THE LATE PEAKING AFTERGLOW OF GRB 100418A

Figure 1 shows the BAT light curves. The prompt emission consists of two main overlapping peaks. The lack of obvious emission above 100 keV indicates the soft nature of the burst. The T₉₀ duration is 8 ± 2 s in the 15–150 keV band. The time-integrated spectrum during the prompt phase is best fit by a single power law with the photon index Γ = 2.16 ± 0.25. There is no significant improvement in χ² for the fit for a cutoff power-law model over a simple power-law model. The 15–150 keV fluence is (3.4 ± 0.5) × 10⁻⁷ erg cm⁻². The ratio of the fluence in the 25–50 keV band to the fluence in the 50–100 keV band is 1.12 ± 0.29, which indicates that GRB 100418A is probably a member of the X-ray rich class of GRBs (Sakamoto et al. 2008). The 1 s peak flux is (5.1 ± 1.5) × 10⁻⁸ erg cm⁻² s⁻¹ in the 15–150 keV band.

Zoom In Zoom Out Reset image size

Although E_peak cannot be measured directly from the observed BAT spectrum, E_peak is very likely to be located at the lower energy range of BAT since the BAT photon index of 2.16 is very close to the typical high-energy photon index of the Band function (Kaneko et al. 2006). Using the E_peak–Γ relation (Sakamoto et al. 2009) and the BAT photon index of 2.16, we find E_peak is 29⁺²₋₂₇ (1σ) keV.

To estimate the isotropic gamma-ray energy release E_γ,iso, we fit the BAT-time-integrated spectrum with the Band function (Band et al. 1993) by fixing the low-energy photon index to 1.0, and then calculated E_γ,iso in the 1 keV–10 MeV band (cosmological rest frame). The best estimate of E_γ,iso is 9.9^+6.3_−3.4 × 10⁵⁰ erg. We applied the same approach to estimate the 1 s peak luminosity L_γ,iso using the BAT 1 s peak spectrum. Our estimated L_γ,iso is 2.1^+1.1_−0.6 × 10⁵⁰ erg s⁻¹ (in the 1 keV–10 MeV band in the rest frame).

The XRT started observations of GRB 100418A 71 s after the BAT trigger, collecting data in Windowed Timing (WT) settling mode while the spacecraft was slewing to the burst location. Swift settled on the burst 79 s after the trigger and the XRT autonomously localized the afterglow on-board taking in the image of the field. The astrometrically corrected X-ray position (Goad et al. 2007; Evans et al. 2009) of R.A., decl. = 256.36313, +11.46140, equivalent to α = 17^h05^m2715, δ = +11°27'411 (equinox 2000.0) with an estimated uncertainty of 19 (radius, 90% confidence including systematic error), updated from the position reported by Osborne et al. (2010), was derived from the full, on-ground data set, using the XRT–UVOT alignment and matching UVOT field sources to the USNO–B1 catalog. Settled WT mode data were collected from T₀ + 86 s to T₀ + 174 s; subsequently, the XRT switched into photon counting (PC) mode for the rest of the observations, from T₀ + 182 s to T₀ + 3 Ms, for a total observing time of 231.8 ks.

The XRT data were reduced with the standard software (xrtpipeline v0.12.4) applying the default filtering and screening criteria (HEASOFT 6.8, Swift tools 3.5), using the latest CALDB 3.6 files released in 2010 June.

The X-ray light curve in the 0.3–10 keV energy band was extracted using the methods described in Evans et al. (2007, 2009). The overall light curve can be modeled by a double-broken power law, as shown in Figure 2. The early steep phase is fitted with a decay index α = 4.18^+0.16_−0.14. The end of the steep decay at approximately T₀ +  600 s is followed by a shallow phase that bears resemblance to the afterglow brightening observed by the UVOT. If one assumes that the observed count rate behavior is not due to superimposed X-ray flares, this afterglow phase can be fitted with a power law of index α = −0.23^+0.13_−0.14 out to approximately T₀ + 66 ks. The late afterglow decays with an index of α = 1.37^+0.13_−0.11. The double-broken power-law fit is statistically marginally acceptable, with a χ² of 100.69 for 66 degrees of freedom (χ²_red of 1.52), an indication of the complexity of the X-ray emission likely due to the presence of mini-flares. A broken power-law fit of the light curve, with the exclusion of the early steep decay is statistically more acceptable (χ² of 38.82 for 25 degrees of freedom) and gives fit parameters consistent with the reported values of the double-broken power law.

