# Prospects of Gravitational Wave Follow-up Through a Wide-field Ultra-violet
Satellite: a Dorado Case Study

Bas Dorsman GRAPPA, Anton Pannekoek Institute for Astronomy and Institute of
High-Energy Physics, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands  Geert Raaijmakers GRAPPA, Anton Pannekoek
Institute for Astronomy and Institute of High-Energy Physics, University of
Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands  S. Bradley
Cenko Astroparticle Physics Laboratory, NASA Goddard Space Flight Center, Mail
Code 661, Greenbelt, MD 20771, USA  Samaya Nissanke GRAPPA, Anton Pannekoek
Institute for Astronomy and Institute of High-Energy Physics, University of
Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands  Nikhef,
Science Park 105, 1098 XG Amsterdam, The Netherlands  Leo P. Singer
Astroparticle Physics Laboratory, NASA Goddard Space Flight Center, Mail Code
661, Greenbelt, MD 20771, USA  Mansi M. Kasliwal Division of Physics,
Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  Anthony L. Piro The Observatories of the Carnegie Institution for
Science, 813 Santa Barbara St., Pasadena, CA 91101, USA  Eric C. Bellm DIRAC
Institute, Department of Astronomy, University of Washington, 3910 15th Avenue
NE, Seattle, WA 98195, USA  Dieter H. Hartmann Department of Physics and
Astronomy, Clemson University, Clemson, SC 29634-0978  Kenta Hotokezaka
Research Center for the Early Universe, Graduate School of Science, University
of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  Kamilė Lukošiūtė GRAPPA, Anton
Pannekoek Institute for Astronomy and Institute of High-Energy Physics,
University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

(Received XXXXXX; Revised XXXXXX)

###### Abstract

The detection of gravitational waves from binary neuron star merger GW170817
and electromagnetic counterparts GRB170817A and AT2017gfo kick-started the
field of gravitational wave multimessenger astronomy. The optically red to
near infra-red emission (‘red’ component) of AT2017gfo was readily explained
as produced by the decay of newly created nuclei produced by rapid neutron
capture (a kilonova). However, the ultra-violet to optically blue emission
(‘blue’ component) that was dominant at early times ($\lesssim$ 1.5 days)
received no consensus regarding its driving physics. Among many explanations,
two leading contenders are kilonova radiation from a lanthanide-poor ejecta
component or shock interaction (cocoon emission). In this work, we simulate
AT2017gfo-like light curves and perform a Bayesian analysis to study whether
an ultra-violet satellite capable of rapid gravitational wave follow-up, could
distinguish between physical processes driving the early ‘blue’ component. We
find that a Dorado-like ultra-violet satellite, with a 50 deg2 field of view
and a limiting magnitude (AB) of 20.5 for a 10 minute exposure is able to
distinguish radiation components up to at least 160 Mpc if data collection
starts within 3.2 or 5.2 hours for two possible AT2017gfo-like light curve
scenarios. Additional sensitivity and additional filters may allow for a
longer acceptable response time. We also study the degree to which parameters
can be constrained with the obtained photometry. We find that, while ultra-
violet data alone constrains parameters governing the outer ejecta properties,
the combination of both ground-based optical and space-based ultra-violet data
allows for tight constraints for all but one parameter of the kilonova model
up to 160 Mpc. These results imply that an ultra-violet mission like Dorado
would provide unique insights into the early evolution of the post-merger
system and its driving emission physics. In addition, this study shows that
ultra-violet plus optical multi-wavelength detections provide complementary
constraints, jointly covering a broader range of early ejecta properties.

Bayesian statistics (1900), Gravitational waves (678), Model selection (1912),
Neutron stars (1108), Nucleosynthesis (1131), Ultraviolet astronomy (1736)

††software: Python language (Oliphant, 2007), NumPy (van der Walt et al.,
2011), SciPy (Jones et al., 2001–), Matplotlib (Hunter, 2007; Droettboom et
al., 2018), IPython (Perez & Granger, 2007), Jupyter (Kluyver et al., 2016),
Dynesty (Speagle, 2020), Corner.py (Foreman-Mackey, 2016), GNU Parallel
(Tange, 2021)

tablenum SIXtablenum

1D

    one dimensional
2D

    two dimensional
3D

    three dimensional
BH

    black hole
BHNS

    black hole neutron star
BBH

    binary black hole
BNS

    binary neutron star
CBC

    compact object coalescence
CCD

    charge coupled device
DETC

    Dorado exposure time calculator
DOS

    Dorado observation scheduler
EFE

    Einstein field equations
$\lambda_{\mathrm{eff}}$

    effective wavelength
EM

    electromagnetic
EoS

    equation of state
ETC

    exposure time calculator
FoV

    field of view
GCN

     $\gamma$-ray Coordinates Network
GRMHD

    general-relativistic magneto-hydrodynamic
GRB

     $\gamma$-ray burst
GW

    gravitational wave
GWB

    gravitational wave background
HLV

    Advanced LIGO and Virgo
HLVK

    Advanced LIGO, Virgo and KAGRA
HPDI

    highest posterior density interval
IR

    infra-red
ISM

    interstellar medium
LIGO

    Laser Interferometer Gravitational-Wave Observatory
LCO

    Las Cumbres Observatory
$d_{L}$

    luminosity distance
LVC

    LIGO-Virgo collaboration
MMA

    multi-messenger astronomy
NIR

    near infra-red
NS

    neutron star
PPD

    posterior probability distribution
S/N

    signal-to-noise ratio
SDSS

    Sloan Digital Sky Survey
SED

    spectral energy density
sGRB

    short $\gamma$-ray burst
SED

    spectral energy distribution
ToO

    target of opportunity
UV

    ultra-violet
UVO

     ultra-violet (UV) and optical
UVOIR

     UV, optical and infra-red (IR)

