Angular Size and Parallax in Astronomy
Overview:
We have studied the daily movements in the sky of the stars and the yearly movement of the Sun with a “Star and Planet Locator.” Using the Locator as a tool, we have also studied the stars and orbital positions of planets in the sky for a specific year, date, and time of day. In the preceding Modules, we have described how when we look at distant objects in the sky, we only see angular positions the celestial sphere. This module explores this topic in more depth, literally! It explores the dependence of angular size and parallax on distance from us.
Materials Needed:
You will need:
A sharp pin or very sharp mechanical pencil
Scissors,
Masking tape,
A ruler (with both inches and cm)
Card-stock paper (alternatively a manila folder and paste) and
A computer printer to make the devices described here and complete the module. To make the devices more durable, the pattern is best if printed on card-stock. Alternatively, the paper pattern could be pasted onto a manila folder page and then cut out and folded. Use a ruler and a blunt pencil to score along straight dashed lines in the pattern prior to folding. Then tape the corners together. Use a sharp pin or mechanical pencil to punch the small sighting hole in the protractor pattern. Finally, a standard piece of 18x24 inch poster board is needed.
Introduction
For objects within about 50 feet, we estimate distances with our binocular vision, in which the brain continually compares the two views of our eyes and estimates distances from the difference between the two. You can see the different views by holding your finger a short distance in front of your face and closing each eye in turn.
The finger appears to jump back and forth against the background of more distant objects.
Beyond our immediate neighborhood, the two views become nearly identical and our eyes only detect the angular size of objects. We cannot directly estimate distances or sizes of far-away objects. However, the eye and brain indirectly estimate such things unconsciously using the relation between angular size, physical size and distance. If an object is familiar to you, i.e. you know its actual size e.g. in cm, you can use this knowledge to estimate the object’s distance. Conversely, one can estimate physical size if the distance is known. Figure 1 shows the connection between the angular size, physical size and distance.
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Figure 1. Relation between angular size, physical size and distance.
Here is an example of the angle, size and distance relation. Assume you see a
UFO far up in the sky. Since you don’t know much about UFO’s, you can’t estimate its distance. Suppose a door opens on the UFO and you see a friend, you can estimate the distance of your friend and also the size of the UFO. Alternatively, if the distance is known along with the angular size, then the physical size can be estimated A protractor is used to measure angles in degrees. Imagine if you put a protractor with its center at your eye. To measure the angle between the two sides of an object, the zero degree point on the protractor would be placed in the direction of one side and the angle of the direction to the other side would be read off the protractor edge. With a regular protractor, this is not possible because the numbers and divisions are not only too close to seeing clearly, but they are also edge-on to the eye. A pinhole protractor like that shown in Figure 2 solves these problems in a convenient inexpensive way. A pattern for the protractor is shown in Figure 3. Print this page out on regular paper (or preferably card-stock) to cut out and construct your own pin-hole protractor.
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Figure 2. Sketch of the assembled pinhole protractor after the pattern is printed, cut out, then folded and taped. A small hole must be punched at the apex (near edge of table) for viewing the angular scale.
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Figure 3. The pattern to use in making a pin-hole protractor
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Outline
In this activity, we will make a pinhole protractor, which can measure angles with precision, to study the angular size/distance and parallax/distance mathematical formulae.
In Part I we will use the instrument to make measurements and calculate physical size from angular size and distance in an everyday situation. Then we will see how astronomers use similar measurements and calculations to study the universe. In Part II we will see how astronomers have used parallax, which works like binocular vision, to determine distances of objects in our Solar System and of neighboring stars in our
Galaxy.
Constructing and Using a Pinhole Protractor
Materials:
● thin cardboard, manila file folder or other stiff paper
● photocopies or print out of the pinhole protractor pattern from Figure 3 along with extra print outs of this exercise.
● transparent tape
● a calculator
● ruler, meter stick or yard stick to measure distances
● a roll of masking tape
● a sheet of poster board
Steps to Make the Pinhole Protractor:
● Glue or paste your copy of the pinhole protractor pattern onto the manila folder or thin cardboard sheet with the printed pattern outside.
● Cut along the solid outline with scissors.
● Place the cut-out pattern on a table with the printed pattern facing you.
● Fold upward toward you along the dotted lines and tape the solid line edges together to make a triangular box with one curved side.
● Carefully use a pin or a sharp pointed pencil to punch a small hole at the cross-marked location where the two straight sides of the box meet.
