Estimation of the tilt of the stellar velocity ellipsoid from RAVE and implications for mass models
Identifieur interne : 002920 ( Istex/Corpus ); précédent : 002919; suivant : 002921Estimation of the tilt of the stellar velocity ellipsoid from RAVE and implications for mass models
Auteurs : A. Siebert ; O. Bienaymé ; J. Binney ; J. Bland-Hawthorn ; R. Campbell ; K. C. Freeman ; B. K. Gibson ; G. Gilmore ; E. K. Grebel ; A. Helmi ; U. Munari ; J. F. Navarro ; Q. A. Parker ; G. Seabroke ; A. Siviero ; M. Steinmetz ; M. Williams ; R. F. G. Wyse ; T. ZwitterSource :
- Monthly Notices of the Royal Astronomical Society [ 0035-8711 ] ; 2008-12-01.
Abstract
We present a measure of the inclination of the velocity ellipsoid at 1 kpc below the Galactic plane using a sample of red clump giants from the RAdial Velocity Experiment (RAVE) Data Release 2. We find that the velocity ellipsoid is tilted towards the Galactic plane with an inclination of . We compare this value to computed inclinations for two mass models of the Milky Way. We find that our measurement is consistent with a short scalelength of the stellar disc (Rd≃ 2 kpc) if the dark halo is oblate or with a long scalelength (Rd≃ 3 kpc) if the dark halo is prolate. Once combined with independent constraints on the flattening of the halo, our measurement suggests that the scalelength is approximately halfway between these two extreme values, with a preferred range 2.5–2.7 kpc for a nearly spherical halo. Nevertheless, no model can be clearly ruled out. With the continuation of the RAVE survey, it will be possible to provide a strong constraint on the mass distribution of the Milky Way using refined measurements of the orientation of the velocity ellipsoid.
Url:
DOI: 10.1111/j.1365-2966.2008.13912.x
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Siebert</name><affiliation><mods:affiliation>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</mods:affiliation></affiliation><affiliation><mods:affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: siebert@astro.u-strasbg.fr</mods:affiliation></affiliation><affiliation><mods:affiliation></mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: siebert@astro.u-strasbg.fr</mods:affiliation></affiliation></author><author><name sortKey="Bienayme, O" sort="Bienayme, O" uniqKey="Bienayme O" first="O." last="Bienaymé">O. Bienaymé</name><affiliation><mods:affiliation>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</mods:affiliation></affiliation></author><author><name sortKey="Binney, J" sort="Binney, J" uniqKey="Binney J" first="J." last="Binney">J. 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Freeman</name><affiliation><mods:affiliation>Australian Natianal University, Canberra, Australia</mods:affiliation></affiliation></author><author><name sortKey="Gibson, B K" sort="Gibson, B K" uniqKey="Gibson B" first="B. K." last="Gibson">B. K. Gibson</name><affiliation><mods:affiliation>University of Central Lancashire, Preston</mods:affiliation></affiliation></author><author><name sortKey="Gilmore, G" sort="Gilmore, G" uniqKey="Gilmore G" first="G." last="Gilmore">G. Gilmore</name><affiliation><mods:affiliation>Institute of Astronomy, Cambridge</mods:affiliation></affiliation></author><author><name sortKey="Grebel, E K" sort="Grebel, E K" uniqKey="Grebel E" first="E. K." last="Grebel">E. K. Grebel</name><affiliation><mods:affiliation>Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Heidelberg, Germany</mods:affiliation></affiliation></author><author><name sortKey="Helmi, A" sort="Helmi, A" uniqKey="Helmi A" first="A." last="Helmi">A. Helmi</name><affiliation><mods:affiliation>Kapteyn Astronomical Institut, Groningen, the Netherlands</mods:affiliation></affiliation></author><author><name sortKey="Munari, U" sort="Munari, U" uniqKey="Munari U" first="U." last="Munari">U. Munari</name><affiliation><mods:affiliation>Astronomical Observatory of Padova in Asiago, Asiago, Italy</mods:affiliation></affiliation></author><author><name sortKey="Navarro, J F" sort="Navarro, J F" uniqKey="Navarro J" first="J. F." last="Navarro">J. F. Navarro</name><affiliation><mods:affiliation>University of Victoria, Victoria, Canada</mods:affiliation></affiliation></author><author><name sortKey="Parker, Q A" sort="Parker, Q A" uniqKey="Parker Q" first="Q. A." last="Parker">Q. A. Parker</name><affiliation><mods:affiliation>Anglo-Australian Observatory, Sydney, Australia</mods:affiliation></affiliation><affiliation><mods:affiliation>Macquary University, Sydney, Australia</mods:affiliation></affiliation></author><author><name sortKey="Seabroke, G" sort="Seabroke, G" uniqKey="Seabroke G" first="G." last="Seabroke">G. Seabroke</name><affiliation><mods:affiliation>Institute of Astronomy, Cambridge</mods:affiliation></affiliation><affiliation><mods:affiliation>e2v Centre for Electronic Imaging, Planetary and Space Sciences Research Institute, The Open University, Milton Keynes</mods:affiliation></affiliation></author><author><name sortKey="Siviero, A" sort="Siviero, A" uniqKey="Siviero A" first="A." last="Siviero">A. Siviero</name><affiliation><mods:affiliation>Astronomical Observatory of Padova in Asiago, Asiago, Italy</mods:affiliation></affiliation></author><author><name sortKey="Steinmetz, M" sort="Steinmetz, M" uniqKey="Steinmetz M" first="M." last="Steinmetz">M. Steinmetz</name><affiliation><mods:affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</mods:affiliation></affiliation></author><author><name sortKey="Williams, M" sort="Williams, M" uniqKey="Williams M" first="M." last="Williams">M. Williams</name><affiliation><mods:affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</mods:affiliation></affiliation><affiliation><mods:affiliation>Australian Natianal University, Canberra, Australia</mods:affiliation></affiliation></author><author><name sortKey="Wyse, R F G" sort="Wyse, R F G" uniqKey="Wyse R" first="R. F. G." last="Wyse">R. F. G. Wyse</name><affiliation><mods:affiliation>Johns Hopkins University, Baltimore, MD, USA</mods:affiliation></affiliation></author><author><name sortKey="Zwitter, T" sort="Zwitter, T" uniqKey="Zwitter T" first="T." last="Zwitter">T. Zwitter</name><affiliation><mods:affiliation>Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Monthly Notices of the Royal Astronomical Society</title><title level="j" type="abbrev">Monthly Notices of the Royal Astronomical Society</title><idno type="ISSN">0035-8711</idno><idno type="eISSN">1365-2966</idno><imprint><publisher>Blackwell Publishing Ltd</publisher><pubPlace>Oxford, UK</pubPlace><date type="published" when="2008-12-01">2008-12-01</date><biblScope unit="volume">391</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="793">793</biblScope><biblScope unit="page" to="801">801</biblScope></imprint><idno type="ISSN">0035-8711</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0035-8711</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract">We present a measure of the inclination of the velocity ellipsoid at 1 kpc below the Galactic plane using a sample of red clump giants from the RAdial Velocity Experiment (RAVE) Data Release 2. We find that the velocity ellipsoid is tilted towards the Galactic plane with an inclination of . We compare this value to computed inclinations for two mass models of the Milky Way. We find that our measurement is consistent with a short scalelength of the stellar disc (Rd≃ 2 kpc) if the dark halo is oblate or with a long scalelength (Rd≃ 3 kpc) if the dark halo is prolate. Once combined with independent constraints on the flattening of the halo, our measurement suggests that the scalelength is approximately halfway between these two extreme values, with a preferred range 2.5–2.7 kpc for a nearly spherical halo. Nevertheless, no model can be clearly ruled out. With the continuation of the RAVE survey, it will be possible to provide a strong constraint on the mass distribution of the Milky Way using refined measurements of the orientation of the velocity ellipsoid.</div></front></TEI><istex><corpusName>oup</corpusName><author><json:item><name>A. Siebert</name><affiliations><json:string>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</json:string><json:string>Astrophysikalishes Institut Potsdam, Potsdam, Germany</json:string><json:string>E-mail: siebert@astro.u-strasbg.fr</json:string><json:null></json:null><json:string>E-mail: siebert@astro.u-strasbg.fr</json:string></affiliations></json:item><json:item><name>O. Bienaymé</name><affiliations><json:string>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</json:string></affiliations></json:item><json:item><name>J. Binney</name><affiliations><json:string>Rudolf Peierls Centre for Theoretical Physics, Oxford</json:string></affiliations></json:item><json:item><name>J. Bland-Hawthorn</name><affiliations><json:string>Anglo-Australian Observatory, Sydney, Australia</json:string></affiliations></json:item><json:item><name>R. Campbell</name><affiliations><json:string>Astrophysikalishes Institut Potsdam, Potsdam, Germany</json:string><json:string>Macquary University, Sydney, Australia</json:string></affiliations></json:item><json:item><name>K. C. Freeman</name><affiliations><json:string>Australian Natianal University, Canberra, Australia</json:string></affiliations></json:item><json:item><name>B. K. Gibson</name><affiliations><json:string>University of Central Lancashire, Preston</json:string></affiliations></json:item><json:item><name>G. Gilmore</name><affiliations><json:string>Institute of Astronomy, Cambridge</json:string></affiliations></json:item><json:item><name>E. K. Grebel</name><affiliations><json:string>Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Heidelberg, Germany</json:string></affiliations></json:item><json:item><name>A. Helmi</name><affiliations><json:string>Kapteyn Astronomical Institut, Groningen, the Netherlands</json:string></affiliations></json:item><json:item><name>U. Munari</name><affiliations><json:string>Astronomical Observatory of Padova in Asiago, Asiago, Italy</json:string></affiliations></json:item><json:item><name>J. F. Navarro</name><affiliations><json:string>University of Victoria, Victoria, Canada</json:string></affiliations></json:item><json:item><name>Q. A. Parker</name><affiliations><json:string>Anglo-Australian Observatory, Sydney, Australia</json:string><json:string>Macquary University, Sydney, Australia</json:string></affiliations></json:item><json:item><name>G. Seabroke</name><affiliations><json:string>Institute of Astronomy, Cambridge</json:string><json:string>e2v Centre for Electronic Imaging, Planetary and Space Sciences Research Institute, The Open University, Milton Keynes</json:string></affiliations></json:item><json:item><name>A. Siviero</name><affiliations><json:string>Astronomical Observatory of Padova in Asiago, Asiago, Italy</json:string></affiliations></json:item><json:item><name>M. Steinmetz</name><affiliations><json:string>Astrophysikalishes Institut Potsdam, Potsdam, Germany</json:string></affiliations></json:item><json:item><name>M. Williams</name><affiliations><json:string>Astrophysikalishes Institut Potsdam, Potsdam, Germany</json:string><json:string>Australian Natianal University, Canberra, Australia</json:string></affiliations></json:item><json:item><name>R. F. G. Wyse</name><affiliations><json:string>Johns Hopkins University, Baltimore, MD, USA</json:string></affiliations></json:item><json:item><name>T. Zwitter</name><affiliations><json:string>Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia</json:string></affiliations></json:item></author><subject><json:item><value>stars: kinematics</value></json:item><json:item><value>galaxy: fundamental parameters</value></json:item><json:item><value>galaxy: kinematics and dynamics</value></json:item></subject><arkIstex>ark:/67375/HXZ-CX6VGRQ8-9</arkIstex><language><json:string>unknown</json:string></language><originalGenre><json:string>research-article</json:string></originalGenre><abstract>We present a measure of the inclination of the velocity ellipsoid at 1 kpc below the Galactic plane using a sample of red clump giants from the RAdial Velocity Experiment (RAVE) Data Release 2. We find that the velocity ellipsoid is tilted towards the Galactic plane with an inclination of . We compare this value to computed inclinations for two mass models of the Milky Way. We find that our measurement is consistent with a short scalelength of the stellar disc (Rd≃ 2 kpc) if the dark halo is oblate or with a long scalelength (Rd≃ 3 kpc) if the dark halo is prolate. Once combined with independent constraints on the flattening of the halo, our measurement suggests that the scalelength is approximately halfway between these two extreme values, with a preferred range 2.5–2.7 kpc for a nearly spherical halo. Nevertheless, no model can be clearly ruled out. With the continuation of the RAVE survey, it will be possible to provide a strong constraint on the mass distribution of the Milky Way using refined measurements of the orientation of the velocity ellipsoid.