##  1 Introduction

The LIGO-Virgo collaboration (LVC) can now regularly detect and study
gravitational waves from the final moments of inspiraling compact object
mergers (Abbott et al., 2020). At least some compact binary mergers involving
a neutron star (NS) are expected to produce electromagnetic (EM) counterparts,
depending on mass-ratio and spins (for a review, see e.g. Nakar, 2020). The
discovery of a binary neutron star (BNS) merger GW170817 (Abbott et al.,
2017a) and subsequent EM counterparts, the $\gamma$-ray burst (GRB) GRB170817A
(Goldstein et al., 2017; Savchenko et al., 2017) and , optical and  (UVOIR)
counterpart AT2017gfo (e.g., Abbott et al., 2017b, c; Arcavi et al., 2017;
Chornock et al., 2017; Coulter et al., 2017; Kasliwal et al., 2017; Lipunov et
al., 2017; Shappee et al., 2017; Soares-Santos et al., 2017; Valenti et al.,
2017), kick-started the era of GW multimessenger astronomy. AT2017gfo
exhibited a featureless thermal spectrum, peaking in the near-UV at $\sim$ 1
day (‘blue’ component). After this time the blue component faded while the
optically red and near infra-red (NIR) emission (‘red’ component) shifted
further towards the IR (e.g., Arcavi et al., 2017; Cowperthwaite et al., 2017;
McCully et al., 2017; Nicholl et al., 2017; Shappee et al., 2017).

It has long been predicted that the ejecta of BNS or black hole neutron star
(BHNS) mergers can host a thermal transient powered by the radioactive decay
of heavy elements formed by rapid neutron captures (r-process), and these
astrophysical transients have been named macronovae or kilonovae (Lattimer &
Schramm, 1974; Li & Paczyński, 1998; Kulkarni, 2005; Metzger et al., 2010).
While the red component of AT2017gfo is consistent with predictions from
several kilonova models (Kasen et al., 2013; Tanaka & Hotokezaka, 2013;
Rosswog et al., 2017; Wollaeger et al., 2018) the blue component is hotter,
more luminous, and faster rising than what is predicted by the aforementioned
models (Drout et al., 2017; Kasliwal et al., 2017). One way to explain the
full light curve evolution is by considering the presence of multiple distinct
ejecta components (e.g., Kasen et al., 2017; Tanaka et al., 2017; Villar et
al., 2017). In such a scenario, ejecta with a higher electron fraction
($Y_{e}\gtrsim$ 0.25) would produce material with a lower lanthanide fraction
and subsequently brighter, bluer, day-long emission. Such an ejecta component
with higher electron fraction could correspond to neutrino driven or
magnetized winds from a long lived NS central remnant (e.g., Shibata et al.,
2017; Metzger et al., 2018; Nedora et al., 2021) but in the case of AT2017gfo
such winds would have difficulties explaining the high velocities inferred for
the early emission. Still, Waxman et al. (2018) found that a single ejecta
component kilonova model, but with a uniform time dependent opacity
($\kappa\propto t^{\gamma}$), is consistent with the available data.
Alternatively, other model ingredients may be driving the blue component such
as a precursor powered by free neutron decay (Metzger et al., 2015; Gottlieb &
Loeb, 2020) or shock interaction, also known as cocoon emission (Kasliwal et
al., 2017; Piro & Kollmeier, 2018).

One critical aspect that was missing in the observations of AT2017gfo was UV
data in the first hours. Swift/UVOT obtained their first exposure in the
ultraviolet band ($\sim$ 200 – 300 nm) at $\Delta t=0.6$ days (15 hours) after
the GW trigger (Evans et al., 2017) of GW170817. Early time UV data would
provide the essential diagnostic to identify the main mechanism or combination
of mechanisms that power the UV and optically blue radiation in the first few
hours (Arcavi, 2018). One way to obtain such data would be by having a large
field of view (FoV) UV satellite on stand-by to rapidly follow up on GW
triggers to find the target in the GW localization sky area while it is still
bright in UV in the first few hours. Two missions that have been proposed to
fulfill this purpose are ULTRASAT (Sagiv et al., 2014) and UVEX (Kulkarni et
al., 2021). In Section 5, we discuss the applicability of our findings to
these currently planned UV missions. We also note that recently Chase et al.
(2022) performed a kilonova detectability study for a wide range of wide-field
instruments.

Performing early UV follow-up of GW events was also an objective of Dorado, a
mission concept that was submitted to the 2019 NASA Astrophysics Mission of
Opportunity call for proposals. This study assumes Dorado’s follow-up and
observational capabilities when simulating UV photometry. Summary mission
specifications of Dorado, ULTRASAT and UVEX are given in Table 1 and we
discuss Dorado and the simulation of photometric data in detail in Section
3.3.

Table 1: Mission specifications of mission concept Dorado and proposed missions ULTRASAT and UVEX. From left to right, the table presents the 5$\sigma$ limiting magnitude (AB) for given exposure time, response time, FoV, and orbit type. LEO, GEO, and HEO are low earth, geo-synchronous, and high earth orbit respectively. Mission | 5$\sigma$ (AB) |  | Response
---
time
$\Omega$ (deg2) | Orbit
Dorado |  | 20.5
---
(10 min. exp.)
$\sim$ 30 min. | 50 | LEO
ULTRASAT |  | 22.3
---
(15 min. exp.)
$\sim$ 30 min. | 200 | GEO
UVEX |  | 25
---
(15 min. exp.)
$\lesssim$ 3 hr. | 12 | HEO

In this study, we perform a Bayesian analysis to examine to what degree
satellite-based UV photometry could distinguish between two competing emission
models for AT2017gfo-like blue emission. The two models considered are
r-process nucleosynthesis (kilonova) and shock interaction (cocoon emission).
Currently, these two options are prominent in the literature, and the data of
AT2017gfo allows for either a pure shock interaction or kilonova scenario to
explain the blue component. In addition, we study to what degree the data
constrains parameters of either model. We also perform an analysis with
ground-based optical and joint  and optical (UVO) data to examine the added
value of early UV photometry. Finally, we study how delayed observation
affects model selection and parameter estimation for satellite-based UV.

