Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.eso.org/~rsiebenm/agn_models/Johnson13.pdf Äàòà èçìåíåíèÿ: Fri Jul 17 12:04:52 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:39:22 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: arp 220

MNRAS 436, 2535­2549 (2013)

doi:10.1093/mnras/stt1758

Advance Access publication 2013 October 9

SATMC:

Spectral energy distribution Analysis Through Markov Chains

S. P. Johnson,1 < G. W. Wilson,1 Y. Tang1 and K. S. Scott2
1 2

Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA North American ALMA Science Center, National Radio Astronomy Observatory, Charlottesville, VA 22903, USA

Accepted 2013 September 14. Received 2013 September 12; in original form 2013 August 5

ABSTRACT

We present the general purpose spectral energy distribution (SED) fitting tool SED Analysis Through Markov Chains (SATMC). Utilizing Monte Carlo Markov Chain (MCMC) algorithms, SATMC fits an observed SED to SED templates or models of the user's choice to infer intrinsic parameters, generate confidence levels and produce the posterior parameter distribution. Here, we describe the key features of SATMC from the underlying MCMC engine to specific features for handling SED fitting. We detail several test cases of SATMC, comparing results obtained from traditional least-squares methods, which highlight its accuracy, robustness and wide range of possible applications. We also present a sample of submillimetre galaxies (SMGs) that have been fitted using the SED synthesis routine GRASIL as input. In general, these SMGs are shown to occupy a large volume of parameter space, particularly in regards to their star formation rates which range from 30 to 3000 M yr-1 and stellar masses which range from 1010 to 1012 M . Taking advantage of the Bayesian formalism inherent to SATMC, we also show how the fitting results may change under different parametrizations (i.e. different initial mass functions) and through additional or improved photometry, the latter being crucial to the study of high-redshift galaxies. Key words: methods: statistical ­ techniques: photometric ­ galaxies: fundamental parameters ­ galaxies: high-redshift ­ submillimetre: galaxies.

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1 I NTR O DUCTION The complete panchromatic spectral energy distribution (SED) of a source encodes a wealth of information concerning its age, mass, metallicity, dust/gas content, star formation rate (SFR), star formation history (SFH) and more. Various emission mechanisms account for the apparent shape and typical features found in SEDs. For example, dust reprocessing results in attenuation of optical/ultraviolet (UV) photons produced from stellar populations whose energy is then re-emitted at infrared (IR) wavelengths while specific species of ions and molecules produce emission/absorption features across the electromagnetic spectrum. Unfortunately, our ability to extract information from observations is hampered both on the observational and theoretical sides. As observers, we try to make due with either coarse, broad-band sampling of a source's SED or with highresolution sampling of a small portion of the SED through spectroscopy. Theorists, on the other hand, must make difficult decisions regarding which physics, spatial scales and evolutionary histories to include in the creation of their SED libraries or synthesis models. Within the literature, SED models can be subcategorized into two main types. Empirical models are derived for particular classifications based on a subset of similar sources (e.g. Arp 220 and
E-mail: spjohnso@astro.umass.edu

M82 for starburst galaxies, the quasar mean template, etc.). These models offer the simplest approximation of a source's SED and are generally preferred when only sparse photometry is available or for coarse estimates of basic properties (e.g. luminosity, SFR, colours). The underlying assumption behind empirical models, namely that all sources of that `type' have the same SED, is difficult to verify and so their use to probe all but the grossest properties is limited. Theoretical models are constructed from sets of physical and radiative processes believed to be the dominant contributors to the emission of a source. Functionally, these again are divided into two classes. Pre-computed template libraries for particular source types are the most widely used (e.g. Calzetti et al. 2000; Efstathiou, Rowan-Robinson & Siebenmorgen 2000; Siebenmor¨ gen et al. 2004; Siebenmorgen & Krugel 2007; Gawiser 2009; Michalowski, Watson & Hjorth 2010 and references therein). Since they are pre-computed, these libraries allow the rapid exploration of a pre-defined parameter space at the expense of being limited to the resolution and scope of the parameter space provided by the authors. For more generalized applications, SED synthesis packages are now available that offer a wider set of input parameters and the exploration of a continuous parameter space [e.g. stellar population synthesis codes such as GALAXEV (Bruzual & Charlot 2003) and panchromatic galaxy synthesis codes like GRASIL (Silva et al. 1998) and CIGALE (Noll et al. 2009)]. Of course, one may use these

