Parallax & Stellar Distances NAME:
Distance Measurement Methods
A. Parallax from Images
In this exercise, you will determine the distances to stars that appear in a pair of images taken 3 months apart. Some stars are in different positions in the two images. You will also determine the minimum distance to those stars for which there is no measurable parallax.
Take a look at the two star field images. You can assume that the patch of sky in these images is near the plane of the ecliptic. (Note that these images are actually artificial constructions to make it possible to measure many stellar parallaxes in one image. In reality, stars that are close enough to us to have measurable parallaxes wouldn’t be so close together on the sky.) The images are “negatives,” so that stars appear black on a white background. The size of a star in the images is determined not by the star’s physical dimensions, but by its brightness. The brighter stars appear bigger than faint ones.
Step 1: Compare the two star images carefully, looking for stars that appear in different locations in the two images. You may find it helpful to lay image #1 on top of image #2, and hold them up to a light. Be careful to align the two images well (most stars will line up perfectly). You can also try using a ruler to check if separations or alignments between stars have changed.
Step 2: Mark each star that has moved noticeably with an arrow on image #1 • Draw the arrows in BELOW the stars, so they point “up” at the stars. Label each star with a number, letter or name.
Step 3: For each star that moves, estimate its center by eye as best you can on image #1，and make a small dot there with a pen or pencil. Then lay image #1 on top of image #2, and look for where each star moved to in image #2. Make a small dot on image #1 at the location where the star is centered in image #2. You should now have two dots on image #1 associated with each star that moves.
Step 4: Use a ruler to measure how far each star has shifted in the three-month period between image #1 and image #2. Make the measurements in millimeters (mm), and record each measurement in the second column of Table 1 on your worksheet.
Step 5: Convert your measurements in millimeters to an angular size using the following conversion:
1 mm = 0.1 arc-second
The resulting shift in units of arc-seconds is the known as the “parallax” of the star. Record the parallax values in the third column of Table 1.
Step 6: Compute the distances to each of the stars that moved noticeably. Use the parallax formula:
d = 1/p
Here d is the distance in parsecs (pc) and p is the parallax angle measured in arc- seconds. For example, a star with a parallax of 0.5 arcsec has a distance of 1/(0.5) = 2.0 pc. Record your results in the fourth column of Table 1.
Step 7: Compute the distances to each of the stars in light years (ly), and record your results in Table 1. Use the conversion:
1 pc = 3.26 ly
Step 8: Estimate the smallest apparent shift you hypothetically could have measured.
That is, how far in mm would a star need to have shifted for you to be able to notice the shift? What does this smallest measurable shift correspond to in arc-seconds? Then compute the distance to a star that has this value of the parallax, following steps 5 and 6 above. Record your answers on the worksheet, and answer the following questions.
Q1: Look at your results from Step 6 (distance column in Table 1) above. What, if anything, can be said about the distances of stars that didn’t seem to move from image #1 to image #2? [For example, “they must be closer than.”.” or “they must be farther away than….” Fill in the blank with a number.]
Q2: Suppose you observed the same field of stars 3 months after image #2 was taken. Describe in words how you would expect the positions of stars to appear. Include a sketch if that would help explain what you would see.
Q3: Suppose you observed the field once more 9 months after image #2 was taken. Describe in words how you would expect the apparent positions of stars to compare to those observed in image #1 and/or image #2.
Q4: Put aside the images you’ve been working on for a moment. Imagine that all the stars in the galaxy have the same intrinsic brightness, like light bulbs all of the same wattage. Stars would then appear brighter or fainter depending on whether they were relatively close or relatively far away from the Earth. ]f this were the case, which stars would appear brighter as seen from Earth: those with large parallaxes, or those with small parallaxes? Explain.
Q5: Now consider the results of your analysis of the two images. Are the results consistent with the hypothesis that all stars in the Galaxy have the same intrinsic brightness? Why or why not? If not, give at least one specific example from this exercise that proves your point.
TABLE 1: Stars with Measurable Parallaxes
Amount of shift (mm)
Smallest possible measurable shift: mm
Smallest possible measurable parallax: arcsec
Distance to star for which a parallax could just barely be measured: pc