arXiv subjects

Hans-Walter Rix

Publications and source records attributed to Hans-Walter Rix.

At least 253 records · Page 14Linked to original sources

arXiv2016

Chemical tagging can work: Identification of stellar phase-space structures purely by chemical-abundance similarity

Chemical tagging promises to use detailed abundance measurements to identify spatially separated stars that were in fact born together (in the same molecular cloud), long ago. This idea has not yielded much practical success, presumably because of the noise and incompleteness in chemical-abundance measurements. We have succeeded in substantially improving spectroscopic measurements with The Cannon, which has now delivered 15 individual abundances for ~100,000 stars observed as part of the APOGEE spectroscopic survey, with precisions around 0.04 dex. We test the chemical-tagging hypothesis by looking at clusters in abundance space and confirming that they are clustered in phase space. We identify (by the k-means algorithm) overdensities of stars in the 15-dimensional chemical-abundance space delivered by The Cannon, and plot the associated stars in phase space. We use only abundance-space information (no positional information) to identify stellar groups. We find that clusters in abundance space are indeed clusters in phase space. We recover some known phase-space clusters and find other interesting structures. This is the first-ever project to identify phase-space structures at survey-scale by blind search purely in abundance space; it verifies the precision of the abundance measurements delivered by The Cannon; the prospects for future data sets appear very good.

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arXiv2016

A Synoptic Map of Halo Substructures from the Pan-STARRS1 3π Survey

We present a panoramic map of the entire Milky Way halo north of dec~-30 degrees (~30,000 deg^2), constructed by applying the matched-filter technique to the Pan-STARRS1 3Pi Survey dataset. Using single-epoch photometry reaching to g~22, we are sensitive to stellar substructures with heliocentric distances between 3.5 and ~35 kpc. We recover almost all previously-reported streams in this volume and demonstrate that several of these are significantly more extended than earlier datasets have indicated. In addition, we also report five new candidate stellar streams. One of these features appears significantly broader and more luminous than the others and is likely the remnant of a dwarf galaxy. The other four streams are consistent with a globular cluster origin, and three of these are rather short in projection (<10 degrees), suggesting that streams like Ophiuchus may not be that rare. Finally, a significant number of more marginal substructures are also revealed by our analysis; many of these features can also be discerned in matched-filter maps produced by other authors from SDSS data, and hence they are very likely to be genuine. However, the extant 3Pi data is currently too shallow to determine their properties or produce convincing CMDs. The global view of the Milky Way provided by Pan-STARRS1 provides further evidence for the important role of both globular cluster disruption and dwarf galaxy accretion in building the Milky Way's stellar halo.

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arXiv2016

ALMA Spectroscopic Survey in the Hubble Ultra Deep Field: Survey Description

We present the rationale for and the observational description of ASPECS: The ALMA SPECtroscopic Survey in the Hubble Ultra-Deep Field (UDF), the cosmological deep field that has the deepest multi-wavelength data available. Our overarching goal is to obtain an unbiased census of molecular gas and dust continuum emission in high-redshift (z$>$0.5) galaxies. The $\sim$1$'$ region covered within the UDF was chosen to overlap with the deepest available imaging from HST. Our ALMA observations consist of full frequency scans in band 3 (84-115 GHz) and band 6 (212-272 GHz) at approximately uniform line sensitivity ($L'_{\rm CO}\sim$2$\times$10$^{9}$ K km/s pc$^2$), and continuum noise levels of 3.8 $μ$Jy beam$^{-1}$ and 12.7 $μ$Jy beam$^{-1}$, respectively. The molecular surveys cover the different rotational transitions of the CO molecule, leading to essentially full redshift coverage. The [CII] emission line is also covered at redshifts $6.0<z<8.0$. We present a customized algorithm to identify line candidates in the molecular line scans, and quantify our ability to recover artificial sources from our data. Based on whether multiple CO lines are detected, and whether optical spectroscopic redshifts as well as optical counterparts exist, we constrain the most likely line identification. We report 10 (11) CO line candidates in the 3mm (1mm) band, and our statistical analysis shows that $<$4 of these (in each band) are likely spurious. Less than 1/3 of the total CO flux in the low-J CO line candidates are from sources that are not associated with an optical/NIR counterpart. We also present continuum maps of both the band 3 and band 6 observations. The data presented here form the basis of a number of dedicated studies that are presented in subsequent papers.