</abstract><qualityIndicators><score>9.184</score><pdfWordCount>7861</pdfWordCount><pdfCharCount>42045</pdfCharCount><pdfVersion>1.5</pdfVersion><pdfPageCount>9</pdfPageCount><pdfPageSize>595 x 782 pts</pdfPageSize><refBibsNative>true</refBibsNative><abstractWordCount>182</abstractWordCount><abstractCharCount>1071</abstractCharCount><keywordCount>3</keywordCount></qualityIndicators><title>Estimation of the tilt of the stellar velocity ellipsoid from RAVE and implications for mass models</title><genre><json:string>research-article</json:string></genre><host><title>Monthly Notices of the Royal Astronomical Society</title><language><json:string>unknown</json:string></language><issn><json:string>0035-8711</json:string></issn><eissn><json:string>1365-2966</json:string></eissn><publisherId><json:string>mnras</json:string></publisherId><volume>391</volume><issue>2</issue><pages><first>793</first><last>801</last></pages><genre><json:string>journal</json:string></genre><subject><json:item><value>Papers</value></json:item></subject></host><namedEntities><unitex><date><json:string>5000</json:string><json:string>6500</json:string><json:string>1000</json:string><json:string>2008-11-27</json:string></date><geogName></geogName><orgName><json:string>Space Sciences Research Institute, The Open University, Milton Keynes</json:string><json:string>Netherlands Research School for Astronomy, the Natural Sciences and Engineering</json:string><json:string>Astrophysical Institute Potsdam, the Australian Research Council, the German Research</json:string><json:string>Research, the Particle Physics and Astronomy</json:string><json:string>Johns Hopkins University</json:string><json:string>National Institute for Astrophysics</json:string><json:string>The Johns Hopkins University</json:string><json:string>Australian Natianal University</json:string><json:string>Slovenian Research Agency</json:string><json:string>Macquary University</json:string><json:string>University of Victoria</json:string><json:string>University of Central Lancashire</json:string><json:string>Institute of Astronomy, Cambridge</json:string><json:string>Netherlands Organization for Scienti</json:string><json:string>University of Ljubljana</json:string></orgName><orgName_funder></orgName_funder><orgName_provider></orgName_provider><persName><json:string>U. Munari</json:string><json:string>Sky Survey</json:string><json:string>K. C. Freeman</json:string><json:string>A. Siebert</json:string><json:string>F. G. Wyse</json:string><json:string>T. Zwitter</json:string><json:string>F. Navarro</json:string><json:string>A. Siviero</json:string><json:string>B. K. Gibson</json:string><json:string>E. K. Grebel</json:string><json:string>A. Parker</json:string><json:string>A. Helmi</json:string></persName><placeName><json:string>Groningen</json:string><json:string>Germany</json:string><json:string>Ljubljana</json:string><json:string>Australia</json:string><json:string>Heidelberg</json:string><json:string>UK</json:string><json:string>Canada</json:string><json:string>Slovenia</json:string><json:string>Victoria</json:string><json:string>Canberra</json:string><json:string>Bologna</json:string><json:string>USA</json:string><json:string>Strasbourg</json:string><json:string>Baltimore</json:string><json:string>Basel</json:string><json:string>MD</json:string><json:string>Velocity</json:string><json:string>Cambridge</json:string><json:string>Italy</json:string><json:string>Sydney</json:string><json:string>Netherlands</json:string></placeName><ref_url></ref_url><ref_bibl><json:string>Lynden-Bell 1962</json:string><json:string>Kuijken & Gilmore 1989a</json:string><json:string>Grillmair & Johnson 2006</json:string><json:string>Battaglia et al. 2005</json:string><json:string>Martinez-Delgado et al. (2007)</json:string><json:string>Read & Moore 2005</json:string><json:string>Odenkirchen et al. 2003</json:string><json:string>Famaey et al. (2005)</json:string><json:string>Seabroke et al. (2008)</json:string><json:string>Famaey et al.</json:string><json:string>Schlegel, Finkbeiner & Davis 1998</json:string><json:string>Binney 1983</json:string><json:string>Ollongren 1962</json:string><json:string>Ibata et al. 2001</json:string><json:string>Ojha (2001)</json:string><json:string>Hawley (1996)</json:string><json:string>Seabroke & Gilmore 2007</json:string><json:string>Amendt & Cuddeford 1991</json:string><json:string>Newberg et al. (2006)</json:string><json:string>Chereul et al. 1998</json:string><json:string>Majewski et al. (2003)</json:string><json:string>Binney & Tremaine 1987</json:string><json:string>Veltz et al. (2008)</json:string><json:string>Alves 2000</json:string><json:string>Hori & Lui 1963</json:string><json:string>Famaey et al. 2005</json:string><json:string>Helmi 2004</json:string><json:string>A. Siebert et al.</json:string><json:string>Ibata et al. (2001)</json:string><json:string>Cignoni et al. (2008)</json:string><json:string>Fellhauer et al. 2006</json:string><json:string>Zwitter et al. 2008</json:string><json:string>Law et al. (2005)</json:string><json:string>Newberg et al. 2002</json:string><json:string>Seabroke et al. 2008</json:string><json:string>Fellhauer et al. (2006)</json:string><json:string>Statler 1989</json:string><json:string>Belokurov et al. (2006)</json:string><json:string>Juric et al. (2008)</json:string><json:string>Zhao, Qiu & Zhang 2000</json:string></ref_bibl><bibl></bibl></unitex></namedEntities><ark><json:string>ark:/67375/HXZ-CX6VGRQ8-9</json:string></ark><categories><wos><json:string>science</json:string><json:string>astronomy & astrophysics</json:string></wos><scienceMetrix><json:string>natural sciences</json:string><json:string>physics & astronomy</json:string><json:string>astronomy & astrophysics</json:string></scienceMetrix><scopus><json:string>1 - Physical Sciences</json:string><json:string>2 - Earth and Planetary Sciences</json:string><json:string>3 - Space and Planetary Science</json:string><json:string>1 - Physical Sciences</json:string><json:string>2 - Physics and Astronomy</json:string><json:string>3 - Astronomy and Astrophysics</json:string></scopus></categories><publicationDate>2008</publicationDate><copyrightDate>2008</copyrightDate><doi><json:string>10.1111/j.1365-2966.2008.13912.x</json:string></doi><id>DCC5AB80FD5FE4FDE4D62204536B7BC5B9B98A5B</id><score>1</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/DCC5AB80FD5FE4FDE4D62204536B7BC5B9B98A5B/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/DCC5AB80FD5FE4FDE4D62204536B7BC5B9B98A5B/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/DCC5AB80FD5FE4FDE4D62204536B7BC5B9B98A5B/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">Estimation of the tilt of the stellar velocity ellipsoid from RAVE and implications for mass models</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher scheme="https://publisher-list.data.istex.fr">Blackwell Publishing Ltd</publisher><pubPlace>Oxford, UK</pubPlace><availability><licence><p>© 2008 The Authors. Journal compilation © 2008 RAS</p></licence><p scheme="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-GTWS0RDP-M">oup</p></availability><date>2008-11-27</date></publicationStmt><notesStmt><note type="research-article" scheme="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</note><note type="journal" scheme="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">Estimation of the tilt of the stellar velocity ellipsoid from RAVE and implications for mass models</title><author xml:id="author-0000"><persName><forename type="first">A.</forename><surname>Siebert</surname></persName><email>siebert@astro.u-strasbg.fr</email><email>siebert@astro.u-strasbg.fr</email><affiliation>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</affiliation><affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</affiliation><affiliation></affiliation></author><author xml:id="author-0001"><persName><forename type="first">O.</forename><surname>Bienaymé</surname></persName><affiliation>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</affiliation></author><author xml:id="author-0002"><persName><forename type="first">J.</forename><surname>Binney</surname></persName><affiliation>Rudolf Peierls Centre for Theoretical Physics, Oxford</affiliation></author><author xml:id="author-0003"><persName><forename type="first">J.</forename><surname>Bland-Hawthorn</surname></persName><affiliation>Anglo-Australian Observatory, Sydney, Australia</affiliation></author><author xml:id="author-0004"><persName><forename type="first">R.</forename><surname>Campbell</surname></persName><affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</affiliation><affiliation>Macquary University, Sydney, Australia</affiliation></author><author xml:id="author-0005"><persName><forename type="first">K. C.</forename><surname>Freeman</surname></persName><affiliation>Australian Natianal University, Canberra, Australia</affiliation></author><author xml:id="author-0006"><persName><forename type="first">B. K.</forename><surname>Gibson</surname></persName><affiliation>University of Central Lancashire, Preston</affiliation></author><author xml:id="author-0007"><persName><forename type="first">G.</forename><surname>Gilmore</surname></persName><affiliation>Institute of Astronomy, Cambridge</affiliation></author><author xml:id="author-0008"><persName><forename type="first">E. K.</forename><surname>Grebel</surname></persName><affiliation>Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Heidelberg, Germany</affiliation></author><author xml:id="author-0009"><persName><forename type="first">A.</forename><surname>Helmi</surname></persName><affiliation>Kapteyn Astronomical Institut, Groningen, the Netherlands</affiliation></author><author xml:id="author-0010"><persName><forename type="first">U.</forename><surname>Munari</surname></persName><affiliation>Astronomical Observatory of Padova in Asiago, Asiago, Italy</affiliation></author><author xml:id="author-0011"><persName><forename type="first">J. F.</forename><surname>Navarro</surname></persName><affiliation>University of Victoria, Victoria, Canada</affiliation></author><author xml:id="author-0012"><persName><forename type="first">Q. A.</forename><surname>Parker</surname></persName><affiliation>Anglo-Australian Observatory, Sydney, Australia</affiliation><affiliation>Macquary University, Sydney, Australia</affiliation></author><author xml:id="author-0013"><persName><forename type="first">G.</forename><surname>Seabroke</surname></persName><affiliation>Institute of Astronomy, Cambridge</affiliation><affiliation>e2v Centre for Electronic Imaging, Planetary and Space Sciences Research Institute, The Open University, Milton Keynes</affiliation></author><author xml:id="author-0014"><persName><forename type="first">A.</forename><surname>Siviero</surname></persName><affiliation>Astronomical Observatory of Padova in Asiago, Asiago, Italy</affiliation></author><author xml:id="author-0015"><persName><forename type="first">M.</forename><surname>Steinmetz</surname></persName><affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</affiliation></author><author xml:id="author-0016"><persName><forename type="first">M.</forename><surname>Williams</surname></persName><affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</affiliation><affiliation>Australian Natianal University, Canberra, Australia</affiliation></author><author xml:id="author-0017"><persName><forename type="first">R. F. G.</forename><surname>Wyse</surname></persName><affiliation>Johns Hopkins University, Baltimore, MD, USA</affiliation></author><author xml:id="author-0018"><persName><forename type="first">T.