------------------------------------------------------------------------ ms.tex ApJL, accepted Content-Type: multipart/mixed; boundary="------------020806040703070401060500" X-MailScanner-Information: Please contact the postmaster@aoc.nrao.edu for more information X-MailScanner-ID: m8FLk4T0009592 X-MailScanner: Found to be clean X-MailScanner-SpamCheck: not spam, SpamAssassin (not cached, score=-4, required 5, autolearn=disabled, RCVD_IN_DNSWL_MED -4.00) X-MailScanner-From: leo@astro.ucla.edu X-Spam-Status: No This is a multi-part message in MIME format. --------------020806040703070401060500 Content-Type: text/plain; charset=ISO-8859-15; format=flowed Content-Transfer-Encoding: 7bit %arXiv:0809.2580 --------------020806040703070401060500 Content-Type: text/plain; name="ms.tex" Content-Transfer-Encoding: quoted-printable Content-Disposition: inline; filename="ms.tex" %% The command below calls the preprint style %% which will produce a one-column, single-spaced document. %% Examples of commands for other substyles follow. Use %% whichever is most appropriate for your purposes. %% \documentclass[12pt,preprint]{aastex} %% manuscript produces a one-column, double-spaced document: %\documentclass[manuscript]{aastex} %% preprint2 produces a double-column, single-spaced document: % \documentclass[preprint2]{emulateapj} %preprint2 %% Sometimes a paper's abstract is too long to fit on the %% title page in preprint2 mode. When that is the case, %% use the longabstract style option. % \documentclass[preprint2,longabstract]{aastex} %% If you want to create your own macros, you can do so %% using \newcommand. Your macros should appear before %% the \begin{document} command. %% %% If you are submitting to a journal that translates manuscripts %% into SGML, you need to follow certain guidelines when preparing %% your macros. See the AASTeX v5.x Author Guide %% for information. \newcommand{\vdag}{(v)^\dagger} \newcommand{\myemail}{leo@astro.ucla.edu} %% You can insert a short comment on the title page using the command bel= ow. %\slugcomment{Not to appear in Nonlearned J., 45.} %% If you wish, you may supply running head information, although %% this information may be modified by the editorial offices. %% The left head contains a list of authors, %% usually a maximum of three (otherwise use et al.). The right %% head is a modified title of up to roughly 44 characters. %% Running heads will not print in the manuscript style. \shorttitle{A 600 min NIR lightcurve of Sgr~A*} \shortauthors{Meyer et al.} %% This is the end of the preamble. Indicate the beginning of the %% paper itself with \begin{document}. \begin{document} %% LaTeX will automatically break titles if they run longer than %% one line. However, you may use \\ to force a line break if %% you desire. \title{A 600 minute near-infrared lightcurve \\ of Sagittarius A*} %% Use \author, \affil, and the \and command to format %% author and affiliation information. %% Note that \email has replaced the old \authoremail command %% from AASTeX v4.0. You can use \email to mark an email address %% anywhere in the paper, not just in the front matter. %% As in the title, use \\ to force line breaks. \author{L. Meyer\altaffilmark{1}, T. Do, A. Ghez, M. R. Morris} Los Angeles, CA 90095-1547} \affil{Universit\"at zu K\"oln, Z\"ulpicher Str. 77, 50937 K\"oln} \author{G. B\'{e}langer} \affil{ESA/ESAC, PO Box 78, 28691 Villanueva de la Ca\~{n}ada, Spain} \author{R. Sch\"odel} \affil{Instituto de Astrof\'{i}sica de Andaluc\'{i}a, Camino Bajo de Hu\'= {e}tor 50, 18008 Granada, Spain} %% Notice that each of these authors has alternate affiliations, which %% are identified by the \altaffilmark after each name. Specify alternat= e %% affiliation information with \altaffiltext, with one command per each %% affiliation. \altaffiltext{1}{Supported by a fellowship within the Postdoc-Program of = the German Academic Exchange Service (DAAD).