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arXiv2016

The Panchromatic Hubble Andromeda Treasury XV. The BEAST: Bayesian Extinction and Stellar Tool

We present the Bayesian Extinction And Stellar Tool (BEAST), a probabilistic approach to modeling the dust extinguished photometric spectral energy distribution of an individual star while accounting for observational uncertainties common to large resolved star surveys. Given a set of photometric measurements and an observational uncertainty model, the BEAST infers the physical properties of the stellar source using stellar evolution and atmosphere models and constrains the line of sight extinction using a newly developed mixture model that encompasses the full range of dust extinction curves seen in the Local Group. The BEAST is specifically formulated for use with large multi-band surveys of resolved stellar populations. Our approach accounts for measurement uncertainties and any covariance between them due to stellar crowding (both systematic biases and uncertainties in the bias) and absolute flux calibration, thereby incorporating the full information content of the measurement. We illustrate the accuracy and precision possible with the BEAST using data from the Panchromatic Hubble Andromeda Treasury. While the BEAST has been developed for this survey, it can be easily applied to similar existing and planned resolved star surveys.

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arXiv2016

The Gaia-ESO survey: Hydrogen lines in red giants directly trace stellar mass

Red giant stars are perhaps the most important type of stars for Galactic and extra-galactic archaeology: they are luminous, occur in all stellar populations, and their surface temperatures allow precise abundance determinations for many different chemical elements. Yet, the full star formation and enrichment history of a galaxy can be traced directly only if two key observables can be determined for large stellar samples - age and chemical composition. While spectroscopy is a powerful method to analyse the detailed abundances of stars, stellar ages are the "missing link in the chain", since they are not a direct observable. However, spectroscopy should be able to estimate stellar masses, which for red giants directly infer ages provided their chemical composition is known. Here we establish a new empirical relation between the shape of the hydrogen line in the observed spectra of red giants and stellar mass determined from asteroseismology. The relation allows to determine stellar masses and ages with the accuracy of 10-15%. The method can be used with confidence for stars in the following range of stellar parameters: 4000 < Teff < 5000 K, 0.5 < log g < 3.5, -2.0 < [Fe/H] < 0.3, and luminosities log L/LSun < 2.5. Our analysis provides observational evidence that the Halpha spectral characteristics of red giant stars are tightly correlated with their mass and therefore their age. We also show that the method samples well all stellar populations with ages above 1 Gyr. Targeting bright giants, the method allows to obtain simultaneous age and chemical abundance information far deeper than would be possible with asteroseismology, extending the possible survey volume to remote regions of the Milky Way and even to neighbouring galaxies like Andromeda or the Magellanic Clouds already with present instrumentation, like VLT and Keck facilities.

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arXiv2016

On Galactic density modeling in the presence of dust extinction

Inferences about the spatial density or phase-space structure of stellar populations in the Milky Way require a precise determination of the effective survey volume. The volume observed by surveys such as Gaia or near-infrared spectroscopic surveys, which have good coverage of the Galactic mid-plane region, is highly complex because of the abundant small-scale structure in the three-dimensional interstellar dust extinction. We introduce a novel framework for analyzing the importance of small-scale structure in the extinction. This formalism demonstrates that the spatially-complex effect of extinction on the selection function of a pencil-beam or contiguous sky survey is equivalent to a low-pass filtering of the extinction-affected selection function with the smooth density field. We find that the angular resolution of current 3D extinction maps is sufficient for analyzing Gaia sub-samples of millions of stars. However, the current distance resolution is inadequate and needs to be improved by an order of magnitude, especially in the inner Galaxy. We also present a practical and efficient method for properly taking the effect of extinction into account in analyses of Galactic structure through an effective selection function. We illustrate its use with the selection function of red-clump stars in APOGEE using and comparing a variety of current 3D extinction maps.

