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Measuring Distances
Introduction
• Trigonometric Parallax
• Spectroscopic Parallax
• Cepheid Variables
• Type Ia Supernovae
• Tully-Fisher Relationship
• Hubble’s ‘Law’
Trigonometric Parallax
While fundamental, this method is very limited in terms of how
far away one can use it. For large numbers of star, groundbased measurements of parallax do not get beyond about 20
pc, although on individual objects this can be extended to
about 100 pc (with much effort).
The HIPPARCOS satellite extended reliable measurements out
to 100 pc for many thousands of stars, but this still
compares badly with the 8000 pc to the center of our
Galaxy.
Clearly we need other methods if we are to be able to measure
distances to most objects in our Galaxy let alone to other
galaxies.
Improving the Accuracy
Spectroscopic Parallax
• Spectroscopic parallax has nothing to do with parallax
– It makes the assumption that stars of exactly the same type
(e.g. have nearly the same spectra) will have the same
brightness
– If we can measure the distance to one star of this type, and
can measure the brightness of both stars, then we can
compute the distance to the second star
b1/b2 = d22/d12
The distance values are squared because brightness falls off as an inverse square
We try to compare Main Sequence Stars
Spectroscopic Parallax
• The key to this method is the identification of
`identical stars'. This is done by a careful
comparison of their spectra, checking for the
same features. This is one of the reasons for the
classification of stars into `spectral types'.
Spectroscopic parallax is a very useful method,
but the values it gives are rather approximate in
general, and very substantial errors are possible if
a normal star is mistaken for a giant or vice versa.
However, it can give us distances almost anywhere
within our own Galaxy and to nearby galaxies
Cepheid Variables
• Cepheid variables are high mass stars in a
late phase of their evolution during which
they become unstable, and start to pulsate.
During the pulsations, they expand and
contract with oscillation periods of order a
few days, and this is visible as a change in
their brightness
• The remarkably useful feature of Cepheid
variables is that their luminosity is fixed
by their pulsation periods
Cepheid Variables
Light curves of some Cepheid variables in the galaxy NGC 3109.
Their brightness indicates a distance of for this galaxy of 1.4 Mpc.
The plot shows B-magnitude versus fraction of a pulsation period.
Cepheid Variables
The period-luminosity relation
was first calibrated by Henrietta
Levitt in 1912
The period-luminosity relation
for Cepheids.
Visual luminosity or power is
measured by absolute
magnitude MV
Cepheid Variables
• Cepheids from the ground can be seen to about 1
Mpc (i.e. out to M31 at 750 Kpc). HST has
extended this out to 20 Mpc to measure the
distance of galaxies in the Virgo cluster.
• The discovery of Cepheids in M31 by Hubble was
one of the key discoveries in establishing the
nature of galaxies which many had thought until
that point to be part of our own Galaxy.
• Cepheids continue to be one of the cornerstones
of astronomy today.
Type Ia Supernovae
• A certain type of exploding star called a Type Ia supernova
appears to follow a fairly consistent light-curve, peaking at
an absolute magnitude of about Mv  -19. This makes them
23.8 magnitudes more luminous than the Sun, equivalent to a
factor of 1023.8/2.5 = 3.3 x 109 .
• These are now playing an important role in modern cosmology
as there are large projects dedicated to finding such
supernovae in very distant galaxies, of order away. Again the
principle is the identification of an object that does not
change its nature over large distances. These are often
called standard candles.
• Note that both Cepheids and Type Ia supernovae have to be
corrected for extinction just as for spectroscopic
parallaxes
SN 1987
Tully-Fisher Relation
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Developed in the 1970's and
1980's by Tully and Fisher.
The method has since been
improved by several workers and
is now one of the more accurate
secondary distance indicators for
the Universe.
The method relies on the fact
that there is a relationship
between the rate at which a
spiral galaxy spins and its
intrinsic luminosity.
The sense of the relation is that
the faster a galaxy spins, the
more luminous is the galaxy.
Tully-Fisher Relation
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Due to the spin of the galaxy, an observer will see part of the
galaxy approaching them and part of the galaxy running away.
This causes the emission from the galaxy to show redshifted,
blueshifted, and no-shifted emission.
The motion will thus cause a narrow line, e.g., a line due to
some element like hydrogen, to be smeared out and to appear
broad to the external observer. The broader the line, the
faster the galaxy must be spinning.
The gas and stars are in orbit in the galaxy, so from Kepler’s
Law, the more mass, the higher the spin rate
Since the observable light we see is produced by stars, it does
not take a stretch of the imagination to infer that the more
massive a galaxy is, the brighter it is likely to be.
This is borne out by the Tully-Fisher relation, the empirically
deduced relation between spin-rate and luminosity.
Hubble’s Law
• The dominant motion in the universe is the smooth expansion
known as Hubble's Law.
• Recessional Velocity = Hubble's constant times distance
V = Ho D
where, V is the observed velocity of the galaxy away from us,
usually in km/sec; H is Hubble's "constant", in km/sec/Mpc and D
is the distance to the galaxy in Mpc.
In 1929, Hubble estimated the value of
the expansion factor, now called the
Hubble constant, to be about 500
km/sec/Mpc. Today the value is still
rather uncertain, but is generally believed
to be in the range of 45-90 km/sec/Mpc
Hubble’s Law
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While in general galaxies follow the
smooth expansion, the more distant
ones moving faster away from us,
other motions cause slight deviations
from the line predicted by Hubble's
Law.
Few of the points fall exactly on the
line. This is because all galaxies have
some additional residual motion in
addition to the pure expansion.
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This is referred to as the "cosmic
velocity dispersion" or "cosmic
scatter" and is probably due to the
fact that the gas clouds that formed
the galaxies all had some small
additional motion of their own.
The recessional velocity of a galaxy
at a particular distance inferred
from Hubble's law is called the
"Hubble velocity".
This diagram shows a typical plot of
distance versus recessional velocity,
with each point showing the relationship
for an individual galaxy.
Hubble’s Law
• About in the middle of the diagram, there are a bunch of
galaxies that appear to be at about the same distance but
are spread out a lot in the velocity direction.
• This feature suggests the presence of a large cluster of
galaxies, like the Virgo Cluster.
• In addition to their "Hubble velocities", these galaxies have
an extra velocity caused by their orbital motion around the
center of the cluster.
– Because clusters of galaxies are very massive, this orbital
velocity can be very large, more than 1000 km/s.
• Therefore in the vicinity of nearby clusters of galaxies, we
cannot use Hubble's law to determine accurately the
distance to the galaxy.
The Distance Ladder