We present a method to constrain the potential of the non-axisymmetric
components of the Galaxy using the kinematics of stars in the solar
neighborhood. The basic premise is that dynamical substructures in phase-space
(i.e. due to the bar and/or spiral arms) are associated with families of
periodic or irregular orbits, which may be easily identified in orbital
frequency space. We use the "observed" positions and velocities of stars as
initial conditions for orbital integrations in a variety of gravitational
potentials. We then compute their characteristic frequencies, and study the
structure present in the frequency maps. We find that the distribution of
dynamical substructures in velocity- and frequency-space is best preserved when
the integrations are performed in the "true" gravitational potential.Comment: 2 pages, 4 figures, to appear in the proceedings of "Assembling the
  Puzzle of the Milky Way", Le Grand Bornand (Apr. 17-22, 2011

Antoja Castelltort, Teresa

Helmi, A.

Monari, G.

Reyle, C

Robin, A

Schultheis, M

English

arXiv

We present a method to constrain the potential of the non-axisymmetric components of the Galaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamical substructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodic or irregular orbits, which may be easily identified in orbital frequency space. We use the “observed” positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitational potentials. We then compute their characteristic frequencies, and study the structure present in the frequency maps. We find that the distribution of dynamical substructures in velocity- and frequency-space is best preserved when the integrations are performed in the “true” gravitational potential

Antoja T.

Monari G.

Helmi A.

Directory of Open Access Journals

EPJ Web of Conferences

Constraining the Milky Way potential using the dynamical kinematic substructures

We present a method to constrain the potential of the non-axisymmetric components of the Galaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamical substructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodic or irregular orbits, which may be easily identified in orbital frequency space. We use the "observed" positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitational potentials. We then compute their characteristic frequencies, and study the structure present in the frequency maps. We find that the distribution of dynamical substructures in velocity- and frequency-space is best preserved when the integrations are performed in the "true" gravitational potential.</p