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arXiv2016

The stellar population structure of the Galactic disk

The spatial structure of stellar populations with different chemical abundances in the Milky Way contains a wealth of information on Galactic evolution over cosmic time. We use data on 14,699 red-clump stars from the APOGEE survey, covering 4 kpc <~ R <~ 15 kpc, to determine the structure of mono-abundance populations (MAPs)---stars in narrow bins in [a/Fe] and [Fe/H]---accounting for the complex effects of the APOGEE selection function and the spatially-variable dust obscuration. We determine that all MAPs with enhanced [a/Fe] are centrally concentrated and are well-described as exponentials with a scale length of 2.2+/-0.2 kpc over the whole radial range of the disk. We discover that the surface-density profiles of low-[a/Fe] MAPs are complex: they do not monotonically decrease outwards, but rather display a peak radius ranging from ~5 kpc to ~13 kpc at low [Fe/H]. The extensive radial coverage of the data allows us to measure radial trends in the thickness of each MAP. While high-[a/Fe] MAPs have constant scale heights, low-[a/Fe] MAPs flare. We confirm, now with high-precision abundances, previous results that each MAP contains only a single vertical scale height and that low-[Fe/H], low-[a/Fe] and high-[Fe/H], high-[a/Fe] MAPs have intermediate (h_Z~300 to 600 pc) scale heights that smoothly bridge the traditional thin- and thick-disk divide. That the high-[a/Fe], thick disk components do not flare is strong evidence against their thickness being caused by radial migration. The correspondence between the radial structure and chemical-enrichment age of stellar populations is clear confirmation of the inside-out growth of galactic disks. The details of these relations will constrain the variety of physical conditions under which stars form throughout the MW disk.

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arXiv2016

Action-based Dynamical Modelling for the Milky Way Disk

We present RoadMapping, a full-likelihood dynamical modelling machinery that aims to recover the Milky Way's (MW) gravitational potential from large samples of stars in the Galactic disk. RoadMapping models the observed positions and velocities of stars with a parametrized, three-integral distribution function (DF) in a parametrized axisymmetric potential. We investigate through differential test cases with idealized mock data how the breakdown of model assumptions and data properties affect constraints on the potential and DF. Our key results are: (i) If the MW's true potential is not included in the assumed model potential family, we can - in the axisymmetric case - still find a robust estimate for the potential, with only <~ 10% difference in surface density within |z| <= 1.1 kpc inside the observed volume. (ii) Modest systematic differences between the true and model DF are inconsequential. E.g, when binning stars to define sub-populations with simple DFs, binning errors do not affect the modelling as long as the DF parameters of neighbouring bins differ by < 20%. In addition, RoadMapping ensures unbiased potential estimates for either (iii) small misjudgements of the spatial selection function (i.e., <~ 15% at the survey volume's edge), (iv) if distances are known to within 10%, or (v) if proper motion uncertainties are known within 10% or are smaller than delta mu <~ 1 mas/yr. Challenges are the rapidly increasing computational costs for large sample sizes. Overall, RoadMapping is well suited to making precise new measurements of the MW's potential with data from the upcoming Gaia releases.

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arXiv2016

Accelerated Fitting of Stellar Spectra

Stellar spectra are often modeled and fit by interpolating within a rectilinear grid of synthetic spectra to derive the stars' labels: stellar parameters and elemental abundances. However, the number of synthetic spectra needed for a rectilinear grid grows exponentially with the label space dimensions, precluding the simultaneous and self-consistent fitting of more than a few elemental abundances. Shortcuts such as fitting subsets of labels separately can introduce unknown systematics and do not produce correct error covariances in the derived labels. In this paper we present a new approach -- CHAT (Convex Hull Adaptive Tessellation) -- which includes several new ideas for inexpensively generating a sufficient stellar synthetic library, using linear algebra and the concept of an adaptive, data-driven grid. A convex hull approximates the region where the data lie in the label space. A variety of tests with mock datasets demonstrate that CHAT can reduce the number of required synthetic model calculations by three orders of magnitude in an 8D label space. The reduction will be even larger for higher-dimensional label spaces. In CHAT the computational effort increases only linearly with the number of labels that are fit simultaneously. Around each of these grid points in label space an approximate synthetic spectrum can be generated through linear expansion using a set of "gradient spectra" that represent flux derivatives at every wavelength point with respect to all labels. These techniques provide new opportunities to fit the full stellar spectra from large surveys with 15-30 labels simultaneously.