Zoom In Zoom Out Reset image size

We have extracted time-resolved spectra of the X-ray afterglow in the energy range 0.3–10 keV, performing the analysis using xspec v12.5.1n. The spectrum from the WT mode data, selecting events with grades 0–2 and binned with 20 counts per bin was fitted with an absorbed power law (using the X-ray absorption models PHABS and ZPHABS), fixing the Galactic absorption to the value of 4.8 ×10²⁰ cm⁻² (Kalberla et al. 2005). The best fit yields a steep photon index of 4.33^+0.28_−0.25 and an intrinsic N_H of 3.3 ± 0.6 × 10²¹ cm⁻² at a redshift of 0.6235, in excess of the Galactic value, with a χ² of 47.5 (42 degrees of freedom). The PC spectrum during the plateau phase, binned to one count per bin and using Cash statistics (Cash 1979) can be fitted with an absorbed power law with a photon spectral index of 2.09^+0.25_−0.34 and an intrinsic absorption of 2.0^+1.6_−1.8 × 10²¹ cm⁻².

To investigate the possibility of spectral evolution in the WT mode observations, we extracted an early WT spectrum from T₀ + 84 s to T₀ + 103 s and a late spectrum from T₀ + 103 s to T₀ + 174 s. The absorbed power-law fits to the spectra indicated a spectral softening during the WT steep decay, yielding photon indices of 4.03 ± 0.15 and 4.63 ± 0.16 for the early and late spectra, respectively, with the intrinsic absorption tied to the best-fit value of the combined data. The evolution of the photon index Γ, interpolated from the hardness ratio measurements as described in Evans et al. (2010), is shown in the bottom panel of Figure 2. A likely explanation for the measured early softening is the passage of E_peak through the XRT energy band, as discussed by Butler & Kocevski (2007) and suggested for this burst by the low E_peak value derived from the BAT data. Alternatively, the difference in photon indices between the WT and the PC fits and the soft WT spectrum could be due to the presence of a blackbody component during the WT observations. The best fit to the entire WT mode data set with an absorbed power law plus blackbody model yielded a photon index of 2.46^+1.05_−1.12, a temperature of 0.12 ± 0.02 keV, and an intrinsic N_H of 1.2^+1.5_−1.1 × 10²¹ cm⁻², with a χ² of 32.5 (40 dof) and a ratio of the flux of the blackbody component over the total flux in the 0.3–10 keV energy band equal to 0.625. The improved χ² is suggestive of a curved spectrum, but the χ²_red of 0.81 is indicative of an issue of overfitting of the data with a spectral model with too many free parameters. We report the best-fit results with the power law and power law plus blackbody models in Table 1.

Finally, we extracted three spectra from the PC data at specific times of the afterglow emission, during the early plateau phase around T₀ +  6 ks, at the peak of the UVOT light curve and during the late decay to produce broadband spectral energy distribution (SED) in combination with UVOT observations. The results of the combined fits are presented in Section 2.4.

The HEASOFT¹⁴ (v6.9) and Swift software (v3.6, Build 25) tools and the latest calibration products were used to analyze the UVOT data. The photometry was performed following the methods described by Poole et al. (2008).

The best Swift position for the afterglow was determined using the 1746 s exposure that started at T₀ + 51217 s, when the afterglow was near its peak intensity. The tool uvotdetect determined a position of α = 17^h05^m2711, δ = +11°27'425 (equinox 2000.0) with an estimated uncertainty of 043 (90% containment including systematic error). The uncertainty is dominated by systematics (Breeveld et al. 2010). This position was used for all UVOT analysis reported in this paper. It supercedes the preliminary UVOT position reported by Siegel & Marshall (2010).

