Download Measure Parallax Lab - the Home Page for Voyager2.DVC.edu.

Measuring Distance Using Parallax
Parallax is the apparent motion of a stationary object
due to a change in the observer’s position. In astronomy
parallax is the gold standard (best) way to measure
distances. That is, until it doesn’t work any more. The
accuracy of parallax measurements depends very
strongly on the distance itself, as you will find out from
your work. In this exercise you will use parallax to
determine distances to three different objects chosen to
be at VERY different distances. You will relate the
accuracy of your measurements to the accuracy of the
distance measurement.
Angle B
Angle C
Angle C
Baseline,
Historical Background: The Greeks knew that if a
Distance between points
distant object is viewed from two different points, the
where observations are made
direction to it will change, as is shown in the figure.
They (and you) could figure out all the sides of a triangle
whose corners are at the observing positions and the nose of the animal given the length of the baseline
and two angles. This procedure is used for finding distances and for surveying and measuring land.
The Greeks tried to measure the parallax of the Sun, the stars, the planets etc. They did not get accurate
parallax measurements for any of these objects. Since the stars did not seem to move AT ALL over the
year they thought that the Earth stands still in the middle of a sphere of stars.
Why couldn’t the Greeks measure the parallax of the stars?
The problem is that as something gets further away, the apparent motion due to parallax gets smaller.
Eventually it becomes too small to measure. The distance which is becomes too far to measure depends
on the equipment and care used.
Materials:
Ruler or tape measure,
protractor, calculator, paper, tape or chalk to
mark the baseline.
Procedure: You will measure the distance
to three objects and then estimate the
accuracy of these measurements. The idea
for measuring the distances is to measure
one side and two angles of a triangle, then
you will use this information (and math) to
figure out the distance to an object. The
angles and distances to measure are shown
in the picture. You may use EITHER type of
protractor.
Protractor
Measure 3 times
to nearest 0.1 deg
Protractor, different type
Because you are exploring the effect of
distance on measurement accuracy, choose
3 objects at different distances, one at a
distance of about 5 times the baseline, one
at about 10 times, and one Much further..
Guess the distances to decide what to
measure.
Decide on baselines to use to measure your objects. The baseline must be AT LEAST 200 cm. You
MAY use the same baseline for all three. You must be able to see the object from both ends of the
baseline. Mark the ends of the baseline with tape or chalk and be certain to have something to show the
direction from one end to the other (a fence or railing is good).
Measuring Distance Using Parallax
1
Measure the length of the baseline to the nearest millimeter (0.1 centimeter). So the number of
centimeters in the length that you measure will not normally end with a zero. You must have three
different values for the length. (The length does not change, but if you measure with enough precision,
your measurement errors will cause the values to be different.)
Look at the object from each end of the baseline. Measure the angle between the baseline and the
direction to the object by putting the center of a protractor at the end of the baseline, lining up the
zero degree line with the baseline, Looking across the center of the protractor to the distant object
and marking the spot on the protractor where the line of sight crosses the protractor. Decide
whether the angle is acute(<90deg) or obtuse(>90deg) before reading the protractor so that the
many scales on a protractor do not confuse you.
SKETCH your set ups. Show your baseline, some landmarks in the background (like trees or buildings)
and identify the objects which you are measuring. Do NOT try to measure of f the angles and sides but
o
DO be sure to show which angles are acute (less than 90 ) and which are obtuse.
Record your measurements in the following table. Be sure that the baseline is measured in centimeters to
the nearest millimeter and that the angles are measured and computed to the nearest 1/10 degree.
Object # 1
Object # 2
Object #3
Name of Object
Baseline Length
Middle Value
Largest Value
Baseline Length
Smallest Value
Baseline Length
Angle B
Middle Value
Angle B
Largest Value
Angle B
Smallest Value
Angle C
Middle Value
Angle C
Largest Value
Angle C
Smallest Value
Angle A Middle Value
o
(180 -Middle B-Middle
C)
Largest A Value
o
(180 -Smallest BSmallest C)
Smallest A Value
o
(180 -Largest BLargest C)
Getting the distance when you know two angles and a side
Your measurements for each object amount to two angles and the included side (the baseline) for a
triangle. Your observation points are at two of the vertices and the object which you are trying to measure
is the third. This is enough information that, there is only one triangle which corresponds to the
measurements. When the triangle is completed, the position of the distant object will be determined. But
drawing is awkward and introduces its own errors.
Measuring Distance Using Parallax
2
You could draw the triangle to scale and measure the distance on the scale picture. This is comparatively
easy to think about, but the process of drawing introduces additional errors and the triangles are long and
skinny.
The “distance” to the object depends on whether the distance is defined to be from one of the observation
points, from the baseline or from some other point. In astronomical measurements, the center of the earth
is often used for nearby objects and the center of the Earth’s orbit is used for distant objects. For your
experiment, you will find side b
and call that the distance. As
the distance gets large, it
Angles B and C are known
doesn’t matter much which side
you choose for the distance.
