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A&A
Volume 679, November 2023
Article Number A103
Number of page(s) 10
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/202346674
Published online 28 November 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Gamma-ray bursts (GRBs) are energetic explosions occurring at cosmological distances. There are mainly two different phases in GRBs, the prompt emission stage and the subsequent afterglow phase (Piran 2004; Mészáros 2006; Kumar & Zhang 2015). The prompt emission usually peaks at a few hundred keV (Fishman & Meegan 1995; Gruber et al. 2014), while the afterglow can be observed in a much broader wavelength, ranging from soft X-rays to radio waves (Mészáros & Rees 1997; Sari et al. 1998; Zhang et al. 2006; Kann et al. 2010; Geng et al. 2016; Alexander et al. 2017; O’Connor et al. 2023). Generally speaking, the afterglow can last for a period ranging from days up to years, whereas the prompt emission typically lasts for less than a few minutes. Despite the short duration of the prompt emission, the total energy released in γ-rays is enormous. For long-duration GRBs, the typical isotropic energy of the prompt emission is ∼1053 erg (Xu et al. 2023). Recently, the discovery of GRB 221009A has once again refreshed our perception of the energetics of GRBs (Ren et al. 2023; An et al. 2023; O’Connor et al. 2023; Sato et al. 2023; Yang et al. 2023). This event was found to have an isotropic energy of ∼1055 erg, which places it as the most energetic GRB in history (An et al. 2023; Yang et al. 2023).

For a typical GRB, both the prompt emission and the afterglow phases can be observed. However, there is a special kind of burst that has a missing prompt emission but an afterglow that can still be observed. Such transients are called orphan afterglows (Rhoads 1997; Huang et al. 2002; Nakar et al. 2002; Zou et al. 2007; Gao et al. 2022). Orphan afterglows are most likely produced in two cases. First, if the initial Lorentz factor of an outflow is significantly less than ∼100, the prompt emission will be too faint to be observed. Such explosions are also known as “dirty fireballs” or “failed GRBs” (Paczyński 1998; Dermer et al. 1999; Huang et al. 2002; Xu et al. 2012). Second, if the outflow is highly collimated (not isotropic), it is hard to detect the prompt emission when the explosion is viewed off-axis, that is, the viewing angle is larger than the half-opening angle of the jet (Rhoads 1997; Nakar et al. 2002). In this case, the majority of γ-ray radiation from the GRB cannot be observed due to the relativistic beaming effect. However, the subsequent softer emission will be less beamed and can be visible to the observer in the afterglow stage (Rhoads 1997; Huang et al. 2002). These transients are called off-axis orphans.

We note that there exists a special class of softer GRBs, namely, X-ray flashes. Their peak photon energy falls in X-ray ranges and is thus significantly less than that of normal GRBs in the prompt emission phase. X-ray flashes can also be explained by the dirty fireball model (Huang et al. 2002) or the off-axis model (Yamazaki et al. 2002), which means they may be closely connected with orphan afterglows (Urata et al. 2015).

Assuming that all the orphan afterglows come from off-axis GRBs, we can estimate the half-opening angle of GRBs through the ratio between the observed orphan afterglow rate and the GRB rate (Rhoads 1997; Paczyński 2000). However, as pointed out by Huang et al. (2002), there should be many orphan afterglows coming from failed GRBs since the baryon loading issue exists widely in popular progenitor models of GRBs (Piran 2004). As a result, this method may not be straightforward.

On the other hand, searching for orphan afterglows is not an easy task (Grindlay 1999; Greiner et al. 2000; Levinson et al. 2002; Rau et al. 2006; Gal-Yam et al. 2006; Khabibullin et al. 2012; Berger et al. 2013; Ho et al. 2018, 2022; Huang et al. 2020; Leung et al. 2023). So far, only six orphan afterglow candidates have been reported. One of them was found at the radio wavelength by the Very Large Array (VLA): FIRST J141918.9 + 394036 (Law et al. 2018; Mooley et al. 2022). The evolution of its brightness and the lack of associated GRB suggest that it might be an off-axis orphan afterglow (Mooley et al. 2022). The other five transients were all detected in optical bands. They are PTF11agg (Cenko et al. 2013; Wang & Dai 2013; Wu et al. 2014), discovered by the Palomar Transient Factory (Law et al. 2009), and AT2019pim (Kool et al. 2019), AT2020blt (Ho et al. 2020; Andreoni et al. 2021; Sarin et al. 2022), AT2021any (Ho et al. 2022; Gupta et al. 2022), and AT2021lfa (Ho et al. 2022; Lipunov et al. 2022), discovered by the Zwicky Transient Facility (ZTF; Bellm et al. 2019).

With the currently available observational data, it is still a challenge to derive the physical parameters and reveal the nature of an orphan afterglow. A major problem is the lack of the trigger time of the unseen GRB associated with the afterglow. Usually, the trigger time is estimated by fitting the observed light curve with a particular model (Ho et al. 2020, 2022; Gupta et al. 2022). Recently, Sarin et al. (2022) used the last non-detection information from an upper limit in the r band to constrain the trigger time of AT2020blt. However, the non-detection may result from a bad seeing or other interferences and thus cannot provide decisive information on the trigger time.

In this study, we present an in-depth study on AT2021any, an orphan afterglow candidate found by ZTF. The trigger time, together with other parameters, is derived by fitting the observed light curves. An efficient code is developed for this purpose. Synchrotron emission and synchrotron self-Comptonization (SSC) are considered in our modeling. The effect of synchrotron self-absorption is also included.