} %% Mark off your abstract in the ``abstract'' environment. In the manuscr= ipt %% style, abstract will output a Received/Accepted line after the %% title and affiliation information. No date will appear since the autho= r %% does not have this information. The dates will be filled in by the %% editorial office after submission. \begin{abstract} We present the longest, by a factor of two, near-infrared lightcurve from= Sgr~A* -- the supermassive black hole in the Galactic center. Achieved b= y combining Keck and VLT data from one common night, which fortuitously h= ad simultaneous Chandra and SMA data, this lightcurve is used to address = two outstanding problems. First, a putative quasi-periodicity of $\sim$= 20\,min reported by groups using ESO's VLT is not confirmed by Keck obser= vations. Second, while the infrared and mm-regimes are thought to be = related based on reported time lags between lightcurves from the two wave= length domains, the reported time lag of 20\,min inferred using the Keck = data of this common VLT/Keck night only is at odds with the lag of $\sim1= 00$\,min reported earlier. With our long lightcurve, we find that (i) th= e simultaneous 1.3 millimeter observations are in fact consistent with a = $\sim100$\,min time lag, (ii) the different methods of NIR photometry use= d by the VLT and Keck groups lead to consistent results, (iii) the Lomb-S= cargle periodogram of the whole NIR lightcurve is featureless and follows= a power-law with slope -1.6, and (iv) scanning the lightcurve with a sli= ding window to look for a transient QPO phenomenon reveals for a certain = part of the lightcurve a 25\,min peak in the periodogram. Using Monte Car= lo simulations and taking the number of trials into account, we find it t= o be insignificant. =20 \end{abstract} %% Keywords should appear after the \end{abstract} command. The uncomment= ed %% example has been keyed in ApJ style. See the instructions to authors %% for the journal to which you are submitting your paper to determine %% what keyword punctuation is appropriate. \keywords{black hole physics, Galaxy: center} %\facilities{VLT, Keck, SMA} %% From the front matter, we move on to the body of the paper. %% In the first two sections, notice the use of the natbib \citep %% and \citet commands to identify citations. The citations are %% tied to the reference list via symbolic KEYs. The KEY corresponds %% to the KEY in the \bibitem in the reference list below. We have %% chosen the first three characters of the first author's name plus %% the last two numeral of the year of publication as our KEY for %% each reference. %% Authors who wish to have the most important objects in their paper %% linked in the electronic edition to a data center may do so by tagging= %% their objects with \objectname{} or \object{}. Each macro takes the %% object name as its required argument. The optional, square-bracket=20 %% argument should be used in cases where the data center identification %% differs from what is to be printed in the paper. The text appearing=20 %% in curly braces is what will appear in print in the published paper.=20 %% If the object name is recognized by the data centers, it will be linke= d %% in the electronic edition to the object data available at the data cen= ters =20 %% %% Note that for sources with brackets in their names, e.g. [WEG2004] 14h= -090, %% the brackets must be escaped with backslashes when used in the first %% square-bracket argument, for instance, \object[\[WEG2004\] 14h-090]{90= }). %% Otherwise, LaTeX will issue an error.=20 \section{Introduction} The near-infrared (NIR) regime at high angular resolution has proven to b= e of great value in Galactic center research. The proper motion of stars = detected in this waveband demonstrates the existence of a supermassive bl= ack hole (BH) at the center of our Galaxy: Sagittarius~A* \citep[Sgr~A*; = see, e.g.,][]{eckigenzel, ghez98, gheznature,ghez,genzel00,rainer1}. In 2= 003, NIR emission associated with Sgr~A* was detected \citep{genzel,ghez0= 4}, which is important since it is the most underluminous BH accretion sy= stem observed thus far (with a bolometric luminosity nine orders of magni= tude lower than its Eddington luminosity).=20 The NIR emission is highly variable: intensity changes by factors $\leq 1= 0$, lasting between about 10 and 100 min, occur at least 4 times a day, a= nd are highly polarized \citep[e.g.][]{ecki1,ecki2,ich2,ich,tuan,trippe}.= Simultaneous NIR and X-ray observations have revealed that each X-ray fl= are is accompanied by a NIR flare with zero time lag, but not vice versa = \citep{ecki04,ecki1,ecki08,belanger05,hornstein07}.