DERIVATION OF A LARGE ISOTOPIC DIFFUSE SKY EMISSION COMPONENT AT 1.25 AND 2.2 μm FROM THE COBE/DIRBE DATA

EBL in the near-infrared (IR) supposedly comprises integrated light emitted from galaxies, quasars, and possible particle decay. Hence, the near-IR EBL is a potentially important physical indicator of star formation history and the unknown radiation processes throughout the history of the universe.

The lower limit of the near-IR EBL is the brightness of the integrated galaxy light (IGL) derived from the deep galaxy counts, such as those detected in the Hubble Deep Field (HDF) by Madau & Pozzetti (2000) or the Subaru Deep Field (SDF) by Totani et al. (2001). On the other hand, the upper limit of the EBL is estimated from observations of high-energy γ-ray sources, assuming the property of intrinsic spectra of the objects (e.g., Aharonian et al. 2006; Albert et al. 2008; Meyer et al. 2012). These γ-rays interact with the EBL photons to create positron-electron pairs. At 1–$2\;\mu {\rm{m}},$ the results of these two methods converge to the same intensity (i.e., ∼10–$20\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$).

Direct measurement of the EBL is hampered by the intense foreground emission, contributed by airglow, zodiacal light (ZL), and integrated starlight (ISL). In previous studies, the ZL and the ISL were subtracted from the sky brightness measured by the space telescope. According to Matsumoto et al. (2005), who reported on Infrared Telescope in Space (IRTS), and Wright & Reese (2000), Cambrésy et al. (2001), and Levenson et al. (2007), who analyzed the Diffuse Infrared Background Experiment (DIRBE) aboard the Cosmic Background Explorer (COBE) satellite, the intensity of the EBL in the near-IR can be several times higher than that of the IGL and that estimated from the high-energy γ-ray observations. If these findings are true, we require additional sources other than normal galaxies. The excess light might originate from distant Population-III (Pop-III) stars which cannot be spatially resolved by recent observations. However, Inoue et al. (2013) calculated the theoretical contribution of light from Pop-III stars to the EBL, and showed that it is smaller than the IGL contribution by 2–3 orders of magnitude. Therefore, if the excess emission is real, we must seek other candidate sources.

Previous studies have revealed that diffuse Galactic light (DGL) comprises starlight scattered off by the interstellar dust grains (e.g., Elvey & Roach 1937; Henyey & Greenstein 1941; van de Hulst & de Jong 1969; Mattila 1979). The DGL contains information on the size distribution of interstellar dust grains and the interstellar radiation field (ISRF) that illuminates them. In the optically thin limit, the intensity of the far-IR $100\;\mu {\rm{m}}$ emission is expected to be proportional to the DGL intensity (Brandt & Draine 2012). Therefore, the DGL has been quantitatively analyzed by correlating the diffuse optical light with the far-IR emission (e.g., Laureijs et al. 1987; Guhathakurta & Tyson 1989; Paley et al. 1991; Zagury et al. 1999; Matsuoka et al. 2011; Brandt & Draine 2012; Ienaka et al. 2013). Although the DGL is worthy of study, it constitutes a foreground emission in the EBL measurements, and must therefore be removed before analyzing the EBL.

Thus far, the DGL and its contribution to the total sky brightness have not been quantified in the near-IR. Leinert et al. (1998) suggested that the near-IR DGL is limited to low Galactic latitudes ($| b| \lt 5^\circ$), where the dust column is sufficiently dense to enhance the intensity of the scattered light. In contrast, Arai et al. (2015) derived the mean DGL spectrum at 0.95–$1.65\;\mu {\rm{m}}$ using the low-resolution spectrometer (LRS) on the Cosmic Infrared Background ExpeRiment (CIBER) in several local regions of high Galactic latitude ($| b| \gtrsim 30^\circ$). Although their result is consistent with the DGL spectra modeled by Brandt & Draine (2012), whether their result is applicable to the general wide field of the sky is not clarified. In addition, Tsumura et al. (2013b) derived the DGL spectrum at relatively low Galactic latitudes ($5^\circ \lesssim | b| \lesssim 15^\circ$) at 1.8–$5.3\;\mu {\rm{m}}$ from data collected in the low-resolution prism spectroscopy mode of the Infra-Red Camara (IRC) onboard the AKARI satellite. To elucidate the general properties of the DGL and the isotropy of the EBL, these studies must be supplemented by measurements of the near-IR DGL and its contribution to the sky brightness over a wide field.

To measure the EBL and DGL at 1–$2\;\mu {\rm{m}},$ we analyze data acquired by the DIRBE aboard the COBE spacecraft. For previous studies of the diffuse IR components, the DIRBE observed the all sky from 1.25 to $240\;\mu {\rm{m}}$ in 10 bands. These diffuse components included the ZL, thermal emission from interstellar dust, and the EBL (Arendt et al. 1998; Dwek et al. 1998; Hauser et al. 1998; Kelsall et al. 1998). Arendt et al. (1998) subtracted the contribution of the ISL from the sky brightness using the DIRBE Faint Source Model (FSM), which is based on the Wainscoat et al. (1992) and Cohen (1993, 1994, 1995) "SKY" models. As shown in Table 4 of Arendt et al. (1998), wherein no entries appear at 1.25 and $2.2\;\mu {\rm{m}},$ the intensity of the $100\;\mu {\rm{m}}$ emission is not correlated with diffuse light at those wavelengths. However, according to Leinert et al. (1998), the Galactic component observed by the DIRBE in these bands undoubtedly contains a scattered light contribution. The missing DGL may have been caused by the poor precision of the DIRBE FSM which does not reproduce the actual astrometry and photometry of Galactic stars.

Since its release, the 2MASS Point Source Catalog (PSC) has been used for the starlight subtraction in several measurement studies of the near-IR EBL (e.g., Gorjian et al. 2000; Wright & Reese 2000; Cambrésy et al. 2001; Levenson et al. 2007). However, because their analyses were limited to small regions of low far-IR intensity at high Galactic latitude ($| b| \gt 40^\circ$), these authors ignored the DGL contribution.

