{'fraction_non_alphanumeric': 0.03517450786394186, 'fraction_numerical': 0.0250759100678926, 'mean_word_length': 4.411113162266937, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Re-examination of the Distribution of Galactic Free Electrons

We present a list of 109 pulsars with independent distance information compiled from the literature. Since the compilation of Frail&Weisberg, there are 35 pulsars with new distance estimate and 25 pulsars for which the distance or distance uncertainty have been revised. We used this data to fit a smooth, axisymmetric, two disk model of the distribution of galactic electrons. The two exponential model components have mean local midplane densities at the solar circle of 2.03e-2 cm^-3 and 0.71e-2 cm^-3, and scale heights of 1.07 and 0.053 kpc. The thick component shows very little radial variation, while the second has a radial scale length of only a few kiloparsecs. We also examined a model which varies as sech^2(x), rather than exp(-x), in both the radial and vertical direction. We prefer this model with no midplane cusp, but find that the fit parameters essentially describe the same electron distribution. The distances predicted by this distribution have a similar scatter as the more complex model of Taylor&Cordes. We examine the pulsars that deviate strongly from this model. There are two regions of enhanced dispersion measure, one of which correlates well with the Sagittarius-Carina spiral arm. We find that the scatter of the observed dispersion measure from the model is not fit well by either a normal or log-normal distribution of lump sizes, but may be caused instead by the uncertainties in the distances.


Introduction
One of the most important discoveries in the study of the interstellar medium is the realization that the warm ionized medium (WIM) is a major component of our galaxy; it has a thick distribution and it is not localized around ionization sources. Reynolds (1989) used the dispersion measures (DM = n e dl) towards pulsars with known distance to measure the scale height of the WIM in the Milky Way, and showed that it is much larger than that of the bulk of the neutral hydrogen.
If the distance to a pulsar is known, this can be used with its DM to constrain models for the spatial distribution of the free electrons. The most popular model is that derived by Taylor & Cordes (1993, TC), which also used the observed scattering measures to a different set of pulsars to refine the parameters of the model. Their most important contribution was the addition of non-axisymmetric elements, i.e., spiral arms defined by the locations of HII regions (Georgelin & Georgelin 1976). Their main justification is the observed asymmetry in the DM vs. galactic longitude plots. They also incorporated the unusually high DM observed towards the Gum Nebula.
Models of this kind are used frequently to determine distances to pulsars. TC claim that their model yields distances accurate to 25%. But, since its publication, the set of pulsars with independent distance measurement has increased, some distances have been revised, and pulsars with forbidden DM (higher than the asymptotic value predicted by TC) have been observed. In addition, observations of the the angular broadening of radio sources has been used to constrain the electron density in the Galactic center (Lazio et al. 1999;Lazio & Cordes 1998a,b) and the scalelength of the distribution in the anti-center direction (Lazio & Cordes 1998c,d). Finally, the recent completion of the Wisconsin Hα Mapper survey of diffuse galactic Hα emission with onedegree angular resolution and ∼ 10 km s −1 velocity resolution (Reynolds 1998;Haffner 2000) will allow the development of more complex models. These observations will allow for a reassessment of the location of galactic spiral arms (Georgelin & Georgelin 1976;Ruseil et al. 1998;Georgelin et al. 2000), as well as the discovery and placement of large angular-scale HII regions, such as the Gum Nebula.
In this work, we present an updated list of pulsars of known distance. We then use this data to constrain a new axisymmetric model for the free electron distribution, and show how the Taylor-Cordes model and the new axisymmetric model fare in predicting distances to the pulsars. We also consider to what degree the available data constrain the lumpiness of the warm ionized medium. Incorporation of non-axisymmetric effects, such as the galactic spiral arms and individual nebulae can subsequently be incorporated using the WHAM data and more recent radio recombination line surveys of HII regions.

