Rotation Curves of Spiral
Galaxies
Image of the Andromeda galaxy (M31) with rotation curve superimposed (Turner,2000)
1
Margutti Raffaella
Index
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Definition
Historical background
Measuring rotation velocities
1. Preliminaries: the doppler effect
2. Emission lines
3.Analysis of the observational data
4.Instrumental resolution
5.Dependence on observational methods
Observational properties of rotation curves
1.Universal properties of rotation curves
2.Morphology & RC
3.Luminosity & RC
4.Environment & RC
5.Evolution & RC
Astrophysical importance of rotation curves
References
2
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)
3
Historical background
1914 Slipher discovered the rotation of M31 and Sombrero galaxy
 Wolf detected “inclined” lines in M81 nuclear spectrum
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1939 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
increase from inner regions to outer ones.
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1954 Schwarzschild reanalyzed the velocities of M31 and fit with a model of
constant M/L
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1959de Vaucouleurs affirmed that “ the rotation velocity decrease with 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 available at
that time.)
De Vaucouleurs and Schwarzschild’s authority make it possible for astronomers to
ignore the strange RC identified by Babcock, Oort and Mayall.

1960The modern era begins with Page and Burbidge & Burbidge. Their
observations were based on Hα and [NII] emission lines. In the same period we
have also the first radio observations of neutral hydrogen which showed a slowly
falling RC for M31 and a flat one for M33.
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2005 A rich variety of techniques are nowadays available ( Hα ,HI, [NII],
[SiII],CO, masers emission lines ...), together with higher signal-to-noise velocities
and high spatial resolution.
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4
Measuring rotation velocities
1.Preliminaries: the doppler effect
Hypothesis:
1.Axially symmetric distribution of matter
2.Circular orbits of matter
3.Galactic disk with circular shape.
cylindrical
coordinates are
suitable
The doppler effect can be used to derive the radial velocity (Vr) of
something:
c  light velocity
Δλ  λr-λe : Doppler shift
EMISSION line
λ
ABSORPTION line Problem: too weak also in nearby galaxies
5
Edge-on spiral galaxies:
R*
P*
We have:
R
Θ(R)
α
P
Observer
Θ(R)  Rotation velocity
of a generic point P on the
disk plane
u  Velocity of a generic
point P of the disk due to
its rotational motion on the
galactic disk ,as seen by
the observer
V Velocity of the whole
galaxy as seen by the
observer , in the direction
of the observer
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The complete doppler shift of a single line emitted by matter which is going
toward the observer is then:
For lines emitted by matter which is going away from the observer:
If the cloud of the emitting material is exactly at the tangential point (in other words: P is
exactly in P*):
From the measurement of the two quantities above we are
now able to find V and θ(R*) !!
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Notes:
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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 α angle (see previous slide).
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
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 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
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:
For galaxies which are NOT edge-on we have to correct for the angle between the
line of sight and the galaxy disk.
Because of inclination, the circular disk of radius RD will look like an ellipsis of
axis :
a=RD
b=RD cos(i)
In the previous discussion it’s
enough to substitute u (slide 5)
with  u sin(i)
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2.Emission lines
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Hα and optical measurements:
-Emission lines traditionally employed: Hα,[NII],[SII];
- Strong emission lines of Hα and [NII] can be more easily detected and
measured than weak broad H and K absorption lines.
-[Si VI]: new technique employed for the first time in the study of NGC1068
(Tecza et al.2000);
-For a limited number of nearby galaxies, RC can be produced from velocity
of individual HII regions.
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HI line
-Powerful tool to obtain kinematics of spiral galaxy because of its radial
extent: 3 or 4 times greater than that of the visible disk.
-Problem : spatial resolution  Thanks to instrumental improvements in the
last 20 years problems of low spatial resolution are now important only
near the nucleus .
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
CO line
-Employed rotational transitions lines of CO:
115.57 Ghz millimeter wave range
230.50 Ghz millimeter wave range
-Powerful tool to study the inner disk and central regions of spiral galaxies, for
extinction is negligible at CO wavelength.
-CO lines are emitted from molecular clouds (the central parts of disks are
usually dominated by molecular gas) . These clouds are often associated
with star formation regions emitting Hα CO is a good alternative to Hα
and HI in the inner regions, where HI is often weak or absent.
-Major interests in current interferometer observations of CO line emission in
nuclear regions are: detailed orientation of the nuclear molecular disk
(NMD) and circumnuclear torus; detection of non-circular motion in NMD.
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Masers lines
Radial velocity observations of masers lines ( SiO,H2O,OH), allows us to
measure the kinematics of stellar components and gas clouds in the disk
and bulge of our Galaxy
11
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 data
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
spectrum are an INTEGRAL along the line of sight through the galaxy.
Only recently the quality of the data has permitted the deconvolution of
various components. A few procedures nowadays employed are described
below.
3.1 Intensity-Weighted-Velocity Method
Intensity-weighted velocities are defined by:
Where I(v) is the intensity profile at a given radius as a
function of the radial velocity.
Rotational velocity is then given by:
Where Vsys  systematic velocity of the galaxy
i angle between the normal to the plane of the galaxy and the
line of sight.
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3.2 Centroid-Velocity (CV) and Peak-Intensity-Velocity (PIV) Method