C 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society


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where the product runs over the M observed data points D whose individual variances are given by i2 and the function f ( d | x ) represents the model observations d for the set of x parameters. Note, however, that this prescription of L assumes that the observations have Gaussian distributed errors, an assumption we will return to later in Section 3.1. In the following sections, we detail specific MCMC features that define the basic operation of SATMC.

packages to construct one's own template libraries (e.g. Michalowski et al. 2010) for specific applications as well. In both cases, as the generality of the underlying physics increases to provide relevance to a wider class of sources, so does the number of free parameters in the models. The unavoidable existence of correlations in these parameters insists that simply finding a best-fitting parametrization is no longer sufficient. Rather, we now require tools that properly reveal the complexities and correlations in the adopted model parameter space. With this in mind, here we present the general purpose MCMCbased SED fitting code SED Analysis Through Markov Chains or 1 SATMC. Monte Carlo Markov Chain (MCMC) techniques are a set of methods based in the Bayesian formalism which enable efficient sampling of multidimensional parameter spaces in order to construct a distribution proportional to the probability density distribution of the input parameters, known as the posterior parameter distribution or simply the posterior. The posterior identifies the nuances in parameter space, including any possible correlations, making MCMCs particularly useful for exploring SED models and libraries with high dimensionality. MCMC-based SED fitting codes are relatively new but have been growing in popularity (e.g. Sajina et al. 2006; Acquaviva, Gawiser & Guaita 2011; Serra et al. 2011; Pirzkal et al. 2012). In addition to deriving the posterior for bestfitting parameter and confidence level estimation, SATMC includes many features to aid in improving performance and allows users to easily incorporate additional knowledge and constraints on parameter space in the form of priors. Additionally, SATMC versions exist in both IDL and PYTHON and both are modular and straightforward to use in any wavelength regime and for any class of sources. This paper is organized as follows. We start by detailing the MCMC algorithm and the basic process for MCMC-based SED fitting. We then provide case examples displaying the versatility and accuracy of SATMC compared to standard least-squares methods. As part of this demonstration, we also present a set of SEDs derived from SATMC when used in conjunction with the SED synthesis routine GRASIL to a sample of X-ray selected starburst galaxies. These fits highlight the key advantages obtained from the MCMC-based approach and their agreement with similar classes of composite SEDs.

2.1 MCMC acceptance and convergence Within the literature, there are a variety of sampling algorithms one may use to construct an MCMC (e.g. Metropolis­Hastings or Gibbs sampling; Metroplois et al. 1953; Hastings 1970; Geman & Geman 1984; Geyer 1992; Chib & Jeliazkov 2001; Verde et al. 2003; Mackay 2003). For SATMC, we employ the Metropolis­Hastings algorithm which works as follows: (i) Generate a proposal distribution q ( x i |xi -1 ) from which the candidate steps x i will be drawn. (ii) Calculate the acceptance probability according to the likeli(i )(i - hood ratio ( = min(1, PPxx-|1DDqqxx|ixi1 |1x)i ) )). (i | )( - (iii) Draw a uniformly distributed random number u from 0 to 1 and accept the step if u < , reject otherwise. (iv) Repeat for the next step. Though the exact choice of q ( x i |xi -1 ) is arbitrary, it is common to adopt an nD -dimensional multivariate-normal distribution 2 N (, ) with = x I where I is the identity matrix, x is the variance of each parameter in x and N (, ) | |- 2 e-
1 1 2

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( x -)

-1

( x -)

(3)

2

SATMC

: T HE MCMC ALGORITHM

The primary motivation for SATMC is to provide a means for efficient sampling of a parameter space with nD free parameters in order to derive parameter estimates and their associated confidence intervals. This is accomplished by sampling an nD -dimensional surface proportional to the probability density function of the parameters given the data P ( x | D), also referred to as the posterior parameter distribution. We determine the posterior through Bayes theorem P ( x | D ) P ( D | x )P ( x ), (1)

