astrophysics introductory course ii- chapter 14: spiral galaxies

8/3/2019 Astrophysics Introductory Course II- Chapter 14: Spiral Galaxies

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Astrophysics Introductory Course II SS 2010

Chapter 14

Spiral Galaxies

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14.1 Components of Spiral Galaxies

Bulges: metal poor to super metal rich stars, rotation approximately equals dispersion (seechapter on elliptical galaxies), can be triaxial or bar-shaped, especially in Sb, Sc types.The surface brightness profiles of bulges usually follow an r 1/n-Sersic model well:

with

where re is the half-light or effective radius which contains half the projected light and

I e ist the surface brightness at re (flux/square-arcsec). The total luminosity is given by:

The special case of n=4 is called the de Vaucouleurs profile.

Disks: metal rich stars, ordered rotation, rotation velocity much higher than velocity dis-persion, star formation, HI, H2-gas, molecular clouds, dust, hot gas (heating by star

formation and supernovae). On average exponential surface brightness profile:

I(R) = I o exp(-R/Ro) and Ltot = 2! I o Ro2

where R is the cylindrical radius, R0 is called the scale length of the disk and I 0 is the

central surface brightness of the disk. Deviations from exponential profile are common,partly due to real mass density variations (spiral arms), partly wavelength dependent(star formation).

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with


disk-bulge(r 1/4) decompositions, Baggett et al 1998




Baryonic halo: metal poor (factor 10 to 1000 lower metallicity than the sun) stars andglobular clusters with on average little/no systematic rotation around galaxy center, awide variety of orbits. Typical ages for globular clusters in the Milky Way are 12Gyr to15Gyr, they seem to be the oldest objects in spirals (consistent with their metallicity).The stars show the following density distribution in the Milky Way (only measured there

so far):

where R! is the radius of the sun’s orbit. For the globular clusters we have, e.g. in the

Milky Way):

Furthermore, the baryonic halo contains low density X-ray gas and low density HI/HII-clouds (mass is insignificant).

Dark Matter Halo: dominates mass of typical spiral outside of 2 or 3 scale length,(maybe) flattened, density usually follows:

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An ultra-deep image of the halo of M31 with HST ACS




14.2 Rotational properties of spiral galaxies

14.2.1 Rotation curve of a thin disk

Apart from the central bulge, most of the stars in spirals are concentrated in a relatively

thin disk. Therefore, the shape of the outer rotation curves should be determined by thepotential and in turn the density distribution of the disks. For completeness, the derivationof the rotation curve for an infinitely thin, axisymmetric mass sheet with arbitrary densitydistribution is given on the next three pages (in blue). A simple analytical approximation for the rotation curve of a thin exponential disk is the following:

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To derive the rotation curve of an infinitely thin disk, we first have to solve the Laplace-equation(in cylindrical coordinates) to obtain the potential outside the disk and then link this solution to themass distribution in the disk:

By separation of variables:

we get:

with a constant k. These two differential equations are solved by:

where I0(kR) is the cylindrical Bessel function of 0th order. The boundary conditions are chosen

such that $ % 0 for z % inf. or R % inf. and $(z = 0) remains finite. Consequently, for arbitrary k:

However, in the presence of an infinitely thin, spatially finite and axisymmetric mass distribution, k

is constrained by the requirement that the potential difference between a location just above and just below the sheet is determined by the surface mass density of the sheet. The relation betweenthis potential difference and the mass surface density is determined by Gauss theorem:

Z ( z ) = Z 0e!k | z | Z

0= const

I ( R) = I 0(kR)

"k ( R, z ) = Z

0e!k | z |

I 0(kR)

is a solution of !

!2" = 0 in cylindrical coordinates.

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Integrating the Poisson equation over a small volume including the mass sheet:

gives:

If we integrate over all surface elements, only the two elements parallel to the mass sheetcontribute to the integral, as all others vanish. Therefore we can write:

where we have defined I k = " !dz to be the mass surface density. Inserting our solution from

above with Zo=1, we get for the limit of z % 0:

The surface density corresponding to the solution !k(z,R) therefore is:

!"+

! z #!"#

! z

$

% &'

( ) d

2S = 4* GI

k ( R)d

2S

!"±

! z = lim

z +0#+

!"k

! z = !kI

0(kR)

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Due to the linearity of Poisson’s equation in $ and &, we can use superposition to obtain moregeneral solutions:

and

to derive the potential of an arbitrary very thin density distribution. If I(R) is given, we have to

invert:

The integral equals a Hankel transformation, with its inversion given by:

Setting I(R) = I0exp(" R/R0), we can derive ! and vc2 = R·#!/#R of the exponential disk (I0, I1, K0,

K1 = modified Bessel functions), see Binney/Tremaine for details:

with y = 1/2·R/R0 and, as usual: I(R) = I(0)exp(" R/R0).