UVOT photometry was performed using a 3'' radius aperture centered on the above position. A nearby source-free region centered at α = 17^h05^m2563, δ = +11°27'059 with a 9'' radius was used for background. The standard aperture correction was used to correct for source photons falling outside the aperture. The tool uvotproduct, which rebins data to produce statistically significant data points when needed, was used to produce the light curve for the white filter reported in Table 2 and plotted in Figure 3. The count rate is approximately constant out to 10 ks, but observations after T₀ + 50 ks found a brighter, decaying source. For an assumed power-law decay model for the data before T₀ + 10 ks, the best-fit decay index is −0.15^+0.21_−0.27. A two-component model with a constant plus a power-law decay was fit to the later data as shown in Figure 3. The best-fit decay index was 1.12 ± 0.10, and the best-fit constant was 0.106 ± 0.024 counts s⁻¹ (m_white = 22.7^+0.3_−0.2). The constant component is most likely due to the host galaxy (Malesani 2010). Figure 4 shows the count rate in the white filter, the XRT count rate, and their ratio.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

UVOT used the white filter for the vast majority of the observing time because, for sources with low extinction such as GRB 100418A, it is by far the most sensitive. Low-resolution UVOT spectra using the six filters were obtained for the only two intervals when there were statistically significant detections with multiple filters. All six filters were also used for observations near T₀ +  1 ks, but the exposures were short and only upper limits and marginal detections were obtained. The UVOT spectra were then combined with the contemporaneous XRT spectra to produce broadband SEDs (Figure 5). In addition, results from the UVOT white filter were combined with the XRT spectrum during the interval when the afterglow was the brightest (T₀ + 51 ks to T₀ + 92 ks). The UVOT data points were produced with the tool uvot2pha on summed images using the same source and background regions as for the UVOT light curves. The spectral fits were done using the tool xspec and the latest UVOT spectral response files (version 104) in CALDB. Table 3 provides a summary of the spectral data.

Zoom In Zoom Out Reset image size

As described above, UVOT and XRT spectra were accumulated for three separate time intervals (Table 3) and then fit with xspec. Because of the limited statistical significance of the individual spectral data points, only relatively simple spectral models were used. Each model assumed a power-law continuum with absorption and extinction in both the Milky Way and the host galaxy. The equivalent hydrogen column density in the Milky Way was always set to 4.8 × 10²⁰ cm⁻² (Kalberla et al. 2005), and the extinction in the Milky Way was based on a reddening E_B-V = 0.07 mag (Schlegel et al. 1998). The xspec models zphabs and zdust were used to model absorption and extinction in the host galaxy. A Small Magellanic Cloud-like extinction curve was assumed.

In all cases, a power-law model with a photon index Γ of ∼2 provides a good description of the data. The values for the absorption and extinction in the host galaxy were determined only for the spectra accumulated between 5.1 and 7.3 ks after the trigger. This fit provides weak evidence for X-ray absorption in the host galaxy with an n_H of ∼1.6 × 10²¹ cm⁻² and extinction corresponding to E_B-V = 0.056 mag. These values were then used for the other two spectra.