The present paper reanalyzes the all-sky map created by DIRBE for the purpose of evaluating the DGL at 1.25 and $2.2\;\mu {\rm{m}}$ and measuring the EBL. Calculating the contribution of the ISL collected by the 2MASS PSC over a wide field of high Galactic latitudes ($| b| \gt 35^\circ$), which includes both low and high $100\;\mu {\rm{m}}$ emission intensity fields, we find a positive linear correlation between the $100\;\mu {\rm{m}}$ emission and the diffuse near-IR light at both 1.25 and $2.2\;\mu {\rm{m}}.$ This means that the near-IR DGL which was ignored in previous DIRBE analyses, is extracted even at high Galactic latitudes. In fields with small DGL components, subtracting the DGL from the isotropic emission does not remove the excess brightness against the IGL at 1.25 and $2.2\;\mu {\rm{m}},$ consistent with the previous studies.

The remainder of this paper is organized as follows. Section 2 briefly describes the analyzed DIRBE data, and Section 3 shows how we estimate the contribution of each near-IR component. In this section, the total sky brightness is decomposed into the ZL, DGL, ISL, and isotropic emission components by a ${\text{}}{\chi }^{2}$ minimum analysis. Section 4 presents the fitting and evaluates the uncertainty in each component. Section 5, compares our fitting results with those of other studies. A summary is presented in Section 6.

Throughout this paper, the surface brightness is expressed in ${\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$ or $\mathrm{MJy}\;{\mathrm{sr}}^{-1}.$ The conversion formula between these units is

$$\begin{eqnarray}&&\nu {I}_{\nu }({\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1})=\left[3000/{\text{}}\lambda (\mu {\rm{m}})\right]\;{{\rm{I}}}_{\nu }\;(\mathrm{MJy}\;{\mathrm{sr}}^{-1}).\end{eqnarray} \tag{ 1 }$$

The intensity ${F}_{i}(\mathrm{Obs})$ of the DIRBE $\epsilon =90^\circ$ map is evaluated by the brightness model ${F}_{i}(\mathrm{Model}),$ where the subscript "$i$" refers to one of two bands (1.25 or $2.2\;\mu {\rm{m}}$). Within these bands, the sky brightness is assumed as a linear combination of four components: the ZL, DGL, ISL, and isotropic emission including the EBL. The ${F}_{i}(\mathrm{Model})$ is thus given by

$$\begin{eqnarray}&&{\text{}}{F}_{i}(\mathrm{Model})={F}_{i}(\mathrm{ZL})+{F}_{i}(\mathrm{DGL})+{F}_{i}(\mathrm{ISL})+{F}_{i}(\mathrm{Iso})\end{eqnarray} \tag{ 2 }$$

where ${F}_{i}(\mathrm{ZL}),$ ${F}_{i}(\mathrm{DGL}),$ ${F}_{i}(\mathrm{ISL}),$ and ${F}_{i}(\mathrm{Iso})$ denote the intensities of the ZL, DGL, ISL, and isotropic emission, respectively. These four terms are modeled as follows.

3.1.1. Zodiacal Light

The ZL term F_i(ZL) is defined as

$$\begin{eqnarray}&&{F}_{i}(\mathrm{ZL})={a}_{i}{F}_{i}(\mathrm{Kel})\end{eqnarray} \tag{ 3 }$$

where ${a}_{i}$ is a free parameter and ${F}_{i}(\mathrm{Kel})$ is the ZL brightness estimated by the Kelsall model (Kelsall et al. 1998). The Kelsall model is a parameterized physical model fitted to the time variation of the sky brightness measured by the DIRBE. To evaluate the scaling factor of the Kelsall model versus the DIRBE data, we adopt the free parameter ${a}_{i}.$ If the Kelsall model completely reproduces the seasonal variation of the ZL brightness observed by the DIRBE, the parameter ${a}_{i}$ will equal 1.0.

3.1.2. Diffuse Galactic Light

In previous studies of DGL measurements in the optical and near-IR, the intensity of the $100\;\mu {\rm{m}}$ emission was correlated with that of the diffuse light (e.g., Matsuoka et al. 2011; Ienaka et al. 2013; Arai et al. 2015). In the optically thin region, the extinction of the DGL is small and the correlation is reportedly linear, consistent with theoretical expectation (see Brandt & Draine 2012). Theoretically, a linear correlation is expected because optically thin fields dominate at high Galactic latitudes in the 1.25 and $2.2\;\mu {\rm{m}}$ bands.

Here, we adopt the diffuse $100\;\mu {\rm{m}}$ emission map created by Schlegel et al. (1998), hereafter "SFD," which is widely used in the correlation analyses. To match the spatial resolutions of the SFD and DIRBE maps, we apply an 8 × 8 pixel binning to the SFD map. The DGL term ${F}_{i}(\mathrm{DGL})$ is then defined as

$$\begin{eqnarray}&&{F}_{i}(\mathrm{DGL})={b}_{i}{F}_{100}\end{eqnarray} \tag{ 4 }$$

where ${b}_{i}$ is a free parameter and F₁₀₀ is the interstellar $100\;\mu {\rm{m}}$ intensity, defined as follows;

$$\begin{eqnarray}{F}_{100}=\left\{\begin{array}{ll}{F}_{\mathrm{SFD}}-0.8\;\mathrm{MJy}\;{\mathrm{sr}}^{-1} & ({F}_{\mathrm{SFD}}\geqslant 0.8\;\mathrm{MJy}\;{\mathrm{sr}}^{-1})\\ 0.0\;\mathrm{MJy}\;{\mathrm{sr}}^{-1} & ({F}_{\mathrm{SFD}}\lt 0.8\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}).\end{array}\right.\end{eqnarray} \tag{ 5 }$$

