Spatial Variations and Breaks in the Optical–Near-infrared Spectra of the Pulsar and Pulsar Wind Nebula in Supernova Remnant 0540–69.3

The top left panel of Figure 2 shows an image obtained by summing over the reddest quarter of the continuum wavelength range (7236–7992 Å, extracted as described in Section 3). We use a Hubble Space Telescope (HST) image (F547M from 2005, shown in De Luca et al. 2007) as guidance to identify and exclude the field stars in the following analysis. The continuum flux image in the top left panel of Figure 2 shows a radial decrease of flux levels, with the bright pulsar being in the center. This image also shows clear signs of a more luminous region in the SW region around A, which has been identified as a hotspot or blob in several earlier studies (e.g., De Luca et al. 2007; Lundqvist et al. 2011). In our analysis, this region will be referred to as the "blob region." Additionally, a jet-like feature (e.g., Gotthelf & Wang 2000) is thought to be located in the NW, though we cannot confirm this from the continuum flux image (Figure 2 top left panel, region around B). However, we call this the "jet region."

As described in Section 3, we also deconvolved the reddest quarter of the MUSE continuum emission. The result is shown in the top right panel of Figure 2. Generally, the deconvolved PWN appears more asymmetric than the corresponding nonconvolved image (top left panel of Figure 2). As expected, the deconvolution process results in more prominent point sources (pulsar and field stars) and reveals possible substructures in the inner parts of the PWN. The most significant substructure being the arc-like feature in the east. Similarly, the blob region in the SW appears more accentuated after the deconvolution.

Deconvolving a diffuse source like PWN 0540 is uncertain, because deconvolution algorithms are predominantly highlighting point sources. This behavior may lead to artificial features in the final image. Preventing these artifacts propagating to the subsequent analyses and results, we manage the MUSE PSF wavelength dependence by convolution (as described in Section 3). An image of the entire convolved MUSE continuum range is presented in the bottom left panel of Figure 2. Despite the decreased spatial resolution due to convolution, we can still identify all the features that are present in the nonconvolved image (top left panel of Figure 2). Additionally, the areas dominated by background emission are more apparent in the convolved image due to the background subtraction performed before the convolution process. As mentioned in Section 3, we use the convolved MUSE data for all subsequent spectral analyses.

As an integral-field spectrograph, every spaxel in the MUSE data contains a spectrum. We are thus able to treat every spaxel separately and get a continuum spectrum for each spaxel. After constructing the continuum spectra, we fit them with a single-PL model F_ν ∝ ν^−α , where α is the spectral index.

In Figure 5, we present example fits (to demonstrate the quality of the fits) from three different spaxels, which we refer to as pulsar, A (located in the blob region, bottom left panel of Figure 2), and B (located in the jet region). The example fits show that the continuum emission from each of these spaxels from different regions of the PWN is well described by a single PL in the MUSE wavelength range. We thus proceed to fit all the spaxels in the nebula with the single-PL model. We note that the example fit from the pulsar spaxel shows hints of a broken PL (bPL), and we investigate this further in Section 4.2, where we study the entire pulsar region instead of a single spaxel.

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As can be seen in the bottom right panel of Figure 2, the spectral index has clear spatial variation across the remnant. We mask regions where α < 0, its uncertainty σ_α > 0.2, and where there are bright field stars. As a general trend, the spectral index is high (i.e. soft) in the central region, α ∼ 1.1, and decreases (i.e. hardens) toward the edges of the central nebula, down to α ∼ 0.1. This means that, in general, the spectral index hardens radially while going toward the outer regions of the central part of the remnant.

We note that in the south, close to the blob region (and spaxel A), some residual emission of the bright field star remains after the masking (bottom right panel of Figure 2). The spectral index of spaxel A in Figure 5 might therefore be slightly harder than the "pure" PWN emission at that location (by ∼0.04), though we note that the spectrum is still well described by the PL model.

We detect several structures that have different spectral indices, for instance, an axis spanning from the NE to the SW (including the blob region) can be identified by having a nearly constant spectral index α ∼ 1.1–1.2. This, in turn, could be an indication of a part belonging to a torus-like structure, which is also visible as a bright region in the flux image, especially in the SW (Figure 2, bottom left panel). We refer to this structure as the "torus" hereafter. Previous studies have found evidence for the torus in the optical and X-rays (Gotthelf & Wang 2000; Morse 2003).

