Off-Axis Color Characteristics of Binary Neutron Star Merger Events: Applications for Space Multi-Band Variable Object Monitor and James Webb Space Telescope

Abstract

With advancements in gravitational wave detection technology, an increasing number of binary neutron star (BNS) merger events are expected to be detected. Due to the narrow opening angle of jet cores, many BNS merger events occur off-axis, resulting in numerous gamma-ray bursts (GRBs) going undetected. Models suggest that kilonovae, which can be observed off-axis, offer more opportunities to be detected in the optical/near-infrared band as electromagnetic counterparts of BNS merger events. In this study, we calculate kilonova emission using a three-dimensional semi-analytical code and model the GRB afterglow emission with the open-source Python package afterglowpy at various inclination angles. Our results show that it is possible to identify the kilonova signal from the observed color evolution of BNS merger events. We also deduce the optimal observing window for SVOM/VT and JWST/NIRCam, which depends on the viewing angle, jet opening angle, and circumburst density. These parameters can be cross-checked with the multi-band afterglow fitting. We suggest that kilonovae are more likely to be identified at larger inclination angles, which can also help determine whether the observed signals without accompanying GRBs originate from BNS mergers.

1. Introduction

In recent years, it has become generally accepted that at least some short gamma-ray bursts (sGRBs) originate from binary neutron star (BNS) mergers or black hole (BH)-neutron star (NS) mergers [1,2]. These events are thought to be accompanied by optical/near-infrared emissions, such as kilonovae and afterglows [3,4]. The combined detection of electromagnetic and gravitational waves can improve the identification of pre-merger stars, confirm the central engine of sGRBs, and constrain the equation of state of neutron stars [5,6]. The LIGO/Virgo gravitational wave detectors have inferred a BNS merger rate of $R_{\mathrm{BNS}}=105.5_{-83.9}^{+190.2}{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ [7]. However, up to now, except for GRB170817A/GW170817/AT2017gfo, no other BNS merger events have been confirmed by both gravitational waves and their associated electromagnetic counterparts.

Observations of GRB170817A/GW170817/AT2017gfo confirmed by a simultaneous detection that BNS mergers can produce kilonovae, which are generated by the decay of r-process elements [2,8]. In recent years, several kilonova candidates have been proposed (see, e.g., Jin et al. [9,10,11,12], Troja et al. [13,14], Ascenzi et al. [15], Fong et al. [16], Rastinejad et al. [17], Levan et al. [18], Dai et al. [19]). To explain these events, numerous analytical and semi-analytical models, as well as simulations, have been proposed. Some models and analyses support spherically symmetric ejecta (e.g., Hotokezaka and Nakar [20] and Sneppen et al. [21]), while other simulations suggest significant three-dimensional structures, with emissions varying depending on the viewing angle (e.g., Heinzel et al. [22] and Shrestha et al. [23]). Due to the single detection associated with a BNS merger, it is impossible to rule out any of these models.

Afterglow is an important electromagnetic emission associated with gamma-ray bursts [24]. It can be observed across a broad wavelength range, from X-rays to radio waves, and can persist for periods ranging from days to years. Several public open-source codes have been developed to calculate afterglow emissions and effectively explain observations, such as BoxFit [25], Jetsimpy [26], and afterglowpy [27]. In this study, we calculate afterglow emissions using the Gaussian jet model of afterglowpy, which has been used to fit and explain numerous afterglows associated with GRBs (e.g., Li et al. [28], Zhu et al. [29]).

Since kilonova performs as thermal emission with continuously changing color, while afterglows maintain nearly constant color in the optical/near-infrared bands, these events can be identified by their color characteristics using only two filters. Zhu et al. [30] combined kilonova light curves simulated by POSSIS [31] with afterglow models. They proposed that the color evolution of a kilonova is distinct compared to other transients, providing a simple method to identify kilonova candidates in observations.

