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# Math in the News: Issue 84

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### Math in the News: Issue 84

1. 1. Math in the News Issue 84 Comet ISON Approaches In this issue we look at the approaching ISON Comet.
3. 3. Comet ISON • As ISON approaches the sun, it will lose mass. • As it emerges from around the sun, there are three possible outcomes for the comet’s trajectory.
4. 4. Scenario 1 • ISON emerges from the sun’s orbit intact (diminished in size) and continues on its journey.
5. 5. Scenario 2 • ISON doesn’t survive its encounter with the sun and is vaporized entirely.
6. 6. Scenario 3 • ISON fragments and emerges from the sun’s orbit as a scattered collection of small comet fragments.
7. 7. Sketching the Path • Let’s assume Scenario 1 and look at the trajectory that the comet might take.
8. 8. A Parabolic Path • The path of Comet ISON can be approximated with a parabola. • A parabola is the locus of points equidistant from the focus and a segment perpendicular to the directrix.
9. 9. A Parabolic Path • At any point on the parabola the distance to the focus is congruent to the vertical distance to the directrix.
10. 10. A Parabolic Path • The path of the comet is approximated by the shape of the parabola, and the sun is at the focus of this parabola.
11. 11. A Parabolic Path • The perihelion distance can be used to find the equation of the parabola in vertex form.
12. 12. A Parabolic Path y = a(x − h) + k 2 • This is the vertex form of the equation for a parabola, where (h, k) define the coordinates of the vertex.
13. 13. A Parabolic Path y = ax 2 • If you assume the vertex of the parabola is at the origin, the equation is simplified.
14. 14. A Parabolic Path 1 a= 4f • To solve for a, use this equation, where f is the focal distance, or in this case the perihelion distance.
15. 15. A Parabolic Path 1 a= 4f • NASA estimates Comet ISON will come within 730,000 miles from the sun. Let’s express this distance in Astronomical Units (AU). (1 AU is the distance from Earth to the sun.)
16. 16. A Parabolic Path 1 AU f = 93,000,000 miles 730,000 miles f ≈ 0.0079 AU 1 a= 4f 1 = 4 • 0.0079 ≈ 31.65 • NASA estimates Comet ISON will come within 730,000 miles from the sun. Let’s express this distance in Astronomical Units (AU).
17. 17. A Parabolic Path y = 31.65x 2 • This is the equation of the parabola, and it is scaled for Astronomical Units.
18. 18. A Parabolic Path y = 31.65x 2 • The graph of the parabola shows where it intersects the Earth’s orbit (y = 1 AU) as it leaves the sun’s orbit.
19. 19. A Parabolic Path y = 31.65x 2 • NASA estimates that Comet ISON may come as close as 0.5 AU to Earth on its way away from the sun. Assuming it does, it should be a dramatic event in the night sky!