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CONSTRAINING SPINNING DUST PARAMETERS WITH THE WMAP FIVE-YEAR DATA

We characterize spinning dust emission in the warm ionized medium (WIM) by comparing templates of Galactic dust and Hα with the five-year maps from the Wilkinson Microwave Anisotropy Probe (WMAP). The Hα-correlated microwave emission deviates from the thermal bremsstrahlung (free–free) spectrum expected for ionized gas, exhibiting an additional broad bump peaked at ∼40 GHz which provides ∼20% of the peak intensity. We confirm that the bump is consistent with a modified Draine and Lazarian spinning dust model, though the peak frequency of the emission is somewhat lower than the 50 GHz previously claimed. This frequency shift results from systematic errors in the large-scale modes of the three-year WMAP data which have been corrected in the five-year data release. We show that the bump is not the result of errors in the Hα template by analyzing regions of high free–free intensity, where the WMAP K-band map may be used as the free–free template. We rule out a pure free–free spectrum for the Hα-correlated emission at high confidence: ∼27σ for the nearly full-sky fit, even after marginalizing over the cosmic microwave background cross-correlation bias. We also extend the previous analysis by searching the parameter space of the Draine and Lazarian model but letting the amplitude float. The best fit for reasonable values of the characteristic electric dipole moment and density requires an amplitude factor of ∼0.3. This suggests that small polycyclic aromatic hydrocarbons in the WIM are depleted by a factor of ∼3.


INTRODUCTION
Following formulation of a physical model (Draine & Lazarian 1998b, hereafter DL98), spinning dust has been invoked to explain the dustcorrelated microwave emission seen by the Wilkinson Microwave Anisotropy Probe (WMAP) from 23-41 GHz (de Oliveira-Costa et al. 2004;Finkbeiner 2004;Finkbeiner, Langston, & Minter 2004;Boughn & Pober 2007;Dobler & Finkbeiner 2008b). Given the WMAP data at 94 GHz, the dust-correlated emission at 23-41 GHz was anomalously high to be explained by a purely thermal emission mechanism. Bennett et al. (2003) attempted to obviate the inconsistency by suggesting that the first year data from WMAP did not require a spinning dust component to construct an internally consistent foregrounds model, so long as the dust-correlated emission was actually dustcorrelated synchrotron at low frequencies. However, when the 3-year WMAP data (23-94 GHz) were combined with lower frequency data from Green Bank (8,14 GHz;Finkbeiner, Langston, & Minter 2004), Tenerife (10,15 GHz;de Oliveira-Costa et al. 2004), and other telescopes (19 GHz;Boughn & Pober 2007), it became clear that the dust-correlated emission at ∼ 20 GHz did not behave spectrally like synchrotron emission, and was instead consistent with a spinning dust spectrum.
The principal source of ambiguity in interpreting the low frequency WMAP data as evidence for spinning dust was that the spectrum of the dust-correlated emission falls from 23-41 GHz. The implication is that either the peak in the spectrum is near or below the lowest WMAP band or that the emission is synchrotron, and it is only with data at lower frequency that the spinning dust spectrum is recovered. In the DL98 spinning dust models however, the peak frequency of the spinning dust emission varies depending on model parameters (e.g., the grain size, geometry, ambient density, etc.), and at higher temperatures with smaller grain size is expected to be ∼ 40 GHz. This regime is termed the "warm ionized medium" (WIM, T ∼ 8000 K) in DL98 compared to the "cold neutral medium" (CNM, T ∼ 100 K) emission that peaks at ∼ 20-30 GHz and is thought to be the source of the dust-correlated emission at low frequencies in WMAP. Dobler & Finkbeiner (2008b, hereafter DF08b) showed that the ambiguity is eliminated and that the turnover in the spinning dust spectrum is recoverable within the WMAP frequency range using Hα-correlated emission as a tracer of WIM spinning dust emission. Typically, Hα is used as a tracer of free-free (thermal bremsstrahlung) emission from ∼ 10 4 K gas in regions where extinction from dust is minimal. However, because a map of Hα represents an emission measure (density squared integrated along the line of sight) and because spinning dust emission is generated via dust grains that are spun up by collisions with ions (which should also be roughly proportional to density squared integrated along the line of sight), the Hα emission should also spatially trace WIM spinning dust emission. In the analysis of DF08b, the peak frequency of the emission was found to be ∼ 50 GHz, which was slightly higher than the ∼ 40 GHz predicted by DL98.
In this paper, we show that the evidence for spinning dust in the Hα-correlated emission has strengthened with the release of the 5-year data, due to both decreased noise as well as corrections made to large-scale fluctuations that were present in the 3-year data.

TEMPLATE FITS
Microwave emission mechanisms in the WMAP frequency range can be separated into four broad categories: free-free, thermal dust, spinning dust, and synchrotron emission. 4 Free-free emission is generated by electron-ion collisions which produce thermal bremsstrahlung in ionized gas; thermal dust refers to electric dipole emission from thermal fluctuations (e.g., lattice vibrations) in the electric charge distribution in the grain; and spinning dust is electric dipole radiation from the smallest dust grains which are excited into rotational modes through a variety of collisional mechanisms. Synchrotron emission consists of a soft component (T ∝ ν −βS with β S ≈ 3) originating from supernova shock accelerated electrons which spiral around the Galactic magnetic field and a hard component (β ≈ 2.5) centered on the Galactic center and extending roughly 20 degrees. This hard component has been termed the "haze" and its origin remains uncertain (Finkbeiner 2004;Dobler & Finkbeiner 2008a).
Each of these emission mechanisms is approximately traced by maps of the sky at other frequencies, described below. This external information about the spatial structure of the foregrounds makes possible a multi-linear fit of the spectrum of each foreground. This multi-linear regression may be performed over the whole sky, or in selected regions to study the spectral variation of each component. We now briefly describe the templates used, and the fitting method (see Dobler & Finkbeiner 2008a, for more details).