In the east at distances ∼ 10 and ∼20 from the pulsar, we identify two regions with a slightly higher spectral index (α ∼ 1.2) overlapping with the torus. In addition, to the west of the pulsar, a larger region of significantly higher spectral index of α ∼ 1.2–1.8 can be observed. This region is overlapping with a field star in the SW but otherwise seems to belong to the PWN.

Additionally, the region in the NW (surrounding B) with α ∼ 0.6–0.8 could be a feature of a jet, as suggested by, e.g., Gotthelf & Wang (2000) and Petre et al. (2007). This kind of a structure is not apparent in the flux levels in the bottom left panel of Figure 2 as mentioned above. Also, the pulsar, the brightest point source within the central regions of the PWN, is not apparent in the spectral index map.

We study the relation between the continuum flux and the spectral index in Figure 6. The pulsar region, a 3 × 3 spaxel square around the pulsar with α ∼ 1.0 and normalized flux > 0.8, can be distinguished from the other regions by its high flux. As expected for a point source, the spectral index in the pulsar region remains constant.

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The torus, the region that in flux images is mainly apparent in the SW (overlapping A) but continues also ∼10 to the NE, has an approximately constant spectral index (α ∼ 1.1), while the normalized-flux values vary (∼0.1–0.8). This results in a plateau-like feature in the α-normalized-flux space as can be seen in Figure 6.

Considering the spaxels with 1.2 < α < 1.35 in the high-α region reveals that this subregion most likely belongs to the torus. This connection can be made by studying the plateau-like shape in Figure 6, where the top of the plateau is smoothly formed by spaxels belonging to the 1.2 < α < 1.35 subregion. The rest of the spaxels in the high-α region in the west and SW, where α > 1.35, behave like a background region; low flux values (≲0.1) with a highly variable spectral index.

At these low flux levels, Figure 6 shows a wide range of continuum slopes, ranging between α ∼ 0 and α ∼ 2. We interpret these spaxels, also located at the edges of the PWN, to suffer from systematic effects and thus consider them more uncertain, even though the formal fit uncertainties are σ_α < 0.2. Finally, for the rest of the PWN, the spectral index and flux values seem to correlate when going radially outward from the edges of the torus to the the edges of the PWN.

L21 present 3D reconstructions of several emission lines from the ejecta in SNR 0540. Here we compare the line emission from [O iii] λ5007 and [S iii] λ9069 with our results for the (deconvolved) continuum emission, presented in Figure 7. Overall, the continuum emission is stronger than the line emission in the center of the PWN. However, there is also significant overlap in some regions, especially on the western side, which may be caused by projection effects.

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The [O iii] λ5007 emission is more extended in the east–west direction than the [S iii] λ9069 emission. This is clear when compared to the continuum emission (middle panels of Figure 7), for instance, the two [O iii]-emitting blobs in the east are located close to the edges of the continuum-emitting region (as seen along the line of sight). In the case of [S iii] λ9069, the brightest part of the corresponding blob seems to have a more central location when compared to the continuum-emitting region (along the line of sight). In the west, the brightest continuum emission (the blob region) and the brightest emission from both lines are roughly overlapping. This agrees with the results in Lundqvist et al. (2011), where the emission of the [S ii] lines is mainly concentrated to the SW overlapping with the continuum emission in that region (the [S ii] λ λ 6716, 6731 and [S iii] λ9069 are very similar; see L21).

We also estimate a possible jet direction, ∼25° to the west from the north and inclined by ∼10° away from the observer, according to the NW cavity/hole angle of the 3D emission line maps. Also Lundqvist et al. (2022, in their Figure 6) report similar evidence for the jet direction by following the cavities of the [O iii] λ5007 emission line provided by Sandin et al. (2013). The cavities of the emission lines in the SE (L21), in turn, seem to be approximately antiparallel to the jet in the NW. In addition to the emission line cavities in the possible jet (and counterjet) direction, both emission lines have less emission at the center, in the NE, and significantly low emission farther in the SW.

The emission line maps can also be compared to the spatial variation of the spectral index (Figure 7, right panels). Noticeable is that the brightest line-emitting regions do not exactly overlap with the hardest (low-α) continuum emission. In the SW, the brightest part of the [O iii] emission is located in the softest (high-α) continuum emission belonging to the torus. Also, the ring-like structure that is seen in the SW of the [S ii] emission is not apparent in the spectral index map, possibly due to a field star blocking the line-of-sight view. Interestingly, the cavities occurring in the NE in the line emission maps seem to be filled with a spur of softer spectral index of the NE arc.