The Pulsar Data Set
A list of 109 pulsars with distance information was gathered from a number of sources and are compiled in Table 1, and presented in order of increasing distance. Of this list, four are in the Large or Small Magellanic clouds. Of the remainder, 76 have both upper and lower distance limits; 20 have only lower limits, and 9 have only upper limits. This dataset is ∼ 50% larger than the data used by TC. Of the 109 pulsars, there are 35 new distance determinations since the compilation of Frail & Weisberg (1990, FW90), which provided the bulk of the measurements used in the TC model, 25 objects in which there have been revisions in either the distance or distance uncertainty of the pulsar, and 49 objects whose distance estimates remained unchanged.
Distance estimates come from a variety of methods, which we briefly summarize here.
Kinematic distances (68 pulsars): The majority of pulsar distance measurements come from the combination of 21 cm absorption combined with an axisymmetric, kinematical model for galactic rotation (Fich, Blitz, & Stark 1989). FW90 re-evaluated all the distance measurements up to that time using this model (with corrections for pulsars towards the Perseus arm), and a uniform set of criteria for converting absorption velocities to distance. These criteria have been adhered to in subsequent work. Probably the largest source of systematic error is due to the non-circular "streaming" motions in the vicinity of spiral arms.
Association with globular clusters (17 pulsars): The next most common distance determination method come from association of a pulsar with a globular clusters of known distance. Table 1 only lists one pulsar per globular cluster; when more than one pulsar is known, the variation in dispersion measure is small. Since the compilation of FW90, the distances to globular clusters have been considerably refined due to improved color-magnitude diagrams and shifts in the assumptions about the luminosity of RR Lyrae stars. As a result, some distances estimates have been revised by more than a factor of two since FW90 (Harris 1996, with online updates at http://physun.mcmaster.ca/∼harris/mwgc.dat). The uncertainty in the distance modulus of these clusters was assumed to be σ = 0.1 + 0.4E B−V magnitudes. More heavily reddened clusters have poorer data since they have greater problems with field contamination and crowding (Harris 1999).

Trigonometric parallax (8 pulsars):
Potentially the most reliable distances come from interferometric measurements of annual parallax. However, there are several practical difficulties arising from ionospheric effects and a scarcity of nearby calibrators for positions. Improvements in the techniques have led to changes in the published distances by more than a factor of two. The distance estimate for B0950+08 increased from 130 pc ) to 280 pc , while the distance estimate for B1919+06 decreased from 3.3 kpc ) to 1.2 kpc (Chatterjee et al 2000). These changes were much larger than the stated uncertainties in the measurements. Accurate estimates are vital if pulsars are to be used as probes of the structure of the local interstellar medium.
Association with other galaxies (4 pulsars): Four pulsars have been associated with the Mag-ellanic Clouds, three in the LMC, one in the SMC. These pulsars are valuable in constraining the electron density in the Galactic halo. However, an unknown fraction of the dispersion measure must arise in the host galaxy, so their utility in constraining the Galactic free electron column density is compromised.
Timing parallax (5 pulsars): Distances to millisecond pulsars have also been estimated using variations in arrival time of the pulses. There is a annual change in the pulse arrival time whose magnitude is given by ∆t = r 2 cos 2 θ/(2cd) where r is the Earth-Sun distance, θ is the angle between the line of sight and the ecliptic plane, and d is the distance (Ryba & Taylor 1991). This variation is ∆t = 1.2µ sec for d = 1 kpc. This level of timing accuracy has been reached for only a few pulsars.
Period derivative distances (2 pulsars): Bell & Bailes (1996) have shown that in many cases, the observed orbital period derivative of binary pulsars is dominated by term of the formṖ b /P b = v 2 /(cd). If one uses the predictions of general relativity to derive the intrinsic period derivative, knowledge of the proper motion of the pulsar then allows for an accurate estimate of the distance. This method has only been applied to two pulsars to date.