Our paper is organized as follows. First, the multi-wavelength observational data of AT2021any are presented and described in Sect. 2. The physical models used to fit the data are then introduced in Sect. 3. In Sect. 4, the fitting results of AT2021any with different models are presented, and the goodness of fit is compared. Finally, our conclusions and a brief discussion are presented in Sect. 5.

2. Observational data of AT2021any

The orphan afterglow candidate AT2021any (ZTF21aayokph) was detected at 06:59:45.6 UTC on January 16, 2021, by ZTF (Ho et al. 2021). The brightness was r = 17.92 ± 0.06 mag when it was first recorded. The most recent non-detection was only 20.3 min before the first detection (Ho et al. 2022), which gives a limiting magnitude of r > 20.28 mag. No associated GRB was recorded during the period between the last non-detection and the first detection (Ho et al. 2022; Gupta et al. 2022). The object faded rapidly in the r band, with a fading rate of 14 mag day−1 during the first 3.3 h. The extinction-corrected color index was found to be g − r = (0.25 ± 0.19) mag (Ho et al. 2022). These observations strongly indicate that AT2021any is a promising orphan afterglow candidate. The redshift of AT2021any was later determined as z = 2.5131 ± 0.0016 through spectroscopic observations (de Ugarte Postigo et al. 2021; Ho et al. 2022). AT2021any was subsequently followed by a variety of optical facilities (see Ho et al. 2022; Gupta et al. 2022 for more information). The multi-wavelength optical photometry data are collected and presented in Table 1. We note that the AB magnitudes in Table 1 have not been corrected for the extinction of the Milky Way.

Table 1.

Optical photometry of AT2021any.

The object was followed in X-rays by Swift-XRT (Ho & Zwicky Transient Facility Collaboration 2021). The observations were performed at three different epochs, with a total exposure time of 8.2 ks. X-ray emission was detected only at the first epoch. The unabsorbed flux density is estimated as 3.30 × 10−13 erg cm−2 s−1, with a neutral hydrogen column density of NH = 8.12 × 1020 cm−2 (Willingale et al. 2013) and an assumed photon index of Γp = 2 (see Table 2).

Table 2.

X-ray afterglow data of AT2021any.

The transient was observed in radio by VLA (Perley et al. 2021; Ho et al. 2022). Eight epochs of observations were performed from 4.90 days to 75.77 days after the discovery of AT2021any. The radio data obtained by Ho et al. (2022) are collected and listed in Table 3.

Table 3.

Radio afterglow data of AT2021any.

3. Dynamics and emission mechanisms of gamma-ray bursts

In this section, we briefly describe the dynamic evolution and radiation process of relativistic outflows that produce GRBs. The dynamics of the outflow can be depicted by the following equations (Huang et al. 2000a, 2006; Geng et al. 2013; Xu et al. 2022):

d R d t = β c Γ ( Γ + Γ 2 − 1 ) , $$ \begin{aligned}&\frac{\mathrm{d}R}{\mathrm{d}t} = \beta c \Gamma (\Gamma + \sqrt{\Gamma ^{2} - 1}),\end{aligned} $$(1)

d m d R = 2 π R 2 ( 1 − cos θ j ) n m p , $$ \begin{aligned}&\frac{\mathrm{d}m}{\mathrm{d}R} = 2\pi R^{2} (1 - \cos \theta _{\mathrm{j}}) n m_{\rm p},\end{aligned} $$(2)

d Γ d m = − Γ 2 − 1 M ej + ϵ r m + 2 ( 1 − ϵ r ) Γ m · $$ \begin{aligned}&\frac{\mathrm{d}\Gamma }{\mathrm{d}m} = -\frac{\Gamma ^{2} - 1}{M_{\rm ej} + \epsilon _{\rm r} m + 2(1 - \epsilon _{\rm r}) \Gamma m}\cdot \end{aligned} $$(3)

Here, R is the shock radius in the GRB rest frame, c is the speed of light, and Γ is the Lorentz factor of the outflow with β = Γ 2 − 1 / Γ $ \beta = \sqrt{\Gamma ^{2}-1}/ \Gamma $. The term t is the observer’s time, m is the swept-up mass of the interstellar medium (ISM), θj is the half-opening angle of the outflow, and mp is the proton’s mass. Finally, n is the number density of the surrounding ISM. For a homogeneous ISM, we took n as a constant. As for a wind ISM, we have n = Ar−2, where A is a coefficient depending on the mass loss rate and the speed of the wind (Chevalier & Li 1999; Dai & Lu 2001; Wu et al. 2003; Ren et al. 2023). The term Mej is the initial mass of the ejecta, and ϵr is the radiative efficiency. In this study, we note that when calculating the dynamical evolution of a structured jet, we divided the whole jet into many small segments. For a segment located at angle θ, Eq. (3) should then be modified as d m d R = sin θ d θ d ϕ R 2 n m p $ \frac{\mathrm{d}m}{\mathrm{d}R} = \sin \theta \mathrm{d}\theta \mathrm{d}\phi R^{2} n m_{\mathrm{p}} $ in order to calculate the swept-up mass, where dθ is the angular length of the segment and dϕ is its width.

The lateral expansion of the outflow is neglected in our calculations. There are two reasons for this. First, the lateral expansion does not significantly affect the brightness of the afterglow, especially at relatively early stages. In fact, according to the study by Huang et al. (2000a), the effect of lateral expansion becomes discernable only when t > 107 s. In the case of AT2021any studied here, the afterglow is generally detected in a time range of t ≤ 107 s. So the omittance of this effect does not bring in marked errors. Another reason is that the inclusion of lateral expansion leads to too much complexity in the calculations for a structured jet. It means that we would need to consider the lateral expansion of each segment. The interaction between adjacent segments would also need to be modeled. Thus, it would lead to further difficulties in determining the equal arrival time surface. As mentioned above, since the omittance of the lateral expansion only results in very limited errors, we ultimately decided not to include it in the current study for simplicity.