In Figure 8, we overlay the HST linear polarization vectors from Lundqvist et al. (2011) onto the MUSE spectral index map from Figure 2 (bottom right panel). The polarization is low $\left(\sim 10 \% \right)$ in the central parts. We note that along the torus, the direction of polarization is different compared to the rest of the PWN, the angle being approximately perpendicular to the NW–SE axis (although Lundqvist et al. 2011 report variations of the degree and angle of polarization along the torus on smaller scales). Generally, the degree of polarization increases toward the outer parts of the remnant, tracing the spatial hardening of the spectral index.

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Here we study the wider wavelength range from UVB to NIR available with X-shooter. Figure 9 shows a fit to the X-shooter PWN spectrum extracted from the nebula region, spanning 528 around the pulsar, see Figure 1. A single-PL fit, which describes the MUSE wavelength range well, is not sufficient when inspecting this wider wavelength range provided by X-shooter. Consequently, we fit the X-shooter spectra with a bPL model

$$\begin{eqnarray}\begin{array}{c}{F}_{\nu }=\left\{\begin{array}{cc}A\cdot {\left({\textstyle \tfrac{\nu }{{\nu }_{{\rm{b}}}}}\right)}^{-{\alpha }_{1}}, & \text{}{\rm{i}}{\rm{f}}\text{}\,\nu \lt {\nu }_{{\rm{b}}},\\ A\cdot {\left({\textstyle \tfrac{\nu }{{\nu }_{{\rm{b}}}}}\right)}^{-{\alpha }_{2}}, & \text{}{\rm{i}}{\rm{f}}\text{}\,\nu \gt {\nu }_{{\rm{b}}},\end{array}\right.\end{array}\end{eqnarray} \tag{ 1 }$$

where A is the amplitude, ν_b is the break frequency, and α₁ and α₂ the low- and high-frequency spectral indices, respectively. We note that this bPL fit describes the data well, see the residuals in Figure 9 (and fit results for both PL and bPL models in Table 1), though there are some remaining discrepancies that may be caused by the systematic effects discussed in Section 5. We compare the relative goodness of the single-PL and bPL fits with an F-test. The F-test yields a value >100 with a p-value of ≪0.01 in favor of the bPL. The best-fit bPL model consists of a flatter slope in the lower-frequency part of the spectrum (${\alpha }_{1}={0.84}_{-0.01}^{+0.01}$) that breaks at ${\mathrm{log}}_{10}\left({\nu }_{{\rm{b}}}\right)={14.67}_{-0.02}^{+0.03}$ Hz to a slightly steeper slope at the higher frequencies (${\alpha }_{2}={1.03}_{-0.02}^{+0.03}$).

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Table 1. Summary of the Fit Results for the PSR and PWN Continuum Emission

		PL Model		bPL Model
		${\mathrm{log}}_{10}\left({F}_{\nu }\right)$ a	α	${\mathrm{log}}_{10}\left({F}_{\nu }\right)$ a	${\mathrm{log}}_{10}\left({\nu }_{{\rm{b}}}\right)$	α1	α2	Δα
					(Hz)
MUSE
4800–8000 Å	PSR b	−28.02	1.022 ± 0.006	−28.06	14.628 ± 0.002	0.00 ± 0.05	1.166 ± 0.007	1.17 ± 0.05
	PWN c	−26.30	1.033 ± 0.004
X-shooter
3200–22500 Å	PWN b	−26.57	${0.889}_{-0.001}^{+0.001}$	−26.57	${14.67}_{-0.02}^{+0.03}$	${0.84}_{-0.01}^{+0.01}$	${1.03}_{-0.02}^{+0.03}$	${0.19}_{-0.03}^{+0.03}$

Notes.

^a Flux (${\mathrm{log}}_{10}\left({F}_{\nu }\right)$, in units of erg s⁻¹ cm⁻¹ Hz⁻¹) evaluated at ${\mathrm{log}}_{10}\left({\nu }_{{\rm{b}}}\right)=14.50\,\mathrm{Hz}$. ^b Extraction region defined in Figure 1. ^c Elliptical extraction region defined in Mignani et al. (2012).