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arXiv2016

Constructing Polynomial Spectral Models for Stars

Stellar spectra depend on the stellar parameters and on dozens of photospheric elemental abundances. Simultaneous fitting of these $\mathcal{N}\sim 10-40$ model labels to observed spectra has been deemed unfeasible, because the number of ab initio spectral model grid calculations scales exponentially with $\mathcal{N}$. We suggest instead the construction of a polynomial spectral model (PSM) of order $\mathcal{O}$ for the model flux at each wavelength. Building this approximation requires a minimum of only ${\mathcal{N}+\mathcal{O}\choose\mathcal{O}}$ calculations: e.g. a quadratic spectral model ($\mathcal{O}=2$) to fit $\mathcal{N}=20$ labels simultaneously, can be constructed from as few as $231$ ab initio spectral model calculations; in practice, a somewhat larger number ($\sim 300-1000$) of randomly chosen models lead to a better performing PSM. Such a PSM can be a good approximation only over a portion of label space, which will vary case by case. Yet, taking the APOGEE survey as an example, a single quadratic PSM provides a remarkably good approximation to the exact ab initio spectral models across much of this survey: for random labels within that survey the PSM approximates the flux to within $10^{-3}$, and recovers the abundances to within $\sim 0.02$ dex rms of the exact models. This enormous speed-up enables the simultaneous many-label fitting of spectra with computationally expensive ab initio models for stellar spectra, such as non-LTE models. A PSM also enables the simultaneous fitting of observational parameters, such as the spectrum's continuum or line-spread function.

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arXiv2016

Mapping the Monoceros Ring in 3D with Pan-STARRS1

Using the Pan-STARRS1 survey, we derive limiting magnitude, spatial completeness and density maps that we use to probe the three dimensional structure and estimate the stellar mass of the so-called Monoceros Ring. The Monoceros Ring is an enormous and complex stellar sub-structure in the outer Milky Way disk. It is most visible across the large Galactic Anticenter region, 120 < l < 240 degrees, -30 < b < +40 degrees. We estimate its stellar mass density profile along every line of sight in 2 X 2 degree pixels over the entire 30,000 square degree Pan-STARRS1 survey using the previously developed MATCH software. By parsing this distribution into a radially smooth component and the Monoceros Ring, we obtain its mass and distance from the Sun along each relevant line of sight. The Monoceros Ring is significantly closer to us in the South (6 kpc) than in the North (9 kpc). We also create 2D cross sections parallel to the Galactic plane that show 135 degrees of the Monoceros Ring in the South and 170 degrees of the Monoceros Ring in the North. We show that the Northern and Southern structures are also roughly concentric circles, suggesting that they may be a wave rippling from a common origin. Excluding the Galactic plane, we observe an excess stellar mass of 4 million solar masses across 120 < l < 240 degrees. If we interpolate across the Galactic plane, we estimate that this region contains 8 million solar masses. If we assume (somewhat boldly) that the Monoceros Ring is a set of two Galactocentric rings, its total stellar mass is 60 million solar masses. Finally, if we assume that it is a set of two circles centered at a point 4 kpc from the Galactic center in the anti-central direction, as our data suggests, we estimate its stellar mass to be 40 million solar masses.

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arXiv2016

The Cannon 2: A data-driven model of stellar spectra for detailed chemical abundance analyses

We have shown that data-driven models are effective for inferring physical attributes of stars (labels; Teff, logg, [M/H]) from spectra, even when the signal-to-noise ratio is low. Here we explore whether this is possible when the dimensionality of the label space is large (Teff, logg, and 15 abundances: C, N, O, Na, Mg, Al, Si, S, K, Ca, Ti, V, Mn, Fe, Ni) and the model is non-linear in its response to abundance and parameter changes. We adopt ideas from compressed sensing to limit overall model complexity while retaining model freedom. The model is trained with a set of 12,681 red-giant stars with high signal-to-noise spectroscopic observations and stellar parameters and abundances taken from the APOGEE Survey. We find that we can successfully train and use a model with 17 stellar labels. Validation shows that the model does a good job of inferring all 17 labels (typical abundance precision is 0.04 dex), even when we degrade the signal-to-noise by discarding ~50% of the observing time. The model dependencies make sense: the spectral derivatives with respect to abundances correlate with known atomic lines, and we identify elements belonging to atomic lines that were previously unknown. We recover (anti-)correlations in abundance labels for globular cluster stars, consistent with the literature. However we find the intrinsic spread in globular cluster abundances is 3--4 times smaller than previously reported. We deliver 17 labels with associated errors for 87,563 red giant stars, as well as open-source code to extend this work to other spectroscopic surveys.

