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Rotation Curves of Spiral Galaxies

Image of the Andromeda galaxy (M31) with rotation curve superimposed (Turner,2000)

Margutti RaffaellaMargutti Raffaella

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IndexIndex DefinitionDefinition Historical backgroundHistorical background Measuring rotation velocitiesMeasuring rotation velocities 1. Preliminaries: the doppler effect1. Preliminaries: the doppler effect 2. Emission lines2. Emission lines 3.Analysis of the observational data3.Analysis of the observational data 4.Instrumental resolution4.Instrumental resolution 5.Dependence on observational methods5.Dependence on observational methods Observational properties of rotation curvesObservational properties of rotation curves 1.Universal properties of rotation curves1.Universal properties of rotation curves 2.Morphology & RC2.Morphology & RC 3.Luminosity & RC3.Luminosity & RC 4.Environment & RC4.Environment & RC 5.Evolution & RC5.Evolution & RC Astrophysical importance of rotation curvesAstrophysical importance of rotation curves ReferencesReferences

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Definition:Definition:

A rotation curve (RC) of a galaxy is defined as the trace of velocities on a position-velocity (PV) diagram along the major axis, corrected for the angle between the line of sight and the galaxy disk. (See figure)

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Historical backgroundHistorical background 19141914 Slipher discovered the rotation of M31 and Sombrero galaxy Slipher discovered the rotation of M31 and Sombrero galaxy Wolf detected “inclined” lines in M81 nuclear spectrumWolf detected “inclined” lines in M81 nuclear spectrum 19391939 Babcock (and later Mayall in 1951) extended M31 RC to almost 2° from Babcock (and later Mayall in 1951) extended M31 RC to almost 2° from

the nucleus. His mass model for M31 showed the mass-to-light (M/L) ratio to the nucleus. His mass model for M31 showed the mass-to-light (M/L) ratio to increase from inner regions to outer ones.increase from inner regions to outer ones.

19541954 Schwarzschild reanalyzed the velocities of M31 and fit with a model of Schwarzschild reanalyzed the velocities of M31 and fit with a model of constant M/Lconstant M/L

19591959de Vaucouleurs affirmed that “ the rotation velocity decrease with de Vaucouleurs affirmed that “ the rotation velocity decrease with increasing distance from the center and tends asymptotically toward Kepler’s third increasing distance from the center and tends asymptotically toward Kepler’s third law”. (It’s important here to remark that high signal-to-noise velocities were NOT law”. (It’s important here to remark that high signal-to-noise velocities were NOT available at that time.)available at that time.)

De Vaucouleurs and Schwarzschild’s authority make it possible for astronomers to De Vaucouleurs and Schwarzschild’s authority make it possible for astronomers to ignore the strange RC identified by Babcock, Oort and Mayall.ignore the strange RC identified by Babcock, Oort and Mayall.

19601960The modern era begins with Page and Burbidge & Burbidge. Their The modern era begins with Page and Burbidge & Burbidge. Their observations were based on Hobservations were based on Hαα and [NII] and [NII] emission lines. In the same period we emission lines. In the same period we have also the first radio observations of neutral hydrogen which showed a slowly have also the first radio observations of neutral hydrogen which showed a slowly falling RC for M31 and a flat one for M33.falling RC for M31 and a flat one for M33.

20052005 A rich variety of techniques are nowadays available ( A rich variety of techniques are nowadays available ( HHαα ,HI, [NII], ,HI, [NII], [SiII],CO, [SiII],CO, masers emission lines ...), together with masers emission lines ...), together with higher signal-to-noise velocities higher signal-to-noise velocities and high spatial resolution.and high spatial resolution.

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MeasuringMeasuring rotation velocitiesrotation velocities

Hypothesis:Hypothesis:1.1.Axially symmetric distribution of matter cylindricalAxially symmetric distribution of matter cylindrical2.2.Circular orbits of matter coordinates areCircular orbits of matter coordinates are3.3.Galactic disk with circular shape. suitable Galactic disk with circular shape. suitable

The doppler effect can be used to derive the radial velocity (VThe doppler effect can be used to derive the radial velocity (Vrr) of ) of something:something:

1.Preliminaries: the doppler effect1.Preliminaries: the doppler effect

c light velocityΔλ λr-λe : Doppler shift

EMISSION line

λ

ABSORPTION line Problem: too weak also in nearby galaxies

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RR*

P* Pα Observer

Edge-on spiral galaxies:Edge-on spiral galaxies:

Θ(R)ΘΘ(R) (R) Rotation Rotation velocity velocity of a generic point P of a generic point P on the on the disk disk planeplane

u u Velocity of a generic Velocity of a generic point P of the disk due to point P of the disk due to its rotational motion on its rotational motion on the galactic disk ,as seen the galactic disk ,as seen by the observerby the observer