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We thus find strong indications for a spectral break in the UVB–NIR wavelength range. There are certain notable caveats related to the NIR part of the spectrum, for example, it being less constrained due to the excluded wavelength intervals with poor atmospheric transmission, and having more uncertain flux levels as a result of the different slit widths. More discussion on the X-shooter results and their significance can be found in Section 5.1.3.

PWNe exhibit axisymmetric structures with equatorial toroidal and polar jet-like features, as is observed, e.g., for the Crab by Mori et al. (2004). Models based on latitude dependence of the pulsar energy flux and magnetization of the pulsar wind have been successful at predicting these kinds of structures (see Gaensler & Slane 2006 and references therein). Previous observations of PWN 0540 have shown evidence for a torus, detected in the optical and X-rays (e.g., Gotthelf & Wang 2000; Morse 2003), and a possible jet (e.g., Gotthelf & Wang 2000; Petre et al. 2007), which our MUSE observations reveal in greater detail.

In the spectral index map (Figure 2, bottom right panel) we clearly see the torus, which is distinctively softer (α ∼ 1.1) than the surrounding emission (α ∼ 0.6–0.9). It spans from NE to SW with an extent ∼ 45 and is slightly off of the diagonal NE–SW line. At the LMC distance ∼ 50 kpc, the major axis of the torus is ∼1.1 pc.

In the convolved flux image (Figure 2, bottom left panel) the torus is not as prominent. A bright region on the SW side can be detected (∼80% of the SNR 0540 maximum flux from the pulsar), but on the NE side the torus is dimmer (∼30% of the maximum flux) and blends in with the surroundings. The prominence of the SW part of the torus is even more clear in the deconvolved flux image (Figure 2, top right panel). The SW emission region has a more elongated shape (main emission reaching ∼30 away from the pulsar), while, in the NE part, the main emission comes from a smaller region only ∼16 from the pulsar. In general in the flux images, the torus seems to follow the diagonal NE–SW direction.

The torus is most likely composed of magnetic fields and shocked relativistic particles. In X-rays, the torus exhibits a similar or slightly larger size as in the optical, the major axis being 40–60 (e.g., Lundqvist et al. 2011), which corresponds to ∼1.0–1.5 pc at the LMC distance. The size of the torus is also comparable in the NIR (Mignani et al. 2012) and thus no significant variation in the torus (or general nebula) size is observed in different wavelengths.

Furthermore, the PWN 0540 torus can be distinguished from the surrounding emission in polarization data as well. The linear polarization angle along the PWN 0540 torus (pointing NW) differs from the rest of the PWN (pointing NE), as seen in Figure 8. Polarization traces the magnetic field configuration and therefore further highlights the difference between the bulk PWN and the torus.

Previous works (De Luca et al. 2007 and Lundqvist et al. 2011) have observed high brightness variability in the SW parts of the torus. This blob in the SW has been either moving or fading between different epochs (e.g., Lundqvist et al. 2011). The deconvolved MUSE fluxes also reveal a bright blob in the SW region (∼18 SW from the pulsar, top right panel of Figure 2). However, this blob is not clearly separated from the "background" torus in the convolved MUSE fluxes nor in the X-shooter data.

The deconvolved MUSE continuum flux image (Figure 2, top right panel) also reveals an arc in the east that is connected to the NE part of the torus. This arc has a radius ∼ 15 and spans from the NE to the SE. We find this arc to trace the softest spectral indices in the east (see the flux contours on the bottom right panel of Figure 2) and it also seems to cut through two especially soft regions where α ∼ 1.2. It is possible that this arc could be an artifact originating from the deconvolution process, since it is located relatively close to a strong point source (the pulsar), which the deconvolution process enhances. However, we believe that the arc is most likely real, since Mignani et al. (2012) have observed a similar arc in the NIR.

Accompanying the toroidal structure, a jet-like feature in the NW part of central SNR 0540 has been observed in X-rays by, e.g., Gotthelf & Wang (2000) and Petre et al. (2007). Additionally in the optical, Serafimovich et al. (2004) reported possible jets both in the NW and SE. Along with these results, our MUSE observations provide further evidence for a jet. In the MUSE spectral index map (Figure 2, bottom right panel) a region 25° from north to the west appears like a jet protruding from the central parts of the torus. This structure features the hardest spectrum in the whole PWN (α ∼ 0.8–0.5) although it does not clearly stand out in the flux image. We detect no significant counterjet in the SE neither in the spectral index map nor in the flux image, possibly due to bright field stars occupying this region. Assuming the jet inclination inferred from the 3D morphology is correct (∼10° away from the observer), the lack of a counterjet cannot be due to relativistic beaming. We also see increased polarization degree in the jet region (∼30%–40%) compared to the central PWN polarization (∼10%–20%), see Figure 8.

