CHANG-ES. III. UGC 10288—AN EDGE-ON GALAXY WITH A BACKGROUND DOUBLE-LOBED RADIO SOURCE

Estimates of the flux densities of the various sources are given in Table 5. All flux densities were measured from the primary-beam-corrected C-array L-band (Figure 1(b)) and D-array C-band (Figure 2(a)) maps. For both bands, the total flux densities of UGC 10288 and CHANG-ES A were measured together. Then the flux density of CHANG-ES A was estimated from a measurement within the two dashed lines shown in Figure 2(a), a width of 34''. The flux density of UGC 10288 alone was then measured in a region of equivalent width to the east of CHANG-ES A along the galaxy's disk. Assuming that this measured flux density is approximately equivalent to the flux density of UGC 10288 within the box occupied by CHANG-ES A, it was then subtracted from the CHANG-ES A measurement as a correction for the contribution of the galaxy's disk within the dashed lines. At L band, this correction amounted to 4% of the flux density of CHANG-ES A, and 22% of the flux density of UGC 10288. At C band, the correction constituted 4% of the flux density of CHANG-ES A and 32% of the flux density of UGC 10288. In addition, at L band, the totals were then summed and compared to the combined, blended emission from all three sources, i.e., UGC 10288, CHANG-ES A, and CHANG-ES B as measured in the D-array L-band image (Figure 1(a)); the totals agree to within 0.5%.

Table 5. Derived Properties of UGC 10288 and CHANG-ES A

Property	UGC 10288	CHANG-ES A
R.A. (J2000)a	16h14m2480	16h14m2328 ± 004
Decl. (J2000)a	−00°12m271	−00°12m11.6 ± 05
SL (mJy)b	4.4 ± 0.5	22 ± 2
SC (mJy)b	1.53 ± 0.05	12.0 ± 0.2
αtotalc	−0.76 ± 0.11	−0.44 ± 0.08

Notes. ^aU10288 position from NED. For CHANG-ES A, the position has been measured from the highest resolution data. ^bFlux densities in L or C bands, measured from the primary-beam-corrected images as described in Section 3. Uncertainties represent variations from adjusting the box size containing the emission. ^cGlobal spectral index according to S_L/S_C = (ν_L/ν_C)^α, where ν_L and ν_C are the L-band and C-band central frequencies, respectively.

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The global spectral index of UGC 10288 was calculated from the flux densities as just described, and the result is given in Table 5. The two sources are both clearly dominated by non-thermal emission.

It is remarkable that UGC 10288 reveals extra-planar emission (see the next two subsections) when the total flux density is only a few mJy. This leads us to consider the star formation rate (SFR) of the galaxy, since halo emission is often associated with such activity.

UGC 10288's SFR_IRAS of 1.3 M_☉ yr⁻¹ as given in Table 1, has been calculated from Infrared Astronomical Satellite (IRAS) data (Paper 1), but, like the radio continuum, the IR emission will certainly be contaminated by emission from CHANG-ES A.

A closer inspection of the IRAS flux densities, moreover, suggests foreground contamination as well. Given the observed IRAS 60–100 μm flux density ratio of UGC 10288 (f_(60 μm)/f_(100 μm) = 0.236), the galaxy appears to be extremely cold. In fact, this flux density ratio is ≈20% smaller (colder) than that for the most extreme (coldest) galaxy included in the Dale & Helou (2002) spectral energy distribution models of normal star-forming galaxies. If CHANG-ES A is indeed a background radio galaxy, we naively would not expect it to be colder than all galaxies in the local universe. If anything, we would expect a warmer temperature arising from hot dust being powered by the accreting black hole. Therefore, it seems that the most likely explanation for the cold dust temperature is significant contamination by difficult-to-remove foreground cirrus emission that is contributing to the IRAS flux densities.

