INTERSTELLAR MEDIUM

Magnetic seismology of interstellargas clouds: Unveiling ahidden dimensionAris Tritsis1,2,3* and Konstantinos Tassis1,3*

Stars and planets are formed inside dense interstellarmolecular clouds by processes imprintedon the three-dimensional (3D) morphology of the clouds. Determining the 3D structure ofinterstellar clouds remains challenging because of projection effects and difficulties measuringthe extent of the clouds along the line of sight.We report the detection of normal vibrationalmodes in the isolated interstellar cloud Musca, allowing determination of the 3D physicaldimensions of the cloud.We found that Musca is vibrating globally, with the characteristicmodes of a sheet viewed edge on, not the characteristics of a filament as previously supposed.We reconstructed the physical properties of Musca through 3D magnetohydrodynamicsimulations, reproducing the observed normal modes and confirming a sheetlike morphology.

Astronomical objects are seen in two-dimensional projection on the plane ofthe sky. This is particularly problematicfor studies of the interstellar medium (ISM),because the three-dimensional (3D) struc-

ture of interstellar clouds encodes informationregarding the physical processes (such as mag-netic forces, turbulence, and gravity) that domi-nate the formation of stars and planets. We seeka solution to this problem by searching for res-onant magnetohydrodynamic (MHD) vibrationsin an isolated interstellar cloud and by analyz-ing its normal modes. Normal modes have beenused extensively to describe and analyze varioussystems in the physical sciences, from quantummechanics and helioseismology to geophysicsand structural biology. Normal modes have beenobserved in the ISM in two small pulsating con-densations (Bok globules) located inside two mo-lecular clouds (interstellar clouds dense enoughto allow the formation of molecular hydrogen)(1, 2). Further applications have been limited be-cause molecular clouds usually exhibit a complexmorphology, including filamentary structures,as a result of turbulent mixing and shock inter-action (3, 4).Recent wide-field radio observations of mo-

lecular clouds (5) have unveiled the presence ofwell-ordered, quasi-periodically spaced elonga-tions, termed striations, on the outskirts of clouds.The thermal dust continuum emission survey ofnearby molecular clouds by the Herschel SpaceObservatory has shown that striations are a com-mon feature of clouds (6–10), often associated withdenser filaments (7–11) inside which stars are

formed. Complementary polarimetric studieshave revealed that striations are always wellaligned with the cloud’s magnetic field projectedonto the plane of the sky (5, 7–12).From a theoretical perspective, the only viable

mechanism for the formation of striations in-volves the excitation of fast magnetosonic waves(longitudinal magnetic pressure waves) (13). Com-pressible fast magnetosonic waves can be excitedby nonlinear coupling with Alfvén waves (incom-pressible transverse waves along magnetic fieldlines) and/or perturbations created by self-gravityin an inhomogeneous medium. These magneto-sonic waves compress the gas and form orderedstructures parallel to magnetic field lines, in agree-ment with observations of striations (5, 7–12).Once magnetosonic waves are excited, they

can be reflected in regions of varying Alfvén speed(defined as nA ¼ B=

ffiffiffiffiffiffiffiffi4pr

p, where B is the mag-

netic field and r is the density of the medium),

setting up normal modes, just like vibrations ina resonating chamber. In regions where stria-tions appear to be unassociated with denser struc-tures (such as in H I clouds), this resonatingchamber may be the result of external pressureconfinement by a more diffuse, warmer medium.However, boundaries can also be naturally created,in the case of a contracting self-gravitating cloud,as a result of steep changes in density and mag-netic field that in turn lead to sharp variationsin the velocity of propagation of these waves (14).Any compressible fast magnetosonic waves ex-cited during the formation of the cloud will thenbe trapped, thus resulting in striations in thevicinity of denser structures.Fast magnetosonic waves traveling in both

directions perpendicular to the magnetic fieldare coupled (13). By considering a rectangularbox, we can express the spatial frequency k ofeach normal mode (n, m) as

knm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipnLx

� �2

þ pmLy

� �2s

ð1Þ

where the ordered component of the magneticfield is considered to be along the z axis and Lxand Ly are the lengths of the box along the x andy axes, respectively, with n andm being integersranging from zero to infinity. By considering arotation matrix, we can show that the spatialfrequencies seen in the power spectra of cutsperpendicular to the long axis of striations areindependent of the orientation of the cloud (14).We analyzed these magnetohydrodynamic

striations seen in Musca (designated G301.70-7.16), a molecular cloud located ~150 to 200 pcfrom Earth (15, 16). Because of its elongated andordered morphology and its low column density(the integrated volume density along the line ofsight), Musca is considered to be the prototypeof a filamentary (cylindrical) molecular cloud(9, 17–20) and is used as a comparison by manytheoretical models. Musca has been mapped by