</forename><surname>Zwitter</surname></persName><affiliation>Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia</affiliation></author><idno type="istex">DCC5AB80FD5FE4FDE4D62204536B7BC5B9B98A5B</idno><idno type="ark">ark:/67375/HXZ-CX6VGRQ8-9</idno><idno type="DOI">10.1111/j.1365-2966.2008.13912.x</idno></analytic><monogr><title level="j">Monthly Notices of the Royal Astronomical Society</title><title level="j" type="abbrev">Monthly Notices of the Royal Astronomical Society</title><idno type="pISSN">0035-8711</idno><idno type="eISSN">1365-2966</idno><idno type="publisher-id">mnras</idno><idno type="PublisherID-hwp">mnras</idno><imprint><publisher>Blackwell Publishing Ltd</publisher><pubPlace>Oxford, UK</pubPlace><date type="published" when="2008-12-01"></date><biblScope unit="volume">391</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="793">793</biblScope><biblScope unit="page" to="801">801</biblScope></imprint></monogr></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2008-11-27</date></creation><abstract><p>We present a measure of the inclination of the velocity ellipsoid at 1 kpc below the Galactic plane using a sample of red clump giants from the RAdial Velocity Experiment (RAVE) Data Release 2. We find that the velocity ellipsoid is tilted towards the Galactic plane with an inclination of . We compare this value to computed inclinations for two mass models of the Milky Way. We find that our measurement is consistent with a short scalelength of the stellar disc (Rd≃ 2 kpc) if the dark halo is oblate or with a long scalelength (Rd≃ 3 kpc) if the dark halo is prolate. Once combined with independent constraints on the flattening of the halo, our measurement suggests that the scalelength is approximately halfway between these two extreme values, with a preferred range 2.5–2.7 kpc for a nearly spherical halo. Nevertheless, no model can be clearly ruled out. With the continuation of the RAVE survey, it will be possible to provide a strong constraint on the mass distribution of the Milky Way using refined measurements of the orientation of the velocity ellipsoid.</p></abstract><textClass><keywords scheme="keyword"><list><head>keywords</head><item><term>stars: kinematics</term></item><item><term>galaxy: fundamental parameters</term></item><item><term>galaxy: kinematics and dynamics</term></item></list></keywords></textClass><textClass><keywords scheme="Journal Subject"><list><head></head><item><term>Papers</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2008-11-27">Created</change><change when="2008-12-01">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/DCC5AB80FD5FE4FDE4D62204536B7BC5B9B98A5B/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus oup, element #text not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="utf-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//NLM//DTD Journal Publishing DTD v2.3 20070202//EN" URI="journalpublishing.dtd" name="istex:docType"></istex:docType><istex:document><article article-type="research-article">
<front>
<journal-meta>
<journal-id journal-id-type="hwp">mnras</journal-id>
<journal-id journal-id-type="publisher-id">mnras</journal-id>
<journal-title>Monthly Notices of the Royal Astronomical Society</journal-title>
<abbrev-journal-title>Monthly Notices of the Royal Astronomical Society</abbrev-journal-title>
<issn pub-type="ppub">0035-8711</issn>
<issn pub-type="epub">1365-2966</issn>
<publisher>
<publisher-name>Blackwell Publishing Ltd</publisher-name>
<publisher-loc>Oxford, UK</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.1111/j.1365-2966.2008.13912.x</article-id>
<article-categories>
<subj-group>
<subject>Papers</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>Estimation of the tilt of the stellar velocity ellipsoid from RAVE and implications for mass models</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Siebert</surname>
<given-names>A.</given-names>
</name>
<xref ref-type="aff" rid="a1">1</xref><xref ref-type="aff" rid="a2">2</xref><xref ref-type="corresp" rid="c1">*</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Bienaymé</surname>
<given-names>O.</given-names>
</name>
<xref ref-type="aff" rid="a1">1</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Binney</surname>
<given-names>J.</given-names>
</name>
<xref ref-type="aff" rid="a3">3</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Bland-Hawthorn</surname>
<given-names>J.</given-names>
</name>
<xref ref-type="aff" rid="a4">4</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Campbell</surname>
<given-names>R.</given-names>
</name>
<xref ref-type="aff" rid="a2">2</xref><xref ref-type="aff" rid="a5">5</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Freeman</surname>
<given-names>K. C.</given-names>
</name>
<xref ref-type="aff" rid="a6">6</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Gibson</surname>
<given-names>B. K.</given-names>
</name>
<xref ref-type="aff" rid="a7">7</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Gilmore</surname>
<given-names>G.</given-names>
</name>
<xref ref-type="aff" rid="a8">8</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Grebel</surname>
<given-names>E. K.</given-names>
</name>
<xref ref-type="aff" rid="a9">9</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Helmi</surname>
<given-names>A.</given-names>
</name>
<xref ref-type="aff" rid="a10">10</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Munari</surname>
<given-names>U.</given-names>
</name>
<xref ref-type="aff" rid="a11">11</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Navarro</surname>
<given-names>J. F.</given-names>
</name>
<xref ref-type="aff" rid="a12">12</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Parker</surname>
<given-names>Q. A.</given-names>
</name>
<xref ref-type="aff" rid="a4">4</xref><xref ref-type="aff" rid="a5">5</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Seabroke</surname>
<given-names>G.</given-names>
</name>
<xref ref-type="aff" rid="a13">13</xref><xref ref-type="aff" rid="a8">8</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Siviero</surname>
<given-names>A.</given-names>
</name>
<xref ref-type="aff" rid="a11">11</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Steinmetz</surname>
<given-names>M.</given-names>
</name>
<xref ref-type="aff" rid="a2">2</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Williams</surname>
<given-names>M.</given-names>
</name>
<xref ref-type="aff" rid="a2">2</xref><xref ref-type="aff" rid="a6">6</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Wyse</surname>
<given-names>R. F. G.</given-names>
</name>
<xref ref-type="aff" rid="a14">14</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Zwitter</surname>
<given-names>T.</given-names>
</name>
<xref ref-type="aff" rid="a15">15</xref>
</contrib>
</contrib-group>
<aff id="a1"><label>1</label>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</aff>
<aff id="a2"><label>2</label>Astrophysikalishes Institut Potsdam, Potsdam, Germany</aff>
<aff id="a3"><label>3</label>Rudolf Peierls Centre for Theoretical Physics, Oxford</aff>
<aff id="a4"><label>4</label>Anglo-Australian Observatory, Sydney, Australia</aff>
<aff id="a5"><label>5</label>Macquary University, Sydney, Australia</aff>
<aff id="a6"><label>6</label>Australian Natianal University, Canberra, Australia</aff>
<aff id="a7"><label>7</label>University of Central Lancashire, Preston</aff>
<aff id="a8"><label>8</label>Institute of Astronomy, Cambridge</aff>
<aff id="a9"><label>9</label>Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Heidelberg, Germany</aff>
<aff id="a10"><label>10</label>Kapteyn Astronomical Institut, Groningen, the Netherlands</aff>
<aff id="a11"><label>11</label>Astronomical Observatory of Padova in Asiago, Asiago, Italy</aff>
<aff id="a12"><label>12</label>University of Victoria, Victoria, Canada</aff>
<aff id="a13"><label>13</label>e2v Centre for Electronic Imaging, Planetary and Space Sciences Research Institute, The Open University, Milton Keynes</aff>
<aff id="a14"><label>14</label>Johns Hopkins University, Baltimore, MD, USA</aff>
<aff id="a15"><label>15</label>Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia</aff>
<author-notes>
<corresp id="c1">*E-mail: <email>siebert@astro.u-strasbg.fr</email></corresp>
</author-notes>
<pub-date pub-type="ppub">
<day>01</day>
<month>12</month>
<year>2008</year>
</pub-date>
<pub-date pub-type="epub">
<day>27</day>
<month>11</month>
<year>2008</year>
</pub-date>
<volume>391</volume>
<issue>2</issue>
<fpage>793</fpage>
<lpage>801</lpage>
<history>
<date date-type="received"><day>28</day><month>08</month><year>2008</year></date>
<date date-type="accepted"><day>2</day><month>09</month><year>2008</year></date>
</history>
<copyright-statement>© 2008 The Authors. Journal compilation © 2008 RAS</copyright-statement>
<copyright-year>2008</copyright-year>
<abstract><p>We present a measure of the inclination of the velocity ellipsoid at 1 kpc below the Galactic plane using a sample of red clump giants from the RAdial Velocity Experiment (RAVE) Data Release 2. We find that the velocity ellipsoid is tilted towards the Galactic plane with an inclination of <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu1.gif"></inline-graphic></inline-formula>. We compare this value to computed inclinations for two mass models of the Milky Way. We find that our measurement is consistent with a short scalelength of the stellar disc (<italic>R</italic><sub>d</sub>≃ 2 kpc) if the dark halo is oblate or with a long scalelength (<italic>R</italic><sub>d</sub>≃ 3 kpc) if the dark halo is prolate. Once combined with independent constraints on the flattening of the halo, our measurement suggests that the scalelength is approximately halfway between these two extreme values, with a preferred range 2.5–2.7 kpc for a nearly spherical halo. Nevertheless, no model can be clearly ruled out. With the continuation of the RAVE survey, it will be possible to provide a strong constraint on the mass distribution of the Milky Way using refined measurements of the orientation of the velocity ellipsoid.</p>
</abstract>
<kwd-group>
<kwd>stars: kinematics</kwd>
<kwd>galaxy: fundamental parameters</kwd>
<kwd>galaxy: kinematics and dynamics</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="ss1">
<title>1 INTRODUCTION</title>
<p>Our understanding of Galactic stellar populations and kinematics makes regular progress with the advent of new large Galactic stellar surveys providing distances, photometry, radial velocities or proper motions. Our Galaxy is, at present, the only place where we can probe the 6D phase space of stellar positions and velocities. For instance, the Galactic 3D potential can be probed through the orbits of the Sagittarius stream (<xref ref-type="bibr" rid="b27">Ibata et al. 2001</xref>; <xref ref-type="bibr" rid="b40">Newberg et al. 2002</xref>; <xref ref-type="bibr" rid="b24">Helmi 2004</xref>; <xref ref-type="bibr" rid="b47">Read & Moore 2005</xref>; <xref ref-type="bibr" rid="b21">Fellhauer et al. 2006</xref>) or Palomar 5 tidal tails (<xref ref-type="bibr" rid="b42">Odenkirchen et al. 2003</xref>; <xref ref-type="bibr" rid="b22">Grillmair & Dionatos 2006</xref>; <xref ref-type="bibr" rid="b23">Grillmair & Johnson 2006</xref>) or through the kinematics of halo stars (<xref ref-type="bibr" rid="b3">Battaglia et al. 2005</xref>). At smaller scales, the potential can also be analysed through the force perpendicular to the galactic plane (<xref ref-type="bibr" rid="b45">Oort 1960</xref>; <xref ref-type="bibr" rid="b30">Kuijken & Gilmore 1989a</xref>,<xref ref-type="bibr" rid="b31">b</xref>,<xref ref-type="bibr" rid="b32">c</xref>,<xref ref-type="bibr" rid="b33">1991</xref>; <xref ref-type="bibr" rid="b12">Crézé et al. 1998</xref>; <xref ref-type="bibr" rid="b52">Siebert, Bienaymé & Soubiran 2003</xref>; <xref ref-type="bibr" rid="b25">Holmberg & Flynn 2004</xref>) or through the coupling between the three components of the velocity in the solar neighbourhood (<xref ref-type="bibr" rid="b6">Bienaymé 1999</xref>).</p>
<p>Here, we concentrate on the question of the orientation of the velocity ellipsoid that is known to be tightly related to the shape and symmetry of the galactic potential (<xref ref-type="bibr" rid="b35">Lynden-Bell 1962</xref>; <xref ref-type="bibr" rid="b44">Ollongren 1962</xref>; <xref ref-type="bibr" rid="b26">Hori & Lui 1963</xref>; <xref ref-type="bibr" rid="b2">Amendt & Cuddeford 1991</xref>).</p>