for generating proposed steps (e.g. Gelman, Roberts & Gilks 1995; Roberts, Gelman & Gilks 1997; Roberts & Rosenthal 2001; Atchade & Rosenthal 2005). Following these works, it is found that the distribution N (, ) should be tuned for optimal performance between the time necessary to reach a stable solution (referred to as the burn-in period) and sampling of the posterior, which may be achieved by adjusting N (, ) until the resulting chains have acceptance rates of 23 per cent in the limit of large dimensionality. We therefore base our proposal distribution around an adaptive covariance matrix (similar to Acquaviva et al. 2011) such that samples are drawn from the multivariate-normal distribution now given by N ( x i -1 , ). To obtain an acceptance rate of 23 per cent, 2 we start by initializing the covariance matrix for each chain as x I, where x is set proportional to the input parameter ranges. After a period of steps, we then calculate directly from the chain over the previous interval, i.e. = E [( x - E [ x ])( x - E ( x ))T ], (4)

where P ( x ) represents our current knowledge of the parameters or priors and P ( D| x ) is the probability of the data given the model parameters, often referred to as the likelihood L. We shall define the general form of the likelihood according to
M

P ( D| x ) = L =
i -1

exp

(Di - f (di | x )) 2i2

2

,

(2)

1

www.ascl.net/1309.005

where E [ x ] is the expectation value or weighted average of x . Following each period of steps, we compute the acceptance rate and scale if the acceptance rate is too high (>26 per cent) or too low (<20 per cent). The covariance matrix is continuously updated until the target 23 per cent acceptance is reached, which allows to take on the shape of the underlying posterior to readily identify and account for possible correlations in the parameters. Once the target acceptance rate has been reached, we then check for chain convergence to determine when the burn-in period is complete. In the MCMC literature, there are many approaches to determine convergence (e.g. Gelman & Rubin 1992; Raftery & Lewis 1992; Gilks, Richardson & Spiegelhalter 1996); we opt for


SED Analysis Through Markov Chains
the Geweke diagnostic (Geweke 1992), which compares the average and variance of samples obtained in the first 10 per cent and last 50 per cent of a chain segment. If the two averages are equal (within the tolerance set by their variances), then the chain is deemed to be stationary and convergence is complete. We check both the acceptance rate and convergence, verifying the acceptance rate before checking for convergence, over a default period of 1000 steps until both have been satisfied. If necessary, the covariance matrix is modified to maintain the target acceptance rate. Once both criteria are fulfilled, burn-in is completed and the chain(s) are set to continue to provide sampling of the posterior. 2.2 Parallel tempering When constructing the posterior distribution, we must be careful to sample all of the nuances of parameter space as the posterior distribution is not guaranteed to be a smooth or even continuous function of the free parameters. As with all fitting approaches, the existence of local maxima in the posterior requires that either we known a priori the general location of the global maximum or we implement a sampling technique capable of increasing the probability of finding it. For example, one could choose to initialize a large number of chains that cover random locations throughout parameter space. While this approach is nearly guaranteed to find the global maximum, it is also extremely inefficient. SATMC utilizes a technique known as `Parallel Tempering' (see review by Earl & Deem 2005) which, in analogy to simulated annealing, uses several chains, each with progressive modifications to likelihood space parametrized as a statistical `temperature', to search parameter space and exchange information about the posterior at each chain's location. For a given chain at temperature T, the likelihood is `flattened' according to LT = L1/(1+T) . The tempered chains thus distort likelihood space, exchanging sensitivity to the details within posterior for the general shape, such that chains with higher temperatures will accept more steps, and thus sample larger regions of parameter space. By coupling these tempered chains, we allow `colder' chains to access areas of parameter space they may have otherwise been unable to reach. The process for handling tempered chains works as follows: (i) Temperatures are assigned to chains in a progressive manner with one chain designated the fiducial `cold' chain with T = 0 (e.g. T = [0, 10, 100, 1000]). (ii) Chains are allowed to progress for a set number of steps (SATMC uses three iterations of acceptance/convergence or 3000 steps by default). (iii) A `swap' of chain state information is proposed using the Metropolis acceptance algorithm = min(1,
1/(1+Tj ) (P ( x i | D)) (P ( x j | D))1/(1+Ti ) 1/(1+Tj ) (P ( x i | D))1/(1+Ti ) (P ( x j | D))

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Figure 1. Simple visualization of transitions between tempered chains. Transitions may occur either between the cold and tempered chains (top) or between adjacent pairs (bottom). The former allows for rapid sampling of parameter space to quickly reach global maxima while the latter improves sampling around a single location to avoid local maxima in the construction the posterior parameter distribution. In this example, chain transitions are proposed after every 100 steps; in practice, these transitions are proposed every 3000 steps.