VV Velocity of the whole Velocity of the whole galaxy as seen by the galaxy as seen by the observer , in the direction observer , in the direction of the observer of the observer

We have:We have:

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The complete doppler shift of a single line emitted by matter which is going The complete doppler shift of a single line emitted by matter which is going towardtoward the observer is then: the observer is then:

For lines emitted by matter which is going For lines emitted by matter which is going awayaway from the observer from the observer:

If the cloud of the emitting material is exactly at the tangential point (in other words: P is If the cloud of the emitting material is exactly at the tangential point (in other words: P is exactly in P*):exactly in P*):

From the measurement of the two quantities above we are From the measurement of the two quantities above we are now able to find V and now able to find V and θθ(R*) (R*) !!!!

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Notes:Notes:

The importance of finding an emitting cloud at the tangential point lies in the The importance of finding an emitting cloud at the tangential point lies in the fact that we have NO information available from the observations about the fact that we have NO information available from the observations about the αα angle (see previous slide).angle (see previous slide).

The spectrum of the radiation detected is not (obviously) made up of a single The spectrum of the radiation detected is not (obviously) made up of a single narrow line. We have instead a “continuum” with a lot of peaks narrow line. We have instead a “continuum” with a lot of peaks superimposed. Every peak is associated with an emitting cloud. This fact, superimposed. Every peak is associated with an emitting cloud. This fact, consequence of the thermal chaotic motion of atoms in the clouds and of the consequence of the thermal chaotic motion of atoms in the clouds and of the presence of more than one cloud along the line of sight, complicates the presence of more than one cloud along the line of sight, complicates the situation. On the other hand, it’s possible to use this complication to get situation. On the other hand, it’s possible to use this complication to get information about the tangential point:information about the tangential point:

From equation above: the peak associated with the maximum of Δλ’ is produced by radiation emitted by a cloud in P*(tangential point). (R* is fixed, θ is assumed to be a NON increasing function .This is really the case!)

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Inclined galaxies:Inclined galaxies:

For galaxies which are NOT edge-on we have to correct for the angle between the For galaxies which are NOT edge-on we have to correct for the angle between the line of sight and the galaxy disk. line of sight and the galaxy disk.

Because of inclination, the circular disk of radius RD will look like an ellipsis of Because of inclination, the circular disk of radius RD will look like an ellipsis of axis :axis :

a=RDa=RD

b=RD cos(i) b=RD cos(i)

In the previous discussion it’s In the previous discussion it’s enough to substitute u (slide 5) enough to substitute u (slide 5) with with u sin(i) u sin(i)

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2.Emission lines2.Emission lines HHαα and optical measurementsand optical measurements::

-Emission lines traditionally employed: H-Emission lines traditionally employed: Hαα,[NII],[SII];,[NII],[SII]; - Strong emission lines of H- Strong emission lines of Hαα and [NII] and [NII] can be morecan be more easilyeasily detecteddetected and and

measuredmeasured than weak broad H and K absorption lines.than weak broad H and K absorption lines. -[Si VI]: new technique employed for the first time in the study of NGC1068 -[Si VI]: new technique employed for the first time in the study of NGC1068 (Tecza et al.2000);(Tecza et al.2000); -For a limited number of nearby galaxies, RC can be produced from velocity -For a limited number of nearby galaxies, RC can be produced from velocity

of individual HII regions.of individual HII regions.

HI lineHI line

-Powerful tool to obtain kinematics of spiral galaxy because of its radial -Powerful tool to obtain kinematics of spiral galaxy because of its radial extent: 3 or 4 times greater than that of the visible disk.extent: 3 or 4 times greater than that of the visible disk.

-Problem : spatial resolution -Problem : spatial resolution Thanks to instrumental improvements in the Thanks to instrumental improvements in the last 20 years problems of low spatial resolution are now important only last 20 years problems of low spatial resolution are now important only near the nucleus .near the nucleus .

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CO lineCO line

-Employed rotational transitions lines of CO:-Employed rotational transitions lines of CO: 115.57 Ghz 115.57 Ghz millimeter wave rangemillimeter wave range 230.50 Ghz 230.50 Ghz millimeter wave rangemillimeter wave range-Powerful tool to study the inner disk and central regions of spiral galaxies, for -Powerful tool to study the inner disk and central regions of spiral galaxies, for

extinction is negligible at CO wavelength.extinction is negligible at CO wavelength.-CO lines are emitted from molecular clouds (the central parts of disks are -CO lines are emitted from molecular clouds (the central parts of disks are

usually dominated by molecular gas) . These clouds are often associated usually dominated by molecular gas) . These clouds are often associated with star formation regions emitting Hwith star formation regions emitting Hαα CO is a good alternative to CO is a good alternative to HHαα and HI in the inner regions, where HI is often weak or absent.and HI in the inner regions, where HI is often weak or absent.