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1Department of Physics and Institute of Theoretical andComputational Physics, University of Crete, P.O. Box 2208, 71003Heraklion, Crete, Greece. 2Research School of Astronomy andAstrophysics, Australian National University, Canberra, ACT 2611,Australia. 3Institute of Electronic Structure and Laser and Instituteof Astrophysics, Foundation for Research and Technology–Hellas,P.O. Box 1527, 71110 Heraklion, Crete, Greece.*Corresponding author. Email: tassis@physics.uoc.gr (K.T.); aris.tritsis@anu.edu.au (A.T.)

Fig. 1.The Musca molecular cloud.This Herschel 250-mm dust emission map of the Musca molecularcloud [with intensity expressed in megajanskies (MJy) per steradian] shows both striations andthe dense elongated structure. The green rectangle marks the region where we have performed ournormal-mode analysis, and the blue arrow shows the mean direction of the magnetic field projectedonto the plane of the sky (9). Grid lines show equatorial coordinates.

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Herschel as part of the Gould Belt Survey (9) andexhibits clear striations oriented perpendicu-larly to the main body of the cloud. We have re-analyzed the archival data (9); Fig. 1 shows theHerschel Spectral and Photometric Imaging Re-ceiver 250-mm dust emission map of Musca. Wehave considered cuts perpendicular to the longaxis of striations inside the green rectangle inFig. 1 in order to study their spatial power spectra.We have verified that our selection does not in-troduce biases by considering cuts perpendicularto the long axis of the striations and studying theirspatial power spectra in other regions as well (14).The normalized power spectra from each

cut and the distribution of the identified peaksare shown in Fig. 2, A and B, respectively. FromEq. 1 and the assumption that Lx is the largestdimension of the cloud, the smallest possiblewave number is obtained for (n, m) = (1, 0).Thus, the first peak in Fig. 2B has to correspondto (n, m) = (1, 0), yielding Lx = 8.2 ± 0.3 pc. Thisvalue is consistent with the observed size ofthe cloud on the plane of the sky, which is var-iously reported to be from 6.5 to 7.85 pc whenscaled to our adopted cloud distance of 150 pc(9, 18, 21). The second peak could correspond toeither (n, m) = (0, 1) or, in the case of a cylindri-cal cloud withLx≫Ly (and thusk0:1≫1), to (n,m) =(2, 0). However, with Lx ~ 8 pc, the (n,m) = (2, 0)peak is expected at k ~ 0.8 (pc)−1, much higherthan the actual location of the second peak. Thus,this second peak has to correspond to (n, m) =(0, 1). By inserting (n, m) = (0, 1) and the valueof the second peak in Eq. 1, we deduce the hid-den, line-of-sight dimension Ly to be 6.2 ± 0.2 pc,comparable to the largest dimension of thecloud. The other normal modes with their un-certainties determined through error propaga-tion are predicted analytically by inserting thesevalues for Lx and Ly into Eq. 1 and are over-plotted in Fig. 2B. Therefore, Musca, previouslyconsidered to be a prototypical filamentary cloud,is instead a sheetlike structure seen edge on.In Fig. 2B, we plot all the normal modes up to