In the above expression, ${F}_{\mathrm{SFD}}$ is the $100\;\mu {\rm{m}}$ intensity of the SFD map. Lagache et al. (2000) reported the EBL at $100\;\mu {\rm{m}}$ as $0.78\;\pm \;0.21\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}.$ Matsuoka et al. (2011) correlated the intensities of the SFD map and optical diffuse light observed by Pioneer 10/11, and revealed a clear break around $0.8\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}.$ Based on these results, we assumes $0.8\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}$ for the EBL at $100\;\mu {\rm{m}},$ and subtract this amount from the region of ${F}_{\mathrm{SFD}}\geqslant 0.8\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}$ on the SFD map to obtain the $100\;\mu {\rm{m}}$ brightness associated with the interstellar dust. In the region of ${F}_{\mathrm{SFD}}\lt 0.8\;\mathrm{MJy}\;{\mathrm{sr}}^{-1},$ we set ${F}_{100}=0.0\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}.$

3.1.3. Integrated Starlight

To estimate the ISL of each region, we created integrated brightness maps of the 2MASS sources. The 2MASS PSC contains the photometry of approximately 470,000,000 objects covering $99.998\%$ of the sky, with accurate detections below the completeness limits ${\text{}}J=15.8$ and ${\text{}}{K}_{s}=14.3\;\mathrm{mag}$ (Cutri et al. 2003; Skrutskie et al. 2006).⁷

To convert the magnitudes of the 2MASS sources into DIRBE flux densities, we require the zero magnitude. As discussed by Cambrésy et al. (2001), the difference between the filters of the DIRBE and the 2MASS is negligible compared with the other uncertainties; hence, they need not be corrected. In addition, Levenson et al. (2007) correlated the integrated brightness of the 2MASS PSC sources against the DIRBE intensity in 40 high Galactic latitude regions. They reported common zero magnitudes at 1.25 and $2.2\;\mu {\rm{m}}$ of 1467 and $540\;\mathrm{Jy},$ respectively. Accordingly, these values are adopted as the zero magnitudes in the following analysis.

We must also apply the DIRBE beam at 1.25 and $2.2\;\mu {\rm{m}}$ to the 2MASS sources. An effective DIRBE beam for acquiring a daily map at $1.25\;\mu {\rm{m}}$ is illustrated in panel (a) of Figure 2. This beam profile measures the relative response of the DIRBE to a point source, and includes the sky scanning and data sampling effects. In the present analysis, the averaged beam should reflect the observation period of 6 months rather than the daily beam, because the intensity of the $\epsilon =90^\circ$ maps is the average of dozens of observations. As illustrated in panel (b) of Figure 2, the averaged beam shapes for the $\epsilon =90^\circ$ maps are estimated by averaging the daily beam profiles. Similar average profiles are obtained in both bands (1.25 and $2.2\;\mu {\rm{m}}$), with FWHMs of $\sim 1^\circ .$

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Since the beam shapes should not largely depend on the location in the CSC projection sky map (COBE DIRBE Explanatory Supplement 1998), we assume that each 2MASS source isotropically transfers its flux to the nearest 13 pixels on the map according to the averaged beam shape (panel (b) of Figure 2). Applying this scheme to the 2MASS sources at brightnesses below the completeness limit ($J\lt 15.8$ and ${K}_{s}\lt 14.3\;\mathrm{mag}$), we calculate the integrated brightness of each pixel at high Galactic latitudes ($| b| \gt 20^\circ$). According to the Explanatory Supplement to the 2MASS All Sky data Release and Extended Mission Products (Cutri et al. 2003), Galactic extinction below Galactic latitudes of 35° renders the stellar color redder than the intrinsic color. Since this effect attenuate the EBL and disrupt the linear combination of the fitting model (Equation (2)), we limit the following analysis to the high Galactic latitude regions ($| b| \gt 35^\circ$). Panels (b) and (b') of Figure 1 are integrated brightness maps of the 2MASS sources at $| b| \gt 20^\circ$ at 1.25 and $2.2\;\mu {\rm{m}},$ respectively. For comparison, the FSM map used by the DIRBE team (Arendt et al. 1998) is also shown (see panels (c) and (c') of Figure 1). In contrast to the FSM maps, wherein the surface brightness smoothly changes across the sky, the 2MASS-derived maps show clear fluctuations reflecting the astrometry and photometry of the actual sources.

In terms of the integrated brightness of the 2MASS sources (${F}_{i}(2\mathrm{MASS});$ see Figure 2), the total ISL term ${F}_{i}(\mathrm{ISL})$ is defined as

$$\begin{eqnarray}&&{F}_{i}(\mathrm{ISL})={c}_{i}{F}_{i}(2\mathrm{MASS})\end{eqnarray} \tag{ 6 }$$

where ${\text{}}{c}_{i}$ is a free parameter representing the integrated brightness of stars fainter than the limiting magnitude of the 2MASS. This formula assumes that the integrated brightness of the fainter stars and the brighter sources below the 2MASS detection limit have the same spacial distribution. In previous studies using the 2MASS data for star subtraction (e.g., Cambrésy et al. 2001; Wright 2001), the analyzed region was sufficiently small to assume isotropic ISL of fainter stars; thus the contributions of fainter stars were subtracted by star-counts models (e.g., Jarrett, SKY model). In contrast, the present analysis covers a wide field of the high-latitude Galactic sky, where the ISL of fainter stars and brighter sources should have the same spatial distribution. This justifies our use of Equation (6), which is free from the uncertainties introduced by the star-counts model. The systematic features of this simple model are discussed in Section 5.2.

The 2MASS PSC should contain faint galaxies that are not resolved as extended sources. The contributions of these faint objects should be isotropic in the sky and should be included in the EBL. Wright (2001) estimated that galaxies with magnitudes ${K}_{s}\lt 14.3\;\mathrm{mag}$ contribute approximately order of 0.12 and $0.14\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$ at 1.25 and $2.2\;\mu {\rm{m}},$ respectively. To derive the EBL intensity, we apply these small corrections to the isotropic term ${d}_{i}$ after the fitting procedure.

3.1.4. Isotropic Emission

Since the isotropic emission is assumed to be independent of the region, the term ${F}_{i}(\mathrm{Iso})$ is defined as

$$\begin{eqnarray}&&{F}_{i}(\mathrm{Iso})={d}_{i}\end{eqnarray} \tag{ 7 }$$

where ${d}_{i}$ is a free parameter.