Given these issues, an alternative approach is to use the calibrated, known radio continuum–far-IR (FIR) relation to estimate the SFR of UGC 10288 from its radio continuum emission as estimated in Section 3.1.1 with CHANG-ES A's flux density subtracted. For this, we adopt the calibrations of Murphy et al. (2011) which use a Kroupa initial mass function and a supernova cutoff of 8 M_☉. The SFR estimated from the thermal radio continuum requires an electron temperature which we take to be 10⁴ K. The SFR estimated from the non-thermal radio continuum requires a knowledge of the non-thermal spectral index, α_NT, whereas we have, from the observed flux densities of Table 5, a global spectral index of α = −0.76, using the convention (S_ν ∝ ν^α).²²

However, we can estimate α_NT by combining the thermal and non-thermal calibrations of Murphy et al. (2011, their Equations (11) and (14)) and the constraint that the measured flux density is the sum of the non-thermal and thermal flux densities at any frequency, i.e.,

$$\begin{equation} \frac{S_{\nu _1}}{S_{\nu _2}} = \left(\frac{\nu _1}{\nu _2}\right)^{\alpha _{{\rm NT}}} \frac{ \left[0.144\,(\nu _1)^{-\alpha _{{\rm NT}}-0.1}\,+\,1\ \right]}{\left[0.144\,(\nu _2)^{-\alpha _{{\rm NT}}-0.1}\,+\,1\ \right]}, \end{equation} \tag{ 1 }$$

where S_ν is the total flux density at frequency ν, and these calibrations assume a linear dependence of SFR on both thermal and non-thermal luminosity. The use of this equation infers a thermal/non-thermal fraction which has a dependence on both frequency and α_NT.

Using our measured flux densities of Table 5 at two frequencies, we can solve Equation (1) to find α_NT = −1.04. This leads to a thermal to total and non-thermal to total flux density in L band of (S_T/S)_1.5 = 0.17 and (S_NT/S)_1.5 = 0.83, respectively. In C band, the results are (S_T/S)_6.0 = 0.44 and (S_NT/S)_6.0 = 0.56, respectively.

Finally, with the SFR calibration (Equation (15) of Murphy et al. 2011), we find SFR = 0.51 M_☉ yr⁻¹.

We caution that, if the non-thermal luminosity, L_NT ∝ SFRⁿ, where n > 1 (e.g., n = 1.3; Niklas & Beck 1997), then there would be a weak dependence on SFR on the right-hand side of Equation (1); the adjustment, however, does not significantly perturb the above results.²³

This newly determined SFR is considerably lower than the original SFR which was found from IRAS values and should be  more accurate since it corrects for the contribution of CHANG-ES A and also uses a spectral index determined from the data. If, instead, we use the global FIR–radio continuum relation (Equation (17) of Murphy et al. 2011) based on earlier calibrations (de Jong et al. 1985; Helou et al. 1985), then SFR = 0.39 M_☉ yr⁻¹. In either case, the radio-derived SFRs are clearly much lower than the IRAS-based estimate of 1.3 M_☉ yr⁻¹.

In summary, the SFR based on IRAS photometry (Table 1) neither corrects for the strong contribution from CHANG-ES A nor for contamination from foreground cirrus. We suggest that a value between 0.4 and 0.5 M_☉ yr⁻¹ is a more accurate estimate of the true SFR in UGC 10288. Note that the range given here is dominated by the assumptions that have been made, rather than by a formal error determination. This galaxy is now one of the lower SFR galaxies in the CHANG-ES sample (Paper 1).

Figure 6 shows the combined C+D array C-band map of UGC 10288 overlaid on the λ3.6 μm image from the Spitzer Space Telescope.²⁴ These radio data have the lowest rms of all combined data sets (Table 4) as well as high resolution. Comparing with the Spitzer image, which was chosen because it should show both foreground stars as well as background quasars, should reveal how much radio emission (especially in regions of apparent halo features) is associated with point-like sources in the displayed field.

We searched for sources for which both the IR and radio emission are point-like and for which the radio emission is sufficiently separated from the disk of UGC 10288 to allow a discrete measurement of flux density. The process was then repeated for the combined B+C array L-band map (not shown), although the latter image has almost four times higher rms noise than at C band for approximately equal beam sizes. The resulting sources are labeled in Figure 6 and their properties listed, along with SDSS identifiers (where available) in Table 6.

Let us consider the region that is encompassed by the box shown in Figure 6 (16^h14^m259 ⩽ R.A. ⩽ 16^h14^m289; −00°12'21'' ⩽ decl. ⩽−00°11'40''). This box contains significant "halo" emission in the form of a large extended arc-like feature visible in the all-data image of Figure 5(b). This arc extends to a height of 21'' (3.5 kpc) above the plane. The flux density contained within this box is S_4.1 GHz = 450 ± 80 μJy.²⁵ Only one identified point-like source (Source 2) is located within this region whose flux density, extrapolated to 4.1 GHz, is <16 μJy (Table 6). Consequently, although we cannot conclude that the arc-like feature represents a single coherent structure, its flux density does not result from background sources. Hereafter, we will refer to this features as the "arc" for simplicity.