For NS-NS merger events with small inclination angles, both prompt emission and afterglow are expected to be observed. However, some afterglow-like signals have been detected in the absence of accompanying GRBs, known as orphan afterglows [32,33,34,35]. These signals have been attributed to phenomena like dirty fireballs or failed GRBs [36], while others may stem from GRBs viewed at off-axis angles [37]. When the observer’s viewing angle greatly exceeds the half-opening angle of the jet, neither prompt emission nor early-time afterglow can be detected. As the Doppler factor $\mathcal{D}$ increases gradually, the afterglow exhibits a rising phase until it peaks and then behaves like a typical afterglow [38]. If the event arises from an NS-NS merger, it may also produce a kilonova. Comparing the color evolution with kilonova models could therefore help identify the origin of orphan afterglows.

The Chinese-French Space Multi-band Variable Object Monitor (SVOM)1 launched on 22 June 2024. The Visible Telescope (VT) channels of SVOM, as described by Fan et al. [39] and Pan et al. [40], are capable of observing color evolution in the optical band following a merger event. SVOM-VT’s primary objective is to detect and observe visible emissions after gamma-ray bursts (GRBs), aiming to achieve a limiting magnitude of +22.5 Mv and locate approximately 60 GRBs per year. The Near-Infrared Camera (NIRCam) on board the James Webb Space Telescope (JWST)2 is ideally suited to detect merger events of compact stars in the near-infrared band [41]. Chen and Liang [42] predicted that JWST could detect kilonovae resembling AT2017gfo up to approximately 50 days after a merger event occurring 200 Mpc away. In recent years, JWST detected a new kilonova candidate associated with GRB 230307A [18,43], which may also originate from compact star mergers. JWST NIRCam offers eight filters with general purpose, including F070W, F090W, F115W, F150W, F200W, F277W, F356W, and F444W, which can be utilized to detect kilonovae.

In this work, we predict the color evolution of BNS events using a semi-analytical kilonova code and afterglowpy. In Section 2, we estimate the heating timescale of kilonovae based on previous simulations, describe the method to calculate kilonova light curves, and grid the parameter space for afterglow. In Section 3, we demonstrate how afterglow influences kilonova observations at various inclination angles and identify the optimal time intervals to observe kilonova color evolution in the bands of SVOM VT and JWST NIRCam, providing a reference for follow-up observations of GRBs and gravitational wave events with these instruments. Finally, we summarize our results and discuss potential directions and applications for future research.

2. Light Curves and Color Evolution

2.1. Kilonova

We employed a three-dimensional semi-analytical code [44] to calculate kilonova light curves. The model considers three ejecta components: dynamical ejecta, neutrino-driven ejecta, and viscosity-driven ejecta. Figure 1 illustrates the structure of these ejecta. Assuming that all kilonova emissions originating from NS-NS mergers are similar to AT2017gfo, we applied the parameters from a fitting model of AT2017gfo. In this model, the dynamical ejecta, neutrino-driven ejecta, and viscosity-driven ejecta have masses of 0.018 $M_{\odot }$, 0.02 $M_{\odot }$, and 0.02 $M_{\odot }$, respectively, with mean opacities of 12.02 ${\mathrm{cm}}^2{\mathrm{g}}^{-1}$, 0.57 ${\mathrm{cm}}^2{\mathrm{g}}^{-1}$, and 2.03 ${\mathrm{cm}}^2{\mathrm{g}}^{-1}$, respectively [44].

