document: scattering , absorption , and reradiation of photons by dust grains affect the propagation of starlight and play a key role in regulating the energy balance in the interstellar medium . 
   studies of these processes have provided a great deal of insight into the physical and chemical properties of dust grains ( for reviews of dust properties , see mathis 1990a ; witt 2003 ) . 
   it is also important to disentangle the effects of intervening dust from many kinds of astronomical observations , ranging from photometry and spectroscopy of galactic stars and external galaxies to mapping small - scale anisotropy in the cosmic microwave background radiation ( cmbr ) . 
   the degree of absorption and scattering by dust depends , in general , on the wavelength of the incident radiation ( whitford 1958 ) . 
   this dependence can be quantified in terms of the normalized dust extinction law @xmath20 ( or reddening law ) , where extinction is the sum of absorption plus scattering . 
   the ratio of total to selective extinction , @xmath21 , is a commonly - used measure of the slope of the extinction / reddening law . using data along many different galactic sight - lines , cardelli , clayton , & mathis ( 1989 , hereafter ccm ) found a tight correlation between the overall shape of the reddening law and the slope @xmath0 . 
   ccm devised an @xmath0-based , one - parameter family of empirical fitting functions to characterize the observed range of extinction law shapes ; this parameterization was later refined by fitzpatrick ( 1999 ) . 
   galactic dust displays a wide range of behavior . while diffuse interstellar dust is observed to have @xmath22 _ on average _ 
   ( savage & mathis 1979 ; ccm ) , this value is by no means universal 
   . significant deviations from this canonical @xmath22 value ( and corresponding differences in the overall shape of the extinction / reddening law ) are known to exist for a variety of interstellar dust clouds , and these variations appear to be generally correlated with environment ( fitzpatrick 1999 ) . 
   for example , @xmath0 has been found to range from 4 to 5 in dense molecular clouds ( mathis 1990b ; larson , whittet , & hough 1996 ; whittet et  al . 
   2001 ) , whereas there are indications that more diffuse , high - latitude cirrus clouds may have @xmath0 values as small as @xmath23 ( fitzpatrick & massa 1990 ; larson et  al .  1996 
   ; szomoru & guhathakurta 1999 ) . 
   studies based on stellar photometry from the optical gravitational lens experiment ( ogle ) and massive compact halo objects ( macho ) projects point to @xmath0 being substantially smaller than  3 in the direction of the galactic bulge ( popowski 2000 ; popowski , cook , & becker 2003 ; udalski 2003 ; sumi 2004 ; popowski 2004 ) . 
   barbaro et  al . 
   ( 2001 ) find departures from the ccm parameterization in the ultraviolet portion of the galactic extinction law . 
   it is worth noting that the above studies sample dust at a wide range of distances from the sun : some probe the diffuse interstellar medium in its immediate vicinity whereas others , especially those at low galactic latitudes , can probe dust at much larger distances well outside the solar neighborhood ( e.g. ,  the rcra cloud studied by szomoru & guhathakurta 1999 ) . 
   some studies have suggested that @xmath0 variations are tied to variations in the size distribution of dust grains from one line of sight to another . in diffuse 
   cirrus clouds , a relative abundance of small grains might explain the steep rise of the extinction curve into the ultraviolet ( ccm ; larson et  al.1996 ) . 
   by contrast , a high abundance of large grains , which grow readily by coagulation in dense molecular clouds , may explain the larger than average values of @xmath0 observed in these regions ( whittet et  al . 
   whittet et  al 
   .  suggest that , in dense molecular clouds , @xmath0 remains close to the standard value of  3.1 except for lines of sight with unusually high extinction ( @xmath24 ) . 
   this may indicate a more complicated relationship between @xmath0 and @xmath25 . 
   moreover , it has been argued that chemical composition can also play a role in determining the shape of the extinction law ( rhoads , malhotra , & kochanski 2004 ) .    to improve our understanding of the dependence of @xmath0 on environment , 
   measurements are necessary along many sight - lines through a broad range of cloud types . 
   clusters with differential extinction / reddening have long been used to measure the @xmath0 of intervening dust ( mihalas & routly 1968 ) . 
   the traditional method of spectral typing individual cluster stars ( e.g. ,  on the morgan - keenan system ) , in conjunction with photometric measurements , allows for a direct and precise measurement of extinction and reddening . 
   however , this method of measuring @xmath0 requires high - quality , flux - calibrated spectra and a good understanding of the effects of metallicity on the energy output of stars . 
   our study focuses instead on the use of broad - band photometry in light of the fact that an extensive suite of photometric data sets is currently available .    in this paper , we propose three  methods for measuring the @xmath0 of dust in the foreground of differentially - reddened globular clusters . 
   the first two  methods are closely related to each other and rely on accurate three - band optical photometry of cluster stars . 
   the third relies on two - band optical photometry and a map of the dust thermal emission ( schlegel , finkbeiner , & davis 1998 , hereafter sfd ) and is particularly well suited to wide - field data . 
   all three  methods are based on the premise that the width of the blue horizontal branch ( hb ) is minimized when the appropriate @xmath0 value is used to correct the photometry for extinction and reddening . 
   the methods are applied to ngc  4833 , a low - latitude galactic globular cluster with variable extinction across its face . 
   realistic monte carlo simulations of the cluster data set are used to estimate the accuracy with which @xmath0 can be recovered . 
   nearly two decades ago , low et  al . 
   ( 1984 ) noticed diffuse background emission in the infrared astronomical satellite ( iras ) 60 and 100@xmath26 m maps and termed it ` infrared cirrus ' due to its complex texture . 
   the cirrus has been associated with thermal emission from dust grains in the diffuse interstellar medium ( beichman 1987 ) . combining the angular resolution of the iras observations with the photometric accuracy of the cosmic background explorer ( cobe)/diffuse infrared background experiment ( dirbe ) data , sfd derived all - sky maps of the dust column density and mean temperature . 
   one of the primary uses of these maps has been to correct extragalactic sources for extinction and reddening . 
   the sfd maps show structure in the cirrus down to the smallest angular scales resolved by iras ( @xmath27 ) . 
   gautier et  al . 