How to Use the Pinhole Protractor to Measure Angles:
Looking at your pinhole protractor and Figure 2, you can see that the protractor angle scale is placed on the curved strip turned at right angles to a circular arc with its center at a small hole where the eye is placed. Under normal daytime lighting outdoors or indoors, this pinhole enables one to get a reasonably clear view not only of the distant object, but also of the up-close angular scale. Try taking your eyeglasses off when using the pinhole protractor. The pinhole actually takes over the function of the lens of the eye!
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To make clear how the protractor works, note the ordinary protractor and its scale printed on the flat bottom of the instrument. By means of outwardly extended lines you can see that 0 o , 5 o , etc. on the flat protractor correspond to the values on the curved strip.
To measure angles with the protractor follow the steps:
● Put one eye very close to the pinhole on the outside of the protractor and lock through the pinhole
● Turn the protractor so that 0 o
on the scale is in the direction of one side of the object.
● Keeping the 0 o
in the direction of one side of the object, read off the coordinate of the other side on the scale. This number is the angle between the two in degrees. Similarly one can measure the angle between two small objects.
Part I. Angular Size, Physical Size and Distance
The angular – physical size – distance relation is diagramed in Figure 1. The angular size is given by the formula:
Angular size o
= 57.3
o
( physical size / distance)
The angular size is expressed here in degrees. The physical size and distance must be in the same units i.e. inches or cm. This formula is not good for large angular sizes.
However, for angles less than about 20 o
it will give good results.
For those interested here is how the formula arises, the straight line physical size is approximated by the arc of the partially visible large circle partially shown in Figure 1. This large circle’s radius is the
“formula distance” which is approximately equal to the actual distance to the middle of the object. The fraction the physical size is of this large circle is nearly equal to the fraction the angular size is of the total circle 360 o , This relation can be solved for the angular size giving the formula above.
You can try to answer these questions now.
1. According to the formula, Angular size o
= 57.3
o
( physical size / distance), how should angular size of the object change as the distance increases ? A drawing of angular size of a given object at various distances should help. a)increases b)decreases c)stays the same, 10 cm, at all three distances
2.At which of these three positions (20 cm, 40 cm, 60 cm) should the calculated angular sizes of a 20 cm object calculated from the formula,
Angular size o
= 57.3
o
( physical size / distance ) ,
best agree with accurately measured angular sizes. a) 60 cm b) 40 cm c) 20 cm d) Equally bad at all these.
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A Trial Verification of the Formula
We will test the above formula by measuring the angular size of an ordinary 10 cm object at various distances and then compare the result to the angular size computed from the distance and physical size of the object via the following steps:
See Figure 4 for the experiment set-up verifying the formula. A sheet of poster board should be fastened down on a table using masking tape with one edge along one edge of the table. Use three sheets of paper taped to the sheet of poster board as shown in
Figure 4. A printed template sheet matching the first of the three Figure 4 sheets is given at the end of this module. Go through the following procedure checking the first page template to make sure the dimensions are correct. Then prepare sheets two and three yourself. The edge of the poster board and table edge are on the left side of the Figure 4.
Mark the first with a mid-page observation point marked in yellow as shown in the figure. Extend a dashed from the observation point perpendicularly outward across the three sheets as shown in Figure 4.
Figure 4. Set-up for angular size, physical size, and distance. Vertical red bar represents the 10 cm target which will be placed symmetrically on the dashed line at 20 cm, 40 cm and 60 cm. This target must also be at right angles to the bar. Upper sketch shows the angle subtended by the object at 40 cm by an observer at the left end of the dashed line.
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As shown in the Figure 4 illustration , use your ruler to measure out from the observation point 20 cm marking this first object point with an “x.” Again from the observation point measure out a 40 cm distance to the second object point and mark it with an “x.” Finally, from the observation point measure out a 60 cm distance to the third object point and mark it with an “x.”
Having set up the pages with the mid-page observation point, dashed line and object locations “x’s” , you should now construct the object which will be placed at the three x positions on the dashed line. The template pattern is shown in the upper half of
Figure 5 below. At the end of this module, there is a full size to-scale template which you can print out to make the object. However, since some printers may shrink or expand the template, you should go through all the steps below with your ruler in hand to check that the dimensions are OK.
Draw on an 8.5x11 inch sheet of paper. Mark a light solid line as shown half way along and perpendicular to the page’s long dimension. Make two parallel dashed vertical lines as shown, 5 cm on each side of this to make a 10 cm wide target. Make a heavy black line 8 cm down along long dimension. Then a similar dashed line 4 cm down. Cut along the long heavy line. Fold along long dashed line. Then fold each of the dashed edges of the 10 cm target so as the make a triangle and tape the two back edges together. The bottom half of Figure 5 is a drawing of the resulting target and how it should be set on the line from the observer. The 10 cm face should be symmetrically placed on and 90 o to the dashed line with the folded and taped apex also on that line as shown in the bottom of
Figure 5..