-Major -Major interestsinterests in in current interferometer observations of CO line emission in current interferometer observations of CO line emission in nuclear regions are: detailed orientation of the nuclear molecular disk nuclear regions are: detailed orientation of the nuclear molecular disk (NMD) and circumnuclear torus; detection of non-circular motion in (NMD) and circumnuclear torus; detection of non-circular motion in NMD.NMD.

Masers linesMasers lines

Radial velocity observations of masers lines ( SiO,HRadial velocity observations of masers lines ( SiO,H22O,OH), allows us to O,OH), allows us to measure the kinematics of stellar components and gas clouds in the disk measure the kinematics of stellar components and gas clouds in the disk and bulge of our Galaxyand bulge of our Galaxy

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Position-velocity diagram along the major axis of the edge-on galaxy NGC3079 in the CO 115.27 Ghz emission line.

Lower panel: composite rotation curve produced by combining the CO results and HI data.

(Sofue & Rubin,2001)

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Logarithmic RC of the Milky Way (thick line),NGC4258 (thin line) and M31 (dashed line).

Innermost rotation velocities are Keplerian velocities calculated for massive black holes.

Observational method used are indicated with horizontal lines. It’s important to note that rotational curves are obtained combining CO data for the central regions, optical for disks and HI for outer disk and halo. (Sofue & Rubin 2001).

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3.Analysis of the observational data3.Analysis of the observational dataThe analysis of the observational data has continued to evolve during the past The analysis of the observational data has continued to evolve during the past years as the quality of the data has improved. Emission lines at a point of the years as the quality of the data has improved. Emission lines at a point of the spectrum are an spectrum are an INTEGRALINTEGRAL along the line of sight through the galaxy. along the line of sight through the galaxy.Only recently the quality of the data has permitted the Only recently the quality of the data has permitted the deconvolutiondeconvolution of of various components. A few procedures nowadays employed are described various components. A few procedures nowadays employed are described below. below. 3.1 Intensity-Weighted-Velocity Method3.1 Intensity-Weighted-Velocity Method

Intensity-weighted velocities are defined by:Intensity-weighted velocities are defined by:

Where I(v) is the intensity profile at a given radius as a Where I(v) is the intensity profile at a given radius as a function of the radial velocity.function of the radial velocity.

Rotational velocity is then given by:Rotational velocity is then given by:

Where VWhere Vsys sys systematic velocity of the galaxy systematic velocity of the galaxy

i i angle between the normal to the plane of the galaxy and the angle between the normal to the plane of the galaxy and the

line of sight.line of sight.

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3.2 Centroid-Velocity (CV) and Peak-Intensity-Velocity (PIV) Method3.2 Centroid-Velocity (CV) and Peak-Intensity-Velocity (PIV) Method

In outer galactic disk, the line profiles can be assumed to be symmetric In outer galactic disk, the line profiles can be assumed to be symmetric around the peak-intensity value around the peak-intensity value the intensity weighted method can be the intensity weighted method can be approximeted by a centroid velocity of half-maximum values of a line approximeted by a centroid velocity of half-maximum values of a line profile (CV), or alternatively by the velocity at which the intensity attains profile (CV), or alternatively by the velocity at which the intensity attains its maximum (PIV). Unpublished tests (by V. Rubin) show that centroid its maximum (PIV). Unpublished tests (by V. Rubin) show that centroid velocities of weak emission lines are characterized by less scatter.velocities of weak emission lines are characterized by less scatter.

ProblemsProblems arise when these methods are used also for arise when these methods are used also for innerinner regions, where regions, where the line profiles are NOT simple but superimpositions of outer and inner the line profiles are NOT simple but superimpositions of outer and inner disk components. Tests indicate that CV and PIV methods often disk components. Tests indicate that CV and PIV methods often underestimate the true rotation velocity (See Sofue & Rubin,2000). The underestimate the true rotation velocity (See Sofue & Rubin,2000). The same situation occurs for edge-on galaxies (here the line profiles is nothing same situation occurs for edge-on galaxies (here the line profiles is nothing but the superimposition of profiles arising from ALL radial distances but the superimposition of profiles arising from ALL radial distances sampled along the line of sight).sampled along the line of sight).

