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1.70 ±0.10

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Much more consistent and accurate results come from radio astronomy— using radio waves instead of visible light. Each October the Sun moves across the image of the quasar labeled 3C279 seen from Earth. Radio astronomers use this occultation to measure the change in angle of the signal as the source approaches Sun, crosses the edge of Sun, and moves behind Sun. This change in angle is measured with respect to the unde-flected signal from another quasar, labeled 3C273, that is about 10 degrees away from Sun as seen from Earth. The accuracy of angle measurement is increased by employing an experimental technique called very long baseline interferometry (VLBI) that uses widely separated dish antennas. A recent observation by D. E. Lebach and collaborators employed one dish— actually a pair of dishes—in Big Pine, California, and another pair of dishes in Westford, Massachusetts. Their observational results correspond to a gravitational deflection 0.9996 ± 0.0008 times that predicted by general relativity. This result straddles the value 1.00000, for which agreement between theory and experiment would be perfect. See the references at the end of this project.

7 Computing Einstein Rings

The first Einstein ring ever observed is shown in Figure 14 on page 5-25. We see an Einstein ring when light from a distant source is deflected toward us around all sides of an intermediate gravitating object (Figure 5). The Einstein ring appears only when the source, the intermediate object, and the observer lie along the same straight line.

The results of preceding sections allow us to predict the approximate angular diameter of the Einstein ring for a point source. The geometric construction is shown in Figure 6. In computing deflection by Sun we wanted to use Sun's radius R. Equation [7] allowed us to eliminate the impact parameter b. The light that enters our eye to form the Einstein ring of a distant star, however, may not be the light that grazes the surface of the intermediate dark object. But equation [7] tells us that for r » 2M we have simply R = b. Then equation [15] becomes

We use additional approximations in the analysis of Figure 6. A more or less complete list follows. These approximations are generally justified by the fact that the distances of source and observer from the intermediate object (defined in Figure 6) are literally astronomical compared with the distance from the intermediate object at which deflections take place.

Figure 5 Schematic diagram of the formation of an Einstein ring Not to scale—and the intermediate object need not be halfway between the distant star and Earth, as shown here

Intermediate

Distant source

Figure 5 Schematic diagram of the formation of an Einstein ring Not to scale—and the intermediate object need not be halfway between the distant star and Earth, as shown here

Light

Light

Earth

fsrc

Figure 6. Schematic diagram, not to scale, for the derivation of the angle of observation 8obs for the Einstein ring at the eye of the observer. This figure defines the quantities rsrc and robs.

fsrc

M rcbs Observer

Source

Figure 6. Schematic diagram, not to scale, for the derivation of the angle of observation 8obs for the Einstein ring at the eye of the observer. This figure defines the quantities rsrc and robs.

1. Assume that rsrc » 2M and robs » 2M and rsrc » b and rQbs »

2. Use Euclidean geometry.

3. Assume that deflection occurs at a single point near the intermediate object. This is reasonable, since the distance fromM to the source and the distance from M to the observer are both literally astronomical compared with the distance along the trajectory over which deflection takes place.

4. Measure the impact parameter b vertically, as shown in Figure 6, rather than perpendicular to the incoming ray of light as b is usually defined.

5. The sine or tangent of a small angle is approximately equal to the value of that angle in radians.

Apply approximations 1 and 5 to equation [38] on page 5-22, where the angle 6sheii is measured with respect to the radially outward direction. The angles called 0 in Figure 6 are defined with respect to the radially inward direction, but the two have the same sine. (Recall the Caution in Sample Problem 3 on page 5-20.) Call the result 0obs for "observation angle" and the radial distance from the intermediate object r0^s. Then from Figure 6,

Use the subscript "src" for "source" and write down the relation between angles shown at the upper right of Figure 6. The result is

QUERY 13 Einstein ring angle. Substitute equation [16] into [18] and use equation [17] to eliminate b. Show that the result is

AMr src robs(robs + rsrc)

where the subscript "Ein" refers to the Einstein ring.

8 Microlensing

The visible stars in rotating galaxies do not have enough total mass to hold these galaxies together; these galaxies should fly apart as they rotate. But these galaxies do not fly apart. This and other lines of evidence convince observers that galaxies contain much more mass than can be counted in the visible stars they contain. This largely unknown presumed presence earns the name dark matter. Individual dark matter objects have been whimsically named MAssive Compact Halo Objects, abbreviated MACHOs, which presumably occupy the halo surrounding galaxies.

By what means could we detect this dark matter in, say, our own galaxy? In 1986 Bohdan Paczynski proposed as a practical procedure a method that had long been known only as a theoretical possibility. The method depends on the deflection of starlight by a dark object. When a dark object lines up between Earth and a distant visible star, the dark object focuses light that would otherwise miss Earth (Figure 5). The result is an increase in the amount of light received from the distant star, even when the equipment is not able to resolve the structure of the Einstein ring. This increase in focused light is called microlensing. Microlensing can be used to study objects in our galaxy that do not emit sufficient light to be detected by other means. Paczynski's proposal started a whole new field of observation, and dozens of microlensing events have been detected.

The great difficulty with microlensing is that, at any given time, only one star in 2 million or so will have its image augmented due to microlensing. As a result, Paczynski predicted, microlensing would be a rare event requiring the monitoring of many stars simultaneously.

