††thanks: current address: National Renewable Energy Laboratory, Golden, CO
80401, USA

# Topological phase transition in an all-optical exciton-polariton lattice

M. Pieczarka<EMAIL_ADDRESS>ARC Centre of Excellence in Future
Low-Energy Electronics Technologies and Nonlinear Physics Centre, Research
School of Physics, The Australian National University, Canberra, ACT 2601,
Australia  Department of Experimental Physics, Wrocław University of Science
and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland  E. Estrecho
ARC Centre of Excellence in Future Low-Energy Electronics Technologies and
Nonlinear Physics Centre, Research School of Physics, The Australian National
University, Canberra, ACT 2601, Australia  S. Ghosh  Division of Physics and
Applied Physics, School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore  M. Wurdack  ARC Centre
of Excellence in Future Low-Energy Electronics Technologies and Nonlinear
Physics Centre, Research School of Physics, The Australian National
University, Canberra, ACT 2601, Australia  M. Steger  Department of Physics
and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA  D. W.
Snoke  Department of Physics and Astronomy, University of Pittsburgh,
Pittsburgh, PA 15260, USA  K. West  Department of Electrical Engineering,
Princeton University, Princeton, NJ 08544, USA  L. N. Pfeiffer  Department of
Electrical Engineering, Princeton University, Princeton, NJ 08544, USA  T. C.
H. Liew  Division of Physics and Applied Physics, School of Physical and
Mathematical Sciences, Nanyang Technological University, Singapore 637371,
Singapore  A. G. Truscott  Laser Physics Centre, Research School of Physics,
The Australian National University, Canberra, ACT 2601, Australia  E. A.
Ostrovskaya<EMAIL_ADDRESS>ARC Centre of Excellence in Future
Low-Energy Electronics Technologies and Nonlinear Physics Centre, Research
School of Physics, The Australian National University, Canberra, ACT 2601,
Australia

††preprint: topo

Topological insulators [1] are a class of electronic materials exhibiting
robust edge states immune to perturbations and disorder. This concept has been
successfully adapted in photonics [2, 3, 4], where topologically nontrivial
waveguides [5, 6, 7] and topological lasers [8, 9, 10, 11, 17] were developed.
However, the exploration of topological properties in a given photonic system
is limited to a fabricated sample, without the flexibility to reconfigure the
structure in-situ. Here, we demonstrate an all-optical realization of the
orbital Su-Schrieffer-Heeger (SSH) model [18] in a microcavity exciton-
polariton system [19, 20, 21, 22], whereby a cavity photon is hybridized with
an exciton in a GaAs quantum well. We induce a zigzag potential for exciton
polaritons all-optically, by shaping the nonresonant laser excitation, and
measure directly the eigenspectrum and topological edge states of a polariton
lattice in a nonlinear regime of bosonic condensation. Furthermore, taking
advantage of the tunability of the optically induced lattice we modify the
intersite tunneling to realize a topological phase transition to a trivial
state. Our results open the way to study topological phase transitions on-
demand in fully reconfigurable hybrid photonic systems that do not require
sophisticated sample engineering.

Microcavity exciton polaritons (polaritons therein), hybrid quasiparticles
resulting from strong coupling of excitons and photons in a semiconductor
microcavity [19, 20, 21, 22], have emerged as a perfect platform for numerous
applications in nonlinear and topological photonics [23, 24, 4, 25]. These
interacting bosons combine a very low effective mass inherited from cavity
photons with repulsive interactions inherited from excitons, allowing for
bosonic condensation at elevated temperatures.

Taking advantage of the photonic part of a polariton, one can modify the
planar microcavity by various fabrication techniques and realize polariton
trapping potentials, as well as a lattice of coupled traps [26, 27, 25, 17].
Additionally, the TE-TM polarized modes splitting in a planar cavity results
in an effective spin-orbit interaction for polaritons enabling realizations of
topological and flat-band systems [29, 25, 27, 28]. Nevertheless, this
technological approach has a major practical drawback as once the sample is
made, there is little or no room for modification of its properties. This
limits the applications of polariton-based photonic topological devices, where
active control is highly desirable [30, 31, 32, 33].

The solution to this issue lies in using the excitonic component of the
polariton for engineering the trapping potential. Under nonresonant optical
excitation above the semiconductor material bandgap, the pumping laser creates
a high-energy excitonic reservoir [34], which acts as a non-Hermitian
potential that replenishes and repels polaritons due to exciton-polariton
interactions [35, 36, 26]. The excitonic potential can therefore be shaped via
spatially structured laser excitation.

In this work, we employ a spatially structured laser beam, imaged via a
microscope objective onto a planar microcavity sample with embedded GaAs
quantum wells, see Fig. 1(a). Spatial structuring is achieved by selectively
reflecting the laser beam from a programmed digital micromirror device (DMD)
(see Methods). The spatial distribution of the pump effectively creates a
chain of coupled circular traps in the zigzag geometry, first proposed for the
realization of low-dimensional topological systems in plasmonics and
nanophotonics [12, 13, 14, 15, 16]. Exciton polaritons trapped in this
optically induced chain realize the orbital version of the Su-Schrieffer-
Heeger (SSH) model, previously demonstrated in etched samples with coupled
micropillars [17, 27, 37].

Figure 1: Realization of an orbital SSH Hamiltonian with a nonresonant optical
excitation. a, Simplified scheme of the experimental setup for creating
exciton polaritons in an optically-induced trapping potential. Inset presents
the spatial distribution of the laser pump reflected from the sample. Dark
areas correspond to polariton traps with the trap diameter $D_{trap}\approx
5.9~{}\mu$m, arranged into a zigzag chain. White scale bar corresponds to 10
$\mu$m. b,d, Sketch of the orbital SSH model for (b) topologically trivial and
(d) nontrivial cases, realized with the trapped $p$-modes of different
orientations $A,D$. c,e Eigenenergies of a tight-binding Hamiltonian model
corresponding to the $N=6$ chain in (b,d). Edge states within the gap are
indicated with an arrow in panel (e). Inset shows the probability density
distribution of the edge state.

The SSH chain [18] is the simplest realization of a topological insulator in
one dimension (1D). Here, we focus on the SSH model realized with the
$p$-modes of each circular trap, coupled as shown in Fig. 1(b,d). In contrast
to linear chains with alternating distances between the lattice sites [8, 9,
38], the orthogonality of the $p$-modes makes the SSH model valid for
$d_{A}=d_{D}=d$, where $d_{A}$ and $d_{D}$ are the lattice constants in the
antidiagonal ($A$) and diagonal ($D$) directions [17]. This is because the
tunneling amplitudes in these two directions, $t_{A}$ and $t_{D}$, are
different due to the collinear or orthogonal alignment of the $p$-modes with
the axis linking the consecutive traps. As a result, two configurations exist
in finite length chains with an even number of lattice sites, with the
staggering order being trivial $t_{D}>t_{A}$ for diagonal, $p_{D}$, and non-
trivial $t_{D}<t_{A}$ for antidiagonal, $p_{A}$, mode orientations. The two
different configurations (phases) are presented in Fig. 1(b,d). They are
characterized by phase winding numbers $\mathcal{W}=0$ for trivial and
$\mathcal{W}=1$ for topological one (see Methods). The eigenenergies of a
tight binding Hamiltonian model for finite sized chains investigated in this
work are presented in Fig. 1(c,e). The normal phase in Fig. 1(c) is
characterised by a spectrum with a trivial bandgap. The topologically
nontrivial phase, presented in Fig. 1(e), differs from Fig. 1(c), as two
eigenvalues close to zero energy emerge in the bandgap. The corresponding
eigenstates are strongly localized at the edges of the chain [see inset in
Fig. 1(e)] and are an indicator of a topological phase. In short chains, the
spectra are discrete and the edge states can have energies slightly different
from zero, depending on the ratio of the tunnelling amplitudes $t_{D}/t_{A}$.
Nevertheless, the topological properties of the system are maintained.

