# Exocomet models in transit: light curve morphology in the optical – near
infrared wavelength range

Szilárd Kálmán Konkoly Observatory, HUN-REN Research Centre for Astronomy and
Earth Sciences, Konkoly Thege 15-17, 1121 Budapest, Hungary  HUN-REN CSFK, MTA
Centre of Excellence, Budapest, Hungary  HUN-REN-ELTE Exoplanet Research
Group, Szombathely, Szent Imre h. u. 112., H-9700, Hungary  ELTE Eötvös Loránd
University, Doctoral School of Physics, Budapest, Pázmány Péter sétány 1/A,
H-1117, Hungary  Gyula M. Szabó HUN-REN-ELTE Exoplanet Research Group,
Szombathely, Szent Imre h. u. 112., H-9700, Hungary  ELTE Eötvös Loránd
University, Gothard Astrophysical Observatory, Szombathely, Szent Imre h. u.
112., H-9700, Hungary  Csaba Kiss Konkoly Observatory, HUN-REN Research Centre
for Astronomy and Earth Sciences, Konkoly Thege 15-17, 1121 Budapest, Hungary
HUN-REN CSFK, MTA Centre of Excellence, Budapest, Hungary  Institute of
Physics and Astronomy, ELTE Eötvös Loránd University, Budapest, Hungary
Szilárd Kálmán<EMAIL_ADDRESS>
###### Abstract

Following the widespread practice of exoplanetary transit simulations, various
presumed components of an extrasolar system can be examined in numerically
simulated transits, including exomoons, rings around planets, and the
deformation of exoplanets. Template signals can then be used to efficiently
search for light curve features that mark specific phenomena in the data, and
they also provide a basis for feasibility studies of instruments and search
programs. In this paper, we present a method for exocomet transit light curve
calculations using arbitrary dust distributions in transit. The calculations,
spanning four distinct materials (carbon, graphite, pyroxene, and olivine),
dust grain sizes ($100$ nm – $300$ nm, $300$ nm – $1000$ nm, and $1000$ nm –
$3000$ nm) encompass light curves in VRJHKL bands. We also investigated the
behavior of scattering colors. We show that multicolor photometric
observations are highly effective tools in the detection and characterization
of exocomet transits. They provide information on the dust distribution of the
comet (encoded in the light curve shape), while the color information itself
can reveal the particle size change and material composition of the transiting
material, in relation to the surrounding environment. We also show that the
typical cometary tail can result in the wavelength dependence of the transit
timing. We demonstrate that multi-wavelength observations can yield compelling
evidence for the presence of exocomets in real observations.

Exocomets(2368) — Comet tails(274) — Multi-color photometry(1077) — Radiative
transfer simulations(1967)

††facilities: Exoplanet Archive††software: RADMC-3D (Dullemond et al., 2012),
optool (Dominik et al., 2021)

##  1 Introduction

Comets, mostly known as spectacular sojourners on our skies (Kronk et al.,
2017), have been counted for a long time among the most ancient reservoirs of
solid material and ice in the Solar System. As witnesses of the formation of
the outer Solar System (e.g. Donn, 1963; Encrenaz, 1984; Vidmachenko &
Steklov, 2013), their orbits can evolve through gravitational perturbations,
causing them to migrate to the inner Solar System – in a process that
presumably happens very similarly in the extrasolar systems as well (e.g.
Graham et al., 2007; Matrà et al., 2017). Comets are believed to be one of the
possible sources of water and nitrogen (e.g. Marty, 2012; Altwegg et al.,
2015; Poch et al., 2020) and possibly even organic material on Earth (e.g.
Fray et al., 2016; d’Ischia et al., 2021). They are therefore invoked as
essentials for the volatiles and habitability of Earth and, based on this
analogy, also on other planetary systems (e.g. Kral et al., 2018). Thus, these
objects are also important from an astrobiological perspective (Welsh, 2019).

Compared to the $\sim 4150$ known exoplanetary systems at the time of writing,
there are only a few systems in which exocometary activity has been detected.
There are two main approaches to the detection of extrasolar comets:
photometry and spectroscopy (Pavlenko et al., 2021). In the circumstellar disk
of the $\beta$ Pictoris system, Ferlet et al. (1987) detected time-variable
features in Ca ii H & K spectral lines, which are attributed to the so-called
Falling Evaporating Bodies (FEBs Beust et al., 1994; Lagrange-Henri et al.,
1989; Beust & Morbidelli, 2000). Given that FEBs were first detected and are
perhaps the most studied in the $\beta$ Pictoris system (Beust & Valiron,
2007; Tobin et al., 2019), these spectroscopic transients are also known as
$\beta$ Pic-like phenomena. Kiefer et al. (2014b) identified 493 individual
exocomets in $\beta$ Pictoris through the study of these features. The large
number of comets detected allowed for a clustering of these objects, yielding
two groups of distinct origins and properties (Kiefer et al., 2014b; Roberge,
2014). Observations of FEBs are not limited to the $\beta$ Pic system.
Redfield (2007) detected time-dependent absorption features in the Na i D
doublet of HD 32297, which is related to the debris disk surrounding the star.
Other well-known examples include HD 172555 (Kiefer et al., 2014a; Grady et
al., 2018), HD 100546 (Grady et al., 1997), HD 24966, HD 38056, HD 79469 and
HD 225200 (Welsh & Montgomery, 2018). As a further step in interpreting these
spectroscopic observations, Kennedy (2018) proposed a method to retrieve some
of the orbital elements of the exocomets in question. Rebollido et al. (2020)
carried out a large spectroscopic survey of (suspected) exocomet host stars,
finding that $17\%$ of stars showing near infra-red excess allow for a
detection of FEB-like events, suggesting a weak link between the evolution of
a protoplanetary disk and the features attributed to exocomets.

