AY202a Galaxies & Dynamics

Lecture 2: Basic Cosmology, Galaxy

Morphology

COSMOLOGY is a modern subject:

The basic framework for our current

view of the Universe rests on ideas and

discoveries (mostly) from the early 20th

century.

Basics:

Einstein’s General Relativity

The Copernican Principle

Fundamental Observations & Principles

Fundamental Observations: The Sky is Dark at Night (Olber’s P.)

The Universe is Homogeneous on

large scales (c.f. the CMB)

The Universe is generally Expanding

The Universe has Stuff in it, and the

stuff is consistent with a hot

origin: Tcmb = 2.725o

Basic Principles:

• Cosmological Principle: (aka the Copernican principle). There is no preferred place in space --- the Universe should look the same from anywhere The Universe is HOMOGENEOUS and ISOTROPIC.

Principles:

Perfect Cosmological Principle: The Universe is also the same in time.

The STEADY STATE Model (XXX)

Anthropic Cosmological Principle:

We see the Universe in a preferred state(time etc.) --- when Humans can

exist

Principles:

Relativistic Cosmological Principle: The Laws of Physics are the same

everywhere and everywhen

(!!!) absolutely necessary (!!!) And we constantly check these

Mathematical CosmologyThe simplest questions are Geometric.

How is Space measured?

Standard 3-Space Metric:

ds2 = dx2 + dy2 + dz2

= dr2 +r2dθ2 + r2sin2θd2

In Cartesian or Spherical coordinates in

Euclidean Space.

Now make our space Non-Static, but

“homogeneous” & “isotropic”

ds2 = R

2(t)(dx

2 + dy2 + dz

2)

And then allow transformation to a more general geometry (i.e. allow non-Euclidean geometry) but keep isotropic and homogeneous:

ds2 = (1+1/4kr2) -2 (dx2+dy2+dz2)R2(t)

where r2 = x2 + y2 + z2, and k is a

measure of space curvature.

Note the Special Relativistic

Minkowski Metric

ds2 = c2dt2 – (dx2 +dy2 + dz2)

So, if we take our general metric and add the 4th (time)

dimension, we have:

ds2 = c2dt2 – R2(t)(dx2 +dy2 + dz2)/(1+kr2/4)

or in spherical coordinates and simplifying,

ds2 = c2dt2 – R2(t)[dr

2/(1-kr

2) + rdθsinθd

which is the (Friedman)-Robertson-Walker Metric, a.k.a. FRW

• The FRW metric is the most general, non-static, homogeneous and isotropic metric. It was derived ~1930 by

Robertson and Walker and perhaps a little earlier by Friedman.

R(t), the Scale Factor, is an unspecified function of time (which is usually assumed to be continuous)

and k = 1, 0, or -1 = the Curvature Constant For k = -1 or 0, space is infiniteinfinite

Rasin Bread Analogy

K = +1

Spherical

c < r

K = -1

Hyperbolic

c > r

K = 0

Flat

c = r

What about the scale factor R(t)?

R(t) is specified by Physics

we can use Newtonian Physics (the Newtonian approximation) but now General Relativity holds.

Start with Einstein’s (tensor) Field Equations

Gg and

GRgR

Where

is the Stress Energy tensor

R is the Ricci tensor

g is the metric tensor

G is the Einstein tensor

and R is the scalar curvature

RgR = g

is the Einstein Equation

The vector/scalar terms of the Tensor Equation

give Einstein’s Equations:

(dR/dt)2/R

2 + kc

2/R

2 = 8Gc2+c

2/3

energy density CC

2(d2R/dt

2)/R + (dR/dt)

2/R + kc

2/R

2

= -8GPc3+c2

pressure term CC

And Friedman’s Equations:

(dR/dt)2

= 2GM/R + c2R

2/3 – kc

2

So the curvature of space can be found as kc

2 = Ro2[(8G/3)o – Ho

2]

if = 0 (no Cosmological Constant)

or

(dR/dt)2/R2 - 8Go /3 =c

2/3 – kc

2/R2

which is known as Friedman’s Equation

Critical DensityGiven

kc2 = Ro

2 [(8G/3)o – Ho2]