<p>In spite of the long interest in this problem, measuring observationally the orientation of the velocity ellipsoid outside of the galactic plane has proven to be very difficult. This is due mainly to the absence of reliable distances away from the Solar neighbourhood. Despite this limitation, the first stellar stream detected within the Milky Way halo towards the north Galactic pole by <xref ref-type="bibr" rid="b36">Majewski, Munn & Hawley (1996)</xref> shows a velocity tilt, the ellipsoid being inclined towards the Galactic plane. This tilt could result from the expected velocity correlation induced by a spheroidal potential if these stars had similar integrals of motion (<xref ref-type="bibr" rid="b5">Bienaymé 1998</xref>). However, we note that this stream is not detected locally in the RAdial Velocity Experiment (RAVE) data (<xref ref-type="bibr" rid="b51">Seabroke et al. 2008</xref>).</p>
<p>Building realistic Galactic potentials shows that the main axis of the velocity ellipsoid, at 1 kpc above the Galactic plane, points in the direction of the <italic>z</italic>-axis of symmetry of the Galaxy towards a point located at 5 to 8 kpc behind the Galactic centre: for instance from numerical orbit computations (<xref ref-type="bibr" rid="b7">Binney 1983</xref>; <xref ref-type="bibr" rid="b30">Kuijken & Gilmore 1989a</xref>) or applying to the <xref ref-type="bibr" rid="b10">Carlberg & Innanen (1987)</xref> Galactic potential the <xref ref-type="bibr" rid="b2">Amendt & Cuddeford (1991)</xref> formulae. Such estimates of the velocity ellipsoid tilt are necessary for an accurate determination of the asymmetric drift<xref ref-type="fn" rid="fn1">1</xref> (<xref ref-type="bibr" rid="b8">Binney & Tremaine 1987</xref>), and for a correct measurement of the force perpendicular to the Galactic plane (<xref ref-type="bibr" rid="b53">Statler 1989</xref>).</p>
<p>In this paper, we study the 2D velocity distribution perpendicular to the Galactic plane for a sample of red clump stars from the RAVE survey (<xref ref-type="bibr" rid="b54">Steinmetz et al. 2006</xref>; <xref ref-type="bibr" rid="b57">Zwitter et al. 2008</xref>). These stars are selected between 500 and 1500 pc below the Galactic plane and provide a measurement of the tilt of the velocity ellipsoid at ≃1 kpc. In <xref ref-type="sec" rid="ss2">Section 2</xref>, we present the selection of the sample while <xref ref-type="sec" rid="ss3">Section 3</xref> focuses on the measurement of the inclination and possible biases. Finally, in <xref ref-type="sec" rid="ss4">Section 4</xref> we compare our measurement to computed inclinations for two extreme classes of mass models and we discuss possible outcomes of this measurement.</p>
</sec>
<sec id="ss2">
<title>2 SELECTION OF THE SAMPLE</title>
<p>Our sample is drawn from the second data release of the RAVE survey (<xref ref-type="bibr" rid="b57">Zwitter et al. 2008</xref>) containing about 50 000 stellar radial velocities and 20 000 measurements of stellar parameters. We focus on red clump giants towards the South Galactic pole to maximize the distance from the plane and to minimize the interstellar extinction. Hence, we select our targets in a cone with <italic>b</italic> < −60°, and we use a colour–magnitude criterion following <xref ref-type="bibr" rid="b55">Veltz et al. (2008)</xref> to select our candidate red clump stars: Two-Micron All-Sky Survey (2MASS) <italic>J</italic>−<italic>K</italic> colour within 0.5–0.7 and <italic>K</italic> < 9.3.</p>
<p>This colour–magnitude cut selects mainly red clump stars whose luminosity function (LF) is well defined and approximately Gaussian: <italic>M</italic><sub><italic>K</italic></sub>=−1.6 ± 0.03. Also, the red clump LF is narrow, the dispersion of the Gaussian LF being 0.22 mag in the <italic>K</italic> band, and nearly independent of the metallicity (<xref ref-type="bibr" rid="b1">Alves 2000</xref>). It makes this population particularly suited to study the kinematics of stars away from the solar neighbourhood as reliable distance estimates can be obtained. Also the extinction in the <italic>K</italic> band remains low, 〈<italic>A</italic><sub><italic>K</italic></sub>〉= 0.007 mag with a maximum extinction of <italic>A</italic><sub><italic>K</italic></sub>= 0.05 mag for this region of the sky (<xref ref-type="bibr" rid="b49">Schlegel, Finkbeiner & Davis 1998</xref>). Hence, extinction does not contribute significantly to our error budget: the average error on the distance is less than 1 per cent with a maximum value of ∼2 per cent for the limiting magnitude of our sample.</p>
<p>The selection criterion, retaining only the objects with a proper motion value in the RAVE catalogue, restricts the sample to 763 red clump candidates spanning a distance interval from the Sun of 500 to 1500 pc. A small fraction of these selected stars are dwarfs or subgiants. According to the photometric and kinematic modelling of the SGP and NGP by <xref ref-type="bibr" rid="b55">Veltz et al. (2008)</xref>, we can estimate that, at the limiting magnitude of our sample <italic>M</italic><sub><italic>K</italic></sub>= 9.3, 75 per cent of the sample are red clump stars, 10 per cent dwarfs and 15 per cent subgiants. Brighter than this limit, the fraction of red clump stars is larger and the quoted fractions are upper limits to our contamination fraction.</p>
<p>We clean our sample further using a kinematic selection. We select stars based on their velocities with the following criteria <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu2.gif"></inline-graphic></inline-formula> and <italic>V</italic> < 100 km s<sup>−1</sup>. This selection enables us to remove the nearby dwarfs whose distance is overestimated by a factor of 14 due to their fainter absolute magnitude, hence an overestimation of their velocities. The resulting sample contains 580 red clump candidates in the direction of the South Galactic pole whose distribution in velocity space is presented in <xref ref-type="fig" rid="f1">Fig. 1</xref>. In this figure, the contours depict the distribution of the original sample smoothed by the individual errors while the dots show the location in velocity space of the remaining 580 stars after the velocity selection.</p>
<fig position="float" id="f1">
<label>Figure 1</label>
<caption>
<p>Selection of the red clump sample in <italic>U</italic>, <italic>V</italic>, <italic>W</italic> velocity space. The contours show the distribution of the 763 red clump candidates belonging to the original sample, smoothed by the individual errors, while the dots represent the location in the velocity space of the 580 stars in the final sample. The contours encompass 90, 70, 50 and 30 per cent of the total sample.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f1.gif"></graphic>
</fig>
<p>We test our selection criteria using the second year observation from the RAVE survey (as a reminder, RAVE Data Release 2 (DR2) contains the first year of observation – i.e. DR1 – and the second year of observation). For these objects, RAVE provides the measurements of the stellar parameters including an estimate of the gravity. The sample selected from second year data contains 294 stars with log <italic>g</italic> measurements, with 231 stars matching the velocity criteria. The histograms of log <italic>g</italic> for each subsample are presented in <xref ref-type="fig" rid="f2">Fig. 2</xref>, where the black histogram presents the distribution of log <italic>g</italic> for the 294 second year stars and the dashed histogram presents the subsample matching our velocity criteria. The red clump giants span a large range in gravity depending on their metallicity: log <italic>g</italic>= 2.08 for the metal-poor, low-mass end and reaches up to log <italic>g</italic>= 3 for the high-mass, metal-rich red clump objects (<xref ref-type="bibr" rid="b56">Zhao, Qiu & Zhang 2000</xref>). This figure clearly indicates that our velocity criteria are efficient for rejecting dwarf stars (with high log <italic>g</italic>) but also remove a small fraction of stars with lower log <italic>g</italic>, primarily subgiants and giants on the ascending branch and also a few red clump stars. Nevertheless, these objects have large velocities and fall in the tails of the velocity distribution. Therefore, they affect only marginally the measurement of the inclination, as our measurement is driven by the larger number of stars in the bulk of the velocity distribution.</p>
<fig position="float" id="f2">
<label>Figure 2</label>
<caption>
<p>Top panel: histogram log <italic>g</italic> for the 294 stars with log <italic>g</italic> measurements in RAVE DR2. Black line full subsample, dashed line subsample matching the velocity criteria. The red clump giants cover the region in log <italic>g</italic>= 2–3 depending on the metallicity or mass. A conservative estimate of RAVE standard error on log <italic>g</italic> is 0.5 dex. Bottom panel: fraction of stars rejected by the velocity criterion as a function of log <italic>g</italic>.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f2.gif"></graphic>
</fig>
<p>It is worth noting that due to the uncertainties in RAVE log <italic>g</italic> measurements which are 0.5 dex for a typical RAVE star (<xref ref-type="bibr" rid="b57">Zwitter et al. 2008</xref>), it is not possible to obtain a firm estimate of the contamination in our sample, nor to use the RAVE log <italic>g</italic> estimates to refine our sample. Also, log <italic>g</italic> measurements are only available for less than half of our sample as stellar parameters cannot be estimated from the spectra collected during the first year of operation of RAVE. Nevertheless, considering a 0.5 dex error on log <italic>g</italic>, we estimate that the contamination using the velocity criteria is reduced to ≃10 per cent which is to be compared to more than 20 per cent without the velocity criteria. We will detail the effect of this contamination on our measurement in the next section.</p>
</sec>
<sec id="ss3">
<title>3 MEASURING THE TILT</title>
<p>The tilt angle δ of the 2D velocity distribution is given by the relation <disp-formula id="m1">
<label>1</label>
<graphic xlink:href="mnras0391-0793-m1.gif"></graphic>
</disp-formula>
where σ<sup>2</sup><sub><italic>UW</italic></sub>, σ<sup>2</sup><sub><italic>U</italic></sub> and σ<sup>2</sup><sub><italic>W</italic></sub> are the velocity distribution moments. In the local velocity coordinates, the velocity in the radial direction is given by the <italic>U</italic> component of the velocity vector (positive towards the Galactic Centre) and the vertical velocity by the <italic>W</italic> component of the vector positive towards the North Galactic pole, while the <italic>V</italic> component is positive towards the Galactic rotation (not used in <xref ref-type="disp-formula" rid="m1">equation 1</xref>).</p>
<p>The computation of the inclination is straightforward for a sample with small and homogeneous errors. Nevertheless, to lower the effect of foreground dwarfs and giants and the contamination due to high-velocity stars, we make use of a velocity cut-off to select our sample. In this case, as our errors in the <italic>U</italic> and <italic>V</italic> velocity directions are large, a direct measurement of the tilt angle may be subject to bias and our selection criteria must be studied as our error budget may not be dominated by the size of the sample (see <xref ref-type="sec" rid="ss3-1">Section 3.1</xref>).</p>
<p>Also, the local velocity ellipsoid is not a smooth distribution and clumps are present on both small and large scales in the velocity space (see e.g. <xref ref-type="bibr" rid="b16">Dehnen 2000</xref>; <xref ref-type="bibr" rid="b14">Chereul, Crézé & Bienaymé 1998</xref>; <xref ref-type="bibr" rid="b15">Dehnen 1998</xref>; <xref ref-type="bibr" rid="b19">Famaey et al. 2005</xref>). These substructures prevent determining the age–velocity dispersion relation in the <italic>U</italic> and <italic>V</italic> directions (<xref ref-type="bibr" rid="b50">Seabroke & Gilmore 2007</xref>) and may also influence the measured tilt angle. We will discuss the effect of such substructures on our measurement in <xref ref-type="sec" rid="ss3-2">Section 3.2</xref>.</p>