The templates
For fitting purposes, we distinguish between the soft synchrotron component originating from supernovashock accelerated electrons, and a harder synchrotron component in the inner Galaxy, possibly with a different physical origin. Soft synchrotron emission is well traced by the Haslam et al. (1982) 408 MHz map. Though the spectral index of this emission is expected to vary slightly from place to place in the sky, La Porta et al. (2008) showed that from 408 to 1420 MHz, T ∝ ν −βS with β S ≈ 3. A similar spectral index was derived by Page et al. (2007) using polarization in the WMAP data.
The WMAP data, even in the first year, contained evidence of a harder synchrotron component in the inner Galaxy, which has become known as the "haze" because of its relatively featureless morphology (Finkbeiner 2004). Dobler & Finkbeiner (2008a, hereafter DF08a) interpret the haze as a separate physical component in 4 Draine & Lazarian (1999) point out that if interstellar dust contains an appreciable fraction of ferrimagnetic or ferromagnetic materials, thermal fluctuations in the magnetization would generate appreciable amounts of magnetic dipole emission at frequencies ν < 100GHz, which would be a fifth type of CMB foreground. However Cassasus et al. (2008) find that this type of emission is inconsistent with the morphology of cm emission from the ρ Oph cloud, implying that magnetic materials are not abundant in interstellar dust. Thus, we do not consider it further in this paper; we will use the term "thermal" emission to refer specifically to emission from thermal fluctuations in the electric dipole moments of grains.
the template fit described in DF08a and below, and the spectrum of this emission was found to be T ∝ ν −βH with β H ≈ 2.5. The template we use for this component is T ∝ 1/r gal where r gal is the distance from the galactic center in degrees. Though systematics related to contamination of any CMB template by residual foregrounds leads to uncertainty in the derived soft and hard synchrotron spectra, we note that neither the presence nor the spectrum of the haze has any significant effect on the conclusions in this paper. For a detailed study of microwave synchrotron in the Galaxy see Dobler & Finkbeiner (2008c, in preparation).
Our thermal dust template is the Finkbeiner, Davis, & Schlegel (1999, hereafter FDS99) two-component model of thermal dust, evaluated at 94 GHz. The FDS99 map uses column densities and temperatures from IRAS/ISSA (Wheelock et al. 1994) and COBE/DIRBE (DIRBE Exp. Supp. 1995) given by Schlegel, Finkbeiner, & Davis (1998;SFD), constrained to fit COBE/FIRAS data in the microwave and sub-mm (FIRAS Exp. Supp. 1997). The preferred FDS99 model (model 8) gives I ν ∝ ν 1.7 B ν (T dust ). Converting this to antenna temperature, T ∝ ν βD with β D ≈ 1.55 in the WMAP bands for typical T dust values. We use the same template for the diffuse, cold neutral medium (CNM) spinning dust which has a peak frequency ∼ 20 GHz.
For a free-free and WIM spinning dust template we use the Hα map assembled by Finkbeiner (2003) using data from three surveys: VTSS (Dennison et al. 1998), SHASSA (Gaustad et al. 2001), and WHAM (Haffner et al. 2003). The template is corrected for dust extinction assuming the dust and ionized gas are uniformly mixed along the line of sight. This approximation fails in regions of very high dust column density, so we mask out regions in our fit where the SFD extinction at Hα is A(Hα) ≡ 2.65E(B − V ) > 1 mag. Additionally, we mask out all point sources listed in the WMAP5 point source list as well as the LMC, SMC, M31, Orion-Barnard's Loop, NGC 5090, and the HII region around ζ Oph. This mask covers 22.2% of the sky and is shown in Figure 2 below. Roughly 1/4 of the masked pixels are point sources.

CMB Estimators
Finally, we also need an estimator for the CMB since it contributes a large variance at WMAP frequencies (and in fact constitutes the biggest source of noise when deriving foreground properties). We use the six CMB estimators described in DF08a and summarize their features in Table 1. As noted in DF08a (see also Hinshaw et al. 2007), every CMB estimator is contaminated to varying degrees by foregrounds which leads to a systematic bias in the inferred foreground spectra. In the case of "internal linear combination" (ILC) type estimators, in which the WMAP maps in each band b are weighted and summed with weight coefficients ζ b that are chosen to approximately cancel the foregrounds while preserving unity response to the CMB, the bias is proportional to the chance spatial correlation of the true foregrounds (as opposed to the templates) with the CMB (see DF08a).
The spectral cross correlation coefficients given in e.g. Fig. 5 depend on the CMB estimator used, but because we fit out a CMB spectrum in the interpretation of the correlation spectra (see §5), the final results for the Hα- 2.3. Template fit procedure To infer the spectra of the individual foreground components, we use the multi-linear regression template fit described in DF08a. We construct the template matrix P , whose columns consist of the templates described above, and derive the spectrum of the template correlated emission, a i b , where a i denotes the spectrum of foreground i and a b is the vector of correlation coefficients for band b. For each band, we solve the matrix equation, for a b , where w b is the WMAP map for band b. To determine the best fit a b in Equation 1 we evaluate the "pseudoinverse" P + (see DF08a) and note that a b = P + w b minimizes the quantity ∆ 2 ≡ p |P a b − w b | 2 where the sum is over unmasked pixels. Dividing P and w b by σ b (the mean noise in each band 5 , as in Bennett et al. 2003;Hinshaw et al. 2007), this solution for a b minimizes, The total χ 2 = b χ 2 b for the fit is given by the sum over the 5 WMAP bands. We also define the residual maps With our template fits, the spectrum of each foreground emission component a i is left completely unconstrained, though it is assumed to be constant across the sky. To appropriately account for the CMB, we also force a CMB = 1 in units of thermodynamic temperature.

FOREGROUND CROSS-CORRELATION SPECTRA
Our foreground spectrum fits are characterized by the cross-correlation coefficient a as well as the formal error in the fit in each band. These errors take into account both the noise in each WMAP band and morphological correlations between the different templates. By our definition of a above, the derived (cross-correlation) spectra are in I ν units of kJy/sr per template unit -Rayleighs for Hα-correlated emission and mK for FDS99correlated emission. Both the data and templates are mean-subtracted over unmasked pixels to remove sensitivity to zero-point levels in the maps. Figure 1 shows our derived cross-correlation spectra for the Hα-and FDS99-correlated microwave emission from 23 to 94 GHz for our six CMB estimators. The dust-correlated spectrum exhibits the familiar thermal tail from 94 to 61 GHz and then the rise from ∼50 to 23 GHz from anomalous emission. While the statistical error bars on the spectra are very small due to both the high sensitivity of WMAP as well as the large sky coverage (∼ 150,000 pixels for HEALPix N side = 128), the systematic effects from the contamination of the CMB estimators by foregrounds are significant.