   ( 1992 ) described the structure in the iras data in terms of its angular power spectrum , @xmath28 where @xmath29 is the fourier power , @xmath30 is the reciprocal of the angular scale , and @xmath31 is the spectral index . similarly , guhathakurta & cutri ( 1994 ) examined both iras 100@xmath26 m maps and optical ccd surface photometry of reprocessed starlight from dust grains : they found that power on small scales was dominated by stars and galaxies , and since it was impossible to completely separate out these components , they merely cleared out the obvious compact `` objects '' ( stars and galaxies ) and smoothed the residual image over arcminute scales ; the resulting power spectrum had an index @xmath32 . 
   more recently , kiss et  al .  ( 2003 ) observed 13  fields with infrared space observatory ( iso)/isophot and found that the power spectrum index varies from field to field over the range @xmath33 for angular scales larger than @xmath34 . 
   this work extends the finding of @xmath35 by herbstmeier et  al . 
   ( 1998 ) in their earlier isophot - based study . 
   these last two  studies indicate that @xmath36 is not universal .    in this study 
   , the scatter in the hb of the differentially - reddened globular cluster ngc  4833 is used to measure the angular power spectrum of dust on scales smaller than the angular resolution of the thermal emission maps ( e.g. ,  iras , dirbe , isophot ) . for uncrowded data such as ours , 
   the smallest angular scale down to which this method can be applied is set by the surface density of hb tracers which in turn determines the typical nearest - neighbor separation ; in crowding - limited situations , the smallest angular scale is a few times the size of the stellar point spread function : few arcseconds for ground - based images and sub - arcsecond for _ hubble space telescope _ 
   images . 
   because interstellar dust causes extinction and reddening of starlight and reprocesses the energy , dust patchiness has a profound impact on the accuracy of many astronomical measurements even at high galactic latitudes . thermal emission from cold dust 
   can confuse cmbr anisotropy measurements . 
   measurements of large - scale structure in galaxy surveys may be influenced by spatial non - uniformities in the foreground dust . 
   studies of stellar populations are at risk because measurements of @xmath37 can be impacted by reddening . 
   extinction can affect the photometry of distance indicators and thereby produce systematic errors in the distance - scale ladder . 
   it is therefore important to quantify extinction / reddening variations on the smallest scales possible . while the sfd study does an excellent job of characterizing moderate- to large - scale dust variations , their maps miss power on scales smaller than a few  arcminutes . 
   +   + the photometry of ngc  4833 is presented in  2 . 
   the construction of simulated cluster data sets in discussed in  3 . 
   the @xmath0 measurement methods are outlined in  4 , along with their application to ngc  4833 and simulated data sets to estimate the accuracy of each method . in  5 , we describe the angular power spectrum of cirrus in the direction of ngc  4833 . 
   possible future extensions of this work are discussed in  6 . 
   the main conclusions of this paper are summarized in  7 . 
   the globular cluster ngc  4833 is located near the equatorial plane of our galaxy ( @xmath38 ) in a dusty region of the constellation musca . the cluster has a mean reddening of @xmath39  mag and reddening variations of order @xmath40  mag across the field ( melbourne et  al . 
   figure  1 of melbourne et  al 
   .  shows an image of the dusty features in the region around the cluster including the coalsack nebula . 
   the johnson - cousins @xmath1 stellar photometry of ngc  4833 used in this study are drawn from melbourne et  al . 
   a detailed description of the data set is given in that paper so only a brief discussion appears here . 
   the ccd images of the cluster were obtained in 1996 using the 0.9-m telescope at the cerro tololo inter - american observatory . 
   the data reduction was carried out with the daophot  ii ( stetson 1994 ) stellar photometry package . 
   figure  [ fig : fullcmd ] shows the ( @xmath4 ,  @xmath5 ) and ( @xmath6 ,  @xmath5 ) cmds ( left and right panels , respectively ) for the full @xmath41 field . 
   boxes are drawn in the former cmd around the 260  objects that are considered to be candidate cluster hb stars for the purposes of this study ( possible non - hb interlopers in the sample are discussed in  4.1.1 ) . 
   extinction / reddening variations across the field affect the width of the principal sequences of the cmds . 
   figure  [ fig : radcmd ] shows that the hb is relatively tight and narrow for the central @xmath42 region of the cluster , but increases in width for stars distributed over a larger area in the outer region of the cluster . 
   this suggests that extinction / reddening variations across the field , rather than photometric errors , are the cause of the large scatter in the hb ; if anything , photometric errors tend to be slightly worse in the inner region of the cluster because of crowding . 
   a color - color diagram of cluster hb stars is plotted in the top panel of figure  [ fig : cc ] , while the bottom panel shows the same after correcting for reddening on a star - by - star basis . 
   the @xmath43 value for each star is taken from the sfd dust map , and an assumed @xmath0 value of  3.0 is plugged into the ccm parameterization of the dust extinction law ( eqn . 
   [ eqn : cardel ] ) to derive @xmath44 and the corresponding @xmath45 value . 
   the reddening - corrected data ( and the uncorrected data , for that matter ) show a linear relationship between @xmath4 and @xmath6 . a least - squares fit ( solid line ) 
   is made to the reddening - corrected data : @xmath46 this equation , in conjunction with the two - polynomial fiducial fit to the hb in the ( @xmath4 ,  @xmath5 ) cmd , is used to simulate @xmath1 photometry of hb stars (  3.2 ) . 
   figure  [ fig : photerr ] shows the typical @xmath47 photometric errors in the @xmath1 bands for ngc  4833 stars as a function of their apparent visual magnitude ( melbourne et  al . 
   the photometric errors , estimated using the daophot  ii software package ( stetson 1994 ) , range from @xmath48  mag at the bright end of the hb ( @xmath49 ) to @xmath50  mag at the faint end ( @xmath51 ) , and contribute noticeably to the width of the hb . 
   the characteristic photometric error is defined to be that at @xmath52 , @xmath53  mag , and this is what is referred to throughout the rest of this paper . 
   a key aspect of our study is the use of simulated cluster data sets to fine tune and test the @xmath0 measurement methods (  4 ) . 
   these simulated data sets are designed to mimic the general properties of the ngc  4833 data set as closely as possible . 