This paper is structured as follows: Section 2 describes the kilonova and
shock interaction models. Section 3 describes the Bayesian theoretical
framework and the simulation of light curves and photometric data. Section 4
presents the results of the Bayesian analysis. Section 5 presents our main
findings, implications for planned UV missions, discusses caveats for this
work, and concludes.

##  2 Radiation Models

This section provides a summary of the physics of the two radiation models and
discusses the model parameters.

###  2.1 Nucleosynthesis Powered Model

The first radiation model is the Hotokezaka & Nakar (2020) model for radiation
powered by $\beta$-decay of radioactive elements produced through r-process
nucleosynthesis (kilonova). This is a semi-analytic model, based on the Arnett
model (Arnett, 1982). The latter models the light curves for supernovae and as
such includes the radioactive decay of 56Ni and 56Co. Hotokezaka & Nakar
(2020) applies to kilonovae however, and solves the time evolution of
$\beta$-decay chains for all elements up to a certain mass $A$ to get the
radioactive power of each decay chain. For simulations performed here elements
up to $A=209$ are included. Here, we do not include the heating due to
$\alpha$-decay and fission because their contributions are rather small in the
early times, where kilonovae are expected to be bright in UV. Thermalization
is treated similar to analytic methods presented by Kasen & Barnes (2019);
Waxman et al. (2019), with injection energies of each decay product specified
per decay chain.

The model assumes isotropic geometry where radial density structure of the
ejecta is modeled as a function of time $t$ and velocity $v$, given by

$\rho(t,v)=\rho_{0}(t)\left(\frac{v}{v_{\rm min}}\right)^{-n}\quad(v_{\rm
min}\leq v\leq v_{\rm max}),$ (1)

where $\rho_{0}(t)$ ensures that the full $M_{\rm ej}$ is retrieved when the
density $\rho(t,v)$ is integrated over the full velocity range [$v_{\rm
min},v_{\rm max}$]. Parameter $n$ defines how steeply the power-law density
distribution falls off. The ejecta are modeled here as finely discretized mass
shells, and a notable feature is that radiative transfer is improved by
accounting for radiation escape from mass shells with different expansion
velocities in the case where the diffusion time is long compared to the
dynamical time.

We also note that the model uses a concentric piece-wise opacity distribution.
While the Hotokezaka & Nakar (2020) model allows up to two opacities, our
adaptation expands the model to allow for an arbitrary number of $m$ zones. An
array of $m+1$ velocities defines the borders of these expanding opacity
zones, specifying each transition velocity. Nevertheless, in this work we use
only two opacity zones, with inner opacity $\kappa_{\rm in}$ and outer opacity
$\kappa_{\rm out}$. The velocities are $v_{\rm min}$, $v_{\rm transition}$ and
$v_{\rm max}$. Note that bound-bound transitions of heavy elements dominate
the kilonova opacities, which in reality vary with the wavelengths and ejecta
conditions such as temperature and density (e.g., Tanaka et al., 2020;
Banerjee et al., 2020). Here, however, we assume opacity to be constant with
time and wavelengths for simplicity. An effective radius and temperature is
calculated such that the model produces effective blackbody radiation. Our
adaptation of the model is publicly available on
GitHub111https://github.com/Basdorsman/kilonova-heating-rate/.

###  2.2 Shock Interaction Powered Model

The second radiation model is an analytical shock interaction model derived by
Piro & Kollmeier (2018). It follows analytical work presented in Nakar & Piran
(2017) and includes some added details to facilitate comparison with
AT2017gfo. In this model, a relativistic GRB jet punches through ejecta
material, depositing some of its energy to the ejecta, and thereby inflating
it to create a shock heated cocoon. The light curve is powered by cooling
emission of this shock-heated material. Such a model could also represent a
shock driven by a short-lived magnetar (Metzger et al., 2018). Similar to the
kilonova model, this model contains no angular dependence and emits effective
blackbody radiation.

The shock model assumes an initial shock radius $R$ corresponding to the
radius of the ejecta at the time when the GRB first punches through. For
GW170817, $R$ is $\sim$ $${10}^{10}$-${10}^{11}$$ $\mathrm{c}\mathrm{m}$,
corresponding to a jet traveling at $\sim$ $c$ taking about $\sim$ 1.7 seconds
to emerge, as derived from the delay time between GW170817 and GRB170817A
(Abbott et al., 2017b; Goldstein et al., 2017; Savchenko et al., 2017). The
shocked envelope is assumed to have energy distributed with respect to
velocity as

$\frac{dE}{dv}\propto v^{-s},$ (2)

where it is assumed that $s>-1$. The parameter $s$ encodes ignorance on how
exactly the jet deposits its energy in the ejecta. As in Piro & Kollmeier
(2018), we assume $s=3$ which is derived from AT2017gfo. The luminosity $L$,
effective temperature $T_{\mathrm{eff}}$ and effective radius
$r_{\mathrm{eff}}$ are given by