.

(iv) The new set of chains is then allowed to run until the next swap of chain state information. This process of coupling individual chains with a Metropolis­ Hastings acceptance algorithm was initially proposed by Geyer (1991) and is referred to as a Metropolis Coupled Monte Carlo Markov Chain or MC3 . When applying temperatures to individual chains, the sampling of each chain will vary due to the distortions of likelihood space; i.e. a chain at a higher temperature will accept more steps than a chain with an identical proposal distribution at lower temperature. To maintain a reasonable sampling of the tempered likelihoods, each chain is treated individually for the purposes of acceptance and convergence testing. Note, however, that updating the tempered chains occurs only with the acceptance and

convergence testing of the cold chain. This ensures that we obtain a proper sampling of the posterior without being influenced by the otherwise distorted likelihood space. In parallel tempering techniques, one can specify whether the potential swaps occur only between the cold and any tempered chain (0, j) or between adjacent chains (i, i ± 1) (see Fig. 1). The former allows for rapid sampling and mixing of the chains to determine the global maximum and is thus used during the burn-in period. The latter passes state information down through the tempered chains so that the cold chain may access a more representative region of parameter space; this method is used after burn-in to properly sample parameter space around the maximum likelihood. In this case, chains progress for a set length (500 steps by default) whereafter the chain transitions are proposed. Should a swap be made with the cold chain after burn-in is complete, we re-compute the acceptance rate and convergence as outlined in the previous section to verify proper sampling. Due to the modified acceptance rate and misshapen likelihood space of the tempered chains, it is generally undesirable to use them in reconstructing the posterior parameter distribution. If there have been no swaps to the cold chain after 10 iterations, the temperatures of all chains are set to 0 and the MCMC is allowed to continue until


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sufficient samples of the posterior distribution around the maximum have been obtained. 3
SATMC

SED S PECIFIC F EATURES

The methods presented in Section 2 are generalized for any MCMCbased algorithm. Here, we detail specific modifications and methods utilized in SATMC for SED fitting. 3.1 Observations and upper limits In order to perform a fit, SATMC requires at least two sets of information: (1) a file containing the M observations including their wavelengths (i ) or frequencies ( i ), observed fluxes (fobs,i ) and corresponding uncertainties (obs,i ) and (2) the model libraries. Following from equation (2), we define the likelihood for a given set of observations and models as
M

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L=
i -1

exp -

(fobs,i - fmod,i ) 2 2fobs,i

2

(5)

or similarly
M

ln(L) =
i -1

-

(fobs,i - fmod,i )2 , 2 2fobs,i

Figure 2. Example likelihood distribution for upper limits. Here, the upper limit has a value of 3 mJy (dotted vertical line). The remaining lines represent the likelihood distribution for the simple step function (solid), one-sided Gaussians assuming the upper limit is at the 5 level with a cut-off at 1.8 mJy (dashed) and 0 mJy (dot­dashed) and one-sided Gaussians with the upper limit at the 3 level and cut-offs at 2 mJy (triple dot­dashed) and 0 (long dash).

(6)

where fmod,i are the model fluxes which for narrow passbands are interpolated model SED values at the observed frequency/wavelength. Alternatively, fmod,i may be calculated from a set of passbands with a given filter response [p( ) as provided by the user] as fmod,i =
max min

SED( )pi ( ) d
max min

pi ( )d

,

(7)

where min and max are taken from the filter files. In cases where there is no detection but only an upper limit, SATMC provides a parametrized probability distribution to allow the user to reflect their confidence in the underlying statistics of the observations. A simple approach for incorporating upper limits is to assume a step function accepting all models that fall below the limit and rejecting all that lie above. While attractive in its simplicity, this method harshly truncates regions of parameter space and improperly weights the upper limit's contribution to L. To partially alleviate these issues, SATMC implements a one-sided Gaussian distribution so that models with fluxes above the upper limit may contribute to the posterior with a small but non-zero probability (see also Feigelson & Nelson 1985; Isobe, Feigelson & Nelson 1986; Sawicki 2012). The one-sided Gaussian is defined by 1 if fmod < fco (8) L 2 exp - (fco -f2mod ) if fmod fco , 2
co

