Have you got your workbook with you Have you got your workbook with you? Learning Astronomy by Doing astronomy Activity 1 – Mathematical and Scientific Methods step 7 – Combine Scaling, Small-Angle Formula, Scientific Notation, and Measurement Uncertainties Let’s get started!

If galaxies look to be the same type, then these galaxies may be assumed to be similar in actual size. Therefore, if one of the galaxies has a smaller angular size than the other, then the apparently smaller galaxy is farther away from us. Here are two spiral galaxies that look similar and thus could be similar in actual size. However, the galaxy on the right has a smaller angular size. If these images were taken on identical telescopes, identical set ups, and identical cameras and detectors, then the galaxy on the right must be farther away from us.

Here are the 3 galaxies used in Activity 1, Step 7 Here are the 3 galaxies used in Activity 1, Step 7. We have added a “ghost” ruler over each galaxy and have indicated what we think are the extents of each one. Where do you start?

Finding the scale factor for the galaxy images Each image in Figure 1.5 is 600 arcsec on a side. Using the ruler in Figure 1.6, measure the length of the white line in one of the images and calculate the scale for all of the images. Your measurements will be different than the ones we made here! We get the angular size of the image from knowing about our telescope and our detector. The number of arc seconds per pixel depends on the focal length of the telescope and the size of each pixel of the detector. Stay tuned for more!

5.1 cm=600 arc seconds 1 cm=118 arc seconds Round off your answers! You don’t need more than 1 decimal place here. One would be justified to say that 5 cm = 600 arc seconds to make the calculations even easier.

5.5 cm × 118 arcsec cm =649 arcsec It would have been completely alright if you noticed that this galaxy filled up most of the image, and so was approximately 600 arc seconds in angular size. For demonstration purposes, we have done a separate measurement and came up with 5.5 cm.

Converting arc seconds to radians Activity 1 uses a value of 600 arc seconds for the angular size of the example galaxy. We have changed the number to our measurement here of 649 arc seconds to give you another example of what to do.

27. The angular size of a galaxy is inversely proportional to its distance from us. How much farther away from us is the farthest galaxy compared to the nearest? _________ USE RATIOS! SO much easier than using actual distances, especially since we do not at this point in the activity KNOW any actual distances. Go to the next slide.

Galaxy B is about 1.3 times farther away than Galaxy A. Galaxy A (left) Angular size: 0.0031 radians Galaxy B (right) Angular size: 0.0024 radians Relative distances: 0.0031 0.0024 =1.29 Galaxy B is about 1.3 times farther away than Galaxy A. How can we get ACTUAL distances?

Using the small angle formula to get actual distances 𝑑= 𝑠 𝜃 Using the small angle formula to get actual distances 𝑠=100,000 light years is the angular size in radians for each galaxy 𝜃 For the galaxies in our example here, where the angular sizes are 0.0031 and 0.0024 radians, we find that Galaxy A is 3.23× 10 8 ly away and Galaxy B is 4.17× 10 8 ly away. 100,000 light years s 𝜃 Distance to Galaxy A: 𝑑= 100000 0.0031 =323,000,000 𝑙𝑦 Distance to Galaxy B: 𝑑= 100000 0.0024 =417,000,000 𝑙𝑦

Your turn: 28. Distance to galaxy A____________ light years; B____________ light years; and C___________ light years. 29. Which galaxy has the most uncertainty in your measurement of its angular diameter? ________ Which one has the least? _____ Explain your answer, using the appropriate techniques we have covered in this activity. You should now have enough foundation to finish all of Activity 1 as well as being prepared for the calculations in the rest of the activities for this quarter. Our use of mathematics – the language of the Universe – has revealed a universe that is unimaginably large and complex, and ultimately the most fascinating thing we might ever study!