$\displaystyle L$
$\displaystyle=$9.5\text{\times}{10}^{40}$\kappa_{0.1}^{-3/5}M_{0.01}^{2/5}v_{0.1}^{8/5}R_{10}t_{\mathrm{day}}^{-4/5}\mathrm{\;erg\
s}^{-1},$ (3) $\displaystyle T_{\mathrm{eff}}$
$\displaystyle=$6.2\text{\times}{10}^{3}$\kappa_{0.1}^{-7/30}M_{0.01}^{1/60}v_{0.1}^{1/15}R_{10}^{1/4}t_{\mathrm{day}}^{-8/15}\mathrm{\;K},$
(4) $\displaystyle r_{\mathrm{eff}}$
$\displaystyle=$3\text{\times}{10}^{14}$\kappa_{0.1}^{1/6}M_{0.01}^{1/6}v_{0.1}^{2/3}t_{\mathrm{day}}^{2/3}\mathrm{\;cm},$
(5)

where $M_{0.01}$ is the shocked mass in units of $0.01M_{\odot}$,
$\kappa_{0.1}$ is the opacity in units of
$0.1\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{g}$, $v_{0.1}$ is the
minimum ejecta velocity in units of 0.1 c, $R_{10}$ is the initial shock
radius in units of ${10}^{10}\text{\,}\mathrm{c}\mathrm{m}$, and
$t_{\mathrm{day}}$ denotes the days elapsed since merger.

##  3 Methodology

We perform a Bayesian analysis to quantify the degree to which various
photometric data distinguishes the radiation models and allows for parameter
estimation. This section elaborates on our methodology and is structured as
follows: Section 3.1 provides a short summary of the relevant theory of
Bayesian model selection and parameter estimation. The Bayesian analysis takes
as input observed data and, in this study, that data is simulated in two
steps. Firstly, Section 3.2 defines ‘fiducial’ (i.e. AT2017gfo-like) light
curves computed with the models introduced in Section 2. Secondly, Section 3.3
lays out how an observation of these light curves is simulated, producing
simulated photometric data.

###  3.1 Bayesian framework

Bayesian model selection and parameter estimation (for an in-depth review, see
e.g. Sivia & Skilling, 2006) rely on some data $D$, and a model $M$ described
by a set of parameters $\theta$. The posterior probability is given by

$P(\theta|D,M)=\frac{P(D|\theta,M)P(\theta|M)}{P(D|M)}=\frac{\mathcal{L}(\theta)\pi(\theta)}{\mathcal{Z}},$
(6)

where $\mathcal{L}(\theta)=P(D|\theta,M)$ is the likelihood function and
$\pi(\theta)=P(\theta|M)$ is the prior. The evidence (marginal likelihood)
$\mathcal{Z}$ is given by

$\mathcal{Z}=P(D|M)=\int_{\Omega_{\theta}}\mathcal{L}(\theta)\pi(\theta)\mathrm{d}\theta,$
(7)

with the integral taken over the whole domain $\Omega_{\theta}$ of $\theta$.
For a given set of data $D$, we can quantify how it supports one model $M_{A}$
compared to another $M_{B}$ by the ratio of evidences. This is also known as
the Bayes factor:

$\mathcal{B}^{A}_{B}=\frac{P(D|M_{A})}{P(D|M_{B})}=\frac{\mathcal{Z}_{A}}{\mathcal{Z}_{B}}.$
(8)

Using Bayes’ rule, we can relate the Bayes factor to the posterior ratio:

$\frac{P(M_{A}|D)}{P(M_{B}|D)}=\frac{P(D|M_{A})}{P(D|M_{B})}\frac{P(M_{A})}{P(M_{B})}=\mathcal{B}_{B}^{A}\frac{P(M_{A})}{P(M_{B})},$
(9)

where $P(M_{i})$, $i=A,B$ is the prior probability of each model. Equation 9
shows that the posterior ratio is equivalent to the prior ratio ‘updated’ by
the Bayes factor, because the Bayes factor is the term that contains all
information regarding the new data. In other words, the Bayes factor
quantifies the ‘distinguishability power’ of a set of data. A value of
$\mathcal{B}^{A}_{B}>1$ means that $M_{A}$ is more strongly supported by the
data under consideration than $M_{B}$. Kass & Raftery (1995) provide a guide
for interpretation of Bayes factors and judge that $\mathcal{B}^{A}_{B}>100$
(or $\log_{10}(\mathcal{B}^{A}_{B})>2$) can be regarded as being “decisive”.
We will assume this number as a threshold for confident model selection.

###  3.2 Fiducial light curves

There is no consensus in the early evolution of light curves of BNS mergers.
While one EM counterpart of a BNS merger was recorded with AT2017gfo, this
data set lacks UV data before 15 hours (Evans et al., 2017) and optical data
before 10 hours (Drout et al., 2017). Even if earlier data was available,
UVOIR radiation from BNS mergers more generally is also expected to be diverse
depending on various binary and post-merger properties. We discuss this
diversity and how it affects the conclusions of this study in more detail in
Section 5. Being limited to photometric data of AT2017gfo, the simulated light
curves for the Bayesian study are based on AT2017gfo.

We employ both radiation models discussed in Section 2 to simulate the
photometric data. Additionally, we use the kilonova model twice with different
$v_{\rm max}$ and $\kappa_{\rm low}$, which allows for additional coverage of
early UV brightness allowed by photometric data obtained from AT2017gfo.
Figure 1 shows these three ‘fiducial light curves’, and they are discussed in
more detail in the remainder of this section.

Figure 1: Comparison of AT2017gfo-like light curves in the Dorado UV band
($185215\text{\,}\mathrm{n}\mathrm{m}$) for kilonova models and shock
interaction model at luminosity distance
$d_{L}=$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$. The data points
correspond to 10 minute exposures that occur with a cadence of 97 minutes.
There are fourteen data points in total for all light curves that are observed
with identical timing. For reference, the dashdotted line indicates the timing
of the first UV data recorded of AT2017gfo by Swift/UVOT at 15 hours (Evans et
al., 2017).