Synchrotron radiation from shock-accelerated electrons is involved in GRB afterglows (Sari et al. 1998). We use a prime (′) superscript to denote the quantities in the shock comoving frame, while those without a prime are in the observer frame. In the fast-cooling regime, the flux Fν′ at frequency ν′ is

F ν ′ = F ν ′ , max { ( ν ′ ν c ′ ) 1 3 , ν ′ < ν ν c ′ , ( ν ′ ν ν c ′ ) 1 2 , ν ν c ′ < ν ′ < ν min ′ , ( ν ′ ν min ′ ) − p 2 ( ν ν c ′ ν min ′ ) 1 2 , ν min ′ < ν ′ < ν ′ max , $$ \begin{aligned} F_{\nu^{\prime }} = F_{\nu^{\prime },\mathrm{max}}\left\{ \begin{array}{ll} (\frac{\nu^{\prime }}{\nu^{\prime }_{\mathrm{c}}})^{\frac{1}{3}},&\nu^{\prime }< \nu^{\prime }_{\mathrm{c}}, \\ (\frac{\nu^{\prime }}{\nu^{\prime }_{\mathrm{c}}})^{\frac{1}{2}},&\nu^{\prime }_{\mathrm{c}}< \nu^{\prime }< \nu^{\prime }_{\mathrm{min}}, \\ (\frac{\nu^{\prime }}{\nu^{\prime }_{\mathrm{min}}})^{-\frac{p}{2}}(\frac{\nu^{\prime }_{\mathrm{c}}}{\nu^{\prime }_{\mathrm{min}}})^{\frac{1}{2}},&\nu^{\prime }_{\mathrm{min}}< \nu^{\prime }< \nu^{\prime }_{\mathrm{max}}, \end{array} \right. \end{aligned} $$(4)

where Fν′, max is the peak flux density (Sari et al. 1998) and p is the power-law index characterizing the distribution of electrons. The terms ν c ′ $ \nu^\prime_{\rm c} $, ν min ′ $ \nu^\prime_{\rm min} $, and ν max ′ $ \nu^\prime_{\rm max} $ are characteristic frequencies corresponding to the cooling Lorentz factor ( γ c ′ $ \gamma^\prime_{\rm c} $), the minimum Lorentz factor ( γ min ′ $ \gamma^\prime_{\rm min} $), and the maximum Lorentz factor ( γ max ′ $ \gamma^\prime_{\rm max} $), respectively. Here, γ max ′ $ \gamma^\prime_{\rm max} $ is the Lorentz factor of the highest energy electrons that could be accelerated by the shock. For these electrons, the power of acceleration is counterbalanced by the energy loss rate due to synchrotron radiation in the magnetic field. It can be approximately calculated as γ max ′ ≃ 10 8 ( B ′ 1 G ) 0.5 $ \gamma^{{\prime}}_{\mathrm{max}} \simeq 10^8(\frac{B{{\prime}}}{1 \mathrm{G}})^{0.5} $, where B′ is the strength of the magnetic field in the comoving frame (Dai & Lu 1999; Huang et al. 2000b). The term ν max ′ $ \nu^\prime_{\rm max} $ is then connected with γ max ′ $ \gamma^\prime_{\rm max} $ as ν ′ max = Γ γ max ′ 2 e B ′ / 2 π m e c $ {\nu{{\prime}}_{\text{max}}}= \Gamma \gamma_{\mathrm{max}}^{\prime 2} e B{{\prime}} / 2 \pi m_{\mathrm{e}} c $, where e is the electron charge and me is the mass of electrons.

In the slow-cooling regime, when ν min ′ $ \nu^\prime_{\rm min} $ <  ν c ′ $ \nu^\prime_{\rm c} $ <  ν max ′ $ \nu^\prime_{\rm max} $, we have

F ν ′ = F ν ′ , max { ( ν ′ ν min ′ ) 1 3 , ν ′ < ν min ′ , ( ν ′ ν min ′ ) 1 − p 2 , ν min ′ < ν ′ < ν ν c ′ , ( ν ′ ν ν c ′ ) − p 2 ( ν ν c ′ ν min ′ ) 1 − p 2 , ν ν c ′ < ν ′ < ν ′ max . $$ \begin{aligned} F_{\nu^{\prime }} = F_{\nu^{\prime },\mathrm{max}}\left\{ \begin{array}{ll} (\frac{\nu^{\prime }}{\nu^{\prime }_{\mathrm{min}}})^{\frac{1}{3}},&\nu^{\prime }< \nu^{\prime }_{\mathrm{min}}, \\ (\frac{\nu^{\prime }}{\nu^{\prime }_{\mathrm{min}}})^{\frac{1-p}{2}},&\nu^{\prime }_{\mathrm{min}}< \nu^{\prime }< \nu^{\prime }_{\mathrm{c}}, \\ (\frac{\nu^{\prime }}{\nu^{\prime }_{\mathrm{c}}})^{-\frac{p}{2}}(\frac{\nu^{\prime }_{\mathrm{c}}}{\nu^{\prime }_{\mathrm{min}}})^{\frac{1-p}{2}},&\nu^{\prime }_{\mathrm{c}}< \nu^{\prime }< \nu^{\prime }_{\mathrm{max}}. \end{array} \right. \end{aligned} $$(5)