Hα and Dust Correlated Emission
As identified in DF08a and DF08b, the spectrum of Hα-correlated emission does not follow the free-free power law as expected. Instead, there is a bump in the spectrum with a peak frequency ν p ≈ 40 GHz. This bump is present for all CMB estimators. We note that it cannot be generated by a contamination of the CMB estimator by foregrounds (since this bias has the spectrum of the CMB) and DF08b argued that this bump is most easily explained by a WIM spinning dust component that is traced by the Hα map. We address both of these considerations in detail below.

Changes from 3-year to 5-year data
There are two notable differences when comparing our results using the 3-year versus 5-year WMAP data. First, the inferred peak frequency ν p ≈ 40 GHz is significantly lower than the 50 GHz seen in the 3-year WMAP data (see DF08b). This shift in peak frequency is primarily due to spurious large-scale power that was present in the 3-year data but has been corrected in the 5-year data release. Figure 2 shows that the amplitude of these fluctuations can be quite large. In particular, the V band dipole is clearly apparent and has peak values from ∼ −7µK to ∼ 10µK.
As described above, we have mean subtracted the maps over unmasked pixels to perform the template fits, and so our results are not sensitive to zero point offsets. However if there is spurious low ℓ power, as in the 3-year data, our fit results will be affected. Hinshaw et al. (2008) point out that these large-scale fluctuations in the 3-year data were due to imperfect characterization of the instrument gain. While the amplitudes of the fluctuations are not large enough to have a significant impact on CMB analyses, Hα-correlated emission mechanisms (free-free and WIM spinning dust) are sufficiently subdominant at WMAP frequencies that ∼ 10µK offsets become impor- -Cross-correlation spectra for the Hα (left), and FDS99 dust (right) templates using various estimators of the CMB anisotropy (see §2.2). Although the formal error in the fit is small, in both cases the unknown degree of CMB contamination introduces a highly covariant uncertainty. In each case, the spectra are the same up to addition of a CMB spectrum of unknown amplitude.
tant. Because the 61 GHz band (V band) was the most affected, and because the WIM spinning dust peak frequency is in the range 30-50 GHz, it is not surprising that our best fit peak frequency has shifted. Had the spurious low-ℓ power been morphologically similar in all bands, the peak frequency would have been similar between the 3-and 5-year data. The problem was exacerbated by the CMB estimators, which necessarily contained some complicated combination of these fluctuations.
The second difference between the 3-and 5-year analyses is that, for a given CMB estimator, the ILC coefficients (defined in Equation 4 below) have changed. Although this does not affect our interpretation of the fit results (see §5), the spectrum of Hα-correlated emission has also changed by more than the formal fit uncertainties. Since this is due primarily to changes in the coefficients used when constructing a CMB estimator, the extent of the contamination of these estimators by foregrounds has changed from year three to year five. For example, in the case of a simple ILC which minimizes the variance over unmasked pixels (CMB3) with respect to the ILC coefficients ζ b , where is the ILC map in thermodynamic mK and the ILC coefficients ζ b are constrained to sum to unity to preserve response to the CMB. These coefficients are sensitive not only to the large-scale fluctuations mentioned above, but also to the noise in the measurements themselves since where c, f b , and n b are the CMB, total foregrounds in each band, and noise in each band respectively. Thus the signal variance in the noise also contributes to the determination of ζ b . This is illustrated by the limit of large measurement noise, in which case the minimum variance linear combination would have 3.3. Hα as a tracer of spinning dust For the 3-year WMAP data, DF08b presented an argument for why the Hα map should trace spinning dust emission at WMAP frequencies. Briefly, the Hα intensity is proportional to the emission measure (EM), n 2 e dℓ ≡ EM, where n e is the electron density and the integral is along the line of sight. We will show below that evaluating the DL98 spectrum with parameters appropriate for the WIM, the total emission is indeed roughly proportional to n 2 e . That is, including other (de-)excitation mechanisms such as plasma drag, far-infrared emission from the grains, etc., the behavior of the total emission with density implies that the Hα EM map should trace a component of the spinning dust emission in the Galaxy.

SPINNING DUST MODEL
In this section, we summarize the spinning dust model used in our analysis of the 5-year Hα-and dust-correlated spectra below. The model for excitation mechanisms and dipole moment distributions is described in detail in DL98. We find that the most relevant parameters are the density, n H and the typical electric dipole moment of 1 nm grains, µ 0 .
We use the updated grain size distribution given in Draine & Li (2007). Our grain size distribution is a double lognormal distribution characterized by a mean and width for the small and large grain populationsa = 4 and 20Å and σ a = 0.4 and 0.55 respectively (Draine & Li 2007). This size distribution has a lower cutoff of a min = 3.55Å which corresponds to N atoms = 20 Carbon atoms. The dipole moment distribution for the grains consists of three delta functions so that 50% of the grains have µ = µ 0 (a/1 nm) 3/2 , 25% have µ half as large, and 25% have µ twice as large.
The spinning dust emission in the DL98 model comes Updated beams and gains lead to differences in the plane, however large scale gradients (and in some cases dipoles) are seen in each band. Center: WMAP5−WMAP3 masked with the pixels that are used in the multi-linear regression fit (see §2). Right: WMAP5−WMAP3 smoothed to 30 • FWHM and masked. Here the large-scale fluctuations are clearly evident. In particular, the peaks range from ∼ −8 µK to ∼ 15 µK. All maps were mean subtracted over unmasked pixels prior to subtraction and smoothing.