The FEB-detections listed above rely on spectroscopic time series
observations, which are complicated by nature. Lecavelier Des Etangs et al.
(1999) proposed the charateristic “round triangle” shape of exocometary
transits, which are thought to be photometric counterparts of the FEBs (Zieba
et al., 2019), and subsequently constructed a library of various transit
shapes (Lecavelier Des Etangs, 1999). Lecavelier Des Etangs et al. (1999)
present a light curve model constructed of two components: extinction and
forward scattering, both of which are based on physical modeling of cometary
activity. The total contribution of these effects depends strongly on the
grain size of the dust (and gas) emitted from the comet as well as on its dust
production rate. The shapes of the transit light curves that result are
colloquially called “sharkfins”. These transit events are naturally detectable
in long baseline datasets of survey-type space observatories, such as Kepler
(Borucki et al., 2010), TESS (Ricker et al., 2015, Transiting Exoplanet Survey
Satellite) or the upcoming PLATO (Rauer et al., 2014, PLAnetary Transits and
Oscillations of stars). TESS also revealed several hundred so-called dippers
(Tajiri et al., 2020), which are generally young stars characterized by
transients with variable depth and lasting characteristically for 0.5–1 days.
These transients are thought to have multiple origins, from the disk warps and
accretion columns of T tauri stars (represented by rather periodic signals) to
the transit of giant exocomets in the line of sight (represented by aperiodic
components). Strøm et al. (2020) also discussed the possibility of
interstellar comets as the origin of the signals which would naturally be
single-occurance events. Although the low rate of periodicity in a system is a
diagnostic of the presence of exocomets, there is still a need for independent
evidence for the cometary origin (Tajiri et al., 2020; Gaidos et al., 2022;
Gaidos, 2022).

The detection of exocomet transits (Rappaport et al., 2018; Zieba et al.,
2019; Kennedy et al., 2019; Lecavelier des Etangs et al., 2022; Kiefer et al.,
2023) often relies on pattern search methods111Private communication with S.
Zieba (including the box-fitting approach of Kennedy et al., 2019, that has to
include a false-positive rejection stop due to exoplanets, binary stars etc.),
which are known to be sensitive only to patterns that were used for training.
In case of the box-fitting approach, this sensitivity arises at the false-
positive rejection step. Most “exocomet transit” potometric transients indeed
represent a sharkfin-like feature, but these are the features the methods are
looking for. With a sophistically trained pattern generator, we can derive
template light curves of transiting exocomets that we can include in the
actual search and interpretation efforts.

In this paper, we utilize a simple model to generate synthetic exocometary
transit light curves. We put comet-like model distributions into transit and
calculate the radiation transfer as a function of the angle of the incident
backlight. We present template light curves from these transit simulations
from optical to near-infrared (NIR) wavelengths. We also explore the “color of
comet dust” (e.g. Kolokolova et al., 2001; Levasseur-Regourd et al., 2007)
when transiting a star, comparing the effect of dust size and material
dependence along the examined wavelengths.

The paper is structured as follows. In Sect. 2 we describe the exocomet
models, as well as the light curve simulations (including both extinction and
forward scattering). In Sect. 3, we present a library of light curves that are
made publicly available. In Sect. 4, we place our models into context and
consider the possible implications regarding the future of the hunt for
exocomets.

##  2 Methods

###  2.1 Radiative transfer modeling

Figure 1: Density distribution of the comet models in this study. The figure
panels show the comet with a narrow tail, the comet with a fan-like tail and
the comet without a tail. The 2D images show the comet models as seen in
reflected light in sky projection, together with the projected densities on
the y and z axis (top and bottom right panels).

Radiative transfer modeling of dust structures, assumed to be responsible for
the obscuration of stellar light in transient phenomena, is performed using
RADMC-3D (Dullemond et al., 2012). RADMC-3D is a freely available open source
code package for diagnostic radiative transfer calculations. It calculates,
for a given geometrical distribution of gas and/or dust, what its images
and/or spectra look like when viewed from a certain angle, including optical
depth calculations, both in scattered light and in thermal emission. To
perform exocomet light curve synthesis, we use RADMC-3D 2.0 and a volume of
2563 cells in Cartesian coordinates, with a box edge size adjusted to the
actual density distribution (see Sect. 2.3), typically in the order of 0.05
au. The center of the coordinate system is the center of the simulation
volume. The central star is located at a distance of $r_{h}$ from the center
of the coordinate system on the x-axis, outside the simulation volume. We
considered two cases: a ‘solar-like’ star (with a photospheric temperature of
5780 K a radius of $R_{\odot}=6.96\times 10^{5}$ km and $\log g=4,5$) and,
since most exocomet detections have been made around A stars (e.g. the $\beta$
Pic system, Zieba et al., 2019; Lecavelier des Etangs et al., 2022), a star
with a photospheric temperature of $8200$ K, $R_{\star}=1.7R_{\odot}$ and
$\log g=4.0$. Although stellar luminosity is known to greatly influence the
formation of cometary comae and tails (Lecavelier Des Etangs, 1999), we used
the same dust distributions for both stars.

The materials used for radiative transfer modeling are characterized by their
composition and size distribution, as explained below. We utilize the optool
package (Dominik et al., 2021) to obtain the complex dust particle opacities,
using materials from optool’s material library, and assuming a specific grain
size distribution. We consider different types of materials for our dust
structures in the different simulations, but we assume that the whole
simulation volume is homogeneous material-wise. Anisotropic scattering is
treated by applying the Henyey-Greenstein function (option scattering_mode_max
= 2 in RADMC-3D), and using the scattering opacity and $g$ anisotropy
parameters obtained for the specific material with optool. To be able to
disentangle the effect of grain size, we use three size distributions. The
number of particles ($N$) within these ranges is given by

$N\propto a^{-q}.$ (1)