Figure 2: Position-resolved spectrum and density distribution of exciton
polaritons in the SSH chain. a, Experimentally measured spectrum along the
chain with $N=6$ sites, integrated over the orthogonal direction. b,
Corresponding spectrum obtained by numerical simulations of the mean-field
open-dissipative model, Eq. (4) in Methods. c,d, Experimental spectrum
measured at a position (c) in the centre of the chain, and (d) at the left
edge of the chain. Shaded areas represent the result of fitting with
Lorentzian lines. $p$-bands and $d$-band peaks are in semitransparent blue and
the edge mode peak is coloured in semitransparent red. e-i, Experimental
images of the exciton-polariton emission corresponding to the spatial density
distributions taken at the energies of the (e) upper $p$-band, (g) edge states
(middle of the bandgap) and (i) the lower $p$-band. f-j, Density distributions
obtained by numerical simulations of the model Eq. (4) corresponding to the
panels (e-i). The scale bar corresponds to 10 $\mu$m and the direction of the
energy gradient in the sample is indicated with an arrow.

We investigate the optically-induced SSH chain at pump powers slightly above
the exciton-polariton condensation threshold. In this regime, the condensate
occupies high-symmetry points of the momentum-space band structure (for
details, see the data in Fig. S1 of the Supplementary Information). The
narrowing of the linewidth of the exciton-polariton emission above the
condensation threshold allows us to resolve the ground and excited bands with
the corresponding bandgaps. The low-energy $s$-band is highly populated due to
efficient energy relaxation of polaritons towards the ground state [39]. We
also observe nonzero occupation of excited states forming a higher-energy
band. The corresponding position-resolved spectrum is presented in Fig. 2(a).
The intense signal from the $s$-band at low energies is clearly separated from
the $p$-band by a bandgap. It indicates that the created potential confines
the excited states and realizes the physics discussed in Fig. 1. More
importantly, the $p$-band is split into two sub-bands corresponding to the
bonding (in-phase) and anti-bonding (out-of-phase) coupling between the
lattice sites (in analogy to electron orbitals in molecules). The gap between
the two $p$-bands is larger than the linewidth, which allows us to directly
identify the localized states at the edges of the SSH chain. The energies of
these states lie inside the $p$-band gap, as indicated by the arrows in Fig.
2. The spectral cross-sections in the bulk (i.e., in the middle) and at the
edge of the chain are presented in Figs. 2(c,d), where the shaded areas
present the result of fitting the spectrum with Lorentzian lines. Existence of
the in-gap edge modes confirmed by these measurements is a signature of a
topological phase of the SSH lattice [1, 27, 17, 37]. We note that the
occupation of all bands is not spatially homogeneous, as the sample is
characterized with a intrinsic linear energy gradient due to the spatially
varying thickness of the cavity (see Methods), oriented antidiagonally with
respect to the $x$-axis in the presented data. This effect is captured in our
simulations of the full open-dissipative mean-field model [see Methods, Eq.
(4)], with the results presented in Fig. 2(b), showing an excellent agreement
with the experimental observations. We emphasize that the simulations are
based on experimentally determined parameters of the sample (see Methods). The
energy gradient does not change the topology of the system, as the polariton
states at the individual lattice sites are hybridized, which is reflected in
the opening of the bandgap (see also Fig. S1).

The spatial distribution of the exciton-polariton density in the chain is
obtained by real-space spectral tomography, which enables selective real-space
imaging of the polariton emission intensity at a given energy (see Methods).
The experimental results are presented in Fig. 2(e,g,i) together with the
results of numerical modelling of Eq. (4) in Fig. 2(f,h,j). The lower $p$-band
shows a characteristic spatial distribution of a bonding state, where the
$p$-modes from neighbouring traps overlap [Fig. 2(i,j)]. The upper $p$-band
shows an antibonding character with pronounced density dips between the traps
indicative of the nodes in the probability density distribution [Fig. 2(e,f)].
Finally, the edge states are composed of antidiagonal $p$-mode configuration
forming the topologically nontrivial realization of the SSH model, Fig.
2(g,h).

To test the robustness of the topologically protected edge states, we
imprinted different chains at different positions on the sample with similar
detunings. The edge state for the $N=6$ chain similar to that in Fig. 2, but
with a different orientation with respect to the energy gradient, is presented
in Fig. 3(a) with the corresponding numerical simulation shown in Fig. 3(b).
For this orientation, the diagonal $p$-mode configuration is the topologically
nontrivial one. Similarly to the $N=6$ case, the orbital SSH model with an odd
number of sites supports two edge states, however each edge state comes from a
different $p$-mode configuration [17, 27, 37]. This is clearly seen in our
experimental data for a chain with $N=5$ sites, presented in Fig. 3(c), in
agreement with simulations in Fig. 3(d). These results demonstrate the
insensitivity of the all-optical realization of a topological SSH model to the
orientation of the energy gradient and local disorder of the sample.

Figure 3: Topological edge states in different lattice realizations. a,c,
Experimental spatial density distributions of the edge state for (a) a chain
of $N=6$ sites with a different orientation compared to the chain in Fig. 2
and (c) a chain of $N=5$ sites. b,d, Results of numerical simulations of the
model Eq. (4) corresponding to the cases in (a) and (c). The scale bar
corresponds to 10 $\mu$m and the direction of the energy gradient is indicated
with an arrow. Figure 4: Measurement of the topological phase transition in
SSH chains. a,b, Spatial distributions of the excitation intensity reflected
from the sample for the modified chains in the (a) topological and (b) the
trivial phase. The scale bar corresponds to 10 $\mu$m. c,d, Experimental
spectra corresponding to the cases presented in (a,b). e,f, Numerically
computed spectra corresponding to the experimental measurements in (c,d). g,
Intensity of the exciton-polariton edge state at the right end of the modified
chains as a function of $d_{D}/d_{A}$. Inset shows the intensity distributions
measured in the middle of the $p$-band gap. h, Winding number $\mathcal{W}$
calculated from the tight-binding Hamiltonian, including the energy gradient
and disorder [see Methods, Eqs. (1) and (3)], as a function of $t_{A}/t_{D}$.
The trivial phase is coloured in blue and the topological phase is coloured in
red in (g,h).

To demonstrate an optically-driven topological phase transition in our system,
we modified the topology of the $N=6$ chain by changing the ratio between the
tunneling amplitudes from $t_{A}/t_{D}>1$ to $t_{A}/t_{D}<1$ in the $N=6$
chain. The coupling between the nearest-neighbour sites depends on the
potential barrier amplitude as well as on the distance between the traps [40].
Therefore, we increase the coupling $t_{A}$ or $t_{D}$ by reducing the trap
separations in $A$ or $D$ directions, while keeping all other parameters
constant. In this way, we modified the SSH Hamiltonian for $p_{A}$
configuration. Fig. 4(a,b) presents the intensity distribution of the laser
excitation reflected from the sample for the chains of modified dimerisations.
The chain maintaining the topological phase is presented in Fig. 4(a) and the
chain in the trivial phase is shown in Fig. 4(b). In both cases, the lattice
constant in one of the directions was reduced by 20%. The measured real-space
spectra for these geometries are shown in Fig. 4(c,d). One observes an
increase of the topological and the trivial bandgaps between the $p$-bands in
comparison to the unmodified chain: $\Delta E_{\text{topo}}=174\pm
10~{}\mu\text{eV}$ for $d_{D}/d_{A}=0.8$ and $\Delta E_{\text{triv}}=218\pm
8~{}\mu\text{eV}$ for $d_{D}/d_{A}=1.25$ in comparison to $\Delta
E_{\text{topo}}=151\pm 9~{}\mu\text{eV}$ for $d_{D}/d_{A}=1.0$, from Fig. 2.
Growth of this value for the modified chains is a direct manifestation of the
control over the tunneling amplitudes, as the bandgap in the SSH model is
proportional to $|t_{D}-t_{A}|$ (full set of values is presented in
Supplementary Information). The geometrical modification leading to the change
of the tunneling amplitudes influences the $s$-band as well, which is now
clearly split, and signatures of edge modes are visible in Fig. 4(c). The
asymmetry of the $p$-band gap values is caused by the energy gradient, which
influences the sensitivity of $t_{A}$ and $t_{D}$ to the trap separation
parameter. Additional input to the tunneling amplitudes comes from the
intrinsic TE-TM splitting and weak birefringence of the sample [41, 33].