REM is a 60-cm diameter fast-reacting telescope with 10° s⁻¹ pointing speed located at the Cerro La Silla premises of the European Southern Observatory, Chile (Zerbi et al. 2001; Chincarini et al. 2003; Covino et al. 2004a, 2004b). The telescope hosts REMIR, an infrared imaging camera, and ROSS, an optical imager and slitless spectrograph. The two cameras observe simultaneously the same field of view of 10'×10' thanks to a dichroic. The telescope observed the field of GRB 100418A from April 19 03:25 UT to 04:34 UT (about 22 ks to 27 ks after the burst), as soon as the field was observable from Chile, as part of an automatic late-time monitoring program. The sequence of observations consisted in the NIR of a sequence of 300  s exposures in the z', J, H, and Ks filters, and in the optical a sequence of 60 s exposures in the V, R, and I filters. Data were reduced in a standard way and calibrated by a suitable number of well-exposed Two Micron All Sky Survey objects in the field¹⁵ for the NIR and Landolt standard stars in the optical. The afterglow was clearly detected in only one frame in the NIR, at 04:14 UT (about 25 ks after the burst) with magnitude H = 14.30 ± 0.20. Analysis of the remaining frames did not reveal any source at the afterglow position with 3σ upper limit in the best frame at H>15.2. There were no UVOT observations at the time of the REM observations, but the expected count rate in the UVOT white filter can be calculated for a specified broadband spectrum. For the best-fit spectrum for T₀ between 5.1 and 7.3 ks (row 1 in Table 3), a count rate of 12.7 s⁻¹ for H = 14.30 is expected. For the best-fit spectrum for no extinction in the host galaxy, a count rate of 22.9 s⁻¹ is expected. Both rates are significantly higher than the extrapolation to 6.9 hr of the best fit to white count rates at later times (Figure 3). This and the lack of detections in other frames suggest that the detection is due to a flare.

GRB 100418A has very low observed prompt energy and luminosity. E_γ,iso is 9.9^+6.3_−3.4 × 10⁵⁰ erg, which is lower than any of the 29 GRBs in the sample of Ghirlanda et al. (2004) except for GRB 980425 and two X-ray flashes. It is also near the low end of the distribution for both Swift and pre-Swift bursts compiled by Butler et al. (2010). In a similar fashion, L_γ,iso is 2.1^+1.1_−0.6 × 10⁵⁰ erg s⁻¹, which is less than any of the 11 GRBs in the sample of Yonetoku et al. (2004) used to establish the E_peak–L_γ,iso relationship. This is indicative of the burst being viewed outside the main emission cone, or being intrinsically less energetic.

The X-ray and optical afterglows of GRB 100418A both have a slow rising (plateau) phase and a subsequent power-law decay phase. They very likely originate from the same emission region and within the same spectral regime for t>T₀ + 5 ks for the following reasons: (1) the ratio of the XRT count rate to UVOT white filter count rate is nearly constant (Figure 4); (2) the UVOT–XRT broadband spectra are consistent with a single power law with the energy index β ∼ 1.0 (Table 3); (3) the UVOT white band and XRT light curve peaks are almost simultaneous. The time (t_p) of the peak count rates in the Swift observations lies between ∼50 ks and ∼95 ks in both the optical and X-ray bands. During the gap in the Swift observations between T₀ + 7 ks and T₀ + 50 ks, ground-based observations (Bikmaev et al. 2010) indicate a rapid increase in the optical flux peaking at about T₀ + 10 ks. We attribute this peak to a flare on top of the plateau emission. Such flares are common in the X-ray band and occasionally seen in the optical band (e.g., Greiner et al. 2009).

3.2.1. Plateau Phase

Both the X-ray and optical light curves show a long period of time during which the flux is changing slowly. The start and stop times are somewhat uncertain. For the optical band, the plateau begins at least as early as the first observation at T₀ + 87 s and ends no later than the peak in the UVOT light curve at T₀ + 51 ks. With no Swift observations between T₀ + 7 ks and T₀ + 51 ks, the plateau phase could end anywhere between these two times. In the X-ray band, the transition to the plateau phase is at T₀ + 600 s, and the plateau appears to end somewhere in the 50–100 ks range. The plateau phase is common in the X-ray band, but the duration in GRB 100418A is toward the high end of the distribution (Evans et al. 2009). The plateau phase in GRB 100418A has some similarities to that of the much brighter GRB 060729 (Grupe et al. 2007). For both bursts, the phase begins at ∼500 s after a steeply falling initial X-ray light curve, and lasts until ∼50 ks in both the X-ray and optical bands. In contrast, no optical flare is seen in GRB 060729, and the brightest flux is seen early, rather than late, in the plateau phase of GRB 060729.