With no cosmological constant, k = 0 if

(8G/3)o = Ho2

So we can define the “critical density” as

ρcrit = 3H02/ 8πG = 9.4 x 10-30 g/cm3

for H=70 km/s/Mpc

COSMOLOGICAL FRAMEWORK:

The Friedmann-Robertson-Walker

Metric + the Cosmic Microwave Background

= THE HOT BIG BANG

Λ

Cosmology is now the search for three numbers + the geometry:

1. The Expansion Rate = Hubble’s Constant H0

2. The Mean Matter Density = Ω (matter) = ΩM

3. The Cosmological Constant = Ω (lambda)= ΩΛ

4. The Geometric Constant k = -1, 0, +1

Nota Bene: H0 = (dR/dt)/R

Taken together, these numbers describe the geometry of space-time and its evolution. They also give you the Age of the Universe.

.

The best routes to the first two are in the Nearby Universe:

H0 is determined by measuring distances and redshifts to galaxies. It changes with time in real FRW models so by definition it must be measured locally.

(matter) is determined locally by

(1) a census, (2) topography, or (3) gravity versus the velocity field (how things move in the presence of lumps).

Other Basics Units and Constants:

Magnitudes & Megaparsecshttp://www.cfa.harvard.edu/~huchra/ay145/constants.html

http://www.cfa.harvard.edu/~huchra/ay145/mags.html

For magnitudes, always remember to think about central wavelength, band-pass and zero point. E.g. Vega vs AB.

Surface brightness (magnitudes per square arcsecond), like magnitudes, is logarithmic and does not “add”.

Why are magnitudes still the unit of choice?

Coordinate Systems2-D: Celestial = Equatorial (B1950, J2000) (precession, fundamental grid) Ecliptic Alt-Az (observers only) Galactic (l & b) Supergalactic (SGL & SGB)3-D: Heliocentric, LSR Galactocentric, Local Group CMB Reference Frame (bad!)

Galactic Coordinates

Tied to MW.B1950 (Besselian year)

NGP at 12h49m +27.4o

NCP at l=123o b=+27.4o

J2000 (Julian year)

NGP at 12h51m26.28s +27o07’42.01” NCP at l=122.932o b=27.128o

Supergalactic Coordinates

Supergalactic Coordinates

Equator along supergalactic plane

Zero point of SGL at one intersection with the Galactic Plane

NSGP at l = 47.37o, b=+6.32o

J2000 ~18.9h +15.7o

SGB=0, SGL=0 at l = 137.37o b = 0o

Lahav et al 2000, MNRAS 312, 166L

Galaxy Morphology“Simple” observable propertiesClassification goal is to relate form to physics.First major scheme was Hubble’s “Tuning Fork

Diagram” (1) Hubble’s original scheme lacked the missing link

S0 galaxies, even as late as 1936 (2) Ellipticity defined as e = 10(a-b)/a ≤ 7 observationally (3) Hubble believe that his sequence was an

evolutionary sequence.(4) Hubble also thought there were very few Irr gals.

Hubble types now not considered evolutionary although

there are connnections between morphology and evolution.

Hubble types have been considerably embellished by Sandage, deVaucouleurs and van den Bergh, etc.

(1) Irr Im (Magellanic Irregulars)

+ I0 (Peculiar galaxies)

(2) Sub classes have been added, S0/a, Sa, Sab, Sb …

(3) S0 class well established (DV L+, L0 and L-)

(4) Rings, mixed types and peculiarities added

(e.g. SAbc(r)p = open Sbc with inner ring and peculiarities)

•

S. van den Bergh introduced two additional

schema:

(1) Luminosity Classes --- a galaxy’s appearance is related to its intrinsic L.