Previous studies have shown that the bolometric light curve and temperature evolution of AT2017gfo can be well-explained by decay heating from elements with solar r-process abundances [20,45]. For simplification in our calculations, we regard energy injection in the form of [45,46], as follows:

$$\dot{Q}\left(t\right)={ϵ}_{\mathrm{th}}{ϵ}_{Y_e}{\dot{ϵ}}_0{(t/\mathrm{day})}^{-\mathit{s}},$$

(1)

where ${\dot{ϵ}}_0=1.58\times {10}^{10}\mathrm{erg}{\mathrm{g}}^{-1}{\mathrm{s}}^{-1}$, $s=1.3$, and the thermalization efficiency ${ϵ}_{\mathrm{th}}$ is approximated to be 0.5 [4] before the ejecta transitions to the nebular phase. The term ${ϵ}_{Y_e}$ is an electron-fraction-dependent factor as given by [46,47]

$${ϵ}_{Y_e}=\left\{\begin{array}{cc}0.5+2.5{[1+e^{4(t/\mathrm{day}-1)}]}^{-1},\hfill & {\mathit{Y}}_{\mathit{e}}\ge 0.25,\hfill \\ 1,\hfill & {\mathit{Y}}_{\mathit{e}}<0.25.\hfill \end{array}\right.$$

(2)

Comparing the simplified assumption with the heating rate of solar r-process abundant, they exhibit good agreement, particularly in the early stages. The flux density of thermal radiation can be calculated by [46]

$$F_{\lambda }(\lambda ,t)\approx {\displaystyle \frac{2}{h^4{c}^3{D}_{\mathrm{L}}^2}}{\int \int }_S{\displaystyle \frac{{\mathcal{D}}^3{(hc/\mathcal{D}\lambda )}^5}{exp(hc/\mathcal{D}{k}_{\mathrm{B}}\lambda T_{\mathrm{mesh}}^{ij})-1}}d{s}^{ij},$$

(3)

where h is the Planck constant, $D_{\mathrm{L}}$ is the luminosity distance, $k_{\mathrm{B}}$ is the Boltzmann constant, $T_{\mathrm{mesh}}^{ij}$ is the temperature at the mesh grid, $d{s}^{ij}$ is the infinitesimal projected photosphere area, and $\mathcal{D}=1/[\Gamma (1-\beta cos\Delta \theta \left)\right]$ is the Doppler factor, where $\Gamma =1/\sqrt{1-{\beta }^2}$, $\beta =v/c$, and $\Delta \theta$ is the angle between the moving direction and the line of sight.

The emission before approximately 1 day post-merger remains unclear. Some studies have suggested that thermal bomb shock heating at a very early time may enhance luminosity [48,49], while others have proposed that exceptionally high opacity could result in faint early light curves [50]. Currently, distinguishing and parameterizing kilonova emission at these very early stages presents challenges.

From approximately 1 day to 1 week after the merger, the heating rate is primarily dominated by the $\beta$-decay of elements with atomic mass numbers ranging from 85 to 140 [20]. Although there is no spectral analysis confirming the abundance of elements in the kilonova ejecta, and no evidence indicating whether $\alpha$-decay and the fission of heavier elements contribute to kilonova emission, these processes could significantly impact the prediction of light curves in the later stages, such as ²²³Ra, ²²⁵Ac, and ²⁵⁴Cf [51]. Additionally, at a late time, kilonovae enter a nebular phase, where line emissions become significant and deviate from black-body-dominated emission. $\beta$-decay energy emerges in the form of electrons, gamma-rays, and neutrinos [45,52]. The heating timescales for electrons and gamma-rays, as given by Pognan et al. [53], for a uniform spherically symmetric ejecta are

$$t_e\approx 150{\left({\displaystyle \frac{M_{\mathrm{ej}}}{0.05M_{\odot }}}\right)}^{2/3}{\left({\displaystyle \frac{v_{\mathrm{ej},\max }}{0.1c}}\right)}^{-2}\mathrm{day},$$

(4)

$$t_{\gamma }\approx 1.3{\left({\displaystyle \frac{M_{\mathrm{ej}}}{0.05M_{\odot }}}\right)}^{1/2}{\left({\displaystyle \frac{v_{\mathrm{ej},\max }}{0.1c}}\right)}^{-1}\mathrm{day},$$

