Transcript lecture1_1 - UMass Astronomy

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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:
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Space
Time
Mass
Electric Charge
More on Measurements
• Which statement is correct?
• A: A person whose weight is 170.000 lb is heavier than
a person whose weight is 120.000 lb.
• B: A person whose weight is 120.001 lb. is heavier than
a person whose weight is 120.000 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/sec2)
• 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…
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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/sec2,
… so after 11 sec we expect the velocity to be
• Vexpected = (9.806 meter/sec2) x (11 sec) = 107.866 meter/sec
• But we measure v=115 meter/sec, not 107.866 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 subdivision 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 = 7.725 ±
0.025 cm
• 0.025 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 ± εstop) – (t_start ± εstart)
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 = 0.00000007, 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 = 0.036 (3.6%)
The uncertainty of velocity measures was
Δv = 10 meter/sec, so:
v = 115 ± 10 meter/sec
ε = 0.087 (8.7%)
Thus, vexpected = 107.866 meter/sec is well
within the error bar. There is no
observable discrepancy with the theory.
Gravity is true, after all!
In Summary
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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”)
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There are two types of errors:
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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
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
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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