The Cosmological Distance Ladder
It's not perfect,
but it works!
First, we must determine
how big the Earth is.
This is done with
principles used by
surveyors.
But first let’s talk about the stars at night. If you live in Texas
and take a long exposure photograph towards the north, the
stars will appear to rotate around the North Celestial Pole,
which is close to the star Polaris.
The reason we have seasons is that the axis of rotation of
the Earth is tilted to the plane of its orbit around the Sun.
The northern hemisphere it tilted the most towards the
Sun on June 21st, which is the first day of summer. The
Sun is highest in the sky at local noontime on that date.
In the 3rd century BC
Eratosthenes obtained
the first estimate of
the circumference of
the Earth. He used
two observations of
the Sun on the first
day of summer.
We can use a pointed column like the Luxor obelisk in Paris
(or a stick) to cast a shadow of the Sun to determine our latitude.
Plots of the end of the
shadow of a gnomon,
as obtained in College
Station, TX, on the
first day of winter
(top), one day after
the fall equinox (middle),
and on the first day of
summer (bottom).
After measuring
the length of
the shadow on
two key days,
this is what
we found.
Some simple geometry/trigonometry gives us the
elevation angle of the Sun above the horizon:
On June 21st hmax = 82o 15.9 arcmin.
On December 21st hmax = 36o 06.6 arcmin.
90o minus the average of these two values
gives us the latitude of College Station, namely
30o 48.7 arcmin. According to Google Earth,
the right answer is 30o 37.2 arcmin. Knowing that
we are in the Central time zone, measuring the time of the
maximum height of the Sun gives us our longitude.
We found 96o 06.5 arcmin west of Greenwich.
From a gnomon experiment done on September 3, 2006,
I determine the latitude and longitude of a location in
South Bend, IN. I found that South Bend and College
Station are 13.78 degrees apart along a great circle arc.
I drove from South Bend to College Station
and found that the two locations are 1263 miles apart.
But, that’s a squiggly route, not the distance along a
great circle arc. Using a map and map tool, I determined
that the great circle distance was 940 miles.
360/13.78 X 940 = 24,557 miles, our estimate for the
circumference of the Earth. The “right answer” is
24,901 miles, so our value was 1.4% too small.
Determining the distance to the Moon
But first we must measure the angular size of the Moon.
Because the Moon’s orbit around the Earth is not circular,
the angular size of the Moon varies over the course of
the month. Likewise, the Sun’s angular size varies over
the course of the year.
A simple device for determining the angular size of the Moon.
Eyeball measures
of the angular size
of the Moon over
35 orbital periods
of the Moon. The
mean angular size
is 31.18 arcmin, a
little over half a
degree.
Aristarchus (310-230 BC)
cleverly figured out how to
use the geometry of a lunar
eclipse to determine the
distance to the Moon in
terms of the radius of the
Earth. Here s is the angular
radius of the Sun, t is the
angular radius of the Earth’s
shadow at the distance of
the Moon, PM is the “parallax”
of the Moon, and PS is the
“parallax” of the Sun.
It is not too difficult to show that s + t = PM + PS .
We know that the Sun has just about the angular size of
the Moon, because when the new Moon occasionally
eclipses the Sun, a total solar eclipse only lasts a few
minutes. But we can also measure the angular size of
the Sun when it near the horizon and we are looking
through a lot of the Earth’s atmosphere. From two such
observations I found s = 15.4 +/- 0.4 arcmin.
Ptolemy (2nd century AD) asserted that the Earth’s shadow is
about 2.6 times the angular size of the Moon. Here you can
see that he is basically right.
On June 15, 2011, there was a total lunar eclipse visible
in Europe. I downloaded 6 images of the Moon and, using
a ruler and compass, determined that the Earth’s shadow
was 2.56 +/- 0.03 times the size of the Moon.
My value for the angular radius of the Earth’s shadow at
the distance of the Moon is (31.18/2) X 2.56 ~ 39.19 arcmin.
From the distance to the Sun (see later) we find that PS is
small, only 0.14 arcmin.
The “parallax” of the Moon turns out to be
PM = 15.4 + 39.19 – 0.14 = 55.17 arcmin .
Since sin(PM) = radius of Earth / distance to Moon, it
follows that the Moon’s distance in Earth radii is 1/sin(PM).
On June 15, 2011, we found that the Moon was 62.3 Earth
radii distant. The true range is 55.9 to 63.8 Rearth.
Next, we must determine the scale of the solar system.
Copernicus (1543) correctly determined the relative
sizes of the orbits of the known planets. But we
needed to know the Astronomical Unit in km.
One way to determine this is to observe a transit
of Venus across the disk of the Sun from a variety
of locations spread out over the Earth. (Using the
Earth as a baseline, we determine the distance.)
These transits have occurred in 1761, 1769, 1874,
1882, and 2004. The next one occurs in 2012.
The big problem
with these observations
is related to the thick
atmosphere of Venus.
A much easier way to measure the scale of the solar
system is to determine the orbit of an asteroid that comes
close to the Earth. Then, using two telescopes at different
locations of the Earth, take two simultaneous images of
the asteroid against the background stars. The asteroid
ill be in slightly different directions as seen from the
two locations.
Asteroid 1996 HW1,
imaged from Socorro, NM,
on July 24, 2008 at
08:27:27.8 UT.
The same asteroid imaged
on the same date at the
same time, but 1130 km
to the west, in Ojai, CA.