(5)

where $M_{\mathrm{ej}}$ is the mass of ejecta and $v_{\mathrm{ej},\max }$ is the maximum velocity. Thermalization becomes inefficient beyond this timescale. We estimate the time scale based on the systematic numerical relativity simulation result of Radice et al. [54], which are listed in Table 2 of their paper. All cases with ejecta mass having velocity $v_{\mathrm{ej}}\ge 0.6c$ are less than ${10}^{-4}{M}_{\odot }$, so we assume $v_{\mathrm{ej}}<0.6c$. This suggests that $t_{\gamma }<1$ day in all cases. Figure 2 shows the distribution of $t_e$, calculated using Equation (4), for all 54 cases (excluding 2 cases without ejecta). In 42.6% of the cases, $\beta$-decay thermalization becomes inefficient after 7 days, and in 90.7% of the cases, it becomes inefficient after 14 days. If the kilonova model considers only the thermal emission, the predicted flux will become inaccurate once the ejecta transitions to the nebular phase. Modeling a nebular phase ejecta remains a challenge. Thus, it is preferable to parameterize kilonova emission between approximately 1 day to 2 weeks after the merger.

In this work, we estimate the light curves of nebular phase ejecta by considering a power-law-decreasing thermalization efficiency, ${ϵ}_{\mathrm{th}}\propto t^{-1.5}$ [44] after a critical time. This critical time, occurring about a week after the merger of GW170817 [55], is fitted to be approximately 6.12 days. Line emission and ionization are ignored in our calculations.

2.2. Afterglow

Non-thermal afterglow emission can influence color evolution when combined with kilonova in an NS-NS merger event. We calculate the afterglow using the open-source Python package afterglowpy. Considering the inclination angles, we apply the numerical computation of the Gaussian jet model. Using the central values of the fitting results [27] for AT2017gfo as the baseline, we examine the combined model with the following parameters: ${\theta }_{\mathrm{obs}}=0.4,\mathrm{rad}$, ${log}_{10}{E}_0=52.96$, ${\theta }_{\mathrm{c}}=0.066,\mathrm{rad}$, ${\theta }_{\mathrm{w}}=0.47,\mathrm{rad}$, ${log}_{10}{n}_0=-2.7$, $p=2.168$, ${log}_{10}{ϵ}_e=-1.42$, and ${log}_{10}{ϵ}_B=-3.96$, where ${\theta }_{\mathrm{obs}}$ is inclination angle in radians, $E_0$ is isotropic-equivalent energy in erg, ${\theta }_{\mathrm{c}}$ is a half-opening angle of jet core in radians, ${\theta }_{\mathrm{w}}$ is a truncation angle, $n_0$ is circumburst density in ${\mathrm{cm}}^{-3}$, p is electron energy distribution index, ${ϵ}_e$ is energy fraction in electrons, and ${ϵ}_B$ is energy fraction in magnetic field.

Since afterglow is a non-thermal emission, distinguishing kilonova from afterglow by color evolution is straightforward. Dashed lines in Figure 3 show the g-r, g-i, and g-H color evolution calculated by the kilonova model using parameters from AT2017gfo. As the temperature decreases, the color of thermal emission becomes redder. Dotted lines in Figure 3 represent the color of the afterglow, which remains almost constant. The solid lines illustrate the combined color evolution of kilonova and afterglow. Given the large number of data points from different telescopes, we average observed data within a day as a single representative point for each band, shown as dots with error bars in Figure 3. This indicates that the model can reasonably describe the color evolution of AT2017gfo. The color evolution can be divided into three stages. The first stage is dominated by kilonova emission. Non-thermal emission from GW170817/GRB170817A/AT2017gfo was not detected until nine days after the GRB, when the first X-ray emission was observed [56], followed by an increase in brightness at all wavelengths. In the next stage, the afterglow gradually becomes significant, and the color starts to decline. Finally, in the third stage, the afterglow dominates completely, and the color becomes constant. For comparison, we also show the color evolution of various types of supernovae, including Ia SN2004eo [57], Ib SN2015ap, IIb SN2016gkg [58], and Ic SN2016coi [59] in Figure 3. Our results are consistent with Zhu et al. [30], indicating that the color characteristics of kilonovae are significantly different from those of supernovae. This distinction allows for the identification of kilonovae from observations in just two bands, if they are not disturbed by afterglow.

