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exercise1.5a_EllipseDrawing-key - From this equation we can...

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Astronomy 103 Class Exercise – Drawing an Ellipse - Key This diagram shows a method for drawing ellipses using a loop of string, a pencil, and two thumbtacks to represent the foci. A piece of cardboard placed behind the paper will allow the tack to hold the paper in a more stable manner. Let’s work through the planning stages of applying this method to draw an accurate scale model of the orbit of the dwarf planet Pluto. The orbit has a semi-major axis of a = 40 AU and an eccentricity of e = 0.25. Let’s use a scale of 1 AU in the real world equals 1 cm in our drawing so that the ellipse that we draw will have a = 40 cm. To apply this method we need to know how far apart to place the thumbtacks and how long to make the loop of string. Thumbtack Separation We know that: – Let’s first calculate c – the distance from the center of the ellipse to one focus. Draw in and notate the distances c and a in the diagram above.
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Unformatted text preview: From this equation we can calculate that c must be equal to: c=(0.25)(40cm)=10 cm The distance between the two thumbtacks is: 2(c)=2(10 cm)=20cm String Length – Let’s calculate the exact string length needed (and then we can add a little bit to tie a knot). We can think of this in terms of the triangle shown above and add up the length of the three sides. We already know the bottom side. To get the other two sides we need to: 1) Note that the triangle above is an isosceles triangle since our pencil is at the end of the semi-minor axis b. Therefore the two unknown sides are equal. 2) Remember the definition of an ellipse – the sum of the distances from the two foci is the same for all points on the ellipse and equal to 2a. Thus, the distance from a focus to the pencil (the length of one unknown side) is: 40 cm Conclusion The entire length of the string is: 2a + 2c = 2(40 cm)+2(10 cm)=100cm c e c ea a = a c b...
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