<p>Finally, if our selection criterion is efficient at rejecting the foreground stars, only the tails of the velocity distribution are affected by the velocity cut-off. As their space velocities are overestimated, such foreground objects will impact on the measurement of the inclination. We will discuss this particular point in <xref ref-type="sec" rid="ss3-3">Section 3.3</xref>.</p>
<sec id="ss3-1">
<title>3.1 The effect of errors and velocity cut-off</title>
<p>Our velocity errors in the cardinal directions are not homogeneous because the <italic>U</italic> and <italic>V</italic> components of the velocity vector are dominated by the proper motion contribution while the <italic>W</italic> component is primarily measured from the RAVE radial velocity. <xref ref-type="fig" rid="f3">Fig. 3</xref> presents the distribution of errors for our sample in the <italic>U</italic>, <italic>V</italic> and <italic>W</italic> components as full, dashed and dotted lines. It is clear that the mode of the velocity error distributions for the <italic>U</italic> and <italic>W</italic> components, the ones we are primarly interested in, differ by a factor of 4: ≃5 km s<sup>−1</sup> for the <italic>W</italic> component while for the <italic>U</italic> component the distribution peaks at ≃20 km s<sup>−1</sup>.</p>
<fig position="float" id="f3">
<label>Figure 3</label>
<caption>
<p>Distribution of the velocity errors in our sample; full line <italic>U</italic> component, dashed line <italic>V</italic> component, dotted line <italic>W</italic> component. The difference between these three distributions arises due to the relative contribution of proper motion and distance errors to the radial velocity errors for the velocities along the three cardinal directions.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f3.gif"></graphic>
</fig>
<p>This large difference results in an anisotropic smoothing of the observed velocity ellipsoid which, combined with our velocity criterion, biases the measurement of the inclination towards a lower value. This bias is due to the structure of <xref ref-type="disp-formula" rid="m1">equation (1)</xref> where the error anisotropy results in an extra component <italic>E</italic> on measured velocity dispersions. If we consider only the extra term on the <italic>U</italic> component, we have σ<sub><italic>U</italic></sub>(measured)<sup>2</sup>=σ<sub><italic>U</italic></sub>(true)<sup>2</sup>+<italic>E</italic><sup>2</sup> and the cross-term <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu3.gif"></inline-graphic></inline-formula> being the correlation coefficient. With this notation, it is clear that the contribution of the additional error term is larger for the denominator than it is for the numerator, hence producing an underestimate of the true inclination. For comparison, a linear fit would not be biased due to the asymmetry of the errors but unfortunately it is more sensitive to outliers. To overcome this problem, we use a Monte Carlo sampling of the velocity error distributions. We add a random velocity term to the <italic>V</italic> and <italic>W</italic> components, degrading the accuracy of the two velocity components, so that the resulting error distributions match the <italic>U</italic> velocity error distribution. For the <italic>U</italic> velocity, it is randomly drawn from its original error distribution. This procedure enables us to obtain isotropic error distributions for all three components, degrading the two best distributions to the level of the least accurate distribution. The inclination is then computed using <xref ref-type="disp-formula" rid="m1">equation (1)</xref> after applying the velocity criterion.</p>
<p>We tested this procedure on a simple velocity ellipsoid model using the RAVE error laws and standard velocity dispersions for the Galactic old disc population, leaving aside the thick disc: σ<sub><italic>U</italic></sub>= 31 and σ<sub><italic>W</italic></sub>= 17 km s<sup>−1</sup>. The size of the sample was set to 1000 data points and we varied the inclination of the ellipsoid from 1 to 20°. The results of this test are shown in <xref ref-type="fig" rid="f4">Fig. 4</xref> where the direct measurement is presented as a dotted line and the Monte Carlo determinations by the open circles with error bars for one random realization of a velocity ellipsoid. The one-to-one relation between the original and recovered angles is sketched by the dashed line. Below 2–4° for the inclination, depending on the realization of the ellipsoid, both methods predict the same inclination but above this threshold, the Monte Carlo sampling recovers the proper value of the angle. On the other hand, the direct measurement, applying <xref ref-type="disp-formula" rid="m1">equation (1)</xref>, always underestimates the true angle with a bias rising with the tilt value. This test clearly indicates that the Monte Carlo sampling of the errors is best suited to measure the tilt of the velocity ellipsoid, while direct measurements using <xref ref-type="disp-formula" rid="m1">equation (1)</xref> are subject to strong bias in the case of heterogeneous error laws. We also note that the value of the bias depends strongly on the random sampling of the ellipsoid, with a bias varying between 2 and 4° at 7°. This spread in the bias value becomes larger as the tilt value increases and indicates that even with a proper model to estimate the bias, it can hardly be used to correct the direct measurement.</p>
<fig position="float" id="f4">
<label>Figure 4</label>
<caption>
<p>Test of the Monte Carlo method to measure the tilt angle of the velocity ellipsoid. The original versus recovered tilt angle are presented for one realization of a velocity ellipsoid having σ<sub><italic>U</italic></sub>= 31 and σ<sub><italic>W</italic></sub>= 17 km s<sup>−1</sup> and sampled using 1000 data points. The RAVE error laws for the velocities are used. The dashed line shows the one-to-one relation while the dotted line is a direct measurement using <xref ref-type="disp-formula" rid="m1">equation (1)</xref>. The results from the Monte Carlo sampling of the error laws are depicted by the open circles and the error bars are the standard deviation of 5000 resampling for each value of the tilt angle.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f4.gif"></graphic>
</fig>
<p>The procedure is then applied on the RAVE sample and the result is presented in <xref ref-type="fig" rid="f5">Fig. 5</xref> which shows the distribution of inclinations in degrees obtained by sampling the error distribution 25 000 times. The mean inclination measured is <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu4.gif"></inline-graphic></inline-formula> with a standard deviation of <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu5.gif"></inline-graphic></inline-formula>. If the anisotropy of the error distributions was not taken into account, the measured inclination would have been <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu6.gif"></inline-graphic></inline-formula> or <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu7.gif"></inline-graphic></inline-formula> too low. The 2D representation of the velocity ellipsoid inclination is shown in <xref ref-type="fig" rid="f6">Fig. 6</xref>. The colour-coding follows the density of stars per bin in the region of the (<italic>U</italic>, <italic>W</italic>) space, where the 2D distribution of the (<italic>U</italic>, <italic>W</italic>) velocities has been convolved by the individual errors. The measured inclination and 1σ errors are presented as white lines (full line for the mean value and dotted lines for the errors).</p>
<fig position="float" id="f5">
<label>Figure 5</label>
<caption>
<p>Distribution of the measured inclination of the velocity ellipsoid per <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu18.gif"></inline-graphic></inline-formula> bin. This distribution is obtained using a Monte Carlo sampling of the error distribution. The mean inclination is found to be <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu19.gif"></inline-graphic></inline-formula> with a standard deviation of <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu20.gif"></inline-graphic></inline-formula>. The grey line is a Gaussian function with identical parameters.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f5.gif"></graphic>
</fig>
<fig position="float" id="f6">
<label>Figure 6</label>
<caption>
<p>Velocity distribution in the (<italic>U</italic>, <italic>W</italic>) plane from our sample after sampling the error distribution. The measured inclination and 1σ range are presented by the full and dashed white line. The colour-coding follows the density per bin.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f6.gif"></graphic>
</fig>
</sec>
<sec id="ss3-2">
<title>3.2 Effect of substructures</title>
<p>Substructures such as the Hyades, Pleiades or the Hercules groups are well-known features of the local velocity ellipsoid, and are easily seen in the velocity space obtained from the <italic>Hipparcos</italic> mission (see e.g. <xref ref-type="bibr" rid="b14">Chereul et al. 1998</xref>; <xref ref-type="bibr" rid="b17">Dehnen & Binney 1998</xref>; <xref ref-type="bibr" rid="b19">Famaey et al. 2005</xref>). These structures may have a wide range of origins such as the disruption of clusters, resonances associated with the bar or spiral arms (see e.g. <xref ref-type="bibr" rid="b16">Dehnen 2000</xref>; <xref ref-type="bibr" rid="b18">De Simone, Wu & Tremaine 2004</xref>; <xref ref-type="bibr" rid="b19">Famaey et al. 2005</xref>; <xref ref-type="bibr" rid="b20">Famaey, Siebert & Jorissen 2008</xref>; <xref ref-type="bibr" rid="b39">Minchev & Quillen 2008</xref>). Nevertheless, the average velocity error in our sample does not allow us to distinguish these substructures.</p>
<p>To test the influence of these velocity substructures on the tilt determination, we use the local sample from <xref ref-type="bibr" rid="b19">Famaey et al. (2005)</xref>. This sample provides not only accurate velocity vectors for about 6500 stars in the solar neighbourhood, it also provides an estimate of the relation of a star to the identified velocity substructures. This allows us to separate the background ellipsoid from the known overdensities.</p>
<p>We first estimate the fraction of stars in substructures in the <xref ref-type="bibr" rid="b19">Famaey et al. (2005)</xref> sample as a function of <italic>z</italic>, the height above the Galactic plane. The number of stars in structures is larger closer to the plane: 36 per cent of the stars are in structures in the 0–200 pc interval while 25 per cent are found in structures between 200 and 500 pc. This drop in number of objects in structures is sharp as seen from <xref ref-type="table" rid="t1">Table 1</xref>, the fraction in objects in structures being lowered by over a factor of 2 between 0 and 500 pc. As our sample covers a distance below the plane from 500 to 1500 pc, we extrapolate this behaviour at higher <italic>z</italic> to estimate the contamination arising from substructures in our sample. Using a conservative extrapolation, we estimate the contamination in our sample to be lower than 7 per cent.</p>
<table-wrap id="t1">
<label>Table 1</label>
<caption><p>Fraction and number of stars in structures in the <xref ref-type="bibr" rid="b19">Famaey et al. (2005)</xref> sample as a function of height above the Galactic plane.</p></caption>
<table>
<thead>
<tr>
<td><italic>z</italic> (pc)</td>
<td><italic>N</italic><sub>tot</sub></td>
<td>Fraction in structures</td>
<td>Number in Hya/Plei</td>
<td>Number in Sirius</td>
<td>Number in Hercules</td>
</tr>
</thead>
<tbody>
<tr>
<td>0–100</td>
<td>1361</td>
<td>0.40</td>
<td>127</td>
<td>63</td>
<td>125</td>
</tr>
<tr>
<td>100–200</td>
<td>1337</td>
<td>0.33</td>
<td>111</td>
<td>49</td>
<td>131</td>
</tr>
<tr>
<td>200–300</td>
<td>844</td>
<td>0.28</td>
<td>46</td>
<td>27</td>
<td>94</td>
</tr>
<tr>
<td>300–400</td>
<td>422</td>
<td>0.22</td>
<td>1</td>
<td>20</td>
<td>37</td>
</tr>
<tr>
<td>400–500</td>
<td>177</td>
<td>0.19</td>
