Astronomy 1 Cosmology
Dr. I. Waddington
University of Bristol 2002/2003 Copyright c 2003 University of Bristol.
These lecture notes are for the exclusive use of members of the Astronomy 1 class and must not be distributed outside the University of Bristol.
Figures from journals published by the American Astronomical Society (AAS) and the Royal Astronomical Society (RAS) are reproduced for educational use with permission from the publishers. Astronomy 1: Cosmology 2002/2003
Astronomy 1: Cosmology
Lecturer: Dr. I. Waddington, Room 4.21b, [email protected]
Dates: weeks 21–23
Lecture notes: www.star.bristol.ac.uk/iw/ast1/ and in the library
Practical: week 23 (May 8)
Problem sheet/tutorial: week 23
Additional reading: M. Zeilik & S. A. Gregory, Introductory Astronomy & Astrophysics, ISBN: 0-03-006228-4, chapters 22, 25, 26 D. J. Raine & E. G. Thomas, An Introduction to the Science of Cosmology, ISBN: 0-7503- 0405-7 (excellent book, goes well-beyond this course but worth considering) Steven Weinberg, The First Three Minutes, ISBN: 0-00-654024-4 (popular account, little maths)
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Introduction
The Goal of Cosmology
• to explain the structure & history of the universe as a whole
⇒ the ‘Hot Big Bang’ theory
The Key Observations & Physics
• the universe is expanding ⇒ gravitational dynamics (General Relativity) ⇒ approximate model using Newtonian gravity
• the universe is filled with thermal radiation ⇒ quantum physics, thermodynamics
• galaxies & large-scale structures evolve ⇒ astrophysics
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The Expanding Universe
Olbers’ Paradox
Why is the night sky dark?
Surface brightness of a star F L/(4πd2) L Σ = = = ∗ Ω A/d2 4πA ⇒ independent of the distance d n stars per pc3, number of stars in element of a cone dN = nV = nr2dΩdr
Covering factor of element
= dN × covering factor of a star
πR2 = nr2dΩdr × r2dΩ
= πnR2dr
Total covering factor Z d Z d C = πnR2dr = πnR2 dr = πnR2d 0 0
When C = 1 every line-of-sight in dΩ ends on a star ⇒ surface brightness along every line-of-sight is Σ∗ 2 8 Occurs when d = 1/(πnR ). Take R = R = 6.96 × 10 m and
number of stars in Local Group n ∼ (distance to next galaxy cluster)3
1012 ∼ (107 × 3.09 × 1016)3
∼ 3 × 10−59m−3
Page 2 Astronomy 1: Cosmology 2002/2003 then d ∼ 2 × 1040 m ∼ 6 × 1023 pc
Takes t = d/c ∼ 2 × 1024 years for their light to reach us
Sky is dark ⇒ either the universe is (i) smaller than d or (ii) younger than t
Distance Ladder
How big is the universe?
Solar System: Astronomical Unit, i.e. the mean Sun–Earth separation, calculated geo- metrically by measuring the distance to Venus (using radar) when it is at greatest elon- gation. 1 AU = 1.50 × 1011 m.
Stars: Trigonometrical parallax. Moving cluster parallax. Spectroscopic parallax: star’s spectrum gives its absolute magnitude (MV ). The distance (d, in parsecs) is found from its apparent magnitude (mV ):
mV − MV = 5 log d − 5. [1.1]
Distances of up to 1000 pc, i.e. our own Galaxy. Depends on measurement of the AU.
Nearest galaxy: RR Lyrae variable stars. All have the same luminosity or absolute magnitude (MV ) so their apparent magnitude (mV ) can be used to find their distance (d). Distance to the Large Magellanic Cloud (LMC) – 50 kpc (50 × 103 pc). Depends on the distance to RR Lyrae variables in our own galaxy.
Nearby galaxies: Cepheid variable stars. Period-luminosity relationship:
P ∝ L0.9
(V -band). Measure mV & P , find L and thus MV from the P-L relation, and then the distance from mV − MV . P-L relation calibrated in the LMC and so depends on distance to the LMC. Distances up to 20 Mpc (20 × 106 pc).
Distant galaxies: Type Ia Supernovae (SNe Ia). Carbon-oxygen white dwarfs, located in binary systems. Luminosity of the light curve at its peak (MV ) is constant for all SNe Ia. Measure light curve and apparent magnitude at the peak (mV ), and find the distance from mV − MV . MV is found from SNe in nearby galaxies whose distance is known from Cepheid measurements. Distances up to ∼500 Mpc.
Distant galaxies: ‘Standard rods’. Object of known size D (e.g., half-light radius of giant elliptical galaxy). Measure angular size θ ⇒ distance r = D/θ
Modern variant uses Fundamental Plane relation between D, σ & L
Distances up to ∼500 Mpc
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Figure 1.1 r dΩ
dr
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Figure 1.2 Supernova light curve at z = 0.458 from Perlmutter et al. (1995, ApJ, 440, L41; c AAS)
R mag qo=0.5 22.00 300
σ c 22.25 200 22.50 qo=0 22.75 23.00
Relative Flux 100 23.50 24.00
0
0 50 100 150 Day (JD - 2448707)
Hubble’s Law
Redshift of a galaxy: λ − λ v z = obs emit ' . [1.2] λemit c Only valid for v c, in which case it is just like a Doppler Shift.
Hubble found that the speed of a galaxy v depends on its distance d: Hubble’s Law:
v = H0d [1.3]
H0 is Hubble’s constant – measures the rate of expansion of the universe.
Best measurements of H0 come from HST observations of SNe Ia, and the current range −1 −1 −1 −1 of values is 60–80 km s Mpc ; we will assume H0 = 65 km s Mpc .
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Use Hubble’s Law to estimate the age of the universe: d d 1 τH = = = . [1.4] v H0d H0
9 τH is the Hubble time and it has a value of 15 Gyr (1 Gyr = 10 years).
Figure 1.3 Measurement of Hubble’s constant from Freedman et al. (2001, ApJ, 553, 47; c AAS)
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Summary
Olbers’ Paradox: darkness of the night sky implies the universe: (i) is not infinite in extent (ii) has not always existed
Distance Ladder: from the Solar System to distant galaxies is a step-by-step process
Typa Ia Supernovae: luminosity (or MV ) at peak of light curve is a known constant; find distance d by measuring mV at the peak:
mV − MV = 5 log d − 5
Hubble’s Law: the universe is expanding:
v = H0d
−1 −1 Hubble’s constant: H0 = 65 km s Mpc Hubble Time: estimate the age of the universe: 1 τH = H0
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