TABLE 2
Environment from polycyclic aromatic hydrocarbons (PAHs) of very small size, N atom ∼ 100 or less. Small asymmetries in the grain geometry result in non-zero dipole moments that are taken to scale with the number of atoms as µ ∝ √ N atom . As these grains are spun up by various excitations processes (see DL98), they emit electric dipole radiation with total power P ∝ ω 4 , where ω/2π is the angular rotation frequency and is tens of GHz. 6 The parameters which characterize the environment and grain properties, with values appropriate for the WIM and CNM (cold neutral medium), are summarized 6 An improved treatment of the spinning dust problem by Ali-Hamoud & Hirata has made a few refinements of the DL98 model but those modifications do not significantly affect the results of this paper (Ali-Hamoud & Hirata priv. comm.)  Decreasing µ 0 also increases the peak frequency but decreases the total power per Hα. Note that in this part of (n H , µ 0 ) parameter space, the shape of the spectrum is nearly unchanged, but shifts in frequency and amplitude. Middle panels: Contours of the peak frequency (left) and peak amplitude (right) in the (n H , µ 0 ) plane. Lower panels: Same, but for CNM parameters. See the discussion in §4 for details.
in Table 2. The top two panels of Figure 3 shows example spinning dust spectra for WIM parameters while holding the density n H fixed while varying the characteristic dipole moment µ 0 and vice-versa. These spectra are in units of I ν /I Hα , or total intensity per intensity at Hα with T gas = 3000 K (our justification for using this gas temperature, which is lower than the commonly used 8000 K, is given in §5.1). For fixed µ 0 , as n H is increased, the spectrum shifts up in frequency with very little change in overall power per Hα. Since I Hα ∝ n 2 H in fully ionized environments, the total spinning dust intensity goes roughly like n 2 H . This point will be explored in more detail shortly, but it illustrates that an EM map like the Hα map should trace spinning dust emission.
For fixed n H , as µ 0 is increased Figure 3 shows that the spectrum increases in amplitude and decreases in peak frequency. This can be understood in terms of the radiated power per grain which goes as µ 2 : grains with larger dipole moments radiate away power faster and are thus harder to spin up leading to a lower peak frequency, but they also emit more total power leading to an increased amplitude.
The lower four panels of Figure 3 are contour plots showing the effects of varying n H and µ 0 on the amplitude and peak frequency of the spinning dust spectrum. Contours are shown for both the WIM and CNM conditions. The contours of peak frequency show that, if the amount of dust per H is assumed unknown (i.e., the amplitude is allowed to vary), then there is a strong degeneracy in the peak frequency between n H and µ 0 . The implication of this strong degeneracy is that a measurement of the spinning dust spectrum with sparse frequency sampling that only gives information about the peak frequency (as is the case with the WMAP data below, see Figure 7) cannot uniquely constrain these two parameters. On the other hand, if the grain abundance is assumed known, than the degeneracy is broken and the parameters are more tightly constrained. Figure 4 shows the behavior with n H of the emissivity per EM for both the total emission, j ν dν, as well as νj ν evaluated at the five WMAP frequencies. The parameters of the WIM and CNM environments are also given in Table 2 while those for the reflection nebula (RN) and photo-dissociation region (PDR) are shown on the plot. In the limit of low n H , all four environments exhibit roughly j ν ∝ n H behavior. In this limit, the dominant excitation and de-excitation mechanism are absorption and emission of infrared photons and thus the total emission scales linearly with density. This is in contrast to the argument presented in DF08b who argued that in the lowest density regions, the grains are in the episodic limit and so the total emission should scale as n 2 H . That argument neglected the importance of photon interactions.
While the linear n H scaling persists through all densities shown here for the RN and PDR regions, the WIM and CNM both flatten to j ν ∝ n 2 H behavior at n H ∼ 0.01 and 5 cm −3 respectively. In this regime, the dominant spin up mechanism is collisions with ions and the n 2 H scaling persists for four orders of magnitude in n H for the WIM (∼ 10 −2 − 10 2 cm −3 ). As the density increases further in the CNM, the spectrum turns over and again goes roughly like j ν ∝ n H . At these high densities, both spin up and damping are dominated by thermal processes (collisions with ions and atoms). However, it is interesting to note that over the density ranges shown in Table 2, both the WIM and CNM models scale as n 2 H indicating that a map that scales with EM like the Hα map should trace WIM and CNM spinning dust emission.

THREE COMPONENT FOREGROUND SPECTRA
Given that the Hα map should trace both the freefree and a spinning dust emission component, and given the fact that our foreground spectra are necessarily contaminated by a component with the spectrum of the CMB (flat in thermodynamic ∆T ) from the CMB crosscorrelation bias described in DF08a and §2, we interpret the Hα-correlated emission in Figure 1 as a three component spectrum. As in DF08b, we fit a free-free plus CMB plus DL98 WIM spinning dust model, to the measured cross-correlation spectrum I ν . Here, plc is the "planck correction" factor which converts thermodynamic ∆T to antenna temperature at frequency ν (FIRAS Exp. Supp. 1997). We choose the parameters n H and µ 0 , generate a spinning dust spectrum, and minimize χ 2 ≡ i (I ν − I mod ν ) 2 /σ 2 i , where σ i are the errors in the fit in each band i, over the parameters F 0 , C 0 , and D 0 . We allow D 0 to float in the fit because of uncertainty about the PAH size distribution and abundance in the WIM. We concentrate on CMB5, but our results are minimally changed for the other CMB estimators, it is only the value of C 0 which is significantly affected. Figure 5 shows the results of our three component fit for the Hα-correlated emission. From the figure, it is clear that a spinning dust spectrum with n H and µ 0 set to 0.15 cm −3 and 3.5 debye respectively (and with the WIM parameters in Table 2) fits the data remarkably well. This value for µ 0 is comparable to the earlier estimates in DL98. This spectrum has a peak frequency of ν p = 37.2 GHz and D 0 fit coefficient of 0.34. It is interesting to note that the value of D 0 is less than unity. This may indicate either that PAHs are depleted in the diffuse WIM by a factor of ∼ 3 or that the grain size distribution differs from that in the CNM from which the values of the lognormal mean and width of our grain size distribution The Hα-correlated spectrum from 23 to 94 GHz (points) using CMB5 and the three-component fit to the data (dashed line). The three components, free-free, spinning dust, and CMB, are shown separately. The amplitude of the free-free component yields Tgas ≈ 3000 K, while the amplitude of the spinning dust component is less than unity indicating that either PAHs are depleted in the WIM or the grain size distribution is altered. Lower: A zoom in of the spectrum showing that, although the bump is subtle, the statistical significance is very high.
From the amplitude of the F 0 fit coefficient, we can derive a gas temperature in the WIM which we find to be T gas ≈ 3000 K. This is the origin of the choice of T gas in the WIM spinning dust model. This value for T gas is much lower than the temperature inferred from emission line ratios (e.g., see Madsen et al. 2006). We stress that this low gas temperature is not the result of assigning some of the intensity to a spinning dust component. Ignoring the bump and merely fitting the data with a freefree only spectrum, still gives a gas temperature below 5000K. Using a technique similar to ours, Davies et al. (2006) also report anomalously low gas temperatures based on the free-free to Hα ratio in the WMAP 1-year data and DF08b found similar temperatures in the 3-year data. The origin of this discrepancy is unclear, though it may result from temperature variations in the WIM along the line of sight as discussed in Heiles (2001). 7 We will explore implications of a two-component temperature model in future work. Lastly, we find that the total intensity in Hα-correlated spinning dust emission is roughly 25% of that in free-free emission at 41 GHz.
The upper panel in Figure 5 shows that the bump in the spectrum is indeed subtle, but the zoom in beneath reveals that it is highly statistically significant. Since we minimize χ 2 over three amplitudes (F 0 , D 0 , and C 0 ), set the values of n H and µ 0 by hand, and fit to five data points, the number of degrees of freedom is formally zero. However, Figures 6 and 7 show that the χ 2 rises quickly as we deviate from these parameters. In the left panel 7 Another possibility is that our template fit of the spectrum is dominated by the strongest Hα features which naturally have a lower temperature.
of Figure 6, ∆χ 2 contours are shown in the (D 0 , C 0 ) plane and it is clear that the null hypothesis that the Hα-correlated emission is simply a linear combination of free-free plus a CMB bias (i.e., D 0 = 0), is ruled out at very high confidence. In fact we find that the D 0 = 0 case is ruled out at ∼ 27σ. The right panel shows χ 2 contours in the (D 0 , ν p ) plane as ν p is varied by hand. This parameter is less well constrained but is broadly consistent with ν p ∼ 35-40 GHz. Figure 7 shows χ 2 and D 0 contours in the (n H , µ 0 ) plane for the full-sky fit. The χ 2 contours show that there is a long narrow valley in the χ 2 surface reflecting the degeneracy in peak frequency shifts resulting from varying n H and µ 0 (see Figure 3). Since varying µ 0 affects the overall amplitude of the spinning dust component in units of I ν /I Hα while n H does not (because of the density squared behavior of the spinning dust emission at these n H ), the value of D 0 changes as we move along the narrow valley of the χ 2 surface. It is important to note that small deviations from these parameters quickly lead to large χ 2 values. Further, given this set of environmental conditions and grain sizes, D 0 = 1 is ruled out; again, implying either PAH depletion or modified grain size distributions in the WIM.