To predict color evolution under more general conditions, we expanded the range of parameters based on observed afterglow data. Becerra et al. [60] collected data from 227 gamma-ray bursts observed by the TAROT3, COATLI4, and RATIR5 telescopes. They provided cumulative distribution functions for six constrained parameters in Figure 9 of their paper, including $E_0$, ${\theta }_c$, $n_0$, p, ${ϵ}_e$, and ${ϵ}_B$. Based on their results, we gridded the parameter space for afterglow as shown in

Table 1. Gridded afterglow parameters constrained by result of Becerra et al. [60].

Parameters	Meanings	Values
${\theta }_{\mathrm{obs}}$	Viewing angle in radians	$[0,{\displaystyle \frac{\pi }{12}},{\displaystyle \frac{\pi }{6}},{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{3}}]$
${log}_{10}{E}_0$	Isotropic-equivalent energy in erg	[49,50,51,52,53,54]
${\theta }_{\mathrm{c}}$	Half-opening angle of jet core in radians	[0.01,0.10,0.20,0.30]
${log}_{10}{n}_0$	Circumburst density in cm−3	[−4,−3,−2,−1,0,1]
p	Electron energy distribution index	[2.1,2.2,2.3,2.4,2.5]
${log}_{10}{ϵ}_e$	Energy fraction in electrons	[−3,−2,−1]
${log}_{10}{ϵ}_B$	Energy fraction in magnetic field	[−5,−4,−3,−2,−1]

. Our tests indicated that the truncation angle ${\theta }_{\mathrm{w}}$ has little influence on color evolution in off-axis conditions, so we assumed ${\theta }_{\mathrm{w}}=3{\theta }_c$ for our calculations. All other parameters were set to their default values in afterglowpy.

3. Results

By combining kilonova and afterglow observations, it is possible to identify NS-NS merger events using two-filter observations in the optical and near-infrared bands. Our goal is to predict the color evolution detectable by SVOM VT and JWST NIRCam, and to provide a reference for optimizing observation strategies. Additionally, we find that the observation windows are related to some parameters, which will help constrain these parameters and can also be cross-checked with the multi-band afterglow fitting. More details will be provided in the following subsections.

3.1. SVOM VT

The cameras of the SVOM VT cover two wavelength ranges: the blue channel (450 to 650 nm) and the red channel (650 to 1000 nm). This enables the detection of color evolution in the electromagnetic counterparts of BNS merger events. To predict these detections, we calculate light curves for kilonova and afterglow at the central frequencies of each channel, 550 nm and 825 nm respectively. Assuming the kilonova is similar to AT2017gfo, we use the same parameters, varying only the viewing angles. Figure 4 shows the light curves of an AT2017gfo-like kilonova with viewing angles of $0^{\circ }$, ${15}^{\circ }$, ${30}^{\circ }$, ${45}^{\circ }$, and ${60}^{\circ }$. For comparison, we also plot the g- and z-band observational data of AT2017gfo. As the viewing angle increases, the brightness decreases.

The parameter space for the afterglow is listed in Table 1. We examine the influence of each parameter by varying them individually. Figure 5 shows the color evolution with parameters matching AT2017gfo, except for the single variable being altered according to Table 1. For comparison, we also plot the g-z and r-z colors of AT2017gfo. The color evolution shows a plateau at around one week, which can be explained by the different light curve trends of bluer and redder bands, as shown by the g- and z-band of AT2017gfo in Figure 4. The plateau appears visibly when the selected bands are appropriate. This result is also seen in the simulation of Zhu et al. [30]. The plateau is clearly evident in the bands of SVOM VT, making it useful for identifying multiple components through future observations.