Spectroscopic parallax of binary companion (1 pulsar):
There is one case in which the binary companion of a pulsar is a ∼ 10 M ⊙ Be star . In this case, spectroscopic parallax was used to estimate the distance.
X-ray luminosity distance (1 pulsar): There is one distance estimate for B0656+14 based upon the identification of the X-ray counterpart together with a model of thermal X-ray emission from the neutron star (Golden & Shearer 1999). As will be seen, this pulsar ends up being an outlier in our model. As a result, we are not convinced that this method is reliable.
Trigonometric parallax of optical counterpart (0 pulsars): If the optical counterpart of a pulsar can be identified, then ground based or Hubble Space Telescope observations could yield a parallax estimate. This technique has been used to determine the distance to the neutron star Geminga (Caraveo et al. 1996). However, we have not included Geminga in our list because it is unclear if it has a reliable radio signal. The search for optical counterparts of pulsars has been relatively unsuccessful to date (Caraveo 2000). Still, we think this method holds some promise, particular for pulsars with the very lowest dispersion measures like J0108-1431 which has DM = 1.83 cm −3 pc (Tauris et al. 1994).
Scattering screen distance (0 pulsars): It has been suggested that the transverse velocity of a pulsar derived using models of interstellar scintillation can be combined with measurements of proper motion to constrain the distance to the pulsar (Gupta 1995;Deshpande & Ramachandran 1998;Cordes & Rickett 1998). Application of this model requires a knowledge of the distribution of electron density and scattering properties along the line of sight, and as a result is principally useful for pulsars which lie behind HII regions of known distance or pulsars well above the disk of the Galaxy.
Cross-checks (7 pulsars): There are six pulsars for which two independent methods have been applied for distance determination. In each case, the distances estimate agree within the stated errors, although in two cases the agreement is marginal. Such checks are important since they test the reliability of the individual methods. We summarize these results here. B1929+10: This pulsar has three discrepant measures for trigonometric parallax, π = 21.5 ± 0.3 mas , π < 4 mas (Backer & Sramek 1982), and π = 5.0 ± 1.5 mas (Campbell 1995). The kinematic distance is d < 1.6 kpc (Weisberg, Rankin, & Boriakoff 1987). We have adopted the most recent parallax distance, which is consistent with the kinematic distance. B0833-45: The distance to the Vela SNR is given as d = 250 ± 30 pc (Cha, Sembach, & Danks 1999), while recent VLBI parallax gives d = 316 +37 −29 pc (Legge 2000). While these uncertainties do not overlap, the uncertainties in stellar distances may be slightly underestimated. B1855+09: Timing parallax distance to this pulsar was given as d = 0.83 +0.66 −0.24 pc (Ryba & Taylor 1991), later refined to d = 0.91 +0.34 −0.20 pc (Kaspi, Taylor, & Ryba 1994). This agrees marginally well with the kinematic distance limits d lower = 1.6 ± 0.5 to d upper = 2.0 ± 0.4 . B1800-21: The kinematic distance limits to this pulsar are d lower = 4.0 ± 0.6 kpc and d upper = 4.9 ± 0.3 kpc which agree with the kinematic distance to the SNR G8.7-0.1, also established kinematically (Finley &Ögelman 1994). B1758-23: The kinematic distance limits to this pulsar are d lower = 3.5 ± 0.9 kpc and d upper = 6.9 ± 0.1 kpc which agree with the kinematic distance to W28, also established kinematically (Frail, Kulkarni, & Vasisht 1994). B1937-21: Kinematic distance limits are d lower = 4.6 ± 1.9 kpc and d upper = 14.8 ± 0.9 kpc , which agrees with the timing parallax distance of d > 3.6 kpc (Kaspi, Taylor, & Ryba 1994). B1534+12: The period derivative distance to this binary pulsar is d = 1.08 ± 0.15 kpc, which is consistent with the timing parallax limit of d > 0.67 kpc (Stairs et al. 1999) Figure 1 shows the spatial distribution of the pulsars in our sample with both upper and lower distance limits projected onto the Galactic plane, while Figure 2 shows a plot of their dispersion measure as a function of galactic longitude. Note that in Figure 2, there is no clear evidence for the asymmetry in maximum dispersion measure around l = 0 • , which is present in the complete set of pulsars including those of unknown distance.