In these situations we need a different method In these situations we need a different method

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3.3 Envelope-Tracing (ET) Method3.3 Envelope-Tracing (ET) Method

This method makes use of the so-called terminal velocity defined by the velocity at which the This method makes use of the so-called terminal velocity defined by the velocity at which the intensity becomes equal to: intensity becomes equal to:

IImaxmax maximum intensity, maximum intensity,

IIlclc Intensity corresponding to the lowest contour level Intensity corresponding to the lowest contour level

(usually (usually ≈ 3 rms noise)≈ 3 rms noise)

ηη usually taken to beusually taken to be 0.2-0.5. For 0.2-0.5. For ηη= 0.2 = 0.2 this equation this equation defines a 20% level of the intensity profile at a given position: defines a 20% level of the intensity profile at a given position: IItt ≈ 0.2 I ≈ 0.2 Imaxmax

The rotation velocity is defined by :The rotation velocity is defined by :

Where Where σσISMISM and and σσobsobs are the velocityare the velocity

dispersion of the interstellar gas and the dispersion of the interstellar gas and the velocity resolution if the observations velocity resolution if the observations respectively. Usually respectively. Usually σσISMISM ≈ 7-10 Km s≈ 7-10 Km s-1-1, ,

while while σσobsobs depends on instruments.depends on instruments.

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The The importanceimportance of the ET method lies in the fact that both finite beam width of the ET method lies in the fact that both finite beam width and disk thickness along the line of the sight cause confusion of gas with and disk thickness along the line of the sight cause confusion of gas with smaller velocities than the terminal one, which often results in a lower rotation smaller velocities than the terminal one, which often results in a lower rotation velocity in the former two methods. velocity in the former two methods.

ProblemsProblems : ET method is ill-defined when applied to the innermost part of a PV : ET method is ill-defined when applied to the innermost part of a PV diagram. This is a consequence of the fact that the two sides of the nucleus diagram. This is a consequence of the fact that the two sides of the nucleus have a discontinuity principally due to the instrumental resolution (usually have a discontinuity principally due to the instrumental resolution (usually large with respect to the velocity gradients. We will deal with this topics later).large with respect to the velocity gradients. We will deal with this topics later).

This discontinuity is avoided by stopping the ET method at a radius This discontinuity is avoided by stopping the ET method at a radius corresponding to the telescope resolution. In the nuclear zone the RC is corresponding to the telescope resolution. In the nuclear zone the RC is approximeted by a straight line crossing the nucleus at zero velocity (this approximeted by a straight line crossing the nucleus at zero velocity (this would imply a “solid body” rotation, here probably nothing but a poor would imply a “solid body” rotation, here probably nothing but a poor approximation to the true motions near the centre!)approximation to the true motions near the centre!)

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3.4 Iteration Method 3.4 Iteration Method

We are going to give here only a simple description of this method. (See We are going to give here only a simple description of this method. (See Takamiya & Sofue for more information ).Takamiya & Sofue for more information ).

An initial rotation curve RC0 is adopted from a PV diagram (PV0) obtained An initial rotation curve RC0 is adopted from a PV diagram (PV0) obtained using one of the methods described above. Using this RC0 and an observed using one of the methods described above. Using this RC0 and an observed radial distribution of intensity of the line used in the analysis, a PV diagram, radial distribution of intensity of the line used in the analysis, a PV diagram, PV1, is constructed. The difference between PV1 (calculated diagram) and PV1, is constructed. The difference between PV1 (calculated diagram) and PV0 is used to correct the initial RC (RC0) to obtain a corrected one (RC1). PV0 is used to correct the initial RC (RC0) to obtain a corrected one (RC1). Then from RC1 and the distribution of intensity a new PV (PV2) is derived. Then from RC1 and the distribution of intensity a new PV (PV2) is derived. PV2 is used to obtain the next iteration curve, RC2, and so on.PV2 is used to obtain the next iteration curve, RC2, and so on.

This iteration is repeated until the difference between PVi and PV0 becomes This iteration is repeated until the difference between PVi and PV0 becomes minimum and stable. RCi is adopted as the most reliable RC.minimum and stable. RCi is adopted as the most reliable RC.

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4.Instrumental Resolution4.Instrumental Resolution In the reality observers have to deal with In the reality observers have to deal with problems arising from finite resolution of problems arising from finite resolution of instruments employed, scattering, extinction of instruments employed, scattering, extinction of the signal due to dusty nuclear disks (this the signal due to dusty nuclear disks (this problem is lessened at the CO lines problem is lessened at the CO lines wavelength).wavelength).

In this figure you can find a simulation of the In this figure you can find a simulation of the effect of beam-smearing (which arises from the effect of beam-smearing (which arises from the finite instrumental resolution) on a PV diagram. finite instrumental resolution) on a PV diagram.