The first fiducial light curve we call the ‘default’ light curve and is
computed with the kilonova model using the same model parameters as Hotokezaka
& Nakar (2020) to fit AT2017gfo. They found the resulting light curve to
reasonably fit to the bolometric light curve and temperature data of AT2017gfo
(taken from Arcavi, 2018; Waxman et al., 2018). The input parameters as well
as prior distributions for the Bayesian analysis are given in Table 2.

Table 2: Input parameters to the kilonova and shock interaction models. On the left hand side, input parameter symbols are given along with units in parentheses. The prior density of the parameters are given in the third column, where the U(min, max) notation is used to indicate the minimum and maximum values of a uniform probability distribution. In the right hand column, the fiducial values for the fiducial light curves are given. Parameter (Unit) | Description | Prior Density | Fiducial value
---|---|---|---
Default [Lower Early Opacity] Kilonova Model
$M_{\mathrm{ej}}$ (M⊙) | Ejecta mass | U(0.01, 0.1) | 0.05
$v_{\mathrm{min}}$ (c) | Minimum ejecta velocity | U(0.05, 0.2) | 0.1
$v_{\mathrm{max}}$ (c) | Maximum ejecta velocity | U(0.3, 0.8) [U(0.21, 0.8)] | 0.4 [0.23]
$n_{\mathrm{ej}}$ | Power law index of ejecta density distribution | U(3.5, 5) | 4.5
$v_{\mathrm{transition}}$ (c) | Transition velocity between high and low $\kappa$ | U($v_{\mathrm{min}}$, $v_{\mathrm{max}}$) | 0.2 [0.2]
$\kappa_{\mathrm{high}}$ (cm2/g) | Effective grey opacity for $v\leq v_{\mathrm{\kappa}}$ | U(1, 10) | 3
$\kappa_{\mathrm{low}}$ (cm2/g) | Effective grey opacity for $v\geq v_{\mathrm{\kappa}}$ | U(0.1, 1) [U(0.01, 0.1)] | 0.5 [0.04]
Shock Interaction Powered Model
$M_{\mathrm{sh}}$ (M⊙) | Shocked ejecta mass | U(0.005, 0.05) | 0.01
$v_{\mathrm{sh}}$[c] | Shocked ejecta minimum velocity | U(0.1, 0.3) | 0.2
$R_{0}$ (1010 cm) | Initial shock radius | U(1, 10) | 5
$\kappa_{\mathrm{sh}}$ (cm2/g) | Effective grey opacity of shocked ejecta | U(0.1, 1) | 0.5

The second fiducial light curve is called the ‘lower early opacity’ light
curve. Its inclusion here is motivated by recent work by Banerjee et al.
(2020) who found light curves with early UV brightness that peak 1.5–2
magnitude brighter than the ‘default’ model. Rather than using a (multiple
zone) gray opacity model, they performed a radiative transfer simulation using
opacities calculated by taking into account atomic structures, including
highly ionized light r-process elements ($Z=20-56$). The ‘lower early opacity’
light curve is a least squares fit with our kilonova model to both their
simulated peak brightness and real Swift/UVOT data from AT2017gfo in AB
magnitude. For this fit, we only allowed $v_{\rm transition}$, $v_{\rm max}$,
$\kappa_{\rm low}$ to vary, which minimizes the change on the late time light
curve while allowing for a good fit with the early UV light curve of Banerjee
et al. (2020). The resulting $v_{\rm transition}$, $v_{\rm max}$, $\kappa_{\rm
low}$ are shown in Table 2, but we find that $v_{\rm transition}$ remains the
same. We also note that the AB magnitude of the ‘lower early opacity’ light
curve at 0.1 day is brighter by $\sim$ 1 mag than the ‘light
r-process+Sm+Nd+Eu’ model of Banerjee et al. (2022), where they perform a
radiative transfer simulation accounting for the expansion opacities of highly
ionized lanthanide elements. Thirdly and lastly, a fiducial light curve is
produced by the shock interaction model. We use the fit to AT2017gfo provided
by Piro & Kollmeier (2018) assuming $s=3$. These parameters too are given in
Table 2.

###  3.3 Simulation of photometric data

Using all the fiducial light curves, we simulate photometric data as observed
by Dorado in UV and by Las Cumbres Observatory (LCO) in the optical r-band.
While more optical bands (such as u, g, and i) are available, we found that
their inclusion minimally affected model selection and parameter estimation
(compared to just the r-band) and omitted them in exchange for reduced
computational time.

Dorado was a mission concept (Singer et al., 2021) consisting of a SmallSat
(slightly larger than a 12U CubeSat) spacecraft equipped with a 13 cm
refractive (7 element) telescope with a 50 deg2 FoV. The instrument adopts a
single (fixed) band-pass over the wavelength range from
$185215\text{\,}\mathrm{n}\mathrm{m}$. The Dorado camera employs delta-doped
charge coupled device (CCD) detectors to provide surface passivation and
reflection-limited response over the UV bandpass (Nikzad et al., 2017). The
spacecraft was designed to accommodate a wide range of low-Earth orbits and
given ride-share availability we assume a noon-midnight sun synchronous
(polar) orbit with a 600 km altitude for the simulations described here.

For the simulation of Dorado photometric data, representative observational
cadences were derived using the dorado-
scheduling222https://github.com/nasa/dorado-scheduling/ software package.
dorado-scheduling can generate realistic observational sequences for
gravitational-wave localizations, including exclusion constraints (Sun, Moon,
and Earth limb), as well as satellite down time for passages through the South
Atlantic Anomaly. The software can define optimized observing strategies based
on the GW localization region and distance, as well as a (position-dependent)
exposure time calculator (see below). However, for these simulations we
assumed the first data point is observed at $\Delta t=72$ min post merger (to
allow for time to uplink to the spacecraft), and a fixed exposure time of 10
minutes every orbit (97 minute cadence) on a single pointing.