The cavity gradient influences the occupation of the topological edge states
in the measured spectrum in Fig. 4(c), with one of the edge states dominating
the spectrum. On the other hand, the spectrum in Fig. 4(d) shows no signatures
of the edge states, as expected for a topological phase transition to a
trivial configuration with $\mathcal{W}=0$.

The flexibility of our all-optical potential allows us to tune the lattice
geometry parameters continuously. Thus, we performed a series of measurements
to pinpoint the topological phase transition in the chain. Direct measurement
of the winding number $\mathcal{W}$ is challenging in our experimental
configuration [43], hence we measure the intensity distribution at the mid-gap
and observe the appearance of the edge states as a signature of the
topological phase, see Fig. 4(g). The transition to a trivial state occurs
around $d_{D}/d_{A}\approx 0.9-0.95$, where we observe an abrupt change in the
intensity at the edge of the chain [there is a clear asymmetry in the edge
modes’ occupations, see inset in Fig. 4(g)]. We reproduce this observation by
calculating the winding number in the tight-binding Hamiltonian, including the
potential gradient and random disorder (see Methods), which softens the
transition threshold and moves it away from the point $t_{A}/t_{D}=1$, Fig.
4(h), as observed in the experiment.

To summarize, we have demonstrated an all-optically driven topological phase
transition in a fully reconfigurable optically-induced orbital SSH lattice
created in an open-dissipative exciton-polariton system. The transition is
controlled by fine-tuning the strength of tunneling between the lattice sites.
We emphasize that implementing this kind of control would typically require
fabrication of many different samples with different implementations of
lattice geometries. Moreover, we have demonstrated the robust topologically
protected edge states in a regime, where exciton polaritons are condensed and
nonlinear (density-dependent) effects could begin to play a role [44].
Combined with the experimental control of the gain and loss (linewidth) of the
trapped polariton condensates [36] our system represents an attractive,
flexible platform for further studies of topological effects in a nonlinear
and non-Hermitian hybrid photonic system.

## Methods

Sample and experiment The sample used in the experiment is a high-quality
GaAs-based microcavity with a long cavity photon lifetime exceeding 100 ps
[42]. The cavity of the length $3\lambda/2$ is enclosed between distributed
Bragg reflectors with 32 (top) and 40 (bottom) Al0.2Ga0.8As/AlAs layer pairs.
The active region is made of 12 GaAs/AlAs quantum wells (QWs) of 7 nm nominal
thickness positioned in three groups at the maxima of the confined photon
field. The measured Rabi spliting is about $\hbar\Omega=15.9\pm 0.1$ meV. The
experiments are done at slightly varying positions on the sample that
correspond to the same photon-exciton detuning $\Delta=-0.43$ meV. All results
are obtained with a microcavity kept in a continuous flow helium cryostat,
ensuring the sample temperature of 7-8K.

The nonresonant optical excitation was provided by a continuous-wave (CW) Ti:
Sapphire laser (M Squared SolsTiS), tuned to the cavity reflectivity minimum
above the QW bandgap. The Gaussian laser beam was transformed and focused to a
top-hat distribution by a shaping lens (Eksma Optics GTH-5-250-4) and imaged
onto the digital micromirror device (DMD). The shape of the lattice potential
was encoded on the DMD which reflected the laser selectively and then imaged
onto the sample via set of lenses and a microscope objective [36].
Photoluminescence from the sample was collected with the same objective in the
reflection geometry and imaged with a set of confocal lenses onto the slit of
the spectrometer (Princeton Instruments IsoPlane 320), equipped with a 2D
sensitive CCD (Andor iXon Ultra 888). The imaging lens in front of the
spectrometer was mounted on a motorized stage, enabling the spectral
tomography of the emission. The tomography was done by collecting a set of
spectral images, scanning the full real-space emission by moving the image
with respect to the entrance slit.

Theory. The SSH model is described by a tight-binding Hamiltonian:

$\hat{H}_{SSH}=\sum_{i}\left(t_{D}\hat{a}^{\dagger}_{i}\hat{a}^{\prime}_{i}+t_{A}\hat{a}^{\dagger}_{i+1}\hat{a}^{\prime}_{i}+h.c.\right)$
(1)

where $\hat{a}_{i}$ and $\hat{a}^{\prime}_{i}$ are the annihilation operators
in the $i^{\text{th}}$ unit cell of a lattice. The eigenfunctions of the
Hamiltonian $\hat{H}_{SSH}$ are given by $(\pm e^{-i\varphi(k)},1)$, with $k$
being the wave vector. The winding number $\mathcal{W}$ is defined by,

$\mathcal{W}=\frac{1}{2\pi}\int_{\text{BZ}}dk,\frac{\partial\varphi(k)}{\partial
k}$ (2)

corresponding to the Zak phase $\mathcal{Z}=\pi\mathcal{W}$, which is a 1D
lattice equivalent of the geometric Berry phase. In an ideal SSH model,
$\mathcal{W}=1$ for $t_{D}<t_{A}$ and $\mathcal{W}=0$ for $t_{D}>t_{A}$. In
our system, the lattice is perturbed by a potential gradient and disorder. To
include these effects, we consider a Hamiltonian given by:

$\hat{H}=\hat{H}_{SSH}+\sum_{i}\left(V_{2i-1}\hat{a}^{\dagger}_{i}\hat{a}_{i}+V_{2i}\hat{a}^{\prime\dagger}_{i}\hat{a}^{\prime}_{i}\right).$
(3)

The potential energy is given by $V_{i}=v_{i}+v_{0}(i-N/2)$, where $v_{0}$ is
a constant, $v_{i}$ is a random value representing disorder in the system, and
$N$ is the number of unit cells in the lattice. For our simulations
$v_{0}=0.2/N$, $v_{i}$ is randomly distributed within the interval $[\pm
0.05]$, and energies are expressed in the unit of $\text{max}(t_{A},t_{D})$.

Within the mean-field approximation, exciton polaritons in optically induced
potentials can be described by a driven-dissipative equation for the polariton
wavefunction, $\psi(x,y,t)$:

$\displaystyle i\hbar\frac{\partial\psi}{\partial
t}=\left[-\lambda\frac{\hbar^{2}}{2m}\nabla^{2}+iP+V(x,y)+\alpha|\psi|^{2}\right]\psi$
(4)

where $\lambda=(1-i\lambda_{0})$, $\lambda_{0}$ is a phenomenological
parameter describing the energy relaxation [45], and $m$ is the effective mass
of the polaritons. The net gain $P(x,y)=P_{0}(x,y)-\gamma$ is given by the
difference between the pump strength (gain) $P_{0}(x,y)$ and the loss $\gamma$
(linewidth). The parameter $\alpha=(\alpha_{R}-i\alpha_{I})$ represents the
polariton-polariton interaction $\alpha_{R}$ and nonlinear decay $\alpha_{I}$.
$V(x,y)$ accounts for a real part of the potential (the SSH lattice) induced
by the repulsive interactions of the polaritons with an optically injected
excitonic reservoir, as well as for the energy gradient in the cavity. For the
numerical simulations, we consider $\alpha_{R}=\alpha_{I}$ and a
transformation $\psi\to\psi/\sqrt{\alpha_{R}}$ to obtain an equation
describing the exciton polaritons: $i\hbar\partial\psi/\partial
t=[-(\lambda\hbar^{2})/(2m)\nabla^{2}+iP(x,y)+V(x,y)+(1-1i)|\psi|^{2}]\psi$.
The loss and effective mass parameters are obtained from the experimental data
in [42]. Considering the long lifetime of exciton polaritons in our GaAs-based
sample, we use a linewidth $\gamma=5.5~{}\mu\text{eV}$ corresponding to a
lifetime $120$ps. We take $m=7.6\times 10^{-5}m_{e}$, where $m_{e}$ is the
mass of an electron. Other parameters were chosen as
$\lambda_{0}=0.05~{}\text{meV}$ and $\max(P_{0})=10.5~{}\mu\text{eV}$, to fit
the phenomenology of the present experiment.

