THE HUBBLE
LAW
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Edwin P. Hubble (1889-1953) observes with the Schmidt
telescope on Mount Palomar, California.
I. INTRODUCTION
One of the most interesting things about the universe is that virtually
all the galaxies in it (with the exception of a few nearby ones) are moving
away from the Milky Way. [Galaxies are gigantic groups of thousands of
millions of stars, the Milky Way is the galaxy in which we live.] This
curious fact was first discovered in the early 20th century by astronomer
Vesto Slipher, who noted that the light from most galaxies was redder than
expected. This so-called redshift was interpreted as the effect of the
galaxy moving away from us resulting in a Doppler shift of its light. [Doppler
shift also occurs with sound - you must have noticed that the note of a
car engine sound is initially high at it approaches you then abruptly becomes
lower as it passes by and starts to move away.]
In the 1920's, Edwin Hubble measured the distances of the galaxies for
the first time, and when he plotted these distances against the speed at
which each galaxy was moving away he noted something strange: The further
a galaxy was from the Milky Way, the faster it was moving away. Was there
something special about our place in the universe that made us a centre
of cosmic repulsion?
Astrophysicists readily interpreted Hubble's relation as evidence of
a universal expansion. The distance between all galaxies in the universe
was getting bigger with time, like the distance between currants in a rising
fruit cake. An observer on ANY galaxy, not just our own, would see all
the other galaxies travelling away, with the furthest galaxies travelling
the fastest.
This was a remarkable discovery. The expansion is believed today to
be a result of a "Big Bang" which occurred between 10 and 20 billion years
ago, a date which we can calculate by making measurements like those of
Hubble. The rate of expansion of the universe tells us how long it has
been expanding. We determine the rate by plotting the velocities of galaxies
against their distances, and determining the slope of the graph, a number
called the Hubble constant, Ho ("H-nought"), which tells us how fast a
galaxy at a given distance is receding from us. So Hubble's discovery of
the correlation between velocity and distance is fundamental in understanding
the evolution of the universe.
Using modern techniques, we will repeat Hubble's experiment. The technique
we will use is fundamental to cosmological research these days. Even though
Hubble's first measurements were made three-quarters of a century ago,
we have still only measured the velocities and distances of a small fraction
of the galaxies we can see, and so we have only a small amount of data
on whether the rate of expansion is the same in all places and in all directions
in the universe. The redshift-distance relation thus continues to help
us map the universe in space and time.
Hubbles's Law
Hubble's Law states that the recessional velocity of a galaxy is proportional
to its distance from us; the equation looks like this:
v = Ho * d,
where v is the galaxy's velocity (in km/sec), d is the
distance to the galaxy (in megaparsecs; 1 Mpc = 1 million parsecs
= 3 million light years), and Ho is the proportionality constant,
called "The Hubble constant". This equation is telling us that a galaxy
moving away from us twice as fast as another galaxy will be twice as far
away.
The velocity is relatively easy for us to measure
using the Doppler effect. An object in motion will have its radiation (i.e.
light) shifted in wavelength according to the following formula:
,
Wavelengths are usually measured in Angstroms (Å). The speed
of light has a constant value of 300,000 km/sec. The quantity on the left
side of the equation above is usually called the redshift, and is
denoted by the letter z. [Note this is the non-relativistic form
of the Doppler formula which is only true for cases where v/c is much less
than 1.]
So, we can determine the velocity of a galaxy from its spectrum:
we simply measure the wavelength shift of a known absorption line and solve
for v. [Absorption lines are sharp dips in the light coming from
an object. They occur at particular wavelengths depending on which element
is absorbing the light.] Example:
A certain absorption line that is found at 5000Å in
the lab is found at 5050Å when analyzing the spectrum of a particular
galaxy. We then conclude that this galaxy is moving with a velocity v =
(50/5000) * c = 3000 km/sec away from us.
A trickier task is to determine a galaxy's distance, since we must
rely on more indirect methods. One may assume, for instance, that all galaxies
are the same physical size, no matter where they are. This is known as
"a standard ruler" or "standard yardstick" assumption. To determine the
distance to a galaxy one would only need to measure its apparent
(angular) size, and use the following approximation for small angles: a
= s / d,
where a is the measured angular size (in radians!), s
is the galaxy's true size (diameter), and d is the distance to the
galaxy (where s and d are both measured in the same length
units]. Hence if galaxies were all intrinsically the same size then the
farther away they were the smaller they would appear and we could use their
apparent size to estimate their distance.
II. MEASUREMENTS
It is very important that when measuring the sizes of the galaxies you
ensure that your browser window is maximised.
For 12 of the 27 galaxies in the sample
provided you are to make two measurements [choose any 12 you prefer]:
-
The angular size of the galaxy is measured by using its image.