   since we are also interested in applying these @xmath0 measurement methods to a broad range of data , the main set of simulations are constructed over a two - dimensional grid of parameter space : three _ input _ @xmath0 values and three  levels of photometric precision . at each of these 9 ( @xmath54 ) grid points , 
   100  monte - carlo realizations are carried out 
   ,  a total of 900  simulated clusters are constructed each containing 260  hb stars . 
   the rest of this section describes the step - by - step procedure used to construct each simulated data set . 
   three  input @xmath0 values , 2.0 , 3.0 and 4.0 , are chosen for the simulated cluster data sets . 
   these mimic a range of extinction law properties for the foreground dust and span the range of values observed in lines of sight with moderate to light reddening (  1.1 ) . 
   the ccm parameterization of the galactic extinction laws : @xmath55 is adopted here . for a given choice of @xmath0 , the visual extinction @xmath25 
   is translated to its corresponding @xmath43 and @xmath45 color excesses . 
   each simulated star is assigned @xmath56- and @xmath5-band photometry via a random drawing from a fiducial hb curve in the cmd . 
   the fiducial is based on stitching together two  second - order polynomials that are fit to the extinction / reddening - corrected ( @xmath4 ,  @xmath5 ) hb of ngc  4833 , where the @xmath43 correction for each star is obtained from the sfd dust map and @xmath0 is assumed to be  3.0 . 
   examples of such two - polynomial fiducial hbs are shown in figures  [ fig : av_method ] and [ fig : schlegel_method ] . 
   the `` sfd - corrected '' cluster hb used for the fiducial fit is similar to the one shown in the left - middle panel of figure  [ fig : cmd6 ] , except for the fact that the one illustrated has had the mean extinction / reddening reapplied to it . 
   when drawing 260  stars from the fiducial , the run of stellar density along the simulated hb is made to mimic that observed along the cluster hb . 
   this is done by dividing the fiducial into sections and demanding that the number of stars in each section match that in the corresponding section of the cluster hb . 
   simulated @xmath57-band photometry is obtained from the linear fit to the ( @xmath4 ,  @xmath6 ) color - color diagram of cluster hb stars ( fig . 
   [ fig : cc ] , eqn . 
   [ eqn : cc ] ) .      the width of the hb can be used to quantify the effect of extinction / reddening variations . 
   figure  [ fig : hbscat ] shows the rms scatter of hb stars in ngc  4833 as a function of their @xmath5 magnitude ( solid line ) . 
   the rms scatter is calculated based on the _ shortest distance _ of each hb star from the fiducial curve ( i.e. ,  distance to the nearest tangent point on the curve ) . 
   the fiducial consists of two  second - order polynomials fit to hb stars within @xmath58 of the cluster center . 
   the dotted line in figure  [ fig : hbscat ] is a model prediction for the hb scatter that would result from photometric errors alone , the dashed line shows the effect of large - scale extinction / reddening variations from sfd plus photometric error , and the dot - dashed line the effect of small- and large - scale extinction / reddening variations plus photometric error . 
   it is clear from the scatter at the bright end of the hb ( @xmath59 ) that only the dot - dashed line matches the ngc  4833 data . 
   the following subsections describe in turn the simulations behind the three  model lines .      a special set of 100  photometric - error - only ( `` @xmath60-only '' ) simulations containing _ no _ reddening variations is constructed for illustration purposes only ; this set is not used in any of the @xmath0 determination exercises described in  4 . 
   each monte carlo realization in this special set involves three  steps : ( * 1 * )  an artificial hb is constructed as described in  3.2 above ; ( * 2 * )  the mean extinction and reddening of the cluster , @xmath61 and @xmath39 ( based on @xmath62 ) , are applied to all simulated stars to bring them into the same range of apparent @xmath1 magnitudes as the ngc  4833 hb stars ; and ( * 3 * )  gaussian photometric errors are added assuming @xmath63 . 
   the details of step  ( 3 ) are given in  3.4 below . 
   the dotted line in figure  [ fig : hbscat ] is the scatter measured in @xmath64  simulated hb stars . 
   the mismatch with the ngc  4833 data ( solid line ) demonstrates that photometric errors alone can not explain the width of the cluster s hb and there must be additional sources of scatter in the hb . 
   ( the dotted line is much smoother than the solid line because it is based on 100@xmath65 as many stars . ) 
   the obvious additional source of hb scatter is differential extinction / reddening across the cluster . 
   large - scale variations based on the sfd map are explored in this section . 
   their maps provide @xmath43 color excesses as a function of sky position , but have an angular resolution of @xmath66 and therefore contain power only on scales larger than this . 
   the sfd map includes ngc  4833 and the region around it , and indicates that the mean reddening along this line of sight is @xmath67  mag with variations on the order of @xmath40  mag . 
   following the procedure described above for the `` @xmath60-only '' simulations (  3.3.1 ) , another special set of 100  simulations is constructed containing only large - scale extinction / reddening variations and photometric errors . 
   again , these special `` sfd+@xmath60 '' simulations are created for illustrative purposes alone . 
   the only difference is in step  ( 2 ) of the procedure : instead of applying the same ( mean ) extinction and reddening values to all simulated hb stars , each star is randomly assigned the sky position of one of the 260  hb stars in ngc  4833 and the corresponding sfd @xmath43 value , and associated @xmath25 and @xmath45 values based on @xmath62 , are applied to the @xmath1 magnitudes of the simulated star . by assigning actual sky positions to the simulated hb stars , 
   we ensure that they mimic the spatial distribution of ngc  4833 s hb stars and range of large - scale extinction / reddening variations across the cluster data set . 
   the dashed line in figure  [ fig : hbscat ] , based on @xmath64  simulated hb stars , shows the scatter resulting from a realistic amount of large - scale extinction / reddening variations and photometric error . 
   even this line falls a little short of explaining the hb scatter observed in the cluster ( solid line ) . 
   the most probable cause of this shortfall is the coarse resolution of the sfd map which averages over structure in the dust distribution on angular scales smaller than @xmath66 . 
   small - scale variations are incorporated into the simulated data sets by adding random gaussian fluctuations with an amplitude proportional to the sfd reddening value at that location : @xmath68\ ] ] where @xmath69 is a random number ( for the @xmath9-th star ) drawn from a gaussian distribution with zero mean and width @xmath70 . 
   this brings us back to the main set of simulations used for testing the @xmath0 determination methods in  4 . 
   they are constructed using a procedure similar to that used for the special set of `` sfd+@xmath60 '' simulations (  3.3.2 ) but with two changes . 
   first , the sfd reddening values in step  ( 2 ) of the procedure are replaced by @xmath71 from eqn . 