There is also a feature extending to the north on the west side of the galaxy (at R.A. ≈ 16^h14^m215, decl. ≈−00°12'12'') as shown in Figure 5(b), but there are no point sources meeting our criteria in this region at all.

We caution that if a background source displays double-lobed radio structure, then it could be displaced with respect to the IR source and might have been missed in this search, as would objects with weak IR flux and those that are blended with the disk. However, even if we increased our point-like source flux density estimate by a factor of two, its contribution to the flux density of halo features in UGC 10288 would still be negligible. By including other array configurations in the maps, in particular short spacings, the sensitivity to extended halo features is enhanced but there should be no change in sensitivity to point sources.

It is beyond the scope of this paper to examine in detail the nature of the point-like sources. SDSS identifiers, where available, indicate that three of the four identified sources are stars, which would be an interesting result given the detected radio emission. However, one of these sources (Source 5, SDSS J161424.84−001318.3), although labeled as a star, has SDSS colors (u − g = +0.18, g − r = −0.18, r − i = 0.45, and i−z = −0.07) which are entirely consistent with those of a quasar over a range of redshift (Pâris et al. 2012, their Figure 28).

It is apparent that halo radio emission is associated with UGC 10288. Here, we appeal to ancillary data, where available, to shed further light on the high-latitude features.

Few resolved images of UGC 10288 are available at other wavebands but, of those available, we show several overlays in Figures 7 and 8. Figure 7 shows our all-array and all-frequency radio continuum images of Figure 5 superimposed on the Hα image of Rand (1996). In Figure 8, we show contours from the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) at λ12 μm and λ22 μm over Hα emission and our L-band combined array image, respectively. In each overlay, the spatial resolutions of the two images are identical.

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Because radio emission from UGC 10288 is blended with that of CHANG-ES A, it is difficult to disentangle which source is responsible for specific features observed at locations close (in projection) to that background source. For example, a horizontal feature appears to emanate from the northern lobe of CHANG-ES A toward the east (see R.A. ∼ 16^h14^m25^s, decl. ∼−00°12'06'' in Figure 7(b)). There is also a projection which extends to the south of CHANG-ES A's southern radio lobe (R.A. ∼ 16^h14^m23^s, decl. ∼−00°12'43'' in Figure 7(b) and farther south as a narrower radio extension in Figure 8(b)). Neither of these have obvious counterparts at other wavebands, although the southern feature appears to lie between two Hα extensions (see below).

The IR emission does not distinguish between UGC 10288 and background sources either, since, for example, the bright background radio source, CHANG-ES B, to the west of CHANGES-A (marked in Figure 6) is also an IR emitter at both λ12 and 22 μm.

As we have no way to determine the origin of radio continuum features near CHANG-ES A, we will concentrate instead on other features that are farther from this region and are more likely to be associated with UGC 10288. The Hα emission is helpful in this regard, since it clearly has an origin in UGC 10288 itself.

We must also note that each of the images suffers from some low-level artifacts.

For the radio images, the map contains residual side lobes which, for the most part, we have included in measuring and quoting rms map values. However, residual side lobes from a source 63 to the east of UGC 10288 are likely responsible for the horizontal feature at the southeast side of Figure 7(b) (R.A. ∼ 16^h14^m30^s, decl. ∼−00°12'50''), and residual side lobes from another point source 62 to the southwest (SW) may be responsible for the faint gray scale NE–SW feature visible on the west side of Figure 8(b) (R.A. ∼ 16^h14^m17^s, decl. ∼−00°12'15'').

Similarly, in the Hα image, it has been noted that the galaxy's bulge has been oversubtracted (Rand 1996). And both WISE images show broad (arcmin scale) NE–SW striping which may be producing or contributing to the broad southward extension at R.A. ∼ 16^h14^m22.5^s, decl. ∼−00°13'15'' (Figure 8(a)).

