## Parallax Measuring Device

Lab Instructor: Lab Section:

Parallax Measurement

Student Worksheet

1. [40 pts] Taking Your Parallax Measurements Refer to Figure 3 and Section 3 of the lab

PDF For instructions on how to take your parallax measurements using the UTK Parallax

Measuring Device (PMD). On the PMD, the two posts are separated by 20 inches, which

serves as the baseline for the parallax measurement. Use the left–post to align a distant

object and a background reference. Then using the right–post measure the angle of the

apparent displacement due to parallax.

As you move your object to the three distance positions, directly measure the distances

using the provided measuring tape. When you move the object to a new position, extend

the tape measure from the point directly below your PMD under the table to the point

directly below the distant object. You are aiming for roughly Position 1 = 10 ft, Position 2

= 20 ft, and Position 3 = 30 ft. Record your direct measurements in Table 1.

Table 1. Direct Measurements of Distance

Measure distances in inches, to the nearest half–inch.

Position 1 (in) Position 2 (in) Position 3 (in)

As you measure the angles you will use to determine the observed parallax, record in

Table 2 found on the next page.

Table 2. Your Parallax Measurements & Calculated Distances

Record angles to the nearest tenth–degree,

distances to the nearest half–inch.

Angle 1 ( ◦ ) Angle 2

( ◦ ) 2???? ( ◦ ) ???? ( ◦ ) D (in)

Position 1 (Approximately 10 ft)

Trial 1

Trial 2

Trial 3

Position 2 (Approximately 20 ft)

Trial 1

Trial 2

Trial 3

Position 3 (Approximately 30 ft)

Trial 1

Trial 2

Trial 3

2. [4 pts] In our experiment, we used a baseline of 20 inches. What would happen if the

vantage points were farther apart? For example, how would our measured parallaxes

change if we used a baseline of 100 inches instead?

(Note that there is no wrong answer to this question. The point is to take a guess, and then

to verify or to disprove it.)

3. [5 pts] Repeat the experiment with the object at Position 3, but this time use a baseline

distance of 40 inches by shifting your PMD 20 inches to the right after selecting your

landmark on the left upright post. Record the data for the increased baseline below

Angle 1 ( ◦ ) Angle 2 ( ◦ ) 2???? (◦ ) ???? (◦ )

Baseline of 40 in. at

Position 3

4. [3 pts] For an object at a fixed distance, how does observed parallax change as the

baseline increases?

5. [3 pts] Calculate the distance of the object at Position 3 using the equation below.

???? = 20 in

tan (????)

How does the distance compare to the distance you calculated for your three trials at

position 3?

6. [2 pts] Which remote distance calculations do you trust more – the ones made with the

smaller or larger baseline distance? Why?

7. [10 pts total] In this question, you are going to analyze your distance and measured

parallax data to confirm the relationship between distance and parallax.

a. [1 pts] Calculate the average parallax your group measured for each of the three

object positions (~10 ft, 20 ft, and 30 ft)

Position 1 Average:

Position 2 Average:

Position 3 Average:

b. [4 pts] In a Google Spreadsheet, make a plot of Parallax vs. Distance using your

direct measurements of distance (Table 1) for the x–axis (the independent variable)

and your average measured parallax for the three positions on the y–axis.

Share this spreadsheet with your instructor.

c. [1 pt] Fit a power–series to your data and display the equation. Since you put your

average parallax values on the u–axis, use p for your variable name instead of y.

Since you put the directly measured distances on the x–axis, use D instead of x.

Write your equation here

d. [4 pts] In your own works, describe the relationship between distance and parallax.

That is, how does the parallax change as the distance changes.

8. [3 pts] Estimate the uncertainty in your measurement of the object’s apparent shift (your

measured values for 2????). For example, do you think your recorded measurements could be

off by ten degrees? One degree? One tenth of a degree?

What is your reasoning for your uncertainty estimate?

9. [3 pts] Based on your estimate of the uncertainty in the angular measurements of 2????,

estimate the uncertainty in your measurements of the object distances. For example, do

you think the distances are different from the directly measured distances by 1 inch, 1 foot,

several feet, etc.?

What is your reasoning for your distance uncertainty measurement? Your answer should

reference your answer to Question 8.

(Note that there is no wrong answer to this question. The point is to take a guess, and then

to verify or to disprove it.)

10. [3 pts] Calculate the average distances determined using parallax using your three trials

for Position 1, Position 2, and Position 3

Position 1 Average:

Position 2 Average:

Position 3 Average:

11. [5 pts] Calculate the absolute error of your distances determined using parallax compared

to the directly measured distance. Use your average distances from your parallax

measurements for the “Parallax Distance.” The vertical bars in the equation indicate

absolute value. That means ignore the negative sign if you get a negative number.

???????????????????????????????? ???????????????????? (%) = 100 ∗ :???????????????????????????????? ???????????????????????????????? − ???????????????????????????????? ????????????????????????????????

???????????????????????????????? ???????????????????????????????? :

Absolute Error for Position 1:

Absolute Error for Position 2:

Absolute Error for Position 3:

12. [3 pts] For which measurement was your error the greatest? Do you think there is a reason

why this measurement had the greatest error?

13. [6 pts] Compare the distances that you calculated for each position using the parallax

method to the distances that you measured directly at the beginning of the experiment.

How well did the parallax technique work? Justify your answer using your absolute errors

calculated in Question 11.

14. [5 pts] Now that we have some understanding of the accuracy and precision of

determining distances using parallax, let’s see how this works for real parallax data for a

star.

Here you will use Equation 3, ???? = !

“ , where the distance D is in parsec and parallax p is in

arcsec.

The star Canopus has a measured parallax of ???? = 0.01055 ± 0.00056 arcsec. The

0.01055 arcsec is the best estimate for the parallax measurement. The ± 0.00056 arcsec

(plus and minus 0.00056 arcsec) is the 1–sigma (1 standard deviation) uncertainty in the

parallax measurement. Recall that in science, we can never measure something exactly.

We always list our best estimate and the wiggle room.

a. [2 pts] What is our best estimate for the distance, in parsec, to Canopus?

b. [1 pt] Using the minus part of the uncertainty (i.e., 0.01055 – 0.00056 arcsec),

what is the distance to Canopus?

c. [1 pt] Using the plus part of the uncertainty (i.e., 0.01055 + 0.00056 arcsec),

what is the distance to Canopus?

d. [ 1 pt] Estimate the uncertainty in the distance to Canopus.

15. [5 pts] Relate the parallax experiment you did in today’s lab to the way that parallax is

used by astronomers to measure the distances to nearby stars. Your answer should include

what your two vantage points represented and how you measured the parallax angle

corresponds to what astronomers observe to determine the parallax of a star.