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arXiv2016

Ages of massive galaxies at $0.5 < z < 2.0$ from 3D-HST rest-frame optical spectroscopy

We present low-resolution near-infrared stacked spectra from the 3D-HST survey up to $z=2.0$ and fit them with commonly used stellar population synthesis models: BC03 (Bruzual & Charlot, 2003), FSPS10 (Flexible Stellar Population Synthesis, Conroy & Gunn 2010), and FSPS-C3K (Conroy, Kurucz, Cargile, Castelli, in prep). The accuracy of the grism redshifts allows the unambiguous detection of many emission and absorption features, and thus a first systematic exploration of the rest-frame optical spectra of galaxies up to $z=2$. We select massive galaxies ($\rm log(M_{*} / M_{\odot}) > 10.8$), we divide them into quiescent and star-forming via a rest-frame color-color technique, and we median-stack the samples in 3 redshift bins between $z=0.5$ and $z=2.0$. We find that stellar population models fit the observations well at wavelengths below $\rm 6500 Å$ rest-frame, but show systematic residuals at redder wavelengths. The FSPS-C3K model generally provides the best fits (evaluated with a $χ^2_{red}$ statistics) for quiescent galaxies, while BC03 performs the best for star-forming galaxies. The stellar ages of quiescent galaxies implied by the models, assuming solar metallicity, vary from 4 Gyr at $z \sim 0.75$ to 1.5 Gyr at $z \sim 1.75$, with an uncertainty of a factor of 2 caused by the unknown metallicity. On average the stellar ages are half the age of the Universe at these redshifts. We show that the inferred evolution of ages of quiescent galaxies is in agreement with fundamental plane measurements, assuming an 8 Gyr age for local galaxies. For star-forming galaxies the inferred ages depend strongly on the stellar population model and the shape of the assumed star-formation history.

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arXiv2016

Is Draco II one of the faintest dwarf galaxies? First study from Keck/DEIMOS spectroscopy

We present the first spectroscopic analysis of the faint and compact stellar system Draco II (Dra II, M_V=-2.9+/-0.8, r_h=19^{+8}_{-6} pc), recently discovered in the Pan-STARRS1 3π survey. The observations, conducted with DEIMOS on the Keck II telescope, establish some of its basic characteristics: the velocity data reveal a narrow peak with 9 member stars at a systemic heliocentric velocity =-347.6^{+1.7}_{-1.8} km/s, thereby confirming Dra II is a satellite of the Milky Way; we infer a velocity dispersion with σ_{vr}=2.9+/-2.1 km/s (<8.4 km/s at the 95% confidence level), which implies log_{10}(M_{1/2})=5.5^{+0.4}_{-0.6} and log_{10}((M/L)_{1/2})=2.7^{+0.5}_{-0.8}, in Solar units; furthermore, very weak Calcium triplet lines in the spectra of the high signal-to-noise member stars imply [Fe/H]<-2.1, whilst variations in the line strengths of two stars with similar colours and magnitudes suggest a metallicity spread in Dra II. These new data cannot clearly discriminate whether Draco II is a star cluster or amongst the faintest, most compact, and closest dwarf galaxies. However, the sum of the three --- individually inconclusive --- pieces of evidence presented here, seems to favour the dwarf galaxy interpretation.

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arXiv2016

Triangulum II: a very metal-poor and dynamically hot stellar system

We present a study of the recently discovered compact stellar system Triangulum II. From observations conducted with the DEIMOS spectrograph on Keck II, we obtained spectra for 13 member stars that follow the CMD features of this very faint stellar system and include two bright red giant branch stars. Tri II has a very negative radial velocity ( =-383.7^{+3.0}_{-3.3} km/s) that translates to ~ -264 km/s and confirms it is a Milky Way satellite. We show that, despite the small data set, there is evidence that Tri II has complex internal kinematics. Its radial velocity dispersion increases from 4.4^{+2.8}_{-2.0} km/s in the central 2' to 14.1^{+5.8}_{-4.2} km/s outwards. The velocity dispersion of the full sample is inferred to be σ_{vr}=9.9^{+3.2}_{-2.2} km/s. From the two bright RGB member stars we measure an average metallicity <[Fe/H]>=-2.6+/-0.2, placing Tri II among the most metal-poor Milky Way dwarf galaxies. In addition, the spectra of the fainter member stars exhibit differences in their line widths that could be the indication of a metallicity dispersion in the system. All these properties paint a complex picture for Tri II, whose nature and current state are largely speculative. The inferred metallicity properties of the system however lead us to favor a scenario in which Tri II is a dwarf galaxy that is either disrupting or embedded in a stellar stream.