In the case of ν max ′ $ \nu^\prime_{\rm max} $ <  ν c ′ $ \nu^\prime_{\rm c} $, we have

F ν ′ = F ν ′ , max { ( ν ′ ν min ′ ) 1 3 , ν ′ < ν min ′ , ( ν ′ ν min ′ ) 1 − p 2 , ν min ′ < ν ′ < ν ′ max . $$ \begin{aligned} F_{\nu^{\prime }} = F_{\nu^{\prime },\mathrm{max}}\left\{ \begin{array}{ll} (\frac{\nu^{\prime }}{\nu^{\prime }_{\mathrm{min}}})^{\frac{1}{3}},&\nu^{\prime }< \nu^{\prime }_{\mathrm{min}}, \\ (\frac{\nu^{\prime }}{\nu^{\prime }_{\mathrm{min}}})^{\frac{1-p}{2}},&\nu^{\prime }_{\mathrm{min}}< \nu^{\prime }< \nu^{\prime }_{\mathrm{max}}. \end{array} \right. \end{aligned} $$(6)

We use the Compton parameter Y to denote the ratio of the inverse Compton scattering luminosity with respect to the synchrotron luminosity. It can be calculated as

Y = − 1 + 1 + 4 ϵ r , e ϵ e ϵ B 2 · $$ \begin{aligned} Y = \frac{-1+\sqrt{1+4\epsilon _{\mathrm{r,e}} \epsilon _{\mathrm{e}} \epsilon _{B} }}{2}\cdot \end{aligned} $$(7)

Here, ϵr, e is the fraction of the electron energy that was radiated, while ϵe is the fraction of thermal energy carried by electrons, and ϵB is the ratio of magnetic field energy to the total energy (Sari & Esin 2001; Wei et al. 2006).

The effect of synchrotron self-absorption is also considered in our calculations. As a result, the observed flux should be corrected by multiplying a factor of f(τν′) = (1 − e−τν′)/τν′, where τν′ is the optical depth. The self-absorption coefficients and the optical depth are calculated by following Wu et al. (2003) and Geng et al. (2016).

The observed flux density at frequency ν can be calculated as

F ν ( Θ ) = ( 1 + z ) D 3 f ( τ ν ′ ) F ν ′ , $$ \begin{aligned} F_{\nu }(\Theta ) = (1+z)\mathcal{D} ^{3} f(\tau _{\nu ^{\prime }}) F_{\nu^{\prime }}, \end{aligned} $$(8)

where Θ stands for the angle between the velocity of emitting material and the line of sight. The Doppler factor is shown with 𝒟 = 1/[Γ(1 − βcosΘ)] and ν′ = (1 + z)ν/𝒟. To calculate the observed flux Fν(t) at a given time t, we integrated the emission power over the equal arrival time surface (EATS), which is determined by

t = ( 1 + z ) ∫ 1 − β cos Θ β c d R · $$ \begin{aligned} t=(1+z)\int \frac{1-\beta \cos \Theta }{\beta c}\mathrm{d}R\cdot \end{aligned} $$(9)

4. Multi-wavelength fitting of the afterglow

In this section, we present multi-wavelength fittings of AT2021any considering two different models. First, we use a simple top-hat jet model, which has a constant energy per solid angle and a uniform Lorentz factor within the range of the jet. Second, we consider a structured jet model with a Gaussian profile (Kumar & Granot 2003; Troja et al. 2018; Geng et al. 2019; Lamb et al. 2021). In this case, the distribution of the kinetic energy is taken as E(θ)= E k,iso exp(− θ 2 / θ c 2 ) $ E(\theta)= {E_{{\rm k,iso}}} \exp(-\theta^{2}/{\theta^{2}_{{\rm c}}}) $ at angle θ, and the profile of the Lorentz factor is assumed to take the form of Γ(θ)=( Γ 0 −1)exp(− θ 2 /2 θ c 2 )+1 $ \Gamma(\theta)=(\Gamma_0 - 1)\exp(-\theta^{2}/2{\theta^{2}_{{\rm c}}}) + 1 $. Here, Ek,iso is the isotropic equivalent energy on the jet axis (θ = 0), and Γ0 is the corresponding Lorentz factor. The term θc stands for the half-opening angle of the jet core. For convenience, we assumed that the jet is cut off at an angle of θj, that is, E(θ) = 0 and Γ(θ) = 1 for θ > θj. In other words, θj effectively denotes the edge of the structured jet.

The observed broadband X-ray flux data were used to derive the flux at the frequency of ν = 1 × 1018 Hz. To do so, a photon index of Γp = 2 was applied (Ho et al. 2022). In the optical bands, we considered a Galactic extinction of E(B − V) = 0.0575 mag (Schlafly & Finkbeiner 2011) to convert the observed magnitudes to flux densities. The radio afterglow data in four different bands were also used in our fitting, namely, the S(3 GHz), C(6 GHz), X(10 GHz), and Ku(15 GHz) bands.

We used the Markov chain Monte Carlo (MCMC) algorithm to get the best-fit results for the multi-wavelength afterglow of AT2021any. We introduced a new parameter, the shift time of the light curve (ts), in order to find the most probable trigger time of the GRB associated with AT2021any. Here, ts is defined as the interval between the GRB trigger time and the beginning of the first observation. For simplicity, we considered a constant ISM density.