University of Groningen

University of GroningenConstraining the Milky Way potential using the dynamical kinematic substructuresMonari, G.; Antoja Castelltort, Teresa; Helmi, A.Published in:ASSEMBLING THE PUZZLE OF THE MILKY WAYDOI:10.1051/epjconf/20121907007IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Monari, G., Antoja Castelltort, T., & Helmi, A. (2012). Constraining the Milky Way potential using thedynamical kinematic substructures. In C. Reyle, A. Robin, & M. Schultheis (Eds.), ASSEMBLING THEPUZZLE OF THE MILKY WAY Article 07007 (EPJ Web of Conferences; Vol. 19). E D P SCIENCES.https://doi.org/10.1051/epjconf/20121907007CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-12-2025EPJ Web of Conferences 19, 07007 (2012)DOI: 10.1051/epjconf/20121907007C© Owned by the authors, published by EDP Sciences, 2012Constraining the Milky Way potential using the dynamicalkinematic substructuresG. Monaria, T. Antoja and A. HelmiKapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen,The NetherlandsAbstract. We present a method to constrain the potential of the non-axisymmetric components of theGalaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamicalsubstructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodicor irregular orbits, which may be easily identified in orbital frequency space. We use the “observed”positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitationalpotentials. We then compute their characteristic frequencies, and study the structure present in the frequencymaps. We find that the distribution of dynamical substructures in velocity- and frequency-space is bestpreserved when the integrations are performed in the “true” gravitational potential.1. INTRODUCTIONThe bar and the spiral arms are two examples of non-axisymmetric features in the Galaxy. Their exactproperties are not well known. Some of these may be constrained from their effect on the velocitydistribution of stars near the Sun (e.g. [1–4]). Here we consider toy models of the Milky Way thatinclude a bar with pattern speed b = 49 km/(sec kpc), with an angle of 20 degrees with respect to theline Sun – Galactic Center and an axisymmetric logarithmic potential with a 220 km/sec flat rotationcurve, following [1] and [2]. The toy model has been set up by generating initial conditions for Nparticles distributed in a 2D disk embedded in this potential. These have been integrated using theBulirsch-Stoer algorithm in a reference frame co-rotating with the bar with a conservation of JacobiIntegral |EJ /EJ | < 10−9, for sufficient time to have stationary structures in the velocity plane of the“solar neighbourhood”.2. CHARACTERIZATION OF THE SOLAR NEIGHBOURHOODWe focus on the motions and orbits of particles crossing a “solar neighbourhood” volume. To find thefrequencies of an orbit we proceed as follows. We compute the discrete Fourier transform of the timesequence of the radial and azimuthal coordinates of an orbit (cf. [5]). The highest peaks in the Fourierspectra correspond to the orbit’s principal frequencies. When the ratio between the principal frequenciesis a rational number, the orbit is said to be “resonant”. This method is capable also to classify stronglyirregular orbits. The kinematics in the “solar neighbourhood” of our toy model is far from being smooth(fig. 1, left panel): substructures (moving groups) can be observed. These moving groups are populatedby families of resonant or irregular orbits, as shown by the frequency map (fig. 2, left panel).ae-mail: monari@astro.rug.nlThis is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, whichpermits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20121907007EPJ Web of Conferences-100 -50 0 50 100U (kms-1)-100-50050100V (kms-1 )-100-50 0 50 100-100 -50  0  50  100V/(Km/sec)U(Km/sec)1:12:13:14:1IrregularsFigure 1. Left panel: density of “stars” in the velocity plane neighborhood (200 pc) obtained with a wavelettransform (described in [3]), for our toy model of a barred Milky Way. Right panel: “Stars” in the velocity plane ofthe left panel, color-coded according to the orbit families shown in the left panel of fig. 2. 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))1:12:13:14:1Irregulars 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))Wrong potential, faster bar, Ωb=55Km/(sec Kpc)1:1 in the true potential2:1 in the true potential3:1 in the true potential4:1 in the true potentialIrregulars in the true potentialFigure 2. Left: frequency plane relative to the “stars” of fig. 1, obtained with the true potential. The different colorsindicate families of irregular or resonant orbits. Note that these define straight lines in the frequency plane. Right:Example of a frequency map obtained by integrating the observed positions and velocities of “stars” in fig. 1 in adifferent and wrong potential (a faster bar, in this case). The color coding corresponds to the same points of thevelocity plane in the right panel of fig. 1.3. FREQUENCY MAPS TO CONSTRAIN THE POTENTIALWe obtain frequency maps (e.g. fig. 2) by integrating the observed positions and velocities of “stars”in fig.1 in different potentials (i.e. with varying characteristic parameters). The substructure in velocity-space appears to be better maintained in frequency space for the right potential (fig. 2, left panel). In thefrequency maps with “wrong” potentials points originally on resonant lines are now mixed. To quantifythe degree of clustering and order in the 4D space (U , V , R , ) we measure the information entropyfor the various potentials explored. Fig. 2 shows that the structure of this space is quite different fordifferent parameters, and this is also reflected in the value of the entropy.References[1] Dehnen, W. 2000, AJ, 119, 800[2] Fux, R. 2001, A&A, 373, 511[3] Antoja, T., Valenzuela, O., Pichardo, B. et al. 2009, ApJ, 700L, 78A[4] Quillen, A. C. & Minchev, I. 2005, AJ, 130, 576Q[5] Ceverino, D. & Klypin, A. 2007, MNRAS, 379, 115507007-p.2

