Survey – extra credits (1.5pt)! Study investigating general patterns of college students’ understanding of astronomical topics There will be 3~4 surveys this semester. Anonymous survey (the accuracy of your responses will not affect your course grade). But, be accurate, please! Your participation is entirely voluntary. SPARK: Assessments > Survey2 The second survey is due: 11:59pm, March 27 th (Sun.) Questions? - Hyunju Lee or Stephen Schneider Funded by Hubble Space Telescope Education & Public Outreach grant

How is scientific investigation made? Key points: The goal of science is to render our universe understandable and unequivocal -- to discover the underlying principles that govern how things (ultimately everything) work The key to make science unequivocal is to base any comparison on numbers, i.e. make comparisons quantitative The essentials of the scientific method: 1) Discover something and observe/describe it in a quantitative way, i.e. with measurements. 2) Develop an hypothesis to explain what is seen. 3) Apply the hypothesis to new data or to a new, but similar situation, and test if it continues to explain what happens Finally, make a prediction of what else should be observed if the hypothesis were true, and then make measurements to test.

Measurements: How to make “quantitative” sense of the world Is Northampton distant? Is Europe distant? Northampton is distant if you walk; it is not if you drive Europe is distant if you fly on a B747 (about 8 hr); it is not if you use the space shuttle to get there (about 30 min) So, to remove any ambiguity, the distance to a place is best characterized by its true value (its measurement): Northampton: 10 miles Europe (NYC to London): 3,300 miles You decide yourself how long it takes to get there with your vehicle

On Measures All measures in physics (and hence in astronomy) can be made by measuring only four fundamentals quantities: Space Time Mass Electric Charge

More on Measurements Which statement is correct? A: A person whose weight is lb is heavier than a person whose weight is lb. B: A person whose weight is lb. is heavier than a person whose weight is lb. (0.001 lb is the weight of two or three teardrops) (You MUST answer this question, since it is used to determine your attendance!)

More on Measurements Can an observer make measurements of arbitrarily small quantities? N O Then, if small differences in a measures cannot be appreciated, how do I know that what I measure is the true value? I DO NOT! Every measurement is subject to uncertainty, or error. Errors are not mistakes; they are limitations in our ability to carry out the “perfect measurement”, i.e. to measure the true value Errors can be minimized, but never eliminated Infinite knowledge cannot be achieved

We can make sense of the universe in a quantitative way by measuring things For example: gravity. The Law of the Free-Falling Bodies says that all bodies fall with the same acceleration (i.e. regardless of their mass) where g is the acceleration of gravity on Earth In other words: falling bodies, no matter their composition, shape or mass, after falling for a time “t”, all reach the same velocity “v”, whose value must be exactly equal to the value of g multiplied by the value of “t” (anywhere on Earth: g=9.81 m/sec 2 ) Is this true? We have to verify it by measures. If it is not true, then the theory of gravity is wrong and we must find a better theory

So, after falling for the same time all objects should reach the same speed. This means that the law of falling bodies under Earth’s gravity says that we should observe this:

But actually, what we observe is this: Why? Air is the culprit: it creates a large amount of drags that slows down some light objects, which then fall with constant velocity (i.e. they do not keep accelerating faster and faster)

SYSTEMATIC ERRORS This test of the law of accelerating bodies is subject to systematic errors here on Earth. If we want to test it, we have to remove the source of systematic errors The Moon has no air: an ideal place for this experiment, because free from the systematic error introduced by aerodynamic drag.

The Falling Body Experiment on the Moon

SYSTEMATIC ERRORS Any spurious cause that creates systematic deviations of measures (or physical process) from their true value in a given experimental set-up. They always perturb the measure in the same way, either adding or subtracting a given amount relative to the true value The smaller the systematic errors, the more precise the measures Systematic errors are often rather difficult to spot and correct

More On Measures OK, now that we have eliminated systematic errors, let’s try to see whether or not the law of falling bodies under gravity is true. We will let some body fall in a tall vacuum chamber and… measure the free falling time, and find t=11 sec measure the velocity, and find v=115 meter/sec (about 241 MPH) Now, Earth’s acceleration of gravity “g” is g=9.806 meter/sec 2, … so after 11 sec we expect the velocity to be V expected = (9.806 meter/sec 2 ) x (11 sec) = meter/sec But we measure v=115 meter/sec, not meter/sec! OMG, Is Gravity wrong!??