´ Bolzonella, Miralles & Pello 2000) though some recent codes have adopted the Bayesian formalism for using priors (e.g. BPZ, Benitez 2000; EAZY, Brammer, van Dokkum & Coppi 2008). The standard approach in photo-z estimation is to apply an SED library (often limited to a few galaxy types) to find the photo-z and then repeat the fitting using the same (or in some cases different) library fixed at the photo-z to estimate source properties. This approach underestimates the true errors on the photo-z and other fitted parameters. SATMC fits the redshift simultaneously with all other parameters and produces a direct determination of the redshiftparameter probability distributions in addition to the marginalized redshift probability distribution P(z).2 We have tested and improved the photo-z estimation with SATMC in collaboration with the CANDELS team. Dahlen et al. (2013) provides a complete analysis of various photometric redshift estimation techniques for samples of CANDLES galaxies in the GOODS-S field (Giavalisco et al. 2004). Out of the 13 participating groups, SATMC was the only MCMC-based code used to generate photo-z's. Since the tests reported in Dahlen et al., we have reduced the outlier fraction (fraction of sources with (zspec - zphot )/(1 + zspec ) > 0.15) from 9­14 per cent to 3­8 per cent through modification of our input templates, luminosity priors (e.g. Kodama, Bell & Bower 1999; Benitez 2000) and zero-point photometry corrections (e.g. Dahlen et al. 2010). 3.3 Template libraries and SED synthesis routines As MCMC samplers require a continuous parameter space, SATMC allows the user to incorporate SED synthesis routines (e.g. GALAXEV, GRASIL, CIGALE) to generate SEDs at candidate steps and compute the resulting likelihoods. Empirical templates are often defined with a scalable normalization factor as one of the few (if any) free parameters which remains continuous when used with SATMC. Template libraries, unfortunately, rarely offer a fully continuous parameter space, often mixing sets of continuous (e.g. model normalization) and discretely sampled parameters. To create a pseudo-continuous
2 Cosmology is presently fixed in H0 = 70, = 0.7 and M = 0.3.

where fco represents the cut-off transition from flat L = 1 to a Gaussian `tail' with standard deviation co . Parametrizing the probability distribution in this manner allows for a compromise between the simple step function, also available in SATMC, and overinterpreting the shape of the noise distribution at small fluxes, as demonstrated in Fig. 2. 3.2 Photometric redshift estimation Since redshift is just another parameter in a source's SED, SATMC is capable of providing photometric redshift (photo-z) determinations for sources in the context of the input SED models. Traditionally, photo-z codes implement least-squares methods (e.g. HYPERZ;

SATMC

and assumes flat

CDM with


SED Analysis Through Markov Chains
space from such template libraries, SATMC computes the likelihoods of models bracketing the current step according to equations (5) and (8) and applies multilinear interpolation to determine the likelihood of the current proposed step. We emphasize that SATMC may be used for any class of SED models (empirical, template library or synthesis routine), so long as the appropriate interface is constructed by the user. One should note, however, that there is a trade-off between the discretely sampled template libraries or empirical templates and continuous parameter space offered by SED synthesis routines (see also Acquaviva et al. 2011; Acquaviva, Gawiser & Guaita 2012 since the run-time of the SED synthesis routines will dominate the MCMC calculation (e.g. 3+ days run-time with GRASIL versus 3­5 min with a template library). Regardless of which class of SED models one wishes to adopt, empirical and theoretical SED models are commonly derived for a particular physical process and/or over a particular wavelength regime (e.g. Bruzual & Charlot 1993; Efstathiou et al. 2000; Siebenmorgen et al. 2004). To fully reproduce a galaxy's SED, additional components may be required either to complete the wavelength coverage or to include a missing physical process (e.g. adding AGN emission to a star formation template set). SATMC will construct a linear combination of multiple input SED models under the assumption that the underlying physical processes are independent. The MCMC process itself does not change: the combination of two models with N1and N2 free parameters, respectively, is viewed as a single model with N1 + N2 parameters when calculating likelihoods and taking potential steps. 3.4 Inclusion of priors An added feature of SATMC is the ability to include additional information to provide additional weights and constraints to likelihood space. In the Bayesian formalism, this extra information forms the priors of equation (1). For our implementation, we expand the definition of priors from the traditional Bayesian definition to include options for limiting and inherently correlating parameter space. This was deemed necessary for circumstances of fitting multiple template libraries where parameters from each model have the same physical interpretation and thus are not independent (e.g. AV from one model library and optical depth in another) and cases where additional information not available to the fits is available (e.g. restricting the age of a galaxy at a given redshift). 3.5 Application of the posterior A final feature of SATMC lies in the determination of the posterior parameter distribution. As we store the likelihood and location in parameter space for each step, it becomes a simple task to construct parameter confidence intervals and even parameter­parameter confidence contours to examine relative parameter degeneracies (see Fig. 3). Unfortunately, in order to visualize an nD -dimensional parameter space, we must project or `marginalize' parameter space into a one- or two-dimensional form. When marginalizing sets of parameters, the true shape of the posterior will be distorted which may not reveal correlations in higher dimensions. This also leads to a simplification when reporting the confidence intervals on individual parameters as traditional terms such as `1 ' imply Gaussianity in the posterior which is unlikely to exist. Instead, one-dimensional confidence levels are produced from the parameter range where 68 per cent of all accepted steps are contained, marginalized over all other free parameters. Throughout the text, `errors' quoted when derived from SATMC refer to these marginalized parameter ranges. 4 TESTING OF THE M CMC A LGORITHM