   [ eqn : smallscale ] above . 
   second , a range of @xmath0 values (  3.1 ) and photometric error levels (  3.4 ) are used .    with the above choice of gaussian width for the small - scale dust variations , @xmath70 , and for @xmath63 and @xmath62 , the scatter in the simulated hb approaches that of ngc  4833 s hb ( dot - dashed vs.  solid lines in fig . 
   [ fig : hbscat ] ) . 
   this is a reasonable estimate of the amount of smoothing of dust fine - scale structure in the sfd maps (  5.1 ; fig . 
   [ fig : powspec ] ) , and is well within the 10% uncertainty in reddening values estimated by the authors . 
   the final step in the construction of the main set of simulated hb data sets is the inclusion of photometric error . 
   since the photometric uncertainty is a function of stellar brightness , this step must be carried out after the inclusion of ( large- and small - scale ) dust effects so that each star is at the appropriate apparent brightness . 
   gaussian random noise is added to the @xmath1 magnitudes of each star . 
   three  levels of photometric error are simulated : ( 1 )  @xmath63 , with the latter being the daophot - based errors shown in figure  [ fig : photerr ] ; ( 2 )  @xmath72 ; and ( 3 )  @xmath73 . 
   we employ three  methods for measuring @xmath0 in the direction of differentially - reddened globular clusters . 
   each method relies on the premise that the width of the blue hb will be minimized when the appropriate @xmath0 value is used to correct for extinction and reddening . 
   a description of each method and its application to the ngc  4833 and simulated cluster data sets follows . 
   the `` @xmath25  rms '' method relies on three - color photometry of differentially - reddened globular clusters . 
   figure  [ fig : av_method ] shows the ( @xmath4 ,  @xmath5 ) and ( @xmath6 ,  @xmath5 ) cmds of a simulated hb ( left and right panels , respectively ) . 
   a fiducial , consisting of two  second - order polynomials stitched together ( solid curve ;  3.2 ) , is fit to the hb stars within @xmath58 of the cluster center where differential extinction / reddening is relatively small ; this is done independently for the ( @xmath4 ,  @xmath5 ) and ( @xmath6 ,  @xmath5 ) cmds . 
   the relative visual extinction of each star is calculated to be the vertical distance between the star and the point where the reddening vector drawn from the star intersects the fiducial ( fig . 
   [ fig : av_method ] inset ) ; it is designated @xmath74 and @xmath75 for the ( @xmath4 ,  @xmath5 ) and ( @xmath6 ,  @xmath5 ) cmds , respectively . '' 
   is used to denote the relative visual extinction of a star based on a single fiducial / cmd , whereas `` @xmath76 '' is used to denote the _ difference _ between two @xmath3 values , say between two  cmds for a given star or between a pair of stars . ] the @xmath74 measurement is based on an assumed / test value of @xmath0 , the slope of the reddening vector in the ( @xmath4 ,  @xmath5 ) cmd . for the @xmath75 calculation , 
   the ccm parameterization given in eqn . 
   [ eqn : cardel ] is used to translate @xmath0 to @xmath44 , the slope in the ( @xmath6 ,  @xmath5 ) cmd . 
   the difference between the two @xmath3 values is defined to be @xmath77 . for a given test value of @xmath0 , 
   the rms scatter of @xmath78 is computed over all hb stars . 
   this is done in two  iterations , with 3-@xmath79 rejection of outliers after the first iteration to eliminate any non - hb stars in the sample ( e.g. ,  cluster blue stragglers or galactic field main sequence turnoff stars in the foreground  encircled symbols in upper panels of fig .  [ 
   fig : radcmd ] ) . 
   non - hb interlopers are expected to have large deviations from the hb fiducial with effective @xmath3 values that are different between the two  cmds . 
   the calculation of the rms of @xmath78 is repeated for a series of test @xmath0 values between 0.5 and 5.5 in steps of  0.25 . 
   the @xmath3 parameters from the two  cmds are expected to approach each other when the test value approaches the true @xmath0 of the dust in the foreground of the cluster . in other words , 
   the rms of @xmath78 should go through a minimum at the correct @xmath0 value . 
   a polynomial fit is made to the rms of @xmath78 vs.test @xmath0 data points to interpolate to the minimum . 
   figure  [ fig : av_results ] shows the rms scatter in @xmath78 vs.  test @xmath0 value , the result of applying the @xmath25  rms method to ngc  4833 ( filled diamonds ) and three  sample simulated cluster data sets ( open triangles ) . the simulated data sets are all based on an input @xmath0 value of 3.0 , but have different levels of photometric error : 10% , 50% , and 100% that of the ngc  4833 data set (  3.4 ) , with the rms sequences arranged in order of @xmath80 increasing upwards . 
   as expected , each sequence of rms of @xmath78 vs.  test @xmath0 goes through a clear minimum ( large open square ) near @xmath62 . 
   the depth / sharpness of the rms minimum decreases with increasing photometric error and the power of the method diminishes . 
   the results for all @xmath81  simulated cluster data sets are summarized in table  [ tbl : sim ] . for @xmath73 , 
   the @xmath25  rms method reproduces the correct @xmath0 to within @xmath82 , as measured by the standard deviation of measured ( output ) @xmath0 values for 100  simulated data sets . 
   the scatter in output @xmath0 increases to @xmath83 for @xmath63 . 
   the ( absolute ) precision of the method appears to decrease with increasing input @xmath0 , again judging from the scatter of output @xmath0 values , although the fractional error @xmath84 remains roughly constant . 
   it is important to note that the output @xmath0 from this method tends to be biased low for moderate to large photometric errors and input @xmath85 . 
   for example , simulated data sets with input @xmath62 and @xmath63 ( a good match to the real cluster data ) have a mean output @xmath86 . 
   this bias of @xmath87 in @xmath88 is significant compared to the error in the mean , @xmath89 , and must be taken into account in the determination of @xmath0 for an actual cluster data set . 
   the cause of the bias towards low @xmath0 may be explained as follows . 
   photometric error produces scatter in the hb which , unlike the scatter caused by variable extinction / reddening , is _ uncorrelated _ between the two  cmds  i.e . ,  the resulting values of @xmath74 and @xmath75 are different . 
   these errors are particularly important at faint magnitudes where the hb is vertical and the effect is a spread in color . all else 
   being equal , choosing a smaller @xmath0 ( and @xmath44 ) translates these color errors into smaller @xmath3 values and hence to smaller ( @xmath76)@xmath90 differences as well . 
   this biases the rms minimum towards lower @xmath0 values . 
   the rms of @xmath78 for the ngc  4833 data set displays a significant minimum at @xmath91 ( open square associated with filled diamonds in fig . 