TopTop An assumed “true” rotation curve (thick An assumed “true” rotation curve (thick one) comprising a central core, bulge, disk and one) comprising a central core, bulge, disk and halo (dashed curves, from inner to outer halo (dashed curves, from inner to outer respectively). Thin lines: assumed density of respectively). Thin lines: assumed density of molecular (inner) and HI gas (outer) molecular (inner) and HI gas (outer) distributions. distributions.

MiddleMiddle An “observed” PV diagram in CO. An “observed” PV diagram in CO. BottomBottom An “observed” PV diagram in HI. An “observed” PV diagram in HI.

It’s now clear that high resolution is crucial in It’s now clear that high resolution is crucial in order to detect central high velocities and steep order to detect central high velocities and steep rise rise

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5.Dependence on observational methods5.Dependence on observational methods

Results are a function of the techniques of observations and reductions:Results are a function of the techniques of observations and reductions:

Observations from emission lines in the optical, millimeter, and radio Observations from emission lines in the optical, millimeter, and radio regions may NOT sample identical regions along the SAME line of sight.regions may NOT sample identical regions along the SAME line of sight.

Instruments sample at different sensitivities with different wavelength and Instruments sample at different sensitivities with different wavelength and spatial resolution.spatial resolution.

A simple RC is an approximation as a function of radius to the full velocity A simple RC is an approximation as a function of radius to the full velocity field of a disk galaxy. It’s obtained by neglecting small-scale velocity field of a disk galaxy. It’s obtained by neglecting small-scale velocity variations and by averaging and smoothing rotation velocities from both variations and by averaging and smoothing rotation velocities from both sides of the galactic center .sides of the galactic center .

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Observational properties of RCObservational properties of RC

Rotational curves of spiral galaxies have their individualities but they also Rotational curves of spiral galaxies have their individualities but they also share many characteristics. This fact has led to a variety of attempts to share many characteristics. This fact has led to a variety of attempts to categorize their shapes and to establish their statistical properties. (See for categorize their shapes and to establish their statistical properties. (See for example Roberts,1975; Burbidge & Burbidge,1975; Kyazumov, 1984; example Roberts,1975; Burbidge & Burbidge,1975; Kyazumov, 1984; Marquez et al.2003; Evstigneeva, 2001).Marquez et al.2003; Evstigneeva, 2001).

We will look at rotation curves as function of:We will look at rotation curves as function of:Morphology ;Morphology ;Luminosity;Luminosity;Environment;Environment;Evolution;Evolution;

Before giving a detailed description, is really useful to have a look at some Before giving a detailed description, is really useful to have a look at some observed rotation curves.observed rotation curves.

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Photograph, spectra Photograph, spectra and RC for five Sc and RC for five Sc galaxies, arranged in galaxies, arranged in order of increasing order of increasing luminosity from top to luminosity from top to bottom. The vertical bottom. The vertical line in each spectrum is line in each spectrum is continuum emission continuum emission from the nucleus. The from the nucleus. The distances reported are distances reported are based on a Hubble based on a Hubble constant h=0.5.constant h=0.5.

(Binney, Tremaine)(Binney, Tremaine)

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Rotation curves of spiral galaxies from optical measurementsRotation curves of spiral galaxies from optical measurements

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Rotation curves of spiral galaxies from 21-cm radio measurementsRotation curves of spiral galaxies from 21-cm radio measurements

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Rotation Curves of spiral galaxies obtained by combining CO Rotation Curves of spiral galaxies obtained by combining CO data for the central regions, optical for disks, HI for outer disk data for the central regions, optical for disks, HI for outer disk and halo. (Sofue et al.,1999) and halo. (Sofue et al.,1999)

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1.Universal properties of RC’s1.Universal properties of RC’s Inner regions are characterized by a steep rise of the rotation velocity Inner regions are characterized by a steep rise of the rotation velocity

θθ(R). (R). With some approximations we can say thatWith some approximations we can say that here here θθ(R)≈R(R)≈R ((linearlinear).). θθ(R) (R) reaches its maximum of 200-300 Km/s at distances of the order of reaches its maximum of 200-300 Km/s at distances of the order of

0.1 Kpc.0.1 Kpc. Depending on the galaxy, sometimes exists an intermediate region Depending on the galaxy, sometimes exists an intermediate region

(corresponding to the bulge and internal part of the galactic disk) (corresponding to the bulge and internal part of the galactic disk) characterized by irregularities and fluctuations. Usually characterized by irregularities and fluctuations. Usually θθ(R)(R) has a has a minimum at Rminimum at R≈ 1 ≈ 1 Kpc and then rises until it reaches a second maximum at Kpc and then rises until it reaches a second maximum at R R ≈ ≈ 4-10 Kpc .4-10 Kpc .