● Write the 10 cm object width (the same for all observations) in the first row of Table 1 .
● As shown in Figure 4, going outward from the middle of the page, 90 o
to the paper edge at the edge of the table, mark the three 20 cm, 40 cm and 60 cm positions on the dashed line increasingly further away from the observer. Mark these positions with
“X’s” with numbers 1, 2 and 3 written on them going from small to large. Measure these distances (20 cm, 40 cm, 60 cm) and record them in Table 1. Use the same units for the distance (cm) as for object width.
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Fold on dashed lines
Cut along solid black line to make a strip. Fold on dashed.
Cut object template and “Paper Mate Sharpwriter” holder from part of an 8 ½ x 11 inch sheet of paper as shown here.
Figure 5. -Upper: Pattern design for object. Bottom: the object for angular size. A mechanical pencil is a good target for parallax.
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● Set the pin-hole protractor at the edge of the table where the dashed line starts at the first page edge. Measure the angular sizes of the object at each position (1, 2,3) using the pinhole protractor and record them in Table 1. Make sure the object face is at the “x” and is 90 degrees to the dashed line. Put put the protractor’s zero on one side of the object and read off the angular size with the other side. The angular size that you will obtain is in degrees.
● Calculate the formula value for the angular size and enter it in the last column of Table
1. Here is an example calculation for an object 12 cm across 70 cm from you. angular size

57.3
physical size distance

57.3
o

12 / 70
 
9.82
o
The % error =100%x│(observed-theory)/theory│. Supposed you measured angular size=9 o
.in our example. The % error = 100%(9.82-9)/9.82= 8.3%.
Compute the % errors for each and enter them in the last column.
Table 1 – Relation between Distance (cm, etc.), Angular Size and Physical Size
Physical size of object……………………………………………..
Position Measured Angular Size Measured Dist. Formula Ang. Size % error
#1 ……………………… ……………… ...………………. ...............
#2 ……………………… …………….... ...……………… ..............
..
#3 ……………………..... ……………….. ....…………… ................
You can try to answer this question now.
3.
How did your measured angular size of the object change as the distance increases in Module 04 Table 1? a)increases b)decreases c)stays the same, 10 cm, at all three distances
At which of the positions (20 m, 40cm, 60cm) do the calculated angular sizes of the object best agree with the actually measured angular sizes in Module 04 Table 1. This is in terms of percent error. Discuss this in your e-learning “Observing Notebook”
Assignment.
.
Estimating sizes of unknown objects if their distances are known .
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The angular size formula given earlier can be solved for physical size:
Physical size = distance x (angular size o /57.3
o )
The next part of this module will be to estimate a physical size from the known distance and measured angular size of the object: The result will be compared to the actual measured physical size.
● Place the object at 60 cm whose distance is known. Enter this distance in Table 2.
● We will use as physical size the 10 cm object on its template.
● Measure the angular size of the object. Enter this measured angular size and the 10 cm physical size in Table 2.
● Use this measured angular size and the known distance to estimate the physical size of the object. Enter your calculated value in Table 2.
●Here is an example for an angular size =6 o
and an object distance =115 cm.
Estimated physical size = distance x (ang. size o
/57.3
o
) = 115 cm x (6 deg./57.3 deg.) =
12 cm.
If the actual physical size is 13 cm, the % error =100%x│(observed-theory)/theory│=100%(13-12)/13=≈ 8%.
Table 2 – Estimating Physical Size from Angular Size and Distance
True physical size …………………………………………
Measured Angular Size Distance Estimated physical size % Error
…………………… …………………… ……………………. ...................
This use of the “known to estimate the unknown” is a basic process in
Astronomy. In the on-line astronomy lecture course, we discussed how the ancient Greek astronomer Aristarchus estimated that the Sun is at a distance of about 840 times the diameter of the Earth.. This is much smaller than modern better estimates! Using the diameter of the Earth as a unit of distance and taking the angular size of the Sun to be approximately 0.5
o
we can estimate the physical size of the Sun with the same formula we used for Table 2,
Physical size = distance x (angular size o
/57.3
o
) =
840 Earth diameters x ( 0.5 /57.3) ≈ 7 Earth diameters.
Although this is much smaller than modern estimates of the Sun’s size, it is many times bigger than the Earth’s which is qualitatively correct.
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Because the Sun is so many times bigger than the Earth, Aristarchus reached the modern conclusion that things made more sense if the Earth orbited the Sun, not the reverse.
