This is done with

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This is done with principles used by surveyors.
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But first lets 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.
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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.
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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.
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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.
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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).
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After measuring the length of the shadow on two
key days, this is what we found.
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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.
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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, thats 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.
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Determining the distance to the Moon
But first we must measure the angular size of the
Moon.
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Because the Moons orbit around the Earth is not
circular, the angular size of the Moon varies
over the course of the month. Likewise, the
Suns angular size varies over the course of the
year.
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A simple device for determining the angular size
of the Moon.
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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.
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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 Earths shadow at the distance
of the Moon, PM is the parallax of the Moon,
and PS is the parallax of the Sun.
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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 Earths atmosphere. From two
such observations I found s 15.4 /- 0.4 arcmin.
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Ptolemy (2nd century AD) asserted that the
Earths shadow is about 2.6 times the angular
size of the Moon. Here you can see that he is
basically right.
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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 Earths shadow was 2.56 /- 0.03 times
the size of the Moon.
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My value for the angular radius of the Earths
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 .
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Since sin(PM) radius of Earth / distance to
Moon, it follows that the Moons 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.
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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.
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Asteroid 1996 HW1, imaged from Socorro, NM, on
July 24, 2008 at 082727.8 UT.
The same asteroid imaged on the same date at
the same time, but 1130 km to the west, in Ojai,
CA.
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An arc second is a very small angle. The
thinnest human hair (diameter 18 microns) at
arms 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.
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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 wouldnt 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.
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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.
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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.
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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).
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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 constan
t 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.
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