Reanalyze the pulsar scenario to explain the cosmic positron excess considering the recent developments

TL;DR

This paper reexamines pulsar contributions to the cosmic positron excess considering recent observations, proposing PSR B1055-52 as a promising source that aligns with AMS-02 and H.E.S.S. data, challenging previous candidates like Geminga.

Contribution

It introduces a new analysis under the two-zone diffusion model, identifying PSR B1055-52 as a viable source of the positron excess based on recent observational constraints.

Findings

PSR B1055-52 can reproduce AMS-02 positron spectrum features.

The slow diffusion scenario favors B1055-52 as the primary source.

Geminga is no longer a viable candidate under new constraints.

Abstract

As the TeV halos around Geminga and PSR B0656+14 have been confirmed by HAWC, slow diffusion of cosmic rays could be general around pulsars, and the cosmic positron spectrum from pulsars could be significantly changed. As a consequence, the most likely pulsar source of the positron excess, Geminga, is no more a viable candidate under the additional constraint from Fermi-LAT. Moreover, the latest measurement by AMS-02 shows a clear cutoff in the positron spectrum, which sets a strict constraint on the age of the pulsar source. Considering these new developments we reanalyze the scenario in this work. By checking all the observed pulsars under the two-zone diffusion scenario, we propose for the first time that PSR B1055-52 is a very promising source of the positron excess. B1055-52 can well reproduce both the intensity and the high-energy cut of the AMS-02 positron spectrum, and may also explain the H.E.S.S $e^-+e^+$ spectrum around 10 TeV. Moreover, if the slow diffusion is universal in the local interstellar medium, B1055-52 will be the unique reasonable source of the AMS-02 positron spectrum among the observed pulsars.

Figures

Tables

Table 1. Table 1: Candidate pulsars of the positron excess under the two-zone diffusion model.

J2000	Other Name	$t_s$	$r_s$	${\dot{E}}_s$
		(kyr)	(kpc)	(${10}^{34}$ erg s-1)
J0633+1746	Geminga	342	0.25	3.25
J0659+1414	B0656+14	111	0.29	3.81
J1003-4747	B1001-47	220	0.37	3.01
J1057-5226	B1055-52	535	0.09	3.01
J1741-2054	-	386	0.30	0.95
J1745-3040	B1742-30	546	0.20	0.85
J1825-0935	B1822-09	232	0.30	0.46

Equations

$$\frac{\partial N}{\partial t}-\nabla \cdot (D\nabla N)-\frac{\partial }{\partial E}(bN)=Q,$$

$$D(E,r)=\{\begin{array}{rlr}{D}_1(E),& & r<r_{\star }\\ D_2(E),& & r\ge r_{\star }\end{array},$$

$$Q(t,E,r)=q(t,E)\delta (r-r_s),$$

$$q(t,E)=q_0{\left(1+\frac{t}{\tau }\right)}^{-2}{E}^{-\gamma },$$

$${\int }_{E_{\min }}^{E_{\max }}q(t_s,E)EdE=\eta {\dot{E}}_s,$$

Full text

1 Introduction

The well-known cosmic-ray (CR) positron excess, which was first detected by PAMELA (Adriani et al., 2009) and confirmed by AMS-02 (Aguilar et al., 2013) and Fermi-LAT (Ackermann et al., 2012) later, may be interpreted by either astrophysical sources or the dark matter annihilation/decay. The dark matter scenarios are strongly disfavored by the $\gamma$-ray observation of Fermi-LAT on the dwarf galaxies and so on (Ackermann et al., 2015; Liu et al., 2017; Lin et al., 2015). Nearby pulsars, such as Geminga, PSR B0656+14, and Vela pulsar, have been proposed as plausible sources of high-energy positrons as buttressed by quantitative calculations (Hooper et al., 2009, 2017; Yüksel et al., 2009; Yin et al., 2013; Della Torre et al., 2015). It is also possible that the excess is contributed by multiple nearby pulsars (Delahaye et al., 2010).