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Rotation curve of a thin exponential disk, see Binney/Tremaine p.78

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14.2.2 Observational Determination of the Rotation of Spiral Galaxies

The circular speed can be approximately determined as a function of the radius bymeasuring the redshift of the emission lines of the gas contained in the disk.

hot stars ionize gas:! hydrogen emission lines, mainly H

"in the optical.

neutral atomic hydrogen gas:hyperfine structure transition ('p 'e % 'p (e) leads to a 21cm radio line.

• The dynamical mass of a spiral galaxy within the radius r is:

) is a geometry factor. For the same radial density distribution but different flattenings(disk vs. sphere) one has 0.8 < ) < 1.2, see Figure on previous page by Binney/Tremaine.

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H) rotation curves (from Binney/Tremaine p.600)

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neutral hydrogen HI(21cm)-rotation curves of spiral galaxies fr om Bosma, Phd thesis (1979)

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Rotation curve of our Galaxy from Fich et al. ARAA 29, 409 (1991)

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14.3 Dark Matter in Spirals

Figure on the left from van Albada et al.ApJ 295, 305 (1985):The distribution of optical light (negativeimage) and HI gas (contours) in NGC 3198.The HI gas extends far beyond the opticaldisk and allows a determ-ination of therotation curve at large radii. In NGC 3198the mass of the HI is small compared to the

stellar mass. Therefore, the mass of the HIgas cannot explain the rotation curve shownon the next page.

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NGC 3198: optical




see: van Albada et al. ApJ 295, 305 (1985)

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A simple analytical model for dark matter halos

!

!

!




Summary on Rotation Curves and Dark Matter

Main observational fact:Rotation curves remain flat out to radii much larger

than the extent of the optical disk!

From v(R) * constant and centripetal force equation GM(R)/R2 = V2/R, it follows:

For the majority of spiral galaxies no decrease in the circular velocity has been measuredeven beyond radii of 50 kpc to 100 kpc.

M/LB of the observable matter in the galactic disk only:

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14.4 The Tully-Fisher Relation

Relation between circular speed vcirc and luminosity of spiral galaxies:

(The power depends on the wavelength band in which L is measured. The power changes because of differences in M/L, star formation etc.)

From the virial theorem, v2 ~ M/r , and the identity L ~ I·r 2 one obtains:

Evidently the observed Tully-Fisher relation implies: (M/L)"2 ·I"1 ~ const. Indeed, mostluminous spirals have similar mass-to-light ratios and surface brightnesses: M/L ~ constand I ~ const (the latter is called the “Freeman law”).

The Tully-Fisher relation is of fundamental importance for the distance determination of spiral galaxies.

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Tully-Fisher Relation:

Figure from Jacoby et

al.: 1992, PASP 104,599 (this is a veryinteresting review of different extragalacticdistance determinationmethods)

Notation:Vcirc can be deduced

from HI observations;a correction for

inclination is

achieved viameasuring theflattening of theoptical or IR image.

2.0 2.4 2.8 2.0 2.4 2.8 2.0 2.4 2.8 2.0 2.4 2.8

log (2 Vcirc)

-24

-22

-20

-18

-16

8

10

12

14

8

10

12

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14.5 Stability of Disks and Spiral Structure

Disks are relatively fragile objects. Generally, disks with a higher velocity dispersion aremore stable against perturbations. From dynamical analysis, Toomre (1964) obtained thefollowing criterion:Disks are stable if:

with the mass surface density I(r), the epicycle frequency + and the radial dispersion ,r.

The Toomre criterion is the ‘Jean’s Criterion’ for a disk.

Example: Galactic disk in solar neighborhood:

The formation of bars and spiral structure requires that the disks live close to Q~1.

Q !

" r

3.4GI (r )#

>1 Toomre's Q parameter

!




Evolution of a spiral galaxy, with gas accretion. The animated sequence is a self-gravitating

simulation, taking into account stars and gas. New stars are constantly formed from gas, according

to a star formation rate proportional to the local gas density. The old stars are in red, the young

stars in blue, the gas is in yellow. After the development of a spiral, a bar is formed by disk instability in the first 500 million years. This bar soon will weaken, then disappear, but other bars

will replace it, for example the second strong bar at about 7 billion years, at the end of the

animation. The current barred galaxies could be experiencing their third or fourth barred episode.

Frederic Bournaud, Francoise Combes.

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Origin of Spiral Structure

Spiral arms cannot rotate with the local circular velocity in the disk, because then theywould wind up. (The revolution time is much shorter at small radii than at large ones: - is not constant, in general: - ~ 1/r)

% Spiral arms are made up of different stars at different times! % Spiral arms are a wave phenomenon! The wave amplitude is typically 10% of the meandensity. There exist several hypotheses to explain the origin of a spiral wave.

As almost all luminous disk galaxies show spiral structure, one explanation is thatspiral density waves are stationary or quasi-stationary, i.e. long-lived (Lindblad 1963,Lin & Shu 1964-70)

Swing amplification of density perturbations in the disk can bring forth trailing multi-armed spiral patterns (Toomre 1981). In this way spiral structure can formspontaneously from small perturbations.

