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1B11 Cosmic distance scale
Why is it so important to establish a cosmic distance scale?
Measure basic stellar parameters, eg radii, luminosities, masses
Explore the distribution of stars, eg Galactic structure
Calibrate extragalactic distance scale, eg galaxy scales, quasar luminosities, cosmological models
1B11 Direct methods
1. Trigonometric parallax
d
1AU star
Sun
)("1
d(pc)
Ground-based telescopes – measure to ~ 0.01” (d=100pc)
Hipparcos – measured to ~ 0.002” (d=500pc)
Gaia – will measure to 2x10-5 arcsec (d=50,000pc)
1B11 Sun-Earth distance
To measure parallax accurately, we must know the distance from the Earth to the Sun.
1AU
The Earth’s orbit is elliptical. An Astronomical Unit is the size of the semi-major axis.
In order to measure the length of the semi-major axis, we need a nearby planet, a radar and Kepler’s Third Law, which is:
32 aT Where T is the orbital period of a planet, and a is the semi-major axis of it’s orbit.
1B11 Measuring an AU
We can measure the orbital periods of the planets by tracking them across the sky.
32 aT So then we can calculate, in units of the Sun-Earth distance, the distances to all of the planets.
Wait until the Earth and a planet, eg Venus are a known AU distance apart, eg when they’re closest together. Then bounce a radar signal to Venus and back, measure the time it takes, multiply by the speed of light and you have the distance in, eg, km.
At their closest, the Earth and Venus are 0.28AU apart.
1B11 Converting from AU to km
1AU
0.28AU
In this position, it takes 280 seconds for light to bounce back from Venus to the Earth. So the distance in km is:
km104.20km/s103.002
(280s)d 75
km101.50.28
104.201AU 8
7
So,
1B11 Nova expansion
Remember that for the proper motion of a star, , (measured in arcsecs per year), the tangential velocity of the star, vt=4.74d (where d is in parsecs and vt is in km/s).
Instead of a star, consider a nova – an explosion from a star. We assume that the shell thrown off is spherically-symmetric.
d
vt
vr
nova shell
1B11 Distances from nova shells
Using spectroscopy and measuring the Doppler effect:
d
vt
vr
nova shell
c
vr
is the proper motion of the shell due only to its expansion
d4.74v t
and since vt = |vr|, then pc4.74
vd r
1B11 Indirect methods – Cepheid variables
Cepheid period-luminosity relation
time
mag
nitu
de
P~1-50 days
m ~1mag average mag
Cepheid variables are pulsating variable stars with a characteristic lightcurve – named after Cephei.
Henrietta Leavitt (1912) found that the period of variability increased with star brightness.
1B11 Calibrating Cepheids
The period-luminosity of Cepheid variables must be calibrated and this has been done by measuring their parallax using Hipparcos.
1.4P2.8logM 10V For P in days,
The mean magnitude is typically very bright: 6MV so they can be seen at very large distances (Henrietta Leavitt was working on a cluster of stars in the Small Magellanic Cloud). Measuring P provides MV, which gives distance via the distance modulus. This is especially important for calibrating extragalactic distance, eg the Hubble Space Telescope Key Project.
1B11 Spectroscopic distances – H-R diagram
This is a Hertzsprung-Russell diagram – a plot of luminosity against temperature for stars. colour/spectral type => luminosity
temperature, K30000 20000 10000 6000 4000 2000
Lum
inos
ity (
L SU
N)
106
10-4
1
103
10-2
1B11 Distances from H-R diagrams
If an H-R diagram is well-calibrated (ie the temperatures and spectral types are well-known), the absolute luminosities can be derived.
The distances are then calculated by measuring their apparent magnitudes and applying the distance modulus equation (ie (mV-MV)=5log10d-5+AV)).
Spectral distances can be calibrated using trigonometric parallaxes.
1B11 Cluster distances from H-R diagrams
(B-V)
MV
calibrated main sequence
Vertical shift gives (mV-MV)
horizontal shift gives E(B-V) => AV
And the distance modulus is (mV-MV)=5log10d-5+AV.
1B11 Standard candles
If there is a type of object whose intrinsic luminosity we can reliably infer by indirect means, then this is a “standard candle”. We measure its apparent flux, calculate the intrinsic luminosity and the inverse square law gives us the distance.
Globular clusters
Construct the H-R diagram for a globular cluster of stars and find the spectroscopic distance. Also look for variables in the cluster.
Integrated absolute magnitudes can be estimated by assuming MV.
Then you can estimate the distances for globular clusters around other galaxies.
1B11 Novae
Novae
The absolute magnitude of a nova can reach MV ~ -10.
And novae which decline faster, are brighter:
where R2 is the decline rate in magnitudes per day, over the first two magnitudes.
)(R2.31log9.96(max)M 210V
MV
t
t2
m=2m
22 t
mR
Calibrate using novae in our Galaxy.Then you can use the relationship to infer extragalactic distances.
1B11 Supernovae
Type II supernovae : core collapse of massive star
Type Ia supernovae : dumping of matter from secondary star onto a primary star in binary systems.
In Type Ia SN, MV(max) reaches approx –20
with only a small variation between different events.
So as long as you catch the maximum, you have a good standard candle out to very large extragalactic distances.
1B11 Tully-Fisher relationship
In 1977, Tully and Fisher found that the width of the 21cm emission line from a galaxy, was broader when the galaxy was brighter.
wavelengthflu
x
21cm x (1+z)
1B11 Tully-Fisher relationship
They suggested that this is a fundamental physical property, because:
More stars => more mass => higher rotation + brighter
The stars and gas in the galaxy are in orbit, so: Centrifugal force = gravitational force
M(star)v2/R = GM(galaxy)m(star)/R2
M(galaxy) = v2R/G (G=Gravitational constant)
And since luminosity is (probably) proportional to M(galaxy),
Luminosity(galaxy) would be proportional to v2
1B11 Tully-Fisher relationship
The trouble is –
It’s not yet clearly understood why the relation works so well.
It works really well over at least 7 magnitudes (a factor of 600 in luminosity terms).
It implies a cross-talk between the bulge and the disk components in galaxies… we don’t know how the bulge and the disk ”conspire” to produce the same mass to light ratios for such a range of luminosities.
1B11 Hubble’s Law
In 1929, Edwin Hubble discovered that all galaxies (beyond our local group) were moving away from us, and that their recession speed was proportional to their distance.
v = H0d
Where v is the recessional velocity, measured from the redshift, d is the distance to the galaxy and H0 is Hubble’s
constant.
This general expansion of the Universe demonstrated to Einstein that it was not static as he had thought.
H0 is hard to measure, Hubble thought it should be ~500 km
s-1, Mpc-1 – current observations indicate 50-90 km s-1 Mpc-1.
The best measurements indicate ~65 km s-1 Mpc-1
1B11 The age of the Universe
Once the Hubble constant has been measured, if we can assume that the rate of the expansion of the Universe has not changed, then we can calculate the age of the Universe,
Hubble’s law: v = H0d
And in general, d = vt, so t = d/v
Substituting for Hubble’s law, then t = d/(H0d)
So t = 1/H0
And if H0 = 65 km s-1 Mpc-1, then t = 1.3x1010 years.
Maybe expansion was faster in the early stages of the Universe, but there is evidence for a “dark energy” in the Universe which is increasing the expansion rate now.