PWN continuum emission can typically be described with PL models over broad wavelength ranges (e.g., Mitchell & Gelfand 2022). The slope of the continuum spectrum can vary spatially across the PWN, and the canonical PWN model of fainter outer regions emitting softer spectra stems from the idea of synchrotron cooling. This picture is supported by X-ray observations of the Crab (e.g., Mori et al. 2004), where the spectral index varies from α ∼ 0.9 to α ∼ 2 toward the outer regions. Generally in X-rays, the spectral index seems to soften toward the PWN outer boundary in young SNRs (e.g., Hu et al. 2022).

In contrast to this model, we find general spectral hardening in the optical spectra of PWN 0540, with the fainter outer regions being harder (down to α ∼ 0.1) than the brighter inner parts (α ∼ 1.1). This spectral hardening toward the PWN outer boundary is further quantified in Figure 6, where this trend, most prominent in the NE, of dimmer fluxes and lower spectral indices can be seen (diagonal trend below the plateau). However, a local spectral softening toward the outer edge of the PWN in the SW is also observed.

Some earlier optical studies of PWN 0540 have revealed hints of similar spectral hardening toward the outer parts of the PWN. Serafimovich et al. (2004) used HST photometry to measure spectral indices along the NE–SW torus. These findings, also presented in Figure 11, show that, in the NE the spectral indices decrease toward the PWN boundary (from α ∼ 1.5 just NE from the pulsar to α ∼ 0.3 at ∼22 NE from the pulsar). We compare our MUSE results to these values by extracting continuum spectra similarly along the torus. We find that the spectral indices from Serafimovich et al. (2004) mostly overlap with our MUSE results, as shown in Figure 11. However, the measurement uncertainties were too large for Serafimovich et al. (2004) to confirm any possible spatial variation. No other previous measurement in the optical provides as detailed information on spatial variations of the spectrum. Mignani et al. (2012) report minor evidence for spectral hardening away from the pulsar in the NIR and optical by obtaining a softer spectral index (α = 0.70 ± 0.04) for PSR 0540 and a harder spectral index (α = 0.56 ± 0.03) for the whole spatially averaged PWN 0540.

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In X-rays, however, observations show radial softening of the spectral index toward the PWN 0540 outer regions. Petre et al. (2007) used Chandra X-ray observations to map the radial variation of the spectral index by using concentric elliptical annuli as extraction regions. They find that the pulsar (which is strongly affected by pileup) completely contaminates the innermost region within ∼10 radius, but that there is a softening of spectral indices from 10 toward the outer regions of the nebula (from α ∼ 0.4 to α ∼ 1.4). Beyond ∼50, they conclude that a more complex model than a single PL is needed to explain the emission in X-rays.

Also Lundqvist et al. (2011) derived an X-ray spectral index map for PWN 0540 from Chandra data with some emphasis along the torus. Similar to the MUSE and X-shooter results along the torus in Figures 2 and 9, where α ∼ 1.1 and α₂ ∼ 1, respectively, Lundqvist et al. (2011) find an approximately constant spectral index in X-rays (α ∼ 0.8) within the ∼20 radius both to the SW and NE of the pulsar, see Figure 11. Beyond ∼20 and up to an ∼50 distance from the pulsar, they observe softening of the spectral index (α ∼ 0.8–1.8) both to the NE and SW. Thus, comparing the optical and X-ray results reveals that the significant variation (softening in X-rays and hardening in the optical) begins at an ∼2'' radius. The only exception to this picture is the SW part of the torus, where the optical spectral index softens further out (Figures 2 and 11). This particular section of the PWN therefore seems to obey the simple synchrotron cooling picture. However, the flux from this region is low and it is located near the excluded high-α region, which might hint at systematic effects affecting the spectral index results.

The origin of the apparent divergence from the simple synchrotron cooling picture for most of the nebula in the optical is not clear. One consideration is that the convolution (Section 3) does not perfectly correct for the MUSE PSF wavelength dependence. While this introduces a systematic uncertainty in the spectral index map, it is not expected to affect the main conclusions, considering that the spectral index hardening is so prominent (the difference between the inner and outer spectral indices being Δα ∼ 1).