<td>0</td>
<td>7</td>
<td>8</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>In a second step, we test the influence of the substructures on the measured inclination. We cannot use the <xref ref-type="bibr" rid="b19">Famaey et al. (2005)</xref> sample directly, as the presence of pertubations in the plane makes any attempt to disentangle the effect of groups from the effect of the pertubations on the inclination hazardous. Therefore, we proceed as follow. We add an additional population, drawn from a subset of the Famaey et al. sample, to our RAVE sample. This subset is randomly selected from the set of stars belonging to groups in the distance interval 300 to 500 pc. We add it to the RAVE sample varying its fraction relative to the RAVE sample from 1 to 20 per cent. This procedure enables us to mimic as closely as possible the velocity distribution of the groups, which is not homogeneous and strongly varies as a function of distance to the plane. This operation is repeated 25 000 times for each fraction of the contamination to ensure a proper coverage of the possible cases.</p>
<p>The results are shown in <xref ref-type="fig" rid="f7">Fig. 7</xref> where the mean deviation (δ<sub>measured</sub>−δ<sub>true</sub>) in degrees as a function of the fraction of stars in groups is drawn as a thick line. The standard deviation of the repeats is shown as dashed lines. We note that the average tilt for the Famaey group sample used here is <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu8.gif"></inline-graphic></inline-formula>, very different from the value of the inclination in our sample. From this figure, we see that the influence of velocity structures on the measured tilt is low, around <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu9.gif"></inline-graphic></inline-formula> for a contamination of 7 per cent with a standard deviation below <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu10.gif"></inline-graphic></inline-formula>. 7 per cent being an upper limit for the contamination in our sample, we do not expect groups to affect our measurement of the tilt. Indeed, the mean deviation combined with the standard deviation measured from this experiment contributes to not more than <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu11.gif"></inline-graphic></inline-formula>, less than 6 per cent of our estimated errors.</p>
<fig position="float" id="f7">
<label>Figure 7</label>
<caption>
<p>Influence of stars in velocity groups on the measured inclination. A sample of <xref ref-type="bibr" rid="b19">Famaey et al. (2005)</xref> stars belonging to groups is randomly added to the RAVE sample, varying the contamination from 1 to 20 per cent. The tilt is measured following the same procedure as for the pure RAVE sample. The thick line represents the average deviation in degrees (δ<sub>tilt</sub>=δ<sub>measured</sub>−δ<sub>true</sub>) found for 25 000 repeats per contamination fraction. The dashed line is the standard deviation of the repeats.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f7.gif"></graphic>
</fig>
</sec>
<sec id="ss3-3">
<title>3.3 Effect of foreground stars</title>
<p>If, as we saw above, the velocity structures do not influence largely the orientation of the velocity ellipsoid, the foreground stars (dwarfs, subgiants and giants on the ascending branch) can be more problematic. The fact that their velocities are overestimated in the <italic>U</italic> direction, because of the overestimate of their distances, will add a component with low inclination to the observed ellipsoid. The bias due to these objects is an underestimate of the tilt at a given distance.</p>
<p>The contamination by foreground objects is about 10 per cent in our sample (see <xref ref-type="sec" rid="ss2">Section 2</xref>), and a factor of 14 overestimation of the <italic>U</italic> velocity for the dwarfs will render their velocity ellipsoid almost uniform in the velocity interval we consider. The velocity dispersion of this foreground population will be large, over 400 km s<sup>−1</sup> instead of ∼31 km s<sup>−1</sup> for σ<sub><italic>U</italic></sub>, due to the distance overestimate. For the other sources of contamination (subgiants and giants on the ascending branch), the distance is overestimated by a factor of 2 or less, and their impact on the velocity ellipsoid is lower.</p>
<p>To obtain an upper limit of the effect of the foreground population, we rely on a resampling technique, replacing 10 per cent of the sample by a random realization of a thin disc population with no inclination. The overestimate of the distance is then translated into an overestimate of the velocities, and the final inclination of the velocity ellipsoid is measured applying the same procedure as above. The difference between the distribution with and without resampling provides an upper limit on the effect of foreground objects on our measurement. We note here that if adding a population with no tilt is, in principle, similar to the experiment done in <xref ref-type="sec" rid="ss3-2">Section 3.2</xref>, here we incorporate the distance overestimation. Furthermore, the added test stars are not restricted to the region in velocity space of the groups.</p>
<p><xref ref-type="fig" rid="f8">Fig. 8</xref> presents the results of this resampling. The black histogram shows the distribution of the 25 000 measurements while the grey Gaussian curve is the Gaussian representation of the distribution obtained in <xref ref-type="fig" rid="f5">Fig. 5</xref>. The presence of a population with no inclination does produce an observable bias in this experiment: we observe a shift of the mode of the distribution. Nevertheless, this bias is small, the measured offset is <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu12.gif"></inline-graphic></inline-formula>, much lower than the standard deviation of the distribution while it is a <italic>worst-case scenario</italic>. Indeed, here we did consider only the contamination by foreground dwarf stars while our real contamination is a mixture of dwarfs and subgiants. In the latter case, the overestimate of the distance is much lower, as these stars have a mean distance from the plane that is larger. Hence, their impact on the velocity ellipsoid orientation will be lower than for dwarfs. We also note that the presence of a foreground population renders the distribution non-Gaussian, adding a tail to the low tilt angle part of the distribution which is not observed in <xref ref-type="fig" rid="f5">Fig. 5</xref>.</p>
<fig position="float" id="f8">
<label>Figure 8</label>
<caption>
<p>Results from the resampling study. 10 per cent of the red clump sample has been replaced by a population of foreground objects with no tilt and velocities in the <italic>U</italic> direction overestimated by a factor of 14. The black histogram is the distribution of measured inclination for this new sample following the same procedure as for <xref ref-type="fig" rid="f5">Fig. 5</xref> while the grey Gaussian curve is the distribution of inclination without the resampling from <xref ref-type="fig" rid="f5">Fig. 5</xref>.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f8.gif"></graphic>
</fig>
<p>We can conclude that our estimate of the inclination of the velocity ellipsoid is robust and that the presence of a population of foreground objects does not introduce a significant bias in our measurement. Indeed, combining both the effect of foreground stars and the possible velocity ellipsoid substructures in a <italic>worst-case scenario</italic>, the resulting bias amounts to ∼10 per cent of our errors. At this level, the biases do not affect our conclusions.</p>
</sec>
</sec>
<sec id="ss4">
<title>4 RELATION TO THE MASS DISTRIBUTION IN THE GALAXY</title>
<p>The tilt of the velocity ellipsoid is intimately linked to the mass distribution in the Milky Way and more specifically – if we trust our knowledge of the structure of the Galactic disc – to the flattening of the halo. We start from the mass model of <xref ref-type="bibr" rid="b17">Dehnen & Binney (1998)</xref><xref ref-type="fn" rid="fn2">2</xref> and its revised parameters provided by <xref ref-type="bibr" rid="b9">Binney & Tremaine</xref> (2008, hereafter BT08) in their table 2.3. This revision proposes two models, referred to as models I and II, which match both local and non-local data. These models are the modified versions of the <xref ref-type="bibr" rid="b17">Dehnen & Binney (1998)</xref> models I and IV.</p>
<p>As noted in these references, a crucial parameter is the scalelength of the disc whose value lies in the range 2–3 kpc. The two models are set on the upper and lower bounds for this parameter. Model I presents a mass model with a short scalelength (<italic>R</italic><sub>d</sub>= 2 kpc) that induces a strong contribution from the disc for the potential at the solar radius up to 11 kpc. On the other hand, model II has a larger scalelength (<italic>R</italic><sub>d</sub>= 3.2 kpc) and therefore, the halo contribution to the rotation curve dominates at the Sun location and beyond. This is also seen from the global shape of the potential where for model II the isopotentials are more spherical than for model I (see figs 2.19 and 2.21 of BT08). For a detailed description of these two models, the reader is referred to chapter 2 of BT08.</p>
<p>We use both the models to discuss below the implications of the tilt on the possible models for the mass distribution in the Milky Way, focusing on the flattening of the halo in the two extreme cases. We note here that the region above (below) the plane between 1 and 2 kpc is best suited to separate the two classes of models. Indeed, in this region, the variation of the angle between the Galactic plane and the normal to the isopotentials as a function of the minor-to-major axis ratio <italic>c</italic>/<italic>a</italic><sub>ρ</sub> is maximum. Hence, we expect the difference between the predicted tilt angle to be the largest in the same region. At larger distances from the plane, the potential becomes more spherical and the difference vanishes between the models in terms of variation of the potential and inclination of the velocity ellipsoid.</p>
<p>To measure the tilt of the ellipsoid as a function of the halo flattening, we vary the density minor-to-major axis ratio <italic>c</italic>/<italic>a</italic><sub>ρ</sub> of the halo from 0.6 to 1.7 for each model, the halo density being described by the relation
<disp-formula id="m2">
<label>2</label>
<graphic xlink:href="mnras0391-0793-m2.gif"></graphic>
</disp-formula>
where the flattening <italic>c</italic>/<italic>a</italic><sub>ρ</sub> enters the equation through the parameter <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu13.gif"></inline-graphic></inline-formula> and <italic>z</italic> being the Galactocentric cylindrical coordinates. <italic>a</italic><sub><italic>h</italic></sub> is a scale parameter with <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu14.gif"></inline-graphic></inline-formula> if <italic>m</italic>≪<italic>a</italic><sub><italic>h</italic></sub> and <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu15.gif"></inline-graphic></inline-formula> for large <italic>m</italic>.</p>
<p>While changing <italic>c</italic>/<italic>a</italic><sub>ρ</sub>, we change the halo density in order to keep the rotation curve almost unchanged in the plane. The bulge and disc components are fixed to the values of table 2.3 of BT08, the solar Galactocentric radius is set to 8 kpc. Keeping the rotation curve and the disc/bulge parameters unchanged enables us to study the influence of the halo flattening on the shape of the velocity ellipsoid, as the contribution to the radial force of each component remains largely the same for each case. The corresponding density of the halo for each mass model is reported in <xref ref-type="table" rid="t2">Table 2</xref>.</p>
<table-wrap id="t2">
<label>Table 2</label>
<caption><p>Modification to the mass models I and II of BT08 table 2.3 used in <xref ref-type="sec" rid="ss4">Section 4</xref>. The halo density is given in M<sub>⊙</sub> pc<sup>−3</sup>. The modified models are built, modifying the halo parameters, to keep the rotation curve almost unchanged in the disc. The bulge and disc parameters are fixed to the BT08 values.</p></caption>