For R grater than the optical one rotation curves are extremely regular (no For R grater than the optical one rotation curves are extremely regular (no more fluctuations are detected). The most important feature is that here more fluctuations are detected). The most important feature is that here RC’s are RC’s are FLAT FLAT θθ(R)≈ (R)≈ constant.constant.( The image of Andromeda in the first ( The image of Andromeda in the first slide is a good example of what we are speaking about). This fact has a lot slide is a good example of what we are speaking about). This fact has a lot of implication in the dark matter problem (now a reality thanks to RC of implication in the dark matter problem (now a reality thanks to RC observations). See Roncadelli,2004.observations). See Roncadelli,2004.

These common properties are really universal for R >0.5 optical radius. Inner These common properties are really universal for R >0.5 optical radius. Inner rotation curves have greater individuality (Sofue et al.1999).rotation curves have greater individuality (Sofue et al.1999).

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2.Morphology & RC2.Morphology & RC

1.1 Sa, Sb and Sc Galaxies1.1 Sa, Sb and Sc Galaxies

SaSa The maximum rotation velocities are higher than those of Sb and Sc The maximum rotation velocities are higher than those of Sb and Sc

galaxies with equivalent optical luminosities (See slide 22).galaxies with equivalent optical luminosities (See slide 22).

Sb Sb High central core , including the massive black hole, which causes a High central core , including the massive black hole, which causes a

nonzero velocity very close to the center.nonzero velocity very close to the center. Steep central rise at 100-200 pc often associated with a velocity peak at Steep central rise at 100-200 pc often associated with a velocity peak at

radii radii ≈ 100-300 pc (Sofue et al.,1999).≈ 100-300 pc (Sofue et al.,1999). The maximum is usually followed by a decline to a minimum at 1-2 KpcThe maximum is usually followed by a decline to a minimum at 1-2 Kpc Gradual rise to a maximum at 6 Kpc due to the main diskGradual rise to a maximum at 6 Kpc due to the main disk Nearly flat outer-rotation curve.Nearly flat outer-rotation curve.

We can find these features in the Milky Way (typical Sb galaxy) rotation We can find these features in the Milky Way (typical Sb galaxy) rotation curve. (Showed in previous slides). curve. (Showed in previous slides).

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ScSc Massive Sc galaxies show steep nuclear rises similar to those of Sb Massive Sc galaxies show steep nuclear rises similar to those of Sb

galaxies. Less massive Sc galaxies have more gentle rises.galaxies. Less massive Sc galaxies have more gentle rises. They show flat rotation to their outer edges.They show flat rotation to their outer edges. Low-surface-brightness Sc galaxies have gentle central rises with Low-surface-brightness Sc galaxies have gentle central rises with

monotonically increasing rotation velocities toward the edge, similar to monotonically increasing rotation velocities toward the edge, similar to dwarf galaxies. (Bosma et al.,1988).dwarf galaxies. (Bosma et al.,1988).

Morphologycal Morphologycal typetype

Maximum Maximum velocityvelocity

SaSa 300 Km/s300 Km/s

SbSb 220 Km/s220 Km/s

ScSc 175 Km/s175 Km/s

Median values of VMedian values of Vmax max are are

reported (Roberts 1978,Rubin reported (Roberts 1978,Rubin et al.1985,Sandage 2000)et al.1985,Sandage 2000)

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Barred GalaxiesBarred Galaxies Large scale rotation properties of SBb and SBc galxies are generally Large scale rotation properties of SBb and SBc galxies are generally

similar of those of non-barred galaxies (Sb and Sc).similar of those of non-barred galaxies (Sb and Sc). The study of their kinematics is more complicated (in theory as well as in The study of their kinematics is more complicated (in theory as well as in

practice) because their gas tracers are less uniformly distributed and their practice) because their gas tracers are less uniformly distributed and their iso-velocity contours are skewed in the direction toward the bar.iso-velocity contours are skewed in the direction toward the bar.

Co-line mapping and spectroscopy reveal the presence of NON-circular Co-line mapping and spectroscopy reveal the presence of NON-circular motions in the nuclear molecular bar. From a theoretical point of view it motions in the nuclear molecular bar. From a theoretical point of view it has already been shown that an oval potential such is due to a bar has already been shown that an oval potential such is due to a bar produces shocks of interstellar material, and the shocked gas streams along produces shocks of interstellar material, and the shocked gas streams along the bar in non-circular orbits. (Sorensen et al.1976,Noguchi 1988,Wada & the bar in non-circular orbits. (Sorensen et al.1976,Noguchi 1988,Wada & Habe 1992,1995, Shlosman et al.1990).Habe 1992,1995, Shlosman et al.1990).