Another reason behind this deviation from the simple synchrotron cooling picture could be that there is an additional continuum emission component other than synchrotron that contributes significantly to the optical spectrum. The most likely such contribution is Balmer recombination continuum and two-photon emission, which may be expected at a low level, similar to the case in the Crab Nebula (Veron-Cetty & Woltjer 1993).

However, in this case we expect a spatial correlation between the hard (blue) continuum and the Balmer emission lines in the central part of the remnant, which is not observed. In particular, the MUSE data reveal weak Hβ emission with a similar distribution as the [S iii] emission in Figure 7, but with the most pronounced emission in the northern part of the SW ring structure (see Figure 1 in L21). By contrast, hardening of the optical continuum emission is observed in the outer regions all around the PWN except the SW. Interestingly, the polarization properties shows a similar systematic variation (Figure 8).

Assuming that the optical continuum is indeed dominated by synchrotron emission, the hardening toward the outer regions may be due to reacceleration of particles by the pulsar wind shock at the inner edge of the ejecta, though we note that such a scenario lacks theoretical predictions. This seems to be consistent with the relative location of the continuum- and line-emitting regions (Figure 7), though the interpretation is clearly complicated by projection effects. Another possibility may be that the hardening is a result of time variability of the pulsar wind. In this scenario, the wind further from the pulsar would have been produced at an earlier time, possibly with different properties, which would then result in a spatially variable spectral index.

The main caveat affecting the comparison between the optical and X-rays above is the possible time variability, observed, e.g., by Petre et al. (2007), which makes comparing results from previous observations challenging. In addition, the results may also be affected by line-of-sight effects, where emission, say from the central PWN, is mixed with PWN contributions further away that reside in the foreground or background along the line of sight. This is complicated to account for since PWN 0540 is known to have a highly asymmetric 3D morphology (e.g., Sandin et al. 2013 and L21).

After considering the high spatial resolution MUSE observations, we turn our focus to the wider spectral range provided by X-shooter. We find strong indications that a single-PL model is not sufficient to characterize the PWN 0540 continuum emission in the UVB–NIR (10¹⁴–10¹⁵ Hz) range, and instead fit the continuum emission with a bPL model. In this section, we discuss the implications of these X-shooter results and how they fit into the multiwavelength context from the radio to X-rays.

The main challenge with the X-shooter observations is the connection of the three spectra: NIR, VIS, and UVB. It is clear that the NIR part of the X-shooter spectrum is overall less constrained because of the large gaps in the data (due to atmospheric transmission that affects all ground-based IR observations) and therefore it is especially challenging to connect the NIR and VIS bands perfectly. These challenges add a systematic uncertainty to all spectral index results, which we estimate to be at least 0.1 (for a more detailed discussion, see Appendix A.1). Additional challenges come with the limited field of view of the slit, i.e., it does not cover the entire PWN.

We can study the details of the synchrotron-emitting particles by focusing on the spectral index difference between the low and high frequencies, Δα ≡ α₂ − α₁. According to the simplest synchrotron model with constant energy injection and with a spatially uniform PL distribution for the emitting particles, the difference between the injected (higher-frequency) and cooled (lower-frequency) spectral slopes is Δα = 1/2. The difference between the best-fit low- and high-frequency spectral indices within the X-shooter spectrum in Figure 9 is ${\rm{\Delta }}\alpha ={0.19}_{-0.03}^{+0.03}$, which significantly deviates from the canonical value of 1/2. Generally, since PWN 0540 is clearly not spatially uniform (as is discussed in Sections 5.1.1 and 5.1.2), the exact value of 1/2 is not expected. Additional factors coming from the aforementioned systematic effects at play in the X-shooter fits might also affect the difference of the two spectral indices. However, this Δα result deviates from the canonical value by more than 5σ, which indicates that the break between the NIR and UVB is most likely not caused by synchrotron losses.

Before discussing how the MUSE and X-shooter UVB–NIR results connect to other wave bands, we address caveats that arise from time and spatial variability of the PWN. During 2011, PSR 0540 experienced a sudden increase in its spin-down rate (Marshall et al. 2015), which also caused an increase in the PWN X-ray flux by ∼32%, reported by Ge et al. (2019). It is believed that a change in the pulsar magnetosphere caused the PWN brightening and the X-ray flux values have remained at the same elevated level after the gradual increase between late 2011 and 2016 (the latest observation was performed in 2019). In addition, De Luca et al. (2007) and Lundqvist et al. (2011) have observed changes in the spatial distribution of the optical continuum emission, especially in the SW, during relatively short timescales ∼ 10 yr. These observed changes in X-ray flux levels and optical (spatial) continuum distribution introduce uncertainties in the comparison between the recent observations to the pre-spin-down rate change observations or observations performed ≳10 yr ago.