=20 Recent campaigns that included mm observations, reported a characteristic= time lag between the NIR/X-ray and longer wavelengths that has been inte= rpreted in terms of an expanding synchrotron plasmon. The limited overlap= between data sets, however, has led to debates over the nature of this t= ime lag \citep[e.g.][]{ecki1,yusef06,yusef07,danmultwav}. The characteristics of the NIR emission from Sgr~A* is of particular inte= rest given that there have been reports of periodic modulations with a pe= riod of $\sim$20\,min present in NIR flares detected during VLT observati= ons \citep{genzel,ecki2,ich2,ich,trippe}. In a recent study, however, \ci= tet{tuan} used robust statistical estimators and found no significant pea= ks in the periodograms of the Sgr~A* lightcurves observed with the Keck I= I telescope. Moreover, quasi-periodicities in the X-ray regime claimed t= o be present by \citet{aschenbach1}, are not statistically significant \c= itep{belanger}. In this Letter, we address the questions of a periodic component in the N= IR flux, and the relation of the NIR to the mm-regime, by combining for t= he first time contiguous Keck and VLT data. During that 10 hour session, = there was also simultaneous coverage by Chandra and SMA. These observatio= ns were published by \citet{danmultwav}, but interpreted taking only the = Keck data into account. \section{The data} During the night of 2005 July 30-31, the VLT observed Sgr~A* from 23:05\,= UT to 06:53\,UT using the Natural Guide Star Adaptive Optics system and N= IR camera NACO on UT4\footnote{ESO program 075.B-0093(B).}. Since these o= bservations have not been published before, the observational details are= given here. The detector integration time was 15\,s, and four images were co-added be= fore the data were recorded: the effective resolution is $\sim$1 image/80= \,s. Dithering was used to minimize the effects of dead pixels. The atmos= pheric seeing conditions ranged between $1\arcsec-2.5\arcsec$ (as determi= ned by the differential image motion monitor) during the first two hours,= and $0.5\arcsec-1.25\arcsec$ afterwards (the average Strehl ratio of the= VLT data is 17\%).=20 At 07:00\,UT, the Keck II telescope started its monitoring campaign using= the NIRC2 camera in combination with Laser Guide Star AO. The Keck obser= vations were made by cycling through the H-K-L wavelength filters \citep[= see][for details]{hornstein07}. Since the VLT observations were carried o= ut in K-band, we consider only the K-band subset of the Keck data. Both data sets were reduced in the same standard way, i.e. sky subtracted= , flat-fielded, and corrected for bad pixels. Images with a Strehl ratio= less than 10\% were removed (13 out of 266 VLT images). For every indivi= dual image, the point spread function (PSF) was extracted with the code S= tarFinder by \citet{diolaiti}. Each exposure was deconvolved with a Lucy-= Richard deconvolution and restored with a Gaussian beam.=20 The flux of Sgr A* and other compact sources in the field were obtained v= ia aperture photometry on the diffraction limited images with a circular = aperture of radius $0\arcsec.03$. The background flux density was determi= ned as the mean flux measured with apertures of the same size at five dif= ferent positions in a field located $1\arcsec$ northwest of Sgr~A* that s= hows no individual stars. Photometric calibration was done relative to st= ars in the field with known flux. For the extinction correction we assume= d $A_K =3D 2.8$\,mag \citep{eisendings05}. Estimates of uncertainties wer= e obtained from the standard deviation of fluxes of nearby constant sourc= es. It is noteworthy that this is the first time that Galactic Center VLT and= Keck data have been reduced homogeneously. While the VLT groups mainly u= se deconvolution and aperture photometry, the Keck group uses PSF fitting= without deconvolution.=20 The Keck part of the lightcurve presented in the next section is similar = to that reported by \citet{hornstein07}, where the PSF fitting technique = was applied. After rescaling the lightcurve by a multiplicative factor to= account for a different de-reddening factor and slightly different calib= ration values used by \citet{hornstein07}, the average difference per dat= a point between both methods is only 0.031\,mJy (de-reddened), well withi= n our $1\sigma$-error bar of 0.18\,mJy. Both data reduction methods are = consistent with each other, and no large systematic errors are introduced= by choosing either one.=20 \section{Results} \subsection{The NIR properties of Sgr A*} Figure~\ref{fig1} (left panel) shows the 2005 July 30-31 de-reddened ligh= tcurve. The first 450 min are the VLT data, and the following 130 min are= the Keck data. Note that the error bars are smaller for the Keck data du= e to better seeing conditions at Mauna Kea that night. The corresponding = Lomb-Scargle periodogram for the entire lightcurve is presented in the ri= ght panel of Fig.