B
Baseline length, a is
Your picture may look like
Figure 3. Notice the labels on
the angles and the sides. All of
the angles are known. Two
have been measured at the
observers. The third is found
because:
c
known
Baseline
a
The distant
object is here
A
C
A+B+C= 180 degrees
b
Angle A
= 180 degrees - B - C
Figure 3
There is a theorem (called the
Law of Sines) which says that
a
b
c
=
=
sine A
sine B
sine C .
The lower case letters represent the lengths of the sides of a triangle; the uppercase letters represent the
angles. Side a is across from angle A. Figure 3 shows the situation. The notation “sine A” means to use a
function called “sine” (abbreviated sin) and to use the value for the angle A. The sine of an angle is the
ratio (answer) you get when the angle is part of a right triangle and you divide the length of the side
opposite the angle by the length of the hypotenuse, so the definition is
s in (a n g l e ) 
le n g t h o f s i d e o p p o s it e
le n g t h o f h y p o t e n u s e
Once you know that a triangle is a right triangle, and you know one of the other angles, you know the
shape of the triangle. This occurs because the third angle must make up the 180 degrees total for the
triangle. Since all the angles are known, the shape is fixed, and so are the relative sizes of the sides. The
trigonometric functions just tabulate the ratios of the sides of triangles. You could think of them as tables
which describe the shapes of right triangles.
There is a table of the values of sine at the back of this write up. Your scientific calculator almost certainly
has the sin function. Check out how it works and use the table to verify, Sine of 90 deg is 1. If you don’t
get that from your calculator, check to be certain that the calculator is in degrees mode.
To find the distance, we will use
a
b
=
sine A
sine B
Here a is the baseline, which you know. The angle B is one of the measured angles, which you know. The
angle A is known because A = 180 degrees - B - C, and you known B and C both. The values of sine A
and sine B are found by looking at the table in the back of the write up. Thus you will know the values of all
the quantities except b, the distance from one of the observers to the distant object. This is exactly what
was wanted. So now you can solve the equation and find the distance
Measuring Distance Using Parallax
3
b  sinB
a
sinA
distance sinB
baseline
sinA
Example Problem:
The measurements are as follows:
Baseline = 204.2 cm, 203.7 cm, 205.5 cm
Angle B = 87.2 degrees, 88.1 degrees, 86.9 degrees
Angle C = 85.4 degrees, 85.9 degrees, 85.1 degrees
Let’s just consider the middle values. First compute angle A as
180 degrees= A + B+ C
180 degrees= A + 87.2+85.4
A = 180 degrees – 87.2 degrees – 85.4 degrees
A= 7.4 degrees
So now you are prepared to find sine A and sine B. Your calculator is the easiest way (or you can
interpolate using the table at the back).
sine A = sine 7.4 degrees = 0.128796
sine B = sine 87.2 degrees = 0.9988.
The sines have no units. The lengths of the two sides had the same units as one another and the units
cancel. Using the formula
distance  sinB
and substituting the known values.
baseline
sinA
204.2cm
0.12876
dis tan ce  1583.565cm
dis tan ce  0.9988 
So the distance to the object is found. The numerical value 1583.565 cm is not really correct. We do not
know the contributing quantities well enough to say the distance to the nearest thousandth of a centimeter.
You only measured the baseline, 204.2 cm to four digits, and the variation among the measurements tells
us that the uncertainty is about 0.9cm (so more like 3 significant figures).
The B values in this example vary by around 0.6 degree(half the difference between largest and smallest).
This shouldn’t be expressed as a percentage,since the measurement error probably would have been
about the same number of degrees regardless of the value of b or C But the sines don’t change as much.
Sine (87.2 degrees) =0.998806
Sine (88.1 degrees) =0.99945
Sine (86.9 degrees) =0.998537
So here the variation is about ±0.0005 in the sine. This is like 3 significant figures. So we might expect that
3
the answer should have only 3 significant figures. We would be better off expressing it as 1.58x10 cm.
You will assess extreme values possible with your set of measurements. They may be further off than you
would expect from the number of significant figures.
Take another look at the formula for the distance
Measuring Distance Using Parallax
4
distance  sinB
baseline
sinA
The bigger the values of sine B and the baseline, the bigger the distance that will result. The smaller the
value of sine A, the bigger the distance (since we divide by sine A). So choosing the largest sine B and
baseline and the smallest sine A will result in the largest value of the distance.
Conversely, the smallest sine B, smallest baseline and the largest sine A result in the largest distance.
The table that follows helps you to organize your work to compute these extreme values of the distances.
Use your measurement data to compute the distance to each body. Record your work in the following
table. Be sure to write the units for each of your quantities,
Distance Formula- choose
max or min
Di
st
values to fit formula
Middle baseline*
Middle(sine B)
/Middle(sine A)
#
1
Max baseline*
Max(sine B)
/ Min(sine A)
1
Min baseline *
Min(sine B)
/ Max(sine A)
1
Middle baseline*
Middle(sine B)
Middle(sine A)
2
Max baseline*
Max(sine B)
/ Min(sine A)
2
Min baseline*
Min(sine B)
/Max(sine A)
2
Middle baseline*
Middle(sine B)
/Middle(sine A)
3
Max baseline*
Max(sine B)
/ Min(sine A)
3
Min baseline*
Min(sine B)
/Max(sine A)
3
Baseline
Length
Angle B
Angle A
Sine B
Sine A
Measuring Distance Using Parallax
Distance
5
1) Which of your distance measurements has the greatest percentage of uncertainty?