Fitting an axisymmetric model
Using this data, we fit a two component model of the galactic disc to the pulsar data. The model has the form, where f (x) is either = exp(−x) or = sech 2 (x), and r ⊙ = 8.5 kpc is the galactocentric distance of the Sun. The fit was achieved through a variant of the χ 2 method: we defined the error-of-the-fit  ∆ as, where n is the number of pulsars (76) with both upper and lower distance limits, ν is the number of free parameters in the model (6) Most of the distances to the pulsars we used (41 out of 76) are determined by assuming a kinematic model for the galactic rotation and comparing it to the 21 cm absorption observed towards the pulsar. For these pulsars, we define the distance to be halfway between the minimum and maximum limits. For these pulsars, there is a uniform probability for the location of the pulsar between the distance brackets, as opposed to the distances obtained by parallaxes, for example, which have a gaussian probability distribution for the distance around a preferred value. Therefore, the kinematic distances have an extra 1/ √ 3 factor in the corresponding σ.
An annealing procedure was used to get the best fit for the n 0,1 , r 0,1 and z 0,1 parameters with A = 0. Then, A was adjusted to get ∆ = 1 and a new fit was obtained. The procedure was repeated until convergence was achieved. Outlier pulsars were spotted by a procedure described below, and those common to both functional forms were taken out of the sample. Then, the procedure was repeated and the new fit is the one considered as final. The parameters of the best fits are in Table  2. The results of the fits for the f (x) = sech 2 (x) case is presented in Figure 3. The corresponding density profiles are shown in Figure 4. There is not enough data to distinguish between the two functional forms, but the resulting fit parameters are different in each case. We prefer the sech 2 (x) model because it does not have a midplane cusp, and yields fewer outliers. For this case: n e (r, z) = 1.77 × 10 −2 cm −3 sech 2 (r/15.4 kpc) sech 2 (R ⊙ /15.4 kpc) sech 2 (z/1.10 kpc) + 1.07 × 10 −2 cm −3 sech 2 (r/3.6 kpc) sech 2 (R ⊙ /3.6 kpc) sech 2 (z/0.04 kpc) These results are comparable to the previous axisymmetric model of Cordes et al. (1991), although our thin disk component has a lower midplane density (n = 10 −2 cm −3 vs. n = 20 × 10 −2 cm −3 ) and a shorter scaleheight (h = 40 pc vs. h = 175 pc). We also find a noise parameter of A = 0.30. Savage et al. (1990) did a similar study with a smaller sample of pulsars. The value of the exponential scale height found is consistent with theirs within the error bars, but their intrinsic scatter (1.65 = 1 + A) is larger than ours.
The procedure for spotting the outliers was the following: Consider the values of DM model obtained by integrating n e in the line of sight towards each pulsar to the distance brackets, and call them DM + and DM − . Consider now the values, for each pulsar (the error bars in the bottom panel of Figures 5 and 6 are the values of the square root above). If sign(x + −1) = sign(x − −1), then that pulsar is considered an outlier. As mentioned, the pulsars spotted as outliers for both functional forms are taken out of the sample for the final calculation of the fit.

Deviations from the Smooth Axisymmetric Model
Since observations of HII regions in the Galaxy show that there are clearly inhomogeneities and asymmetries in the distribution of free electrons, we have looked for patterns in the spatial and statistical distribution of our residuals, DM data /DM model . We discuss in turn, the individual outliers, the distribution of residuals with respect to longitude and distance, and the nature of the scatter about our smooth model. In the future, the combination of this data with new radio recombination surveys for distant HII regions and velocity-resolved Hα surveys of more nearby gas will yield a more complicated, but realistic, model.

Outliers
Of the 76 pulsars with both upper and lower limits, 15 are outliers in both the exponential and sech 2 (x) model. These outliers are noted in Table 3, together with the observed dispersion measure, and the dispersion measure that we would predict given the distance, DM ± = (1 ± A)DM model (D). Two of these pulsars have dispersion measure that are lower than one would expect given their distance. The first, B1741-11, with a timing parallax distance, is only 0.36 kpc distant. Given the lumpiness of the local interstellar medium (Cox & Reynolds 1987;Toscano et al. 1999), it is not out of the question for such a low density sightline to arise for such a short distance. The second pulsar with a much lower dispersion measure than expected is B1937+21. This pulsar, which has a kinematic distance of d = 4.6 kpc to 14.6 kpc, has DM = 71 cm −3 pc, while our model yields DM − = 208 cm −3 pc. This yields a mean of electron density of n e < 0.016 cm −3 over at least a  While low DM outliers are difficult to explain, high DM outliers are likely to arise due to the passage the pulsar line of sight through a dense HII region. Of the 13 high outliers, four are associated with SNR (Vela, MSH 15-52 SNR, G308.8-0.1, and W28) and one has a 10 M ⊙ companion (and presumably an associated HII region). Using the ionizing output luminosity tabulated in Osterbrock (1989), the dispersion measure of an HII region around an O9 star, for example, would be DM = 2nR S = 315 cm −3 pc, where R S is the Strömgren radius. The excess dispersion measure, defined as DM excess = DM data − DM + for the 13 high outliers range from DM excess = 7 − 578 cm −3 pc. Thus these lines of sight are consistent with the intersection of the line of sight with discrete HII regions. However, we have searched catalogs of diffuse HII regions (Lockman, Pisano, and Howard 1996) and the WHAM maps (Haffner 2000) for correlations with these northern declination pulsars in this sample, but nothing outstanding was found. Since the majority of these pulsars lie at southern declinations, the high angular resolution Hα maps of Gaustad et al. (1997) will be extremely useful in the future.
There are two outliers for which we suspect the distance estimate may be incorrect. The distance to B0656+14 was obtained using an X-ray luminosity distance. Given the number of assumptions necessary to estimate the X-ray luminosity of a neutron star, we have some concerns about the reliability of this method. The distance to B0823+26 is based on a parallax measurement by Gwinn et al. (1986). Since the other pulsar examined in this study (B0950+08) has had a significant revision in its distance, a reconsideration of this pulsar parallax may be in order.
We have also compared to the 29 pulsars for which there are only upper or lower limits. We found that 26 of the limits are satisfied by the model while B2020+28 (D > 3.1 kpc), B2016+28 (D > 3.2 kpc), and B1818-04 (D < 1.6 kpc) are not. Thus our model satisfies the distance constraints of 91 out of 109 pulsars.