   University of GroningenConstraining the Milky Way potential using the dynamical kinematic substructuresMonari, G.; Antoja Castelltort, Teresa; Helmi, A.Published in:ASSEMBLING THE PUZZLE OF THE MILKY WAYDOI:10.1051/epjconf/20121907007IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Monari, G., Antoja Castelltort, T., & Helmi, A. (2012). Constraining the Milky Way potential using thedynamical kinematic substructures. In C. Reyle, A. Robin, & M. Schultheis (Eds.), ASSEMBLING THEPUZZLE OF THE MILKY WAY [07007] (EPJ Web of Conferences; Vol. 19). E D P SCIENCES.https://doi.org/10.1051/epjconf/20121907007CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 26-04-2023EPJ Web of Conferences 19, 07007 (2012)DOI: 10.1051/epjconf/20121907007C© Owned by the authors, published by EDP Sciences, 2012Constraining the Milky Way potential using the dynamicalkinematic substructuresG. Monaria, T. Antoja and A. HelmiKapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen,The NetherlandsAbstract. We present a method to constrain the potential of the non-axisymmetric components of theGalaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamicalsubstructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodicor irregular orbits, which may be easily identified in orbital frequency space. We use the “observed”positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitationalpotentials. We then compute their characteristic frequencies, and study the structure present in the frequencymaps. We find that the distribution of dynamical substructures in velocity- and frequency-space is bestpreserved when the integrations are performed in the “true” gravitational potential.1. INTRODUCTIONThe bar and the spiral arms are two examples of non-axisymmetric features in the Galaxy. Their exactproperties are not well known. Some of these may be constrained from their effect on the velocitydistribution of stars near the Sun (e.g. [1–4]). Here we consider toy models of the Milky Way thatinclude a bar with pattern speed b = 49 km/(sec kpc), with an angle of 20 degrees with respect to theline Sun – Galactic Center and an axisymmetric logarithmic potential with a 220 km/sec flat rotationcurve, following [1] and [2]. The toy model has been set up by generating initial conditions for Nparticles distributed in a 2D disk embedded in this potential. These have been integrated using theBulirsch-Stoer algorithm in a reference frame co-rotating with the bar with a conservation of JacobiIntegral |EJ /EJ | < 10−9, for sufficient time to have stationary structures in the velocity plane of the“solar neighbourhood”.2. CHARACTERIZATION OF THE SOLAR NEIGHBOURHOODWe focus on the motions and orbits of particles crossing a “solar neighbourhood” volume. To find thefrequencies of an orbit we proceed as follows. We compute the discrete Fourier transform of the timesequence of the radial and azimuthal coordinates of an orbit (cf. [5]). The highest peaks in the Fourierspectra correspond to the orbit’s principal frequencies. When the ratio between the principal frequenciesis a rational number, the orbit is said to be “resonant”. This method is capable also to classify stronglyirregular orbits. The kinematics in the “solar neighbourhood” of our toy model is far from being smooth(fig. 1, left panel): substructures (moving groups) can be observed. These moving groups are populatedby families of resonant or irregular orbits, as shown by the frequency map (fig. 2, left panel).ae-mail: monari@astro.rug.nlThis is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, whichpermits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20121907007EPJ Web of Conferences-100 -50 0 50 100U (kms-1)-100-50050100V (kms-1 )-100-50 0 50 100-100 -50  0  50  100V/(Km/sec)U(Km/sec)1:12:13:14:1IrregularsFigure 1. Left panel: density of “stars” in the velocity plane neighborhood (200 pc) obtained with a wavelettransform (described in [3]), for our toy model of a barred Milky Way. Right panel: “Stars” in the velocity plane ofthe left panel, color-coded according to the orbit families shown in the left panel of fig. 2. 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))1:12:13:14:1Irregulars 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))Wrong potential, faster bar, Ωb=55Km/(sec Kpc)1:1 in the true potential2:1 in the true potential3:1 in the true potential4:1 in the true potentialIrregulars in the true potentialFigure 2. Left: frequency plane relative to the “stars” of fig. 1, obtained with the true potential. The different colorsindicate families of irregular or resonant orbits. Note that these define straight lines in the frequency plane. Right:Example of a frequency map obtained by integrating the observed positions and velocities of “stars” in fig. 1 in adifferent and wrong potential (a faster bar, in this case). The color coding corresponds to the same points of thevelocity plane in the right panel of fig. 1.3. FREQUENCY MAPS TO CONSTRAIN THE POTENTIALWe obtain frequency maps (e.g. fig. 2) by integrating the observed positions and velocities of “stars”in fig.1 in different potentials (i.e. with varying characteristic parameters). The substructure in velocity-space appears to be better maintained in frequency space for the right potential (fig. 2, left panel). In thefrequency maps with “wrong” potentials points originally on resonant lines are now mixed. To quantifythe degree of clustering and order in the 4D space (U , V , R , ) we measure the information entropyfor the various potentials explored. Fig. 2 shows that the structure of this space is quite different fordifferent parameters, and this is also reflected in the value of the entropy.References[1] Dehnen, W. 2000, AJ, 119, 800[2] Fux, R. 2001, A&A, 373, 511[3] Antoja, T., Valenzuela, O., Pichardo, B. et al. 2009, ApJ, 700L, 78A[4] Quillen, A. C. & Minchev, I. 2005, AJ, 130, 576Q[5] Ceverino, D. & Klypin, A. 2007, MNRAS, 379, 115507007-p.2

