# Astro 6590: Galaxies and the Universe ```Astro 6590: Galaxies and the Universe
Lecture 7A:
HW#1&amp;2
Finishing up a few items
Oort’s constants
Force Law in the galactic plane
Then… RG
HW#3 due next Wed, 24 Sep
HW #1
1 dΦ
1 dA
Φ dm m where A(m)≡ dN/dM so
A dm =
1 dA
M - Mo
for a Gaussian LF:
=m
σ2
A dm
dlnA
and for a homogeneously distributed
so &lt;M&gt;m –Mo = - σ2
dm
population
dlnA
dm = 0.6 ln 10 = 1.38
so &lt;M&gt;m = Mo – 1.38 σ2 =&gt; independent of m.
1. Use B&amp;M 3.6.1
Thus, the mean absolute magnitude of a magnitude limited smaple is
smaller (brighter) than that of a volume limited sample by 1.38 σ2
2. This is B&amp;M 3.5 ; it follows from the previous result; For a
randomly distributed population, the distribution of galaxies in M
in a magnitude-limited sample is independent of the survey’s
limiting magnitude.
1
HW#1 (cont)
3.1 Straightforward
3.2 Straightforward: for α &lt; -2
3.3 If the 2nd cluster is 5X more distant, we lose 52 in flux.
If the survey is good to mlim &amp; for first cluster is complete
to M* + 4=&gt;
Mlim = mlim+ 5 – 5 log d = M* + 4
for the 2nd cluster Mlim = mlim+ 5 – 5 log (5d) = M* + 4 – 5log5
3.4 For α = -1.1; what is ratio of “missing” Lint? Define f as:
L
∫0lim Φ(L) LdL
f≡
= fraction of L lost per cluster
∞
∫0 Φ(L) LdL
Nearby cluster loses f=0.037; one at 5d loses 0.517, so the ratio
is 13.97. (Some people defined the measure in different ways.)
3.5 Fraction which lies below L* increases as faint end steepens
3.6 An exercise in reading recent literature and integrating…
HW #2
1. (m-M) = 5 log d -5
(m-M) = 5 log d – 5 + A =&gt; A = 10-3 d
Off by 2X at (m-M)=12.4, dtrue ~ 1.5 kpc
Re
2. Ltot = 2 ∫ 0 I(R) 2πR dR =&gt; straightforward
j(r) r dr
2
2
√r – R
∞
1
dI
dR
and j(r) = ∫
π r dR √R2 – r2
∞
3. Following B&amp;T: I(R) = 2 ∫ R
4. Following B&amp;M: dN/dM = No(M) ζ(M) where ζ (M) is your
favorite IMF. Note that stars of M ≥ 2 M have τMS &lt; 109yrs
and will die. Assume τMS ~ 1010(M/M)-2.5.
Low mass stars dominate because (1) more are formed and
(2) they live longer.
Assume L∝Mx;
Mtot = ∫ M dN/dM dM ~ 1.5 x 109 M
depends
Ltot = ∫ dN/dM L dM ~ 1.3 x 1010 M
on x and IMF
2
Local Standard of Rest (LSR)
Hipparcos results
Uo = 10.00 &plusmn; 0.36 km/s =&gt; radially inwards
Vo = 5.25 &plusmn; 0.62 km/s =&gt; in direction of Galactic rotation
Wo = 7.17 &plusmn; 0.38 km/s =&gt; vertically upwards
Motions of stars near the Sun
• Hipparcos catalog
• Need to account for variations with
stellar type
• See Dehnen &amp; Binney 1998
3
Geometry of solar motion
From
Mihalas &amp;
Binney
R sin α = R cos l – d
Vr = Θ cos α - Θ sin l
= (ω − ω) R sin l
Vt =(ω − ω) R cos l
-ωd
where ω = Θ/R
Galactic Rotation Relations
In the solar neighborhood, d &lt;&lt; R and |R-R|&lt;&lt; R
Vobs = (ω−ω) R sin l
Estimate the variation of (ω−ω) by a first order Taylor
expansion:
(ω−ω) ≈ dω
(R - R)
dR R
dω d Θ
=
dR dR R
dω =
dR R