(2) Anemic Spirals --- very low surface brightness disks that probably result from the stripping of gas

(c.f. Nature versus Nurture debate)

Morgan also introduced spectral typing of galaxies as in stars a, af, f, fg, g, gk, k

Luminosity Classes (S vdB + S&T Cal)

Real scatter much(!) larger

•

Other embellishments of note:

Morgan et al. during the search for radio galaxies introduced N, D, cD

Arp (1966) Atlas of Peculiar Galaxies

Some 30% of all NGC Galaxies are in the Arp or Vorontsov-Velyaminov atlases

Arp and the “Lampost Syndrome”

Zwicky’s Catalogue of Compact and Post-Eruptive Galaxies (1971)

Surface Brightness Effects

Arp (1965)

WYSIWYGNormal galaxieslie in a restrictedRange of SB(aka the Lampost Syndrome)

By the numbers

In a Blue selected, z=0, magnitude limited sample:

1/3 ~ E (20%) + S0 (15%)

2/3 ~ S (60%) + I (5%)

Per unit volume will be different.

also for spirals, very approximately

1/3 A ~ 1/3 X ~ 1/3 B

Mix of types in

any sample depends on selection by color, surface brightness, and even density.

Note tiny fraction of Irregulars

Quantitative Morphology

Elliptical galaxy SB Profiles

Hubble Law (one of four)

I(r) = I0 (1 + r/r0)-2

I0 = Central Surface Brightness

r0 = Core Radius

Problem 4 π ∫ I(r) r dr diverges

De Vaucouleurs R ¼ Law (a.k.a. Sersic profile with N=4)

I(r) = Ie e -7.67 ((r/re) ¼ -1)

re = effective or ½ light radius

I e = surface brightness at re

I0 ≈ e 7.67 Ie ≈ 103.33 Ie ≈ 2100 Ie

re ≈ 11 r0

and this is integrable

[Sersic ln I(R) = ln I0 – kR1/n ]

King profile (based on isothermal spheres fit to Globular Clusters) adds tidal cutoff term

re ≈ r0 rt = tidal radius

I(r) = IK [(1 + r2/rc2)-1/2 – (1 + rt

2/rc2)-1/2 ]2

And many others, e.g.:

Oemler truncated Hubble Law

Hernquist Profile

NFW (Navarro, Frenk & White) Profile

generally dynamically inspired

King profiles

Rt/Rc

Typical numbers

I0 ~ 15-19 in B

<I0> ~ 17

Giant E

r0 ~ 1 kpc

re ~ 10 kpc

Sersic profiles

Small N, less centrally concentrated and steeper at large R

Spiral Galaxies

Characterized by bulges + exponential disks

I(r) = IS e –r/rS

Freeman (1970) IS ~ 21.65 mB / sq arcsec

rS ~ 1-5 kpc, f(L)

If Spirals have DV Law bulges and exponential disks, can you calculate the Disk/Bulge ratio for given rS, re, IS & Ie ?

NB on Galaxy Magnitudes

There are MANY definitions for galaxy magnitudes, each with its +’s and –’s

Isophotal (to a defined limit in mag/sq arcsec)

Metric (to a defined radius in kpc)

Petrosian

Integrated Total etc.

Also remember COLOR

Reading Assignment

For next Wednesday

The preface to Zwicky’s “Catalogue of Compact and Post-Eruptive Galaxies”

and NFW “The Structure of Cold Dark Matter Halos,” 1996, ApJ...463..563

Read, Outline, be prepared to discuss Zwicky’s comments and Hernquist’s profile.

Hubble,1926

Investigated 400 extragalactic nebula in what he though was a fairly complete sample.

Cook astrograph + 6” refractor (!) + 60” & 100”

Numbers increased with magnitude

Presented classification scheme (note no S0)

97% “regular”

Sprials closest to E have large bulges

Some spirals are barred

E’s “more stellar with decreasing luminosity”

mT = C - K log d

23% E 59% SA 15% SB 3% Irr II

(no mixed types)

Plots of characteristics. Fall off at M~12.5

Luminosity-diameter relation

Edge on Spirals fainter

Apparent vs actual Ellipticity -- inclination

Absolute mags for small # with D’s

Calibration of brightest stars ---future use as distance indicators

Masses via rotation, Opik’s methodLog N - M or Log N log S

Space Density 9 x 10-18 Neb /pc3

1.5 x 10-31 g/cc

Universe Size 2.7 x 1010 pc ~ 30000 Mpc

Volume 3.5 x 1032 pc3

Mass 1.8 x 10 57 g = 9 x1022 M_sun