ARTS repository - University of Groningen

   University of GroningenConstraining the Milky Way potential using the dynamical kinematic substructuresMonari, G.; Antoja Castelltort, Teresa; Helmi, A.Published in:ASSEMBLING THE PUZZLE OF THE MILKY WAYDOI:10.1051/epjconf/20121907007IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Monari, G., Antoja Castelltort, T., & Helmi, A. (2012). Constraining the Milky Way potential using thedynamical kinematic substructures. In C. Reyle, A. Robin, & M. Schultheis (Eds.), ASSEMBLING THEPUZZLE OF THE MILKY WAY [07007] (EPJ Web of Conferences; Vol. 19). E D P SCIENCES.https://doi.org/10.1051/epjconf/20121907007CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 14-02-2023EPJ Web of Conferences 19, 07007 (2012)DOI: 10.1051/epjconf/20121907007C© Owned by the authors, published by EDP Sciences, 2012Constraining the Milky Way potential using the dynamicalkinematic substructuresG. Monaria, T. Antoja and A. HelmiKapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen,The NetherlandsAbstract. We present a method to constrain the potential of the non-axisymmetric components of theGalaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamicalsubstructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodicor irregular orbits, which may be easily identified in orbital frequency space. We use the “observed”positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitationalpotentials. We then compute their characteristic frequencies, and study the structure present in the frequencymaps. We find that the distribution of dynamical substructures in velocity- and frequency-space is bestpreserved when the integrations are performed in the “true” gravitational potential.1. INTRODUCTIONThe bar and the spiral arms are two examples of non-axisymmetric features in the Galaxy. Their exactproperties are not well known. Some of these may be constrained from their effect on the velocitydistribution of stars near the Sun (e.g. [1–4]). Here we consider toy models of the Milky Way thatinclude a bar with pattern speed b = 49 km/(sec kpc), with an angle of 20 degrees with respect to theline Sun – Galactic Center and an axisymmetric logarithmic potential with a 220 km/sec flat rotationcurve, following [1] and [2]. The toy model has been set up by generating initial conditions for Nparticles distributed in a 2D disk embedded in this potential. These have been integrated using theBulirsch-Stoer algorithm in a reference frame co-rotating with the bar with a conservation of JacobiIntegral |EJ /EJ | < 10−9, for sufficient time to have stationary structures in the velocity plane of the“solar neighbourhood”.2. CHARACTERIZATION OF THE SOLAR NEIGHBOURHOODWe focus on the motions and orbits of particles crossing a “solar neighbourhood” volume. To find thefrequencies of an orbit we proceed as follows. We compute the discrete Fourier transform of the timesequence of the radial and azimuthal coordinates of an orbit (cf. [5]). The highest peaks in the Fourierspectra correspond to the orbit’s principal frequencies. When the ratio between the principal frequenciesis a rational number, the orbit is said to be “resonant”. This method is capable also to classify stronglyirregular orbits. The kinematics in the “solar neighbourhood” of our toy model is far from being smooth(fig. 1, left panel): substructures (moving groups) can be observed. These moving groups are populatedby families of resonant or irregular orbits, as shown by the frequency map (fig. 2, left panel).ae-mail: monari@astro.rug.nlThis is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, whichpermits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20121907007EPJ Web of Conferences-100 -50 0 50 100U (kms-1)-100-50050100V (kms-1 )-100-50 0 50 100-100 -50  0  50  100V/(Km/sec)U(Km/sec)1:12:13:14:1IrregularsFigure 1. Left panel: density of “stars” in the velocity plane neighborhood (200 pc) obtained with a wavelettransform (described in [3]), for our toy model of a barred Milky Way. Right panel: “Stars” in the velocity plane ofthe left panel, color-coded according to the orbit families shown in the left panel of fig. 2. 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))1:12:13:14:1Irregulars 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))Wrong potential, faster bar, Ωb=55Km/(sec Kpc)1:1 in the true potential2:1 in the true potential3:1 in the true potential4:1 in the true potentialIrregulars in the true potentialFigure 2. Left: frequency plane relative to the “stars” of fig. 1, obtained with the true potential. The different colorsindicate families of irregular or resonant orbits. Note that these define straight lines in the frequency plane. Right:Example of a frequency map obtained by integrating the observed positions and velocities of “stars” in fig. 1 in adifferent and wrong potential (a faster bar, in this case). The color coding corresponds to the same points of thevelocity plane in the right panel of fig. 1.3. FREQUENCY MAPS TO CONSTRAIN THE POTENTIALWe obtain frequency maps (e.g. fig. 2) by integrating the observed positions and velocities of “stars”in fig.1 in different potentials (i.e. with varying characteristic parameters). The substructure in velocity-space appears to be better maintained in frequency space for the right potential (fig. 2, left panel). In thefrequency maps with “wrong” potentials points originally on resonant lines are now mixed. To quantifythe degree of clustering and order in the 4D space (U , V , R , ) we measure the information entropyfor the various potentials explored. Fig. 2 shows that the structure of this space is quite different fordifferent parameters, and this is also reflected in the value of the entropy.References[1] Dehnen, W. 2000, AJ, 119, 800[2] Fux, R. 2001, A&A, 373, 511[3] Antoja, T., Valenzuela, O., Pichardo, B. et al. 2009, ApJ, 700L, 78A[4] Quillen, A. C. & Minchev, I. 2005, AJ, 130, 576Q[5] Ceverino, D. & Klypin, A. 2007, MNRAS, 379, 115507007-p.2