Unfortunately, the world had to wait over a thousand years before Copernicus persuaded other astronomers to take a Sun-centered Solar System seriously.
You can try to answer these questions now.
4. The modern Sun’s distance is l2,000 times the diameter of the Earth.. The modern value is closer to 12000 x Earth’s diameter. Using the diameter of the Earth as a unit of distance and taking the angular size of the Sun to be approximately 0.5
o
estimate the physical size of the Sun with the formula
Physical size = distance x (angular size o
/57.3
o
) = ? Earth diameters. Pick the best answer below a) 1 Earth diameter b) 10 Earth diameters c) 100 Earth diameters d) 1000 Earth diameters e) 12,000 Earth diameters
Part II: Parallax Estimates of Distances
Introduction
We use binocular vision to estimate distances of nearby objects in every-day life.
Each of our two eyes presents a different view of the world. The brain continually compares these two views to estimate distances of nearby objects. Similarly, astronomers use parallax to measure distances of nearby astronomical objects. We have discussed how poorly the distance from the Sun was known. Rather than the views from our two eyes, to estimate the distance of objects in our Solar System, views from opposite sides of the Earth were used. To estimate the much larger distances outside our Solar System, astronomers are forced by the huge distances of stars to use the views from two opposite sides of the Earth’s orbit 186,000,000 miles or 300,000 km apart.
Geometry of Parallax Set-up
The pinhole protractor is used in a table-top parallax measurement as seen in
Figure 4. In this setup, a small pencil represents a “nearby star” or a Solar System object at distance d. There are two observation points at the table edge representing two points on opposite sides of the Earth or Earth’s orbit. A baseline is thus formed of the two
“orbital radii”, 2r, 90 o
to the nearby star’s direction. Figure 6 shows a diagram of the setup. A distant star serves as a “reference star”. This is marked on the right side of the page although it is actually a good deal farther away.
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P
Figure 6. The basic geometry of the parallax set-up
In Figure 6 the angle at the nearby star is the shift angle. Half this angle is called the parallax, P. Just like in the angular size formula: d

57.3
o r
P with the accuracy requirement that the parallax or half of the shift angle be less than 20 o
.
The above formula requires that the parallax be in degrees and d is in the same units as r.
To represent a Solar System object or a neighboring star, we use a a target aa pencil taped to the middle of the 10 cm face of our previously used object as show in
Figure 5.. Alternatively, any small objects that can stand on end and be clearly visible will work well.
The table with the baseline and nearby star should be near one wall of the room and the “distant star” any narrow object as far away as possible at the other end of the room. Also the size of the Earth’s orbital radius should be as small as possible compared to the distance of the reference star. A similar caution applies to the ratio of the distance of the nearby star to the reference star. This is the major drawback to doing the measurements indoors. We advise keeping these ratios at about 1/20 or smaller to keep the error below 20% of the actual distance of the nearby star. A baseline 2r of about 20 cm is good. The nearby star is placed at two different distances, 40 cm and 60 cm. For a setup like the above, d should be in error 10% to 20% depending on how carefully the measurements are taken.
We will use the same three pages attached to the poster board and table as was used for the angular size portion of this module. We repeat Figure 4 again below but this time we emphasize the two parallax observing positions on each side of the dashed line on the left edge. For stars outside the Solar System, the point where the dashed line reaches the paper edge represents the Sun.
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The line extending out from the Sun should have the previously marked nearer 40 cm distance on the second page. The third should have the next distant 60 cm distance marked. Figure 7 shows how the three pages are lined up end to end on the table. The sheets are attached with masking tape on the sheet of poster board on the table. The board is also fastened down using masking tape with the baseline along one edge of the table. You should now add a narrow “distant star” object as far as possible along the extension of the dashed line to the right .
Figure 7. The three sheets laid out on the poster board. Note two Earth positions from which parallax observations are made
Parallax Data and Calculation
Figure 8 shows a sketch of the tabletop set-up with a student observing the star target pencil from one end of the baseline.
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Figure 8. Sketch of student using protractor from one end of baseline in the parallax setup
Depending on the exact position of the distant star, the observer may see two different possible situations for the angles between the distant star and nearby star. Figure 9a shows one possible view of the distant star and two positions of the nearby star and the distant star are shown. The parallax angle is half the difference of these two. Figure 9b shows the other possible view of the nearby star and the distant star. In this case, the parallax angle is half the sum of the angles between the nearby star and the distant star.
So follow this procedure in making observations from the two ends of the 10 inches = 2r baseline.
● In Table 3 record the baseline length along with units and compute r, the distance from one end of the baseline (Earth) to the center (Sun). Also record the measured distances to the nearby and more distant objects.