We set the size distribution exponent $q=3.5$ in all cases (Dohnanyi, 1969).
The size ranges of the three distributions are a = 0.1–0.3 $\mu$m (labeled as
200 nm and corresponding to a mean size of $143.82$ nm), 0.3–1.0 $\mu$m (650
nm, with a mean size of $439.51$ nm) and 1.0–3.0 $\mu$m (2 $\mu$m), which
corresponds to a mean particle size of $1438.2$ nm. We selected these specific
grain size intervals to sample the total scattering cross section provided by
Pawellek et al. (2024), at the lower size ranges of that distribution, as well
as the range where the rapid changes in the total cross section occur. Larger
grains with a $\gtrsim$ 10 $\mu$m cause proportionally smaller extinction and
increased brightness from forward scattering and are not considered in our
present analysis. While in a real comet we likely see a mixture of different
materials, our goal here is to see the effect of a single composition
separately. Therefore, in our calculations, we use four different homogeneous
compositions, in each case using each of the following material compositions
separately: graphite (Draine, 2003), amorphous pyroxene with 70% Mg content
(Dorschner et al., 1995; Henning & Stognienko, 1996), amorphous olivine with
50% Mg content (Dorschner et al., 1995; Henning & Stognienko, 1996), and
carbon grains (Zubko et al., 1996). While the choice of materials is
arbitrary, we note that the carbonaceous (Woodward et al., 2021) and silicate
dust (Wooden et al., 1999) is common in the comets of the Solar System. The
debris disk of $\beta$ Pictoris is also known to host various silicates
(Okamoto et al., 2004; Lu et al., 2022; Rebollido et al., 2024). In the case
of a more realistic scenario when dust grains with different compositions are
mixed, the total extinction / amount of scattered light can be obtained from
the linear combination of the optical depths / scattered light contribution of
the individual materials, assuming that the optical depth of the overall dust
structure remains low ($\tau$ $\ll$ 1) and the radiation field is dominated by
the radiation of the central star even inside the dust structure. As we will
see, this is the case in our simulations. In case of Trans-Neptunian objects
(in a frozen environment) the model spectra usually assume the presence of
compound materials such as ices depleted to solid particles (e.g. Szabó et
al., 2018). However, in case of comet Hale-Bopp, the spectra are well
described as a sum of contributions of simple materials (pyroxene and olivine
in particular Wooden et al., 1999). The lack of ice layer on comet dust in
general seems to be plausible taken the low ice-to-dust ratio on comet layers
into account, and the energetic irradiation field. For this reason, we
restrict our simulations to homogeneous materials.

Extinction and forward–scattering light curves are calculated separately,
assuming that the dust distribution (i.e. the comet) passes in the line of
sight of the central star, orbiting it at a specific distance from the star
(0.1 au on a circular orbit), the orbital plane is seen perfectly edge-on, and
the occultation is central. We therefore neglect the impact of the orbital
inclination (relative to the line of sight), which is known to influence the
exact shape and depth of the transits (Lecavelier Des Etangs, 1999).

For each light curve position, the central star is at a different location
behind the density distribution (simulation volume). We cover the whole
occultation event by changing the observer’s aspect angle by discreet
intervals ($0.1^{\circ}$ during the transit itself,
$0.2^{\circ}$–$0.05^{\circ}$ out-of-transit). This is equivalent to the motion
of the comet in front of the central star. The extinction light curve is
obtained from the optical depth map, which is calculated by RADMC-3D for the
specific observing geometry, convolved with the limb-darkened, resolved disk
of the central star. We utilized the quadratic l imb-darkening rule (Wade &
Rucinski, 1985), where the surface intensity of the star can be expressed as

$\frac{I\left(\gamma\right)}{I(0)}=1-u_{1}\cdot\left(1-\cos\left(\gamma\right)\right)-u_{2}\cdot\left(1-\cos\left(\gamma\right)\right)^{2}.$
(2)

Here, $\gamma$ is the angle between the normal vector on the stellar surface
and the line of sight. We adopted the coefficients $u_{1}$ and $u_{2}$ from
Claret (1998), these are listed in Table 1.

###  2.2 Light curve calculations

The forward-scattered light curve component corresponds to the total intensity
of the surface brightness distribution of the same observing geometry, without
the contribution of the central star (also directly obtained with RADMC-3D).
It is calculated for a point source (assuming blackbody radiation), thus we
neglect the effect of limb-darkening on this component. It can easily be
scaled between the two stellar types (Solar-like and A-type star) by
calculating the ratio of the emitted photons in each star. In this case, the A
star we selected emits $4.1\times$ as many photons (at every wavelength),
yielding $4.1\times$ higher amplitudes for this light curve component, as we
show below. As a final step, we fit a spline function to interpolate both
components of the light curves.

Passband | Solar-like star | A star
---|---|---
$u_{1}$ | $u_{2}$ | $u_{1}$ | $u_{2}$
V (550 nm) | 0.517 | 0.229 | 0.276 | 0.388
R (650 nm) | 0.395 | 0.279 | 0.216 | 0.359
J (1200 nm) | 0.149 | 0.334 | 0.064 | 0.300
H (1650 nm) | 0.058 | 0.375 | 0.031 | 0.276
K (1650 nm) | 0.041 | 0.327 | 0.015 | 0.236
L (3450 nm) | 0.019 | 0.240 | 0.006 | 0.132
Table 1: Limb-darkening coefficients for the two stars considered in the light
curve simulations, in the passbands where these calculations were carried out.

Extinction (defined as absorption plus scattering) efficiency of the dust
grains depends on the size and material of a grain, as well as the phase
angle. As the same amount (mass) of dust may consist of a larger number of
smaller grains or smaller number of larger grains, in a random spatial
distribution smaller grains cover a larger area, with an area increase
proportional to the relative grain size, leading to an overall increased
extinction cross section. However, this is complicated by the grain size
dependence of the extinction efficiency that depends on the dielectric
properties of the grains. This is demonstrated in Fig. 2 (left panel) where we
show the extinction coefficient for pyroxene grains, one of the materials used
in our simulations, for different grain sizes.

Figure 2: Left panel: Extinction coefficient as a function of wavelength for
pyroxene grains with 70% Mg-content (Dorschner et al., 1995; Henning &
Stognienko, 1996). Right panel: $g$ scattering parameter of the Henyey-
Greenstein function as a function of wavelength, for different grain sizes,
for the same material as in the left panel. The curves on both panels
correspond to grain sizes of 200 nm (solid), 650 nm (dashed) and 2 $\mu$m
(dash-dotted).

Scattering by dust grains is typically strongly phase angle dependent in the
visible – near-infrared wavelength range (see e.g. Gordon, 2004). The phase
angle dependence in our simulations is characterized by the $g$ scattering
parameter of the Henyey-Greenstein scattering function – $g\,=\,0$ corresponds
to isotropic scattering, $g\,=\,-1$ dominant backward scattering, and
$g\,=\,1$ dominant forward scattering. An example is presented in Fig. 2
(right panel) for pyroxene grains, which show strong forward scattering for
visible light, and moderate forward scattering for the near-infrared. Due to
our modeling geometry – the dust structure is between us and the star, seen
under a phase angle of $\phi$ $\approx$ 180° – our ’observed’ light is
dominated by forward scattered radiation.