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With the SATMC algorithm as outlined in Sections 2 and 3, we now set out to verify the fitting results by analysing sets of well-known sources and template libraries. Though the examples provided here are for galaxy SED modelling, SATMC makes no distinction between source types and may just as easily be applied to Galactic sources, localized regions within a particular galaxy or spectroscopy of individual sources. We begin with the well-known galaxies Arp 220 ¨ and M82 and the starburst SED library of Siebenmorgen & Krugel (2007, hereafter SK07) which has already been shown to provide reasonable fits to Arp 220 and M82. The SED library consists of over 7000 templates with emission from a starburst parametrized by its total luminosity Ltot , nuclear radius R, visual extinction AV ,ratio of luminosity from OB stars to the total luminosity and hotspot dust density n. The observed SEDs for Arp 220 and M82 were constructed from data available on the NASA/IPAC Extragalactic Database (NED)3 over the 1­1500 m wavelength range. To ensure a meaningful comparison to SK07, we utilize the same photometric data for M82 and Arp 220 (including multiple aperture JHK photometry for M82) such that the only difference lies in the fitting method. Table 1 provides the best-fitting models as obtained by SK07 and SATMC with the models and parameter­parameter confidence contours shown in Fig. 3. Note that while SATMC will sample all of parameter space within that defined by the input template library, it is not possible to extrapolate likelihood information beyond the limits of the templates. This effect is responsible for the apparent truncation of parameter space seen in Fig. 3. Despite the irregular, non-uniform parameter sampling of the SK07 templates, SATMC closely recovers similar values to SK07 for the best-fitting model parameters. The parameter­parameter constraints as shown in Fig. 3 highlight the uncertainty in applying template sets, particularly for Arp 220 where SK07 suggest two likely best-fitting templates. For a more realistic test in determining the physical properties of a galaxy, we turn to the bright, lensed submillimetre galaxy SMM J2135-0102 (Swinbank et al. 2010) and apply the template SED library of Efstathiou et al. (2000, hereafter ERS00). The ERS00 starburst library consists of 44 templates with emission parametrized simply by the age of the starburst and the optical depth of molecular clouds where the new stars are forming. A normalization factor is required with the ERS00 templates to scale the emission from a single giant molecular cloud (on which the templates were formulated) to the entire system. This normalization factor roughly translates into an SFR with the model age according to SFR Norm â e-
t/20 Myr

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as ERS00 assumes an exponentially decaying SFH with an e-folding time of 20 Myr. Applying the templates to the lensingcorrected SED of SMM J2135-0102, we find best-fitting parameters of 1960+282 , 56.6+10..5 Myr and 199.9+0.1.4 for model normal-44 -250 -15 2 ization, age and optical depth, respectively; this model is shown in the left-hand panel of Fig. 4. Using the standard FIR­SFR relation of Kennicutt (1998), the best-fitting parameters imply an SFR of 192+28 M yr-1 . Comparatively, Swinbank et al. fit SMM -25 J2135-0102 with a two-temperature modified blackbody and derived an SFR of 210 ± 50 M yr-1 using Kennicutt (1998), fully consistent with our results.

3

http://ned.ipac.caltech.edu/


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Figure 3. Best-fitting SEDs for M82 (left) and Arp 220 (right) using the S