Transcript Integrative Studies 410 Our Place in the Universe

Measurements, Triangulation &
Conclusion Part I
Performing Experiments
• Experiments must be repeatable – requires careful
control over variables
• Possible outcomes of an experiment:
– The experiment may support the theory
• We then continue to make predictions and test them
– The experiment may falsify the theory
• We need a new theory that describes both the original data and
the results of the new experiment
• Since we cannot do every possible experiment, a
theory can never be proven true; it can only be
proven false
Making Measurements
• Errors
– Random
– Systematic
• With every measurement, it is essential to provide an
estimate of the uncertainty – the likely range of errors
• Example:
– Using a ruler marked in mm, we round to the nearest marking –
at most off by half a division, or 0.5 mm
– Cite a measurement of 15 mm as 15  0.5 mm to indicate that
the real value of the length is likely to be anywhere between
14.5 mm and 15.5 mm
– If a theory predicts a value of 15. 2 mm, then a reading of
15  0.5 mm is in agreement with the theory but a reading of
15  0.1 mm is probably not
Is the uncertainty small or big?
• It depends! If you have a small uncertainty and the
measured length is also small, you might have a
huge uncertainty!
• Use percentages:
– Percent error = (estimated uncertainty)/(result) x 100%
– Example: 51.3 cm ± 0.2 cm gives
– Percent error = (0.2 cm)/(51.3cm) x 100 % = 0.4 %
(This is a pretty small uncertainty)
Is the result precise or accurate or what?
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Two different concepts: precision and accuracy!
High precision means small error
High accuracy means close to an accepted value
Examples:
****
high precision, high accuracy
****
high precision, low accuracy
*
* *
*
*
*
*
accepted value
*
low precision, high accuracy
low precision, low accuracy
When do results agree?
• Results agree, if they are within the error
margins of each other
• Examples:
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O
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O
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values very different, but errors large: agreement!
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O | |
O
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values closer, but errors smaller: no agreement!
Astronomical Distance
Measurements
• Fundamental technique uses
triangulation:
– Objects appear to move with
respect to background if looked
at from different vantage points
• Try looking at you thumb with
only your left, then right eye
• The more the thumb jumps,
the closer it is!
• Measure “jump”, get distance
• See: Link, Link 2
Liu Hui, How to measure
the height of a sea island.
Simple Triangulation
• Use geometry of similar
triangles
• You know everything about
a triangle if you know
– Two sides and an angle
– One side and two angles
• Example: baseline 100ft, angles
90° and 63.4° then distance =
(100ft)(tan 63.4°) = 200ft
Parallax Basics
• The closer the object, the
bigger the parallax (or
parallactic angle)
– Pencil held close (solid lines)
– Pencil held far (dashed lines)
• The farther the object the
harder to measure the small
angle, the more uncertain
the distance
Triangulating the Size of the Earth
• Eratosthenes (ca. 276 BC)
– Measures the radius of the earth to about 20%
Calculation
• Angle is measured to be
7.2 = 360/50
• So distance AlexandriaSyene is 1/50 of Earth’s
circumference
• Baseline can be
measured: 5000 stades
•  Circumference is
23,330 miles (modern
value: 25,000 miles –
only 7% off
Baseline: Bigger = Better
• Can use Earth’s large size for
a 12,700km baseline
• Just wait 12 hours!
Counterargument
or not?
• Objection to
Aristarchus’s model of a
moving Earth: parallax of
stars is not observed
(back then)
• Aristarchus argued
(correctly) that this means
the stars must be very far
away
Distances to the Stars
•
Use even bigger baseline by
waiting ½ year, not ½ day
Baseline: 300 million km
Parallax can be used out to
about 100 light years
The bigger the parallactic
angle, the closer the star!
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A star with a measured parallax
of 1” is 1 parsec away
1 pc is about 3.3 light years
The nearest star (Proxima
Centauri) is about 1.3 pc or 4.3
lyr away
The most important measurement
in Astronomy: Distance!
• The distances are astronomical – of course
• The distance scales are very different
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Solar system: light minutes
Stars: light years
Galaxies: 100,000 ly
Universe: billions of ly
• Need different “yardsticks”
Yardsticks and the Expanding
Universe
• Realizing (measuring) the distances to
objects means realizing how big the
universe is:
– We realized that the solar system is not the
universe
– We realized that our galaxy is not the universe
– We realized that the universe is not static
What can we conclude from
observing patterns in the sky?
• Earth OR Celestial Sphere rotates
• Earth rotates around the Sun OR Sun moves
about Earth
• Moon rotates around the Earth or v.v.?
– Must be former, due to moon phases observed!
• Size of the earth from two observers at
different locations
• Size of moon & moon’s orbit from eclipses