Signal-to-noise estimates were derived using the dorado-
sensitivity333https://github.com/nasa/dorado-sensitivity/, the exposure time
calculator for the Dorado mission. dorado-sensitivity generates realistic
foreground models for zodiacal light (based on spacecraft and target
location), as well as airglow emission (based on location within the orbit).
While the software is capable of incorporating Milky Way dust extinction, this
is not included here. The fiducial light curve is folded through the Dorado
effective area curve to generate the expected source counts, and then compared
with the background (including both foreground and source shot noise). For
reference, for a 10 minute exposure, Dorado’s 5$\sigma$ limiting magnitude for
an isolated point source is typically $\sim$ 20.5 mag (AB).

For ground-based optical observations, we similarly use the same detection
schedule for all simulated events but starting at 12 hours and with a 12 hour
cadence up to 48 hours. To calculate this photometric data we use an exposure
time calculator (ETC) constructed using data of LCO
instruments444https://exposure-time-calculator.lco.global/. LCO is a network
of 25 telescopes at seven sites around the world, sensitive in optical and NIR
wavelengths. Being purpose-built to observe transient events, it robotically
schedules observations and leverages its global network to observe around the
clock and make rapid observations of targets of opportunity, evading potential
local weather limitations (Brown et al., 2013).

##  4 Bayesian Analysis

This study employs Dynesty (Speagle, 2020) for Nested Sampling (Skilling,
2004, 2006). Nested Sampling is a method for simultaneously estimating
posterior probability $P(\theta|D,M)$ (see Equation 6) and evidence
$\mathcal{Z}$ (see Equation 7). The full pipeline for simulation of
photometric data and subsequent Bayesian analysis is publicly available on
Github555https://github.com/Basdorsman/kilonova-bayesian-analysis/. This
section presents the results of the Bayesian analysis, and is structured as
follows: Section 4.1 presents the results of model selection and parameter
estimation via photometric data consisting of satellite-based UV, ground-based
optical, and both. Section 4.2 considers satellite-based UV photometric data
but analyzes the effect of delayed observation of the target.

###  4.1 Model Selection and Parameter Estimation for UV, Optical and Joint
Data

Figure 2: Bayes factors obtained from employing satellite-based UV, ground-
based optical or joint UVO photometric data. Left: Resulting Bayes factors
when attempting to distinguish the shock interaction model from ‘default’
kilonova model. Note here that the results obtained from shock data in UV+r
bands overlap the results from the UV band. Right: Same, but now
distinguishing the shock interaction model from the ‘lower early opacity’
kilonova model. The solid (dotted) lines correspond to Bayes factors resulting
when using simulated data produced by the shock interaction model (kilonova
model). The black horizontal line indicates the $\log_{10}(\mathcal{B})=2$
threshold for decisive evidence.

Figure 2 shows Bayes factors as a function of luminosity distance of the
merger. In all cases up to $160\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$, the
kilonova models can be confidently distinguished from the shock interaction
model using only UV data. Conversely, the optical data is able to distinguish
the models only up to $\sim$ 110 (60) $\mathrm{M}\mathrm{p}\mathrm{c}$ for the
‘default’ (‘lower early opacity’) light curves. Moreover, optical data
produced by the shock interaction light curve presents edge cases at
$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ and beyond this distance the data
is insufficient to distinguish the models. The combined UVO data set allows
for (marginally) better distinguishability than the UV data.

Figure 3: Posterior distributions of the parameters of the ‘default’ kilonova
model, for an AT2017gfo-like event at $d_{L}$ =
$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,
$100\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ and
$160\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. The sub-figures on the diagonal
of the corner plots display the PPD and the fiducial values for the model
parameters (black lines). The sub-figure titles on the diagonal list the 0.16,
0.5, and 0.84 fractional quantiles. The contours in the 2D plots indicate the
2D 1$\sigma$ (39%) and 2$\sigma$ (86%) credible regions.

To get a handle on parameter estimation and its distance dependency, Figure 3
shows the posterior probability distributions (PPD) for the ‘default’ kilonova
model at three selected luminosity distances: $d_{L}$ =
$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$,
$100\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ and
$160\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. Two out of the seven model
parameters: $v_{\mathrm{transition}}$ and $\kappa_{\mathrm{low}}$, are well
constrained by the UV photometric data, but especially so at the most close-by
distance at $40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. We note that these
two parameters in part define the outer opacity zone of the ejecta outflow.
Because the outer opacity zone is bright in UV (for these parameter values
such that the light curves resemble AT2017gfo) we expect these parameters to
be relatively well constrained by the UV photometric data. The remaining 5
parameters are not significantly constrained.

Figure 4: Same as Figure 3 but for the shock interaction model.