4.1. The top-hat jet model

We began our fitting with the top-hat jet model. The numerical results are presented in Table 4, and the corresponding corner plot is shown in Fig. 1. The multi-wavelength observational data points and the best-fit light curves are plotted in Fig. 2. The best-fit value for the initial Lorentz factor of the jet (Γ0) is ∼83, which favors a failed GRB origin. We note that the Lorentz factor is mainly sensitive to the early afterglow, especially its onset. In the case of AT2021any, although the onset of the afterglow was not clearly detected, the most recent non-detection was luckily only 20.3 min before the first detection, and it gives a firm constraint on the onset of the afterglow. This is the reason that the Lorentz factor can be effectively inferred from the observations.

thumbnail Fig. 1.

Parameters derived for AT2021any by using the top-hat jet model. The contour curves show the 1σ − 2σ − 3σ confidence levels. The best-fit parameter values are marked with 1σ uncertainties above the panel of their posterior distribution.

Table 4.

Best-fit results for the top-hat jet model and structured jet model and their corresponding goodness of fit.

From Fig. 1, we observed that the X-ray and optical data are generally well fitted. However, the radio data seem to somewhat deviate from the theoretical light curves. We note that the effect of interstellar scintillation is not included in our calculations. Small-scale inhomogeneities in the ISM can cause scintillation by changing the phase of radio waves. The line of sight to a distant source also shifts as the Earth moves, leading to fluctuations in the radio flux. The scattering effect would also be significant when the radio frequency is smaller than the transition frequency. In fact, Ho et al. (2022) pointed out that interstellar scintillation may have a significant contribution to the radio afterglow of AT2021any. They calculated the transition frequency at the direction of AT2021any and got a result of 15 GHz by using the NE2001 model (Cordes & Lazio 2002). Therefore, the radio light curves we considered here could be largely affected by interstellar scintillation.

Figure 2 shows that the optical light cures have an obvious break at about tb ∼ 0.4 days. Before this time, the temporal index is about αopt, 1 = −0.75 ± 0.1, while it becomes αopt, 2 = −1.33 ± 0.23 after the break time. We note that αopt, 2 is satisfactorily consistent with the result expected for the fireball model (Sari et al. 1998), that is, −(3p − 2)/4 = −1.3 for p ∼ 2.4 from our best fit. It indicates that the optical band has crossed the cooling frequency at around 0.4 days. The theoretical temporal index before the optical band crosses the cooling frequency should be −3(p − 1)/4 = −1.05. It is smaller than the observed value of αopt, 1 = −0.75 mentioned above. The difference may be due to the EATS effect, which is more significant in the early stages of the afterglow.

thumbnail Fig. 2.

Observed multi-wavelength afterglow of AT2021any and best-fit results when using the top-hat jet model (solid curves).

In Fig. 2, the half-opening angle of the jet is 0 . 08 − 0.01 + 0.01 $ 0.08^{+0.01}_{-0.01} $, while the viewing angle is 0 . 03 − 0.01 + 0.01 $ 0.03^{+0.01}_{-0.01} $. It indicates that the jet was essentially observed on the axis. An achromatic jet break can be seen in the optical light curves, and the temporal decay index is around −2 after the jet break. The break time is about 2 days, which is also roughly consistent with the theoretical jet break time of t j ∼ 1.2 ( 1 + z ) 2 E k , iso , 53 1 / 3 n − 1 / 3 θ j , − 1 8 / 3 = 1.96 $ t_{\mathrm{j}} \sim 1.2\frac{(1+z)}{2} E_{\mathrm{k,iso}, 53}^{1/3} n^{-1/3} \theta_{\mathrm{j}, -1}^{8/3} = 1.96 $ days (Sari et al. 1999).

We note that the decay index of our theoretical X-ray light curve is around 1.3 between 0.01 and 1 day. It is obviously steeper than the optical light curves in the same period. This is easy to understand, as the frequencies of X-ray photons are much higher than that of optical photons. As a result, it is in the fast-cooling regime (i.e., the frequency is higher than the characteristic cooling frequency). Therefore, the theoretical decay index should be ∼ − (3p − 2)/4 = 1.3 (Sari et al. 1998), which is well consistent with our numerical results.

The SSC might have some effects on the X-ray afterglow, and we present some further discussions on this issue here. The effect of SSC can be assessed by the Compton parameter Y, which is defined as the ratio of the inverse Compton scattering luminosity with respect to the synchrotron luminosity. As shown in Eq. (7), Y is sensitively dependent on ϵr, e, that is, the fraction of the radiated electron energy. According to Sari & Esin (2001), ϵr, e takes the form of ϵ r , e = ( γ min γ c ) p − 2 $ \epsilon_{\mathrm{r,e}} = (\frac{\gamma_{\mathrm{min}}}{\gamma_{\mathrm{c}}})^{p-2} $, considering that the bulk of electrons are in the slow-cooling regime at the afterglow phase. It can be further expressed as ϵ r , e = ( 1.27 × 10 − 8 p − 2 p − 1 ϵ e ϵ B Γ 4 t ) p − 2 $ \epsilon_{\mathrm{r,e}} = (1.27 \times 10^{-8} \frac{p-2}{p-1} \epsilon_{\mathrm{e}} \epsilon_{B} \Gamma^{4} t)^{p-2} $ (Huang et al. 2000b). Taking Γ ∝ t−3/8 for the adiabatic expansion case (Sari et al. 1998) and combining the best-fit parameters for the top-hat jet discussed here, we obtain ϵr, e ∼ 0.007t−0.2. Consequently, we get Y = − 1 + 1 + 4.76 t − 0.2 2 $ Y = \frac{-1+\sqrt{1+4.76 t^{-0.2}}}{2} $ from Eq. (7). One can see that the Compton parameter decreases from Y = 0.35 at t = 100 s to Y = 0.16 at t = 10 000 s. Therefore, SSC will generally have a negligible effect on the afterglow of AT2021any.