High velocities variations arising from the barred potential.( Simulations of High velocities variations arising from the barred potential.( Simulations of PV diagrams show fluctuations of tens kilometers per second superposed PV diagrams show fluctuations of tens kilometers per second superposed on the usual RC.(Athanassoula &Bureau 1999, Bureau & Athanassoula on the usual RC.(Athanassoula &Bureau 1999, Bureau & Athanassoula 1999).1999).

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Dwarf GalaxiesDwarf GalaxiesOnly within the past decade instrumental improvements have made it Only within the past decade instrumental improvements have made it

possible to study the kinematics of dwarf galaxies, galaxies of low possible to study the kinematics of dwarf galaxies, galaxies of low mass.mass.

Observational problemObservational problem: many of them are small in angular extent, so : many of them are small in angular extent, so observations are subject of important beam smearing.observations are subject of important beam smearing.

Most of the rotation curve shape are similar to those of spiral galaxies.Most of the rotation curve shape are similar to those of spiral galaxies. Dwarfs tend to show increasing outer RC, whereas most massive Dwarfs tend to show increasing outer RC, whereas most massive

galaxies have slightly declining rotation in the outmost part.galaxies have slightly declining rotation in the outmost part. Dwarfs usually show gentle central rises. Sa and Sb types show steeper Dwarfs usually show gentle central rises. Sa and Sb types show steeper

rises and higher central velocities.rises and higher central velocities. Dwarf with higher central light concentrations have more steeply rising Dwarf with higher central light concentrations have more steeply rising

rotation curves.rotation curves.

Irregular galaxiesIrregular galaxies Rotation curve of irregular galaxies usually show peculiar features. Rotation curve of irregular galaxies usually show peculiar features. Some irregular galaxies exhibit normal rotation curves, such as Some irregular galaxies exhibit normal rotation curves, such as

observed for a ring galaxy NGC 660, amorphous galaxy NGC 4631 observed for a ring galaxy NGC 660, amorphous galaxy NGC 4631 and NGC 4945. (Sofue et al.,1999).and NGC 4945. (Sofue et al.,1999).

Polar ring galaxies provide a unique opportunity to probe the rotation Polar ring galaxies provide a unique opportunity to probe the rotation and mass distribution perpendicular to galaxy disc.and mass distribution perpendicular to galaxy disc.

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HI velocity field of the LMC HI velocity field of the LMC (dwarf galaxy) superposed on (dwarf galaxy) superposed on a Ha Hαα image and a position-image and a position-velocity diagram along the velocity diagram along the major axis.major axis.

The ellipse indicates the The ellipse indicates the position of the optical bar.position of the optical bar.

Thick line in PV diagram Thick line in PV diagram traces the rotation curves traces the rotation curves (corrected for the inclination (corrected for the inclination angle of 33°).angle of 33°).

(Sofue, Rubin,2001)(Sofue, Rubin,2001)

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3.Luminosity & RC3.Luminosity & RC

Very luminous galaxies tend to have higher peak velocity. The broad Very luminous galaxies tend to have higher peak velocity. The broad maximum in the disk is followed by a slightly declining RC. maximum in the disk is followed by a slightly declining RC.

Intermediate galaxies have nearly flat rotation across the disk.Intermediate galaxies have nearly flat rotation across the disk.

Less luminous ones tend to show increasing outer RC. At lowest Less luminous ones tend to show increasing outer RC. At lowest luminosities there is more variation in the shape of the RC.luminosities there is more variation in the shape of the RC.

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4.Environment & RC4.Environment & RC

4.1 Cluster galaxies and field galaxies4.1 Cluster galaxies and field galaxies

First of all it’s important to distinguish between galaxies in clusters and field-First of all it’s important to distinguish between galaxies in clusters and field-galaxies. A variety of mechanism can alter the internal kinematics of spiral in galaxies. A variety of mechanism can alter the internal kinematics of spiral in clusters: gas stripping, star stripping, galaxy-galaxy encounters, interaction clusters: gas stripping, star stripping, galaxy-galaxy encounters, interaction with the general tidal field. with the general tidal field.