Spatial Distribution of Residuals
We now consider whether the known asymmetries in the distribution of galactic HII regions is reflected in the current dataset. A plot of DM data /DM model versus galactic location is presented in Figure 5, with the spiral arm positions used by TC overlaid. There seem to be two lines of pulsars with a higher than expected dispersion measure, marked by dashed lines. Some of these pulsars have been discussed by Johnston et al. (2001) as particularly noticeable outliers. One of these groups agrees roughly with the position of one of the spiral arms and has a pitch angle 27 • from the tangent. The other has a pitch angle of 22 • and is not coincident with any of the spiral arms. Given the distance uncertainties for these pulsars, it seems clear that any spiral structure that might exist is only weakly exhibited in this data set.
We have also considered whether there is evidence for a difference in the estimated midplane density if we use only pulsars identified as "interarm pulsars" to estimate the midplane density  Table 3 have filled symbols. There are two regions of high DM data /DM model , at approximately 2 and 5 kpc from the Sun towards the galactic center. One of these regions (noted with dashed lines) coincides with the position of the Sagittarius-Carina arm.
at the solar neighborhood, where those pulsars are marked as such in Figure 1. We found that there was a slight decrease in the derived midplane density, a factor of 2/3, compared to the total dataset. However, some of these pulsars may lie in or beyond the Local Arm, which although not included in the TC model, is known to exist in the Hα data (Reynolds 1983).

Constraints on Clumpiness
We now consider what factors affect the scatter in the relationship between our simple axisymmetric model and the observed data. Figure 6 shows the comparison between the DM model and DM data . DM model takes into account the geometry of the distribution, so it measures the effective integration path. Therefore, we will use it instead of the distance in order to examine the nature of the scatter. An interesting feature in this plot is that the scatter appears to be a fixed fraction of the total dispersion measure.
We considered the possibility that the scatter of DM data about the smooth model might derive from a patchiness of the distribution of electrons in the Galaxy. Such a patchiness of the diffuse ionized medium has been predicted, for example, by Miller & Cox (1993) using the observed locations of O stars in the Solar Neighborhood, and a model for the ISM distribution, to calculate the steady state Strömgren volume distribution and ionization.
What would happen if the warm ionized medium were purely located in discrete lumps (or HII regions)? 1 In that case, we define DM = DM model as the dispersion measure that would result for the average line of sight through some variable number of clumps. The expected total number of lumps intersected along a line of sight would then be n = DM /DM lump . The variance in observed number should also be n so that (DM data − DM ) 2 = nDM 2 lump = DM · DM lump . We can therefore define a quantity, µ, for each pulsar, If the lump sizes are normally distributed, µ should be independent of DM , and its average over a large enough sample of pulsars should be the dispersion measure of the lump. In Figure 7 we plot the running mean of this quantity for both the top-down (from large DM model to small DM model ) and bottom-up sums. There is a strong trend in the lump size estimator with the distance. This could be explained by having two lump populations: small frequent lumps and large rarer lumps. Nearby, we pick only small lumps, yielding a small mean. As we move farther, we pick up more large lumps and the mean value increases. This could explain the steps observed in the bottomup running mean, while the top-down running mean is flatter. We thought that a log-normal distribution with the appropriate shape parameter might have that property, but found that a lognormal distribution could almost reproduce the properties of Figure 5 (constant fractional scatter with increasing DM), but not Figure 7 (µ is not constant).
Another possibility is that the deviations of the observed and modeled DM are not due to statistical noise, but instead fractional errors in the distance measurements. In such case, the variance is proportional to f 2 DM 2 , where f is the approximate fractional error, rather than √ nDM lump . In this case, a plot of ( DM −DM data ) 2 / DM 2 , should be roughly flat, which is verified in Figure 8. The corresponding value is f ∼ = 30%. If distance uncertainties are indeed the main source of scatter, it will be difficult to say anything definitive about the lumpiness of the warm ionized medium based on this type of data.