The radio light curve peaked at about 20 days. The pre-peak temporal index of the radio light curve is about one-half, indicating that the radio emission will reach the peak flux when the observed radio frequency crosses the characteristic frequency of νmin (Zhang 2018). The post-peak radio light curve is dominated by the jet break effect. Here, we further address the effect of synchrotron self-absorption, which is mainly determined by the synchrotron self-absorption frequency (νa). Following Gao et al. (2013), we derived the self-absorption frequency as νa = 1.03 × 109 Hz in the case of νa < νmin < νc. However, in the case of νmin < νa < νc, it is νa = 8.3 × 1012(t/1s)−0.72 Hz. In both cases, we observed that νa is in the radio ranges. Thus, the synchrotron self-absorption mainly affects the radio afterglow light curves and has a negligible effect on the optical and X-ray light curves.

4.2. The structured jet model

We also modeled the observational data with a structured jet. The best-fit results are presented in Table 4, and the corresponding corner plot is shown in Fig. 3. We observed that the best-fit Γ0 is ∼68, which is likely too low to prevent the e± pair formation in the prompt γ-ray phase. The geometry parameters were derived as θc = 0.10 ± 0.01, θ j = 0 . 76 − 0.46 + 0.50 $ {\theta_{\text{j}}}=0.76^{+0.50}_{-0.46} $, and θ obs = 0 . 02 − 0.002 + 0.003 $ {\theta_{\text{obs}}}=0.02^{+0.003}_{-0.002} $. An on-axis viewing angle was still favored here. Figure 3 shows that the error bar of θj is relatively large. This is due to the fact that the radiation from materials outside the jet core contributes little to the observed emissions, which means that the observed flux is insensitive to θj. In fact, at the early stage, we could only see a small fraction of the jet due to the beaming effect. As the jet decelerates, the Lorentz factor decreases, and we were able to see a larger area of the jet. However, the Lorentz factor of the materials outside the jet core are too small to produce significant emissions at later stages, leading to a steep decay in the afterglow light curve. The parameter set of Ek,iso ∼ 5.50 × 1052 erg, p ∼ 2.3, ϵe ∼ 0.17, and ϵB ∼ 0.001 were derived for the structured jet model. These parameter values are similar to those of the top-hat jet model. As for the ambient density n, the structured jet model requires a relatively larger value of n ∼ 0.87 cm−3 as compared to n ∼ 0.16 cm−3 for the top-hat jet model, indicating that a cleaner circum-burst environment is needed for the top-hat jet model. In the framework of the structured jet model, the best-fit shift time is about ts ∼ 1000 s, which is approximately 200 s smaller than that obtained for the top-hat jet model.

thumbnail Fig. 3.

Parameters derived for AT2021any by using the structured jet model. The contour curves show the 1σ − 2σ − 3σ confidence levels. The best-fit parameter values are marked with 1σ uncertainties above the panel of their posterior distribution.

We compared the observational data points with the best-fit light curves of the structured jet model in Fig. 4. Similar to Fig. 2, we observed that the radio data points show significant fluctuations and thus could not be satisfactorily fit by the theoretical light curves (especially in the C band). Again, it may be due to the interstellar scintillation effect. The theoretical r-band optical light curve still possesses a shallow decay, with a timing index of –0.69 ± 0.01 before tb ∼ 0.35 days. This index is slightly larger than the analytical value of −3(p − 1)/4 = −0.975 and may be due to the EATS effect. After tb ∼ 0.35 days, the timing index is −1.33, which is roughly consistent with the analytical result of ∼ − (3p − 2)/4 = −1.225. The jet break time is about 2 days for the structured jet model.

thumbnail Fig. 4.

Observed multi-wavelength afterglow of AT2021any and best-fit results when using the structured jet model (solid curves).

4.3. Comparing the goodness of fit

We assessed the goodness of fit for different models. Two tests were performed for this purpose. First, we used the reduced χ2, which is calculated as

χ 2 / d . o . f . = ∑ i ( log f th , i − log f obs , i ) 2 σ i 2 × 1 d . o . f . , $$ \begin{aligned} \chi ^{2}/\mathrm{d.o.f.}=\sum _i \frac{ (\log f_{\mathrm{th},i} - \log f_{\mathrm{obs},i})^{2} }{\sigma _i^2} \times \frac{1}{ \mathrm{d.o.f.} }, \end{aligned} $$(10)

where the degree of freedom (d.o.f.) is defined as the difference between the number of observational data points and the model parameters. The terms fth, i and fobs, i are respectively the theoretical flux density and the observed flux density at the time of ti, while σi represents the error bar of each data point. The reduced χ2 of each model is presented in Table 4. We found that the structured jet model has a relatively lower reduced χ2, suggesting that it is the preferred model.

