Transcript Cosmic distance scale

1B11
Foundations of Astronomy
Cosmic distance scale
Liz Puchnarewicz
[email protected]
www.ucl.ac.uk/webct
www.mssl.ucl.ac.uk/
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
1AU
star
p
Sun
d
1
d(pc) 
p (" )
Ground-based telescopes – measure p to ~ 0.01” (d=100pc)
Hipparcos – measured p to ~ 0.002” (d=500pc)
Gaia – will measure p to 2x10-5 arcsec (d=50,000pc)
Parallax
1B11 Sun-Earth distance
To measure parallax accurately, we must know the distance
from the Earth to the Sun.
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:
T a
2
3
1AU
Where T is the orbital period of a
planet, and a is the semi-major axis of
it’s orbit.
1B11 Measuring an AU
T a
2
3
We can measure the orbital periods of the
planets by tracking them across the sky.
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:


(280s)
d
 3.00  105 km/s  4.20  107 km
2
4.20  107
8
1AU


1.5

10
km
So,
0.28
1B11 Nova expansion
Remember that for the proper motion of a star, m, (measured
in arcsecs per year), the tangential velocity of the star,
vt=4.74md (where d is in parsecs and vt is in km/s).
nova
shell
d
vr
m
vt
Instead of a star, consider a nova – an explosion from a
star. We assume that the shell thrown off is sphericallysymmetric.
1B11 Distances from nova shells
nova
shell
d
m
vr
vt
vr  
Using spectroscopy and measuring
the Doppler effect:
m is the proper motion of the shell
due only to its expansion
and since vt = |vr|, then
d
c

v t  4.74 md
vr
4.74 m
pc
1B11 Indirect methods – Cepheid variables
magnitude
Cepheid period-luminosity relation
m
~1mag
P~1-50 days
average mag
time
Cepheid variables are pulsating variable stars with a
characteristic lightcurve – named after dCephei.
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.
For P in days,
MV  2.8log10P  1.4
The mean magnitude is typically very bright: MV  6
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
Luminosity (LSUN)
106
103
1
10-2
10-4
30000 20000 10000 6000 4000
temperature, K
2000
This is a Hertzsprung-Russell diagram – a plot of luminosity
against temperature for stars. colour/spectral type => luminosity
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
MV
calibrated main
sequence
Vertical shift
gives (mV-MV)
horizontal shift gives
E(B-V) => AV
(B-V)
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:
MV (max)  9.96  2.31log10 (R2 )
where R2 is the decline rate in magnitudes per day, over the
first two magnitudes.
m
R2 
m=2m
MV
t2
Calibrate using
t2
novae in our Galaxy.
Then you can use the relationship to
infer extragalactic distances.
t
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.
flux
21cm x (1+z)
wavelength
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.