In any following discussion, therefore, we will keep these cautions in mind and discuss results that appear to be robust to such artifacts. Figure 7(a) shows the highest resolution (∼35) radio/Hα overlay with arrows pointing to specific features. Discrete in-disk Hα emission reveals specific sites of unobscured star formation.

Feature 1 is a discrete star-forming region in the disk above which there is a vertical plume of Hα emission (to the north). The Hα plume is more readily seen in the smoothed map of Figure 7(b) and has a λ12 μm counterpart (Figure 8(a)). No radio emission can be confirmed in this plume, but a slight extension visible in the L-band gray scale image of Figure 8(b) is close to its location, slightly displaced to the east.²⁶ That is, the radio extension's location is more closely aligned with the gap²⁷ immediately to the east of star-forming region #1.

Feature 2 refers to several star-forming complexes that are located below the strong northward radio plume seen in Figure 7. The two arrows point to gaps in the Hα in-disk emission, the easternmost one also corresponding to a gap in the in-disk radio emission. Two "prongs" in the radio emission close to the disk (Figure 7(b)) are immediately above these gaps.

The southern Hα extension of feature 3 was originally pointed out by Rand (1996) and is in a region of low brightness in comparison to the star-forming regions to the east and west of it. This feature shows a small northward elongation in the radio at high resolution (Figure 7(a)).

Feature 4 points to another gap in the Hα emission between two star-forming regions. There is a small radio extension to the north that is visible in the high-resolution image (Figure 7(a)) but a much larger corresponding radio continuum feature can be seen at lower resolution (Figure 7(b)).

Finally, feature 5 refers to two southward prongs which were originally identified by Rand (1996) who also noted a "hemispherical" shape formed by these two features together. We have not detected any high-latitude radio emission associated with this feature.

In summary, our high resolution data suggest that radio continuum features are more likely to be seen above in-disk gaps in the Hα emission.

It should be pointed out, of course, that apparent correlations above an edge-on disk could apply to any location along the line of sight (although such indeterminacy is less severe at large galactocentric radii). This means that such apparent correlations do not necessarily represent physical correlations. Nevertheless, the observed trend is worth pointing out, since it is rare that an edge-on galaxy has adequate data to compare Hα and radio continuum results at the same resolution with good sensitivity. A similar anti-correlation has been observed before between in-disk Hα emission and high-latitude radio continuum features in NGC 5775 (Lee et al. 2001). We discuss this further in Section 4.

On broader scales, it is more difficult to associate high-latitude radio emission with in-disk activity. Intriguing features are a southward Hα arc seen in Figure 7(b) at R.A. ∼ 16^h14^m245, decl. ∼−00°12'48'', and a possible companion feature at R.A. ∼ 16^h14^m227, decl. ∼−00°12'42'' between which the southern radio continuum extension, mentioned above, lies. In Figure 8(a), the λ12 μm emission shows several other northward extensions as well as a southward arc-like feature on the east side of the galaxy, and in Figure 8(b), a southward radio extension seen at L band on the far east end of the disk.

We note two other observations regarding the distributions.

The first is the north–south radio asymmetry observed on the east side of the major axis, best seen in Figure 7(b). That is, there is a large arc-like extension toward the north but nothing equivalent to the south.²⁸ This may reflect N/S asymmetries in star-forming regions and/or N/S density asymmetries in the disk. Such asymmetries can also be produced by ram pressure as a galaxy moves through an intergalactic medium; this is possible for UGC 10288, given that it is in a sparse group (Section 1), but we have no independent observations that would argue for or against this speculation.

The second observation is that UGC 10288 does not appear to have a global radio halo, but rather it has discrete high-latitude radio extensions. Table 7 compares beam-corrected exponential scale heights between the 4.1 GHz all-data map (Figure 5(b)) and both the Hα and WISE λ12 μm data at the same (7'') resolution according to the method of Dumke et al. (1995). The fitting has been carried out for data that have been averaged in strips, each 264 = 1.76 s wide, that are labeled in Figure 7(b).²⁹ The values are from single-exponential fits that apply to the disk (for example, for Strip #2 North, the horizontal feature at decl. ∼ −00°12'06'' was avoided). Note that the strips that do not have significant extensions (mainly #2 South but also #1 South and #2 North) have low to modest scale heights in radio continuum (0.46, 0.85, and 0.86 kpc, respectively), whereas the strips containing discrete extensions (#1 North and #4 North/South) are wider in z on average (1.3, 1.4, and 1.1 kpc, respectively), as would be expected (see Section 4 for more discussion).