Figure 5 indicates that the electron energy distribution index p has minimal influence on color evolution. The isotropic-equivalent energy $E_0$, the energy fraction in electrons ${ϵ}_e$, and the energy fraction in the magnetic field ${ϵ}_B$ primarily affect the color evolution at a later time. The parameters that significantly influence the early phase are the half-opening angle of the jet ${\theta }_{\mathrm{c}}$ and the circumburst density $n_0$. When the viewing angle exceeds the jet’s half-opening angle, the afterglow brightness is initially dim and then gradually brightens due to the increasing Doppler factor. A larger viewing angle combined with a narrower jet core results in a weaker afterglow contribution at early times. For the circumburst density $n_0$, a higher density naturally leads to a stronger interaction between the ejecta and the surrounding medium, thereby boosting the afterglow’s luminosity. Since kilonova flux also decreases with increasing viewing angle, it is not straightforward to conclude that a large observation angle favors kilonova identification. Quantitative calculations are therefore essential.

Combining the AT2017gfo-like kilonova and afterglow with parameters listed in Table 1, we obtain 54,000 samples. In the parameter space of ${\theta }_{\mathrm{obs}}$, ${\theta }_{\mathrm{c}}$, and ${log}_{10}{n}_0$, we separated all of the samples into four groups. These groups represent samples for which the color evolution deviates from that of a single kilonova by less than 0.5 magnitudes within two weeks, one week, and two days, and a group that is not easily distinguished, respectively, regardless of changes in the other parameters within the constrained range of Table 1. We noted that the four groups can be separated by three planes that can be described by equation in form of $A{\theta }_{\mathrm{obs}}+B{\theta }_{\mathrm{c}}+{log}_{10}{n}_0+C=0$ in the parameter space, where A, B, and C are free parameters. When a sample satisfies

$$-3.85{\theta }_{\mathrm{obs}}+19.4{\theta }_{\mathrm{c}}+{log}_{10}{n}_0+4.08\le 0,$$

(6)

the color evolution will not deviate from the kilonova model by more than 0.5 magnitudes before 2 weeks from the merger, allowing a 2-week observation window for the kilonova. When the parameters are in the range of

$$-9.8{\theta }_{\mathrm{obs}}+40.5{\theta }_{\mathrm{c}}+{log}_{10}{n}_0+4.1\le 0,$$

(7)

the deviation from the kilonova model remains within 0.5 magnitudes within a 1-week observation window. When

$$-17.5{\theta }_{\mathrm{obs}}+72.3{\theta }_{\mathrm{c}}+{log}_{10}{n}_0+3.28\le 0,$$

(8)

the color evolution will be interfered by afterglow with the deviation less than 0.5 magnitude within 2 days after the merger. Conversely, if

$$-17.5{\theta }_{\mathrm{obs}}+72.3{\theta }_{\mathrm{c}}+{log}_{10}{n}_0+3.28>0,$$

(9)

it becomes challenging to identify a kilonova by color evolution after two days. We present the grouped samples in Figure 6. The green shaded area in each subplot represents the range of color evolution for a sample group, while the blue shaded area indicates the optimal observation window for identifying a kilonova. The columns of sub-figures display results for inclination angles of ${15}^{\circ }$, ${30}^{\circ }$, and ${60}^{\circ }$, respectively. The rows of sub-figures correspond to samples with observation windows of 2 weeks, 1 week, 2 days, and those that are difficult to distinguish, respectively.