(n, m) = (2, 2). We find good agreement betweenthe predicted wave numbers and observationsup to the first few modes, with n or m = 4 cor-responding to physical scales of ~1.6 pc (fig. S1).However, the shape of the cloud is more compli-cated than an idealized rectangle, exhibitinghigher-order structure on smaller scales, so thenormal modes may be better modeled by a rect-angle with rounded edges or an ellipse. Thus,Eq. 1 is an approximation that applies only tothe normal modes with small spatial frequen-cies (i.e., large physical scales). At spatial fre-quencies higher than ~2 (pc)−1, the density ofnormal modes becomes so high that they can-not be identified in either the observations orthe theoretical predictions (the uncertaintiesoverlap for all predicted modes).Through ideal (nondissipative) MHD simu-

lations including self-gravity (14), we have con-structed a 3D model of Musca, including thedense structure and striations in the low-densityparts. In Fig. 3 we show the column density mapfrom our simulation, which reproduces the ob-

served dimensions of the cloud. A 3D representa-tion of the volume density of the model of Muscais shown in Fig. 4. As intuitively expected fromthe normal-mode analysis of the observations,the shape of the cloud is that of a rectangle withrounded edges.The maximum column density in the simula-

tion, from an edge-on view, is 1.9 × 1022 cm−2.For comparison, the maximum column densityderived observationally from the dust emissionmaps (9) is ~1.6 × 1022 cm−2. The maximumvolume number density in the simulation is~2 × 103 cm−3, high enough for molecules to becollisionally excited and therefore observed viatheir rotational emission lines. Molecular lineobservations of the Musca molecular cloud arelimited to CO, including several isotopologues,and NH3 (17–20); the latter is observed only to-ward the densest core of Musca. The numberdensities required to excite CO and NH3 linesare ~102 and 103 cm−3, respectively (22), whichare easily reached in our simulated model of thecloud. To reproduce the observed column densityin any filamentlike geometry, in contrast to thesheetlike structure, the number density has to be~5 × 104 cm−3 or higher (18). This value is wellabove the density threshold for star formationfor clouds in the Gould Belt and a density thresh-old derived specifically for Musca (23). More evi-dent star formation activity would be observed if

Musca was a filament. Moreover, if the 3D shapeof Musca was that of a filament, NH3 would beeasily excited and observed throughout the ridgeof the dense structure.We used a suite of simulations of clouds of

different shapes to validate our analysis andverify that Eq. 1 can be used to extract the cor-rect cloud dimensions (14). In each of our sim-ulations, the known dimensions of the cloudswere recovered by the simulated normal-modeanalysis. In contrast to the distribution of peaksseen in Fig. 2B, in cylindrical clouds (Ly≪ Lx) thefirst few peaks at low spatial frequencies are allmultiples of the first peak. The first few peaksfor cylindrical clouds are due only to the largestdimension of the cloud, resulting in a sparserdistribution of peaks than the sheet geometry(fig. S4). This is both quantitatively and qual-itatively different from the distribution seen inthe Musca data (Fig. 2), strengthening the casethat the intrinsic shape of Musca is sheetlike.Sheetlike structures are common in turbulentclouds, as they may represent planarlike shocksfrom processes such as supernova explosionsor expanding ionization regions or may resultsimply from accretion along magnetic field lines(4, 24, 25).For decades, thedetermination of the 3D shapes

of clouds has been pursued through statisticalstudies (26–28), which do not provide information

Tritsis et al., Science 360, 635–638 (2018) 11 May 2018 2 of 3

A B

Fig. 2. Comparison of observed normal modes with the analytical solution. (A) Normalizedpower spectra (black lines) of cuts through the observations perpendicular to the striations. Peakswe identified are marked with red dots. Fk, normalized spectral power density. (B) Distribution ofpeaks at different spatial frequencies. The red lines depict the values used to derive the dimensionsof the cloud. The blue dashed lines show the rest of the normal modes [up to (n, m) = (2, 0)],predicted analytically from Eq. 1 given the cloud dimensions derived from the first two peaks. Shadedregions indicate the 1s regions of the analytical predictions due to uncertainties in the determinationof the locations of the first two peaks, propagated through Eq. 1. The bin size is comparable to thestandard deviations of the points constituting the first two peaks. Npeaks, number of peaks.

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on a cloud-by-cloud basis. Other proposedmethods(29, 30) rely on complex chemical and/or radia-tive processes and thus depend on numerousassumptions. With its 3D geometry now deter-mined, Musca can be used to test theoreticalmodels of interstellar clouds.

