## Info

What is the value of 0obs when 0O = 0? What is the physical meaning of 80 = 0? What is the physical meaning of the plus or minus in front of your result for 0O = 0?

QUERY 17 Off-axis source: First example. Consider the special case

80 = 80l = 51/29Ein. Using equation [21], find the two resulting values of ®obsi • Why is one value negative? What does this negative value mean physically?

QUERY 18 Off-axis source: Second example. Find the two values of 0Obs2 f°r a sec~ ond undetected angle 0O2 = 121/20Ejn. Compare your results to those obtained in Query 17. Is 0obS2 greater or less than 0obsi? (Careful!)

We need to keep reminding ourselves that equation [21] does not predict numerically correct details of the donut structure in the upper cover image, because robs = 10M for that figure, violating our assumption that robs » that led to [21]. Nevertheless, qualitative features of the upper image make sense when analyzed using the results of Queries 16 through 18. (The "diamond necklace" circling the center of the upper image results from a far greater deflection of light near the black hole and cannot be described at all by the approximate equation [21]. Section 10 discusses the diamond necklace.)

QUERY 19 Features of Einstein donut on the cover. Refer to points labeled A, B, C in Figure 8, which shows a top view of Saturn, its ring, and a distant observer. For each of the points A, B, and C the angle 90bs will have double values. When examining these values, distinguish between angles at which the observer looks to the LEFT of center, as indicated in Figure 8, and angles at which the observer looks to the RIGHT. Use your results for Queries 16-18 to answer the following qualitative questions:

1. For each of the points A, B, C as seen on the LEFT, is the magnitude of ®obs greater than or less than the magnitude of 0O, the angle of observation in the absence of the deflecting mass M?

2. For each of the points A, B, C as seen on the RIGHT, is the magnitude of 90bs greater than or less than the magnitude of 0O?

3. Reading from left to right, what is the order of points A, B, C as seen on the LEFT side of the center of Saturn?

4. Reading from left to right, what is the order of points A, B, C as seen on the RIGHT side of the center of Saturn?

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