Random and Measure Errors No, as we will see, Gravity is NOT wrong. The fact is that every measure is affected by Random and/or Measure errors. These errors are “perturbations” that affect measures in an unpredictable way, sometime in excess, some other time in deficiency, i.e. they can unpredictably add or subtract an unknown amount from the true value These errors are NOT mistakes; they are what makes an infinitely accurate measure impossible. The figure of merit of a measure (accuracy) is the magnitude of the Random/Measure Errors: the smaller the better Measure Errors: the deviation stems from the finite accuracy of the measuring apparatus Random Errors: the deviation stems from the combined action of a large number of many uncontrollable factors

Measure Error A simple case of Measure Error: the read-out error The finiteness of the sub- division of the ruler does not allow measures with arbitrarily small accuracy. All we can say is that the length of the arrow is between 7.70 and 7.75, namely L = ± cm cm is the Measure Error

Random Errors: an Astronomical Example Measure the speed of a High Proper Motion star and verify that is not moving faster than the speed of light To make the measure we need thr travel time and distance traveled DISTANCE: one has to accurately center on the image of the star at the start and end positions and measure the distance that star travels in a given time (here 20 years). Not easy: look at how the center of each star moves due to seeing Barnard Star: a high proper motion star V ≈ 90 km/s (201,367 MPH)

Atmospheric Seeing Ever-changing instantaneous blurring of astronomical images due to the refracting power of the Earth’s atmosphere It introduces a Random Error if one tries to measure the position of features on the surface of the Moon… …or anywhere else in the sky

Random Error: Measuring the period T of the pendulum A simple case of Random Error: the response time of the observer To get the period, we need to measure the time the pendulum takes to complete one full oscillation Depending on whether the observer acts too soon or too late, the measure gets altered by an unknown amount This happens twice: when we start the stopwatch and when we stop it. T = (t_ stop ± ε stop ) – (t_ start ± ε start )

Uncertainty Ultimately, all measurements of physical quantities are subject to uncertainties. Variability in the results of repeated measurements arises because variables that can affect the measurements result are impossible to hold constant. Even if the "circumstances," could be precisely controlled, measures would still have an error associated with them, because measure apparatuses can only be manufactured with finite level of quality (the infinitely accurate instrument is only a theoretical abstraction) Steps can be taken to limit the amount of uncertainty, but it will always be there, no matter how refined (and expensive) technology can be. So, the real goal of an experiment is: reduce the uncertainty in the measures to the degree that is needed to prove or disprove a theory

The quality of a measure: the relative, or fractional, error If I measure the distance from this building to the IS building (75 meter) with an error of 25 meters, I am not doing a great job But if I measure the distance between Earth and the Moon with an error of 25 meters, I am doing a superb job So, what really matters is not the absolute value of the error (the absolute error), but how big is the error relative to the measure I am making. To express the quality of a measure we take the ratio between the error and the measure itself (relative error): In the first case ε=25/75=0.33, namely the error is 33% the value of the measure In the second case ε=25/340,000,000 = , namely the error is 7 million-th of a percent of the measure. If you measure T= 2.9 sec with an error of 0.3 sec, the relative error is ε=0.3/2.9 = 0.1, namely the error amounts to 10% of the value of the measure.

Back to Testing the Law of Free Fall In this experiment: The error of time measures was Δt = 0.4 sec, so: t = 11.0 ± 0.4 sec ε = (3.6%) The uncertainty of velocity measures was Δv = 10 meter/sec, so: v = 115 ± 10 meter/sec ε = (8.7%) Thus, v expected = meter/sec is well within the error bar. There is no observable discrepancy with the theory. Gravity is true, after all!

In Summary The only way to accurately describe the world, and compare observations with theory, is by using measurements Measuring: expressing the strength of a physical quantity by a number Every measurement is affected by uncertainty (also referred to as “error”) Errors are NOT mistakes; they reflect the fact the perfect measurement (the one that yields the true value) cannot be achieved’ In other words: uncertainty can be minimized, but it cannot be eliminated There are two types of errors: SYSTEMATIC: deviations of the measurements from the true value due to some specific disturbance. They only happen in one direction (i.e. perturb the measurement either in excess or in defect). Often hard to spot and eliminate. RANDOM: the combination of a (typically large) number of independent causes that affect the measurements. They can perturb it in a unpredictable way, either in excess of in defect, with no way of knowing. They are the ultimate cause that prevent to achieve the perfect knowledge of a phenomenon. Every physical theory is true to the extent that it explains the observations within the uncertainty of the measurement errors A theory that has been working just fine with relatively large uncertainty, might need to be tossed away and replaced with something else if the accuracy of the measurements improves. A “wrong” theory can also still be used as long as the measures are not accurate enough to reveal discrepancies. For example, Newtonian Gravity and General Relativity There is no need to use the complications of G.R. to describe the orbit of the Earth around the Sun But we need to use G.R. to describe the orbit of stars around a Super-Massive Black Hole