GALAXIES 626

Lecture 20:

Stability of spiral disks and

the formation of spiral arms

Spiral armsDefining feature of spiral galaxies - what causes them?

Observational clues

Seen in disks that containgas, but not in gas poorS0 galaxy disks.

Defined by blue light fromhot massive stars. Lifetimeis << galactic rotation period.

When the sense of the galactic rotation is known, the spiralarms almost always trail the rotation.

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A galaxy that looks identical after a rotation through an angle of 2pi/m radians is said to have m-fold symmetry and usually has m dominant spiral arms;

Most spiral galaxies have two arms and approximate twofold symmetry.

A trailing arm is one whose outer tip points in the direction opposite to galactic rotation, while the outer tip of a leading arm points in the direction of rotation;

In almost all cases the spiral arms are trailing.

The number of arms:

Leading and trailing arms:

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Tightness of spiral arms

How tightly are the spirals wound?

Measured pitch angle as a function of Hubble type for 113 galaxies (Kennicutt 1981).

The pitch angle i of the arm at any radius r is the angle between the tangent to the arm and the circle r=constant.

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Orbits in the disk plane

Circular frequency (angular velocity) is decided by the force-field in the orbit plane, and is related to circular linear velocity as , and is the rotation curve.

for rigid rotation (e.g., homogeneous sphere) is constant;

in general declines with radius: differential rotation

Flat rotation curve (circular linear velocity is constant) .

Ω

Ω

Ω

/V rΩ = ( )V r

( )V r

Differential rotation

First ingredient for producing spiral arms is differential rotation. For galaxy with flat rotation curve:

V R =constant

R =VR

µR−1Angularvelocity

Any feature in the disk will be wrapped into a trailing spiralpattern due to differential rotation:

Tips of spiral arms point away from direction of rotation.

Differential rotation is not enough to explain observed spiral structure. Again assuming a flat rotation curve:

=VR

ddR

=­VR2

Two points in the disk, separated by ∆R in radius, and initially at the same azimuth, will be sheared apart over time. After time t, separated by an angle:

∣ddR

∣D Rt

Return to the same azimuth (one wrapping up) after a time given by:

∣ddR

∣D Rt=2 p

A spiral pattern will be wrapped up on a radial scale ∆R after a time t given by:

DRR

=2 pRVt

Using values appropriate for the Milky Way (R = 8.5 kpc,V = 200 km/s):

DRR

=0 .25 1 Gyrt

This is already a very tightly wrapped spiral. Spiral armswould make an angle of ~2 degrees to the tangent direction.

In real galaxies, arms are not so wrapped up:• Sa spirals, ~5 degrees• Sc spirals, around 10 to 30 degrees

Implies spiral pattern must be continually renewed.

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The winding problem

Possible resolutions?

1. The spiral arms are quite young, and are continuously appearing and dying (chaotic spiral model);

2. The spiral pattern may be a temporary phenomenon resulting from a recent violent disturbance such as an encounter with another galaxy (tidal model);

3. The spiral arms are some sort of wave pattern that propagates through the galactic disk.

Properties of spiral arms can be explained if they are not rotating with the stars, but rather density waves:

• Spiral arms are locations where the stellar orbitsare such that stars are more densely packed.

• Gas is also compressed, possibly triggering starformation and generating population of young stars.

• Arms rotate with a pattern speed which is not equalto the circular velocity - i.e. long lived stars enter andleave spiral arms repeatedly.

• Pattern speed is less than the circular velocity - partially alleviating the winding up problem.

Epicyclic frequency

For a disk of stars, `cold’ means that the stars’ random motions(in radial and vertical direction) are small compared to theircircular velocity.

Define the velocity dispersion in the radial direction by:

sR2=⟨vR

2 ⟩

Define the epicyclic frequency κ via:

k2 R =1R3

ddR

[ R2 2 ]

For a point mass gravitational field, κ = Ω. .

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Epicyclic frequency: frequency of small radial oscillations in circular orbits

Orbits in the disk plane

The motion in the orbit plane is well approximated by the superposition of two motions:

2. retrograde motion at angular frequency around a small ellipse (epicycle ) whose center is called the guiding center; The

length of the semi-axes of the epicycle are in the ratio

3. Prograde motion of the guiding center at angular frequency around a circle.

Ω

κ

/ 2κ Ω

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Orbits in the disk plane

2. Kepler orbit in the solar system : the orbit is closed and elliptical;

/ 1κ Ω =

The elliptical orbit of a planet in the solar system

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/κ Ω

3. However, we may view the motion in a rotating frame at proper angular speed so that the orbit is closed. This angular speed is chosen as

.

pΩ

pnmκΩ = Ω −

Orbits in the disk plane

2. In general the ratio is irrational, so the orbit is unclosed and forms a rosette figure;

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Kinematic density waves

In the frame rotating at pattern speed , the orbits are elliptical:

pΩ

Bar Leading spirals Trailing spirals

The two-arm prevalence

GALAXIES 626 The pattern speed changes as a function of radius (depends on rotation curve) so the kinematic density waves also wind up......

just more slowly.