   University of GroningenConstraining the Milky Way potential using the dynamical kinematic substructuresMonari, G.; Antoja Castelltort, Teresa; Helmi, A.Published in:ASSEMBLING THE PUZZLE OF THE MILKY WAYDOI:10.1051/epjconf/20121907007IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Monari, G., Antoja Castelltort, T., & Helmi, A. (2012). Constraining the Milky Way potential using thedynamical kinematic substructures. In C. Reyle, A. Robin, & M. Schultheis (Eds.), ASSEMBLING THEPUZZLE OF THE MILKY WAY [07007] (EPJ Web of Conferences; Vol. 19). E D P SCIENCES.https://doi.org/10.1051/epjconf/20121907007CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 18-11-2022EPJ Web of Conferences 19, 07007 (2012)DOI: 10.1051/epjconf/20121907007C© Owned by the authors, published by EDP Sciences, 2012Constraining the Milky Way potential using the dynamicalkinematic substructuresG. Monaria, T. Antoja and A. HelmiKapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen,The NetherlandsAbstract. We present a method to constrain the potential of the non-axisymmetric components of theGalaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamicalsubstructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodicor irregular orbits, which may be easily identified in orbital frequency space. We use the “observed”positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitationalpotentials. We then compute their characteristic frequencies, and study the structure present in the frequencymaps. We find that the distribution of dynamical substructures in velocity- and frequency-space is bestpreserved when the integrations are performed in the “true” gravitational potential.1. INTRODUCTIONThe bar and the spiral arms are two examples of non-axisymmetric features in the Galaxy. Their exactproperties are not well known. Some of these may be constrained from their effect on the velocitydistribution of stars near the Sun (e.g. [1–4]). Here we consider toy models of the Milky Way thatinclude a bar with pattern speed b = 49 km/(sec kpc), with an angle of 20 degrees with respect to theline Sun – Galactic Center and an axisymmetric logarithmic potential with a 220 km/sec flat rotationcurve, following [1] and [2]. The toy model has been set up by generating initial conditions for Nparticles distributed in a 2D disk embedded in this potential. These have been integrated using theBulirsch-Stoer algorithm in a reference frame co-rotating with the bar with a conservation of JacobiIntegral |EJ /EJ | < 10−9, for sufficient time to have stationary structures in the velocity plane of the“solar neighbourhood”.2. CHARACTERIZATION OF THE SOLAR NEIGHBOURHOODWe focus on the motions and orbits of particles crossing a “solar neighbourhood” volume. To find thefrequencies of an orbit we proceed as follows. We compute the discrete Fourier transform of the timesequence of the radial and azimuthal coordinates of an orbit (cf. [5]). The highest peaks in the Fourierspectra correspond to the orbit’s principal frequencies. When the ratio between the principal frequenciesis a rational number, the orbit is said to be “resonant”. This method is capable also to classify stronglyirregular orbits. The kinematics in the “solar neighbourhood” of our toy model is far from being smooth(fig. 1, left panel): substructures (moving groups) can be observed. These moving groups are populatedby families of resonant or irregular orbits, as shown by the frequency map (fig. 2, left panel).ae-mail: monari@astro.rug.nlThis is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, whichpermits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20121907007EPJ Web of Conferences-100 -50 0 50 100U (kms-1)-100-50050100V (kms-1 )-100-50 0 50 100-100 -50  0  50  100V/(Km/sec)U(Km/sec)1:12:13:14:1IrregularsFigure 1. Left panel: density of “stars” in the velocity plane neighborhood (200 pc) obtained with a wavelettransform (described in [3]), for our toy model of a barred Milky Way. Right panel: “Stars” in the velocity plane ofthe left panel, color-coded according to the orbit families shown in the left panel of fig. 2. 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))1:12:13:14:1Irregulars 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))Wrong potential, faster bar, Ωb=55Km/(sec Kpc)1:1 in the true potential2:1 in the true potential3:1 in the true potential4:1 in the true potentialIrregulars in the true potentialFigure 2. Left: frequency plane relative to the “stars” of fig. 1, obtained with the true potential. The different colorsindicate families of irregular or resonant orbits. Note that these define straight lines in the frequency plane. Right:Example of a frequency map obtained by integrating the observed positions and velocities of “stars” in fig. 1 in adifferent and wrong potential (a faster bar, in this case). The color coding corresponds to the same points of thevelocity plane in the right panel of fig. 1.3. FREQUENCY MAPS TO CONSTRAIN THE POTENTIALWe obtain frequency maps (e.g. fig. 2) by integrating the observed positions and velocities of “stars”in fig.1 in different potentials (i.e. with varying characteristic parameters). The substructure in velocity-space appears to be better maintained in frequency space for the right potential (fig. 2, left panel). In thefrequency maps with “wrong” potentials points originally on resonant lines are now mixed. To quantifythe degree of clustering and order in the 4D space (U , V , R , ) we measure the information entropyfor the various potentials explored. Fig. 2 shows that the structure of this space is quite different fordifferent parameters, and this is also reflected in the value of the entropy.References[1] Dehnen, W. 2000, AJ, 119, 800[2] Fux, R. 2001, A&A, 373, 511[3] Antoja, T., Valenzuela, O., Pichardo, B. et al. 2009, ApJ, 700L, 78A[4] Quillen, A. C. & Minchev, I. 2005, AJ, 130, 576Q[5] Ceverino, D. & Klypin, A. 2007, MNRAS, 379, 115507007-p.2