Figure 3: Synthetic, monochromatic light curves of a comet with narrow tail,
plotted for the three selected grain size distributions, labelled as $a\sim 2$
$\mu$m (top row), $650$ nm (middle row) and $200$ nm (bottom row), for an A
star. Columns denote the materials used in the simulations: carbon (left),
graphite (second from the left), pyroxene (second from the right), and olivine
(right). Note the change in the scale of the y axis from row to row.

Thermal emission of the dust is also included in the radiative transfer
calculations. The dust temperature is also dependent on the composition and
grain size (see e.g. Henning & Stognienko, 1996), and due to the proximity of
the dust structure to the central star ($r_{h}$ = 0.1 au) relatively high
$T_{d}$ $\approx$ 1000 K dust temperatures are expected. However, as in our
simulations the distance to the star remains constant during the transit and
the optical depths are low ($\tau$ $\ll$ 1), the thermal emission contribution
to the total brightness at a specific wavelength is constant before, during,
and after the transit, meaning that it has no effect on the differential light
curves.

The final “observable” light curve is the sum of the extinction and
(forward–)scattering light curves. The calculated monochromatic light curves
represent six common photometric bands: V (550 nm), R (650 nm), J (1.2
$\mu$m), H (1.65 $\mu$m), K (2.2 $\mu$m) and L (3.45 $\mu$m). We extrapolated
the limb darkening coefficients for the L band (Table 1), which was not
included in Claret (1998). These passbands were selected to represent the
passband of `PLATO` (Rauer et al., 2014; Jannsen et al., 2024) and to sample
Ariel spectrographs (Tinetti et al., 2018, 2021; Szabó et al., 2022), while
avoiding the computationally expensive approach of simulating entire
wavelength ranges of specific instruments.

###  2.3 Density distributions

Figure 4: Same as Fig. 3 but for the comet model with a fan-like tail.

The input to our radiative transfer code is the spatial distribution of grains
in the coma and possibly the tail in the comet. In the current stage, we can
calculate the radiative transfer in a homogeneous environment built up by
grains of a selected material with the same distribution of grain sizes
everywhere in the coma.

First, our idea was to put actual images of selected Solar System comets
directly in transit in front of an arbitrary star. However, this concept had
incurable drawbacks and therefore had to be omitted. The most important issues
were: i) The comets are seen in a 2D projection on the sky, while the
radiative transfer has to be calculated in a 3D environment – including a
third dimension in the distribution would have been mostly speculative rather
than plausible; and ii) The scattering physics in a comet is complex. We see
the actual comets because of the reflection properties of the dust at some
specific illumination geometry, while we see the transiting exocomets because
of the combination of absorption at a different, backlit geometry, together
with components belonging to forward scattering. Without exact knowledge of
material properties (spatially changing dust distributions, etc.) within the
comet, the visible image cannot be plausibly converted into light scattering
by that comet in transit.

Instead of using a combination of 2D images and additional more-or-less
unfounded assumptions, we have chosen to base our model calculations entirely
on mathematical foundations. This approach aligns well with the strength of
our code, allowing us to calculate 3D distributions with homogeneous material
properties. However, to ensure that our models are connected to the appearance
of real comets, we used the visual appearance of observed Solar System comets
as an inspirational source in the model design. We also aimed for a simple
mathematical formalism and ultimately defined 3D distributions that accurately
capture the impression of the chosen comets in their symmetry relationships
and intensity distributions.

Our model parameters are therefore the material properties (which material,
distribution of grain size) and the spatial density distribution. The actual
cometary comae are known to follow a power-law light distribution that belongs
to a spatial matter distribution of $r^{-2}$ in the case of homogeneous and
isotropic matter production (A’Hearn et al., 1984). This distribution has
infinite extension and infinite mass – which does not occur in practice, of
course, due to dust dynamics and dust fragmentation in a radiation
environment. Describing this cutoff is difficult if not impossible
analytically, and in the case of real comets, dust dynamics is calculated with
numerical methods (Fulle, 2004). In this paper, to avoid the issues of
physical interpretation, we postulate that the truncating function is a
Gaussian distribution with finite extension. Although not fitting to any
comet, this truncated model describes a light distribution that resembles most
cometary comae.

For the demonstration presented in this paper, we selected three models with
different weights and symmetry of the tail. Instead of the physical model of
cometary activity presented in Lecavelier Des Etangs et al. (1999) which
conveniently recreates the shape of a classical comet as explained by the so-
called “fountain model” (e.g. Alfven, 1957; Wallis, 1968; Wallis & MacPherson,
1981), we construct dust density distributions that resemble well-known comets
from the Solar System. As a result, we can also study the way the shape of an
exocomet might influence it’s transit light curve.

#### A comet with a narrow tail

We regard one of the fragments of Comet 73P/Schwassmann-Wachmann as the
baseline for our simulations. In this case, the comet is assumed to have a
central coma and a long tail (Fig. 1, left panels) to provide an appearance
similar to that obtained by Reach et al. (2009). We assumed a three
dimensional density distribution in the form of

$\rho=\Bigg{\\{}\begin{array}[]{lr}w_{1}\rho_{0}e^{-r^{2}/\sigma_{1}^{2}}/r^{2}+(1-w_{1})\rho_{0}e^{-(x^{2}+z^{2})/\sigma_{2}^{2}}e^{-y^{2}/\sigma_{f}^{2}}&:y\geq
0\\\
w_{1}\rho_{0}e^{-r^{2}/\sigma_{1}^{2}}/r^{2}+(1-w_{1})\rho_{0}e^{-(x^{2}+z^{2})/\sigma_{2}^{2}}e^{-|y|/\sigma_{t}}&:y<0\\\
\end{array}$ (3)

to represent such a comet, where $r^{2}=x^{2}+y^{2}+z^{2}$. We set
$w_{1}\,=\,0.2$, $\sigma_{1}\,=\,0.0015$ au, $\sigma_{2}\,=\,0.0025$ au,
$\sigma_{f}\,=\,0.0025$ au and $\sigma_{t}\,=\,0.010$ au. This dust
distribution also outlines the limitations of our modeling, and the deviation
from (Lecavelier Des Etangs et al., 1999; Lecavelier Des Etangs, 1999; Zieba
et al., 2019), since the tail of the cometary shape is truncated and we
neglect the change in the distance between the star and each particle of the
cometary tail.