The Gum Nebula
In addition to our nearly full sky template fit, we have also carried out smaller regional fits. Of particular interest are the four regions of the Gum Nebula shown in Figure 4 of DF08b. This is a region that is very bright in Hα but has little in the way of thermal dust emission or synchrotron. The null hypothesis is that the spectrum of this region should correspond almost entirely to a free-free spectrum. Figure 8 shows the three component model fit as well as the contour plots for the four regions of the Gum Nebula. The bump in the Hα-correlated spectrum persists for all four regions, though the spinning dust amplitudes D 0 and CMB correlation bias amplitudes C 0 vary slightly from region to region. There is some evidence that the peak frequency ν p also varies slightly from region to region, but it is the least well constrained parameter and all four regions indicate roughly ν p ∼ 35-45 GHz. From the ∆χ 2 contours in the (D 0 , C 0 ) plane, we find that the null hypothesis of D 0 = 0 is ruled out at high confidence in all regions.
Due to the larger error bars in the template fit (the smaller regions include fewer pixels), the χ 2 valley in the (n H , µ 0 ) plane is somewhat broader than in the full sky fit. Nevertheless, all of the spectra are consistent with a WIM spinning dust spectrum with n H ∼ 0.1 cm −3 , µ 0 ∼ 2 debye, and D 0 ∼ 0.4.

Thermal Dust-Correlated Emission
As shown in Figure 1, dust-correlated emission falls from 23 GHz to 60 GHz and it was this departure from a thermal spectrum which initially lead to speculation that this "anomalous" dust component originated from spinning grains. In this section, we fit our three component model to this data but replace the WIM spinning dust model in Equation 5 with a CNM spinning dust model and the free-free component with a thermal dust spectrum T = T 0 × (ν/94 GHz) βD . We choose β D = 1.7 but The null hypothesis that the Hα-correlated emission follows a free-free power law with a contamination by the CMB spectrum is ruled out to very high significance (χ 2 ∼ 740 for 3 degrees of freedom at D 0 = 0). Right: χ 2 contours in the (D 0 , ν peak ) plane. Though the peak frequency is not constrained to high accuracy, ν peak ∼ 37 GHz is roughly the best fit value. The χ 2 (dashed) and D 0 (solid) contours in the (n H , µ 0 ) plane. There is a deep narrow valley in the χ 2 surface representing the near degeneracy in effects on the spinning dust peak frequency model with increasing n H and decreasing µ 0 . The values n H = 0.15cm −3 and µ 0 = 3.5 debye produce an acceptable fit with D 0 ≈ 0.3. Although D 0 does vary along the narrow valley in χ 2 , D 0 = 1 does not yield an acceptable fit, indicating that either the PAHs are depleted in the WIM or the grain size distribution is altered.
note that our results are not significantly changed for the range 1.5 < β D < 2.
The left hand panel in Figure 9 shows the results of fitting a CNM spinning model to the WMAP dustcorrelated spectrum. The spinning dust model has n H = 11 cm −3 and µ 0 = 5.5 debye. From the figure, it is clear that this particular spinning dust spectrum is not a very good fit to the data (χ 2 = 50.5), especially considering that there are few free parameters. However, the right hand panel of Figure 9 indicates that, although the amplitude is approximately correct (T 0 ≈ 1.0), for the CNM parameters there is no point in parameter space which yields a significantly better fit.
The poor fit is due to the fact that the 23, 33, and 41 GHz data points exhibit roughly powerlaw behavior (T ∝ ν −2.83 ) so that the spinning dust spectrum has too much curvature to fit the data. This powerlaw behavior has lead several authors to misidentify this emission as "dust-correlated synchrotron", however both WMAP polarization data (Kogut et al. 2007) and data at lower frequencies over large areas (Finkbeiner, Langston, & Minter 2004;de Oliveira-Costa et al. 2004) argue strongly against the synchrotron hypothesis. Furthermore, we note that, along any line of sight, there may very well be regions with different environment and grain properties leading to a superposition of spinning dust spectra. This degeneracy between synchrotron and a superposition of spinning dust spectra can only be broken with sufficient frequency coverage. In particular full-sky, high-resolution maps between 5 and 15 GHz could eliminate the ambiguity.
Lastly, we point out that the anomalous dustcorrelated emission in the CNM could originate from an emission mechanism that is neither synchrotron nor spinning dust. For example, Draine & Lazarian (1999) suggest that emission from magnetized dust grain could contribute to the total emission in the lower frequency WMAP bands. However, in the case of the WIM, the fact that the Hα map and the DL98 spinning dust models both scale roughly as density squared suggests that the bump in the Hα-correlated emission originates from spinning dust and not DL99 magnetic dust.  Dobler & Finkbeiner (2008b). The null hypothesis that the Hα-correlated emission is well represented by a free-free spectrum with a CMB contamination is ruled out to high significance in all regions (χ 2 ≥ 14 for 3 degrees of freedom at D 0 = 0). to strongly limit the contribution of magnetic grain materials to the microwave emission from dust.