Predicting Pulsar Distances
One of the principal uses for a model of the galactic free electron distribution model is to predict the distance to pulsars. While we have not yet introduced the effects of asymmetries, spiral structure, and individual H II regions, we have written two FORTRAN routines (one for each functional form tested) that calculate pulsar distances using the model parameters in Table 2. 2 A comparison of the model distances and true distances for our sample of pulsars is given in Figure  9, using the f (x) = sech 2 (x) model. The error bars are obtained by calculating the distance that corresponds to (1 ± A)DM data . We note that no pulsars have a DM higher than the asymptotic limit when the uncertainty associated with our noise parameter A is considered.
In Figure 10, we compare the distances predicted by the TC model and our model with the observed distance constraints. When we consider only those pulsars with allowed DM (smaller than the asymptotic value), the dispersion in our model is similar to the model of TC, but with fewer free parameters.
We note however that the model we have developed is relatively unconstrained for pulsars interior to a galactocentric radius of R ∼ 4 kpc and exterior to 12 kpc. For example, unlike Taylor & Cordes, we have not included a annulus of electron density at R = 4 kpc, which presumably would be associated with the molecular ring. Lazio & Cordes (1998a,c) have discussed how additional information can be used to constrain these regions. We intend to address these issues in the future when we address the non-axisymmetric structure using the Wisconsin Hα Survey.   bottom-up (dotted) running averages are shown. The solid line is the least-squares fit to logarithm of the data. Since the slope is close to zero, this implies that the dispersion is approximately a constant fraction of DM. We suspect that this behavior is due principally to distance uncertainties. Fig. 9.-Comparison of the predicted and measured distances using the f (x) = sech 2 (x) model. The horizontal axis is the pulsar number, sorted by distance. The gray region is the quoted range in measured distances. The starred points are the outlier pulsars identified in Table 3, and the error bars include the effect of the noise parameter, A = 0.3.

Conclusions.
A smooth model for the distribution of galactic free electrons was obtained from a set of 109 pulsars with independent distance information. Although a more complex model incorporating spiral arms might be possible, we do not think that it is well constrained by this pulsar data alone, so we chose to use a simpler and probably more robust functional form. The exponential scale height obtained is consistent with the value quoted in Reynolds (1996). The scatter parameter found (A = 0.3) is smaller than the one found by Savage et al. (1990). This scatter parameter is used to predict a range of confidence in the predicted distances to pulsars.
Of pulsars with both upper and lower distance limits, fifteen pulsars are identified as outliers, with thirteen of these showing excess dispersion measure. Some of these are associated with supernova remnants or known HII regions. There is one very unusual pulsar, B1937 + 21, with an extremely low dispersion measure given its distance. In examining the residuals, we identified two regions of enhanced electron density, one of which corresponds well with the expected position of the Sagittarius-Carina spiral arm.
We found that a simple probabilistic model for a lumpy WIM failed to reproduce the deviations of the observed data from the smooth model. We suspect that the main source of scatter in our model is due to distance uncertainties, although it seems clear that are also occasionally large anomalous dispersion measures associated with HII regions. Some of these are in spiral arms, but their distribution may not be uniform in these arms.
We are grateful to Ron Reynolds, Matt Haffner, Joel Weisberg, and Linda Sparke for useful advice and encouragement, to NASA ATP grant NAG5-8417 for financial support, and to CONACYT-MEXICO for support for GCG.