The study of the positron excess is now entering into a new stage as recent developments change the situation rapidly. First, the TeV $\gamma$-ray observation of HAWC indicates that Geminga and B0656+14 are surrounded by very inefficient diffusion halos (Abeysekara et al., 2017a), and it is possible that the slow-diffusion region is general around pulsars (Linden et al., 2017). If so, the positron contribution from pulsars should be significantly different from that derived by the previous one-zone propagation model. Second, the AMS collaboration has just published the latest positron spectrum using 6.5 years of data (Aguilar et al., 2019). The positron spectrum is unprecedentedly extended to $\sim$ 1 TeV, and a sharp spectral dropoff around 300 GeV is detected for the first time with a significance of more than $3\sigma$. This definitely provides a stronger limit to the high-energy positron source.

In Fang et al. (2018), we perform the first numerical calculation of the positron spectrum of pulsars at the Earth under the two-zone diffusion scenario, in which the diffusion velocity outside the slow-diffusion halo around the pulsars is assumed to be the average value in the Galaxy. In this case, Geminga can naturally explain the positron excess if we take the injection spectral index derived by HAWC (Abeysekara et al., 2017a). However, analysis of the Fermi-LAT data on the Geminga halo shows that the injection spectrum in GeV should be much harder than that provided by HAWC (Xi et al., 2018; Di Mauro et al., 2019). As a result, the conversion efficiency to positrons is significantly depressed, and Geminga is disfavored as the dominant source of the high-energy positrons (Xi et al., 2018; Di Mauro et al., 2019).

In this work, we give a reanalysis of the pulsar candidates to explain the positron excess considering the new experimental developments. We assume the slow-diffusion halo is universal around pulsars and adopt the two-zone diffusion model to all the observed nearby pulsars to calculate their positron fluxes at the Earth. We consider that the positron spectral cutoff detected by AMS-02 should not be a collective effect of pulsars, since even old pulsar wind nebulae (PWNe) can accelerate positrons to $\sim 100$ TeV (Caraveo et al., 2003; Abeysekara et al., 2017a). The spectral break in hundreds of GeV is more likely to correspond to the radiative cooling of the positrons released by the dominant source, which will set a strict constraint to the pulsar age. We attempt to find the promising pulsar(s) that can both explain the intensity and the spectral cut of the the latest positron spectrum of AMS-02.

In the next section, we describe the details of the calculation of the positron spectrum from pulsars. In Section 3, we present the prominent sources of positrons among the observed pulsars under the two-zone diffusion model, and also discuss the impacts from the uncertainties of the model. In Section 4, we give further discussions about the most prominent positron source found in Section 3: PSR B1055-52. We summarize this work in Section 5.

2 Method

2.1 The two-zone propagation model

The propagation of CR electrons and positrons ($e^{\pm}$) is generally described by the diffusion-cooling equation:

[TABLE]

where $D$ is the diffusion coefficient, $b(E)=-{\rm d}E/{\rm d}t$ is the energy-loss rate, and $Q$ denotes the source function. The energy-loss rate takes the form of $b(E)=b_{0}(E)E^{2}$, and we calculate $b_{0}(E)$ following Schlickeiser & Ruppel (2010), where the synchrotron and inverse Compton radiation cooling of $e^{\pm}$ are included. We assume the interstellar magnetic field in the Galaxy to be 3 $\mu$G as the default to calculate the synchrotron term (Minter & Spangler, 1996).

As the diffusion coefficient derived by HAWC is hundreds times slower than the Galactic average value (Aguilar et al., 2016), this slow-diffusion velocity in the vicinity of the pulsars should not be universal in the Galaxy (Hooper et al., 2017). The slow-diffusion halo111Recently, Liu et al. (2019) propose an alternative scenario that the TeV halo of Geminga is not attributed to the strong turbulence, but interpreted by the anisotropy diffusion of the electrons along the local regular magnetic field which is presumed to be aligned with the line of sight towards Geminga. could either be self-induced by the escaping $e^{\pm}$ from the pulsar (Evoli et al., 2018), or simply be a preexisting turbulent region created by the parent supernova remnant of the pulsar (Fang et al., 2019). Thus, the slow-diffusion regions may have a scale of tens of pc considering their possible origins. We write the diffusion coefficient as

[TABLE]

where $r$ is the distance to the pulsar, $D_{1}(E)=10^{26}(E/1{\rm\,GeV})^{0.33}$ cm2 s*-1* is the diffusion coefficient inferred by HAWC, $D_{2}(E)=4.28\times 10^{28}(E/1{\rm\,GeV})^{0.38}$ cm2 s*-1* is the average diffusion coefficient in the Galaxy given by the diffusion-reacceleration model of Yuan et al. (2017), and we assume a spherically symmetrical diffusion for the $e^{\pm}$ released by local pulsars. The size of the slow-diffusion region is set to be 50 pc as the default.