###  .1 Author contribution statement

E.A.O. and A.G.T. supervised and guided this research. M.P. initiated the
research project. M.P. and E.E. designed and performed the experiment. S.G.
and T.C.H.L. performed theoretical analysis and numerical simulations. M.P.
analysed and interpreted the data with the input from E.E., M.W., A.G.T, and
E.A.O. M.S. and D.W.S. designed and characterised the sample, which was grown
by K.W. and L.N.P. M.P. and E.A.O. wrote the manuscript with the input from
all authors.

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# A reference meteor magnitude for intercomparable fluxes

Althea V. Moorhead<EMAIL_ADDRESS>NASA Meteoroid Environment
Office, Marshall Space Flight Center, Huntsville, Alabama 35812, USA  Denis
Vida  Department of Physics and Astronomy, University of Western Ontario,
London, Ontario N6A 3K7, Canada  Institute for Earth and Space Exploration,
University of Western Ontario, London, Ontario N6A 5B8, Canada  Peter G. Brown
Department of Physics and Astronomy, University of Western Ontario, London,
Ontario N6A 3K7, Canada  Margaret D. Campbell-Brown  Department of Physics and
Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada

###### Abstract

The rate at which meteors pass through Earth’s atmosphere has been measured or
estimated many times over; existing flux measurements span at least 12
astronomical magnitudes, or roughly five decades in mass. Unfortunately, the
common practice of scaling flux to a universal reference magnitude of +6.5
tends to collapse the magnitude or mass dimension. Furthermore, results from
different observation networks can appear discrepant due solely to the use of
different assumed population indices, and readers cannot resolve this
discrepancy without access to magnitude data. We present an alternate choice
of reference magnitude that is representative of the observed meteors and
minimizes the dependence of flux on population index. We apply this choice to
measurements of recent Orionid meteor shower fluxes to illustrate its
usefulness for synthesizing independent flux measurements.

##  1 Introduction

Historically, meteor shower activity has usually been measured in terms of
zenithal hourly rate (ZHR), the number of meteors observed per hour by a
visual observer under ideal conditions (Norton & Chitwood, 2008). The ZHR is
related to but distinct from meteor flux (Rendtel, 2022). In recent decades,
however, a number of research groups have begun to report meteor fluxes from
camera networks or meteor radars (some examples include Campbell-Brown, 2004;
Blaauw et al., 2016; Molau, 2022; Vida et al., 2022). These networks vary in
sensitivity and coverage, and thus measure the meteor flux in different size
regimes. For instance, the Southern Argentina Agile MEteor Radar (SAAMER)
measures meteor fluxes down to a limiting equivalent optical magnitude of +9
(Bruzzone et al., 2021), while the NASA All-Sky Fireball Network has been used
to measure the Perseid meteor shower flux to a limiting magnitude of
approximately -3.5 (Ehlert & Erskine, 2020).

If the flux of a single meteor shower is measured simultaneously by several
different networks with different limiting magnitudes, these fluxes can be
combined to inspect the shower’s mass (or magnitude) distribution. For
instance, Campbell-Brown et al. (2016) combined flux data from five different
optical systems as well as the Canadian Meteor Orbit Radar (CMOR) in order to
construct a mass distribution that spanned four decades in mass. Similarly,
Blaauw (2017) combined radar, optical, and lunar impact data to construct a
mass profile for the Geminids.

The determination of a network’s limiting magnitude, however, is not
straightforward, as the detection limit usually varies over the detector’s
field of view (FOV). For example, the distance from which meteors are viewed
tends to vary within the FOV in a way that depends on FOV width and
orientation; if the region surveyed by a portion of the FOV is more distant,
the meteor must be brighter to be detected. One could address this by
discarding all meteors that are too dim to be detected in any part of the
field of view, but this typically results in a large reduction in the overall
count.

The most common way to deal with a variable limiting magnitude is to divide
the FOV into smaller pieces and scale the area of each piece to reflect its
contribution to the overall flux, assuming that meteor magnitudes follow a
power-law distribution (Kaiser, 1960; Koschack & Rendtel, 1990a; Bellot Rubio,
1994; Vida et al., 2022; Molau, 2022). This is known as the effective
collection area method and is described in Section 2.1. Because the areas are
scaled to correspond to a constant reference limiting magnitude, one can in
theory choose any reference magnitude one likes, so long as the power-law
assumption holds over the range of magnitudes considered. A common choice is
to scale the flux to a constant limiting magnitude of +6.5 and/or convert to
ZHR (e.g., Roggemans, 1989; Bruzzone et al., 2021; Molau, 2022; Vida et al.,
2022) to facilitate comparisons with visual rates (Koschack & Rendtel, 1990b):

$\displaystyle f(+6.5)$
$\displaystyle=f(M_{\text{ref}})\,r^{6.5-M_{\text{ref}}}$ (1) ZHR
$\displaystyle=\frac{f(+6.5)\cdot
37\,200~{}\text{km}^{2}}{(13.1\,r-16.5)(r-1.3)^{0.748}}$ (2)

We use $f(M)$ to denote the flux of meteors with absolute magnitudes brighter
than $M$, and we use $M_{\text{ref}}$ to refer to the chosen reference
magnitude. Notice that the above conversion depends on the assumed population
index, $r$; if the flux is based on magnitudes that are very different from
+6.5, the result can be quite sensitive to the choice of $r$ (see Figure 1).
For example, if the meteor flux measured by CMOR is scaled from its limiting
magnitude of +8 to an equivalent ZHR, but ${r=2.0}$ is used for a shower whose
true value is ${r=2.2}$, the ZHR will be inflated by 77%. Note that these
fluxes include all meteors brighter than the given limiting magnitude. This is
distinct from the dimmest apparent magnitude visible to a visual observer in
the center of their visual field, which is also sometimes called the limiting
magnitude (e.g., Koschack & Rendtel, 1990b), but which we call the visual
limit.

Figure 1: Relative error in magnitude-6.5-limited flux and ZHR computed using
an assumed population index of ${r=2.0}$, assuming that the true population
index is ${r=2.2}$. This plot covers limiting magnitudes ranging from those of
patrol radars to fireballs.

Thus, while the use of a common limiting magnitude allows us to compare fluxes
published by different groups using different data, it obscures the
characteristic limiting magnitude of the system ($M_{\text{ref}}$) and is
sensitive to errors in the assumed magnitude distribution. If two independent
flux measurements disagree, it can be unclear whether this is because the data
are truly in conflict or because an incorrect choice of $r$ has been used.
Even when it is clear that two sets of measurements use different values of
$r$, it is usually impossible for the reader to invert that choice unless
magnitude information is also provided (or, in the case of the Video Meteor
Network, a tool is provided that allows the reader to specify $r$; see Section
4.2). For this reason, the Campbell-Brown et al. (2016) and Blaauw (2017)
studies exclusively used data from their own observation networks, which
allowed them to enforce a consistent set of assumptions.

One possible solution to this problem of incompatible flux measurements is the
choice of a reference magnitude that minimizes the dependence of the measured
flux on population index. We derive an expression for such a reference
magnitude in Section 2.2 that is compatible with the widely used “effective
collection area” method of measuring fluxes. In Sections 2.3 and 2.4, we
provide two alternative ways to estimate $M_{\text{ref}}$; these estimates can
be used by researchers to validate their flux calculations, and may in some
cases enable readers to estimate $M_{\text{ref}}$ for published fluxes. In
Section 4, we apply these equations to several independent measurements of
Orionid fluxes to demonstrate their value for synthesizing disparate
measurements.

##  2 Reference limiting magnitude

The most common approach used to derive meteor flux from observed counts is
the effective collection area (ECA) method. This method appears to have
originated with Koschack & Rendtel (1990a) and Koschack & Rendtel (1990b), who
used a similar approach to estimate the number of meteors that can be detected
by a visual observer. Bellot Rubio (1994) adapted the approach to compute
fluxes from camera data. These efforts were predated by Kaiser (1960), who
developed an equivalent approach for radar data.