#### A comet with a wide, fan-like tail

Here, a central dense coma is surrounded by a spherical structure that is
extended in one direction (defined in the $+y$ direction in the simulation
volume), but fades quickly in the opposite direction as in the case of Comet
29P/Schwassmann-Wachmann (Trigo-Rodríguez et al., 2008). In this case, we
assumed the following densitry distribution:

$\rho=\Bigg{\\{}\begin{array}[]{lr}w_{1}\rho_{0}r^{-2}e^{-r^{2}/\sigma_{1}^{2}}+(1-w_{1})\rho_{0}e^{-r^{2}/\sigma_{2}^{2}}&:y\geq
0\\\
w_{1}\rho_{0}r^{-2}e^{-r^{2}/\sigma_{1}^{2}}+(1-w_{1})\rho_{0}e^{-(x^{2}+z^{2})/\sigma_{2}^{2}}e^{-y^{2}/\sigma_{t}^{2}}&:y<0\\\
\end{array}$ (4)

The relative weights of the central and extended components are characterized
by the weight factor $w_{1}$, the core and extended structures are described
by two Gaussians with width parameters $\sigma_{1}$ and $\sigma_{2}$, and the
fading tail is also described with a Gaussian with the width parameter
$\sigma_{t}$ in the —- $y$ direction. We set $w_{h}\,=\,0.5$,
$\sigma_{1}\,=\,0.003$ au, $\sigma_{2}\,=\,0.012$ au and
$\sigma_{t}\,=\,0.006$ au in our calculations. The resultant distribution is
shown in Fig. 1, in the middle panels.

#### A comet without tail

This density distribution is assumed to be fully spherically symmetric, that
is, the density depends only on the radial distance $r$ from the center of the
comet. This density distribution was selected to resemble Comet 17P/Holmes
during its 2007 outburst (e.g. Moreno et al., 2008a, b; Trigo-Rodriguez et
al., 2008). The distribution is constructed as the sum of two 3D Gaussian
profiles, characterized by the width parameters $\sigma_{1}$ and $\sigma_{2}$.
The relative weights of the two Gaussians are determined by the weight factor
$w_{h}$. The central density $\rho_{0}$ is set for each simulation as
described below. The spatial distribution of the density is determined as
follows:

$\rho=w_{h}\rho_{0}e^{-r^{2}/\sigma_{1}^{2}}+(1-w_{h})\rho_{0}e^{-r^{2}/\sigma_{2}^{2}}$
(5)

We applied $w_{h}$ = 0.7, $\sigma_{1}$ = 0.002 au and $\sigma_{2}$ = 0.008 au.

To set up the dust distributions, our requirement was that the maximum optical
depth ($\tau$) in the simulation with the a = 2 $\mu$m grain size is
$\tau_{R}$ = 10-4 in the R band (650 nm) for the solar-like star,
corresponding to the nominal maximum absorption depth222We regard $\tau$ as
the measure of the extinction component only, and AD as the measure of the
combination of both light curve components (extinction and forward
scattering). Thus, in a real observation scenario, only AD can be obtained.
(AD) identified by Lecavelier des Etangs et al. (2022) in the $\beta$ Pictoris
system. This specifies a total dust mass that is then kept constant for the
other grain sizes (a = 200 and 650 nm). These dust masses are listed in Table
2 below. In the following, we refer to the three dust distributions by the
name of the comet that they are meant to resemble.

Comet name | carbon [$10^{11}$ kg] | graphite [$10^{11}$ kg] | pyroxene [$10^{11}$ kg] | olivine [$10^{11}$ kg]
---|---|---|---|---
No tail | 2.96 | 4.14e+11 | 5.49e+11 | 6.98e+11
Fan-like tail | 4.65 | 7.48 | 9.93 | 1.26
Narrow tail | 0.84 | 1.35 | 1.80 | 2.29
Table 2: Total dust masses of the four materials used in the simulations, at
the distance of $r_{h}$ = 0.1 au.

##  3 Results

Figure 5: Same as Fig. 3 but for the comet model without a tail.

Simulated light curves were calculated for each of the three model
distributions in transit, assuming carbon, graphite, pyroxene, and olivine
dust, and with a characteristic grain size in the coma/tail of 2 $\mu$m, 650
nm and 200 nm. Transit light curves were calculated in the 6 predefined
wavelengths (resulting in 12 scenarios for comet each model, at every
wavelength), yielding a 6-color photometric time series in each simulation.
Template light curves are plotted in Figs. 3 – 5 for a star with spectral type
A, and in 12–14 for a Solar-like star.

###  3.1 Light curve morphology

Figures 3 and 12 show the exocomet with a prominent thin tail, which is a
morphologically somewhat similar distribution to what is expected from the
fountain model (e.g. Alfven, 1957; Wallis, 1968; Wallis & MacPherson, 1981),
leading to sharkfin-like transit light curves. Indeed, on a qualitative
comparison, the light curves in Figs. 3 and 12 closely resemble those seen in
Lecavelier Des Etangs et al. (1999) and Lecavelier Des Etangs (1999). Our
simulations also bear close resemblance to observed transit features that are
attributed to exocometary activity, such as in Rappaport et al. (2018); Zieba
et al. (2019); Lecavelier des Etangs et al. (2022), or a disintegrating
exoplanet (Rappaport et al., 2012; Brogi et al., 2012). The light curves seen
in Fig. 12 and especially in Fig. 3 readily reproduce the peak observable
before the transit itself (Rappaport et al., 2018; Zieba et al., 2019), which
is a consequence of the forward–scattered light.

Figs. 3 – 5 and 12–14 unanimously exhibit a common strong dependence of $\tau$
(and, as a consequence, AD) on the dust grain size, where smaller grain sizes
generally yield deeper transits. This is especially true for carbon and
graphite. This is a natural consequence of our calibration process, since all
distributions of individual comets have the same total mass (Table 2). Figs.
6, 7, and 8 represent the measured AD for the three comet models considered.
The well-known wavelength dependence of the scattering cross section (that is
itself $\propto a^{2}$) means that while the light curves of a given material
for a given $a$ are topologically similar to each other, in the regime of
small dust grains ($a\sim 200$ nm), the absorption depth varies considerably
throughout the explored passbands. At the $2$ $\mu$m dust size range, there
are only comparatively minor differences in absorption depths at different
wavelengths. Slightly larger differences in AD can be observed at the $650$ nm
grain sizes, where (unlike in $200$ nm regime) AD increases towards the red
end of the wavelength range (Figs. 6 – 8).