Hα-CORRELATED BUMP
In this section we will address alternative explanations for the origin of the bump in the Hα-correlated emission. The alternatives listed here are not meant to be an exhaustive list of possibilities but rather constitute potentially important features of the maps, our fitting procedure, etc., and thus require careful attention about whether they can artificially generate a bump in our spectrum. For each of these, we argue why the bump is more likely to be explained by spinning dust.

Bandpass Effects
Although the spectra presented in previous sections consisted of five data points, the individual channels of WMAP have broad (∆ν/ν ≈ 0.2) bandpasses that must be taken into account. For a given emission mechanism, the total intensity in a WMAP channel is the spectrum of that emission integrated over the bandpass of that channel, where I tot b is the total intensity in band b and ω(ν) is the bandpass for band b normalized so that ω(ν)dν ≡ 1. Thus, for a free-free spectrum (I ν ∝ ν −0.15 ), the observed spectrum will be slightly modified from a straight powerlaw when plotted against the CMB weighted band centers (see below). We find that the deviations are at  Table 2), and a CMB component. Close inspection reveals that the curvature in this single component spinning dust spectrum is incompatible with the data and over-predicts the amplitude at 61 GHz. Right: The same as Figure 7 except for CNM parameters and fitting to FDS99-correlated emission. Although the amplitude is roughly D 0 = 1, the χ 2 values are unacceptably high. In fact, for the model in the left hand panel, χ 2 = 50.5. Thus, for reasonable CNM parameters, a single component DL98 spinning dust model is ruled out at high confidence. the level of ∼ 1%. Nevertheless, this total intensity, evaluated at each band, is what must be used to fit the derived cross-correlation spectrum.
Additionally, the WMAP data themselves must be converted from thermodynamic ∆T to I ν in each band assuming a CMB weighted band center, 8 where the spectrum of the CMB I CMB ν ∝ ν 2 plc(ν) (i.e., flat in thermodynamic ∆T ). All of the fits presented in previous sections take the bandpass weighting into account, and we find that the changes to the results are also at the ∼ 1% level.

Dust Extinction and Masking Effects
It has long been known that, although a map of Hα is an excellent tracer of free-free emission from ionized gas, it is not perfect. In particular, extinction of Hα by dust makes the map a poor tracer of free-free in regions of high dust column density. To account for this Finkbeiner (2003) (whose map we have used in this paper) apply a dust correction which assumes uniform mixing between the dust and gas. Dickinson et al. (2003) take a slightly different approach by matching slices through the galactic plane to an extinction model plus dust column density model and attempting to directly measure the extinction.
The effects of dust on the Hα map could manifest themselves in two ways. First, recall that our mask is generated by avoiding regions where the extinction due to dust at Hα is E(B − V ) > 1 mag. It is difficult to 8 By design the WMAP receivers measure energy and so the natural data unit is R Iν dν, however since they are then calibrated off of the CMB dipole, the data are presented in thermodynamic ∆T .  envision that our mask could have a sufficiently large impact on our derived spectrum to produce the bump, but it is instructive to consider the possibility. In Figure 10 we show the derived Hα-correlated spectrum for a wide range of dust cuts (i.e., for numerous masks). As can be seen in the figure, the bump feature is robust to variations in the mask.
The second potential issue is that perhaps the Hα map has been over corrected for dust so that it is actually a rather poor tracer of free-free, potentially leading to the bump as a spurious artifact. To test this hypothesis, we performed our template fit with an Hα map that was not corrected for dust extinction at all. Even in this extreme case of under -correction, we find that the bump persists. We conclude that the bump does not result from errors in the extinction correction of the Hα map.

CMB Cross-Correlation Bias
The CMB cross-correlation bias described above (and more extensively in DF08a) is both large and ubiquitous in CMB foreground analyses. Because the effect on the derived spectrum of each component can be substantial, it is important to rule out the possibility that this bias is producing the bump. This possibility represents the "null hypothesis" discussed in §5.1. The basic question is, can the derived cross-correlation spectrum be fit by a simple linear combination of free-free and CMB spectra? As we showed in that section, this D 0 = 0 case is ruled out at very high confidence with a χ 2 per degree of freedom of 245 (for 3 degrees of freedom at D 0 = 0).

Cross-Correlation Between the Templates
If there is any chance spatial cross-correlation between the templates, than this will impact the shape of the derived spectra. Furthermore, if one of the templates is contaminated by Hα morphology or vice-versa, than the spectrum of each foreground will be "contaminated" by the others. Again, a clear example of this would be poor dust correction of the Hα map adding dust-like morphology to that map. Another example would be reflection of Hα photons created in the plane off of dust at high latitudes, which would also imprint a dust-like morphology on the Hα map. There is also the fact that the WIM emits thermal radiation as well and so there is some Hα morphology present in the FDS99 map.
DF08b showed that the chance spatial crosscorrelations between the templates used in the analysis is sufficiently small that there should be relatively little contamination of one foreground spectrum onto another. We explore this point in greater detail in Appendix A, where we specifically address the fact that WIM features appear at a low level in the dust map.