The results offer clear predictions for future SVOM VT follow-up observations of GRBs and gravitational wave events. As mentioned in Section 2.1, the kilonova emission before 1 day is still unclear, and modeling the nebular phase of kilonovae remains challenging, making the observation window before 2 weeks crucial. If the inclination of a merger event is inferred from gravitational waves, and combined with constraints on the range of ${\theta }_{\mathrm{c}}$ and $n_0$, one can estimate the optimal follow-up time required for kilonova detection. For example, in an on-axis event where ${\theta }_{\mathrm{obs}}=0$, Equations (6) and (7) cannot be satisfied, indicating that the window for identifying the kilonova is likely to be less than 2 days. Overall, Equations (6)–(9) indicate that a larger ${\theta }_{\mathrm{obs}}$, a smaller ${\theta }_{\mathrm{c}}$, and $n_0$ tend to result in a longer ideal observation window for kilonovae in optical bands. We noted that when ${\theta }_{\mathrm{obs}}=0$, most samples are difficult to distinguish kilonovae by color evolution, except for a small subset of samples. The samples detectable in the on-axis condition are influenced by the combined effect of all parameters. For ${\theta }_{\mathrm{obs}}=\pi /6$, 8.33% of the samples fall within the range of Equation (6), 25.0% satisfy Equation (7), 37.5% are within the range of Equation (8), and 62.5% align with Equation (9). In the condition of ${\theta }_{\mathrm{obs}}=\pi /3$, 33.3%, 62.5%, 70.8%, and 29.2% of the samples satisfy Equations (6)–(9), respectively. It is evident that off-axis BNS merger events are more likely to be detected by color evolution than on-axis events. This is consistent with GW170817/GRB170817/AT2017gfo, which had non-thermal emission detected starting 9 days after the merger [56].

3.2. JWST NirCam

We computed the kilonova and afterglow emissions across eight filters (F070W, F090W, F115W, F150W, F200W, F277W, F356W, and F444W) of NIRCam, which can be used to detect counterparts of NS-NS mergers. To enhance the clarity of color evolution characteristics, we specifically analyzed the results using the filters F070W and F444W, chosen for their significant wavelength difference.

Similar to SVOM VT, we generated 54,000 samples, assuming an AT2017gfo-like kilonova at various viewing angles combined with afterglow using parameters listed in Table 1. Although Chen and Liang [42] predicted that JWST NIRCam can detect kilonovae up to 50 days at distances of 200 Mpc, the lack of a detailed model for the nebular phase of kilonovae still poses challenges for late-time identification. Our focus remains primarily on the observation window before 2 weeks. We categorized all samples into groups as in the previous subsection, noting overlap primarily between samples with observation windows of 1 week and 2 days. This overlap may be due to our parameter grid division not being precise enough. Therefore, we divided the samples into three groups corresponding to observation windows before 2 weeks, 1 week after the merger, and those that are difficult to distinguish. Similar to samples of SVOM, these groups can also be delineated by planes in the parameter space of ${\theta }_{\mathrm{obs}}$, ${\theta }_{\mathrm{c}}$, and ${log}_{10}{n}_0$. Samples satisfying

$$-4.45{\theta }_{\mathrm{obs}}+22.7{\theta }_{\mathrm{c}}+{log}_{10}{n}_0+4.1\le 0$$

(10)

always have observation windows of at least 2 weeks. For those satisfying

$$-9.8{\theta }_{\mathrm{obs}}+40.95{\theta }_{\mathrm{c}}+{log}_{10}{n}_0+3.6\le 0,$$

(11)

the observation window tends to be at least 1 week. Samples where

$$-9.8{\theta }_{\mathrm{obs}}+40.95{\theta }_{\mathrm{c}}+{log}_{10}{n}_0+3.6>0$$

(12)

indicate that kilonovae are difficult to identify based on color evolution alone. We show the grouped samples in Figure 7. The green shaded area in each subplot represents the range of color evolution for a sample group, while the blue shaded area indicates the optimal observation window for identifying a kilonova. The columns of sub-figures display results for inclination angles of ${15}^{\circ }$, ${30}^{\circ }$, and ${60}^{\circ }$, respectively. The rows of sub-figures correspond to samples with observation windows of 2 weeks, 1 week, and those that are difficult to distinguish, respectively.