A peak between the plateau and late decay phases due to the transition of a characteristic synchrotron frequency in the optical and X-ray band can be excluded, since there is no evidence (Table 3) of spectral evolution around the peak as predicted by the synchrotron forward shock models (for the interstellar medium (ISM) model, see Figure 2 of Sari et al. 1998; for the wind model, see Figure 6 of Zou et al. 2005). Moreover, the difference in the peak times in the optical and X-ray bands is within a factor of ∼2, as stated above. Therefore, the plateau phase and the peak should have a hydrodynamic origin. We also note that the plateau phase cannot be explained in the standard forward shock models through the closure relation. The observed closure relation is α − 1.5β ∼ −1.7, while the predicted relation is α − 1.5β = 0 ± 0.5 in the standard models (e.g., Zhang et al. 2006).

Energy injection model. One possible explanation for the plateau phase is significant continuous energy injection into the forward shock (Zhang et al. 2006). This can be accomplished by long-lasting operation of the central engine, transferring of the Poynting flux energy to the external shock (Dai & Lu 1998), or a range in the Lorentz factors of ejecta such that slower ejecta catch up to the decelerated shock (Rees & Mészáros 1992).

We will start with the first case and assume that luminosity L(t) = L₀(t/t_b)^−q. Table 2 of Zhang et al. (2006) provides the relationship between α and β for various GRB models. Since the spectral indices in the X-ray and optical bands are consistent at this time (Section 2.4), we will assume that all the break frequencies for the synchrotron spectrum are below the X-ray band. In this case, q = 2(α + 1 − β)/(1 + β) = −0.21^+0.15_−0.16 for the measured values, α = −0.23^+0.13_−0.14 (Section 2.2) and β = 0.98 ± 0.09 (Table 3). This value for q is marginally consistent with q = 0 case discussed by Zhang & Mészáros (2001) for a magnetar-like central engine model. The peak in the light curve corresponds to the cessation of energy injection. The isotropic kinetic energy in the late afterglow is therefore E_k,iso,f = E_k,iso,i(t_p/t_i)^1−q ⩾ 10⁵³ erg, where t_p ∼ 50–90 ks is the peak time, the start time of the dominance of the injected energy over the initial energy is t_i < 600 s. We assume that the prompt γ-ray efficiency is ∼50%, so that the initial kinetic energy is comparable to the prompt gamma-ray energy release, E_k,iso,i ∼ E_γ,iso ∼ 10⁵¹ erg.

Models for a range in the Lorentz factors of the ejecta that follow the form of M(>γ) ∝ γ^−s (s>1) in which M is the amount of ejected mass moving with Lorentz factors greater than γ are equivalent to continuous ejection models for the appropriate value of s (Zhang et al. 2006). For our measured values of q, s = 6.4^+1.3_−1.0 for the ISM model. This relatively large s value is disfavored by comparing with the mean value of ∼2.5 and the range of values derived from the previous Swift observations (Table 3 in Nousek et al. 2006). For the wind model, the entire 90% confidence range in q produces negative values for s, which is inconsistent with the model for the range of Lorentz factors.

Off-axis jet model. An alternative explanation for the plateau phase is that the afterglow is being observed from viewing angles slightly outside the jet (e.g., Eichler & Granot 2006). Depending on the structure of the jet and the off-axis angle, the light curve can have a long initial phase with little change in the flux or a delayed, but a rapidly rising initial phase. If the viewing angle is also outside the γ-ray jet, then the observed E_γ,iso and E_peak will also be reduced. The unusually low values found for GRB 100418A may be consistent with this interpretation.

Let θ_j and θ_v be the half-opening angle of a GRB jet and the viewing angle from the jet axis, respectively. In the following, we adopt the point source approximation for the GRB jet. The isotropic gamma-ray energy release observed within the jet cone is

$$\begin{equation} E_{\gamma,\rm iso}^{\rm on}=\displaystyle \frac{\mathfrak {D}(\theta =0)}{\mathfrak {D}(\theta =\theta _v-\theta _j)}E_{\gamma,\rm iso}^{\rm off}\simeq \gamma _0^2(\theta _v-\theta _j)^2E_{\gamma,\rm iso}^{\rm off}, \end{equation} \tag{ 1 }$$