Variations of T gas with Position
Gas temperatures in the interstellar medium are known to vary with position. For example, there is evidence that there is a decrease in T gas towards the Galactic center (Quireza et al. 2006). However, even though relatively small variations in T gas lead to large changes in the Hα to free-free ratio, the effect is simply an amplitude shift (Valls-Gabaud 1998). Thus, when we fit the nearly full sky, we are averaging over many regions with slightly different amplitudes, but all with spectra ∝ ν −0.15 . Thus, our resultant spectrum should be ∝ ν −0.15 indicating that a superposition of many regions with varying T gas would not produce a bump in the average spectrum. 6.6. Model Independent Identification of the Anomalous Emission Rather than attempting to address every possible contamination mechanism that could result in a bump in the Hα-correlated spectrum, it is possible to identify the bump without the use of any template fitting and only using the WMAP data. The top two panels of Figure  11 show the WMAP data at 23 GHz (with CMB5 subtracted) as well as the Hα map. We identify two regions that are bright in Hα and also bright at K band (23 GHz): ζ Oph near (ℓ, b) = (6, 23) degrees and the Gum Nebula near ℓ = 260 and −22 < b < 17 degrees . If the Hα map traces only free-free emission, then subtracting K band scaled by a free-free spectrum from Ka, Q, V, and W band should completely eliminate these features.
The bottom grid in Figure 11 shows K band in the left most column and then the result of subtracting K band scaled by a free-free spectrum from Ka, Q, V, and W bands (columns 2, 3, 4, and 5 respectively) for both the Gum Nebula region (row 1) and the ζ Oph region (row 2). Since the spectrum of free-free is harder than synchrotron and softer than thermal dust, synchrotron emission is highly over-subtracted and thermal dust emission is under-subtracted (note the prominence of thermal dust emission in column 5). What we find is that with a freefree spectrum, there is a clear excess in the southern Gum Nebula region and particularly in ζ Oph. The implication is that, ζ Oph and the Gum Nebula contain an emission component which is harder than free-free from 23 GHz to 41 GHz, but softer than free-free from 41 GHz to 94 GHz. Furthermore, this spectrum is not soft enough at 23-61 GHz to be synchrotron, yet is vastly brighter than the expected thermal dust emission, so it cannot be any of the standard foreground components. Rather, it has a spinning dust type spectrum with peak frequency near ∼ 40 GHz.