In Figure 12 we present spatially averaged results of the PWN 0540 multiwavelength continuum spectrum. The plotted MUSE spectrum is extracted through the same elliptic extraction region (34 × 24 centered at the pulsar but the pulsar excluded) that is used in Mignani et al. (2012). The resulting PL fit gives α = 1.033 ± 0.004. For X-shooter, we used the nebula region defined in Figure 1. We then scaled the flux levels to correspond to the flux levels of the larger Mignani et al. (2012) extraction region, using MUSE to determine the flux ratio between the two regions. As can be seen in Figure 12, the X-shooter and MUSE spectra are overlapping and give similar results for the spectral indices within 1σ (see also Table 1). We note, however, that this nearly perfect agreement might be serendipitous since the extraction regions are different and there are known systematic differences between the instruments (Appendix A).

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Previous results in the optical and NIR by Serafimovich et al. (2004) and Mignani et al. (2012) mostly overlap with the X-shooter and MUSE results. Differences in the slopes and flux levels may occur due to different extraction regions and observational epochs. Additionally, many of the photometric results (e.g., Mignani et al. 2012) seem to be contaminated by emission lines, for instance with the [Fe ii] 1.26 μm line in the NIR and especially [O iii] λ λ4959, 5007 in the optical. In conclusion, all these challenges explain the conflicting results previously obtained for the spectral index in the optical range (Serafimovich et al. 2004; Mignani et al. 2012).

As can be seen in Figure 12, previous works have reported higher flux densities in the IR, especially in the lower-frequency part (Spitzer and AKARI results; Williams et al. 2008; Lundqvist et al. 2020) compared to the optical (VLT/NACO and HST results; Mignani et al. 2012; Serafimovich et al. 2004). A dust component with T ∼ 50 K and M_dust ∼ 3 × 10⁻¹ M_⊙ has been proposed to account for this excess emission at log(frequency) ∼ 13 Hz (Williams et al. 2008 and Lundqvist et al. 2020). Lundqvist et al. (2020) computed a spectral slope of ${\alpha }_{\mathrm{IR}}={0.87}_{-0.08}^{+0.10}$ between the far-infrared dust bump and the NIR/optical.

The X-shooter spatially averaged nebula continuum seems to fit this picture. In the top right inset of Figure 12, we show how the lower-frequency spectral index $\alpha ={0.84}_{-0.01}^{+0.01}$ obtained from a bPL fit to the X-shooter data connects the NIR and IR emission. However, this good agreement may be a coincidence due to the time variability discussed above, in combination with likely contamination by lines in the IR (discussed in Lundqvist et al. 2020), and the different color excess values used. We therefore conclude that dust emission in the IR is possible but cannot be confirmed with our data.

If we assume that the IR fluxes at log(frequency) ∼ 13 Hz are contaminated by line emission, we can study how the UVB–NIR fluxes are connected to the radio band. Since the optical–NIR spectral index ($\alpha ={0.84}_{-0.01}^{+0.01}$, see fit in Figure 12) aligns well with the corresponding value from Lundqvist et al. (2020; $\alpha ={0.87}_{-0.08}^{+0.10}$), we end up with the same conclusion that there has to be at least one spectral break between the IR and the radio, in addition to the spectral break we found between the NIR and UV (at ${\mathrm{log}}_{10}\left({\nu }_{{\rm{b}}}\right)={14.67}_{-0.02}^{+0.03}$ Hz).

For the spectral break between the radio and IR wavelengths, Lundqvist et al. (2020) obtain a location at log(frequency) ∼12.7 Hz with the radio spectral index being α = 0.17 ± 0.02. Similar results were reached by Brantseg et al. (2014), where the spectral break is located at log(frequency) ∼ 13.8 Hz with a radio spectral index of α = 0.16 ± 0.1. The magnitude of the break is consistent with a cooling break.

