Peter
Stewart, William Stewart, Bela Szomi Kralj, Ian Pass, Rajko Sušanj, Damir
Šegon, Marko Šegon, Jeremy Taylor, Yakov Tchenak, Eric Toops, Steve Trone,
Wenceslao Trujillo, Paraksh Vankawala, Martin Walker, Bill Wallace, Didier
Walliang, Jacques Walliang, Christian Wanlin, Tom Warner, Urs Wirthmueller,
Steve Welch, Alexander Wiedekind-Klein, John Wildridge, Bill Witte, Stephane
Zanoni, and Dario Zubović.

## Data availability

The data used in this article are available in the Global Meteor Network
database at https://globalmeteornetwork.org/data/. Readers wishing to
replicate our results should select those trajectories with dates between 10
Dec 2018 and 22 May 2021 (inclusive).

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##  Appendix A Residual analysis

###  A.1 Radiant drift

In Section 3.1, we fit linear trends to the value of a shower’s sun-centered
ecliptic longitude, latitude, and geocentric speed as a function of solar
longitude. Figure 1 presents the results for the Orionid shower; Fig. 8
presents the residuals corresponding to this fit.

Figure 8: Residuals corresponding to our best-fit linear drifts in Orionid
radiant and speed. The top row presents residuals-vs-fits plots for sun-
centered ecliptic longitude (left), ecliptic latitude (center), and geocentric
speed (right). Those meteors that lie more than 2.5∘ from the best-fit lines
appear in blue. The middle row presents normal probability plots of the
residuals, including all data; the bottom row presents normal probability
plots of the residuals, including only points that lie within 2.5∘ of the
best-fit lines.

We can see from the top row of Fig. 8 that there are no obvious signs of non-
linearity in the data, nor are there signs of unequal variance or residual
correlations. Thus, three of the four conditions for using least-squares
fitting appear to be satisfied. However, the normal probability plot in the
middle row of Fig. 8 indicates that the residuals are not normally
distributed.

The distribution of the residuals of the radiant longitude and latitude appear
to have heavier tails than a normal distribution would. This is because there
are two populations contributing to the distribution: the meteor shower and
the sporadic background. We find that if we clip the data to those meteors
that lie within 2.5∘ of our nominal best-fit radiant drift line (which appear
in red in the scatter plots of Fig. 8), the resulting residual distribution
much more closely resembles a normal distribution (see bottom row of Fig. 8).
This cutoff appears to work well for all showers considered in this paper, and
motivates our clip-and-iterate approach to fitting linear trends to the
radiant and speed drifts.

###  A.2 Radiant offset distribution

In Section 3.3, we fit Rayleigh and double Rayleigh distributions to noise-
subtracted histograms of shower member radiant offsets. In Fig. 9, we display
residuals for each shower relative to both single and double Rayleigh fits.

Figure 9: Standardized residuals (that is, residuals divided by one-$\sigma$
uncertainties) corresponding to our best-fit radiant offset dispersion for all
analyzed showers. The top row of each set presents the standardized residuals
as a function of the measured angular offset for both single and double
Rayleigh fits. The bottom row presents normal probability plots of the
standardized residuals; we mark unity with a black dashed line. Residuals
included in the normal probability plots and reduced $\chi^{2}$ values appear
as filled, rather than open, points.

Figure 9 differs from Fig. 8 in several ways: [1] we present standardized
residuals (i.e., $(f_{i}-\hat{f}_{i})/\sigma_{f_{i}}$) so that readers can
easily see the deviation of the residuals from zero in units of
$\sigma_{f_{i}}$; [2] we present residuals-vs-predictor plots rather than
residuals-vs-fits so that readers can see how the standardized residuals have
smaller variances when the angular offset is large; and [3] the dashed black
lines in the normal probability plots are not fits to the data but instead
correspond to unity, since we would ideally expect the standardized residuals
to follow a standard normal distribution.

The unequal variance in standardized residuals between the core of the angular
offset distribution and its tail is due to the limitations of how we estimate
the uncertainty associated with empty bins (see equation 18). The assignment
of a constant uncertainty to empty bins out in the tail of the distribution
results in over-large uncertainties and thus too-small standardized residuals.
We cannot remove the tail from our fitting algorithm: if anything, we are
under-weighting the tail in our fits. However, the standardized residuals in
the tail will not be useful for assessing the goodness of fit or the normality
of residuals; inclusion of the empty bins will drive down the reduced chi-
squared statistic artificially. Thus, for each shower we locate the minimum
value of $i$ for which $f_{i}\leq 0$ (see equation 11) and use only those
values left of this boundary (which we will call $i_{\mathrm{max}}$) to
compute the normal probability plot of the residuals and the reduced chi-
squared statistic:

$\displaystyle\chi^{2}_{\mathrm{red}}$
$\displaystyle=\frac{\sum_{i=1}^{i_{\mathrm{max}}-1}(f_{i}-\hat{f}_{i})^{2}/\sigma^{2}_{f_{i}}}{i_{\mathrm{max}}-1-n_{\mathrm{par}}}$
(29)

where $n_{\mathrm{par}}=2$ for a Rayleigh distribution and 4 for a double
Rayleigh distribution. Table 4 provides reduced chi-squared statistics for all
fits and showers.

shower | $\chi^{2}_{\mathrm{red},\,1}$ | $\chi^{2}_{\mathrm{red},\,2}$ | evidence for
---|---|---|---
code |  |  | double Rayleigh
PER | 3.90 | 0.93 | strong
GEM | 7.13 | 1.27 | strong
ORI | 4.88 | 1.10 | strong
ETA | 5.73 | 1.13 | strong
NTA | 1.65 | 0.64 | weak
STA | 1.73 | 1.00 | weak
LYR | 3.61 | 1.35 | strong
SDA | 2.26 | 1.01 | strong
QUA | 1.06 | 1.22 | none
HYD | 1.81 | 1.59 | none
LEO | 3.66 | 0.83 | strong
CAP | 2.95 | 2.22 | none
Table 4: Reduced chi-squared statistic corresponding to both single and double
Rayleigh fits ($\chi^{2}_{\mathrm{red},\,1}$ and
$\chi^{2}_{\mathrm{red},\,2}$, respectively) to the angular offset
distribution of each shower analyzed. The last column provides a qualitative
assessment of the strength of the evidence in favor of a double Rayleigh fit
over a single Rayleigh fit.

Using the rule of thumb that $\chi_{\mathrm{red}}^{2}>1$ indicates a poor fit
and $\chi_{\mathrm{red}}^{2}\sim 1$ indicates a reasonable fit, we find that a
double Rayleigh distribution provides a much better fit for seven showers
(PER, GEM, ORI, ETA, LYR, SDA, and LEO) and a somewhat better fit for the
Taurids (NTA and STA); readers may come to the same conclusion simply by
reviewing Fig. 4. Three showers (QUA, HYD, and CAP) show no evidence of being
better described by a double Rayleigh, although we note that these three are
among the four least-populated showers in our data set.

Even with the exclusion of the tail of the distribution, some showers show
signs of lack-of-fit or non-normality in the residuals. For instance, the
Orionid residuals trend downward as the angular offset approaches zero, and
the data points in the normal probability plot curve away from the unity line.
In other cases, such as the Geminids, there are too few non-tail bins for us
to test the normality of the residuals. Thus, it may be productive to revisit
the shape of the radiant offset distribution of these showers after additional
years of data are collected.