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arXiv2015

Red giant masses and ages derived from carbon and nitrogen abundances

We show that the masses of red giant stars can be well predicted from their photospheric carbon and nitrogen abundances, in conjunction with their spectroscopic stellar labels log g, Teff, and [Fe/H]. This is qualitatively expected from mass-dependent post main sequence evolution. We here establish an empirical relation between these quantities by drawing on 1,475 red giants with asteroseismic mass estimates from Kepler that also have spectroscopic labels from APOGEE DR12. We assess the accuracy of our model, and find that it predicts stellar masses with fractional r.m.s. errors of about 14% (typically 0.2 Msun). From these masses, we derive ages with r.m.s errors of 40%. This empirical model allows us for the first time to make age determinations (in the range 1-13 Gyr) for vast numbers of giant stars across the Galaxy. We apply our model to 52,000 stars in APOGEE DR12, for which no direct mass and age information was previously available. We find that these estimates highlight the vertical age structure of the Milky Way disk, and that the relation of age with [alpha/M] and metallicity is broadly consistent with established expectations based on detailed studies of the solar neighbourhood.

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arXiv2015

APOGEE chemical tagging constraint on the maximum star cluster mass in the $α$-enhanced Galactic disk

Stars born from the same molecular cloud should be nearly homogeneous in their element abundances. The concept of chemical tagging is to identify members of disrupted clusters by their clustering in element abundance space. Chemical tagging requires large samples of stars with precise abundances for many individual elements. With uncertainties of $σ_{[X/{\rm Fe}]}$ and $σ_{\rm [Fe/H]} \simeq 0.05$ for 10 elements measured for $>10^4$ stars, the APOGEE DR12 spectra may be the first well-suited data set to put this idea into practice. We find that even APOGEE data offer only $\sim 500$ independent volume elements in the 10-dimensional abundance space, when we focus on the $α$-enhanced Galactic disk. We develop and apply a new algorithm to search for chemically homogeneous sets of stars against a dominant background. By injecting star clusters into the APOGEE data set we show that chemically homogeneous clusters with masses $\gtrsim 3 \times 10^7 \, {\rm M}_\odot$ would be easily detectable and yet no such signal is seen in the data. By generalizing this approach, we put a first abundance-based constraint on the cluster mass function for the old disk stars in the Milky Way.

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arXiv2015

Finding, characterizing and classifying variable sources in multi-epoch sky surveys: QSOs and RR Lyrae in PS1 3$π$ data

In area and depth, the Pan-STARRS1 (PS1) 3$π$ survey is unique among many-epoch, multi-band surveys and has enormous potential for all-sky identification of variable sources. PS1 has observed the sky typically seven times in each of its five bands ($grizy$) over 3.5 years, but unlike SDSS not simultaneously across the bands. Here we develop a new approach for quantifying statistical properties of non-simultaneous, sparse, multi-color lightcurves through light-curve structure functions, effectively turning PS1 into a $\sim 35$-epoch survey. We use this approach to estimate variability amplitudes and timescales $(ω_r, τ)$ for all point-sources brighter than $r_{\mathrm{P1}}=21.5$ mag in the survey. With PS1 data on SDSS Stripe 82 as ``ground truth", we use a Random Forest Classifier to identify QSOs and RR Lyrae based on their variability and their mean PS1 and WISE colors. We find that, aside from the Galactic plane, QSO and RR Lyrae samples of purity $\sim$75\% and completeness $\sim$92\% can be selected. On this basis we have identified a sample of $\sim 1,000,000$ QSO candidates, as well as an unprecedentedly large and deep sample of $\sim$150,000 RR Lyrae candidates with distances from $\sim$10 kpc to $\sim$120 kpc. Within the Draco dwarf spheroidal, we demonstrate a distance precision of 6\% for RR Lyrae candidates. We provide a catalog of all likely variable point sources and likely QSOs in PS1, a total of $25.8\times 10^6$ sources.

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