Now the model brightness ${F}_{i}(\mathrm{Model})$ for ${F}_{i}(\mathrm{Obs})$ is given by

$$\begin{eqnarray}&&{F}_{i}(\mathrm{Model})={F}_{i}(\mathrm{ZL})+{F}_{i}(\mathrm{DGL})+{F}_{i}(\mathrm{ISL})+{F}_{i}(\mathrm{Iso})\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&=\;{a}_{i}{F}_{i}(\mathrm{Kel})+{b}_{i}{F}_{100}+{c}_{i}{F}_{i}(2\mathrm{MASS})+{d}_{i}.\end{eqnarray} \tag{ 9 }$$

Prior to fitting, we should remove pixels that might perturb the analysis. To suppress the large photometric uncertainty of bright stars, we mask the pixels around stars brighter than J = 5 and ${K}_{s}=4\;\mathrm{mag}$ on the CSC projection map at 1.25 and $2.2\;\mu {\rm{m}},$ respectively. In addition, we blank out the circular regions around the Magellanic Clouds and probable Galactic extended sources listed in the Explanatory Supplement to the 2MASS All Sky Data Release and Extended Mission Products (Cutri et al. 2003). Furthermore, we select regions with ${F}_{\mathrm{SFD}}\lt 10\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}$ (where the Galactic extinction is assumed negligible) and exclude outliers by applying 2 σ clipping to the integrated intensities of the 2MASS sources ${\text{}}{F}_{i}(2\mathrm{MASS}).$ Approximately 65% of the total pixels in the $| b| \gt 35^\circ$ region in both bands survive these masking procedures.

To determine the parameters ${a}_{i},$ ${\text{}}{b}_{i},$ ${\text{}}{c}_{i},$ and ${\text{}}{d}_{i},$ we minimize the following ${\text{}}{\chi }^{2}$ function in each band;

$$\begin{eqnarray}&&{\text{}}{\chi }_{i}^{2}=\displaystyle \sum _{j}\;\displaystyle \frac{{[{F}_{i}(\mathrm{Obs})-{F}_{i}(\mathrm{Model})]}^{2}}{{\sigma }_{i}^{2}}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&=\displaystyle \sum _{j}\;\displaystyle \frac{{[{F}_{i}(\mathrm{Obs})-{a}_{i}{F}_{i}(\mathrm{Kel})-{b}_{i}{F}_{100}-{c}_{i}{F}_{i}(2\mathrm{MASS})-{d}_{i}]}^{2}}{{\sigma }_{i}^{2}}\end{eqnarray} \tag{ 11 }$$

where "${\text{}}j$" refers to the pixels used in the fitting. The total uncertainty ${\text{}}{\sigma }_{i}$ in each pixel is calculated as follows:

$$\begin{eqnarray}&&{\text{}}{\sigma }_{i}^{2}={\sigma }_{i}{(\mathrm{Obs})}^{2}+{b}_{i}^{2}{\sigma }_{100}^{2}+{c}_{i}^{2}{\sigma }_{i}{(2\mathrm{MASS})}^{2}\end{eqnarray} \tag{ 12 }$$

where ${\text{}}{\sigma }_{i}(\mathrm{Obs}),$ ${\sigma }_{100},$ and ${\sigma }_{i}(2\mathrm{MASS})$ are the standard deviations of the intensities in the $\epsilon =90^\circ$ map, the intensities of the $100\;\mu {\rm{m}}$ emission, and the integrated intensity of the 2MASS sources, respectively. We adopt ${\sigma }_{100}=0.35\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}$ derived by Ienaka et al. (2013). The ${\text{}}{\sigma }_{i}(2\mathrm{MASS})$ at each pixel is calculated identically to the brightness of the 2MASS sources (see Section 3.1.3):

$$\begin{eqnarray}&&{\text{}}{\sigma }_{i}{(2\mathrm{MASS})}^{2}={[-0.4(\mathrm{log}10){10}^{-0.4{\text{}}{m}_{i}}{\sigma }_{{\text{}}{m}_{i}}{F}_{{\text{}}i0}]}^{2}\end{eqnarray} \tag{ 13 }$$

where ${\text{}}{m}_{i},$ ${\text{}}{\sigma }_{{m}_{i}},$ and ${\text{}}{F}_{i0}$ denotes the magnitude of each 2MASS source, the uncertainty of this magnitude, and the zero magnitude derived by Levenson et al. (2007), respectively. Sources lacking a photometric uncertainty entry in the 2MASS PSC are assigned an uncertainty of $0.5\;\mathrm{mag}.$

The parameters determined by the fitting at 1.25 and $2.2\;\mu {\rm{m}}$ in the $| b| \gt 35^\circ$ region are summarized in Table 1. Owing to the large sample size (over 100,000 points), the statistical uncertainty in each parameter is very much smaller than the determined value. In Figure 3, the sky brightness obtained by the DIRBE observations is decomposed into the ZL, DGL, ISL, and isotropic emission according to the linear combination model (Equation (2)). Filled circles represent the weighted means of the points within an arbitrary x-direction bin. In further discussion, these weighted means will be assumed as representative values.

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Table 1.  Results of the Fitting in Each Region