The second test was conducted with the Bayesian Information Criterion (BIC) method (Schwarz 1978). The criterion of BIC is defined as

BIC = − 2 ln L ( P ) + k ln ( N ) , $$ \begin{aligned} \mathrm{BIC} = -2\ln \mathcal{L} (P) + k \ln (N), \end{aligned} $$(11)

where N is the number of observational data points and k is the number of model parameters. Here, P stands for a set of the model parameters, and ℒ is the maximized value of the likelihood function. The likelihood function takes the form of Xu et al. (2021)

L ( P ) = ∏ i 1 2 π σ i exp [ − 1 2 ( log f th , i − log f obs , i ) 2 σ i 2 ] · $$ \begin{aligned} \mathcal{L} (P)=\prod _{i} \frac{1}{\sqrt{2\pi }\sigma _i} \exp \left[-\frac{1}{2} \frac{ (\log f_{\mathrm{th},i} - \log f_{\mathrm{obs},i})^{2} }{\sigma _i^2} \right]\cdot \end{aligned} $$(12)

According to the BIC test, the model that provides the minimum BIC score should be the preferred model. Usually, the BIC score is compared through ΔBIC values (i.e., the difference between the best model and other models). We list the ΔBIC score for each model in Table 4. Again, we observed that the structured jet scenario is better than the top-hat jet scenario.

Radio afterglows are largely affected by interstellar scintillation. The random fluctuation of radio flux may affect the goodness of fit. To avoid the uncertainties caused by this factor, we performed the model fitting by excluding all the radio data. The parameters derived are also presented in Table 4. Figures 5 and 6 show the best-fit light curves for the top-hat and structured jet models, respectively. For the top-hat jet model, the geometry parameters differ significantly from the previous results, that is, when the radio data were included. In this case, the best-fit angles are θ j = 0 . 17 − 0.04 + 0.03 $ {\theta_{\text{j}}}= 0.17^{+0.03}_{-0.04} $ and θ obs = 0 . 12 − 0.04 + 0.04 $ {\theta_{\text{obs}}}= 0.12^{+0.04}_{-0.04} $, and are relatively larger. Still, an on-axis scenario is favored. Other parameters, such as Γ0, Ek,iso, p, n, ϵe, ϵB, and ts, do not change too much. As for the structured jet model, the major difference induced by excluding the radio data concerns the shift time. We obtained a new shift time, ts ∼ 750 s, which is 250 s smaller than the previous value derived when including the radio data. Apart from this difference, the best-fit values for other parameters are essentially similar. Finally, based on both the reduced χ2 test and the BIC test, the structured jet model is still preferred after excluding the radio data; thus, the main conclusion remains unchanged.

thumbnail Fig. 5.

Observed optical and X-ray afterglow of AT2021any, and best-fit results when using the top-hat jet model. We note that the radio afterglow data are not included in this plot since they were interfered by interstellar scintillation (see Sect. 4.3).

thumbnail Fig. 6.

Observed optical and X-ray afterglow of AT2021any, and best-fit results when using the structured jet model. We note that the radio afterglow data are not included in this plot since they were interfered by interstellar scintillation (see Sect. 4.3).

To summarize, according to the above two tests, AT2021any is best described by the structured jet model. This conclusion is supported no matter whether the radio data, which are affected by interstellar scintillation, are included or excluded. Additionally, our results suggest that AT2021any should be an on-axis failed GRB that has an initial Lorentz factor significantly smaller than normal GRBs.

5. Conclusions and discussion

Orphan afterglows are fascinating transients that could provide useful information on the triggering mechanisms of GRBs. By fitting the multi-wavelength afterglow data, we can gain an in-depth understanding of the nature of orphan afterglows. In this study, we applied two different kinds of outflows to the orphan afterglow candidate AT2021any: a top-hat jet and a structured Gaussian jet. We find that the structured Gaussian jet model presents the best fit to the multi-wavelength light curves of AT2021any. According to our modeling, the trigger time of the GRB associated with AT2021any is about 1000 s prior to the first detection.

In the framework of the structured Gaussian jet model, the isotropic kinetic energy of AT2021any was derived as 5.50 × 1052 erg. From this kinetic energy, we could estimate the γ-ray efficiency η of the unseen GRB associated with it. The source was in the field of view of Fermi-GBM but was undetected by the instrument (Ho et al. 2022), which places a firm upper limit on the peak γ-ray flux as ∼1 × 107 erg s−1 cm−2. Consequently, the corresponding upper limit of the peak luminosity is Lp ≲ 5.31 × 1051 erg s−1 for a redshift of z = 2.513. The upper limit of the isotropic γ-ray energy is Eγ, iso = Lp * T90/(1 + z)≲1.51 × 1052 erg for a typical burst duration of T90 = 10 s (for long GRBs). So, the γ-ray efficiency is η = Eγ, iso/(Eγ, iso + Ek,iso)≲21.5%, which is roughly consistent with the result derived by Gupta et al. (2022) (η ≲ 28.6%). We note that the upper limit of the γ-ray efficiency derived here is typical for long GRBs. Some long GRBs could have much higher γ-ray efficiencies, but these can still be explained by the photosphere model (Rees & Mészáros 2005; Pe’er 2008) or the internal-collision-induced magnetic reconnection and turbulence (ICMART) model (Zhang & Yan 2011; Zhang & Zhang 2014).