In outer galactic disk, the line profiles can be assumed to be symmetric
around the peak-intensity value  the intensity weighted method can be
approximeted by a centroid velocity of half-maximum values of a line
profile (CV), or alternatively by the velocity at which the intensity attains
its maximum (PIV). Unpublished tests (by V. Rubin) show that centroid
velocities of weak emission lines are characterized by less scatter.
Problems arise when these methods are used also for inner regions, where
the line profiles are NOT simple but superimpositions of outer and inner
disk components. Tests indicate that CV and PIV methods often
underestimate the true rotation velocity (See Sofue & Rubin,2000). The
same situation occurs for edge-on galaxies (here the line profiles is nothing
but the superimposition of profiles arising from ALL radial distances
sampled along the line of sight).
 In these situations we need a different method
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3.3 Envelope-Tracing (ET) Method
This method makes use of the so-called terminal velocity defined by the
velocity at which the intensity becomes equal to:
Imax  maximum intensity,
Ilc  Intensity corresponding to the lowest contour level
(usually ≈ 3 rms noise)
η  usually taken to be 0.2-0.5. For η= 0.2 this equation
defines a 20% level of the intensity profile at a given position:
It ≈ 0.2 Imax
The rotation velocity is defined by :
Where σISM and σobs are the velocity
dispersion of the interstellar gas and the
velocity resolution if the observations
respectively. Usually σISM ≈ 7-10 Km s-1,
while σobs depends on instruments.
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 The importance 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
smaller velocities than the terminal one, which often results in a lower rotation
velocity in the former two methods.
 Problems : 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 have
a discontinuity principally due to the instrumental resolution (usually 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
corresponding to the telescope resolution. In the nuclear zone the RC is
approximeted by a straight line crossing the nucleus at zero velocity (this would
imply a “solid body” rotation, here probably nothing but a poor approximation
to the true motions near the centre!)
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3.4 Iteration Method
We are going to give here only a simple description of this method. (See
Takamiya & Sofue for more information ).
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
radial distribution of intensity of the line used in the analysis, a PV diagram,
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).
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.
This iteration is repeated until the difference between PVi and PV0 becomes
minimum and stable. RCi is adopted as the most reliable RC.
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4.Instrumental Resolution
In the reality observers have to deal with
problems arising from finite resolution of
instruments employed, scattering, extinction of
the signal due to dusty nuclear disks (this
problem is lessened at the CO lines
wavelength).
In this figure you can find a simulation of the
effect of beam-smearing (which arises from the
finite instrumental resolution) on a PV diagram.
Top An assumed “true” rotation curve (thick
one) comprising a central core, bulge, disk and
halo (dashed curves, from inner to outer
respectively). Thin lines: assumed density of
molecular (inner) and HI gas (outer)
distributions.
Middle An “observed” PV diagram in CO.
Bottom An “observed” PV diagram in HI.
It’s now clear that high resolution is crucial in
order to detect central high velocities and steep
rise
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5.Dependence on observational methods
Results are a function of the techniques of observations and reductions:
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Observations from emission lines in the optical, millimeter, and radio
regions may NOT sample identical regions along the SAME line of sight.
Instruments sample at different sensitivities with different wavelength and
spatial resolution.
 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
variations and by averaging and smoothing rotation velocities from both
sides of the galactic center .
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Observational properties of RC
Rotational curves of spiral galaxies have their individualities but they also
share many characteristics. This fact has led to a variety of attempts to
categorize their shapes and to establish their statistical properties. (See for
example Roberts,1975; Burbidge & Burbidge,1975; Kyazumov, 1984;
Marquez et al.2003; Evstigneeva, 2001).
We will look at rotation curves as function of:
Morphology ;
Luminosity;
Environment;
Evolution;
Before giving a detailed description, is really useful to have a look at some
observed rotation curves.
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Photograph, spectra
and RC for five Sc
galaxies, arranged in
order of increasing
luminosity from top to
bottom. The vertical
line in each spectrum is
continuum emission
from the nucleus. The
distances reported are
based on a Hubble
constant h=0.5.
(Binney, Tremaine)
22
Rotation curves of spiral galaxies from optical measurements
23
Rotation curves of spiral galaxies from 21-cm radio measurements
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Rotation Curves of spiral galaxies obtained by combining CO
data for the central regions, optical for disks, HI for outer disk
and halo. (Sofue et al.,1999)
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1.Universal properties of RC’s
Inner regions are characterized by a steep rise of the rotation velocity θ(R).
With some approximations we can say that here θ(R)≈R (linear).
 θ(R) reaches its maximum of 200-300 Km/s at distances of the order of
0.1 Kpc.
 Depending on the galaxy, sometimes exists an intermediate region
(corresponding to the bulge and internal part of the galactic disk)
characterized by irregularities and fluctuations. Usually θ(R) has a
minimum at R≈ 1 Kpc and then rises until it reaches a second maximum at
R ≈ 4-10 Kpc .
 For R grater than the optical one rotation curves are extremely regular (no
more fluctuations are detected). The most important feature is that here
RC’s are FLAT  θ(R)≈ constant.( The image of Andromeda in the first
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
observations). See Roncadelli,2004.
These common properties are really universal for R >0.5 optical radius. Inner
rotation curves have greater individuality (Sofue et al.1999).
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2.Morphology & RC
1.1 Sa, Sb and Sc Galaxies
Sa