Today, it is widely used within the meteor astronomy community (recent
examples include Blaauw, 2017; Margonis et al., 2019; Molau, 2022; Vida et
al., 2022). We review this method in Section 2.1. We then derive a reference
magnitude in Section 2.2 that minimizes the dependence of flux on population
index.

We do not specify a method for determining the limiting magnitudes of a meteor
observation network; this will depend on the properties of the network and the
meteor detection algorithm. Our assumptions are that the limiting magnitude of
the network has been characterized and that the ECA method is being used to
measure fluxes. We refer readers to Molau et al. (2016) and Vida et al. (2022)
for more information about the calculation of optical limiting magnitudes, and
to Kaiser (1960) for an analogous discussion of radar fluxes. We do, however,
offer an alternate method for estimating the reference magnitude that can be
used to test whether the limiting magnitudes have been correctly calculated
(see Section 2.4).

###  2.1 The effective collection area (ECA) method

Let $f(M_{\text{ref}})$ denote the flux of meteors brighter than reference
magnitude $M_{\text{ref}}$. The observed number of meteors follows a Poisson
distribution with an expected value of

$\displaystyle\langle N(M_{\text{ref}})\rangle$
$\displaystyle=f(M_{\text{ref}})\,(\vec{A}\cdot\hat{v})\,T\,,$ (3)

where $N(M_{\text{ref}})$ is the number of meteors brighter than magnitude
$M_{\text{ref}}$ passing through area $\vec{A}$ within time interval $T$. If
the meteors belong to a shower, they share a common velocity vector $\vec{v}$;
the unit vector $\hat{v}$ is aligned with the meteors’ velocity vector. Note
that $\vec{v}$ is the meteor’s motion through space, and not its apparent
angular motion; the latter plays a role in determining the limiting magnitude
(Kingery & Blaauw, 2017; Molau et al., 2016).

If the collection area is flat and its normal vector points toward zenith,
then the area presented to the shower is

$\displaystyle\vec{A}\cdot\hat{v}$ $\displaystyle=\cos Z_{R}\,,$ (4)

where $Z_{R}$ is the angle between the local zenith and the shower radiant
(also known as the zenith angle). However, meteors entering the atmosphere at
a high angle of incidence encounter a gentler atmospheric density gradient
along their path, which is thought to decrease their peak brightness
(Zvolánková, 1983). Both Zvolánková (1983) and Molau & Barentsen (2013)
suggest that this can be handled by applying an exponent to the zenith angle
term:

$\displaystyle\cos^{\gamma}Z_{R}\,,$ (5)

where the constant $\gamma$ varies between showers but is approximately 1.5-2.

As discussed in Section 1, the limiting meteor magnitude typically varies
within the field of observation. The ECA method handles this by dividing the
field of view into a grid and assuming that the limiting magnitude does not
vary appreciably within a grid cell. If cell $i$ covers area $A_{i}$ and has
limiting meteor magnitude $M_{i}$, then the expected number of meteors in that
cell is

$\displaystyle\langle n_{i}\rangle$
$\displaystyle=f(M_{i})\,A_{i}\,T\cos^{\gamma}\\!Z_{R}\,.$ (6)

Recall from equation 1 that the relationship between limiting magnitude and
flux is a power law; that is,

$\displaystyle f(M_{i})$
$\displaystyle=f(M_{\text{ref}})\,r^{M_{i}-M_{\text{ref}}}\,,$ (7)

where $r$ is the population index. Therefore,

$\displaystyle\langle n_{i}\rangle$
$\displaystyle=f(M_{\text{ref}})~{}r^{M_{i}-M_{\text{ref}}}~{}A_{i}\,T\cos^{\gamma}\\!Z_{R}\,.$
(8)

The sum of a set of independent Poisson random variables is itself a Poisson
random variable, and its expected value is the sum of the expected values of
its components. Thus, the expected total number of observed meteors across all
cells is

$\displaystyle\langle N\rangle$ $\displaystyle=\sum_{i}\langle n_{i}\rangle$
$\displaystyle=f(M_{\text{ref}})\,T\,\left(\sum_{i}A_{i}\,r^{M_{i}-M_{\text{ref}}}\,\cos^{\gamma}\\!Z_{R}\right)\,,$
(9)

where the quantity in parentheses is called the effective collection area:

$\displaystyle A_{\text{eff}}$
$\displaystyle=\sum_{i}A_{i}\,r^{M_{i}-M_{\text{ref}}}\,\cos^{\gamma}\\!Z_{R}\,.$
(10)

Equation 9 can then be rewritten as

$\displaystyle\langle N\rangle$
$\displaystyle=f(M_{\text{ref}})\,A_{\text{eff}}\,T\,.$ (11)

The observed number of events, ${N=\sum_{i}n_{i}}$, is an unbiased estimate
for the rate of the Poisson process. Therefore, we estimate the flux as

$\displaystyle f(M_{\text{ref}})$
$\displaystyle\approx\frac{N}{A_{\text{eff}}\,T}$ (12)

and recover (for example) Equation 1 of Molau (2022).

Note that if $M_{i}$ varies smoothly across the FOV, one could replace the sum
in Equation 10 with an integral. For instance, Kaiser (1960) integrates the
equivalent of $r^{M_{i}}$ along the radar echo line; see Equation 18 of that
paper. However, most optical applications of the ECA methods use the discrete
form presented here.

###  2.2 Derivation of reference magnitude

We seek a reference limiting magnitude that minimizes the dependence of flux
on population index. To obtain this, we set

$\displaystyle\frac{df}{dr}$
$\displaystyle=-\frac{N}{T}\frac{1}{A_{\text{eff}}^{2}}\frac{dA_{\text{eff}}}{dr}=0$
(13)

It can be shown that ${dA_{\text{eff}}/dr}$ is zero when:

$\displaystyle M_{\text{ref}}$
$\displaystyle=\frac{\sum_{i}M_{i}\,A_{i}\,r^{M_{i}}}{\sum_{i}A_{i}\,r^{M_{i}}}$
(14)

We see from this equation that the optimal reference magnitude is a weighted
average of the limiting magnitudes of each cell, and therefore lies within the
network’s limiting magnitude range.

If we multiply both the numerator and denominator of equation 14 by
$f(M_{\text{ref}})$, ${r^{-M_{\text{ref}}}}$, ${\cos^{\gamma}\\!Z_{R}}$, and
$T$, we find that can re-write equation 14 as follows:

$\displaystyle M_{\text{ref}}$ $\displaystyle=\frac{\sum_{i}M_{i}\,\langle
n_{i}\rangle}{\sum_{i}\,\langle n_{i}\rangle}$ (15)

We see that the weights are proportional to the expected number of meteors in
each cell. Thus, $M_{\text{ref}}$ is a weighted average of the limiting
magnitude (which is used by Vida et al., 2022), where the weights are the
expected rates for each cell.

###  2.3 Average limiting magnitude per meteor

The form of equation 15 hints that we can estimate the reference magnitude
using the observed meteor counts:

$\displaystyle M_{\text{ref}}$
$\displaystyle\approx\frac{\sum_{i}M_{i}\,n_{i}}{\sum_{i}n_{i}}$ (16)

Notice that we have replaced the expected counts, $\langle n_{i}\rangle$, with
the observed counts, $n_{i}$. This approximation makes it clear that our
proposed reference magnitude not only lies within the network’s range of
limiting magnitudes, but is in fact the per-meteor mean limiting magnitude.
This makes $M_{\text{ref}}$ a doubly appealing choice in that it is
representative of the limiting magnitude distribution in addition to
minimizing the dependence on $r$.

We suggest that researchers publishing fluxes verify that the approximation
given in equation 16 is similar to the true value given by equation 14. A
large discrepancy could indicate that either the cell areas or population
index are incorrect. We will also show in an upcoming paper that equation 16
provides a reference magnitude that minimizes the covariance between flux and
population index when fitting for both quantities simultaneously.