For variable coefficient problems like Equation (1) and (2), numerical method should be the more straightforward choice. We follow the method given in Fang et al. (2018). The finite volume method is adopted to construct the differencing scheme for the diffusion operator. We emphasize that this is necessary to ensure the conservation of flux for the variable-coefficient diffusion equation and obtain the correct solution.

2.2 Pulsars as the positron source

We assume pulsar as point-like source with continuous $e^{\pm}$ injection, so the source function should be expressed as

[TABLE]

where $r_{s}$ is the distance of the pulsar. The time dependency of $e^{\pm}$ injection is assumed to have the same profile with the spin-down luminosity of the pulsar (Pacini & Salvati, 1973), then

[TABLE]

where $\tau=10$ kyr is the typical the spin-down time scale of the pulsar. Note $\gamma$ is the injection spectral index of the PWN, rather than that of the pulsar. Radio observations indicate that the $e^{\pm}$ spectral index is smaller than 2.0 for most PWNe (Reynolds et al., 2017). Besides, the injection spectral index of Geminga also tends to be hard considering the constraint from the observation of Fermi-LAT (Xi et al., 2018; Di Mauro et al., 2019). So we set $\gamma=1.8$ as the default in the following calculation. The normalization $q_{0}$ is determined by the relation

[TABLE]

where we set $E_{\rm min}=1$ GeV, $E_{\rm max}=100$ TeV, $t_{s}$ and $E_{s}$ are the current age and spin-down luminosity of the pulsar, respectively. Equation (5) means the spin-down energy of the pulsar is converted to the injected $e^{\pm}$ with a proportion of $\eta$.

The $r_{s}$, $t_{s}$, and $E_{s}$ of all the published pulsars can be found in the ATNF catalog222http://www.atnf.csiro.au/research/pulsar/psrcat (Manchester et al., 2005). For the measurement of the pulsar distance, the kinematic method and the trigonometric parallax take the precedences in the ATNF catalog. For the cases in which those methods are unavailable, the pulsar distance is derived from the dispersion measure (DM), which depends on a specific electron distribution model. We should note that for the version later than 1.55 (Nov 2016), the electron distribution model of Yao et al. (2017) (YMW16) is adopted as the default instead of those of Taylor & Cordes (1993) (TC93) and Cordes & Lazio (2002) (NE2001). The distance estimates of some pulsars are significantly changed under the YMW16 model. For example, the distance of PSR J0940-5428 is 0.38 kpc using the YMW16 model, compared with 2.95 kpc based on the NE2001 model; the distance of PSR B1055-52 is 0.09 kpc using the YMW16 model, compared with 0.72 kpc based on the NE2001 model.

3 Result

As mentioned above, we set $\gamma=1.8$ and $r_{\star}=50$ pc as the defaults to calculate the positron spectrum of pulsars at the Earth. Figure 1 shows the contours of the positron flux at 300 GeV as a function of age and distance of pulsars. The yellow shaded area means that the pulsars inside this area can contribute more than half of the measured positron flux of AMS-02 at 300 GeV under the two-zone diffusion model. For comparison, we also draw the blue and green areas corresponding to the one-zone fast-diffusion model ($r_{\star}=0$) and the one-zone slow-diffusion model ($r_{\star}=\infty$), respectively. Any two-zone diffusion scenarios should be the transitions among these three models. In the calculation, we use a uniform initial spin-down luminosity $\dot{E}_{0,s}$ to determine the normalization $q_{0}$, and the log-mean $\dot{E}_{0,s}$ of all the nearby ($<1$ kpc) and relatively young ($<10$ Myr) pulsars are adopted. We assume all the spin-down energy is delivered to $e^{\pm}$ ($\eta=100\%$). The observed pulsars in the ATNF catalog are shown in Figure 1. The pulsars of which the $\dot{E}_{0,s}$ is larger than the log-mean value mentioned above are marked with filled circles, otherwise they will be marked with unfilled ones. Thus, the upper limits of the shaded areas should be in fact somehow higher than those shown in the figure for the filled circles, and lower for the unfilled circles.