The maximum rotation velocities are higher than those of Sb and Sc
galaxies with equivalent optical luminosities (See slide 22).
Sb
High central core , including the massive black hole, which causes a
nonzero velocity very close to the center.
 Steep central rise at 100-200 pc often associated with a velocity peak at
radii ≈ 100-300 pc (Sofue et al.,1999).
 The 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 disk
 Nearly flat outer-rotation curve.
We can find these features in the Milky Way (typical Sb galaxy) rotation curve.
(Showed in previous slides).
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Sc
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Massive Sc galaxies show steep nuclear rises similar to those of Sb
galaxies. Less massive Sc galaxies have more gentle rises.
They show flat rotation to their outer edges.
Low-surface-brightness Sc galaxies have gentle central rises with
monotonically increasing rotation velocities toward the edge, similar to
dwarf galaxies. (Bosma et al.,1988).
Morphologycal
type
Maximum
velocity
Sa
300 Km/s
Sb
220 Km/s
Sc
175 Km/s
Median values of Vmax are
reported (Roberts 1978,Rubin
et al.1985,Sandage 2000)
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Barred Galaxies
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Large scale rotation properties of SBb and SBc galxies are generally
similar of those of non-barred galaxies (Sb and Sc).
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
iso-velocity contours are skewed in the direction toward the bar.
Co-line mapping and spectroscopy reveal the presence of NON-circular
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 produces
shocks of interstellar material, and the shocked gas streams along the bar in
non-circular orbits. (Sorensen et al.1976,Noguchi 1988,Wada & Habe
1992,1995, Shlosman et al.1990).
High velocities variations arising from the barred potential.( Simulations of
PV diagrams show fluctuations of tens kilometers per second superposed
on the usual RC.(Athanassoula &Bureau 1999, Bureau & Athanassoula
1999).
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Dwarf Galaxies
Only within the past decade instrumental improvements have made it
possible to study the kinematics of dwarf galaxies, galaxies of low
mass.
Observational problem: many of them are small in angular extent, so
observations are subject of important beam smearing.
 Most of the rotation curve shape are similar to those of spiral galaxies.
 Dwarfs tend to show increasing outer RC, whereas most massive
galaxies have slightly declining rotation in the outmost part.
 Dwarfs usually show gentle central rises. Sa and Sb types show steeper
rises and higher central velocities.
 Dwarf with higher central light concentrations have more steeply rising
rotation curves.
Irregular galaxies
 Rotation curve of irregular galaxies usually show peculiar features.
 Some irregular galaxies exhibit normal rotation curves, such as
observed for a ring galaxy NGC 660, amorphous galaxy NGC 4631 and
NGC 4945. (Sofue et al.,1999).
 Polar ring galaxies provide a unique opportunity to probe the rotation
and mass distribution perpendicular to galaxy disc.
30
HI velocity field of the LMC
(dwarf galaxy) superposed on
a Hα image and a positionvelocity diagram along the
major axis.
The ellipse indicates the
position of the optical bar.
Thick line in PV diagram
traces the rotation curves
(corrected for the inclination
angle of 33°).
(Sofue, Rubin,2001)
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3.Luminosity & RC