To have the best chance to observe microlensing, one wants a background rich in visible stars to increase the chance that one of them will lie behind some dark object in the foreground. There are several such star-rich backgrounds for Earth observations: the galactic bulge in the center of our galaxy and the so-called Magellanic Clouds, two satellite galaxies of our Milky Way Galaxy visible in the southern hemisphere. (The name Magellanic comes from observation of these clouds by Ferdinand Magellan's crew during the first circumnavigation of the globe, 1519-1522 A.D.)

When a microlensing event occurs, the amount of light received on Earth from the distant star grows and fades over a period of days or weeks and at peak may be many times the normal flux of light from that star.

Microlensing is so recent a technique that path breaking developments occur from month to month. Preliminary results appear to show that the dark matter observed by microlensing does not account for the missing mass of our galaxy—the mass in addition to that of the visible stars needed to keep the galaxy from flying apart. See the references at the end of this project.

QUERY 14 Identifying microlensing events. Discussion questions: How can we distinguish between the increased amount of light from a star due to microlensing and greater light flux due to some internal mechanism of the star? The following inquiries are meant to help answer this question.

Will the increase in light due to microlensing be the same for all colors of the spectrum of light from the star? Suppose a natural increase in light is due to greater burning rate, leading to a higher temperature of the star. Do you expect that this process will result in a changed spectrum of light from the star?

Is a microlensing event likely to occur more than once for the same star? Might an increase in the flux of light due to internal processes occur more than once in a given star?

Do you expect that the time profile of a microlensing event—the curve of light intensity vs. time—can be predicted by astronomers? Is it likely that this time profile will be the same as that for an increase in light due to internal processes?

9 The Einstein Donut

Images on the cover of this book illustrate some gravitational effects of a black hole on the visual appearance of background objects.

The undistorted lower image of Saturn was taken with an infrared camera on the Hubble Space Telescope. In addition to Saturn's familiar rings, you see a dot at the lower left of the image, which is Saturn's moon Dione. Another moon, Tethys, forms a bright spot at the upper right edge of Saturn's round disk. (We brightened the images of these two satellites to make them more obvious.)

Now place the center of a black hole on the direct line of sight between us and Saturn so that we, the observers, are at a radial distance (reduced circumference) r = rDbs = 10M on one side of the black hole. Saturn is at a much greater distance on the other side. We assume that Saturn, black hole, and the viewer are all relatively at rest, and that there is no change in the structure of Saturn (or us!) due to the curvature of spacetime induced by the black hole. The upper image on the cover results. What do we see?

First, Saturn itself is distorted into a donut (torus), to which we give the name Einstein donut. The donut shape results when the source is an apparent disk instead of a point. In what follows we will use a generalization of the Einstein ring equation [19] to explore major features of the donut. But we cannot legitimately calculate numerical values from the result, because its derivation depends on the assumption that rQ^s » 2M, whereas we placed the observer at robs = 1OM to create the upper cover image. This upper image is more dramatic than the one seen by an observer who is at a much greater distance.

Second, we see duplicate distorted images of Saturn's rings. One image lies along the top of the outer surface of the donut, while the other image, reversed right-for-left and top-for-bottom, hugs the inside bottom of the donut.

Third, gravitational deflection smears the image of Saturn's moon Dione and creates a second, faint image just inside the Einstein donut on the upper right.

Fourth, a circular "diamond necklace" of dots shines faintly inside the Einstein donut.

QUERY 15 Donut size. The upper and lower images on the cover are displayed to the same scale. Explain in one sentence why the average diameter of the donut in the upper image is larger than the diameter of Saturn shown in the lower image.

Figure 7 is a generalization of Figure 6 for a source point off the axis between the observer and the center of the black hole. Here 0O is the angle at which the source point would be viewed if the intermediate deflecting mass were removed.

Using Figure 7 and its caption we can repeat the derivation that led to equation [19], but for this new case. This derivation depends on all five assumptions (approximations) given on page D-10. The present derivation is a bit complicated and therefore optional. The result is:

4Mrvc 0Ein

°obs obs obs + src > °obs where, recall, the Einstein ring angle 0Ein (given by equation [19]) is the angle of observation in the special case that the source, deflecting mass, and observer all lie along the same straight line.

Source

Figure 7 Construction figure (not to scale) for the derivation of the observation angle 80bs (equation [20]) for a source off the axis that runs between the observer and the deflecting mass. The angle 0O is the observation angle in the absence of the deflecting mass M The length q and construction angle y are used in the derivation of equation [20] Some labels have been omitted for clarity. For example, the distance from the observer to point A is (to a good approximation) the observer radius robs. The small unmarked angle below the symbol vy on the upper right has the value 80b$ - 80

Figure 7 Construction figure (not to scale) for the derivation of the observation angle 80bs (equation [20]) for a source off the axis that runs between the observer and the deflecting mass. The angle 0O is the observation angle in the absence of the deflecting mass M The length q and construction angle y are used in the derivation of equation [20] Some labels have been omitted for clarity. For example, the distance from the observer to point A is (to a good approximation) the observer radius robs. The small unmarked angle below the symbol vy on the upper right has the value 80b$ - 80

QUERY 16 Observation angle for off-axis source. Equation [20] is a quadratic equation. Show that the solution is

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