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Download Integrative Studies 410 Our Place in the Universe
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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 * * * * * * * 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: | 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 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! • • – – • 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