<table>
<thead>
<tr>
<td rowspan="2" valign="bottom"><italic>c</italic>/<italic>a</italic></td>
<td colspan="2" align="center">ρ<sub>halo</sub></td>
</tr>
<tr>
<td>Model I</td>
<td>Model II</td>
</tr>
</thead>
<tbody>
<tr>
<td>0.6</td>
<td>0.838</td>
<td>0.327</td>
</tr>
<tr>
<td>0.7</td>
<td>0.765</td>
<td>0.293</td>
</tr>
<tr>
<td>0.8</td>
<td>0.711</td>
<td>0.266</td>
</tr>
<tr>
<td>0.9</td>
<td>0.670</td>
<td>0.245</td>
</tr>
<tr>
<td>1.0</td>
<td>0.635</td>
<td>0.229</td>
</tr>
<tr>
<td>1.1</td>
<td>0.608</td>
<td>0.215</td>
</tr>
<tr>
<td>1.2</td>
<td>0.585</td>
<td>0.204</td>
</tr>
<tr>
<td>1.3</td>
<td>0.566</td>
<td>0.195</td>
</tr>
<tr>
<td>1.4</td>
<td>0.548</td>
<td>0.186</td>
</tr>
<tr>
<td>1.5</td>
<td>0.534</td>
<td>0.179</td>
</tr>
<tr>
<td>1.6</td>
<td>0.520</td>
<td>0.172</td>
</tr>
<tr>
<td>1.7</td>
<td>0.510</td>
<td>0.167</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>The inclination of the velocity ellipsoid is computed for a given mass distribution using orbit integration. A single orbit is integrated over 30 rotations using a fourth-order Runge–Kutta algorithm. The initial conditions are drawn from a Shu distribution function matching the local data (<xref ref-type="bibr" rid="b6">Bienaymé 1999</xref>). For each potential, the orbit library contains over 2 million orbits from which we randomly select 10 points per orbit in the last 15 rotations. We further restrict the orbit library to data points matching the interval in <italic>R</italic> and <italic>z</italic> of the RAVE sample, this reduces the size of our final libraries to 6 × 10<sup>4</sup> to a few 10<sup>5</sup> points per library.</p>
<p>We measure the tilt using a Monte Carlo selection of the orbits, requiring that the distribution of the selected orbits matches the selection function of our RAVE sample in the (<italic>R</italic>, <italic>z</italic>) plane. Here, <italic>R</italic> is the Galactocentric radius and <italic>z</italic> is the distance above (below) the plane. This selection function is obtained by convolving the distribution of the RAVE sample in <italic>R</italic> and <italic>z</italic> by their errors for each star. This procedure ensures us that the spatial distribution in <italic>R</italic> and <italic>z</italic> of the RAVE sample is well matched by the orbit selection. We select 5000 orbits using the spatial constraints, about 10 times larger than the observed sample but about 10<sup>2</sup> times less than orbit library size to minimize the probability of the same orbit to be selected twice, and the tilt is measured using the associated velocities and <xref ref-type="disp-formula" rid="m1">equation (1)</xref>. The measurement is repeated 500 times to obtain the mean inclination and dispersion. This procedure is repeated for each orbit library. The convergence of this procedure is tested using 1000, 5000 and 10 000 orbits points for the selection. The result shows a very good stability of the mean inclination, of the order of a few 10<sup>−2</sup> degrees, while the dispersion increases as the number of orbits becomes lower. For the model I with <italic>c</italic>/<italic>a</italic><sub>ρ</sub>= 1.0, we obtain, respectively, an inclination of 9.89 ± 1.09, 9.83 ± 0.44 and 9.85 ± 0.36 for 1000, 5000 and 10 000 orbits, which indicates that the gain in precision above 5000 orbits is limited as the computing time scales linearly with the number of orbits.</p>
<p>The resulting measurements are presented in <xref ref-type="fig" rid="f9">Fig. 9</xref>, left-hand panel where the full horizontal line is our measurement from <xref ref-type="sec" rid="ss3">Section 3</xref> and the horizontal dashed lines are the 1σ limit. The two remaining curves correspond to the measurements obtained from our orbit analysis. The top curve is our prediction for the class of model I of BT08 and the bottom curve for the class of model II. For comparison, the right-hand panel presents for the direct measurement, without correcting the tilt for the velocity error anisotropy. The circles and crosses correspond to Monte Carlo realizations of the RAVE sample using the orbit libraries where the RAVE velocity errors have been applied on the orbit library directly. Circles and crosses are, respectively, for the class of models I and II. The error bars are the standard deviation of 1000 realizations.</p>
<fig position="float" id="f9">
<label>Figure 9</label>
<caption>
<p>Left-hand panel: inclination of the velocity ellipsoid as a function of the halo flattening <italic>c</italic>/<italic>a</italic><sub>ρ</sub> in the RAVE selection function. The full and thin dash–dotted horizontal lines correspond to our measurement and error bars using isotropic error laws. The two dashed curves correspond to the class of models I (top panel) and II (bottom panel) in BT08 for which we varied the halo flattening. Right-hand panel: same as left-hand panel but for a direct measurement. The horizontal line is the direct measurement of the tilt without correction for the velocity error anisotropy. The circles and crosses are for Monte Carlo realization of the RAVE sample using the RAVE velocity error laws for the class of models I (circles) and II (crosses). The error bars are the standard deviation obtained from 1000 realizations.</p>
</caption>
<graphic xlink:href="mnras0391-0793-f9.gif"></graphic>
</fig>
<p>The classes of models show the same general gross properties. The predicted tilt rises as the flattening decreases, reaching a maximum in the prolate halo region. This maximum is expected and varies depending on the details of each model. It is due to the fact that, when <italic>c</italic>/<italic>a</italic><sub>ρ</sub> becomes large, the potential becomes separable in cylindrical coordinates. On the other hand, if <italic>c</italic>/<italic>a</italic><sub>ρ</sub> approaches 0, the problem reduces to the plane-parallel case and the expected inclination at 1 kpc is δ≃ 3°.</p>
<p>The two classes of models provide different estimates for the tilt of the velocity ellipsoid within the limits of our sample. The tilt variation versus <italic>c</italic>/<italic>a</italic><sub>ρ</sub> is larger for prolate models than it is for oblate models. For a density flattening of 0.6, the difference is only 2°, while for slightly prolate models the difference reaches up to 5°. Also, in the oblate case, the expected inclination rises more quickly than for the prolate case. If one compares the predictions for the two classes of models to the measured inclination, a clear tendency is present: the more massive the halo is, the more prolate it must be to match the tilt of the velocity ellipsoid at 1 kpc. The two extreme cases ‘separater’ around <italic>c</italic>/<italic>a</italic><sub>ρ</sub>≃ 0.9: while for a massive disc <italic>c</italic>/<italic>a</italic><sub>ρ</sub>≤ 0.9 is necessary to reproduce the tilt, <italic>c</italic>/<italic>a</italic><sub>ρ</sub>≥ 0.9 is needed for a massive halo in the 1σ limit. More specifically, the measured orientation of the velocity ellipsoid is consistent with a short scalelength of the disc if the halo is oblate, while in the other case – a scalelength of the disc of the order of 3 kpc – the measured value for the tilt implies that the halo must be prolate.</p>
<p>Using a direct measurement, applying the velocity errors on Monte Carlo realization of the RAVE sample has the benefit of reducing the errors by a factor of <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu16.gif"></inline-graphic></inline-formula> (<xref ref-type="fig" rid="f9">Fig. 9</xref>, right-hand panel). Nevertheless, the bias increases with the inclination, see <xref ref-type="fig" rid="f4">Fig. 4</xref>, which results in the difference between the two models being lower. This direct measurement indicates that low values of <italic>c</italic>/<italic>a</italic><sub>ρ</sub> are marginally inconsistent with the measured tilt, with <italic>c</italic>/<italic>a</italic><sub>ρ</sub> > 0.7 being preferred even so the same general conclusions hold for both modelling technique. We note, however, that applying the errors on the orbit library (direct method) and correcting the anisotropy of the velocity errors (unbiased measurement) produces slightly different predictions for the tilt. This is partly due to the fact that the <italic>U</italic> and <italic>V</italic> velocities are computed from the knowledge of distances and proper motions. Hence, <italic>U</italic> and <italic>V</italic> errors increase with distance which is not taken into account in the Monte Carlo simulation for the direct measurement, the RAVE sample being too small to estimate properly the error laws as a function of distances. This effect is also reduced in the unbiased measurement but is still present since the correcting term is added to the true error, hence distant stars will still have on average larger errors while nearer objects will have on average smaller errors.</p>
<p>Further constraints on the minor-to-major axis ratio are also available from independent studies. For example, the flattening of the dark halo has been estimated from the shape of the Sagittarius dwarf tidal stream. A value <italic>c</italic>/<italic>a</italic><sub>ρ</sub> > 0.7, with a preferred flattening of <italic>c</italic>/<italic>a</italic><sub>ρ</sub>≃ 1, is obtained by <xref ref-type="bibr" rid="b27">Ibata et al. (2001)</xref> and <xref ref-type="bibr" rid="b37">Majewski et al. (2003)</xref> using, respectively, carbon stars and M-giants from the 2MASS survey along the orbit of Sagittarius. <xref ref-type="bibr" rid="b28">Johnston, Law & Majewski (2005)</xref> give even stronger constraint 0.75 < <italic>c</italic>/<italic>a</italic><sub>ρ</sub> < 1.1 at a 3σ level, with oblate haloes strongly favoured if precession of Sgr's orbit is considered. In contrast, <xref ref-type="bibr" rid="b24">Helmi (2004)</xref> and <xref ref-type="bibr" rid="b34">Law, Johnston & Majewski (2005)</xref> demonstrated that only Galactic potentials with prolate haloes could reproduce the velocity trends in the leading debris with a preferred axis ratio <italic>c</italic>/<italic>a</italic><sub>ρ</sub>= 5/3. However, <xref ref-type="bibr" rid="b34">Law et al. (2005)</xref> explored a wide variety of Galactic potentials but failed to find a single orbit that can fit both the velocity trends and the sense of precession.</p>
<p>Looking back at the solar neighbourhood, if the Sgr stream is orbiting in oblate and spherical potentials, <xref ref-type="bibr" rid="b34">Law et al. (2005)</xref> and <xref ref-type="bibr" rid="b38">Martinez-Delgado et al. (2007)</xref> both predict that the Sun is currently bathing in a stream of debris from Sgr, passing both inside and outside the solar circle. Models orbiting in prolate potentials are on the other hand inconsistent with this prediction. <xref ref-type="bibr" rid="b4">Belokurov et al. (2006)</xref>, <xref ref-type="bibr" rid="b41">Newberg et al. (2006)</xref> and <xref ref-type="bibr" rid="b51">Seabroke et al. (2008)</xref> all provide strong evidence for the absence of Sgr debris in the solar neighbourhood. <xref ref-type="bibr" rid="b21">Fellhauer et al. (2006)</xref> argue that the origin of the bifurcation in the Sgr stream is only possible if the halo is close to spherical, as the angular difference between the branches is a measure of the precession of the orbital plane. This suggests that the absence of the Sgr stream near the Sun is consistent with nearly spherical and prolate Galactic potentials and seemingly inconsistent with oblate potentials. However, recently <xref ref-type="bibr" rid="b48">Ruzicka, Palous & Theis (2007)</xref> studied the Magellanic System – Milky Way interaction using test particle simulations and compared them to H <sc>i</sc> observations. They concluded that <italic>c</italic>/<italic>a</italic><sub>ρ</sub> < 1 values (oblate halo) are preferred and allow a better match to H <sc>i</sc> observations.</p>