Figure 4 shows the PPD for the shock interaction model. Even for
$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ the parameters are constrained
only slightly within prior ranges. As is evident from the figure, this lack in
constraining of parameters is due to parameter degeneracy in this model. We
find the following degeneracy in the model: consider a set of parameters
$M_{1}$, $\kappa_{1}$ and $R_{1}$ that produce some $L_{1}$,
$T_{\mathrm{eff},1}$ and $r_{\mathrm{ph},1}$. There exists another set of
parameters, $M_{2}=\phi^{-1}M_{1}$, $\kappa_{2}=\phi\kappa_{1}$ and
$R_{2}=\phi R_{1}$, where $\phi$ is some constant. Substituting these into
Equations 3, 4 and 5, all factors $\phi$ cancel and we are left with
$L_{1}=L_{2}$, $T_{\mathrm{eff},1}=T_{\mathrm{eff},2}$ and
$r_{\mathrm{ph},1}=r_{\mathrm{ph},2}$. This means that the blackbody spectrum
is identical for different parameters. Because of this degeneracy, parameters
cannot be individually constrained even if observing in multiple bands. We
note however, that in the case of AT2017gfo, the $R_{\mathrm{shock}}$ could be
independently constrained from the time delay between GWs and GRB. In such a
case $\kappa$ would become constrained, but a degeneracy remains between
$M_{\rm shock}$ and $v_{\rm shock}$ ($M_{2}=\phi^{-4}M_{1}$ and $v_{2}=\phi
v_{1}$).

Figure 5 shows the results for the ‘default’ kilonova model again (same as
$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ in Figure 3), but now also
including a comparison with ground-based optical photometric data and with the
combined data of both bands. In isolation ground-based optical observations
constrain $M_{\mathrm{ejecta}}$ and $\kappa_{\mathrm{high}}$ more accurately
and precisely than the UV. For
$v_{\mathrm{min}},v_{\mathrm{transition}},v_{\mathrm{max}},\kappa_{\mathrm{high}}$
and $n$, performance of either single band is comparable, but constraints on
$\kappa_{\mathrm{low}}$ are much worse for the optical than UV. However, in
combination ground-based optical and UV are complementary to each other,
providing significant improvement compared to using either band in isolation.
In that case, all parameters except $n$ are well constrained.

Figure 5: Similar to Figure 3, but now comparing results for parameter
estimation using ground-based optical photometric data (solid), using
satellite-based UV photometric data (dashdot) and both (dashed) for an event
at $40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$.

###  4.2 Delayed detection

The time it takes from GW trigger to first on-target exposure (effective
response time) should be as short as possible to capture the physics that
govern the early post-merger system. This is especially relevant for the UV
compared to optical and IR because the emission rises and fades more rapidly
($\sim$ hours compared to $\sim$ days and $\sim\,$weeks respectively). Figure
6 shows the results for model selection using satellite-based UV photometric
data, where the data set as a whole was shifted forwards in time with various
delays as indicated in the figure. Note that the cases for data at
$1.2\text{\,}\mathrm{h}$ are identical to the UV data in Figure 2. For
distances up to $160\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$, it is possible
to distinguish the models if the first data is collected up to 5.2 (3.2) hours
after GW trigger for the ‘default’ (‘lower early opacity’) light curves. For
$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ and beyond, the models cannot be
distinguished if the first data is collected after 13.2 hours.

Figure 6: Bayes factors obtained from model selection using satellite-based UV
photometric data. The data is shifted in time such that the first data point
is obtained at $1.2$, $3.2$, $5.2$ and $13.2$ hours after GW trigger. The
distinction between left and right, dotted and solid lines as well as the
black horizontal line are explained in the caption of Figure 2.

Figure 7 shows the results for parameter estimation for increasingly delayed
photometric data. Note that the results for $1.2\text{\,}\mathrm{h}$ are
identical to $40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ in Figure 3. All
parameters experience worsening constraints for increasing response time. Of
the two parameters that were previously well constrained for satellite-based
UV photometric data, $v_{\mathrm{transition}}$ is still well constrained even
for data starting at $25.2\text{\,}\mathrm{h}$ while $\kappa_{\mathrm{low}}$
is not well constrained beyond $13.2\text{\,}\mathrm{h}$. We posit that
$v_{\mathrm{transition}}$ may be well constrained even with such delayed data
because it affects both opacity zones, while $\kappa_{\mathrm{low}}$ only
plays a role in defining the quicker fading outer opacity zone.

Figure 7: Similar to Figure 3, but now with the satellite-based UV photometric
data at $40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ shifted in time such that
the first data point is obtained at $1.2$, $13.2$ and $25.2$ hours.

##  5 Discussion and Conclusion

We have performed model selection and parameter estimation, assuming
AT2017gfo-like ‘blue’ emission produced by either kilonova or shock
interaction radiation and assuming observational capabilities of a UV
satellite (Dorado) and a ground-based observation network (LCO). Our main
findings are threefold:

Firstly, our results suggest that satellite-based UV photometric data, unaided
by ground-based optical follow-up would be sufficient to distinguish these
models for an event up to at least
$160\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ for all considered scenarios.
Conversely, ground-based optical photometric data only selects models up to
$\sim$ $110\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$ in the most optimistic
scenario and has considerably worse performance ($\sim$
$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$) in other scenarios. Combined UVO
photometric data only marginally improves model selection beyond what is
possible with only UV data.

Secondly, we find that combined UVO photometric data allows the full set of
parameters (save $n$, which defines steepness of the ejecta velocity profile)
of our kilonova model to be well constrained. In comparison, photometric data
from either UV or optical alone constrains smaller subsets of the parameters
and with reduced accuracy and precision. This result suggests that multiple-
wavelength detections are essential for constraining early ejecta geometry and
opacity.

Thirdly, we find that the data allows to discern between the models if first
data is observed no later than 3.2 (5.2) hours after GW trigger for the
‘default’ (‘lower early opacity’) light curves up to
$160\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. Moreover, after
$13.2\text{\,}\mathrm{h}$ the models can no longer be discerned beyond
$40\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$. These results indicate that rapid
on-target observations on the order of a few hours is necessary for
distinguishing the kilonova from shock interaction models through UV
photometric data.

To place these results in broader context, we comment on prospects for planned
wide-field UV instruments to detect and characterize EM counterparts during
the O5 observation run of GW observatories Advanced LIGO, Virgo and KAGRA
(HLVK) in 2024 and beyond. We focus here on O5, because the launch dates of
ULTRASAT (2025) and UVEX ($\sim$ 2028) are in part overlapping with O5.