Band	Region	Number of	${a}_{i}$	${\text{}}\nu {b}_{i}$	${\text{}}{c}_{i}$	${\text{}}\nu {d}_{i}$
($\mu {\rm{m}}$)	(deg)	Pixels	(dimensionless)	$({\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{MJy}}^{-1})$	(dimensionless)	$({\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1})$
1.25	$| b| \gt 35$	116578	1.0087 ± 0.0001	4.79 ± 0.02	1.0238 ± 0.0003	60.03 ± 0.08
2.2	$| b| \gt 35$	119394	1.0450 ± 0.0002	1.49 ± 0.01	1.0333 ± 0.0004	27.54 ± 0.04
1.25	$| b| \gt 35,0\lt {\text{}}l\lt 60$	18557	1.0017 ± 0.0004	8.67 ± 0.06	1.0206 ± 0.0009	64.51 ± 0.25
2.2	$| b| \gt 35,0\lt {\text{}}l\lt 60$	18852	1.0350 ± 0.0006	2.79 ± 0.03	1.0389 ± 0.0011	27.90 ± 0.13
1.25	$| b| \gt 35,60\lt {\text{}}l\lt 120$	19714	0.9956 ± 0.0003	1.91 ± 0.05	1.0276 ± 0.0008	64.25 ± 0.18
2.2	$| b| \gt 35,60\lt {\text{}}l\lt 120$	20309	1.0286 ± 0.0004	0.94 ± 0.03	1.0409 ± 0.0010	29.15 ± 0.10
1.25	$| b| \gt 35,120\lt {\text{}}l\lt 180$	19582	0.9993 ± 0.0003	5.60 ± 0.04	1.0236 ± 0.0009	68.51 ± 0.20
2.2	$| b| \gt 35,120\lt {\text{}}l\lt 180$	20213	1.0534 ± 0.0005	1.36 ± 0.02	1.0288 ± 0.0011	27.79 ± 0.10
1.25	$| b| \gt 35,180\lt {\text{}}l\lt 240$	20275	1.0229 ± 0.0003	4.07 ± 0.04	0.9986 ± 0.0008	55.97 ± 0.20
2.2	$| b| \gt 35,180\lt {\text{}}l\lt 240$	20748	1.0492 ± 0.0005	1.26 ± 0.02	1.0196 ± 0.0011	28.88 ± 0.11
1.25	$| b| \gt 35,240\lt {\text{}}l\lt 300$	19516	1.0224 ± 0.0002	0.64 ± 0.06	1.0223 ± 0.0008	53.92 ± 0.19
2.2	$| b| \gt 35,240\lt {\text{}}l\lt 300$	20011	1.0593 ± 0.0004	$-0.07\pm 0.04$	1.0306 ± 0.0010	26.12 ± 0.10
1.25	$| b| \gt 35,300\lt {\text{}}l\lt 360$	18934	1.0030 ± 0.0003	5.58 ± 0.05	1.0421 ± 0.0008	58.62 ± 0.23
2.2	$| b| \gt 35,300\lt {\text{}}l\lt 360$	19261	1.0372 ± 0.0005	2.04 ± 0.03	1.0524 ± 0.0010	25.87 ± 0.12

Note. The symbols in the column headings are defined in Section 3. Error in each component is the statistical uncertainty derived by the fitting.

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As shown in panels (b) and (b') of Figure 3, the diffuse near-IR light is positively linearly correlated with the interstellar $100\;\mu {\rm{m}}$ emission at both 1.25 and $2.2\;\mu {\rm{m}},$ and the correlations are significant. This indicates that the DGL component certainly exists, even at high Galactic latitudes ($| b| \gt 35^\circ$). Moreover, the linear correlation continues through the low to high $100\;\mu {\rm{m}}$ intensity region (${F}_{100}\lesssim 9\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}$), indicating that the $100\;\mu {\rm{m}}$ emission also well-tracks the DGL at these wavelengths. This trend was discovered owing to the wide sky coverage of the DIRBE maps, which have wide dynamic range of the $100\;\mu {\rm{m}}$ intensity. In conclusion, by combining the precise star subtraction from the 2MASS PSC data with wide-field coverage of the sky, we can find the near-IR DGL at 1.25 and $2.2\;\mu {\rm{m}}$ even at low column density.

As shown in panels (a), (a'), (c), and (c') of Figure 3, the ZL and ISL are also decomposed from the sky brightness with high linearity. Therefore, the assumed linear combination model of the sky brightness is appropriate for our purpose.

Although the residuals ${F}_{i}(\mathrm{Obs})-{F}_{i}(\mathrm{Model})$ in each panel of Figure 3 appears to be functions of ${F}_{i}(\mathrm{Kel}),$ ${F}_{100},$ and ${F}_{i}(\mathrm{ISL}),$ they deviate from the best-fit line by no more than ±10 and $\pm 5\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$ at 1.25 and $2.2\;\mu {\rm{m}},$ respectively. As shown in Table 2, these deviations are within ±2% of the typical DIRBE intensity ${F}_{i}(\mathrm{Obs})$ in both bands. Therefore, the large sample size of high quality DIRBE data has enabled a very precise analysis. The possible origins of the systematic features in the residuals are discussed in Section 5.2.

Table 2.  Typical Intensity of Each Component Determined by the Fitting at $| b| \gt 35^\circ$

Component (${\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$)	$1.25\;\mu {\rm{m}}$	$2.2\;\mu {\rm{m}}$
${F}_{i}(\mathrm{ZL})={a}_{i}{F}_{i}(\mathrm{Kel})$	544 ± 199	208 ± 74
${F}_{i}(\mathrm{DGL})={b}_{i}{F}_{100}$	4.87 ± 6.06	1.50 ± 1.87
${F}_{i}(\mathrm{ISL})={c}_{i}{F}_{i}(2\mathrm{MASS})$	175 ± 80	66.2 ± 34.7
${F}_{i}(\mathrm{Iso})={d}_{i}$	60.03 ± 0.08	27.54 ± 0.04
${F}_{i}(\mathrm{Obs})$	787 ± 220	304 ± 87

Note. Except for ${F}_{i}(\mathrm{Iso}),$ each component is represented by its average and standard deviation of the samples in the $| b| \gt 35^\circ$ region, where the regions around the 2MASS sources brighter than J = 5 and K_s = 4 mag at 1.25 and $2.2\;\mu {\rm{m}}$ are masked, respectively.

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The statistical uncertainty of the parameters determined in the minimum ${\chi }^{2}$ analysis is not the only uncertainty in each component. Other uncertainties include the parameter variation among different regions, the absolute gain of the DIRBE, the uncertainty in the faint galaxies in the 2MASS PSC, and the systematic uncertainty in the Kelsall model.