DISCUSSION
We have repeated the multi-linear regression template fit outlined in Dobler & Finkbeiner (2008a, DF08a) on the 5-year WMAP data and found that the anomalous "bump" at ∼ 40 GHz in the Hα-correlated emission spectrum persists and has in fact become more pronounced. The most significant changes in the spectrum from 3-year to 5-year data come from large scale modes (particularly a large dipole at 61 GHz V band) of amplitude ∼ 10 µK that were corrected in the 5-year data.
We interpret this bump as a "warm ionized medium" (WIM) spinning dust component that is traced by the Hα map. We find that for WIM parameters, the total spinning dust emission per grain scales roughly as the ambient ionized gas density squared (see Figure 4). Therefore, a map of emission measure (EM = n 2 e dℓ) like the Hα map would be expected to trace this component.
Any derived foreground spectrum is subject to the CMB correlation bias (see DF08a) from chance spatial correlations of the foreground emission with the CMB. Thus, we fit a three component model consisting of freefree, WIM spinning dust, and CMB spectra to the derived Hα-correlated emission in which the amplitude of each component is allowed to float. The best fit model gives a value for the ion density of the ambient medium of n H ∼ 0.15 cm −3 and a characteristic dipole moment for the grains of ∼ 3.5 debye referenced to 1 nm grains. However, there is a strong degeneracy between these two parameters as they have similar effects on the peak frequency. The Hα map. First column: A cut out of K band around the Gum Nebula (row 1) and ζ Oph (row 2). Second through fifth column: K band subtracted from Ka, Q, V, and W bands using a free-free ν −2.15 spectrum. Clearly there is residual emission in both the southern Gum Nebula and the ζ Oph cloud using a free-free spectrum indicating the need for a separate component. This component is harder than free-free from 23 to 41 GHz and softer than free-free from 41 to 94 GHz, which is indicative of a spinning dust spectrum. Although there are significant regions of over-subtraction (due to the substantial synchrotron component at low frequencies), the Hα-correlated emission is clearly under-subtracted by a free-free spectrum.
The amplitude of the spinning dust spectrum must be reduced by a factor ∼ 0.3 compared to the model to fit the data, possibly indicating that PAHs are depleted in the WIM. From the amplitude coefficient of the free-free component, we use the Hα to free-free ratio to infer a gas temperature of ∼ 3000 K. Near the peak frequency at 41 GHz, the total spinning dust emission is roughly 20% of the free-free emission. We also find that the bump persists in smaller regional fits of the Gum Nebula as well with roughly the same amplitude. There is some evidence that the peak frequency varies with region, but this parameter is the most poorly constrained in our analysis.
As in numerous other studies, we also find the now familiar rise in the thermal dust -correlated spectrum from 41 to 23 GHz from anomalous dust-correlated emission. An attempt to fit a three component model of thermal dust, "cold neutral medium" (CNM) spinning dust, and CMB spectra to the dust-correlated emission yields a poor fit. However, we expect that, along any line of sight, there will be multiple regions with different environmental properties so that the low frequency dust-correlated emission may represent a superposition of spinning dust spectra. Although other possibilities exist for this emission (magnetic dust and "dust-correlated synchrotron" being two), we point out that since the Hα map is a density squared map, and since WIM spinning dust emission goes roughly as density squared, we believe it is very likely that the bump in the Hα-correlated emission represents a spinning dust component. Magnetic dust emission would not correlate with Hα in this way.
We have shown that the bump cannot be explained by numerous possible systematic uncertainties in our fit. For example, the bump cannot be the result of the CMB correlation bias because a fit to the spectrum with a two component free-free plus CMB model yields a very bad χ 2 = 740 with three degrees of freedom. It also cannot be due to poor correction for dust extinction of the Hα map since the bump persists even if the template fit is performed with a completely uncorrected map, nor is it sensitive to our choice of mask for which we cut on dust extinction of Hα. Variations in gas temperature with position and cross-correlation between the templates used in the fit are also incapable of producing the bump.
Perhaps the most striking visualization that the bump is not an artifact of the fitting procedure is by subtracting the K band WMAP data scaled with a free-free spectrum from Ka, Q, V, and W bands. This reveals that in regions that are very bright in K band and Hα which were thought to be completely free-free dominated at these frequencies (e.g., the Gum Nebula and ζ Oph), there is a positive residual at 33, 41, and 61 GHz. Scaling by a free-free plus bump spectrum instead removes the residual. The implication is that there is emission coming from these regions that has a spectrum that is harder than free-free from 23-41 GHz and then softer than freefree from 41-94 GHz, which resembles a spinning dust spectrum.
The interpretation of the dust-correlated anomalous emission as spinning dust (or magnetic dust) has been plagued by the fact that ancillary data sets at lower frequencies are needed to distinguish that emission from synchrotron. Here we have shown that a turnover in the spinning dust spectrum is completely recoverable within the WMAP frequency range. The only surprise is that it correlates with the Hα map. In hindsight, this is not very surprising since the Hα map traces density squared emission, though it does suggest that there is no "true" template for the spinning dust. Rather, it is potentially ubiquitous and probably will not correlate precisely with any map of the sky.
We acknowledge informative discussions with Simon Cassasus, Clive Dickinson, Joanna Dunkley, Ben Gold, Gary Hinshaw, Al Kogut, and David Spergel. We thank Matt Haffner and Carl Heiles for insights on gas temperatures in the warm ionized medium, and Alex Lazarian, Chris Hirata, and Yacine Ali-Hamoud for stimulating discussions on excitation mechanisms in spinning dust models. Some of the results in this paper were derived using HEALPix (Górski et al. 1999;Calabretta & Roukema 2007). This research made use of the IDL Astronomy User's Library at Goddard 9 . GD and DPF are partially supported by NASA LTSA grant NAG5-12972. BTD is partially supported by NSF grant AST-0406883.
9 Available at http://idlastro.gsfc.nasa.gov   Note that δ ∼ 4 × 10 −5 mK/R is roughly required to appropriately remove Hα morphology from FDS99 as shown in Figure 12. As expected, as δ is increased, the spectrum from Figure 5 is contaminated with FDS99-correlated spectrum in Figure 9 (see text, Equation A8). Middle: the spectrum of Hα-correlated emission assuming that FDS99 exactly traces thermal dust emission with T dust = FDS99 × (ν/94 GHz) 1.7 . For δ = 0 mK/R, the bump in the Hα-correlated spectrum represents the difference of WIM and CNM spectra, for δ = 4 × 10 −5 mK/R, the spectrum is almost entirely WIM, and as δ is increased further, the bump represents an addition of WIM plus increasingly more CNM spectra. Indeed, the peak frequency moves to the lower CNM value as δ gets large. Right: The χ 2 (dashed) and D 0 contours in the (n H , µ 0 ) plane for the spectrum shown in the middle panel with δ = 4 × 10 −5 mK/R. With this small level of contamination, our results for the best fit n H , µ 0 , and D 0 values are almost completely unchanged (cf., Figure 7).
A.1. Hα as an imperfect tracer of free-free For the template fits the equation we are trying to solve, neglecting the CMB and the associated cross correlation bias and considering only two foregrounds, is for a 1 and a 2 , where H is the true Hα map which we assume exactly traces free-free emission, D is the dust map, t 1 and t 2 are the true amplitudes (at an arbitrary frequency), and H ′ is the observed Hα map (which may be contaminated in some way). By multiplying Equation A1 by H ′ and solving for a 1 , we obtain where again, H is the Hα map, and L is the Kalberla et al. (2005) H I map. With this notation, δ 0 has the units of mK/R. We solve the above equation over unmasked pixels (where the mask consists of the point source mask described in §2.1 plus a masking of all pixels for which the dust extinction at Hα is 2.65E(B − V ) > 0.5 mag) and find that δ 0 ∼ 4.5 × 10 −5 mK/R. Although this value varies slightly from region to region (Heiles et al. 1999), the contamination is very small and extremely subtle (see Figure 12). Nevertheless, based on our value of δ 0 , Hα morphology does indeed "contaminate" the FDS99 map since the large grains in the WIM emit thermal radiation at 94 GHz as well.
To calculate the affect on the inferred spectrum, we can use the same procedure as §A.1 but now taking D ′ → D+δ 0 H. It is straightforward to show that the recovered Hα-correlated spectrum becomes, That is, the inferred Hα-correlated spectrum is contaminated by the FDS99-correlated spectrum at a level that is given by δ 0 . Assuming that H exactly traces both the free-free emission and the WIM spinning dust emission and that D exactly traces CNM emission (thermal and spinning dust), the implication is that the bump in the inferred spectrum of the Hα-correlated emission represents the WIM spinning dust spectrum minus some amount of the CNM spinning dust spectrum, determined by how much the Hα morphology has leaked into the FDS99 map via large grain thermal emission in the WIM. The left panel of Figure 13 shows the level of contamination of the Hα-correlated spectra by taking D ′ = FDS99−δ × Hα as a tracer of CNM dust. As expected from Equation A8, for large values of δ, the Hα map is being over-subtracted from FDS99 and the resultant spectrum is highly contaminated by a CNM thermal plus spinning dust spectrum (cf., Figures 5 and 9). For δ = 0 mK/R on the other hand, some of the WIM spinning dust bump is being absorbed in the D ′ -correlated spectrum and so the bump in the Hα-correlated spectrum represents a WIM spinning dust minus some CNM spinning dust spectrum. Based on the fit in Equation A7, the δ = δ 0 ≈ 4 × 10 −5 mK/R case is a nearly pure mixture of free-free plus CMB plus WIM spinning dust spectrum.
However, since the WIM does also emit thermal dust radiation, if we assume that FDS99 is a perfect representation of the thermal emission at 94 GHz, then removing the WIM (Hα) morphology from FDS99 implies that the Hαcorrelated spectrum also contains that thermal emission. We can estimate the amplitude of this effect by assuming that the WIM thermal emission has a ν β dependence on frequency with β = 3.7 (in intensity units, β = 1.7 in antenna temperature) and adding δ × H × (ν/94 GHz) 3.7 back into the Hα-correlated spectrum. This is equivalent to setting, The resultant spectra are shown in the middle panel of Figure 13. In this case, the bump is ideally due to only spinning dust emission from the WIM for δ = 4 × 10 −5 mK/R. For larger values of δ the CNM spinning dust spectrum begins to leak into the fit so that by δ = 10 × 10 −5 mK/R, the spinning dust bump has moved towards the lower CNM value, and the amplitude has increased since the WIM and CNM spectra are now both present in the data. Lastly, the right panel of Figure 13 shows χ 2 and D 0 contours in the (n H , µ 0 ) plane for a free-free plus CMB plus WIM spinning dust fit to the δ = 4 × 10 −5 mK/R spectra in the middle panel of Figure 13. These contours indicate that the best fit values of n H and µ 0 are not significantly affected by the presence of WIM morphology in the FDS99 map for our estimate of δ = 4 × 10 −5 mK/R. Nevertheless, the exact level of PAH depletion measured by our technique (D 0 ) will depend on precisely how one estimates the WIM leakage in FDS99. We defer a detailed analysis of this to future work.