Simulations of GW detections in O5 were done by Petrov et al. (2022). From
their simulated data set (Singer, 2021) we compute that of BNS events within
160 Mpc, 69% are localized within 100 deg2, which would be easily followed up
by Dorado within a single orbit. The fraction of events that meet both
distance and localization criteria, multiplied by the annual detection rate
(190, from Petrov et al. (2022)) gives an annual detection rate of $\sim$ 3.2.
If we include events up to 400 Mpc, we find that still 46% of events are
localized within 100 deg2, corresponding to an annual rate of $\sim$ 20
events. Thus, for BNS mergers, if we assume similar brightness to AT2017gfo
(see below), we expect an annual detection rate of $\sim$ $3.2$ of which shock
interaction and kilonova radiation could be distinguished and kilonova ejecta
parameters constrained with a Dorado-like satellite. For BHNS mergers, which
have not been simulated here, the expected number of detected EM counterparts
should be much lower, as only a small region of BHNS parameter space will lead
to disruption of the NS (Foucart, 2020). While the simulations performed here
are specific to Dorado, some of the findings also are relevant to currently
planned wide-field UV missions. To start with, the wide-field UV mission
ULTRASAT is more sensitive than Dorado with a limiting magnitude of $\sim$
22.3 (5$\sigma$, AB, 3$\times$$300\text{\,}\mathrm{s}$ exposure) and has a
larger FoV at 200 deg2 (Asif et al., 2021). Because of this, and ULTRASATs
ability to point to a given ToO within 30 minutes (Sagiv et al., 2014), it is
both sensitive and quick enough to provide the UV data for model selection and
constraints as suggested by simulations here. Secondly, UVEX is even more
sensitive with a limiting magnitude of 25 (AB, 5$\sigma$) but has a smaller
FoV of 12 deg2. In addition, UVEX will provide additional detail in the light
curves and spectra through its two (far and near UV) sensors and on-board
spectroscope (Kulkarni et al., 2021). Because this mission has additional
capabilities, applicability of these results to this mission is not
straightforward. For both we recommend separate simulations that accurately
represent their mission parameters to assess capability for model distinction
and parameter estimation.

Finally, we remark on the various simplifying assumptions made in this study
that may affect the robustness of above findings. To start with, throughout
this work we have made the assumption that the EM counterparts of BNS mergers
all are similar to that of AT2017gfo. This introduces a ‘AT2017gfo-bias’ in
our predictions for the ability of a future UV satellite to achieve science
goals. It is as of yet uncertain how representative AT2017gfo is for the
actual kilonova population and O4 and O5 are expected to shed light on this.
This population is expected to be diverse, for example due to binary
properties such as mass ratio and spins, but also post-merger properties such
as remnant outcome (e.g. Kawaguchi et al., 2020) and jet-ejecta interaction
(e.g. Klion et al., 2021). These are expected to have an effect on ejecta
geometry and composition which in turn affects brightness and color of light
curves.

Another assumption underlying this study is that both models assume isotropic
ejecta. However, the photon emission as well as composition and geometry of
the ejecta are not expected to be spherically symmetric (e.g. Metzger, 2017;
Heinzel et al., 2021). Other models that include the inclination angle, such
as the grid of 2D simulations presented by Wollaeger et al. (2021), would
allow for a more representative study covering the angular dependence of UV
light curves. Still, Heinzel et al. (2021) recommend the inclusion of $\sim$ 1
mag uncertainties for kilonova models used in Bayesian studies relating to
inclination angle to capture yet unknown systematic model uncertainties.

We also note that a significant source of uncertainty remains in the nuclear
physics taking place in these high energy events. For example, Zhu et al.
(2022) find that uncertainties in nuclear inputs lead to typically one order
of magnitude variation in inferred nuclear heating, bolometric luminosity and
ejecta mass.

Lastly, although the scope of this study was limited to shock interaction and
kilonova radiation, we note that a more comprehensive study may include
additional radiation scenarios, such as a neutron precursor or winds driven by
a long-lived remnant, and also a combination of emission channels.

In conclusion, these results show that UV data offers a unique window to
distinguish the processes governing the early post-merger system. For Dorado,
rapid follow-up as well as the ability to quickly localize the target within a
few hours catches the quickly fading UV emission, allowing to distinguish
models. We also find that through multi-wavelength observations the kilonova
emission can be constrained up to at least
$160\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$, unlocking a fuller understanding
of the geometry and opacity of the ejecta outflow.

BD is grateful to the whole Dorado team for their development of the mission
concept, which enabled this detailed case study. BD acknowledges support from
ERC Consolidator grant No. 865768 AEONS (PI: Anna Watts).

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  *[UV]: ultra-violet
  *[IR]: infra-red
  *[LVC]: LIGO-Virgo collaboration
  *[NS]: neutron star
  *[EM]: electromagnetic
  *[BNS]: binary neutron star
  *[GRB]: γ-ray burst
  *[]:
  *[UVOIR]: , optical and   ()
  *[GW]: gravitational wave
  *[NIR]: near infra-red
  *[BHNS]: black hole neutron star
  *[FoV]: field of view
  *[UVO]:   () and optical
  *[LCO]: Las Cumbres Observatory
  *[CCD]: charge coupled device
  *[ETC]: exposure time calculator
  *[$d_{L}$]: luminosity distance
  *[PPD]: posterior probability distribution
  *[2D]: two dimensional
  *[GWs]: gravitational wave
  *[HLVK]: Advanced LIGO, Virgo and KAGRA
  *[ToO]: target of opportunity