   [ fig : av_results ] ) , where the error bar is taken directly from the scatter in output @xmath0 for comparable simulated data sets . correcting for the bias in the measured @xmath88 for such simulations , 
   our best guess for the @xmath0 towards ngc  4833 is @xmath92 . 
   the `` @xmath25  slope '' method is closely related to the @xmath25  rms method in that it relies on the same @xmath74 and @xmath75 values determined for each star in  4.1.1 above . instead of computing the difference between the two @xmath3 values as in the @xmath25  rms method , 
   one is plotted against the other and a linear fit made . 
   this is illustrated in figure  [ fig : av_slope_method ] for a simulated cluster data set created with ( input ) @xmath62 ; four  test @xmath0 values are shown . 
   the linear fit in each panel is effectively a bivariate one : one of the dashed lines is a fit treating the @xmath93- and @xmath94-axes as the independent and dependent variables , respectively , and vice versa for the other dashed line ; their slopes are averaged to get the solid line . 
   as expected , the slope of the linear fit approaches unity when the test @xmath0 approaches the true ( input ) @xmath0 value . as in  4.1.1 
   , a polynomial fit is made to the slope vs.  test @xmath0 data to interpolate to unit slope . 
   figure  [ fig : slope_results ] shows results from the application of the @xmath25  slope method : the slope of the relation between @xmath74 and @xmath75 is plotted versus test @xmath0 . 
   the format of this figure is the same as for figure  [ fig : av_results ] , except that a slope of unity ( rather than an rms minimum ) corresponds to the best - fit @xmath0 value . 
   the @xmath25  slope method results for ngc  4833 and simulated data sets are summarized in table  [ tbl : sim ] .    applying this method to @xmath81  simulated data sets 
   , we learn that , like the @xmath25  rms method , the power of the @xmath25  slope method decreases with increasing photometric errors . 
   the scatter in output @xmath0 is @xmath82 for @xmath73 , but increases to @xmath95 for @xmath63 . 
   some of the 9  simulation categories display a small but significant bias ( positive or negative ) in the measured ( output ) @xmath88 , but not the category directly relevant to ngc  4833 . 
   the cause of this bias is not as readily apparent as for the `` @xmath25  rms '' method (  4.1.2 ) . for the ngc 
   4833 data set , the @xmath25  slope method gives @xmath96 . 
   the `` optical / ir '' method is different from the previous two  methods in that it relies on two - color photometry ( i.e. ,  a single cmd which , in our case , is @xmath5 vs.  @xmath4 ) and the sfd reddening map . using a test @xmath0 value , 
   @xmath25 values are derived for all hb stars by scaling the corresponding sfd @xmath43 values . 
   these color and magnitude corrections are applied on a star - by - star basis . 
   figure  [ fig : schlegel_method ] shows the ( @xmath4 ,  @xmath5 ) cmd of a simulated hb before ( left ) and after ( right ) making the extinction / reddening corrections . 
   the latter hb is slightly tighter , but not dramatically so because the sfd reddening values smooth over small - scale dust variations . 
   the rms scatter of the corrected hb stars is computed with respect to a best - fit fiducial curve ( two  second - order polynomials  solid line in fig . 
   [ fig : schlegel_method ] ) based on `` shortest distance '' ( see  3.3 ) . 
   this process is repeated for a range of test @xmath0 values as for the other two  methods . 
   the rms scatter is expected to go through a minimum when the test @xmath0 approaches the true @xmath0 value . as in  4.1.1 
   , a polynomial fit is made to the hb scatter vs.  test @xmath0 data to interpolate to the minimum . 
   figure  [ fig : schlegel_results ] shows results from the application of the optical / ir method : rms scatter of hb stars versus test @xmath0 . 
   the format of this figure is the same as for figure  [ fig : av_results ] . the optical / ir results for ngc  4833 and simulated data sets are summarized in table  [ tbl : sim ] . 
   tests of the method on @xmath81  simulated cluster data sets indicate that the method has a low level of predictive power ( @xmath97 in @xmath0 ) over the full range of photometric errors simulated . 
   this is because the main source of error here is uncertainty in the sfd reddening values , including small - scale dust variations that get smoothed out by the relatively coarse resolution of their map . 
   the optical / ir method gives @xmath98 for ngc  4833 . 
   our best estimate of the extinction law slope in the direction of ngc  4833 is @xmath7 , based on a weighted average of the values from the three  determination methods ( see table  [ tbl : sim ] ) . before averaging , 
   the output @xmath0 value from the @xmath25  rms method is corrected for measurement bias : @xmath99 (  4.1.2 ) . 
   the weights are proportional to @xmath100 ^ 2 $ ] where @xmath101 is the uncertainty in output @xmath0 for a given determination method as derived from the simulations . 
   figure  [ fig : cmd6 ] illustrates the effectiveness of the star - by - star extinction / reddening corrections based on the @xmath0 determination methods described earlier in this section . 
   the top two  panels show the uncorrected ngc  4833 hb in the ( @xmath4 ,  @xmath5 ) and ( @xmath6 ,  @xmath5 ) cmds ( left and right panels , respectively ) . 
   the photometry in the middle two  panels has been corrected for extinction / reddening based on the sfd @xmath43 maps , using a best - fit @xmath62 and the corresponding @xmath44 from eqn . 
