Parallax - Measuring Distance Indirectly

Parallax

Measuring Distance Indirectly

Parallax is the "apparent change in position of distant objects, due to the actual change in position of the observer." The amount of apparent shift in position is called the parallactic shift, q _p, or simply parallax. The angle q _p (in degrees) is measured with respect to a fixed point in space, S, by measuring the directions q ₁and q ₂and forming the difference, see Figure 1.

Figure 1. Determination of the parallactic shift, q _p.

Consider the following situation. If the distance to an object is very much greater than the base line distance AB, it can be shown that the distances AO and BO are approximately the same, see Figure 2. Show that the distance, r (AO or BO) is as follows: (Hint: think of AB as an arc length of a circle of radius R.)

Equation 1

Figure 2. Solution of the "skinny" triangle

The assumption that r>> AB implies that the parallactic shift is also very small. For what value of q _p is r one order of magnitude greater than AB? In applications (notably astronomy) where parallax is used to measure the distance to a distant object, the parallactic shift is usually less than 1°. If q p is measured in seconds of arc, show that r is as follows: (Hint: how many second of arc are there in a degree?)

Equation 2

PROBLEM: The parallactic shift of a neighboring star is approximately 18˛ when the observation base line AB is across the diameter of the Earth. Determine the distance to the star in meters and then...name that star.

MEASURING THE DISTANCE TO THE GYM ROOF
FROM A CROSS CORRIDOR AT WHS

OBJECT: Measure the distance from a window on the third floor of WHS to a white expansion joint on the facia of the gymnasium roof, see Figure 3.

Figure 3. A view toward WHS Cafeteria and Gymnasium

PROCEDURE: There are many ways of determining the parallax. In this experiment the parallax will be determined by very carefully measuring the base line distance AB, the distance between sight points CD, and the distance L between the lines AB and CD, see Figure 4. The sight points, A and B, may be determined by suspending a plum bob in such a way that the plum bob string, window frame and object O, are along the same line of sight.

Figure 4. Measure the Distance from A to O at WHS.

STEP BY STEP:

Step 1. Measure and record the distance between the window frame, CD.

Step 2. Locate the line of sight point A.

Step 3. Locate the line of sight point B.

Step 4. Measure and record the distance between the sight points, AB.

Step 5. Measure and record the ^ distance between line AB and CD.

Step 6. Calculate the angle of parallax, q _p, and determine the distance, r, to the white expansion joint on gym roof facia.

DISCUSSION and CONCLUSION: Answer the following questions in your discussion of the results and conclusions.

1. What are the uncertainties in your data and how do they affect the uncertainty in your results? Based on the answer to the previous question what is the uncertainty in your result?

2. Referring to Figure 4 and using your result for the distance to the expansion joint, what is the ratio of AB to r and how does this value compare to the tan q _p? The significance of this comparison is that the more closely they agree, the better the approximation in equation 1 (or 2). What is the maximum value for q _p, such that the error between (tan q _p) and q _p is Ł 1%? (Note: make sure you express the angle in radian measure and not degrees!) What is your error?

3. Determine the height of the flag pole in front of Waltham High School using indirect methods. What is the uncertainty in your measurement? Discuss what can be done to reduce our uncertainty. (For an additional 8 STARS, Make a more precise measurement of the flag pole with the uncertainty reduced by one order of magnitude or more)

Extra Credit Points = 12 STARS