In all three cases, we see excess light before transit ingress. This is the
contribution of forward scattering, the light coming from the dust grains,
which are seen close to the edge of the star but are still not in transit.
Because forward scattering is efficient at very small deflection angles (Fig.
2), this contribution appears just before the transit (and is well known in
the case of transits of evaporating/disintegrating planets as well (e.g.
Rappaport et al., 2012; Garai, 2018). The amplitude of the forward-scattered
light also shows a clear dependence on both the material composition, the
grain size, and the wavelength, dust distribution, and stellar spectral type
(Fig. 9), with the highest amplitude (on the order of $\approx 90$ ppm)
occurring in the V and R bands, for olivine, in the case of the wide-tailed
comet model. Fig. 9 also clearly shows that carbonaceous materials are less
efficient at forward scattering, implying that the exocomet candidates
observed by (Zieba et al., 2019) are not primarily made of these materials.
Infrared wavelengths yield lower amplitudes for this component of the light
curve. In this regime we observe a complete turnaround of the amplitudes as
the $a=200$ nm grain sized dust shows the highest amplitudes for the forward-
scattered light, for the carbonaceous material.

The amplitude is also influenced by the shape of the comet itself, this
manifests in the fact that the wide-tailed comet (Fig. 1) yields the highest
amplitudes for this light curve component. Both comet models that own a tail
component produce asymmetrical scattered light components, unlike the
spherically symmetric coma alone.

A prominent difference between extinction and forward scattering is that the
latter depends on the direct illumination of the dust. Therefore, the
amplitude of the forward scattering components are scaled to arbitrary
star–comet distances by $\left(r_{h}/1\text{au}\right)^{2}$ as a first-order
approximation (where $r_{h}$ is the actual star–comet distance). In theory,
the transit duration can be used to estimate $r_{h}$ from a real exocomet
transit observation (e.g. Boldog et al., 2023), which could then be used to
put further investigate the cometary composition through the forward-scattered
light. Because the extinction simply occurs due to blocking of a part of the
line of sight, the extinction components (analogous to the transit depths) are
independent of $d$ (e.g. Mandel & Agol, 2002), and depend only on the total
scattering cross section of the dust. In this way, proper modeling of the
forward scattering and the extinction together can also be used to reveal the
star-to-transient distance via the ratio of the extinction to forward
scattering.

Figs. 4–5 and 13–14 show transient light curves for comet models with
increasing circular symmetry, that is, a fan-like tail and no tail at all. The
combined light curve of the single coma (Figs. 5 and 14) also resembles a
grazing transit of an exoplanet (e.g. Mancini et al., 2014). This is an
important observation, because the shape of the transient itself is not a
decisive diagnostics of the comet nature: exocomets (being similar to
17P/Holmes in outburst or other coma-dominated comets such as 1P/Encke for
example) can also produce symmetrical light curve transients. On the other
hand, possibly dust clumps (in a debris disk) unrelated to comets can also
bear asymmetry that results in a more sharkfin-like transient feature. In
these cases, we propose to observe the AD ratios (Figs. 6 – 8), or the
“colors” directly during transit (16 – 19). The wide range in AD that is
observed especially for the smallest grain sizes represented here (Figs. 6 –
8) implies that a spectrophotometric observation can be used to extract
information regarding the material composition of an exocometary transit. This
is discussed below.

The light curves are similar to each other between the two stars (a ‘solar-
like’ and an A-type star) for all three given dust distributions. This is
observed when comparing Figs. 3, 4, and 5 with 12, 13, and 14 , respectively.
One key difference is the Absorption Depth, as the same dust distribution
transiting in front of the A spectral type star considered here (that is
$1.7\times$ larger than the Sun-like star) yield transits that are $\leq
2\times$ shallower at each wavelength. In the cases of exoplanets, the planet
is fully obscuring a portion of the stellar disk, leading to a direct way to
estimate its size based on the transit depth (e.g. Seager & Mallén-Ornelas,
2003), which is analogous to AD. For exocomets, no such straightforward method
can exist to estimate the size based on the absorption depth, as it depends on
a large number of factors, as shown above. We have also repeated the same
calculations with a Solar-like host star, and we show the light curves in
Appendix A.

###  3.2 Color index variations

When comparing Figures 3–5 and 12–14, the dependence of AD on the properties
of the material and the characteristic size of the particles is evident. This
is further emphasized by the direct comparison of AD (Figs. 6, 7, and 7), or
the color variations (Figs. 16 – 19). The asymmetry of the dust distribution
in combination with the (wavelength-dependent) limb darkening of the star
yields $\approx 6$ time brighter or fainter portions of the light curve (Figs.
16 and 15). This is a phenomenon previously known and extensively studied in
the case of solar system comets (e.g. Jewitt & Meech, 1986; Kolokolova et al.,
1997; Luk’yanyk et al., 2019). However, the available time series data for
extrasolar comets are currently monochromatic, so the value of AD ratios and
colors has not been particularly emphasized until now.

Figure 6: Absorption depth as a function of wavelength for the narrow-tailed
comet (Fig. 1, left) for the Solar-like star (left) and the A star (right).
The four materials considered (carbon, graphite, pyroxene, and olivine) are
represented by colors (blue, red, magenta, and grey, respectively), while full
circles, squares, and triangles correspond to the different dust grain sizes
($200$ nm, $650$ nm, and $2$ $\mu$m, respectively). Straight lines are shown
for visualization purposes only.

Figure 7: Absorption depth as a function of wavelength for the narrow-tailed
comet (Fig. 1, left) for the Solar-like star (left) and the A star (right).
The four materials considered (carbon, graphite, pyroxene, and olivine) are
represented by colors (blue, red, magenta, and grey, respectively), while full
circles, squres, and triangles correspond to the different dust grain sizes
($200$ nm, $650$ nm, and $2$ $\mu$m, respectively). Straight lines are shown
for visualization purposes only.

Figure 8: Absorption depth as a function of wavelength for comet without tail
(Fig. 1, right) for the Solar-like star (left) and the A star (right). The
four materials considered (carbon, graphite, pyroxene, and olivine) are
represented by colors (blue, red, magenta, and grey, respectively), while full
circles, squres, and triangles correspond to the different dust grain sizes
($200$ nm, $650$ nm, and $2$ $\mu$m, respectively). Straight lines are shown
for visualization purposes only.