(hint,
per cent error  100 
(value  official value)
Treat the Middle value for each distance as the official.
official value
Negative values do matter, since that means a value is smaller than expected. But a negative value is considered
comparable to a positive one. E.g. A value with -10% error has larger error than one with +9%)
2)
If you tried to measure another object which is even more distant using the same equipment
which you were already using, would your answer become more or less accurate?
Distance
Best Estimate (the
middle value from the
previous table)
Plus (the difference
between the best
estimate and the
largest)
or Minus amount (the
difference between the
best estimate and the
smallest)
#1
#2
#3
3)
How accurately do you think that you could measure the angles in this experiment?
(Hint:: Look at the variation among the repeated measurements of the same angle B or C. The variation is due to your personal
measurement error. Calculate the difference between largest and smallest measurement of the same quantity. If you find half the
difference, that is a measurement of the uncertainty inthe measured value. You have 6 different B and C angles and all of them are
measured the same way. That is like having 6 different samples of your measurement process. Look at all these uncertainties and
choose value that encompasses the general run of values.)
How accurately could you determine the value for the angle A, the angle at the distant object (the one
which you did not measure, but which you computed from the other two)? (a number of degrees please!)
How accurately could you measure the length of the baseline? (a number of centimeters please!)
Which of these factors, the baseline, angle A, or angle B is the largest contributor to the uncertainty in the
distance which you found?
4) Suppose you want to measure the distance to a distant light pole. Would you get a more accurate value
using a 25 foot baseline or one with a 10 foot baseline? Assume that you can get the baseline laid out
with no trouble and that you are using the same equipment that you used for the rest of this experiment.
Extra Credit
The Greeks wanted to know the distance to the Moon and to the stars.
Suppose that they had stationed two persons on opposite sides of the Earth, and both of the persons
observed the Moon against the stars at the same time. Draw a sketch of how two people would set up
this measurement so that they would be as far apart as possible.
Look up how far away the moon is. How large would angle A be (a numerical value please)?
Do you think the Greeks could have observed this change? Why or why not?
What difficulties would there have been in measuring the distance to the Moon with this method?
Measuring Distance Using Parallax
6
Sine Table
o
What if the angle is MORE than 90 ? The value of the sine starts to decrease from its high of 1. The way
o
that it works is that sine x = sine (180 -x). For example
o
o
o
o
sine 95 =sine(180 -95 )=sin 85 .
Your scientific calculator will find sines and relieves the problem of what to do with the tenths of degree).
Check at least one value to be certain that the calculator is in the correct mode. Cheap calculators have
no problem. Some calculators have several modes. Be certain to use DEGREES mode.
Angle,
Degrees
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Sine A
Angle,
Sine A
Degrees
0
34 0.559192903
0.017452406
35 0.573576436
0.034899497
36 0.587785252
0.052335956
37 0.601815023
0.069756474
38 0.615661475
0.087155743
39 0.629320391
0.104528463
40
0.64278761
0.121869343
41 0.656059029
0.139173101
42 0.669130606
0.156434465
43
0.68199836
0.173648178
44
0.69465837
0.190808995
45 0.707106781
0.207911691
46
0.7193398
0.224951054
47 0.731353702
0.241921896
48 0.743144825
0.258819045
49
0.75470958
0.275637356
50 0.766044443
0.292371705
51 0.777145961
0.309016994
52 0.788010754
0.325568154
53
0.79863551
0.342020143
54 0.809016994
0.35836795
55 0.819152044
0.374606593
56 0.829037573
0.390731128
57 0.838670568
0.406736643
58 0.848048096
0.422618262
59 0.857167301
0.438371147
60 0.866025404
0.4539905
61 0.874619707
0.469471563
62 0.882947593
0.48480962
63 0.891006524
0.5
64 0.898794046
0.515038075
65 0.906307787
0.529919264
66 0.913545458
0.544639035
Measuring Distance Using Parallax
Angle,
Sine A
Degrees
66 0.913545458
67 0.920504853
68 0.927183855
69 0.933580426
70 0.939692621
71 0.945518576
72 0.951056516
73 0.956304756
74 0.961261696
75 0.965925826
76 0.970295726
77 0.974370065
78 0.978147601
79 0.981627183
80 0.984807753
81 0.987688341
82 0.990268069
83 0.992546152
84 0.994521895
85 0.996194698
86
0.99756405
87 0.998629535
88 0.999390827
89 0.999847695
89.1
0.999876632
89.2
0.999902524
89.3
0.99992537
89.4
0.999945169
89.5
0.999961923
89.6
0.999975631
89.7
0.999986292
89.8
0.999993908
89.9
0.999998477
90
1
7