4.2.1. Regional Parameter Variations

To understand the parameter variation among different regions with similar dynamic ranges of each component, we divide $| b| \gt 35^\circ$ region into six Galactic longitude fields, i.e., $0^\circ \lt l\lt 60^\circ ,60^\circ \lt l\lt 120^\circ ,120^\circ \lt l\lt 180^\circ ,$ $180^\circ \;\lt l\lt 240^\circ ,240^\circ \lt l\lt 300^\circ ,$ and $300^\circ \lt l\lt 360^\circ .$ In each region, we apply the fitting procedure described in Section 3.2. The results for each region are shown in Table 1. Because each region contains over 10,000 points, the statistical uncertainty in this analysis remains small. Figure 4 presents the parameters obtained in the six regions and in the $| b| \gt 35^\circ$ region as functions of Galactic longitude. Reasonably, the parameters determined by the fitting in the six regions are randomly distributed around the parameters determined in the $| b| \gt 35^\circ$ field, showing some degree of scatter. The standard deviation of the parameter values in the six regions is adopted as a conservative uncertainty and is listed for each parameter in the row "Scatter" in Table 3.

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Table 3.  Uncertainty Budget Associated with Each Parameter

	${a}_{i}$ (dimensionless)		${\text{}}\nu {b}_{i}$ (${\mathrm{nWm}}^{-2}{\mathrm{MJy}}^{-1}$)		${\text{}}{c}_{i}$ (dimensionless)		${\text{}}\nu {d}_{i}$ (${\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$)
Band ($\mu {\rm{m}}$)	1.25	2.2	1.25	2.2	1.25	2.2	1.25	2.2
Statistical	0.0001	0.0002	0.02	0.01	0.0003	0.0004	0.08	0.04
Scatter	0.012	0.012	2.88	0.97	0.014	0.011	5.66	1.37
Gain	⋯	⋯	⋯	⋯	⋯	⋯	1.86	0.85
Galaxies	⋯	⋯	⋯	⋯	⋯	⋯	0.12	0.14
ZL model	⋯	⋯	⋯	⋯	⋯	⋯	15	6
Quadrature sum	0.012	0.012	2.88	0.97	0.014	0.011	16.14	6.21
Result ($| b| \gt 35^\circ$)	1.0087	1.0450	4.79	1.49	1.0238	1.0333	60.03	27.54

Note. Symbols in the column headings are defined in Section 3.

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4.2.2. Uncertainty in the Absolute Gain of DIRBE

4.2.3. Uncertainty Associated with the Faint Galaxies in the 2MASS PSC

4.2.4. Systematic Uncertainty in the Kelsall Model

4.2.5. Total Uncertainty

As shown in Table 3, the parameter ${a}_{i}$ at $1.25\;\mu {\rm{m}}$ (determined as 1.0) lies within the uncertainty limits, indicating that the Kelsall model well-reproduces the time variation of the sky brightness measured by the DIRBE at $1.25\;\mu {\rm{m}}.$ The parameter ${a}_{i}$ at $2.2\;\mu {\rm{m}}$ exceeds 1.0 by approximately $4\%.$ On average, a 4% variation in the Kelsall model corresponds to $8\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1},$ which is slightly larger than the claimed systematic uncertainty in the model ($6\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$). This suggests that the Kelsall model underestimates the ZL intensity in this band. For one thing, in the fitting procedure, the Kelsall model was sampled a sky pixel every $\sim 5^\circ$ or $\sim 10^\circ$ as a spatial grid (not used the all pixels) to avoid the excessive computational requirements. In addition, comparing the parameter values at 1.25 and $2.2\;\mu {\rm{m}},$ some of the parameters determined in the Kelsall model, especially the phase function parameter ${\text{}}{C}_{\mathrm{2,2}}$ and the Albedo ${\text{}}{A}_{2},$ seem anomalous $2.2\;\mu {\rm{m}}$ (see Table 2 of Kelsall et al. 1998). Specifically, at $2.2\;\mu {\rm{m}},$ ${\text{}}{C}_{\mathrm{2,2}}$ is 3 times smaller than at $1.25\;\mu {\rm{m}}$ and ${\text{}}{A}_{2}$ is unnaturally larger than that at $1.25\;\mu {\rm{m}}.$ These results may change the spatial distribution of the ZL brightness in the Kelsall model, and may also explain why ${a}_{i}$ deviate from 1.0 at $2.2\;\mu {\rm{m}}.$

The uncertainty in the parameter ${b}_{i}$ is dominated by scatter among the different regions and exceeds 50% of the result. This large error is attributed to the low typical brightness of the DGL (1–2 orders of magnitude fainter than the other components; Table 2). In that situation, if the SFD intensity spatially correlates with that of the Kelsall model or the ISL to some extent, some of the DGL might be absorbed by the other components in the fitting process. Investigating this possibility is beyond the scope of the present study; instead, we conservatively estimate the uncertainty in the DGL as the scatter in the six regions. Remarkably, the present analysis identified the DGL despite its much lower brightness than that of the other components, by virtue of the well-calibrated all-sky maps of the DIRBE. The DGL results in the optical and near-IR, determined in the present and previous studies, are compared in Section 5.3.

The parameter c_i exceeds 1.0 by 1%–4% in both bands. Assuming that the ISL of the sources brighter and fainter than the 2MASS detection limit have the same spatial distribution, this excess is presumably contributed by the fainter stars. This result also indicates that the zero magnitude derived by Levenson et al. (2007) is appropriate for converting the 2MASS magnitude to the DIRBE flux in the present study.

The determined parameter ${d}_{i}$ is discussed in Section 5.4.

Figure 5 illustrates the residuals ${F}_{i}(\mathrm{Obs})-{F}_{i}(\mathrm{Model})$ derived from the fitting in the $| b| \gt 35^\circ$ region as functions of Galactic latitude $b$ and ecliptic latitude $\beta .$ In general, the dependence of the residuals on Galactic latitude traces the ISL or the DGL, whose intensities are also functions of Galactic latitude. On the other hand, the dependence on ecliptic latitude is expected to measure the accuracy of the ZL model.

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As shown in panels (a) and (a') of Figure 5, the residuals tend to increase toward low Galactic latitudes, suggesting that some component is missed at these latitudes. Similar trends are observed when the residuals are plotted against the integrated intensity of the 2MASS sources (panels (c) and (c') in Figure 3). In this case, the residuals increase toward regions of higher intensity, where the data of the lower Galactic latitude fields are more dominant.