We can observe that the smallest dust grains ($\sim$ 200 nm size range)
produce transit light curves of significantly varying depth with the central
wavelength. This is a “colorful” dust in transmission. In the case of
carbonaceous dust (carbon and grahite) with 200 nm size, the optical V and R
signals are the deepest, and they are very similar to each other, the V-R
color index remains neutral during the transit, while the NIR components
exhibit shallower transit light curves, which are material independent. We can
observe the transit even in the L band as well, unlike in the case of
silicates. When dust has a size range of 2 $\mu$m, the deepest signal is
observed in the L band, the optical transit light curves are the most shallow
ones, but with much less color difference than in the case of small dust. In
between, at around 650 nm dust size ranges, the transit curves follow a very
similar shape with close-to-neutral colors (i.e. near-constant AD, or color
index of 1, Figs. 6–8, and 16 – 19). In summary, the same mass of carbonaceous
dust in transmission is “dark red”, “less dark neutral” and “pale blue” if the
size range is $\sim 200$, $650$ and $2000$ nm, respectively.

In the case of silicates, almost no variation is observed in the K and L bands
in the case of the smallest dust distribution examined. The optical signal can
be quite deep ($\sim 400$–$6000$ ppm in Figure 7, for 29P orbiting a solar-
like star), but still less deep than in the case of carbonaceous materials.
With increasing dust size, AD in the $NIR$ bands also increases. In the case
of the 2$\mu$m dust size, where the size of the scattering particles is
compatible with the wavelength, the deepest signal will be observed in the H
and K bands for pyroxene and olivine, respectively.

The change in color index becomes truly significant when the photometric
transients are detected simultaneously in multiple bands. While differences
can be seen by comparing the light curves, e.g. in Figs. 3 –5, we
quantitatively show the color variation curve in Appendix (B).

Exocomets are typically found in dusty environments, and transients related to
the transit of overdensities in the hot dust cloud can lead to similar,
nonperiodic transients. A substantial distinction between the two dust
overdensity structures lies in the chemical differences and the different size
distributions that can be discerned by multi-band observations. Dust clouds
evolved in a warm environment and represent processed material, unlike
exocomet dust, which originated freshly from a previously frozen environment
and then underwent fragmentation, possibly presenting dust that has a
different composition and grain size distribution than the warm dusty
environment. Consequently, we anticipate that the transients of dusts will be
only accompanied by minor changes in AD ratios at most, while exocomets will
exhibit characteristic AD ratio changes that will also be consistent with our
implemented code. This color information can distinguish between the two
transient scenarios.

Figure 9: Amplitudes of the forward-scattered light for the three dust
distributions shown on Fig. 1 (narrow-tailed comet – left, fan-like-tailed
comet – middle, tailles comet – right) for the A star. Note the change in the
scale of the y axis. Straight lines are shown for visualization purposes only.

###  3.3 Transit timing variations

We also observe a change in the transit minima of the asymmetric dust
distributions (Figs. 12, 3, 13, and 4) at different wavelengths. These are
analogous to the transit timing variations (TTVs) of exoplanets (e.g. Agol et
al., 2005). By extracting the times of minima (Figs. 10 and 11), we show that
there may be significant offset at the different wavelengths. We note that
$t=0$ in our simulations corresponds to the collinearity of the observer, the
center of dust distribution, and the center of the stellar disk. Due to the
asymmetric dust distribution, the minimum of the light curve does not
necessarily occur at $t=0$. This phenomenon occurs because the limb-darkened
stellar disk is convolved by the non-spherical comet tails of Figs. 1. We draw
caution to the K– and L–band minima of the pyroxene-comets with $a=200$ nm
grain sizes, as these show extremely shallow transits, where the determination
of the timings may be misleading, although these show the most significant
discrepancy with the timing data at other wavelengths. We also note that the
transit timing data (Figs. 10 and 11) are estimated on the simulation grid
(thus, only specific values can be recovered). The uncertainties are likewise
estimated from the sampling on the grid.

Figure 10: Extracted times of minima in the case of the narrow-tailed comet
(Fig. 1, left) for the Sun-like star (left) and the A star (right). Straight
lines are shown for visualization purposes only.

Figure 11: Extracted times of minima in the case of the wide-tailed comet
(Fig. 1, middle) for the Sun-like star (left) and the A star (right). Straight
lines are shown for visualization purposes only.

###  3.4 Application to further surveys

The idea we present here is the multiband observation of nonperiodic
photometric transients in stars with FEBs and/or dippers, and the evaluation
of the observations in the light of AD ratios before, after and during the
transients. There will likely be no unique solution for the multiband signals
because the dust scattering scenario is complex and it has degenerations
between the dust properties, the size distribution, and the total mass of dust
in the line of sight. However, changing colors can still be observed and in
favorable cases, the shift of the characteristic dust size can be proven. Here
we review a few surveys where the multiband observations or multiband follow-
ups promise a good possibility for an exocomet program.

#### TESS

Most of the known exocomet transits to date have been discovered by TESS
(Ricker et al., 2015) in the $\beta$ Pic system (Zieba et al., 2019;
Lecavelier des Etangs et al., 2022). Given the apparent aperiodic or
quasiperiodic nature of the transits themselves, continuous, survey-type
photometry over long baselines appears to be the favored strategy. TESS has
the advantage of long time series with almost continuous coverage. This is
almost a compulsory requirement in “exocomet hunting”, because the transients
occur unpredictably and have a duration in the order of 0.5-1 day – making
them very hard to detect from the ground, most promisingly with an observatory
network. TESS represents the largest space photometry factory today, which
makes it a prime instrument in exocomet search as well. However, TESS is a
single-band photometer and, therefore, a targeted follow-up observation is
required for multiband photometry purposes. These follow-up observations may
have a longer cadence than the TESS data: space photometry shows the shape of
the transient with good precision, while the follow-up observations have the
task of measuring precise colors and confirm the presence of possible color
variations, where the cadence can be longer than for the “transit shape”
photometry.

#### CHEOPS

Well-tailored observations (with e.g. CHEOPS; Benz et al., 2021) of individual
stellar systems can also yield interesting results. A transient feature,
related to the debris disk surrounding DE Bootis, has been identified in its
CHEOPS light curve by Boldog et al. (2023). A possible exocometary transit was
identified in the HD 172555 system using its available  CHEOPS observations
(Kiefer et al., 2023). Here, synchronous follow-up observations with multiband
instruments are also required to map the colors during the transients.