~\ref{fig1}. It is consistent with the finding of \citet= {tuan} that the NIR emission of Sgr~A* is described by a single stochasti= c process (other than additive measurement noise) that has a power-law sp= ectrum, $\mathcal{P}\propto f^{-\alpha}$, very similar to the X-ray emiss= ion of AGN. The Lomb-Scargle periodogram in Fig.~\ref{fig1} is a single r= ealization of this process and therefore fluctuates around the spectrum. = We determine the probability density function (PDF) of these fluctuations= around the spectrum at a given frequency empirically with Monte Carlo si= mulations. With the PDF at hand, we can assess the likelihood that peaks = in the periodogram are not due to a fluctuation but rather have a physica= l cause intrinsic to the source.=20 We used two different Monte Carlo based analyses recently developed by \c= itet{tuan} and \citet{belanger} to look for significant peaks in the peri= odogram. These analyses are carried out by first determining the power-la= w index of the spectrum of the stochastic process, and then generating li= ghtcurves \citep[realizations of this process;][]{timmer95} with a sampl= ing function matching those of the data set. Finally, the significance of= each observed periodogram peak is derived from the large simulated refer= ence data set. =20 An accurate determination of the power-law index $\alpha$ of the underlyi= ng stochastic process is important in this approach. \citet{belanger} do = this by first performing several estimates of $\alpha$ by fitting a power= -law to the periodogram made from the lightcurve binned with successively= larger bin times, and then carrying out simulations with the same count = rate and sampling to find the matching curve of $\alpha$ versus bin time.= \citet{tuan} also use MC simulations, but determine the power-law index = using the structure function, which also takes the sampling into account.= =20 Both methods lead to a spectral index of $\alpha=3D1.6\pm 0.05$ (formal f= itting error). Note that the length of the lightcurve allows to sample fr= equencies $< 10^{-4}$\,Hz ($0.006\,\mbox{min}^{-1}$) for the first time, = showing that the power-law extends to this regime. Unfortunately, we cann= ot fully exclude the possibility of spectral leakage from a process with = a spectral index steeper than -1.6. However, our main conclusions in this= paper also hold true for steeper power-law indices. We plan to use more = sophisticated spectral estimators elsewhere. The periodogram in the right panel of Fig.~\ref{fig1} clearly shows no ou= tlying peaks above the underlying spectrum of the power-law process. However, the featurelessness of the periodogram of the whole lightcurve d= oes not rule out the presence of a periodic component in parts of it, as = the mechanism giving rise to such a component may be transient and short = lived. We therefore did a search for periodicities over a restricted rang= e of frequencies by scanning the lightcurve using a sliding window method= where window here means a sub-span of the time series. This consists of = constructing a periodogram for the data subset corresponding to each wind= ow, and assigning a significance to each point based on the probability d= ensity functions derived from the simulations of the whole lightcurve des= cribed above. Fig.~\ref{scan} shows the result of such a scan using a 60\,min window wi= th 5\,min steps. The probability that the most significant peak in each p= eriodogram is due to a statistical fluctuation around the power-law spect= rum is plotted against the start time of the sliding window. The most sig= nificant peak overall (the one with the lowest probability to be due to a= fluctuation) is found in the window starting at minute 385, and we thus = looked at this window subset in more detail.=20 Indeed, the flux between 385 and 445 min looks very similar to the 'sub-f= lare' phenomenology reported by \citet{genzel}, \citet{ecki1,ecki2}, and = \citet{ich2}. These sub-flares are flux peaks superimposed on broader, lo= nger lasting flux excursions and are thought to be the manifestation of t= he claimed quasi-periodicity. \citet{ich2,ich} and \citet{ecki08} showed = that they can be interpreted in terms of a relativistically orbiting spot= whose emission adds to the emission of the accretion flow. The NIR flux = can therefore be described as $F(t)=3DA(t)+M(t)+S(t)$ with $A(t)$ a stoch= astic process with a power-law spectrum as above, $M(t)$ uncorrelated mea= surement noise, and $S(t)$ the deterministic flux of an (evolving) orbiti= ng spot leading to a (quasi-)periodicity. Our MC simulations include $A(t= )$ and $M(t)$ so that a possible quasi-periodic component can be identifi= ed as significant in the periodogram. Fig.