=
1 dΘ Θ
R dR
R2
so that
1
dΘ
Θ
R dR R R2

4
Galactic Rotation Relations
dω
1
dΘ
Θ
- 
=
dR R R dR R R2
To first order, then
dΘ - Θ
Vr =
dR R
R
(R - R) sin l

For d &lt;&lt; R
=&gt;
Θ
R
Vr ≈
-
R - R ≈ d cos l, so
dΘ
dR
d sin l cos l
R
Also, sin l cos l = &frac12; sin 2l, so we can define
Vr = A d sin 2 l
Θ
R
A ≡ &frac12;
and
dΘ
dR
Oort’s Constant A
R
Oort’s Constants
Θ
R
A ≡ &frac12;
Vr = A d sin 2 l
dΘ
dR R
Oort’s Constant A
We can show similarly (B&amp;M 637-641) for proper motions:
proper motion μ = Vt / 4.74 d
where d in pc, μ in arcsec/yr
Vt = d (A cos 2 l + B) where
B ≡-&frac12;
dΘ
dR
Θ
+
R
Notice the identities:
ω =
dΘ
dR
Θ
= A-B
R
= - (A + B)
R
A ≡ - &frac12; R
dω
dR
Oort’s Constant B
R
A and B are known as Oort’s
constants
A = 14.82 &plusmn; 0.84 km/s kpc-1
B = 12.37 &plusmn; 0.64 km/s kpc-1
R
Feast&amp;Whiteoak 1997
5
Oort’s constants
Hipparcos π and μ for 20000 single
stars at d&gt;100pc as fn(SpecType).
3 contributions to local kinematics:
• Motion of Sun relative to
avg motion of *s in S.N.
• Differential velocity field
due to galactic rotation
around Sun (shear)
• Peculiar velocities
Oort’s constants
F. Mignard
• Derives Solar Motion: u, v, w, V, (l,b)apex wrt different stellar
early vs late types. Same with A, less with B. Sun is moving upward
(+z) from the GP 7.2&plusmn;0.2 km/s
6
Force law ⊥ Galactic plane
Kz = -
∂Φ
∂z
Kapteyn 1922;
Oort 1932
Force law ⊥ Galactic plane
7
Force law ⊥ Galactic plane
Force law ⊥ Galactic plane
8
Force Law ⊥ Galactic plane
Force Law ⊥ Galactic plane
9
Force Law in the galactic plane
Bianaym&eacute; et al. 2006
A&amp;A 446, 933
• 203 nearby “red clump”
stars in Hipparcos catalog
(so distances known)
• NGP K-giants
Force Law in the galactic plane
Bianaym&eacute; et al. 2006
A&amp;A 446, 933
Try to fit including DM component
ρeff =&gt; ρeff = 0 works too.
Surface mass density w/in |z|
from the GP (i.e. disk)
Σ0.8kpc = 57-66 Mpc-2
Σ1.1kpc = 57-79 Mpc-2
Σstars = ~40 Mpc-2 Don’t need much
ΣISM = ~13 Mpc-2 DM in disk
P (z-oscill) = 42&plusmn;2 Myr
Dinosaur extinctions =&gt;
P=26 Myr (Raup&amp;Sepkoski
1986)
10
On-going surveys
RAVE: radial velocities for 1,000,000 stars to &lt; 5 km/s
uses 6dF spectrograph on 1.2m UK Schmidt telescope at AAO
Programs under SDSS: (APO 2.5m telescope)
SEGUE-1: 240,000 stars to 4 km/s
SEGUE-2: 350,000 stars to 4 km/s
APOGEE: higher res IR spectra for 100,000 stars =&gt; metallicity
=&gt; Powerful probes of structure and history of MW
11
```