If we use the sample mean of the limiting magnitude (Eq. 16) as a reference
magnitude, then a sensible extension is to use the sample standard deviation
of the limiting magnitude to describe the range of limiting magnitudes within
a meteor detection network:

$\displaystyle s_{\text{ref}}$
$\displaystyle=\frac{\sum_{i}n_{i}(M_{i}-M_{\text{ref}})^{2}}{\left(\sum_{i}n_{i}\right)-1}$
(17)

If the distribution of limiting magnitudes is strongly skewed or multi-model,
researchers could consider quoting another measure of variation, such as the
interquartile range.

###  2.4 Average absolute magnitude

If we assume that $M_{i}$ places a strict upper limit on the absolute meteor
magnitude that can be observed in cell $i$, then the probability of observing
a meteor with absolute magnitude $M^{\prime}$ in that cell is:

$\displaystyle P(M^{\prime}\,|\,M_{i})$ $\displaystyle=\ln
r\times\begin{cases}r^{M^{\prime}-M_{i}}&M^{\prime}<M_{i}\\\
0&M^{\prime}>M_{i}\end{cases}$ (18)

Using this probability distribution, we can calculate the average absolute
magnitude we expect for meteors in that cell:

$\displaystyle\langle M^{\prime}\,|\,M_{i}\rangle$ $\displaystyle=\ln
r\int_{-\infty}^{M_{i}}M^{\prime}\,r^{m-M_{i}}\,dM^{\prime}=M_{i}-\frac{1}{\ln
r}$ (19)

We see that the average observed magnitude in any given cell should be
systematically offset from the limiting magnitude in that cell by a constant
value of $-1/\ln r$. Furthermore, the probability that a meteor is detected in
a cell with limiting magnitude $M_{i}$ is:

$\displaystyle P(M_{i})$ $\displaystyle=\frac{\langle n_{i}\rangle}{\langle
N\rangle}=\frac{A_{i}\,r^{M_{i}}}{\sum_{j}A_{j}\,r^{M_{j}}}$ (20)

We can now combine equations 19 and 20 to obtain the expected absolute
magnitude of a meteor detected by the system:

$\displaystyle\langle M^{\prime}\rangle$ $\displaystyle=\sum_{i}\langle
M^{\prime}\,|\,M_{i}\rangle\,P(M_{i})=M_{\text{ref}}-\frac{1}{\ln r}$ (21)

If we substitute the mean absolute meteor magnitude for the expected absolute
meteor magnitude, equation 21 provides yet another estimate of
$M_{\text{ref}}$:

$\displaystyle M_{\text{ref}}$ $\displaystyle\approx\frac{1}{\ln
r}+\frac{1}{N}\sum_{k}M_{k}^{\prime}$ (22)

Here we use the index $k$ to emphasize that this sum is performed over all
meteors rather than all cells. This estimate depends only on the observed
absolute magnitudes and the population index; it does not depend at all on the
cell-specific limiting magnitudes.

Equation 21 has two important uses. First, it can be compared with equation 16
to determine whether the limiting magnitudes have been computed correctly.
Second, it may be useful for estimating the reference limiting magnitude
associated with a published flux value when only meteor counts and magnitudes
are provided.

##  3 Inapplicability to visual rates

The ECA method as presented in Section 2.1 assumes that every meteor that is
bright enough to be detected in a given cell _will_ be detected. However,
Koschack & Rendtel (1990a) found that there is no hard upper limit on meteor
magnitude when it comes to human observers; instead, the probability of
detection varies with the brightness of the meteor and its angular offset from
the observer’s line of sight. As a result, Equations 14, 16, and 22 cannot be
applied to visual observations; we derive an alternative in this section.

Equations 1 and 2 can be combined:

$\displaystyle f(M_{\text{ref}}\,|\,\text{ZHR},\,r)$ $\displaystyle=$
$\displaystyle\frac{\text{ZHR}}{\text{2840~{}km}^{2}}$
$\displaystyle(r-1.26)(r-1.3)^{0.748}\,r^{M_{\text{ref}}-6.5}$ (23)

If we then set the derivative of this equation with respect to $r$ to zero, we
obtain:

$\displaystyle M_{\text{KR}}$
$\displaystyle=6.5-\frac{r}{r-1.26}-0.748\,\frac{r}{r-1.3}$ (24)

However, equation 23 does not account for observing conditions. If a visual
observer is viewing a hazy or light-polluted sky, they may not be able to
detect meteors down to magnitude +6.5. We seek a reference magnitude that
takes this into account.

We have been unable to reproduce the derivation of equation 2 from the
information provided in Koschack & Rendtel (1990a) and Koschack & Rendtel
(1990b). We nevertheless have attempted to simulate a reasonable range of
magnitudes that is consistent with their general approach. First, we
constructed an empirical profile for airmass vs. elevation angle ($\phi$) by
repeatedly evaluating the following integral for different choices of $\phi$:

$\displaystyle X(\phi)$
$\displaystyle=\int_{0}^{\infty}\rho_{\text{atm}}(h(s,\phi))\,ds$ (25)
$\displaystyle h(s,\phi)$
$\displaystyle=\sqrt{R_{\oplus}^{2}+s^{2}+2R_{\oplus}s\sin\phi}-R_{\oplus}$
(26)

where $R_{\oplus}$ is the Earth’s radius, $\rho_{\text{atm}}$ is the air
density, and the integration variable $s$ is distance from the observer. We
performed this task using the “StandardAtmosphereData” function in Mathematica
(Wolfram Research, Inc., 2023), which defaults to the 1976 United States
Standard Atmosphere model (United States Committee on Extension to the
Standard Atmosphere (1976), COESA).

Next, we divided the visible sky into bins and calculated the elevation angle
and distance corresponding to each bin, assuming that meteors ablate at a
height of 100 km. The limiting magnitude of each bin is then:

$\displaystyle M_{i}$
$\displaystyle=m_{0}-5\ln\frac{d}{100~{}\text{km}}-X(\phi)$ (27)

where $m_{0}$ is the visual limit, the dimmest magnitude of an object visible
to the observer.

Although Koschack & Rendtel (1990a) did not settle on a satisfactory
functional form for the probability of perception, we were able to approximate
Table 4 of Koschack & Rendtel using the following formula:

$\displaystyle p(\delta,\alpha)$
$\displaystyle=\text{expit}\\{-\left(a+b\,\delta+c\tan\frac{\alpha}{2}\right)\\}\,,~{}\text{where}$
(28) $\displaystyle\delta$ $\displaystyle=m^{\prime}-m_{0}<0$ (29)

Here “expit” is the standard logistic function, $m^{\prime}$ is apparent
meteor magnitude, $\alpha$ is the angular separation between the observer’s
line of sight (LOS) and the meteor, and the constants are ${a=2.6240}$,
${b=1.3815}$, and ${c=10.6136}$. This approximation gives us values that are
typically within 0.05 of the correct fractional value.

We can see from Equation 28 that the probability of observing a meteor right
at the limit of detectability (${\delta\sim 0}$) is quite low, even in the
center of the observer’s field of view (${\alpha=0}$, ${p\simeq 0.07}$). The
observer has a much better chance of observing meteors that are several
magnitudes brighter than the detectability limit and that pass near the
direction of their gaze. As a result, the typical magnitude observed will be
quite a bit brighter than the limiting magnitude, especially in the peripheral
visual field. This effect is illustrated in Figure 2.

Figure 2: (Top) The limiting absolute meteor magnitude that can be seen as a
function of the meteor’s azimuth and elevation angle relative to a visual
observer. The black “x” marks the direction of the observer’s gaze, which has
an elevation of 50∘. (Middle) The median absolute meteor magnitude that can be
seen by the observer, assuming a population index of ${r=2.4}$. Even in the
center of the observer’s field of view, the median magnitude is several
magnitudes brighter than the limiting magnitude; this is because observing a
meteor near the limit, while possible, is very unlikely. (Bottom) The number
of meteors observed per unit solid angle. This number has been scaled so that
it is unity at the center of the observer’s field of view; in reality it will
depend on the shower’s ZHR.