~\ref{scan} shows that there is a peak in the periodogram correspondi= ng to the $385 - 445$\,min subset which has a false alarm probability of = only $2\cdot 10^{-5}$ (this corresponds to $4.2\sigma$ in Gaussian equiva= lent terms, but note that the PDF is not Gaussian). This peak occurs at a= frequency of $0.04\, \mbox{min}^{-1}$ (25\,min). However, we have to ask= the question how likely it is that a periodogram peak with such a low pr= obability does not only occur at this window, but in \textit{any} window.= After all, there is \textit{a priori} nothing special about this certain= sub-span of the lightcurve. We therefore count how often a peak with pro= bability $\leq 2\cdot 10^{-5}$ occurs in our simulations in any window: f= or a fixed window length of 60\,min, we find 3120 occurrences for 30,000 = simulations. This implies that the overall false alarm probability of the= peak in the $385 - 445$\,min subset is $0.104$ (corresponding to $1.6\si= gma$). Furthermore, we have to take into account the trials with windows = of different lengths in scanning the data: this immediately yields a fina= l significance of $\leq 1\sigma$. Hence, we conclude that the whole lightcurve is consistent with a pure po= wer-law process and no periodic component is needed. It is important to p= oint out, however, that if we had observed (or analyzed) only the part of= the data between $385 - 445$\,min, our result would have been interprete= d as a $4\sigma$ detection of a 25\,min QPO. This points to an explanatio= n of the contradictory results of \citet{genzel}, \citet{ecki1,ecki2} and= \citet{ich2}, on the one hand, and \citet{tuan}, on the other. Our resul= ts here seem to favor the finding of \citet{tuan} where no periodicity wa= s detected.=20 It is interesting to note that the 25\,min periodogram peak is most signi= ficant for a 60\,min sliding window, implying that only the first two of = the four 'sub-flares' between 380 and 460\,min are sampled. The reason f= or this is probably that the period -- if present -- is evolving over the= four cycles resulting in a periodogram peak which is too wide to be iden= tified as a significant periodic component. This, however, does not exclu= de the presence of an evolving, inwards spiraling bright spot or more com= plicated hydrodynamic instabilities in the accretion flow around Sgr~A* \= citep[e.g.][]{falanga}. \subsection{The connection to the millimeter regime} The Chandra X-ray Observatory and the SMA observed Sgr~A* simultaneously = to this new NIR lightcurve as reported by \citet{hornstein07} and \citet{= danmultwav}. Chandra started its observations at 20:00\,UT (30 July) and = stopped at 08:30\,UT (31 July), so that there are simultaneous X-ray data= for the entire NIR lightcurve seen in Fig.~\ref{fig1}. No X-ray flare oc= curred during the overlap time (23:05\,UT -- 09:00\,UT), see Fig.~6 in \c= itet{hornstein07}, despite the activity in the NIR with flares which are = somewhat stronger than average. This may be due to the higher X-ray backg= round caused by the steady Bondi-Hoyle accretion flow within $1\arcsec$ a= round Sgr~A* or/and a large surface area of the flaring region that leads= to a low synchrotron self-Compton luminosity \citep[see also][]{danmultw= av}.=20 In contrast to the X-ray lightcurve, the SMA 1.3\,mm observations, which = started at 05:28\,UT, show some variability (see Fig.~\ref{fig4}). In par= ticular, they show one flare at 08:20\,UT, roughly 20\,min after the NIR = flare observed with Keck. In their analysis, \citet{danmultwav} took only= the Keck NIR data into account. They interpreted this 20\,min time delay= in terms of an expansion of energetic plasma similar to the model propos= ed by \citet{vanderlaan}. In such a model, the expanding plasma region is= optically thick before, and optically thin after the peak flux. As this = transition is frequency dependent, smaller and later flare peaks are expe= cted at longer wavelengths. The 20\,min lag, however, differs from other = reported time lags. In a second data set from 2006 July, \citet{danmultwa= v} infer a $97\pm17$\,min time lag between an X-ray flare \citep[which is= supposed to be synchronous with the NIR, see][]{ecki1,ecki08} and a mill= imeter flare. \citet{yusef07} infer a lag of $110\pm17$\,min for the same= data. With the full NIR information at hand, the assumption of a correlation be= tween the 08:00\,UT NIR flare and the 08:20\,UT millimeter flare becomes = uncertain. Fig.