Note that the images used in this lab are negatives, so that bright
objects - such as stars and galaxies - appear dark! Note also that there
may be more than one galaxy in the image; the galaxy of interest is always
the one closest to the centre. These images have been extracted from the
Digitized Sky Survey taken with the Palomar Schmidt Telescope.
To measure the size, simply move the mouse and click on opposite ends
of the galaxy along its longest part. [this is because a disk galaxy
may be seen edge-on and so the longest dimension is a truer reflection
of its actual diameter]. You should calculate the angular size of the galaxy
(in milliradians; 1 mrad = 0.057 degrees = 206 arcseconds) by following
the instructions alongside each image; write this number down on your table
of
results. Click here if you want to be reminded
how to use Pythagoras to calculate the lengths of lines.
-
The recession velocity of the galaxy is determined by measuring
the redshift of spectral lines in the galaxy's spectrum. The full
(optical) spectrum of the galaxy is shown at the top of each spectrum page.
These were obtained by Kennicutt and are published in the Astrophysical
Journal Supplement Series. Below the full spectrum is an enlarged portion
of the same spectrum, in the vicinity of the common Calcium H and K absorption
lines.
Measure the wavelength of each of the two lines by using the mouse
pointer as instructed on the spectrum pages. Write this wavelength in your
table of results.
You may wish to include a comment in the appropriate column on your table
of results for cases where you think the measurements were hard to make
perhaps because the lines were hard to identify in the spectrum or because
the galaxy image was faint etc.
III. CALCULATIONS
-
Determine the distance (in Mpc) to each galaxy by making an important assumption:
as stated in the introduction, we may assume that all of these galaxies
are about the same size; from other methods we know that galaxies of the
type used in this lab are about 22 kpc (1 kiloparsec = 1000 pc)
across. We may then find the distance to the galaxies via:
distance (kpc) = size (kpc) / a (rad)
or equivalently, upon multiplying both sides by 1000:
distance (Mpc) = size (kpc) / a (mrad)
remember a is your measurement of the galaxy diameter.
-
For each measured line calculate the redshift z (see
above. The rest wavelength of each line can be found on the spectra.
Take the average redshift of the measured lines for each galaxy, and enter
it in the appropriate column on the table of results. Finally, use this
average redshift to calculate the velocity of the galaxy using the Doppler-shift
formula:
v = c z , where c is the speed of light
Now we can use these data to determine the value of the Hubble constant
and estimate the age of the Universe.
Hubble's Law predicts that the galaxies should lie on a straight line
when plotted on a graph of distance vs. velocity. Make a graph like
this (please use graph paper or a computer plotting package), with
distance
on the x-axis in units of Mpc, and velocity on the y-axis in km/s.
Draw a straight line using a ruler that best fits the points on the graph;
remember that this line must pass through the origin of coordinates
i.e. the point (0,0) at the bottom left of your graph. Can you explain
why?
-
Measure the gradient of your best-fit line (vertical change / horizontal
change) - this is the value of the Hubble constant, in the funny units
of km/sec/Mpc.
-
Your data points probably do not lie along a perfect straight line, and
you will notice that you had to make a guess as to where to draw your line.
One simple way to estimate the uncertainty in the value of Ho is to draw
the steepest reasonable line and
the shallowest reasonable line on the graph, and measure their slopes.
Half
of the difference between these two slopes is your uncertainty in Ho.
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And now for the age of the universe!: If the universe has been expanding
since its beginning at a constant speed, the universe's age would simply
be 1/Ho.
-
First, convert Ho to inverse-seconds (1/sec) by cancelling out the distance
units: 1 Mpc = 3.09*1019 km.
-
The "expansion age" of the universe is then t = 1/Ho.
-
This is a very simple model for the expansion of the universe. A better
model would account for the deceleration caused by gravity. Models like
this predict the age of the universe to be: t = 2/(3Ho). Re-calculate the
age using this relation.
-
How does this value compare to the age of the Sun?
IV. FURTHER QUESTIONS
Theoretically, your plot should be a straight line, but it probably isn't.
Think carefully about the following sources of error and answer the questions
below.
-
One obvious source of error is the assumption we made that all spiral galaxies
have the same diameter. How would an over-estimate or an under-estimate
of a galaxy's diameter affect your estimate of the distance to it? Explain.
-
Another consideration is the fact that galaxies are found in groups. The
motion of these galaxies through space as they revolve around their common
centre is called peculiar motion. How does this peculiar motion
affect your velocity measurement?
-
A third source of error is the error in the measurements that you make.
These three sources of error can be categorized as either "random" or "systematic".
Categorize each of them and explain your reasoning. (Note that some
of the aforementioned sources of error could be both systematic and random.)
This laboratory was originally developed by the Department of Astronomy
at the University of Washington.