On the higher-frequency side, Kaaret et al. (2001) ¹⁰ measured the PWN continuum spectrum in X-rays and found a spectral index α = 1.04 ± 0.18. This result lies within the uncertainty range of the X-shooter spatially averaged higher-frequency spectral index, ${\alpha }_{2}={1.03}_{-0.02}^{+0.03}$, implying that no break between the optical and X-rays would be required. However, the picture is most likely more complicated. In addition to the larger color excess value used in this work, the X-ray fluxes measured by Kaaret et al. (2001) overshoot the optical fluxes, as can be seen in Figure 12. In reality, the situation is even worse, the overshooting being enhanced by the 32% increase in X-ray fluxes (Ge et al. 2019).

In conclusion, the evidence for multiple breaks (a cooling break between log(frequencies) of 12–13 Hz and another break between 14 and 15 Hz) supports the scenario of multiple particle populations at play in PWN 0540, unless the multiwavelength comparison is strongly affected by time variability.

MUSE observations show that the pulsar spectrum in the optical exhibits a similar spectral index (α ∼ 1) to the central PWN spectrum (bottom right panel of Figure 2). However, the statistically preferred model for the pulsar was found to be the bPL model. The bPL best-fit model has a break frequency at ${\mathrm{log}}_{10}\left({\nu }_{{\rm{b}}}\right)$ = 14.628 ± 0.002 Hz, with spectral indices α₁ = 0.00 ± 0.05 for the lower-frequency part and α₂ = 1.166 ± 0.007 for the higher-frequency part of the spectrum.

Previous observations in the optical have yielded slightly harder (smaller α) optical spectral indices for PSR 0540, and no evidence for a spectral break has been reported in this wavelength range. Spectroscopic observations of PSR 0540 in the optical, performed by Serafimovich et al. (2004), resulted in a spectral index $\alpha ={1.07}_{-0.19}^{+0.20}$ (or $\alpha ={\left[1.07-0.2\right]}_{-0.19}^{+0.20}$ to counter the larger color excess value used in this work). Mignani et al. (2012) used photometry in the optical–NIR and report an even harder spectral index of α = 0.70 ± 0.04. A general hardening in the NIR range is compatible with our results, though the values of α clearly differ.

As discussed in Section 5.1, there are many factors that make it challenging to compare our results to the ones provided in the literature. The most significant of these is the spin-down rate change in 2011 (Marshall et al. 2015) related to which the PWN X-ray flux levels were measured to have increased by ∼30%, but the PSR flux in X-rays appear to have stayed unchanged (Ge et al. 2019). How the spin-down rate change has affected pulsar fluxes at lower energies, for example in the optical range, has not been studied.

To our knowledge, Mignani et al. (2019) have performed the only observations after the 2011 spin-down rate change that focus on the PSR 0540 spectrum. They find that the near-ultraviolet and far-ultraviolet band (around 2350 and 1590 Å, respectively) fluxes are completely incompatible with the pre-2011 optical spectra. The spectral index they infer is α ∼ 3, the largest spectral index ever recorded for a pulsar in the UV. Thus, Mignani et al. (2019) deduce that a turnover below ∼2400 Å (or in log(frequency), above ∼15 Hz) is needed to explain the optical–UV spectrum. Our optical spectral index α ∼ 1 follows the spectral index trend reported for other pulsars in the optical, α ∼ 0–1 (Mignani et al. 2007), but is incompatible with the UV slope from Mignani et al. (2019). We are not able to draw any conclusions regarding a possible spectral turnover around a log(frequency) of 15 Hz due to a high noise level in the X-shooter spectra at these frequencies.

On the other hand, Mignani et al. (2019) assumed a color excess value of E(B − V) = 0.20 mag, which complicates the comparison with our results. Taking the color excess value found in this work, E(B − V) = 0.27 mag, and applying it to the results reported in Mignani et al. (2019), we find that the UV spectral index remains at a high value α ∼ 3. This is due to the shape of the extinction curve (2175 Å UV bump), which does not change significantly when varying the color excess values.

In X-rays, the PSR 0540 spectrum seems to be harder than in the optical, at least according to pre-2011 observations. Lundqvist et al. (2011) report α = 0.74 ± 0.01 for the PSR 0540 X-ray spectral index whereas Kaaret et al. (2001) report α = 0.92 ± 0.11. The latter is still ∼3σ from our MUSE higher-frequency spectral index α₂ = 1.166 ± 0.007. We confirm the previous conclusion (Serafimovich et al. 2004; Mignani et al. 2010) that, even without the Mignani et al. (2019) results, the PSR 0540 spectral energy distribution from the optical to the X-rays cannot be described with a single PL.