Very luminous galaxies tend to have higher peak velocity. The broad
maximum in the disk is followed by a slightly declining RC.

Intermediate galaxies have nearly flat rotation across the disk.

Less luminous ones tend to show increasing outer RC. At lowest
luminosities there is more variation in the shape of the RC.
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4.Environment & RC
4.1 Cluster galaxies and field galaxies
First of all it’s important to distinguish between galaxies in clusters and fieldgalaxies. A variety of mechanism can alter the internal kinematics of spiral in
clusters: gas stripping, star stripping, galaxy-galaxy encounters, interaction
with the general tidal field.
There’s a correlation between outer rotation-velocity gradients and distance
from the cluster centre: inner-cluster galaxies show shallower rotation curves
than do outer- cluster galaxies. (Rubin et al.,1988).
 A study of 81 galaxies in the Virgo cluster (Rubin et al.,1999),shows that
about 50% of them have rotation curves identified as disturbed: asymmetrical
rotation on the two sides of the major axis, falling outer rotation curves, inner
velocity peculiarities. For the field galaxies, 74% exhibited rotation curves are
normal.
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4.2 Rotation curves as a function of the overall spatial density of luminosity
The results of the study made by Evstigneeva & Reshetnikov (See
Evstigneeva & Reshetnikov,2001), show that the shape of the rotation curve
doesn’t depend on the overall spatial density of luminosity.

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
distances from the nucleus. This may related to the destruction of the outer
regions of their gaseous disks in gravitational interactions with surrounding
galaxies.
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5.Evolution & RC
We directly observe galaxy evolution by studying galaxies closer to their era of
formation. Modern techniques have made it possible to obtain rotation curves
for distant spirals with z ≈ 1 (thanks to Keck and HST).
The rotation properties are similar to those of nearby galaxies, with peak
velocities between 100 and 200 Km/s.

 Regularly rotating spiral galaxies existed at z ≈ 1.
Spiral galaxy evolution, over the last half of the age of the universe, has not
dramatically altered the rotation kinematics of spiral galaxies.
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Astrophysical importance of rotation
curves
We end this work with some words about the astrophysical
importance of rotation curves.
Rotation curves are tools for several purposes:
 For studying the kinematics of galaxies;
 For inferring the evolutionary histories and the role that
interactions have played;
 For observing evolution by comparing rotation curves in
distant galaxies with galaxies nearby.
 For relating departures from the expected rotation curve form,
to the amount and distribution of dark matter;
36
According to Newtonian dynamics, it’s simple to show that
Where r  distance from the galaxy centre;
M(r)  enclosed mass;
From measurements of Vrot we have some information about the distribution of
mass in the galaxy. In particular:
 Inner regions : The RC rises linearly with galactocentric distance

 Outer regions: The RC is nearly flat  The enclosed mass then rises linearly
with galactocentric distance.

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 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: where we don’t
see SHINING matter.
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