   [ eqn : cardel ] to translate the @xmath4 color excess to @xmath25 and @xmath45 . 
   note , the mean extinction and reddening of the cluster have been added back to the data for ease of comparison with the top panels . 
   the hb is somewhat tighter after the sfd - based correction but the effect is fairly subtle , a result of missing small - scale dust structure in the sfd map and possibly other sources of uncertainty as well . 
   the correction in the bottom two  panels of figure  [ fig : cmd6 ] is based on information derived from @xmath1 photometry of the cluster hb . 
   the relative visual extinction measures derived from the two  cmds , @xmath74 and @xmath75 (  4.1.1 ) , are averaged to obtain a @xmath102 value for each hb star . 
   this is translated to color excesses in the two  cmds using the best - fit @xmath62 and its corresponding @xmath44 ( eqn . 
   [ eqn : cardel ] ) . 
   the dramatic tightening of the hb indicates that the scatter is strongly correlated between ( @xmath4 ,  @xmath5 ) and ( @xmath6 ,  @xmath5 ) cmds 
   e.g . ,  if a star is displaced to the right of / below the hb fiducial in one cmd , it tends to be displaced in the same direction and by the same amount ( or by a proportional amount in color ) in the other cmd . 
   the most natural explanation of this correlated hb scatter is of course differential extinction / reddening in the ngc  4833 data set . 
   there is a handful of stars at each extremity of the hb in the bottom two  panels of figure  [ fig : cmd6 ] that do not follow the tight sequence delineated by the rest of the stars  in fact , their scatter is unchanged relative to the uncorrected cmds in the top panels . 
   this is because the polynomial hb fiducial used to determine @xmath3 does not extend all the way to the ends of the sequence ( to avoid extrapolation of the fit ) ; as a result no @xmath3 measurement or correction is available for these stars . 
   a couple of stars near the middle of the hb continue to show large deviations in the @xmath3-corrected cmds but these are probably non - hb stars ( see  4.1.1 and fig . 
   [ fig : radcmd ] ) .    before closing out this discussion , 
   two additional sources of hb scatter are considered . 
   the first of these is photometric error . 
   errors in @xmath56 and @xmath57 magnitudes produce uncorrelated shifts between the two  cmds . 
   errors in @xmath5 produce shifts with slope @xmath103 in the ( @xmath4 ,  @xmath5 ) cmd , more or less orthogonal to the reddening vector , and shifts of slope @xmath104 in the ( @xmath6 ,  @xmath5 ) cmd , roughly along the reddening vector though its slope @xmath44 is significantly steeper than unity . only at the very top of the hb where it becomes horizontal can @xmath5 magnitude errors mimic differential extinction / reddening effects . to summarize 
   , photometric errors can not explain the correlated scatter seen in ngc  4833 s hb . moreover , as will be demonstrated in  5 , the difference in @xmath3 between pairs of hb stars scales with their angular separation in a manner consistent with the dust power spectrum measured at larger angular separations ; photometric errors have no natural way of explaining this trend either .    another factor to consider 
   is _ intrinsic _ scatter in the hb caused by astrophysical effects such as variations in chemical abundance , age , and mass loss between cluster stars . 
   it is possible that these can produce correlated scatter between the two  cmds and thereby mimic differential extinction / reddening . 
   however , this scenario can not explain the scaling of the @xmath3 differences with angular separation (  5 ) . 
   if there was a radial gradient in intrinsic stellar properties within the cluster , one might expect to see a shift of the hb sequence in the cmd between inner and outer regions . 
   instead , figure  [ fig : radcmd ] shows no such shift ( the increased scatter in the hb in the outer cmd is best explained in terms of dust variations ; see  2.1 ) . 
   in earlier sections , we demonstrated the effects of spatial extinction / reddening variations on the photometry of ngc  4833 . 
   we now turn to a statistical description of the strength of these extinction / reddening variations as a function of angular scale . 
   figure  [ fig : dustmap ] is a map of the _ relative _ visual extinction across the face of ngc  4833 . 
   each point on the map represents the sky position of a hb star in the cluster . 
   the size of each point is scaled by the measured strength of the relative visual extinction @xmath74 at that location ; it varies from @xmath105 to @xmath106  mag ( smallest@xmath107largest ) . 
   as described in  4.1.1 , @xmath74 is the vertical distance between the star s cmd location and the point where the reddening vector drawn from the star intersects the fiducial hb ; the fiducial is a fit to the central region of the cluster , and therefore corresponds to a mean visual extinction of @xmath108 . in keeping with the findings of melbourne et  al . 
   ( 2000 ) , there is a clear overall n - s gradient in @xmath74 across ngc  4833 , but the extinction can vary significantly from point to point even for points relatively close to each other . 
   these variations can be characterized in terms of the angular power spectrum of the foreground dust complex . 
   a direct measurement of the power spectrum is difficult however because of the sparse and irregular spatial sampling of the data points . 
   the following approach is adopted instead . 
   first , the `` variance spectrum '' as a function of angular scale is calculated from the cluster @xmath74 data (  5.1 ) . 
   next , a variance spectrum and conventional power spectrum are measured from the uniformly - sampled sfd maps , and an empirical transformation is calculated from the former to the latter (  5.2 ) . 
   finally , this empirical transformation is applied to the cluster variance spectrum to convert it to a power spectrum , and this allows us to study the composite power spectrum over a wide range of angular scales (  5.3 ) . 
   the best - fit value of @xmath62 for ngc  4833 is used to determine the relative visual extinction corresponding to the @xmath9-th hb star , @xmath109 . the difference in relative @xmath25 between stars @xmath9 and @xmath10 , @xmath110 , and their angular separation @xmath11 
   are then calculated for all possible pairs of hb stars . 
   figure  [ fig : powspec ] ( open diamonds ) shows the variance ( mean square deviation ) of @xmath111 as a function of angular separation @xmath112 in a log - log plot ; the radial bins are chosen so as to include roughly equal numbers of pairs in each bin . 
   the variance decreases with decreasing @xmath112 before levelling off at log@xmath113\bigr]=-2.3 $ ] for @xmath114 . 
   the behavior at low @xmath112 may be attributed to photometric errors and possibly the natural width of the hb .    in order to quantify and compensate for the effect of photometric error on the variance spectrum , we resort to the special set of `` @xmath115-only '' simulations containing _ no _ reddening variations (  3.3.1 ) . 
   the variance of this `` @xmath60-only '' simulated data set , analyzed in the same way as the real cluster data set , is : log@xmath116\bigr]=-2.4 $ ] , and is of course independent of stellar angular separation ( dashed horizontal line in fig . 