Very prominent spiral arms seem to form in interacting spiral galaxies (M51, M81).Interaction obviously can trigger strong density waves.

In ”stochastic” spirals, e.g. NGC2841, self-propagating star formation and differentialrotation can result in a structure similar to spiral arms.

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14.5.1 Spontaneous Spiral Structure

Swing amplification of particle noise (= small perturbations in the density of stars in thedisk) can bring forth trailing multi-armed spiral patterns. Shown on the next pages is anN-body experiment by Josh Barnes with a central bulge (yellow), an exponential disk (blue),and a dark halo (red). Apart from Poissonian fluctuations due to particle noise, this disk isinitially featureless. Later frames, separated by about half a rotation period at threeexponential scale lengths, show the development of transient spiral.

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The perturbation (grey patch) initially has the form of a leading spiral (right), but is sheared into atrailing spiral (left) by the differential rotation of the disk. In a stable disk, random motions normallydisrupt an overdense region before it has time to collapse. Here, however, the specific form of thedensity wave insures that individual stars remain within the overdense region for a substantialfraction of the epicyclic period + (the epicycle and the perturbation rotate in the same direction, sostars stay in the perturbation longer than they would under other conditions). As a consequence, thewave is amplified by a modified form of the Jeans instability.


see: Barnes, J. E. :Institute for Astronomy, University of Hawaii

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14.5.2 Tidal Spiral Structure

Tides between galaxies provoke a two-sided response. Such perturbations, if further swing-amplified in differentially-rotating disks, may produce striking ‘grand-design’ spiralpatterns. In the experiment shown here, an artificial tide was applied by taking the unper-turbed disk above and instantaneously replacing each x velocity with

where k is a constant used to adjust the strength of the perturbation. No perturbation

was applied to the y and z velocities. Frames made after 1.5 rotation periods show thedevelopment of an open, two-armed spiral pattern which becomes more tightly wound withtime.

!




see: Barnes, J. E. :Institute for Astronomy, University of Hawaii

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Interacting spiral galaxy M51(Wendelstein Observatory)


NGC 2841

14.5.3 Self-propagating star formation

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Numerical simulations of stochastic self-propagating star formation in spirals (Gerola &Seiden 1978, ApJ 223, 129). Originally roundish star formation regions get stretched outin differential rotation. Numbers give the time in units of 15 Mio. years, the upper panel is

for the rotation curve of M101, the lower for M81.

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14.5.4 Spiral Structure and Star Formation

Spiral structure and star formation are closely linked via the following sequence of events:

% the spiral density wave compresses the gas in the disk (which enhancesthe strength of the density wave)

% molecular clouds collide and collapse% star formation sets in

Spiral arms are very prominent in the blue light and the H)

line, because of the short-lived

massive blue stars forming in the wake of the density wave.

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M 101. The spiral density wave becomes prominent because it triggerscompression of molecular clouds and star formation (Wendelstein Observatory).

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14.6 Star formation history of spirals

• The H)-line strength and the far-infrared luminosity provide information about

the present star formation rate # in galaxies. The H) strength is directly

proportional to the ionizing flux of massive, hot stars and in turn to the number of these stars which again is a measure of the actual star formation rate (SFR). The far-IR-luminosity comes from warm dust grains which are heated by luminous stars.SFR . and H) luminosity are related via:

• The underlying assumption of this relation is the so-called case B approximation:Photons of the Lyman-continuum are completely absorbed and reemitted as Ly), H),

etc. (see Osterbrock: Astrophysics of Gaseous Nebulae). Ly) is again absorbed by

dust, H) can escape more easily because dust absorbs less at longer wavelengths.

Typical SFRs of spirals are 1...15 M!/yr .

• Colours and H)-equivalent widths provide information about the star formation

history. A high star formation rate in the past will have produced many low-mass redstars which cause the galaxy to appear redder and to have more continuum light. TheH)

-equivalent width (a measure for the strength of the H)

emission relative to the un-

derlying continuum) is an indicator of how many stars are currently formed relative tothe average SF rate over the history of the galaxy.

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from Kennicutt et al. AJ 88, 1098 (1983)

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from Devereux et al. ApJ 350, 25 (1990)

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The equivalent width of H) vs B " V

color can be used to constrain the initialmass function of stars in spirals (figurefrom Kennicutt et al. ApJ No 272, p54,1983) The three evolutionary modelsshown in the figure use different IMFs(from top to bottom):$ ~ m"2

$ ~ m"2.35

$ ~ Miller-Scalo IMF

or:

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• Comparing the H)-flux with colours leads to:

(for a schematic star formation history of different Hubble types see Sandage 1996)

• The relation between the equivalent width of H) compared to a color (e.g. B-V) is an

indicator for the slope of the initial mass function (IMF). Current data showcompatibility with the Salpeter IMF and are probably incompatible with a Miller-ScaloIMF.

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Kennicutt et al. ApJ 272, 54 (1983)

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from: Sandage et al.A&A 161, 89 (1989)very schematic

astrophysics introductory course ii- chapter 14: spiral galaxies

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