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(2014).24. F. Nakamura, Z.-Y. Li, Astrophys. J. 687, 354–375 (2008).25. T. C. Mouschovias, Astrophys. J. 373, 169–186 (1991).26. P. C. Myers, G. A. Fuller, A. A. Goodman, P. J. Benson,

Astrophys. J. 376, 561–572 (1991).27. C. E. Jones, S. Basu, J. Dubinski, Astrophys. J. 551, 387–393 (2001).28. K. Tassis, C. D. Dowell, R. H. Hildebrand, L. Kirby, J. E. Vaillancourt,

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ACKNOWLEDGMENTS

We thank V. Pavlidou, G. Panopoulou, V. Charmandaris, N. Kylafis,A. Zezas, E. Economou, J. Andrews, S. Williams, P. Sell, D. Blinov,I. Liodakis, T. Mouschovias, and the three anonymous refereesfor comments that helped improve this paper. Funding: K.T. andA.T. acknowledge support by the Seventh Framework Programmethrough Marie Curie Career Integration grant PCIG-GA-2011-293531,“Onset of Star Formation: Connecting Theory and Observations.” A.T.acknowledges funding from the European Research Council (ERC)under the European Union’s Seventh Framework Programme(FP/2007-2013)/ERC grant agreement 617001. Usage of the MetropolisHPC Facility at the Crete Center for Quantum Complexity andNanotechnology of the University of Crete, supported by the EuropeanUnion Seventh Framework Programme (FP7-REGPOT-2012-2013-1)under grant agreement 316165, is acknowledged. Authorcontributions: A.T. performed the numerical simulations and theanalysis of the observations and wrote the text. K.T. contributedto the interpretation of the results and the writing of the text.Competing interests: The authors declare no conflicts of interest.Data and materials availability: All observational data used inthis research are from the Herschel Gould Belt Survey project and arepublicly available at http://archives.esac.esa.int/hsa/whsa/#home(observation ID, 1342216012). All simulation outputs and setupfiles are available at https://doi.org/10.6084/m9.figshare.5950360.The FLASH software used in this work was developed in part bythe Advanced Simulation and Computing Program of the NationalNuclear Security Administration, U.S. Department of Energy, atthe Flash Center for Computational Science at the University ofChicago and was obtained from http://flash.uchicago.edu/site/. Theyt analysis toolkit is available at http://yt-project.org/#getyt, andMayavi2 is available at https://github.com/enthought/mayavi.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/360/6389/635/suppl/DC1Materials and MethodsFigs. S1 to S4References (31–43)

15 June 2017; accepted 14 March 201810.1126/science.aao1185

Tritsis et al., Science 360, 635–638 (2018) 11 May 2018 3 of 3

Fig. 3. Column density map of the model of the Musca molecular cloud. The map is an edge-onview of the molecular gas column density from our MHD simulation of a sheetlike structure. Thecolor bar shows the logarithm of the column density. The purple contour marks the region with N(H2) >2 × 1021 cm−2, used to identify Musca’s main, dense filament (9). The magnetic field is along thez axis, and the time of the snapshot since the beginning of the simulation is ~2.7 million years.

Fig. 4. 3D model of the Musca molecular cloud. The model shows the logarithmic 3D volumedensity in our MHD simulation of the Musca cloud. Density isosurfaces are set at 90, 75, 70, and 55%of the logarithm of the maximum number density. Black lines represent the magnetic field.

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Magnetic seismology of interstellar gas clouds: Unveiling a hidden dimensionAris Tritsis and Konstantinos Tassis

DOI: 10.1126/science.aao1185 (6389), 635-638.360Science 

, this issue p. 635Sciencestructure and shows that Musca is a sheet seen edge-on, not a filament as previously assumed.Tassis discovered that the cloud is vibrating with magnetohydrodynamic waves. The pattern of vibrations reveals the 3D 2D projection of them onto the sky. While examining far-infrared observations of the nearby Musca cloud, Tritsis andwhich stars form. Determining the three-dimensional (3D) morphology of these clouds is difficult because we only see a

Molecular clouds are relatively dense assemblies of interstellar dust and gas (mostly molecular hydrogen) fromA vibrating molecular cloud in three dimensions

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