● Place the protractor at the edge of the table with the viewing hole precisely at one end of the baseline as shown in Figure 8. As shown in Figure 9, examine the relative locations of the distant and near stars and record the angle “A” between them. Make the same observations from the other end of the baseline recording the angle “B” between them.
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Figure 9. Two possible views of the nearby star (symbolized by the larger star) against the background, much more distant star (symbolized by the smaller star). In “a” , the nearby star “jumps” from one side to the other of the distant star. The total shift angle
(twice the parallax) is equal to angle A + angle B. In “b” the nearby star “jumps” from a smaller to a larger value on the right of the distant star. The total shift angle is angle A minus the smaller angle B. It is not shown but a similar one sided jump can occur to the left of the distant star.
● Determine which of the two situations of Figure 9 apply and calculate the parallax accordingly, entering it in table 3:
Figure 9a. Parallax equals half the positive difference between angles A and B.
Figure 9b. Parallax equals half the sum of the angular separations A and B.
● Calculate the two distances from the parallax angle and the Earth-Sun distance r in the above formula.
● Compute the error in the distance estimate by subtracting the measured distance from the calculated distance for both smaller and larger nearby star distances. Divide by the measured distance to get percentage errors and record in Table 3.
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Table 3 – Estimating Distances by Parallax
Baselin= 2re =
…………………………; r = ………………………………
Measured Smaller Distance #1 = …………………………………
Object Angle A Angle B Parallax Angle Distance % Error
#1 ……….. .………. ………………. ..………. ……….
Measured Larger Distance #2 = …………………………………
Object Angle A Angle B Parallax Angle Distance % Error
#2 ……….. .………. ………………. ..………. ……….
You can try to answer these questions.
Look at the values you obtained in Table 3 to answer the next two questions.
5.In the Module 4 Table 3 data, when you measured the parallax for a target at a amaller then a larger distance, how did the parallax angle change for the more distant compared to the closer object? a) more distant is smaller b) more distant is larger c) is the same for more distant as for the closer
6.In the Module 04 Table 3 “Estimating Distances by Parallax”, which of your two estimated distances should have a larger percent error, the closer or the more distant object? d) the more distant e) the closer of the two f) Both have the same error.
To infinity and beyond
Although the title of this section is, strictly speaking, impossible, the actual scale of distances in the universe is practically infinite compared to our familiar Earthly scales.
The following problem, which you should work and answer in the open book e-learning
Module 04 quiz will show this. The size of this result really makes humanity and our
Solar System seem insignificant.
The parallaxes of even the closest stars are much smaller than the values we measured in
Module 04. The closest known star outside the solar system, Proxima Centauri, has a parallax of only 0.7725 arcseconds or “. One arc second corresponds to 1/60 of an arc minute. One arc minute corresponds 1/60 of a degree.
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Here is how to convert one arc second to degrees. One arc second is 1/60=0.01666.. arc minute. Converting arc minutes to degrees, 0.016666... /60 = 2.777...x 10 -4 degrees in one arc second.
Using the conversion factor, 0.7725 arc sec x 2.777...x 10 -4 degrees in one arc second =
2.146 x 10
-4
degrees.
7.
For Proxima Centauri’s parallax of only 0.7725 “. corresponding to only 2.146 x
10
-4
degrees. Recall
d= 57.3
o
(r / p o
)
and that the Earth to Sun distance is 1 Astronomical Unit. Calculate the distance in AU of Proxima Centauri. Pick the closest answer below. The distance from the
Sun to the nearest star is about ____ Earth orbital radii a) 250,000 AU b) 25,000 AU c) 2,50 AU d) 250 AU e) 25 AU
8.
Calculate the distance in AU of an imaginary star with a parallax of 1 “ or 1/3600 degree. This is one parsec distance in AU. Recall
d= 57.3
o
(r / p o
) a)206,000 AU b)20,600 AU c)2,060 AU d)206 AU e)20.6 AU
Astronomers use a unit of distance, the parsec, pc, which is the distance of the above imaginary star with parallax p = 1 “ or 1/3600 o
. For this unit of distance, the distance in parsecs and the parallax in “ are related by the simple formula,
D pc
= 1/p”
9.
The nearest star with a parallax 0.7725” is at a distance of a) 0.13 pc b) 1.3 pc c) 2.6 pc d) 3.26 pc e) 206,000 pc
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Page 1 pattern for angular size and parallax . Check that black dots are 10 cm from dashed line. Redraw if they are incorrect.
Yellow dot is observation point for angular size (Part I). Black dots are the two earth points for the parallax observations (Part II).
Continue the dashed line through pages 2 and 3 with your ruler.
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