<p>In <xref ref-type="fig" rid="f9">Fig. 9</xref>, the preferred region by most studies, 0.75 < <italic>c</italic>/<italic>a</italic><sub>ρ</sub> < 1, does not permit us to set strong constraints either on the flattening nor on the mass of the disc. The measured value of the tilt falls between the two classes of models in the allowed region, and the error bars on the RAVE measurement do not permit us to tighten the parameter space reliably. For strongly prolate haloes, as suggested by <xref ref-type="bibr" rid="b24">Helmi (2004)</xref>, the class of model II is preferred while short disc scalelength is marginally rejected at the 2σ level. If one adopts the axis ratio <italic>c</italic>/<italic>a</italic><sub>ρ</sub>= 1 as preferred by <xref ref-type="bibr" rid="b37">Majewski et al. (2003)</xref> or <xref ref-type="bibr" rid="b27">Ibata et al. (2001)</xref>, the value of the tilt is better recovered with a model whose scalelength of the disc lies in the range <italic>R</italic><sub>d</sub>= 2.5–2.7 kpc. Nevertheless, at the 1σ level, large and short values for <italic>R</italic><sub>d</sub> are permitted with this analysis.</p>
<p>Various studies in the literature have used star counts to constrain the scalelength of the thin disc. For example, recently <xref ref-type="bibr" rid="b29">Juric et al. (2008)</xref> measured the scalelength of the stellar disc and found <italic>R</italic><sub>d</sub>= 2.6 kpc (±20 per cent) using Sloan Digital Sky Survey data. Similarly, using data from the Bologna open cluster survey (BOCCE), <xref ref-type="bibr" rid="b11">Cignoni et al. (2008)</xref> found a scalelength in the range 2.25–3 kpc and <xref ref-type="bibr" rid="b43">Ojha (2001)</xref> found 2.8 kpc using the 2MASS survey. These values are in good agreement with our finding but have similarly large error bars. Refining our measurement will provide an independent constraint on the scalelength of the stellar disc.</p>
</sec>
<sec id="ss5">
<title>5 CONCLUSIONS</title>
<p>We measured the tilt of the velocity ellipsoid at ≃1 kpc below the Galactic plane using a sample of red clump giants from the RAVE DR2 catalogue. We find its inclination to be <inline-formula><inline-graphic xlink:href="mnras0391-0793-mu17.gif"></inline-graphic></inline-formula>. Estimates of the effect of contamination by foreground stars and substructures have been shown to be small and their effect on our measured value can be neglected.</p>
<p>We compared this value to predictions from two extreme cases of mass models for the Milky Way proposed by BT08. In the case of a massive disc with a small scalelength (<italic>R</italic><sub>d</sub>= 2 kpc), the inclination is compatible with an oblate halo whose minor-to-major axis ratio <italic>c</italic>/<italic>a</italic><sub>ρ</sub> is lower than 0.9 at the 1σ level. On the other hand, in the case of a massive halo with large disc scalelength (<italic>R</italic><sub>d</sub>≃ 3 kpc), prolate haloes are preferred with <italic>c</italic>/<italic>a</italic><sub>ρ</sub>≥ 1. When a direct measurement is used, low values for <italic>c</italic>/<italic>a</italic><sub>ρ</sub> can be marginally rejected, indicating that <italic>c</italic>/<italic>a</italic><sub>ρ</sub> > 0.7.</p>
<p>When further independent constraints from previous studies are considered, we find that an intermediate value for the disc scalelength <italic>R</italic><sub>d</sub>≃ 2.5–2.7 kpc is preferred for a nearly spherical halo, but no extreme model can be clearly ruled out, due to our large error bars. This range is in good agreement with other studies relying on star count analysis and deep photometric surveys. Nevertheless, these results have large error bars of the same order of as our measurement and cannot be used to further constrain the mass distribution.</p>
<p>RAVE continues to acquire spectra and this work relies on the second data release of the survey. So far, RAVE has collected more than 200 000 spectra, four times the size of the sample used here. With the current observing rate, we can expect to multiply by 10 the size of our sample in the coming years which will allow us to significantly reduce our error bars. By the end of the survey, we will be able to provide a new mass model for the Milky Way galaxy with a constrained scalelength of the disc and minor-to-major axis ratio of the dark halo.</p>
</sec>
</body>
<back>
<fn-group>
<fn id="fn1">
<label>1</label><p>The asymmetric drift is the tendency of a population of stars to lag behind the local standard of rest for its rotational velocity, the lag increasing as a function of age.</p>
</fn>
<fn id="fn2">
<label>2</label><p>The Galactic potentials are computed using the <sc>galpot</sc> program written by W. Dehnen. This program is available within the <sc>nemo</sc> package: <ext-link ext-link-type="uri" xlink:href="http://carma.astro.umd.edu/nemo/">http://carma.astro.umd.edu/nemo/</ext-link>.</p>
</fn>
</fn-group>
<ack>
<p>AS would like to thank the anonymous referee for a very constructive report and his comments that helped clarify this paper. Funding for RAVE has been provided by the Anglo-Australian Observatory, the Astrophysical Institute Potsdam, the Australian Research Council, the German Research foundation, the National Institute for Astrophysics at Padova, The Johns Hopkins University, the Netherlands Research School for Astronomy, the Natural Sciences and Engineering Research Council of Canada, the Slovenian Research Agency, the Swiss National Science Foundation, the National Science Foundation of the USA (AST-0508996), the Netherlands Organization for Scientific Research, the Particle Physics and Astronomy Research Council of the UK, Opticon, Strasbourg Observatory and the Universities of Basel, Cambridge, and Groningen. The RAVE web site is at <ext-link ext-link-type="uri" xlink:href="http://www.rave-survey.org">http://www.rave-survey.org</ext-link>.</p>
</ack>
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</article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>Estimation of the tilt of the stellar velocity ellipsoid from RAVE and implications for mass models</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Estimation of the tilt of the stellar velocity ellipsoid from RAVE and implications for mass models</title></titleInfo><name type="personal"><namePart type="given">A.</namePart><namePart type="family">Siebert</namePart><affiliation>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</affiliation><affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</affiliation><affiliation>E-mail: siebert@astro.u-strasbg.fr</affiliation><affiliation></affiliation><affiliation>E-mail: siebert@astro.u-strasbg.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">O.</namePart><namePart type="family">Bienaymé</namePart><affiliation>Université de Strasbourg, Observatoire Astronomique, Strasbourg, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="family">Binney</namePart><affiliation>Rudolf Peierls Centre for Theoretical Physics, Oxford</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="family">Bland-Hawthorn</namePart><affiliation>Anglo-Australian Observatory, Sydney, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">R.</namePart><namePart type="family">Campbell</namePart><affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</affiliation><affiliation>Macquary University, Sydney, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">K. C.</namePart><namePart type="family">Freeman</namePart><affiliation>Australian Natianal University, Canberra, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">B. K.</namePart><namePart type="family">Gibson</namePart><affiliation>University of Central Lancashire, Preston</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">G.</namePart><namePart type="family">Gilmore</namePart><affiliation>Institute of Astronomy, Cambridge</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">E. K.</namePart><namePart type="family">Grebel</namePart><affiliation>Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Heidelberg, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="family">Helmi</namePart><affiliation>Kapteyn Astronomical Institut, Groningen, the Netherlands</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">U.</namePart><namePart type="family">Munari</namePart><affiliation>Astronomical Observatory of Padova in Asiago, Asiago, Italy</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J. F.</namePart><namePart type="family">Navarro</namePart><affiliation>University of Victoria, Victoria, Canada</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Q. A.</namePart><namePart type="family">Parker</namePart><affiliation>Anglo-Australian Observatory, Sydney, Australia</affiliation><affiliation>Macquary University, Sydney, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">G.</namePart><namePart type="family">Seabroke</namePart><affiliation>Institute of Astronomy, Cambridge</affiliation><affiliation>e2v Centre for Electronic Imaging, Planetary and Space Sciences Research Institute, The Open University, Milton Keynes</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="family">Siviero</namePart><affiliation>Astronomical Observatory of Padova in Asiago, Asiago, Italy</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">M.</namePart><namePart type="family">Steinmetz</namePart><affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">M.</namePart><namePart type="family">Williams</namePart><affiliation>Astrophysikalishes Institut Potsdam, Potsdam, Germany</affiliation><affiliation>Australian Natianal University, Canberra, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">R. F. G.</namePart><namePart type="family">Wyse</namePart><affiliation>Johns Hopkins University, Baltimore, MD, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">T.</namePart><namePart type="family">Zwitter</namePart><affiliation>Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="research-article" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Blackwell Publishing Ltd</publisher><place><placeTerm type="text">Oxford, UK</placeTerm></place><dateIssued encoding="w3cdtf">2008-12-01</dateIssued><dateCreated encoding="w3cdtf">2008-11-27</dateCreated><copyrightDate encoding="w3cdtf">2008</copyrightDate></originInfo><abstract>We present a measure of the inclination of the velocity ellipsoid at 1 kpc below the Galactic plane using a sample of red clump giants from the RAdial Velocity Experiment (RAVE) Data Release 2. We find that the velocity ellipsoid is tilted towards the Galactic plane with an inclination of . We compare this value to computed inclinations for two mass models of the Milky Way. We find that our measurement is consistent with a short scalelength of the stellar disc (Rd≃ 2 kpc) if the dark halo is oblate or with a long scalelength (Rd≃ 3 kpc) if the dark halo is prolate. Once combined with independent constraints on the flattening of the halo, our measurement suggests that the scalelength is approximately halfway between these two extreme values, with a preferred range 2.5–2.7 kpc for a nearly spherical halo. Nevertheless, no model can be clearly ruled out. With the continuation of the RAVE survey, it will be possible to provide a strong constraint on the mass distribution of the Milky Way using refined measurements of the orientation of the velocity ellipsoid.</abstract><subject><genre>keywords</genre><topic>stars: kinematics</topic><topic>galaxy: fundamental parameters</topic><topic>galaxy: kinematics and dynamics</topic></subject><relatedItem type="host"><titleInfo><title>Monthly Notices of the Royal Astronomical Society</title></titleInfo><titleInfo type="abbreviated"><title>Monthly Notices of the Royal Astronomical Society</title></titleInfo><genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><subject><topic>Papers</topic></subject><identifier type="ISSN">0035-8711</identifier><identifier type="eISSN">1365-2966</identifier><identifier type="PublisherID">mnras</identifier><identifier type="PublisherID-hwp">mnras</identifier><part><date>2008</date><detail type="volume"><caption>vol.</caption><number>391</number></detail><detail type="issue"><caption>no.</caption><number>2</number></detail><extent unit="pages"><start>793</start><end>801</end></extent></part></relatedItem><identifier type="istex">DCC5AB80FD5FE4FDE4D62204536B7BC5B9B98A5B</identifier><identifier type="DOI">10.1111/j.1365-2966.2008.13912.x</identifier><accessCondition type="use and reproduction" contentType="copyright">© 2008 The Authors. 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