The probability of perception complicates the determination of the reference
magnitude. The number of meteors we expect to see in a given area is now:

$\displaystyle\langle n_{i}\rangle$ $\displaystyle\propto
r^{M_{i}-M_{\text{ref}}}A_{i}\,P_{i}(r)$ (30)

where $P_{i}$ is the fraction of all meteors we expect to see in that area:

$\displaystyle P_{i}(r)$
$\displaystyle=\int_{-\infty}^{0}p(\delta,\alpha_{i})\cdot r^{\delta}\ln
r\,d\delta$ (31)

Because $P_{i}$ is a function of $r$, it must be included in our calculation
of ${dA_{\text{eff}}/dr}$. Equation 14 is replaced with:

$\displaystyle M_{\text{ZHR}}$
$\displaystyle=\frac{\sum_{i}M_{i}\,A_{i}\,r^{M_{i}}\,P_{i}-r\sum_{i}A_{i}\,r^{M_{i}}\,\frac{dP_{i}}{dr}}{\sum_{i}A_{i}\,r^{M_{i}}\,P_{i}}$
(32)

This equation must be evaluated numerically. We have done so, using a range of
plausible $r$ values, and found that

$\displaystyle M_{\text{ZHR}}-m_{0}$ $\displaystyle\approx M_{\text{KR}}-6.5$
(33)

The two expressions agree to within 0.2 magnitudes between ${r=1.7}$ and
${r=3.0}$. Furthermore, we find that the average apparent magnitude can be
approximated as follows:

$\displaystyle\bar{m}^{\prime}$ $\displaystyle\approx
m_{0}-3.8+2.23\ln(r-1.3)$ (34)

This expression allows us to estimate the visual limit from the mean apparent
magnitude. Assuming the classical +6.5 limit, the mean apparent magnitude will
be approximately +2.9 for ${r=2.4}$.

If we combine equations 24, 33, and 34, we can estimate the reference
magnitude directly from the mean apparent magnitude:

$\displaystyle M_{\text{ZHR}}\approx\bar{m}^{\prime}$
$\displaystyle+3.8-2.23\ln(r-1.3)$
$\displaystyle-\frac{r}{r-1.26}-0.748\,\frac{r}{r-1.3}$ (35)

This approximate reference magnitude has both of our desired qualities: it
minimizes the dependence of flux on $r$ and it is representative of the range
of magnitudes observed. It does, however, require observers to record the
apparent magnitudes of individual meteors; fortunately, the International
Meteor Organization (IMO) Visual Meteor Database (VMDB) includes apparent
magnitudes for a subset of all observations.

Figure 3: Approximate reference magnitude ($M_{\text{ZHR}}$) as a function of
population index ($r$) and mean apparent magnitude ($\bar{m}^{\prime}$),
calculated using Equation 35. For example, if ${\bar{m}^{\prime}=3}$ and
${r=2.4}$, then $M_{\text{ZHR}}$ is between 2.5 and 3.

We would like to point out that our treatment of ZHR and reference magnitude
neglects the difference between visual and camera or photometric magnitude.
For instance, Jacchia (1957) found that visual magnitude estimates were 1-2
magnitudes brighter than photographic magnitude measurements for the same
meteors. This difference, which they call the “color index,” is a function of
photographic magnitude (see Figure 12b of that paper) and is attributed to the
Purkinje effect (see also Protte & Hoffmann, 2020). If we assume that the
perception probability applies to visual magnitude and use another sigmoid
function to describe the color index, we predict an average apparent _visual_
magnitude that is about half a magnitude dimmer than Equation 34.
Unfortunately, Equation 31 becomes quite complicated in this case. A full
treatment of this effect is beyond the scope of this paper; for now, we will
simply consider the uncertainty associated with Equation 35 to be at least 0.5
magnitudes.

We began this section by drawing a distinction between the hard detection
threshold assumed for instrumental detection and the lack of such a threshold
for visual observations. However, instrumental detection thresholds will also
be “softened” to some extent by noise. If the magnitude range of this
softening effect is non-negligible compared to the range of detected
magnitudes in the cell or the range of limiting magnitudes represented in the
network, then the use of the ECA method – and Equations 14, 16, and 22 – may
not be appropriate.

##  4 Application to the Orionids

In this Section, we compare fluxes from four independent meteor observation
networks and demonstrate the benefit of using our proposed reference
magnitude. These networks are the Global Meteor Network (GMN; Vida et al.,
2021), the IMO Video Meteor Network (VMN; Molau & Barentsen, 2014), the IMO
Visual Meteor Database (VMDB; Roggemans, 1988), and the Canadian Meteor Orbit
Radar (CMOR; Jones et al., 2005).

We selected the Orionids for this demonstration because all four networks
assume quite different values of $r$ for this shower (see Table 1). Normally,
this would prohibit a useful synthesis of the data. However, we will attempt
to estimate the reference magnitude of each network and show that this removes
some of the discrepancy in activity. We include the past four years of Orionid
activity; this choice is determined by GMN, whose fluxes date back to 2020.

All three flux networks (GMN, VMN, and CMOR) use the same flux-to-ZHR
conversion (eqs. 1 and 2). Thus, once we determine $M_{\text{ref}}$ for each
network, we can convert the published ZHR values for that network to
$f(M_{\text{ref}})$ by inverting these equations, using each source’s assumed
$r$.

Table 1: Orionid population index ($r$) and reference magnitude for four
independent networks.

network | $r$ | $M_{\text{ref}}$ | $M_{\text{ZHR}}$
---|---|---|---
CMOR | 3.0 | +8.0 |
GMN | 2.2 | +3.9 |
VMN | 2.5 | +2.0 |
VMDB | 2.4 |  | +2.6

Note – The population index ($r$) listed is the value used to calculate meteor
flux or ZHR by each network. We also include our estimated reference magnitude
($M_{\text{ref}}$ or $M_{\text{ZHR}}$) for each network where applicable.

###  4.1 Global Meteor Network (GMN)

GMN meteor fluxes are already reported to the appropriate reference magnitude
(Equation 14; these are called time-area-product or TAP-weighted magnitudes in
Vida et al., 2022). In fact, the GMN flux data files report the reference
magnitude for each observation interval as well as the overall average; the
reference magnitude may vary over time due to changes in observing conditions
(Vida et al., 2022). In Table 1, we report the reference magnitude averaged
over all years.

###  4.2 Video Meteor Network (VMN)

VMN fluxes can be obtained from a web portal111https://meteorflux.org/; these
fluxes are quoted to magnitude +6.5 and are also converted to ZHR. We used the
default options for the Orionids, with one exception: we opted to turn on the
perception coefficient (PC) correction.

The website does not report meteor magnitudes, but it does allow the user to
vary the assumed population index (Molau et al., 2014). This gives us an
opportunity to extract $M_{\text{ref}}$ using yet another approach. Using
equation 1, we calculate:

$\displaystyle\frac{d\ln f(+6.5)}{d\ln r}$
$\displaystyle=\frac{r}{f(M_{\text{ref}})}\frac{df(M_{\text{ref}})}{dr}+6.5-M_{\text{ref}}$
(36)

Because ${df(M_{\text{ref}})/dr=0}$, we can therefore obtain $M_{\text{ref}}$
by repeatedly changing the assumed value of $r$ and regressing the reported
${\ln f(+6.5)}$ against ${\ln r}$. The results of this process are shown in
the top panel of Figure 4. We obtain ${M_{\text{ref}}=1.96}$ in this manner.
The bottom panel of Figure 4 demonstrates that this value does in fact
minimize the variation in flux; when we use $M_{\text{ref}}$, the variation in
flux over ${r=2.3}$ to ${r=2.6}$ is less than 1%, which is much less than the
factor-of-two range we see in $f(+6.5)$. Figure 4 also indicates that there is
some noise in the data, which is probably due to rounding error; we therefore
round our reference magnitude to +2.

Figure 4: (Top) Magnitude-6.5-limited meteor flux output by the VMN web portal
as a function of user-selected population index ($r$). Note that these data
are presented on a log-log scale and the red line represents the best fit line
between ${\ln f(+6.5)}$ and ${\ln r}$. (Bottom) Equivalent flux for
${f(M_{\text{ref}}=+1.96)}$. Note that our calculated $M_{\text{ref}}$
minimizes the variation in $f(M_{\text{ref}})$ with respect to $r$.