An arc second is a very small angle. The thinnest
human hair (diameter ~ 18 microns) at arm’s length
subtends an angle of just about 7 arc seconds.
From our two asteroid images, we found that the asteroid
was shifted 5.05 +/- 0.61 arcsec as viewd from the two
locations. This translates into a distance to the asteroid
of 4.66 X 107 km. From a solution for the orbit of this
asteroid, we determined that it was 0.294 Astronomical
Units distant. This means that the Earth-Sun distance
is roughly 1.59 +/- 0.19 X 108 km. The correct length
of the AU is 1.496 X 108 km.
We determined the size of the Earth. Our value was
1.4% smaller than the true value.
We determined the distance to the Moon. Our value
was 3.3% larger that the known mean distance.
We determine the Earth-Sun distance. Our value was
6% larger than the true value.
Rarely in science do we know the true value of
something we are measuring. Otherwise it wouldn’t
be called research! But we think it was a useful
exercise to determine these first 3 rungs of the
cosmological distance ladder, but distances throughout
the cosmos depend on them.
By measuring
stellar parallaxes
we can determine
the distances to
the nearby stars.
The Hipparcos
satellite measured
many thousands
of parallaxes.
As the stars in the Hyades star cluster (in Taurus) move
through space their proper motions appear to converge
at a particular point on the sky. This is an effect of
perspective. Think of a flock of birds all flying towards
the horizon.
If the angular distance of the cluster from the
apparent convergent point is , then there is simple
relationship between the radial velocity of a star in
the cluster and its transverse velocity:
vT = vR tan()
Since the transverse velocity vT = 4.74 dpc  (where
 is the proper motion in arcsec per year and d is
the distance in parsecs), we can use the moving
cluster method to get estimates of the distance to
all the Hyades stars with measured proper motions and
radial velocities. The average would be the distance
to the cluster.
The method of main sequence fitting
Since main sequence
stars of a particular
temperature all
have the “same”
intrinsic brightness,
a cluster with a main
sequence some
number of magnitudes
fainter than the
Hyades gives us
its distance in terms
of the distance to the
Hyades.
Pulsating stars of known intrinsic brightness
RR Lyrae stars have mean absolute magnitudes of about
MV ~ +0.7. If you know the apparent magnitude and
absolute magnitude of a star, you can determine its
distance using this formula
MV = mV + 5 – 5 log d ,
where the apparent magnitudes have been corrected for
interstellar extinction and d is the distance in parsecs.
If you find a Cepheid variable star with a period of
30 days, it is telling you, “I am a star that is 10,000
times more luminous than the Sun!”
Cepheid variable stars were used by Shapley to determine
the distances to globular clusters in our Galaxy. This
allowed him to determine the distance to the center of
the Galaxy.
In 1924 Edwin Hubble used Cepheids to show that the
Andromeda Nebula is very much like our Milky Way
Galaxy. Both are large ensembles of a couple hundred
billion stars. And the Andromeda Galaxy is a couple
million light years away.
In the 1990's, using the Hubble Space Telescope, we
used observations of Cepheids in somewhat more distant
galaxies to determine their distances.
Perhaps the best standard candles to use for extragalactic
astronomy are white dwarf (Type Ia) supernovae. There
is a relationship between their absolute magnitudes and
the shape of their light curves that allows us to determine
their absolute magnitudes.
For Type Ia
supernovae we
define the “decline
rate” as the number
of magnitudes the
object gets dimmer
in the first 15 days
after maximum
light in the blue.
Fast decliners are
fainter objects.
On the x-axis
is the number of
magnitudes that
a Type Ia supernova gets fainter
in the first 15 days
after maximum
light.
Krisciunas et al. (2003)
In the nearinfrared, Type
Ia supernovae
are even better
standard candles.
Only the very
fastest decliners
are fainter than
the rest.
Similar to a figure in Krisciunas et al. (2004c)
A typical Type Ia supernova at maximum light is
4 billion times brighter than the Sun. As a result,
such an object can be detected halfway across the
observable universe, further than 8 billion light-years
away.
As with RR Lyr stars and Cepheids, if you know the
absolute magnitude and the apparent magnitude of
an object, you can calculate its distance, providing
you properly account for any effects of interstellar
dust.
Because galaxies exert gravitational attraction to each
other, galaxies have “peculiar velocities” on the order
of 300 km/sec. For galaxies at d ~ 42 Megaparsecs,
their recessional velocities are roughly 3000 km/sec
for a Hubble constant of 72 km/sec/Mpc. So one
can get an estimate the galaxy's distance using Hubble's
Law (V = H0 d) with a 10% uncertainty due to the
peculiar velocity. At a redshift of 3 percent the speed
of light (9000 km/sec), the effect of any perturbations
on the galaxy's motion are correspondingly smaller
(roughly 3 percent).
The bottom line is that from a redshift of z = 0.01 to
0.1 (recessional velocity 3000 to 30,000 km/sec)
one can use the radial velocity of a galaxy to determine
its distance.
Beyond z = 0.1 one needs to know the mean density
of the universe and the value of the cosmological
constant in order to get the most accurate distances.
In fact, it was the discovery that distant galaxies
(with redshifts of about z = 0.5) are “too faint”
that implied that they were too far away. This was
the first observational evidence for the Dark Energy
that is causing the universe to accelerate in its
expansion.
For more information on the extragalactic distance
scale, see:
www.astr.ua.edu/keel/galaxies/distance.html