where E^off_γ,iso is the off-axis energy release at a viewing angle θ_v>θ_j, $\mathfrak {D}(\theta)=[\gamma _0(1-\beta _0\cos \theta)]^{-1}$ is the Doppler factor. In this model, E^off_γ,iso is the observed γ-ray energy release and its value is ∼10⁵¹ erg. β₀ and γ₀ are the initial speed (in units of the speed of light) and the Lorentz factor of the jet in the prompt phase. At such early times θ_v − θ_j>1/γ₀ is easily satisfied, which leads to the second equality of the above equation.

The Lorentz factor evolves as γ = γ₀(t/t_dec)^−3/8 for t>t_dec, if the jet is viewed on-axis and the environment is constant ISM. For forward–reverse external shock models, the deceleration time is $t_{\rm dec}=[(3E_{k,\rm iso}/4\pi \gamma _0^2nm_p c^2)^{1/3}/2\gamma _0^2c](1+z)$ in the thin shell case, while t_dec ≃ T₉₀ in the thick shell case (Zhang et al. 2003). The transition from the plateau phase to the steep decay phase at t = t_p corresponds to the Lorentz factor γ ≃ 1/(θ_v − θ_j). Therefore, we get γ₀(θ_v − θ_j) ≃ (t_p/t_dec)^3/8. For the thick shell case, we can directly estimate γ₀(θ_v − θ_j) ∼ 10^1.5, and $E_{\gamma,\rm iso}^{\rm on}\sim 10^{54}$ erg. The intrinsic duration of the prompt emission if we observed the jet on-axis is $T_{90}^{\rm on}\simeq \mathfrak {D}(\theta =\theta _v-\theta _j)/\mathfrak {D}(\theta =0)T_{90}^{\rm off}$ (e.g., Zhang et al. 2009), which is ∼10 ms. Such a short intrinsic T₉₀ and large intrinsic energy output make the off-axis model less favored compared to the energy injection model, albeit we cannot exclude this possibility. Furthermore, the large value of γ₀(θ_v − θ_j) results in a very rapid increase in the plateau phase (Wu et al. 2004), while the observed increase is very slow. This problem may be solved by introducing some structure in the jet profile. For the thin shell case, we can constrain γ₀(θ_v − θ_j) ⩽ 10^1.5, and $E_{\gamma,\rm iso}^{\rm on}\le 10^{54}$ erg, because in this case t_dec>T₉₀. If the prompt gamma-ray efficiency is ∼50%, then the isotropic kinematic energy E_k,iso in the afterglow phase should be of order of 10⁵⁴ erg, which is comparable with the late kinematic energy in the energy injection model.

The energy injection model and off-axis jet model cannot be distinguished with complete confidence by the available observational data of GRB 100418A. While both models produce similar energies, we favor the energy injection model because of the difficulties with the off-axis jet model as discussed above.

3.2.2. Late Decay Phase

After the peak in the UVOT light curve at about T₀ +  51 ks, both the optical and X-ray light curves generally follow a power-law decay model (segment III of the canonical X-ray light curve). This decay can be interpreted with the standard afterglow model. The measured values of α and β give α − 1.5β = −0.36^+0.18_−0.14 in X-ray and α − 1.5β = −0.60^+0.16_−0.13 in the optical, which is consistent with the relationship α − 1.5β = −1/2 predicted at frequencies above all the break frequencies in the standard ISM model. The power-law index of shock accelerated electron energy distribution p can be derived from the late optical to X-ray overall spectral index (β = 1.15, Table 3), i.e., p = 2,  β = 2.3. We can also use the optical and X-ray temporal decay indices to obtain the values of p, which are p = 2.16 ± 0.13 and p = 2.49^+0.17_−0.15, respectively. The above values of p derived in different ways are consistent with each other within 1.5σ error bars. We adopt p = 2.3 in the following calculation.