Aguilar et al. (2019) fit the AMS-02 positron spectrum with a diffuse term and a high energy source term, and the cutoff energy of the source term is $E_{c}=810^{+310}_{-180}$ GeV. If the cutoff is interpreted by the cooling of the positrons released in the early age of a pulsar, we may approximately relate the age of the pulsar to the cooling time $1/[b_{0}(E_{c})E_{c}]$. In Figure 1, we show this age range with the red band for reference.

3.1 Candidate sources of the positron excess

Figure 1 indicates that in the two-zone diffusion scenario the positron excess can only be contributed by nearby ($r_{s}<500$ pc) and middle-aged ($t_{s}\sim 0.1-1$ Myr) pulsars. As $e^{\pm}$ injected by pulsars are trapped in the slow-diffusion region for a long time, $e^{\pm}$ released by young pulsars do not have enough time to arrive at the Earth. Table 1 summarizes the relevant information of the pulsars inside the yellow area of Figure 1. We show the positron spectrum of these pulsars at the Earth in Figure 2, with $\eta=50\%$ for all the pulsars. Obviously, B1055-52 and Geminga are the most prominent positron sources in the energy range of AMS-02 with our default parameters. Moreover, B1055-52 can well reproduce the cutoff of the AMS-02 spectrum, while Geminga is not old enough to fit this spectral fall.

B1055-52, together with Geminga and B0656+14, are called the three musketeers as they have similar ages, spin-down luminosities, and so on (Becker & Truemper, 1997). All of them are $\gamma$-ray pulsars (Abdo et al., 2010), and surrounded by X-ray PWNe (Caraveo et al., 2003; Posselt et al., 2015; Bîrzan et al., 2016; Posselt et al., 2017). However, B1055-52 has rarely been discussed when it comes to the interpretation of the positron excess, because it was once believed to be located much farther according to the previous models of the Galactic electron distribution (1.53 kpc for TC93 and 720 pc for NE2001). The very close distance of 90 pc indicated by the YMW16 model ensures a sufficient positron contribution to the AMS-02 data. High-energy positrons injected at the later age of B1055-52 also have enough time to reach the Earth, which may even explain the highest energy part of the H.E.S.S spectrum333 https://indico.snu.ac.kr/indico/event/15/session/5/contribution/694 as shown in the right of Figure 2. Meanwhile, Mignani et al. (2010) also argued that B1055-52 should be a nearby source, but with a distance of $350\pm 150$ pc which is farther than that given by the latest DM. In this case, a higher conversion efficiency for B1055-52 is required to explain the positron spectrum. In the right of Figure 2, we show the case of $r_{s}=350\pm 150$ pc and $\eta=100\%$ for B1055-52 with the blue band.

Geminga has been widely discussed as the source of the positron excess (Hooper et al., 2009, 2017; Yüksel et al., 2009; Yin et al., 2013; Fang et al., 2018; Profumo et al., 2018; Tang & Piran, 2019) . In Fang et al. (2018), we point out that Geminga can reasonably explain the positron spectrum of AMS-02 in the two-zone diffusion scenario, with $\gamma=2.2$ provided by Abeysekara et al. (2017a). However, $\gamma$-ray measurement of Fermi-LAT around Geminga implies that the $e^{\pm}$ spectrum cannot be extrapolated to GeV energies with $\gamma=2.2$. The spectrum should be harder and the conversion efficiency is constrained to be less than $5\%$, which disfavors Geminga as the main source of positron excess (Xi et al., 2018). While, the $e^{\pm}$ accounting for the AMS-02 spectrum should be mainly released in the early age of Geminga. Considering the proper motion of Geminga and the uncertainties of the diffusion pattern in the early time, Geminga may not be finally excluded as the positron source.