   [ fig : powspec ] ) . 
   the variance due to photometric errors is subtracted from the variance spectrum of ngc  4833 , equivalent to statistical subtraction of errors in quadrature . 
   the resulting variance spectrum is shown as filled diamonds in figure  [ fig : powspec ] . 
   a power - law fit to the corrected variance spectrum : @xmath117 yields an index of @xmath14 , over the radial range @xmath118@xmath119 . 
   the upper end of this range corresponds to the nyquist frequency of the @xmath41 area covered by the ngc  4833 data set . 
   the first radial bin of the corrected variance spectrum ( @xmath120 ) is excluded from the power - law fit because it is strongly affected by uncertainties in the photometric error correction . 
   the cluster variance spectrum is used to make a rough estimate of how much small - scale power is missing from the sfd dust map . the sfd map , like the iras 100@xmath26 m data on which they are based , have an angular resolution of about @xmath121 , or @xmath122 . 
   this point in the cluster variance spectrum corresponds to var@xmath123=4.4\times10^{-3}$ ] . 
   the rms scatter at this point is @xmath124}=6.6\times10^{-2}$ ] . since the mean visual extinction @xmath2 in the direction of ngc  4833 
   , the rms is about 7% of the mean . 
   this provides a good sanity check : it corroborates the result from the analysis of the cluster hb width (  3.3.2 ) that the sfd map is missing small - scale variations on the order of 6% of the reported reddening value . 
   it also gives an indication of how accurately one can correct for extinction / reddening using the sfd map , with general implications for precision photometry in dusty regions of the sky .      in this section , 
   the sfd map is used to investigate the relationship between the variance spectrum and conventional angular power spectrum . 
   this is done by extracting a @xmath125 sfd reddening image centered on ngc  4833 . the @xmath126 reddening values are converted to @xmath25 using the best - fit @xmath0 of 3.0 . 
   the difference in visual extinction between pixels @xmath9 and @xmath10 , @xmath127 , is calculated for all possible pairs of pixels . 
   the variance of @xmath128 is computed as a function of angular separation @xmath112 . 
   the ` @xmath129 ' symbols in figure  [ fig : pwvr ] show @xmath130 for the sfd map , where @xmath19 . 
   the two - dimensional power spectrum of dust fluctuations is computed via a fast fourier transform ( fft ) of the sfd reddening image . 
   this is then azimuthally averaged to produce a one - dimensional angular power spectrum @xmath131 , shown by asterisks in figure  [ fig : pwvr ] . over the radial range @xmath132 to @xmath133 [ @xmath134 to @xmath135 , @xmath136 and @xmath131 
   have comparable logarithmic slopes ( power - law index : @xmath137 ) such that : @xmath138 with a best - fit scale factor of @xmath139 .      the above transformation ( eqn . 
   [ eqn : var_to_ps ] ) is applied to the photometric - error - corrected variance spectrum derived from the ngc  4833 @xmath140 data (  5.1 ) to convert it to an angular power spectrum . 
   the result is shown as filled diamonds in figure  [ fig : pwvr ] . 
   this small - scale power spectrum derived from optical data matches smoothly onto the power spectrum derived from the sfd map at larger angular scales . 
   a power law with an index @xmath141 provides an adequate fit to the data over the full range of projected separations shown in figure  [ fig : pwvr ] , @xmath17 to @xmath142 . 
   our best - fit index @xmath143 is significantly shallower than the index of @xmath144 found by gautier et  al . 
   ( 1992 ) and guhathakurta & cutri ( 1994 ) and at the shallow end of the range of indices found by kiss et  al . 
   ( 2003 ) in their isophot - based study of 13  fields : @xmath145 to @xmath146 ( @xmath82 ) . 
   the shallower the power - spectrum index , the larger the _ fraction _ of power missed by the sfd map on small scales relative to that on larger scales . 
   another possible implication relates to kiss et  al.s finding that shallower @xmath143 values are associated either with low hi column density , @xmath147  @xmath148 , and/or warm dust . 
   for example , in the region of the draco nebula , they find that the map of 90@xmath26 m emission ( dominated by warmer dust ) yields @xmath149 while the 170@xmath26 m map ( dominated by cooler dust ) yields @xmath150 . 
   while ngc  4833 has an extended blue hb , the methods presented in this paper should be usable for clump hbs or the rgb or any other tight cmd feature for that matter . 
   for instance , law et  al . 
   ( 2003 ) used the width of the rgb to measure differential reddening in the direction of three  galactic globular clusters , von braun et  al . 
   ( 2002 ) used the main sequence for their study , and udalski ( 2003 ) used clump stars to probe dust along the line of sight to the galactic bulge . 
   our methods should work for any three  bands , not just @xmath1 . 
   additional bands ( beyond three ) can provide : ( 1 )  a longer wavelength baseline and therefore more leverage in @xmath0 determination ; ( 2 )  a larger number of independent measures of the star - by - star extinction / reddening which can be averaged to beat down the errors ; and ( 3 )  most importantly , an empirical check of the ccm parameterization of the shapes of galactic extinction laws . 
   there is at least one  generalization of the methodology presented here that might be worth exploring in the future . in determining @xmath3 
   , we effectively use @xmath5 magnitude to mark where along the hb sequence the extinction / reddening - corrected star lies . a more general definition of a `` locator '' parameter , to mark the position of the unreddened star along any given cmd sequence , may be particularly useful . 
   for example , multi - wavelength data sets may not have any single observable in common across all of the cmds in question ( unlike the @xmath5 mag for our two  cmds ) , and a color parameter may be more sensitive than @xmath5 mag when using a not - so - vertical cmd feature ( such as the upper portion of ngc  4833 s hb ) . 
   several high - quality , multi - band ( three or more ) , ground- and space - based photometric data sets have long been available for many star clusters , so the methods presented here can be readily implemented to probe the nature of dust along many lines of sight . 
   these include milky way clusters , as well as others near enough to allow clean photometry in relatively uncrowded fields , such as _ hubble space telescope _ 
   images of clusters in , and even somewhat beyond , the magellanic clouds . 
   many of these clusters have detectable extinction / reddening variations across them  e.g . ,  @xmath151  centauri , ngc  6388 , ngc  6441 ( law et  al . 