There’s a There’s a correlation between outer rotation-velocity gradients and distance correlation between outer rotation-velocity gradients and distance from the cluster centrefrom the cluster centre: inner-cluster galaxies show shallower rotation curves : inner-cluster galaxies show shallower rotation curves than do outer- cluster galaxies. (Rubin et al.,1988). than do outer- cluster galaxies. (Rubin et al.,1988). A study of 81 galaxies in the Virgo cluster (Rubin et al.,1999),shows that A study of 81 galaxies in the Virgo cluster (Rubin et al.,1999),shows that about about 50%50% of them have rotation curves identified as disturbed: asymmetrical of them have rotation curves identified as disturbed: asymmetrical rotation on the two sides of the major axis, falling outer rotation curves, inner rotation on the two sides of the major axis, falling outer rotation curves, inner velocity peculiarities. For the field galaxies, velocity peculiarities. For the field galaxies, 74%74% exhibited rotation curves are exhibited rotation curves are normal.normal.

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4.2 Rotation curves as a function of the overall spatial density of luminosity4.2 Rotation curves as a function of the overall spatial density of luminosity

The results of the study made by Evstigneeva & Reshetnikov (See The results of the study made by Evstigneeva & Reshetnikov (See Evstigneeva & Reshetnikov,2001), show that Evstigneeva & Reshetnikov,2001), show that the shape of the rotation curve the shape of the rotation curve doesn’t depend on the overall spatial density of luminositydoesn’t depend on the overall spatial density of luminosity..

The only difference lies in the fact that rotation curves of galaxies in regions The only difference lies in the fact that rotation curves of galaxies in regions with high spatial density of luminosity can be traced out to smaller relative with high spatial density of luminosity can be traced out to smaller relative distances from the nucleus. This may related to the destruction of the outer distances from the nucleus. This may related to the destruction of the outer regions of their gaseous disks in gravitational interactions with surrounding regions of their gaseous disks in gravitational interactions with surrounding galaxies. galaxies.

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5.Evolution & RC5.Evolution & RC

We directly observe galaxy evolution by studying galaxies closer to their era We directly observe galaxy evolution by studying galaxies closer to their era of formation. Modern techniques have made it possible to obtain rotation of formation. Modern techniques have made it possible to obtain rotation curves for distant spirals with z curves for distant spirals with z ≈ 1 (thanks to Keck and HST).≈ 1 (thanks to Keck and HST).

The rotation properties are similar to those of nearby galaxies, with peak The rotation properties are similar to those of nearby galaxies, with peak velocities between 100 and 200 Km/s.velocities between 100 and 200 Km/s.

Regularly rotating spiral galaxies existed at z Regularly rotating spiral galaxies existed at z ≈ 1.≈ 1.

Spiral galaxy evolution, over the last half of the age of the universe, has not Spiral galaxy evolution, over the last half of the age of the universe, has not dramatically altered the rotation kinematics of spiral galaxies.dramatically altered the rotation kinematics of spiral galaxies.

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Astrophysical importance of rotation Astrophysical importance of rotation curvescurves

We end this work with some words about the astrophysical We end this work with some words about the astrophysical importance of rotation curves.importance of rotation curves.

Rotation curves are tools for several purposes:Rotation curves are tools for several purposes: For studying the kinematics of galaxies;For studying the kinematics of galaxies; For inferring the evolutionary histories and the role that For inferring the evolutionary histories and the role that

interactions have played;interactions have played; For observing evolution by comparing rotation curves in For observing evolution by comparing rotation curves in

distant galaxies with galaxies nearby.distant galaxies with galaxies nearby. For relating departures from the expected rotation curve form, For relating departures from the expected rotation curve form,

to the amount and distribution of dark matter;to the amount and distribution of dark matter;

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According to Newtonian dynamics, it’s simple to show that According to Newtonian dynamics, it’s simple to show that

Where r Where r distance from the galaxy centre; distance from the galaxy centre;

M(r) M(r) enclosed mass; enclosed mass;

From measurements of VFrom measurements of Vrotrot we have some information about the distribution of we have some information about the distribution of

mass in the galaxy. In particular:mass in the galaxy. In particular:

Inner regions : The RC rises linearly with galactocentric distanceInner regions : The RC rises linearly with galactocentric distance

Outer regions: The RC is nearly flat Outer regions: The RC is nearly flat The enclosed mass then rises linearly The enclosed mass then rises linearly with galactocentric distance.with galactocentric distance.

It’s enough to have a look at the first image of this work (RC of the It’s enough to have a look at the first image of this work (RC of the Andromeda galaxy), to understand that we NEED MATTER, and to be more Andromeda galaxy), to understand that we NEED MATTER, and to be more precise, DARK matter in order to explain the features of the RC. A simple precise, DARK matter in order to explain the features of the RC. A simple example: the rotation curve is FLAT where we don’t see matter, or better: example: the rotation curve is FLAT where we don’t see matter, or better: where we don’t see SHINING matter.where we don’t see SHINING matter.

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