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?• 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

* * * * low precision, high accuracy

* * * * low precision, low accuracy

accepted value

When do results agree?

• Results agree, if they are within the error margins of each other

• Examples:

| O | | O |

values very different, but errors large: agreement!

| O | | O |

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 Alexandria-Syene 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!

– 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– 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