Transcript Earth and Stars

How to measure the size of the Earth
•Aristotle, the famous Greek natural philosopher, reports that
mathematicians had allegedly evaluated the dimension of Earth at
40.000 stadia, adding: `From their supposition, it follows that the shape
of Earth must be a sphere and also that its size be small relative to the
distance of other celestial bodies.‘
•It was generally agreed
upon that measuring the size
of Earth could be done by
measuring the altitude of a
star from two cities situated
on the same meridian.
•Then, a difference expressed in degrees would be found. If the
distance between the two cities was known, from estimates by
caravaneers for instance, it would then be possible to find the value of
a degree of meridian and hence derive the value of the terrestrial
circumference.
•The stadium, Aristotle's unit length, apparently corresponds to 185
meters, so the value of 74.000 km thus obtained is much too high.
Archimedes, in his treatise on the number of sand grains quotes a value
of 300.000 stadia for the terrestrial circumference. This means that the
measurement must have been attempted several times.
Eratosthenes' measurement
•From his readings, he had learnt
that once a year (on the day of
the Summer solstice), the bottom
of a well situated at Aswan in
Upper Egypt was illuminated by
the Sun;
•However, at Alexandria, this
never happened: obelisks always
cast a shadow;
•He believed that Earth was a
sphere;
•He assumed that Alexandria and Aswan were on the same meridian;
•He knew (or better, he assumed) that the distance between the two
cities was 5,000 stadia (as caravans covered the distance in 50 days
at a rate of 100 stadia a day);
•He postulated that sunrays reached Earth as parallel beams (an
idea that was commonly held by the mathematicians of his time).
Alexandria
Aswan (Syene)
So, on solstice day, he decided to measure the length of the meridian
shadow cast by a gnomon at Alexandria. He found a value of 1/50th of a
circumference (i.e. 7.12°) and derived the value of the terrestrial
circumference: 50 x 5.000 = 250.000 stadia. Although our idea of the
exact value of the stadium (which was not the same at Athens, Alexandria
or Rome) is fairly hazy, this puts the terrestrial circumference at 40.000
km. The result is remarkable, although several errors were introduced in
the calculations:
•The distance between Alexandria and Aswan is 729 km, not 800;
•The two cities are not on the same meridian (the difference in longitude
is 3°);
•Aswan is not on the Tropic of Cancer (it is situated 55 km farther North);
•The angle difference is not 7.12° but 7.5°.
The most extraordinary thing is that the measurement rests on
the estimated average speed of a caravan of camels: one can
certainly do better in the matter of accuracy. Yet, in spite of all
these flaws, it worked fine: around 250 BC, Earth had at last a
size.
Stellar Parallax (Trigonometric Parallax )
Measuring distances to objects within our Galaxy is not always a
straightforward task - we cannot simply stretch out a measuring tape
between two objects and read off the distance. Instead, a number of
techniques have been developed that enable us to measure distances to
stars without needing to leave the Solar System. One such method is
trigonometric parallax, which depends on the apparent motion of nearby
stars compared to more distant stars, using observations made six
months apart.
•A nearby object viewed from two different positions will appear to move
with respect to a more distant background. This change is called
parallax.
•By measuring the amount of the shift of the object's position (relative to
a fixed background, such as the very distant stars) with observations
made from the ends of a known baseline, the distance to the object can
be calculated.
•A conveniently long baseline for measuring the parallax of stars (stellar
parallax) is the diameter of the Earth's orbit, where observations are
made 6 months apart. The definition of the parallax angle may be
determined from the diagram below:
The position of a foreground star is measured when the Earth is at
position A. 6 months later, the Earth has moved around the Sun to
position B - this provides a baseline of 2AU. Compared to the more
distant 'fixed' stars, the foreground star has moved on the sky by an
angle, 2p.
http://sci2.esa.int/interactive/media/flashes/2_1_1.htm
If the parallax angle, p, is measured in arc seconds (arcsec), then the
distance to the star, d in parsecs (pc) is given by:
•The only star with a parallax greater than 1 arcsec as seen from
the Earth is the Sun - all other known stars are at distances greater
than 1 pc and parallax angles less than 1 arcsec. When measuring
the parallax of a star, it is important to account for the star's proper
motion, and the parallax of any of the 'fixed' stars used as
references.
1. Why we can’t use parallax method to measure to distance
for further stars?
The further the distance, the smaller the parallax will be. So
the results won’t be accurate any more.
2. What is the angle difference between Alexandria and
Aswan when we measure the size of the Earth? (Tuff!!!)
It’s 7.12° (or more accurately will be 7.5°
3. What does ‘AU’ stand for?
It stands for Astronomical Unit.