A relativistic fireball with a lower initial Lorentz factor is usually optically thick at the internal shock radius, but it becomes optically thin at the external shock radius. In other words, the photosphere radius of a dirty fireball is much larger than the internal shock radius and is much smaller than the external shock radius. In the case of AT2021any, we have Γ0 ∼ 68, Ek,iso ∼ 5.50 × 1052 erg, and n ∼ 0.87 cm−3 as derived from the structured Gaussian jet model. For two mini shells ejected with a separation time of δt ∼ 10−3 s, the internal shock radius can be obtained as R IS ∼ 2 Γ 0 2 c δ t = 2.69 × 10 11 $ R_{\mathrm{IS}} \sim 2\Gamma_{0}^{2}c\delta t = 2.69 \times 10^{11} $ cm (Rees & Mészáros 1994). The optical depth is dominated by electron scattering for Γ0 < 105 (Mészáros et al. 1993), which can be calculated as τ T = E k,iso σ T 8 π N sh R δ R c 2 m p Γ 0 3 $ \tau_{\mathrm{T}} = \frac{{E_{\text{k,iso}}}\sigma_{\mathrm{T}}}{8\pi N_{\mathrm{sh}} R \delta R c^{2} m_{\mathrm{p}} \Gamma_{0}^{3}} $ (Mészáros & Rees 2000; Zhang 2018). Here, σT is the Thomson scattering cross section of electrons, Nsh stands for the number of all the mini shells, and δR ∼ 3 × 107 cm is their typical width. Taking the burst duration as T90 = 10 s and assuming Nsh ∼ T90/δt = 104, we can obtain the photosphere radius as R ph = E k,iso σ T 8 π N sh δ R c 2 m p Γ 0 3 = 1.09 × 10 13 $ R_{\mathrm{ph}} = \frac{{E_{\text{k,iso}}}\sigma_{\mathrm{T}}}{8\pi N_{\mathrm{sh}} \delta R c^{2} m_{\mathrm{p}} \Gamma_{0}^{3}} = 1.09 \times 10^{13} $ cm for τT = 1. At the same time, the external shock radius is R ext = ( 3 E k,iso 2 π n m p c 2 Γ 0 2 ) 1 / 3 = 1.65 × 10 17 $ R_{\mathrm{ext}} = \left(\frac{ 3 {E_{\text{k,iso}}}}{2\pi n m_{\mathrm{p}} c^{2} \Gamma_{0}^{2}} \right)^{1/3} = 1.65 \times 10^{17} $ cm (Rees & Mészáros 1992; Zhang 2018). Comparing these radii, we found that they satisfy RIS < Rph < Rext, indicating that for AT2021any, the synchrotron radiation of γ-rays in the prompt emission phase is invisible to us due to the optically thick condition, which is consistent with observational constraints. However, the emission in the afterglow phase could be observed since it is optically thin at late stages.

Our model does not include the impact of the reverse shock. Usually, an optical orphan afterglow would be found at the early stage of the burst, when the smoking gun is still relatively bright. At this stage, the optical emission of the afterglow may be affected by the reverse shock (Wu et al. 2003; Wang & Dai 2013). We note that the reverse shock component may help better constrain the physical parameters of an orphan afterglow. Wang & Dai (2013) engaged the reverse shock emission from a post-merger millisecond magnetar to explain the light curves of PTF11agg, another orphan afterglow candidate. They argued that the multi-wavelength light curves can be better fitted by adding a reverse shock component. Nevertheless, in the case of AT2021any, no clear evidence supporting the existence of a reverse shock was spotted. The reason may be that the first detection is about 1000 s after the trigger, as derived from our best-fit result, and the reverse shock is expected to take effect tens of seconds after the burst.

The lack of information on the host galaxy extinction is another factor that may affect the goodness of the multi-wavelength fitting. Sarin et al. (2022) added two additional parameters in their study of the orphan afterglow candidate AT2020blt in order to account for the host galaxy extinction. Those parameters are both related to the hydrogen column density of the host galaxy (Güver & Özel 2009). On the other hand, overestimating the host galaxy extinction could distort the light curve, so the extinction of the host galaxy is a tricky problem in multi-wavelength fitting. Thus, it needs to be considered cautiously.

Acknowledgments

We would like to thank the anonymous referee for helpful suggestions that lead to an overall improvement of the presentation. This study is supported by National Key R&D Program of China (2021YFA0718500), by National SKA Program of China No. 2020SKA0120300, by the National Natural Science Foundation of China (Grant Nos. 12233002, 12041306, 12273113, 12321003), by the Youth Innovations and Talents Project of Shandong Provincial Colleges and Universities (Grant No. 201909118), and by the Youth Innovation Promotion Association (2023331).

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All Tables

Table 1.

Optical photometry of AT2021any.

In the text
Table 2.

X-ray afterglow data of AT2021any.

In the text
Table 3.

Radio afterglow data of AT2021any.

In the text
Table 4.

Best-fit results for the top-hat jet model and structured jet model and their corresponding goodness of fit.

In the text

All Figures

thumbnail Fig. 1.

Parameters derived for AT2021any by using the top-hat jet model. The contour curves show the 1σ − 2σ − 3σ confidence levels. The best-fit parameter values are marked with 1σ uncertainties above the panel of their posterior distribution.
In the text
thumbnail Fig. 2.

Observed multi-wavelength afterglow of AT2021any and best-fit results when using the top-hat jet model (solid curves).
In the text
thumbnail Fig. 3.

Parameters derived for AT2021any by using the structured jet model. The contour curves show the 1σ − 2σ − 3σ confidence levels. The best-fit parameter values are marked with 1σ uncertainties above the panel of their posterior distribution.
In the text
thumbnail Fig. 4.

Observed multi-wavelength afterglow of AT2021any and best-fit results when using the structured jet model (solid curves).
In the text
thumbnail Fig. 5.

Observed optical and X-ray afterglow of AT2021any, and best-fit results when using the top-hat jet model. We note that the radio afterglow data are not included in this plot since they were interfered by interstellar scintillation (see Sect. 4.3).
In the text
thumbnail Fig. 6.

Observed optical and X-ray afterglow of AT2021any, and best-fit results when using the structured jet model. We note that the radio afterglow data are not included in this plot since they were interfered by interstellar scintillation (see Sect. 4.3).
In the text

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