The typical frequency ν_m, cooling frequency ν_c, and X-ray (ν_X = 10 keV/h>max{ν_c, ν_m}) flux density of synchrotron radiation from a GRB jet are (e.g., Granot & Sari 2002)

$$\begin{equation} \nu _m\simeq 3.5\times 10^{15}\epsilon _e^2\epsilon _B^{1/2}E_{k,\rm iso,54}^{1/2}t_5^{-3/2}\;\rm Hz, \end{equation} \tag{ 2 }$$

$$\begin{equation} \nu _c\simeq 4.4\times 10^{11}\epsilon _B^{-3/2}n_0^{-1}E_{k,\rm iso,54}^{-1/2}(1+Y)^{-2}t_5^{-1/2}\;\rm Hz, \end{equation} \tag{ 3 }$$

$$\begin{equation} F_{\nu _X}\simeq 21\epsilon _e^{p-1}\epsilon _B^{(p-2)/4}E_{k,\rm iso,54}^{(p+2)/4}(1+Y)^{-1}t_5^{(2-3p)/4}\;\mu \rm Jy, \end{equation} \tag{ 4 }$$

where _e and _B are the fractions of the shock energy deposited in electrons and magnetic fields, respectively, and n is the number density of ISM. Since in many previous GRB afterglows, _e ≫ _B has been derived (e.g., Panaitescu & Kumar 2002), we approximate the Compton parameter as Y ≃ (_e/_B)^1/2. For model parameters we adopt the notation of X = X_n × 10ⁿ in cgs units.

At the beginning of the decay phase, t ∼ 10⁵ s, both ν_m and ν_c are below the optical R band (ν_R ∼ 5 × 10¹⁵ Hz), which lead to $\epsilon _e^2\epsilon _B^{1/2}E_{k,\rm iso,54}^{1/2}\le 0.1$, and $\epsilon _e\epsilon _B^{1/2}n_0 E_{k,\rm iso,54}^{1/2}\ge 10^{-3},$ respectively. The observed X-ray flux density at hν_X = 10 keV at t = 2 × 10⁵ s is $F_{\nu _X}\simeq 10^{-2}$ μJy, which leads to another constraint on model parameters, i.e., $\epsilon _e^{p-1.5}\epsilon _B^{p/4}E_{k,\rm iso,54}^{(p+2)/4}\simeq 10^{-3}$. The solution of the above three constraints, assuming $E_{k,\rm iso}\simeq 10^{54}$ erg, is 10^4p/3−10 ⩽ _B ⩽ 10^{12(p−2.5)/(3−p)}n^{(4p−6)/(3−p)}₀, or 10⁻⁷ ⩽ _B ⩽ 4 × 10⁻⁴n^4.6₀. The ISM number density should be n ⩾ 0.1 cm⁻³. Therefore, a set of model parameters with $E_{k,\rm iso}\simeq 10^{54}$ erg, n = 1 cm⁻³, p = 2.3, _e = 0.15, and _B = 10⁻⁴ can fit the X-ray and optical light curves in the decay phase, although some parameters cannot be tightly constrained.

Since there is no evidence for a break in the X-ray light curve at late times, any jet break must occur at t_b>2 × 10⁶ s. This is also supported by the optical observation. The late optical afterglow decayed as t^{−1.12±0.10}, until the host galaxy begins to dominate around t ∼ 10⁶ s. The non-detection of a jet break indicates that the jet's half-opening angle is $\theta _j>1/\gamma \sim 14^\circ E_{k,\rm iso, 54}^{-1/8}n_0^{1/8}(t/2\times 10^6\;{\rm s})^{3/8}$, and the collimation-corrected jet's energy is $E_{k,\rm jet}\simeq (\theta _j^2/2)E_{k,\rm iso}\ge 3\times 10^{52}E_{k,\rm iso,54}^{3/4}n_0^{1/4}$ erg. The jet energy is at the high end of the jet energy distribution of known GRB samples and is comparable to the maximum energy release in magnetar models (Cenko et al. 2010 and references therein). The large jet energy disfavors the off-axis model, because the intrinsic duration of ∼10 ms in this model would classify GRB 100418A as a short energetic GRB, which is rare event.