Two Lectures on Observational Cosmology

Report
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
o
origin: Tcmb = 2.725
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 Cosmology
The simplest questions are Geometric.
How is Space measured?
Standard 3-Space Metric:
2
2
2
2
ds = dx + dy + dz
2
2
2
2
2
2
= dr +r d + r sin  df
In Cartesian or Spherical coordinates in
Euclidean Space.
Now make our space Non-Static, but
“homogeneous” & “isotropic” 
2
2
2
2
2
ds = R (t)(dx + dy + dz )
And then allow transformation to a more
general geometry (i.e. allow nonEuclidean geometry) but keep isotropic
and homogeneous:
2
2 -2
ds = (1+1/4kr )
2
2
2
2
2
2
(dx +dy +dz )R (t)
2
2
where r = x + y + z , and k is a
measure of space curvature.
Note the Special Relativistic
Minkowski Metric
2
2
2
2
2
2
ds = c dt – (dx +dy + dz )
So, if we take our general metric and add the 4th (time)
dimension, we have:
2
2
2
2
ds = c dt – R (t)(dx2 +dy2 + dz2)/(1+kr2/4)
or in spherical coordinates and simplifying,
2
2
2
2
2
2
ds = c dt – R (t)[dr /(1-kr ) + r (dq +sin q df )]
2
2
2
2
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 infinite
Rasin Bread Analogy
K = +1
Spherical
c < pr
K = -1
Hyperbolic
c > pr
K=0
Flat
c = pr
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
Gmu = 8pTmn + Lgmu
Gmu = Rmn - 1/2 gmu R
and
Where
Tmn is the Stress Energy tensor
Rmn
is the Ricci tensor
gmu is the metric tensor
Gmu is the Einstein tensor
and R

is the scalar curvature
Rmn - 1/2 gmu R = 8pTmn + Lgmu
is the Einstein Equation
The vector/scalar terms of the Tensor Equation
give Einstein’s Equations:
(dR/dt) /R + kc /R = 8pGe/3c +Lc /3
2
2
2
2
2
2
energy density
2
2
2
2
CC
2
2(d R/dt )/R + (dR/dt) /R + kc /R
= -8pGP/c +Lc
2
3
pressure term
CC
And Friedman’s Equations:
(dR/dt) = 2GM/R + Lc R /3 – kc
2
2 2
2
So the curvature of space can be found as
2
 kc =
2
2
Ro [(8pG/3)ro – Ho ]
if L = 0 (no Cosmological Constant)
or
(dR/dt) /R - 8pGro /3 =Lc /3 – kc
2
2
2
which is known as Friedman’s Equation
2
/R
2
Critical Density
Given
2
2
kc = Ro [(8pG/3)ro – Ho2]
With no cosmological constant, k = 0 if
2
(8pG/3)ro = Ho
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.
W(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 & Megaparsecs
http://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 Systems
2-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 properties
Classification 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 PostEruptive Galaxies (1971)
Surface Brightness Effects
Arp (1965)
WYSIWYG
Normal galaxies
lie in a restricted
Range 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)
-7.67 ((r/re) ¼ -1)
I(r) = Ie e
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 + rt2/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 method
Log 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

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