The phenomenon might stem from the contribution of stars with no entry in the 2MASS PSC, possibly because they were masked by their nearest bright sources. According to the Explanatory Supplement to the 2MASS All-sky Data Release and Extended Mission Products (Cutri et al. 2003), masking around bright stars can filter faint sources from the detection process. Although the masked area in the all-sky averages to $0.25\%$ and $0.43\%$ at 1.25 and $2.2\;\mu {\rm{m}},$ respectively, the fraction of such regions tends to increase toward lower Galactic latitudes as the number density of bright sources increases. In addition, the 2MASS compensated for saturation caused by bright stars by fitting the unsaturated wings of their intensity profiles. This suggests that the 2MASS PSC could have missed faint stars.

The simply modeled ISL term ${F}_{i}(\mathrm{ISL})={c}_{i}{F}_{i}(2\mathrm{MASS})$ might also contribute to the latitude dependence of the residuals. If the integrated intensities of the bright and faint stars (below the detection limit of the 2MASS) have different spatial distributions, the model assumption is not strictly valid. Although the dominant cause of the features in the residuals cannot be determined, the amplitude of the residuals is within±10 and $\pm 5\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$ at 1.25 and $2.2\;\mu {\rm{m}},$ respectively. Within these ranges, the ${d}_{i}$ terms are certainly isotropic. The origin of the residuals' dependence on $b$ requires searching by an all-sky with superior sensitivity and spatial resolution to the 2MASS.

As illustrated in panels (b) and (b') of Figure 5, the dependence of the residuals on ecliptic latitude, especially the turbulence near the ecliptic plane, may reflect the incompleteness of the Kelsall model. When the residuals are plotted against the Kelsall model ${\text{}}{F}_{i}(\mathrm{Kel})$ (panels (a) and (a') of Figure 3), distortion appears in the higher intensity region (comprising fields of lower ecliptic latitudes). Cambrésy et al. (2001), who similarly subtracted the ZL using the Kelsall model, reported the same trend. These results highlight the difficulty in applying the ZL model near the ecliptic plane, where the distribution of the zodiacal dust (including the dust bands and the circumsolar ring) becomes complex.

The effects of these latitude dependences are naturally included in the scatter of the fitting results among the different regions (Figure 4). Therefore, the fluctuations related to Galactic or ecliptic latitude are not added to the uncertainty budget.

We now discuss the spectrum of the parameter ${b}_{i}$ in the optical and near-IR. The four colored curves in Figure 6 are synthetic DGL spectra calculated by Brandt & Draine (2012) based on two estimates of the ISRF continuum and two dust models; namely, the Zubko et al. (2004) and Weingartner & Draine (2001) models, hereafter referred to as ZDA04 and WD01, respectively. The WD01 model assumes that the half-mass grain radius ${a}_{0.5}$ (denoting that 50% of the mass is contributed by grains with radii $a\gt {a}_{0.5}$) is $\sim 0.12\;\mu {\rm{m}}$ for both silicate and carbonaceous grains (Draine 2011). In the ZDA04 model, grains with radii $a\gt 0.2\;\mu {\rm{m}}$ comprises a small proportion of the mass, and the half-mass–radius differs between carbonaceous grains (${a}_{0.5}\sim 0.06\;\mu {\rm{m}}$) and silicate grains (${a}_{0.5}\sim 0.07\;\mu {\rm{m}}$). Consequently, the WD01 creates a redder scattered spectrum than the ZDA04 at far-optical and near-IR wavelengths. The local ISRF continua are estimated either from Mathis et al. (1983) with de-reddening of the original ISRF of the MMP83 (see Brandt & Draine 2012 for details) or from a synthesis model of solar-metallicity star populations (Bruzual & Charlot 2003). The de-reddening correction and Bruzual–Charlot model are hereafter referred to as MMP83 and BC03, respectively. In the latter, the star formation rate is proportional to $\mathrm{exp}(-t/5\;\mathrm{Gyr}),$ where t denotes the star formation timescale in units of $\mathrm{Gyr}.$

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In the optical region, Figure 6 plots the results collected by Pioneer 10/11 (Matsuoka et al. 2011) (open circles) and toward a high Galactic latitude translucent cloud MBM32 (Ienaka et al. 2013) (triangles). The Pioneer 10/11 results were obtained in the same field of the sky as the present analysis ($| b| \gt 35^\circ$). In the near-IR, Figure 6 plots our results (filled red circles), the mean of six small regions from CIBER (Arai et al. 2015) (squares), and the AKARI results (Tsumura et al. 2013b), collected at relatively low Galactic latitudes ($5^\circ \lt | b| \lt 15^\circ$) (asterisks).

Around $1.25\;\mu {\rm{m}},$ our result is marginally consistent with the CIBER results, considering the uncertainties in the measurements. This suggests that results acquired in local regions represent the wider region at high Galactic latitudes. At $2.2\;\mu {\rm{m}},$ our result is consistent with the AKARI results, allowing for the uncertainties. In contrast to our results at high Galactic latitude ($| b| \gt 35^\circ$), the AKARI results were taken at lower galactic latitude ($5^\circ \lt | b| \lt 15^\circ$). Whether the relationship between the parameter ${b}_{i}$ and Galactic latitude results from the wide scatter of ${b}_{i}$ among the different regions (Section 5.1) is difficult to determine.

The green and blue curves show the spectra of the scaled ZDA04 and WD01, respectively. Since the original ZDA04 and WD01 models underestimate the observed ${b}_{i}$ by a factor of 2, these models are arbitrarily scaled to the optical results (Matsuoka et al. 2011; Ienaka et al. 2013) by factors of 1.9 and 1.7, respectively. Ienaka et al. (2013) suggested two possible explanations for this discrepancy: deficient UV photons in the ISRF and underestimation of the assumed albedo of the dust grains in the models. Combining the results in the optical, CIBER, and the present study, we find that the scaled ZDA04 provides a better fitting spectrum than WD01 at $1-2\;\mu {\rm{m}},$ implying a bluing of the DGL spectrum in this wavelength range. In contrast, the spectrum derived from AKARI significantly exceeds the ZDA04 and WD01 spectra at longer wavelengths ($\gtrsim 3\;\mu {\rm{m}}$), possibly because it includes the thermal emission at low Galactic latitudes, whereas the ZDA04 and WD01 spectra include only the scattered light component. At shorter wavelengths ($\lesssim 3\;\mu {\rm{m}}$), the observed DGL can be well-fitted to the model spectra containing the scattered component alone.