More importantly this isn't fully self consistent: the gravity due to the non-

axisymmetric nature of the patterns in the disk has to be taken into account

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Kinematic density waves

How bad is the winding problem?

1. Since is not exactly constant the orientations of different orbits drift at slightly different speeds, so the pattern tends to twist or wind up. (modified

version of the winding problem) . The estimated pitch angle at after is about . Compared with the material arm’s , we have come some way toward resolving the winding problem but still not good enough.

/ 2κΩ −

1010 yrt =

10kpcR =

1.4i = o

0.25i = o

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Theories of spiral structure

1. The Lin-Shu model:

Spiral structure is a quasi-stationary density wave (except for overall rotation). As the wave is amplified gravitationally, energy dissipation in the ISM leads to damping. Eventually the wave reaches a stable, finite amplitude pattern.

The Lin-Shu density wave model can explain most of the features seen in spiral galaxies, i.e., the prevalence of trailing arms and two-armed spirals.

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Lin-Shu theory tightly wound case

normal mode analysis( )

Ri t m k r dr

eω φ ↵ ↵− + + ↵ ↵ ↵ ↵↵″∝

WKBJ approximation to solve the Poisson’s

equation

linear perturbation approach

local dispersion relation( ) 2 2 2 2

02 sm G k k vω κ πΩ − = − Σ +

2 10

0

is the sound speed in a polytropic gasns

dpv nKd

−= = ΣΣ

an axisymmetric background equilibrium

polytropic equation of state+

pattern speed

p mωΩ =

fluid equation set in the disk plane

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Are there other possibilities?

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Theories of spiral structure

1. Tidal arms by a recent encounter

Many of the most beautiful spirals have nearby companion galaxies (i.e., M51). Can the spiral patterns be caused by the tidal force from the companion galaxy?

Toomres showed that tidal models can successfully reproduce most of the features explained by the Lin-Shu hypothesis.

In the tidal model the spiral is a density wave; however, the wave is a transitory one rather than the long-lived QSSS envisaged by Lin and Shu.

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Theories of spiral structure

Model of the M51 system in which the spiral arms are caused by a recent passage of NGC 5195. The scenes on the left are viewed from the Sun, and those on the right are viewed from the side.

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Theories of spiral structure

1. Spiral detonation waves: based on the hypothesis that star formation is self-propagating.

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Theories of spiral structure

1. Driving by bars or asymmetries

Many spirals have central bars or oval distortions, and the spirals usually begin at the tips of the bars. This suggests the bars may be dynamically responsible to the spiral structures.

Spiral structure driven by a central bar in a differential rotating gas disk with no self-gravity.

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Theories of spiral structure

Question: is there any need for quasi-stationary density waves, that is, are there any regular, grand-design spirals with neither bars nor companions?

Kormendy & Norman (1979) examined 54 spiral galaxies with published rotation curves.

Results: 25 had bars; 8 more had close companions; 9 of the remainder had no clear global spiral pattern; and the remaining 12 galaxies are candidates for quasi-stationary density waves, however, they are ragged compared with those with bars and companions. In addition, 10 of the 12 galaxies exhibit almost rigid rotation throughout the spiral pattern, so they might represent material arms rather than density-wave patterns.

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Some examples....

DENSITY-WAVE SPIRAL?

NGC 5364 (Sc, grand-design spirals)

Two very regular spiral arms; one of the most symmetrical galaxies; formed by large-scale global process that involves the whole galaxy; best candidate for Lin-Shu model

Locally generated structures

M33 (NGC 598, Sc)

A member of the Local Group of galaxies; the arms are much broader and less distinct than in NGC 5364; a local, rather than global origin for the spiral structure

BAR-DRIVEN SPIRAL?

NGC 1300 (SBb)

One of the most dramatic barred spirals; spiral arms start at the tips of the bar; driving by bars?

TIDAL MODEL?