   2003 ) , m10 , and m12 ( von braun et  al . 
   the sloan digital sky survey ( sdss , stoughton et  al . 
   2002 ) five - band ( @xmath152 ) data set , with its high photometric accuracy , minimal systematic errors , excellent uniformity / homogeneity , and large areal coverage , is opening up lines of sight towards differentially - reddened star clusters across the entire high - latitude north galactic cap for this type of analysis . 
   near - infrared @xmath153 photometry from the two  micron all - sky survey ( 2mass ) data should also prove very fruitful , both for probing highly - reddened lines of sight and for extending the sdss wavelength coverage . 
   making @xmath0 and power spectrum measurements along many sight - lines should lead to a better understanding of grain formation and chemistry and the effect of interstellar dust on astronomical observations . 
   while our @xmath54 grid of simulations covers some amount of parameter space in photometric error and @xmath0 , it is fairly limited in scope : it can not simply be translated to any arbitrary cluster data set . 
   another data set may be different from the ngc  4833 data set in terms of photometric bandpasses , level of photometric error , areal coverage , detailed morphology of cmd features , degree of extinction / reddening variations , and/or extinction law slope . 
   new simulations must be carried out , matched to the exact parameters of each new cluster data set in question , in order to test the efficacy of the @xmath0 measurement methods presented here . 
   * we have demonstrated the use of three  methods , `` @xmath25  rms '' , `` @xmath25  slope '' , and `` optical / ir '' methods , for determining the dust extinction law slope @xmath0 in the direction of differentially - reddened galactic globular clusters , and have tested the methods on an extensive suite of simulated cluster data sets . 
   * for cluster data sets with low photometric error , @xmath154  mag , the `` @xmath25  rms '' and `` @xmath25  slope '' methods can be used to determine @xmath0 to an accuracy of @xmath1550.3 . for cluster data 
   sets with @xmath156  mag , the two  methods yield @xmath0 to within @xmath83 , with some systematic biases . * the `` optical / ir '' method generally provides relatively imprecise estimates of @xmath0 , @xmath157 , over the full range of photometric errors explored . * 
   combining the results from all three  methods gives a mean extinction law slope of @xmath7 for the line of sight towards the low - latitute galactic globular cluster ngc  4833 . 
   * the scatter in the cluster hb is used to estimate the amount of small - scale structure in the dust complex in the foreground of ngc  4833 . 
   the schlegel et  al . 
   ( 1998 ) iras+dirbe - based map of the dust thermal emission averages over small - scale reddening variations that are @xmath158% of the mean reddening value . 
   * star - to - star variations in relative visual extinction across the face of ngc  4833 provide a measure of the foreground dust angular power spectrum for projected separations in the range @xmath17@xmath18 
   . this small - scale power spectrum derived from cluster optical data matches smoothly onto the larger - scale power spectrum derived from the sfd reddening map of the region . 
   the overall power spectrum is well fit by a power law : @xmath15 , where @xmath30 is the reciprocal of the angular scale @xmath112 , with spectral index @xmath137 . 
   we would like to thank neil balmforth , david burstein , sandra faber , and peter stetson for comments that significantly improved this work . 
   we would also like to thank ata sarajedini for providing the observations of ngc  4833 . 
   cc|c|ccc + & & 2.0 & @xmath159 & @xmath160 & @xmath161 + 0.003 & 10 & 3.0 & @xmath162 & @xmath163 & @xmath164 + & & 4.0 & @xmath165 & @xmath166 & @xmath167 + & & 2.0 & @xmath168 & @xmath169 & @xmath170 + 0.015 & 50 & 3.0 & @xmath171 & @xmath172 & @xmath173 + & & 4.0 & @xmath174 & @xmath175 & @xmath176 + & & 2.0 & @xmath177 & @xmath178 & @xmath179 + 0.030 & 100 & 3.0 & @xmath180 & @xmath181 & @xmath182 + & & 4.0 & @xmath183 & @xmath184 & @xmath185 +   + 0.030 &  &  & @xmath186 & @xmath187 & @xmath188
summary: we present three  methods for measuring the slope of the galactic dust extinction law , @xmath0 , and a method for measuring the fine - scale structure of dust clouds in the direction of differentially - reddened globular clusters . 
   we apply these techniques to @xmath1 photometry of stars in the low - latitude galactic globular cluster ngc  4833 which displays spatially - variable extinction / reddening about a mean @xmath2 . 
   an extensive suite of monte carlo simulations is used to characterize the efficacy of the methods . 
   the essence of the first two  methods is to determine , for an assumed value of @xmath0 , the _ relative _ visual extinction @xmath3 of each cluster horizontal branch ( hb ) star with respect to an empirical hb locus ; the locus is derived from the color - magnitude diagram ( cmd ) of a subset of stars in a small region near the cluster center for which differential extinction / reddening are relatively small . 
   a star - by - star comparison of @xmath3 from the ( @xmath4 ,  @xmath5 ) cmd with that from the ( @xmath6 ,  @xmath5 ) cmd is used to find the optimal @xmath0 . in the third method , @xmath0 is determined by minimizing the scatter in the hb in the ( @xmath4 ,  @xmath5 ) cmd after correcting the photometry for extinction and reddening using the schlegel , finkbeiner , & davis ( 1998 ) dust maps . 
   the weighted average of the results from the three  methods gives @xmath7 for the dust along the line of sight to ngc  4833 . 
   the fine - scale structure of the dust is quantified via the difference , @xmath8 , between pairs of cluster hb stars ( @xmath9,@xmath10 ) as a function of their angular separation @xmath11 . 
   the variance ( mean square scatter ) of @xmath12 is found to have a power - law dependence on angular scale : @xmath13 , with @xmath14 . 
   this translates into an angular power spectrum @xmath15 , with the index @xmath16 for @xmath17@xmath18 , where @xmath19 . 
   the dust angular power spectrum on small scales ( from optical data ) matches smoothly onto the larger - scale power spectrum derived from schlegel et  al.s far - infrared map of the dust thermal emission .