The results provide concise predictions for future JWST observations. Equations (10)–(12) indicate that larger values of ${\theta }_{\mathrm{obs}}$ and smaller values of ${\theta }_{\mathrm{c}}$ and $n_0$ tend to result in longer ideal observation windows for kilonovae in near-infrared bands. We noted that when ${\theta }_{\mathrm{obs}}=0$, most samples are difficult to distinguish kilonovae by color evolution, except for a small subset of samples. Under the condition of ${\theta }_{\mathrm{obs}}=\pi /6$, 12.5%, 33.3%, and 66.7% of the samples satisfy Equations (10)–(12), respectively. For ${\theta }_{\mathrm{obs}}=\pi /3$, 37.5%, 62.5%, and 37.5% of the samples satisfied Equations (10)–(12), respectively.

4. Discussion and Conclusions

We calculated the light curves of a kilonova combined with afterglow using a three-dimensional semi-analytical kilonova model and the open-source Python package afterglowpy. Since the emission of kilonovae at very early and late times is not yet well-modeled, it is better to parameterize kilonovae in the range of 1 day to 2 weeks after the merger. Although both kilonova and afterglow contribute to the observed emission in optical and near-infrared bands, kilonova emission is dominated by thermal processes within this timeframe, while afterglow emission is non-thermal. This distinction can be recognized by color evolution.

We estimated the color evolution of AT2017gfo-like events that could be observed by SVOM VT and JWST NirCam, based on kilonova parameters from Gong et al. [44] and afterglow parameters constrained by Becerra et al. [60]. We observed that viewing angles (${\theta }_{\mathrm{obs}}$), half-opening angles of the jet core (${\theta }_{\mathrm{c}}$), and circumburst density ($n_0$) significantly impact color evolution within the observation window of 1 day to 2 weeks. All samples can be grouped by observation windows of different time intervals in the parameter space of ${\theta }_{\mathrm{obs}}$, ${\theta }_{\mathrm{c}}$, and $n_0$. Equations (6)–(9) correspond to observation windows for SVOM VT of 2 weeks, 1 week, 2 days, and those that are difficult to detect. Similarly, Equations (10)–(12) correspond to observation windows for JWST NirCam of 2 weeks, 1 week, and those that are difficult to detect. The results indicate that more optical/near-infrared counterparts of NS-NS merger events are expected to be detected in off-axis conditions, where GRBs would not be observed. As Figure 3 showed, color evolution of kilonova events differs from a supernova. This approach helps determine whether an observed optical/near-infrared transient originates from a compact star merger, provides rough constraints on the parameters, and allows for cross-checking with X-ray and radio observation results.

Zhu et al. [30] calculated the color evolution of kilonova based on the POSSIS code [31]. In this paper, we calculate the kilonova emission using the three-dimensional model described by Gong et al. [44], in which the wavelength-dependent opacity and the evolution of thermal efficiency are considered, we find that our results are consistent with that of Zhu et al. [30]. In addition, this work provides a simple method to preliminary determine whether the kilonova component is present in the observed data. Usually, the kilonova component can be identified only through multi-wavelength light curves fitting or obtaining the kilonova spectra, which would cost much observation time, while here we propose that we can search for the kilonova component just through measuring the color evolution, which would greatly reduce the observation time. This method is suitable for SVOM VT, a new telescope for GRB follow-up observations. Once SVOM detects a short GRB, VT can quickly perform follow-up observation and preliminarily determine whether there is the kilonova signal. For dim GRBs, JWST may take the follow-up observation to search for the kilonova signal.

In this work, we focus on kilonova emissions originating from NS-NS merger events. However, BH-NS mergers are also expected to produce kilonovae, GRBs, and detectable gravitational waves. To date, no electromagnetic counterparts from BH-NS mergers have been observed, making it challenging to constrain the physical parameters for kilonovae in such events. Furthermore, simulations indicate an anisotropic distribution of ejecta from BH-NS mergers (e.g., Kyutoku et al. [62], Kawaguchi et al. [63]), suggesting that viewing angles along the longitudinal direction should also be considered. Future observations of electromagnetic counterparts related to BH-NS mergers, combined with gravitational wave detections, will allow for further refinement of kilonova models.