Dark Matter and a Distant Supernova Make an Eerie Cross

Dark Matter and a Distant Supernova Make an Eerie Cross A long time ago, in a galaxy far, far away...a massive star exploded. That cataclysm created an object called a supernova  (similar to the one we call the Crab Nebula). At the time this ancient star died, own galaxy, the Milky Way, was just starting to form. The Sun didnt even exist yet. Nor did the planets. The birth of our solar system still more than five billion years in the future. Light Echoes and Gravitational Influences The light from that long-ago explosion sped across space, carrying information about the star and its catastrophic death. ​Now, about 9 billion years later, astronomers have a remarkable view of the event. It shows up in four images of the supernova created by a gravitational lens created by a galaxy cluster. The cluster itself consists of a giant foreground elliptical galaxy collected together with other galaxies. All of them are embedded in a clump of  Ã¢â‚¬â€¹dark matter. The combined gravitational pull of the galaxies plus the gravity of dark matter distorts light from more distant objects as it passes through. It actually shifts the direction of the lights travel slightly, and smears the image we get of those distant objects. In this case, the light from the supernova traveled by four different paths through the cluster. The resulting images we see here from Earth form a cross-shaped pattern called an Einstein Cross (named after physicist Albert Einstein). The scene was imaged by the Hubble Space Telescope. The light of each image arrived at the telescope at   a slightly different time - within days or weeks of each other. This is a clear indication that each image is the result of a different path the light took through the galaxy cluster and its dark matter shell. Astronomers study that light to learn more about the action of the distant supernova and the characteristics of the galaxy in which it existed.   How Does this Work? The light streaming from the supernova and the paths it takes are analogous to several trains that leave a station at the same time, all traveling at the same speed and bound for the same final destination. However, imagine each train goes on a different route, and the distance for each one is not the same. Some trains travel over hills. Others go through valleys, and still others make their way around mountains. Because the trains travel over different track lengths across different terrain, they do not arrive at their destination at the same time. Similarly, the supernova images do not appear at the same time because some of the light is delayed by traveling around bends created by the gravity of dense dark matter in the intervening galaxy cluster. The time delays between the arrival of each images light tell astronomers something about the arrangement of the dark matter around the galaxies in the cluster. So, in a sense, the light from the supernova is acting like a candle in the dark. It helps astronomers map the amount and distribution of dark matter in the galaxy cluster. The cluster itself lies some 5 billion light-years from us, and the supernova is another 4 billion light-years beyond that. By studying the delays between the times that the different images reach Earth, astronomers can glean clues about the type of warped-space terrain the supernova’s light had to travel through. Is it clumpy? How clumpy?   How much is there?   Answers to these questions arent quite ready yet. In particular, the appearance of the supernova images could change over the next few years. Thats because light from the supernova continues to stream through the cluster and encounter other parts of the dark matter cloud surrounding the galaxies.    In addition to the Hubble Space Telescopes observations of this unique lensed supernova, astronomers also used the W.M. Keck telescope in Hawaii to do further observations and measurements of the supernova host galaxy distance. That information will give further clues into conditions in the galaxy as it existed in the early universe.

The Associative and Commutative Properties

The Associative and Commutative Properties There are several mathematical properties that are used in statistics and probability; two of these, the commutative and associative properties, are generally associated with the basic arithmetic of integers, rationals, and real numbers, though they also show up in more advanced mathematics. These properties- the commutative and the associative- are very similar and can be easily mixed up. For that reason, it is important to understand the difference between the two. The commutative property concerns the order of certain mathematical operations. For a binary operation- one that involves only two elements- this can be shown by the equation a b b a. The operation is commutative because the order of the elements does not affect the result of the operation. The associative property, on the other hand, concerns the grouping of elements in an operation. This can be shown by the equation (a b) c a (b c). The grouping of the elements, as indicated by the parentheses, does not affect the result of the equation. Note that when the commutative property is used, elements in an equation are rearranged. When the associative property is used, elements are merely regrouped. Commutative Property Simply put, the commutative property states that the factors in an equation can be rearranged freely without affecting the outcome of the equation. The commutative property, therefore, concerns itself with the ordering of operations, including the addition and multiplication of real numbers, integers, and rational numbers. For example, the numbers 2, 3, and 5 can be added together in any order without affecting the final result: 2 3 5 10 3 2 5 10 5 3 2 10 The numbers can likewise be multiplied in any order without affecting the final result: 2 x 3 x 5 30 3 x 2 x 5 30 5 x 3 x 2 30 Subtraction and division, however, are not operations that can be commutative because the order of operations is important. The three numbers above cannot, for example, be subtracted in any order without affecting the final value: 2 - 3 - 5 -6 3 - 5 - 2 -4 5 - 3 - 2 0 As a result, the commutative property can be expressed through the equations a b b a and a x b b x a. No matter the order of the values in these equations, the results will always be the same. Associative Property The associative property states that the grouping of factors in an operation can be changed without affecting the outcome of the equation. This can be expressed through the equation a (b c) (a b) c. No matter which pair of values in the equation is added first, the result will be the same. For example, take the equation 2 3 5. No matter how the values are grouped, the result of the equation will be 10: (2 3) 5 (5) 5 10 2 (3 5) 2 (8) 10 As with the commutative property, examples of operations that are associative include the addition and multiplication of real numbers, integers, and rational numbers. However, unlike the commutative property, the associative property can also apply to matrix multiplication and function composition. Like commutative property equations, associative property equations cannot contain the subtraction of real numbers. Take, for example, the arithmetic problem (6 – 3) – 2 3 – 2 1; if we change the grouping of the parentheses, we have 6 – (3 – 2) 6 – 1 5, which changes the final result of the equation. What Is the Difference? We can tell the difference between the associative and the commutative property by asking the question, â€Å"Are we changing the order of the elements, or are we changing the grouping of the elements?† If the elements are being reordered, then the commutative property applies. If the elements are only being regrouped, then the associative property applies. However, note that the presence of parentheses alone does not necessarily mean that the associative property applies. For instance: (2 3) 4 4 (2 3) This equation is an example of the commutative property of addition of real numbers. If we pay careful attention to the equation, though, we see that only the order of the elements has been changed, not the grouping. For the associative property to apply, we would have to rearrange the grouping of the elements as well: (2 3) 4 (4 2) 3