The Sc galaxy M51 (NGC 5194) with its companion galaxy NGC 5195

One of the most known spiral galaxies; the spiral structure of M51 may be associated with the companion galaxy; tidal model?

M51 (NGC 5194, Sc)

Theories of spiral structure

1. Chaotic spiral arms: “a swirling hotch-potch of spiral arms”

To explain the flocculent or ragged galaxies rather grand-design spirals, a chaotic model where the pieces of arms are constantly forming and dying but statistically appear an overall pattern, is more reasonable.

Ragged galaxies are more common than grand-design spirals.

Theories of spiral structure

Summary

2. In some cases it appears that the concept of the spiral arm as a density wave is correct;

3. However, there is little or no direct evidence for the hypothesis that the spiral pattern is stationary (i.e., that it will look the same in or so).

910 yr

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What determines the stability

of the disk and whether it will form

spiral arms and undergo star formation?

In isolated disk, creation of a density wave requires an instability. Self-gravity of the stars and / or the gas can provide this.

Simplest case to consider is gas. Imagine a small perturbationwhich slightly compresses part of the disk:

• Self-gravity of the compressed clump will tend to compress it further.

• Extra pressure will resist compression.

If the disk is massive (strong self-gravity) and cold (lesspressure support) first effect wins and develop spiralwave pattern.

For a disk of stars with surface mass density (mass perunit area) Σ, define Toomre’s Q as:

QºksR

3.36GS

Alar Toomre showed that if Q drops to ~1, a disk of stars is unstable to axisymmetric perturbations.

In practice, a spiral pattern generally grows if Q < 1.2 or so.

For the Milky Way’s disk near the Sun, Σ is about 50 Solarmasses per pc2, and κ is about 36 km/s. Stars of Solar massand below have σR = 30 km/s.

Suggests Q ~ 1.4 - close to critical value.

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Global, numerical stability analyses

Global axisymmetric stability (N-body simulation)

Evolution of a Kalnajs disk of 100,000 stars with Q around unity

Toomre’s local stability criterion is also sufficient for global stability to axisymmetric modes

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Global, numerical stability analyses

Bar-like instability

Evolution of the disk after removing the axisymmetric constraint

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Where does this come from?

Stabilization from pressure at small-scales and the rotationat large-scales

Small-scale: Jeans criterion λJ = tσ ff = /(2 G )σ π ρ 1/2

in 2D (disk) = h and h = Σ ρ σ2 / ( 2 G ) ==>π Σ

λJ = σ2 / ( 2 G ) = hπ Σ

Large-scale: Stabilisation by rotational shear

Tidal forces Ftid = d(Ω2 R)/dR R ~ Δ κ2 R Δ

Internal gravity forces of the condensation R Δ(G RΣ π Δ 2)/ RΔ 2 = Ftid ==> Lcrit ~ G / Σ κ2

Lcrit = λJ ==> σcrit ~ G / π Σ κ Q = / σ σcrit > 1

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Stability when several components

Disk of stars and gas, each stabilises or de-stabilises the other

Approximate estimation of the two contributionsUnstable if (k = 2 / ) π λ

(2 G k π Σs)/ (κ2 + k2 σs2) + (2 G k π Σg)/ (κ2 + k2 σg

2) > 1

For low values of k (large ), the stellar component dominates theλinstability; at small scale, the gas dominates by its low dispersion

For maintaining instabilities, gas is required, since it dissipatesStars may be unstable only transiently, since the componentheats up and becomes stable (self-regulation)

For star formation at large-scale Qg is not sufficient

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Critical gas surface density

Often used to justify star formation (Kennicutt 89)Qg ~ σg / κ Σ

gas unstable if > critΣ Σ

Critical density reached for the ultimateHII regions radius

Here, only HI gas

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Star formation rateFor normal disks as for starburst galaxies, the star formation rateappears to be proportional to gas densityBut average on large-scale, the whole disk

Global Schmidt law, with a power n=1.4 (Kennicutt 98) Σ SFR ~ Σ g 1.4

Another formulation works as well Σ SFR ~ Σ g or Ω Σ g/tdyn SFR ~gas density/tff ~ ρ 1.5

or cloud-cloud collisions in ρ 2 (Scoville 00)

may explain the Tully-Fisher relation (Silk 97, Tan 00)L ~ R2 Σ SFR ~ R2 Σ g Virial VΩ 2

~ RΣ

L~V3

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Slope n=1.4Normal galaxies (filled circles)starburst (squares)nuclei (open circles)

Slope 1

GALAXIES 626

Lecture 20:

End