Slipher’s Spectra of Nebulae Lowell Telescope – near Flagstaff Arizona – provided spectra of...
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Transcript of Slipher’s Spectra of Nebulae Lowell Telescope – near Flagstaff Arizona – provided spectra of...
Slipher’s Spectra of Nebulae
Lowell Telescope – near Flagstaff Arizona –provided spectra of ~50 nebulae – showing 95% of them were moving away from Earth at speeds up to 1500km/s
Iron Iron Scandium Sodium
Iron Iron Scandium Sodium
Different Model Universes• 1917 Einstein’s Cosmological Constant
Universe• 1916, 1920 de Sitter’s Empty Universe
Solutions predicted spectra shift• 1922 Friedmann showed family of
solutions based on homogenous isotropic Universe.
• 1930 Einstein de-Sitter Flat Universe• 1936 Robertson-Walker Solutions
Hubble’s Discovery of the Expanding Universe
•1926 Hubble measuring average space density of Galaxies to look at effects of Curvature using static Universe solutions
•1927 Lemaitre showed Hubble Law (velocity proportional to Distance) expected of Friedmann Universes, and demonstrated, using Hubble’s/Slipher Data the Law in nature. Noted age of Universe roughly 1/Hubble Constant
•1928 Robertson predicts linear relationship and claims to see it in plot of Galaxy brightness versus Slipher’s redshift
•1929 Humason announces velocity of NGC 7619 of 3779 km/s
•1929 at National Academy of Science Hubble presents paper announcing Universe is Expanding
•1929-1931 Hubble/Humason extend relation to 20000 km/s
Lemaitre – The unsung hero
• Fought for Belgium starting at 14 in WWI• Seminary in 1923, ordained as a priest• 1923 Visited Eddington in Cambridge England • 1924 Visited Harlow Shapley in Cambridge, Mass• 1925 enrolled in PhD at MIT, but returned to Brussels to work
on it.• 1927 published “homogeneous Universe of constant mass
and growing radius accounting for the radial velocity of extragalactic nebulae”– Independently derived Friedman Equations– Suggested Universe was expanding– Showed it was confirmed by Hubble’s data.
Mathematics accepted by Einstein, but basic idea rejected by Einstein
• 1931 Discussed primeaval atom which everything grew out of – the Big Bang
Apply Gauss’ Law:
Since this holds for any sphere, there is a solution for all spheres, M=0
So need to fake it: Lets assume that all forces cancel out, outside the sphere, and we only have to worry about those forces inside the sphere.
€
GM
r2ˆ r ⋅∫ dA =< g > A = 4πGM
• Within shell of radius r,
– Note as time moves forward, galaxy moves out, (r increases), but Mass “affecting” object is constant.
– Equation of Motion for any Galaxy
30 3
4rM πρ=
constant2
1 2 =−=r
GMmmvE
Critical Universe
• E=0
( )
critG
H
r
rGrH
r
GMmmv
ρπ
ρ
πρ
≡=
=
=
8
3
34
2
1
2
1
20
3
20
2
Critical Density defines line between bound and unbound Universe. Density higher than this, Universe has negative Energy – Bound, lower than this, Universe has positive Energy - Unbound
The density changes in time, so v=H(t)r
General Equation of Motion (non-critical case)
( )
200
20
03
02
200
20
0
03
02
000
2
3
4
234
2
1
3
4
2
34
2
1 at
2
1
rGH
r
rG
dt
dr
rGH
mK
r
mrGrHmKt
Kr
GMmmv
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
⎟⎠⎞
⎜⎝⎛
−⎟⎠
⎞⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
−=
=−
ρπρπ
ρπ
ρπ
M is constant as seen by our galaxy, so I have usedthe current epoch expression for M for all times r.
non-critical case, cont
( )0*
0
2
*
*
20
002
0
0
2
*
*
200
2002
00
2
*
*00
200
20
03
02
*
*00
00*
*
*
*
0*
*
0
*
20
0000*
0*
1D
3
81
3
8
3
4
23
4
2
1
3
4
234
2
1
1
1
1
1
3
8
Ω−=Ω
−⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−⎟⎟
⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟
⎠
⎞⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
⎟⎠⎞
⎜⎝⎛
−⎟⎠
⎞⎜⎝
⎛
=
==
==
==Ω==
τ
ρπρπ
τ
ρππρτ
ρπρπ
τ
ττ
τ
ρπ
ρ
ρτ
d
dD
H
G
r
r
H
G
d
dD
rGH
r
rrG
d
dDHr
rGH
r
rG
dt
dr
d
dDHr
dt
dr
Hdt
dr
rd
dt
dt
dr
dr
dD
d
dD
Hd
dt
rdr
dD
H
GtH
r
rD
crit
Substitution to simplifyThis ODE
€
dD*
dτ *
⎛
⎝ ⎜
⎞
⎠ ⎟
2
− Ω0
D*
= 1− Ω0( )
Again define variables to solve this ODE
ξ =1− Ω0
Ω0
D* τ =1− Ω0( )
3 / 2
Ω0
τ *
dξ
dτ=
dξ
dD*
dD*
dτ *
dτ *
dτ
dξ
dD*
=1− Ω0
Ω0
, dτ *
dτ=
Ω0
1− Ω0( )3 / 2
dξ
dτ=
1− Ω0
Ω0
dD*
dτ *
Ω0
1− Ω0( )3 / 2
dD*
dτ *
=dξ
dτ1− Ω0( )
1/ 2
dξ
dτ
⎛
⎝ ⎜
⎞
⎠ ⎟2
1− Ω0 − Ω0
1− Ω0
ξΩ0
= 1− Ω0( )
dξ
dτ
⎛
⎝ ⎜
⎞
⎠ ⎟2
−1
ξ=
1− Ω0( )1− Ω0
= ±1
€
dξ
dτ
⎛
⎝ ⎜
⎞
⎠ ⎟2
=1
ξ±1
dτ = dξξ
1± ξ
τ = dξξ
1± ξ0
ξ
∫
final substitution! for Bound (-) case
ξ = sin2(η /2) =1
21− cosη( )
dξ = sin(η /2)cos(η /2)dη
τ = sin(η /2)cos(η /2)dηsin2(η /2)
1− sin2(η /2)0
η
∫
τ = sin(η /2)cos(η /2)dηsin2(η /2)
cos2(η /2)=
0
η
∫ sin2(η /2)dη
τ =0
η
∫ 1
2−
1
2cosη
⎛
⎝ ⎜
⎞
⎠ ⎟dη =
1
2η −
1
2sinη
⎛
⎝ ⎜
⎞
⎠ ⎟