~\ref{fig4} shows that there is another equally bright NIR= flare preceding the 8:00\,UT flare, with no counterpart at a 20 min lag = in the 1.3\,mm lightcurve. Therefore, it is equally likely that the mm fl= are is correlated with the preceding flare, as it is with the 8:00\,UT fl= are: substantially altering the interpretation of \citet{danmultwav}.=20 In fact, the huge difference in the reported time lags in the 2005 mm-IR = and the 2006 mm-X-ray data can be resolved with the NIR data presented he= re: if the $\sim$8:20\,UT mm flare (Fig.~\ref{fig4}) is correlated with t= he wide NIR flare at $\sim$6:00\,UT, then the inferred time lag is $\sim$= 140\,min. This lag is much closer to 97\,min/110\,min, than to the 20\,mi= n assumed by \citet{danmultwav}. Note that the 8:00\,UT NIR flare might t= hen be correlated with the SMA flare seen at 10:15\,UT. =20 \section{Conclusions} In this Letter we report on a 600 minute NIR lightcurve of Sgr~A* that co= mbines Keck II and VLT data. We showed that the Lomb-Scargle periodogram = of the overall lightcurve is featureless, and is statistically consistent= with a single stochastic process that has a power-law spectrum of index = $-1.6$. A certain subset of the data has a prominent peak in its correspo= nding periodogram which, however, is not significant when analyzed in the= context of the whole lightcurve.=20 This points to the following dilemma in Sgr~A* research: if a periodic co= mponent exists, it is clearly a weak and transient phenomenon that persis= ts over very few cycles, and probably has an evolving period. As we have = shown, certain parts of longer pure red noise lightcurves can easily mimi= c such a behavior. To be able to distinguish between both scenarios, many= lightcurves are needed and the range of frequencies under consideration = must be narrowed down, e.g. to $0.04 - 0.07\;\mbox{min}^{-1}$ if a notice= able peak continues to occur in this range only. We furthermore showed the difficulty with establishing the simple expandi= ng plasmon model frequently proposed to relate NIR and mm-flares. Sgr~A* = is such an active source in the NIR (and maybe not every NIR flare has a = millimeter counterpart) that very long simultaneous observations are need= ed for a meaningful cross-correlation analysis. \acknowledgements We are very grateful to Dan P. Marrone for providing us with the SMA data= =2E Some of the data presented here were obtained from Mauna Kea observat= ories. We are grateful to the Hawai'ian people for permitting us to study= the universe from this sacred summit. This work was supported by NSF gra= nt AST-0406816. {\it Facilities:} \facility{VLT:Yepun (NACO)}, \facility{Keck:II (NIRC2),= \facility{SMA} } \begin{thebibliography}{}=20 \bibitem[Aschenbach et al.(2004)]{aschenbach1} Aschenbach, B., Grosso, N.= , Porquet, D., Predehl, P. 2004, A\& A, 417, 71A %\bibitem[Baganoff et al.(2001)]{baganoff01} Baganoff, F. K., Bautz, M. 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Rig= ht panel: Lomb-Scargle periodogram of the lightcurve. No smoothing was ap= plied.} \label{fig1} \end{figure} \begin{figure} %\begin{minipage}{8.5cm} \includegraphics[scale=3D.65]{f2.eps} \caption{Sliding window scan of the lightcurve seen in Fig.~\ref{fig1}. E= ach time window has a length of 60\,min and has been shifted by 5\,min fo= r the scan. The abscissa shows the start time of each sliding window. The= ordinate shows the probability that the most significant peak in the cor= responding periodogram is due to a fluctuation around the power-law spect= rum of the red noise stochastic process. A very low probability means tha= t an additional \mbox{(quasi-)periodic} component is needed to explain t= he periodogram peak. See text for more details.} \label{scan} %\end{minipage} \end{figure} \begin{figure} %\hspace{1cm} %\begin{minipage}{8.5cm} \includegraphics[scale=3D.65]{f3.eps} \caption{The SMA 1.3 mm data (upper curve) from \citet{danmultwav} plotte= d over a common time axis with the overlapping part of the NIR lightcurve= (lower curve) shown in Fig.~\ref{fig1}. The flux units are mJy for the N= IR and Jy for the SMA data which have been shifted upwards by 3.5 Jy for = better comparison. Please note that the first four points in the SMA ligh= tcurve are somewhat unreliable (D. Marrone, priv. comm.). The scatter in = these four points seems large compared to the error bars, which indicates= that they should in fact be larger for these points than shown in the pl= ot.} \label{fig4} %\end{minipage} \end{figure} \end{document} --------------020806040703070401060500--