In order to give an overview of the relative spatial distributions of the various sources of emission, we finally display a multi-frequency image (Figure 9) in which several data sets have been overlaid with different colors (see caption). Our radio data show CHANG-ES A and CHANG-ES B in cyan; our masking technique (Rector et al. 2007) allows us to show their optical counterparts (yellowish) at their cores. The disk of UGC 10288 looks quite wide vertically in this picture because of the WISE λ12 μm image. For more information as to the techniques used to combine the data sets, see Rector et al. (2007).

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Figures 3 and 4 show the polarization maps at L and C bands, respectively. We do not detect polarization in the disk of UGC 10288 in the individual observations. Rather, it is the background radio source, CHANG-ES A, that shows polarization. Although a more complete analysis of the polarization-related properties awaits a future paper, a few results are presented here.

For both B-array, L band (resolution ∼ 36) and C-array, C band (resolution ∼ 30), the highest polarization occurs right at the core of CHANG-ES A (peak values and resolutions provided in Table 3). At this location, the percentage polarization at L and C bands is 1.9% and 2.0%, respectively.

North and south of the core, although the polarized intensity is lower, the percentage polarization increases. At the peak polarization position of the C-band data in the northern lobe, for example, the values are 17% and 30% at L and C bands, respectively (R.A. = 16^h14^m237, decl. = −00°11'48''). At the southern peak position, the corresponding values are 6.3% and 31% at L and C bands, respectively (R.A. = 16^h14^m2347, decl. = −00°12'21'').³⁰

The L-band percentage polarization is weaker than at C band, as expected, since more Faraday depolarization occurs at the lower frequency, either intrinsic to the source or contributed via UGC 10288 or both. Moreover, the L-band value is lower still for the southern lobe, compared to the northern, likely because of stronger Faraday depolarization in the (possibly turbulent) disk of UGC 10288. The fractional polarization in the core is much lower than in either lobe; this is not unusual for double-lobed radio sources (e.g., Rudnick et al. 1986; Saikia & Kulkarni 1998) suggesting that much of the core/lobe difference is intrinsic to CHANG-ES A.

UGC 10288 does not lend itself well to global magnetic field calculations, given the interfering presence of CHANG-ES A. However, we have taken two sample positions, one in the disk, and one in the large northern arc on the east side of UGC 10288 visible in the 4.1 GHz map, to estimate the total minimum energy magnetic field strength, B. We use the revised minimum energy formulae given in Beck & Krause (2005) for isotropic fields and adopt a proton-to-electron number density ratio of 100. The positions are marked with crosses in Figure 7(b), the disk position being immediately below the arc.

At the position in the eastern disk, a projected distance of x = 39'' from the center, α = −0.44 ± 0.18 and I_4.1 GHz = 70.4 μJy beam⁻¹. Then α_NT = −0.54 and I_NT = 55.4 μJy beam⁻¹ by Equation (1) and the arguments presented in Section 3.1.2. For a line-of-sight distance of 6.0 kpc (estimated from a 1/e radial scale length) then the minimum energy magnetic field strength is B = 10 ± 3 μG. Here, the error represents measurement error only, which is dominated by the uncertainty in α; it does not include errors that may be associated with the assumptions.

At the position in the northern extension, 15'' above the plane (2.5 kpc in projection), α = −1.00 ± 0.50 (Figure 10), I_4.1 GHz = 36.0 μJy beam⁻¹, we assume that the emission is dominantly non-thermal based on the absence of Hα emission, and take l = 1.5 kpc, which is approximately the width of the arc at this location (9''). Then, provided that the conditions for minimum energy are met for this region of steep spectral index, B = 12 ± 5μG, where, again, the error represents measurement error.

As Figure 10 illustrates, there is considerable variation in spectral index (and its error) and the magnetic field strength will also vary with position. It is also not clear whether the thermal/non-thermal emission ratios follow the expectations of the calibrations of Murphy et al. (2011), point by point. Nevertheless, the approach in which the thermal/non-thermal ratio is tied to an SFR calibration should be an improvement upon previous estimates for other galaxies which usually assume a constant and singular thermal/non-thermal ratio for every point in the galaxy. These preliminary calculations, then, suggest a total magnetic field strength of order 10 μG at the two sample positions selected.