We estimate the EBL intensity from the derived parameter ${\text{}}{d}_{i}.$ This estimate adds the contribution of the faint galaxies appearing in the 2MASS PSC back to ${\text{}}{d}_{i}.$ Assuming the faint-galaxy contribution estimated by Wright (2001) and the uncertainty in ${d}_{i}$ (Table 3), the EBL intensity at 1.25 and $2.2\;\mu {\rm{m}}$ is estimated as 60.15 ± 16.14 and $27.68\pm 6.21\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1},$ respectively.

5.4.1. Isotropy test

5.4.2. Comparison with Other Studies

Figure 7 compares the resultant EBL with those of previous studies. Using the DIRBE data, Cambrésy et al. (2001) derived the EBL at 1.25 and $2.2\;\mu {\rm{m}},$ by subtracting the Galactic stars by 2MASS and the ZL by the Kelsall model. These authors targeted regions with low intensity of the dust emission (DIRBE $240\;\mu {\rm{m}}$ brightness ${I}_{240}\lt 3\;\mathrm{MJy}\;{\mathrm{sr}}^{-1}$). In such regions, the expected DGL brightness at 1.25 and $2.2\;\mu {\rm{m}}$ is $\lesssim 7$ and $\lesssim 2\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1},$ respectively, assuming the parameter ${b}_{i}$ determined in the present study and the conversion factor between the 100 and $240\;\mu {\rm{m}}$ intensities (1.297; see Table 4 of Arendt et al. 1998). For this reason, Cambrésy et al.'s (2001) study ignored the DGL. In addition, the F₁₀₀ histograms (panels (b) and (b') of Figure 3) show that regions of lower $100\;\mu {\rm{m}}$ intensity (and DGL brightness) dominate in the sky. Therefore, the EBL results derived in this study are reasonably consistent with those obtained by Cambrésy et al. (2001), despite the lack of any quantitative DGL evaluation in the latter study. However, Cambrésy et al. (2001) noticed fluctuations in EBL with ecliptic latitude, which they attributed to a small DGL component at $1.25\;\mu {\rm{m}}$ (see Figure 5 of Cambrésy et al. 2001).

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Using the FSM for the starlight subtraction, Hauser et al. (1998) derived the EBL intensity at high Galactic and ecliptic latitudes. Although their results are plotted as 95% confidence upper limits in Figure 7, their direct values are significantly smaller than ours; $33.0\pm 21$ and $14.9\pm 12\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$ at 1.25 and $2.2\;\mu {\rm{m}},$ respectively (see Table 2 of Hauser et al. 1998). This discrepancy can be explained by the following two things related to the ISL evaluation. At first, in converting the magnitudes of the sources into DIRBE flux densities, the present study adopts the zero magnitude of 1467 and 540 Jy at 1.25 and $2.2\;\mu {\rm{m}},$ respectively, but Hauser et al. (1998) used a higher one, i.e., 1547 and 612.3 Jy at 1.25 and $2.2\;\mu {\rm{m}},$ respectively (COBE DIRBE Explanatory Supplement 1998). We used the zero magnitudes derived by Levenson et al. (2007), who correlated the intensity of the 2MASS-derived ISL with that of the DIRBE and corrected the zero magnitude to fit the photometric scale of the 2MASS to that of the DIRBE. Therefore, the zero magnitudes we adopted are suitable to estimate the ISL contribution in the DIRBE data. Next, Wright & Reese (2000) suggested that the Wainscoat et al. (1992) star-counts model, which is the basis of the FSM, overestimates the counts by $\sim 10\%$ in the $6\lt K\lt 10$ range at high Galactic latitudes, compared with the 2MASS. This is within the 10%–15% uncertainty of the FSM, estimated in Arendt et al. (1998). Considering these differences associated with the ISL estimation, the ISL intensity in Hauser et al. (1998) can be higher than that in the present study by ∼15% and $\sim 20\%$ at 1.25 and $2.2\;\mu {\rm{m}},$ respectively. These percentages correspond to ∼26 and $\sim 13\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$ at 1.25 and $2.2\;\mu {\rm{m}},$ respectively, assuming the ISL intensity derived in the present study (Table 2). This overestimation of the ISL in Hauser et al. (1998) well explains the resultant EBL differences between Hauser et al. (1998) and the present study at both bands.

In the EBL measurement, the removal of the ZL from the sky brightness is controversial, as multiple ZL models are available. For instance, Gorjian et al. (2000), Wright (2001), and Levenson et al. (2007) used the ZL model based on Wright (1998), whereas Cambrésy et al. (2001) and the present study adopted the Kelsall model. As noted by Levenson et al. (2007), the ZL intensity at the ecliptic pole at 1.25 and $2.2\;\mu {\rm{m}}$ is ∼22 and $\sim 5\;{\rm{n}}\;{\rm{W}}\;{{\rm{m}}}^{-2}\;{\mathrm{sr}}^{-1}$ lower in the Kelsall model than in Wright's (1998) model, respectively. Consequently, the difference between the two models tends to be larger at $1.25\;\mu {\rm{m}}$ than at $2.2\;\mu {\rm{m}}.$ As shown in Figure 7, the resultant EBL obtained with the Kelsall model can be a few times lower than that obtained with the Wright model especially at $1.25\;\mu {\rm{m}}.$ At $2.2\;\mu {\rm{m}},$ the results of both models converge within their uncertainties. To eliminate the uncertainty introduced by the ZL model itself and its variation, we must observe outside the ZL cloud.

Note that the present decomposition analysis cannot identify where the isotropic emission including the EBL comes from. For example, if the isotropic component associated with the ZL exists, it may contribute to the "EBL" called in this paper. Actually, the Kelsall model was developed to fit to the seasonal variation of the DIRBE sky brightness, ignoring the uniform component if exists. Hauser et al. (1998) also emphasized that the Kelsall model cannot uniquely determine the true ZL signal; in particular an arbitrary isotropic component could be added to the model.

5.4.3. Implications of the Present Results