###  4.3 Visual Meteor Database (VMDB)

Zenithal hourly rates (ZHR) for major showers are published by the
International Meteor Organization (IMO) on their
website.222https://www.imo.net/members/imo_live_shower These ZHRs are
automatically computed from user-supplied data and the IMO does not recommend
their use for scientific applications. We ignored this warning for the
purposes of this demonstration and downloaded Orionid ZHR values for the years
2020 through 2023. The IMO uses ${r=2.4}$ for the “whole campaign.”

Magnitude counts are available to IMO members via the Video Meteor Database
(VMDB).333https://www.imo.net/members/imo_vmdb We downloaded these data for
Orionids between 2020 and 2023 (inclusive). Because visual observations do not
contain distance information, these are apparent magnitudes; we found that the
mean apparent magnitude was +2.75. When we input this value, along with
${r=2.4}$ into equation 35, we obtain a reference magnitude of
${M_{\text{ZHR}}=2.6}$.

###  4.4 Canadian Meteor Orbit Radar (CMOR)

CMOR is a patrol radar; it detects the reflection of an emitted radar pulse
off of the surface of the cylinder of plasma produced by an ablating meteor.
The ability to detect a meteor is therefore not determined by its brightness,
but rather by the electron line density of the trail (usually denoted $q$).
Furthermore, CMOR does not necessarily detect the meteor when its trail is at
its peak electron line density. Instead, it detects a meteor when its velocity
has no radial component relative to the radar station and its trail is
therefore perpendicular to the radar beam.

Electron line density can be converted to an equivalent radar magnitude, but
individual radar magnitudes are not readily available for those meteors
contributing to the Orionid fluxes measured by CMOR. However, the CMOR flux
pipeline accepts population index (in the form of a mass index) as an input
parameter. We therefore varied $r$ in order to extract $M_{\text{ref}}$ using
the approach outlined in 4.2. We found that ${M_{\text{ref}}=+8.0}$, which is
fairly close to the limiting magnitude of +8.4 quoted in Campbell-Brown
(2004).

###  4.5 Synthesis

The raw ZHR values reported for each network are shown in Figure 5. The peak
values are quite discrepant; the CMOR and VMDB data have a peak ZHR of
approximately 15, the VMN data have a peak ZHR of about 30, and the GMN data
have a peak ZHR of about 50.

Figure 5: Raw Orionid ZHR values from four different networks recorded between
2020 and 2024.

We convert ZHR to flux using equation 23 and each network’s chosen value of
$r$. We also fit a double-exponential function (Jenniskens, 1994) to the data
in order to extract the peak value. Some of the data in Figure 5 show signs of
sporadic contamination; for instance, both video networks appear to asymptote
around ${\text{ZHR}=1}$. We therefore simultaneously fit for a constant
background level and subtract it from the flux. The results are shown in
Figure 6.

Figure 6: Background-subtracted Orionid fluxes from four networks; these
fluxes correspond to the reference magnitudes listed in Figure 1. The dark
line superimposed on each data set represents our fitted profile.

Our profile fits allow us to extract the peak ZHR or flux from each data set.
These peak fluxes are then plotted against the reference magnitudes in Figure
7. If the mass distribution does indeed take the shape of a power law, then
this plot should resemble a line. If we fit for $r$ using these four points,
we obtain an overall population index of ${r=2.7}$ (corresponding to a peak
ZHR of 28). On the other hand, the Orionid mass distribution may not resemble
a power law over large ranges in mass. The dynamical simulations of Egal et
al. (2020) predict a relative lack of Orionid particles at sizes that are
roughly comparable with CMOR’s limiting magnitude. In the bottom panel of
Figure 7, we compare our peak fluxes with the relative particle counts shown
in Figure 13 of Egal et al. (2020), extracted using WebPlotDigitizer (Rohatgi,
2022). Masses have been converted into magnitudes using the mass-magnitude
relation of Verniani (1973), and an arbitrary scaling has been applied to the
particle counts to bring them into rough agreement with the data. Both
distributions appear to be plausible matches to the data.

Figure 7: The peak flux of the Orionid meteor shower from four networks; each
point on this plot corresponds to the maximum value of a fitted flux profile
in Figure 6. In the top panel, we compare the data with a constant population
index of ${r=2.7}$; in the bottom panel, we compare the same data with the
mass-magnitude profile of Egal et al. (2020).

Although we have obtained ${r=2.7}$ using the measurements discussed in this
paper, we do not suggest that the meteor astronomy community should
necessarily adopt this value for the Orionid meteor shower. Our intent is
instead to demonstrate how our proposed reference magnitude can be used to
better compare disparate results. When fluxes are quoted to a network-specific
reference magnitude, we can combine independent measurements to probe the
magnitude distribution and reveal any remaining discrepancies. For instance,
if we convert flux back to ZHR using ${r=2.7}$ for all networks, we find that
the discrepancy between networks has been reduced (see Figure 8), but not
eliminated. The spread in peak ZHR values has been reduced by approximately
one-third (see Table 2). The visual rates are still lower than the
instrumentally measured rates but, because we have accounted for differences
in the assumed value of $r$, we now know that this discrepancy must be due to
some other factor.

Figure 8: Orionid ZHR values from all four networks, calculated from $f(M_{\text{ref}})$ or $f(M_{\text{ZHR}})$ using a uniform population index of ${r=2.7}$. We have subtracted the estimated background contamination.  Table 2: Peak ZHR values from each network (second column) and after correcting for differences in $r$ (third column). network | ZHR | ZHR${}_{\text{corr}}$
---|---|---
CMOR | 15.3 | 25.7
GMN | 51.3 | 41.1
VMN | 31.8 | 34.3
VMDB | 15.3 | 16.0

##  5 Conclusion

Currently, meteor fluxes are typically scaled to ZHR or a standard limiting
magnitude of +6.5 prior to publication. We argue that fluxes should be
reported to a network-specific reference magnitude that is equal to a weighted
average of the limiting magnitude of the observing station(s). This choice of
reference magnitude minimizes the dependence of the reported flux on the
assumed population index and therefore enables the synthesis of independent
meteor shower flux measurements. This is particularly useful when the
population index varies with magnitude, as the index that is most appropriate
for computing flux from a particular set of observations may differ from the
average index over a large range of magnitudes.

If researchers choose to publish fluxes using our suggested reference
magnitude, we suggest that they also include some measure of the degree of
variation within their network’s limiting magnitude. We have suggested the
sample standard deviation of the limiting magnitude. If the variation in
limiting magnitude is very large and if there are a sufficient number of
meteor detections, the researcher could also consider subdividing the data
into magnitude ranges (e.g., Molau et al., 2014).

In many cases, fluxes are quoted to a minimum mass rather than limiting
magnitude. We favor the use of a reference magnitude rather than a reference
mass for two reasons. First, magnitude is more closely related to the observed
quantity: the apparent brightness seen by the observer or observing station.
This is reflected in the effective collection area method for measuring
fluxes, which depends only on limiting magnitude. The second reason is that
there are multiple competing formulae for converting magnitude to mass
(Jacchia et al., 1967; Verniani, 1973; Campbell-Brown et al., 2016), and
therefore the conversion of magnitude to mass creates another opportunity to
introduce discrepancies that arise from different assumptions rather than real
differences in the data.

To demonstrate the utility of our reference magnitude, we estimated its value
for several independent measurements of Orionid fluxes between 2020 and 2024.
We included data from four networks, each of which used a different assumed
population index to compute meteor flux or ZHR. We found that the data could
be brought into closer agreement if we assume that the Orionid mass index is
approximately 2.7. This synthesis could potentially be improved if exact,
rather than estimated, reference magnitudes for CMOR and VMN were to be
published. The data also suggest that a meteor observation network of
extremely sensitive cameras with a reference magnitude of approximately +6
(similar to electron-multiplied charge-coupled device, or EMCCD, cameras;
Brown & Weryk, 2020; Mills et al., 2021; Gural et al., 2022) would be a
valuable addition.

This work was supported in part by NASA Cooperative Agreement 80NSSC21M0073
and by the Natural Sciences and Engineering Research Council of Canada.

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