3.2 Uncertainties of the model

Here we discuss the main uncertainties in the calculation above: $\gamma$, $r_{\star}$, and $B$. Observations favor a hard injection $e^{\pm}$ spectrum for PWNe, while if the $e^{\pm}$ are accelerated at relativistic shocks (the termination shocks), theories predict a softer spectrum with $\gamma\geq 2.2$ (Kirk & Duffy, 1999). So we present a case with $\gamma=2.4$ in the left of Figure 3, where the injection spectrum is much softer than the default. B1055-52 provides too much low-energy positrons in this case, while we should note that there is degeneracy between $\gamma$ and $r_{\star}$ in terms of the positron spectrum at the Earth. So if $r_{\star}$ is larger, the spectral shape of B1055-52 may still match the measurement. The spectral shape of Geminga has a better consistency with the AMS-02 data for $\gamma=2.4$, since the spectral cut due to radiative cooling appears at lower energy compared with the case of $\gamma=1.8$. However, due to the constraint from Fermi-LAT, the scenario of $\gamma>2.0$ for Geminga is excluded (Xi et al., 2018).

The size of the slow-diffusion halo around pulsars cannot be generally constrained by observations at present. More TeV halos associated with pulsars need to be confirmed. What is certain is that we have $r_{\star}\geq 20$ pc around Geminga and B0656+14 (Abeysekara et al., 2017a). Besides, if the extend TeV emission around PSR J1826-1334 and J1907+0602 (HESS J1825-137 and MGRO J1908+06 respectively) are attributed to the inefficient diffusion of $e^{\pm}$, the scale of $r_{\star}$ should be at least tens of pc (Khangulyan et al., 2018; Aliu et al., 2014). Huang et al. (2018) also argue that a considerable slow diffusion should be existed around the Vela pulsar to avoid the conflict with the high-energy $e^{-}+e^{+}$ spectrum of H.E.S.S. Moreover, Figure 1 indicates that if $r_{\star}$ is null or too small, pulsars cannot explain the positron flux and the positron spectral cut of AMS-02 simultaneously (for example, the blue shaded area has no overlap with the red area). All these imply that the size of the slow-diffusion region could be considerable for pulsars. However, the upper limit of $r_{\star}$ is more difficult to be decided. In the right of Figure 3, we show the most extreme scenario with $r_{\star}=\infty$, that is, the slow diffusion is universal in the local interstellar medium (ISM). In this case, only very nearby source could explain the high-energy positron spectrum. Obviously, B1055-52 is the best candidate as it is the nearest pulsar in the ATNF catalog and it also has a proper age. Meanwhile, only a very small conversion efficiency of $0.1\%-0.2\%$ is required to explain the data.

We have assumed a typical interstellar magnetic field of 3 $\mu$G in the calculations above. López-Coto & Giacinti (2018) argue that the local magnetic field may be in a range of $3-5$ $\mu$G. If $B>3$ $\mu$G, the cooling cut of pulsars in the positron spectrum should be happened at lower energies compared with Figure 2 and 3, which means the spectral cut of younger pulsars such as B1001+47 may match the cut of AMS-02. However, the spin-down luminosity of these observed younger sources are too low to generate enough positrons. So the conclusion is unchanged and B1055-52 should still be the most likely candidate.

4 Discussion

4.1 The distance of PSR B1055-52

We have shown that B1055-52 is a very competitive source of the positron excess under the two-zone diffusion scenario. An important factor of its dominance is the very close distance of 90 pc. This distance is given by the DM based on the electron density model of YMW16, while the annual parallax of this source has not been measured yet.

The relation of the $\gamma$-ray luminosity $L_{\gamma}$ and $\dot{E}_{s}$ could be a supporting evidence for the distance of B1055-52. Abdo et al. (2010) show the $L_{\gamma}-\dot{E}_{s}$ map for the observed $\gamma$-ray pulsars. Most pulsars with high $\dot{E}_{s}$ are distributed between $L_{\gamma}=\dot{E}_{s}$ and $L_{\gamma}=(10^{33}\,{\rm ergs}^{-1}\dot{E}_{s})^{1/2}$, while there seems to be a break in $\dot{E}_{s}\sim 10^{35}$ erg s*-1*, that is, the $\gamma$-ray efficiency drops faster with the decrease of $\dot{E}_{s}$. This is consistent with the theoretical predictions (Muslimov & Harding, 2003; Zhang et al., 2004). The $\gamma$-ray efficiency of B0656+14 is very low, which has $L_{\gamma}=3.1\times 10^{32}$ erg s*-1* and $\dot{E}_{s}=3.8\times 10^{34}$ erg s*-1*. If the distance of B1055-52 is 90 pc, then B1055-52 is almost coincident with B0656+14 on the $L_{\gamma}-\dot{E}_{s}$ map, with $L_{\gamma}=2.6\times 10^{32}$ erg s*-1* and $\dot{E}_{s}=3.0\times 10^{34}$ erg s*-1*. So we consider $r_{s}=90$ pc is reasonable for B1055-52. First, B1055-52 is similar with B0656+14 in most aspects. Their X-ray PWNe are both very faint or compact (Posselt et al., 2015; Bîrzan et al., 2016), which is different from that of Geminga (Posselt et al., 2017). Second, the distance of B0656+14 is given by the method of trigonometric parallax with very small uncertainty (Brisken et al., 2003), which means the position of B0656+14 on the $L_{\gamma}-\dot{E}_{s}$ map is exact.