#### PLATO

With the 2026 launch of PLATO (Rauer et al., 2014), offering up to more than a
year of uninterrupted ultra-high precision multicolor (Grenfell et al., 2020)
light curves in the Long-duration Observation Phases (Nascimbeni et al.,
2022), further detection of these transits is expected.

#### ARIEL

The upcoming Ariel mission (Tinetti et al., 2021) will be observing individual
exoplanet systems from a target list. Ariel is planned to have the ability of
synchronous visual + infrared photometry (VISPhot, FGS1 and FGS2; Szabó et
al., 2022) together with NIR spectra taken by two low-resolution spectrographs
(NIRSpec and AIRS Tinetti et al., 2021) with coverage up to $7.8$ $\mu$m,
which is also capable of synthetic spectrophotometry. Ariel will be the first
dedicated exoplanet space observatory with multiband abilities, and will be
very much suited for a confirmation of the comet nature of dusty transients.
Moreover, the low-resolution spectra from Ariel can indeed reveal the chemical
constituents of the dusty transients, providing an important constraint to the
deeper modeling and confirming the origin of them.

#### Rubin LSST

LSST will be a large wide-field ground-based survey designed to obtain
repeated images that cover the entire southern celestial hemisphere and the
Ecliptic. The 8.4 m telescope of the Rubin Vera Observatory will be equipped
with a 3.2-gigapixel camera, and six filters (ugrizy) covering the wavelength
range 320-1050 nm (Ivezić et al., 2019). LSST is not designed to collect
photometric data series and, therefore, is not considered as an instrument to
discover dippers or proven exocomet transits. Rather, it will be able to
follow-up up the already known dippers of the southern sky, providing well
calibrated 6 band photometry for 10 years of planned operation, with a pace of
$\sim 4$ days. This can be combined with the uninterrupted space photometry of
the dippers, revealing the transients with very good time resolution. The
combined dataset will be very valuable to study practically any dipper in the
southern sky together with the subtle changes of dust color over a very long
time basis. This will be an unprecedented opportunity to point out the changes
of dust properties during the transient events which are compatible with
exocomets, providing very strong evidence to this emerging scientific field.

##  4 Summary and concluding remarks

The conclusions of this article can be summarized in the following points.

  * •

We have introduced an application based on RADMC-3D that allows simulating the
transits of 3D, $256^{3}$ pixelated comet models in front of stars at any
photometric wavelength, assuming an arbitrary distance from the star, an
arbitrary dust composition, and wavelength of observation.

  * •

Inspired by the comets in the Solar System, we calculated transit models
assuming compositions of carbonaceous (carbon and graphite) and silicate
(pyroxene and olivine) dust, at the central wavelength of the VRJHKL band for
three comet models with a thin tail, a fan-like tail, and no tail.

  * •

The behavior of silicate and carbonaceous comet dust proved to be
distinguishable in simulated VRJKHL transit observations. We also successfully
simulated the dependence of wavelength-scattering cross-section curves on
particle sizes.

  * •

The origin of once-frozen comet dust in the dusty environment of extrasolar
systems can be discerned from the hot/warm circumstellar dust mostly due to
the differing size distribution, and the cometary nature can be confirmed
through the different particle distributions and chemical compositions.
Synchronuous optical+NIR observations are well suited for this task.

  * •

We presented some possible applications in planned space telescope missions in
the present and near future.

Observational astrophysics of dippers and systems showing the FEB phenomenon
and investigating possible exocomets in these systems are an emerging area in
extrasolar astronomy. Our results demonstrated the potential of multiband
observations in this field and point toward promising opportunities.

The project has been implemented with the support provided by the Ministry of
Culture and Innovation of Hungary from the National Research, Development and
Innovation Fund, financed under the SNN-147362 funding scheme. This work was
partly supported by the K-138962 grant and the TKP2021-NKTA-64 excellence
grant of the National Research, Development and Innovation Office (NKFIH,
Hungary). SzK and GyMSz thank the support of the PRODEX Experiment Agreement
No. 4000137122 between the ELTE Eötvös Loránd University and the European
Space Agency (ESA-D/SCI-LE-2021-0025).This research has made use of the NASA
Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National AeronAUtics and Space
Administration under the Exoplanet Exploration Program. Project no. C1746651
has been implemented with the support provided by the Ministry of Culture and
Innovation of Hungary from the National Research, Development and Innovation
Fund, financed under the NVKDP-2021 funding scheme. The generated light curves
will be made publicly available upon acceptance of the paper at
https://osf.io/ndjw6/?view_only=d9f0e84412a14e169e3de044628be6ab.

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##  Appendix A Light curves for a Sun-like star

In this section, we show the light curves calculated in Sections 3 (Figs. 3
–5), but for a Solar-like host star (Figs. 12 –14). The light curves are
morphologically similar, whereas quantitative deviations occur due to the
differing stellar radius and the differing limb darkening (see its discussion
in 2.2).

Figure 12: Same as Fig. 3 but the transits are calculated for solar-like star.
Figure 13: Same as Fig. 4 but the transits are calculated for Solar-like star.
Figure 14: Same as Fig. 5 but the transits are calculated for a Solar-like
star.

##  Appendix B Color variations

In this section, we plot the differential light curves calculated in different
simulation bandpasses. These figures show the color of the total stellar light
after having traveled through the comet dust and suffered a spatially
dependent transfer, including both outward and inward scattering. The figures
are plotted in units of $\rm\mu$mag, and the $V-X$ colors indicate a redder
color towards positive values (see its discussion in Sect. 2.2.

Figure 15: Color variations for the narrow-tailed comet (Fig. 1) orbiting an A
star. Figure 16: Color variations for the narrow-tailed comet (Fig. 1)
orbiting a solar-like star. Figure 17: Color variations for the wide-tailed
comet (Fig. 1) orbiting an A star. Figure 18: Color variations for the wide-
tailed comet (Fig. 1) orbiting a solar-like star. Figure 19: Color variations
for spherically symmetric comet (1) orbiting an A star. Figure 20: Color
variations for spherically symmetric comet (Fig. 1) orbiting a solar-like
star.