A powerful tool, which is potentially available for UGC 10288, is the technique of using the background radio galaxy, CHANG-ES A, to probe the foreground magnetic field. To this end, we computed the classical Faraday rotation measure (RM) map (Figure 11) between the B-array L-band and the C-array C-band data, at 5'' resolution. The maximum 1σ error is ±10 rad m⁻² at the outer edge of each patch, while the 1σ error of the mean RM of Figure 11 is ±2 rad m⁻². The RM ambiguity is ±93 rad m⁻².

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The observed RM values are a superposition from CHANG-ES A, the galaxy UGC 10288, and the foreground of the Milky Way. The lobes of radio galaxies can have high RM values with a large dispersion, so that, when observed with low resolution, the average RM is small. The central sources of radio galaxies (jets) generally do not reveal significant intrinsic RM. The RM of CHANG-ES A's northern lobe (at about 7 kpc projected distance above the plane of UGC 10288, marked "A" in Figure 11) of +30 ± 4 rad m⁻² (average of the two visible regions) agrees well with the foreground RM of the Milky Way (Taylor et al. 2009; Oppermann et al. 2012) at the Galactic coordinates of our radio maps (l = +12°, b = +34°). This indicates that no correction for the RM ambiguity is needed.

The southernmost blob in Figure 11 (marked "C") is located behind the disk of UGC 10288, about 1.5 kpc above the plane, with an observed average value of RM ≃ +62 ± 4 rad m⁻². We interpret the (foreground subtracted) value of RM_i ≃ +32 rad m⁻² as indication of a regular magnetic field in the galaxy disk, pointing toward us. A line-of-sight component of about +2 μG, a thermal electron density of 0.02 cm⁻³, and a path length of 1 kpc can provide the observed RM_i. If the path length is more than 1 kpc in the halo, the field strength and/or the electron density lowers by the same factor. However, as noted earlier (Section 3.1.4), when high-latitude radio emission is observed, the features have generally been discrete with typical sizes of the order of a kpc. In addition, disk exponential scale heights are also typically ≈1 kpc. Consequently, at the location of "C" at 1.5 kpc above mid-plane, we might expect the effective path length of the magnetic field to be approximately that of a typical high-latitude discrete feature.

RM_i ≃ −7 rad m⁻² (observed average value of RM ≃ +23 ± 2 rad m⁻² for both parts) toward the central region of the radio galaxy, at about 3.5 kpc above the plane of UGC 10288 ("B"), indicates that the field direction is reversed with respect to the plane. A thermal density in the halo of 0.01 cm⁻³ requires a regular halo field with a line-of-sight component of about −1 μG, pointing away from us. Again, if the path length is more than 1 kpc in the halo, the field strength and/or the electron density lowers by the same factor. Between disk and halo, no regular field from the galaxy is detected, so that the RM is solely due to the Galactic foreground.

The conclusion that a field reversal is present depends, of course, on the accuracy of the RM measurements, especially region "C" which is present in the total power maps only as diffuse emission between the core of CHANG-ES A and its southern radio lobe. If we use a higher cutoff of 3σ to form the RM map, then the value of "C" becomes 64 rad m⁻¹, rather than the 62 rad m⁻² seen in Figure 11, i.e., within the quoted 1σ uncertainty of ±10 rad m⁻² on the RM at the map edges (see above). A 3σ cutoff also reduces the 1σ uncertainty in RM to ±7 rad m⁻²; however, such accuracy is not required for this conclusion.

A reversal between the azimuthal components of the disk field and halo field may be quite common among galaxies; it has been found from modeling the magnetic field in the Milky Way from extragalactic RMs (e.g., Sun et al. 2008; Jansson & Farrar 2012) and also in the northern part of the galaxy M51 (Fletcher et al. 2011). Dynamo models can hardly explain such reversals if the same dynamo process operates in disk and halo, unless wind outflows allow such "mixed parities" (Moss et al. 2010).

These results are necessarily preliminary. The fact that many spectral channels are available in each of our observations (Table 2) lends itself well to the application of the RM Synthesis technique (Brentjens & de Bruyn 2005) to UGC 10288. This requires that (R.A.–decl.–frequency) cubes be formed for all data sets, and is an approach that will be attempted in the future.