4.2 Could the TeV halo of B1055-52 be detected?

TeV halos around pulsars are strong evidence of slow diffusion. Like Geminga and B0656+14, B1055-52 should also own a TeV $\gamma$-ray halo if it is surrounded by a slow-diffusion region. However, the location of B1055-52 is ($286.0^{\circ}$, $+6.6^{\circ}$) in the Galactic coordinate and ($164.5^{\circ}$, $-52.5^{\circ}$) in the equatorial coordinate, which is out of the sight of HAWC (Abeysekara et al., 2017b). H.E.S.S has performed Galactic plane survey with high quality data recored from 2004 to 2013 (H. E. S. S. Collaboration et al., 2018), while the survey only covers the Galactic latitude from $-3^{\circ}$ to $3^{\circ}$, and the area of B1055-52 is still not included in the survey. Besides, if the radius of the TeV halo is 20 pc, the field angle of the halo should be $25^{\circ}$ for $r_{s}=90$ pc, which is too large for the imaging atmospheric Cherenkov telescopes including H.E.S.S. Thus, a large field-of-view TeV $\gamma$-ray observatory in the southern hemisphere with large field of view, such as the SGSO which is proposed recently (Albert et al., 2019), is necessary to search for the TeV halo around B1055-52.

5 Conclusion

As the slow-diffusion regions around Geminga and B0656+14 have been confirmed by HAWC, slow diffusion around pulsars could be general. In this work, we interpret the latest AMS-02 positron spectrum with identified pulsars under the two-zone diffusion scenario. The intensity and the high-energy spectral cut of the AMS-02 spectrum are required to be explained simultaneously. We consider that the spectral drop is attributed to the cooling of the positrons from the dominate pulsar.

We provide the contours of the positron contribution from pulsar as a function of age and distance of pulsar. Three different diffusion cases are shown, with $r_{\star}=0$, $r_{\star}=50$ pc, and $r_{\star}=\infty$. Any two-zone diffusion scenarios should be the transitions among these models. The contours clearly show that if the scale of the slow-diffusion halo is significant ($\sim 50$ pc), the high-energy positron spectrum tends to be explained by nearby and middle-aged pulsars. Meanwhile, $r_{\star}$ should not be generally null or too small, otherwise no pulsar is able to reproduce the intensity and the spectral cut of the AMS-02 data at the same time.

We scan all the observed pulsars, and find that B1055-52 and Geminga are the most promising candidates of the positron excess, although Geminga is disfavored by the constraint of the $\gamma$-ray observation of Fermi-LAT. B1055-52, which has the similar age and spin-down luminosity with Geminga and B0656+14, was rarely discussed in terms of the positron excess, because it was too far as indicated by the previous DM. The latest DM derives a very close distance of 90 pc for B1055-52, and it becomes the brightest positron source, which may even explain the H.E.S.S $e^{-}+e^{+}$ spectrum around 10 TeV. Besides, even if the slow diffusion is universe in the local ISM, B1055-52 is still able to explain the positron excess and it is the only candidate in this scenario.

We also discuss that the distance of 90 pc can be reasonable for B1055-52, while we suggest the annual parallax measurement for B1055-52 to obtain a more precise distance. The putative TeV halo around B1055-52 cannot be detected by the current experiments, due to the limit of their locations or the field of view. Future wide field-of-view TeV observatory in the southern hemisphere like SGSO may help us to gain insight into the diffusion pattern around B1055-52 and make a certain judgment to the relation between B1055-52 and the positron excess.

This work is supported by the National Key Program for Research and Development (No. 2016YFA0400200) and by the National Natural Science Foundation of China under Grants No. U1738209, 11851303.