University of GroningenConstraining the Milky Way potential using the dynamical kinematic substructuresMonari, G.; Antoja Castelltort, Teresa; Helmi, A.Published in:ASSEMBLING THE PUZZLE OF THE MILKY WAYDOI:10.1051/epjconf/20121907007IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Monari, G., Antoja Castelltort, T., & Helmi, A. (2012). Constraining the Milky Way potential using thedynamical kinematic substructures. In C. Reyle, A. Robin, & M. Schultheis (Eds.), ASSEMBLING THEPUZZLE OF THE MILKY WAY Article 07007 (EPJ Web of Conferences; Vol. 19). E D P SCIENCES.https://doi.org/10.1051/epjconf/20121907007CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 24-12-2025EPJ Web of Conferences 19, 07007 (2012)DOI: 10.1051/epjconf/20121907007C© Owned by the authors, published by EDP Sciences, 2012Constraining the Milky Way potential using the dynamicalkinematic substructuresG. Monaria, T. Antoja and A. HelmiKapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen,The NetherlandsAbstract. We present a method to constrain the potential of the non-axisymmetric components of theGalaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamicalsubstructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodicor irregular orbits, which may be easily identified in orbital frequency space. We use the “observed”positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitationalpotentials. We then compute their characteristic frequencies, and study the structure present in the frequencymaps. We find that the distribution of dynamical substructures in velocity- and frequency-space is bestpreserved when the integrations are performed in the “true” gravitational potential.1. INTRODUCTIONThe bar and the spiral arms are two examples of non-axisymmetric features in the Galaxy. Their exactproperties are not well known. Some of these may be constrained from their effect on the velocitydistribution of stars near the Sun (e.g. [1–4]). Here we consider toy models of the Milky Way thatinclude a bar with pattern speed b = 49 km/(sec kpc), with an angle of 20 degrees with respect to theline Sun – Galactic Center and an axisymmetric logarithmic potential with a 220 km/sec flat rotationcurve, following [1] and [2]. The toy model has been set up by generating initial conditions for Nparticles distributed in a 2D disk embedded in this potential. These have been integrated using theBulirsch-Stoer algorithm in a reference frame co-rotating with the bar with a conservation of JacobiIntegral |EJ /EJ | < 10−9, for sufficient time to have stationary structures in the velocity plane of the“solar neighbourhood”.2. CHARACTERIZATION OF THE SOLAR NEIGHBOURHOODWe focus on the motions and orbits of particles crossing a “solar neighbourhood” volume. To find thefrequencies of an orbit we proceed as follows. We compute the discrete Fourier transform of the timesequence of the radial and azimuthal coordinates of an orbit (cf. [5]). The highest peaks in the Fourierspectra correspond to the orbit’s principal frequencies. When the ratio between the principal frequenciesis a rational number, the orbit is said to be “resonant”. This method is capable also to classify stronglyirregular orbits. The kinematics in the “solar neighbourhood” of our toy model is far from being smooth(fig. 1, left panel): substructures (moving groups) can be observed. These moving groups are populatedby families of resonant or irregular orbits, as shown by the frequency map (fig. 2, left panel).ae-mail: monari@astro.rug.nlThis is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, whichpermits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20121907007EPJ Web of Conferences-100 -50 0 50 100U (kms-1)-100-50050100V (kms-1 )-100-50 0 50 100-100 -50  0  50  100V/(Km/sec)U(Km/sec)1:12:13:14:1IrregularsFigure 1. Left panel: density of “stars” in the velocity plane neighborhood (200 pc) obtained with a wavelettransform (described in [3]), for our toy model of a barred Milky Way. Right panel: “Stars” in the velocity plane ofthe left panel, color-coded according to the orbit families shown in the left panel of fig. 2. 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))1:12:13:14:1Irregulars 0 10 20 30 40 50 0  20  40  60  80  100Ωφ(Km/(sec Kpc))ΩR(Km/(sec Kpc))Wrong potential, faster bar, Ωb=55Km/(sec Kpc)1:1 in the true potential2:1 in the true potential3:1 in the true potential4:1 in the true potentialIrregulars in the true potentialFigure 2. Left: frequency plane relative to the “stars” of fig. 1, obtained with the true potential. The different colorsindicate families of irregular or resonant orbits. Note that these define straight lines in the frequency plane. Right:Example of a frequency map obtained by integrating the observed positions and velocities of “stars” in fig. 1 in adifferent and wrong potential (a faster bar, in this case). The color coding corresponds to the same points of thevelocity plane in the right panel of fig. 1.3. FREQUENCY MAPS TO CONSTRAIN THE POTENTIALWe obtain frequency maps (e.g. fig. 2) by integrating the observed positions and velocities of “stars”in fig.1 in different potentials (i.e. with varying characteristic parameters). The substructure in velocity-space appears to be better maintained in frequency space for the right potential (fig. 2, left panel). In thefrequency maps with “wrong” potentials points originally on resonant lines are now mixed. To quantifythe degree of clustering and order in the 4D space (U , V , R , ) we measure the information entropyfor the various potentials explored. Fig. 2 shows that the structure of this space is quite different fordifferent parameters, and this is also reflected in the value of the entropy.References[1] Dehnen, W. 2000, AJ, 119, 800[2] Fux, R. 2001, A&A, 373, 511[3] Antoja, T., Valenzuela, O., Pichardo, B. et al. 2009, ApJ, 700L, 78A[4] Quillen, A. C. & Minchev, I. 2005, AJ, 130, 576Q[5] Ceverino, D. & Klypin, A. 2007, MNRAS, 379, 115507007-p.2

G. Monari

T. Antoja

A. Helmi

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We present a method to constrain the potential of the non-axisymmetric components of the Galaxy using the kinematics of stars in the solar neighborhood. The basic premise is that dynamical substructures in phase-space (i.e. due to the bar and/or spiral arms) are associated with families of periodic or irregular orbits, which may be easily identified in orbital frequency space. We use the "observed" positions and velocities of stars as initial conditions for orbital integrations in a variety of gravitational potentials. We then compute their characteristic frequencies, and study the structure present in the frequency maps. We find that the distribution of dynamical substructures in velocity- and frequency-space is best preserved when the integrations are performed in the "true" gravitational potential

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Constraining the Milky Way potential using the dynamical kinematic substructures

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