problemset1

# problemset1 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.901: Astrophysics I Spring Term 2006 PROBLEM SET 1 Due: Thursday, February 16 in class Reading: Hansen, Kawaler, & Trimble, Chapter 1. (Note: This refers to the 2nd edition of 2004, not the 1st edition of 1994.) You may also find Carroll & Ostlie, Chapter 7 useful for useful background on stellar binaries. 1. HK&T, Problem 1.1 2. HK&T, Problem 1.2 3. HK&T, Problem 1.3 4. HK&T, Problem 1.6 5. Historical astronomy: Fundamental length scales. Accurately determining distances and sizes in astrophysics remains a fundamental and challenging problem to this day. However, a number of surpris- ingly accurate measurements can be made simply with the naked eye. The ancient Greek philosopher Aristotle (c.384–322 B.C.) was able to deduce that the Earth was spherical from observations like the shape of the Earth’s shadow during lunar eclipses and the changing view of stars in the sky during travel from north to south. A number of fundamental length scales in our solar system were also correctly deduced by the ancient Greeks. (a) Size of the Earth. Eratosthenes (c.276–196 B.C.) deduced the size of the spherical Earth using the following facts: (1) On a particular summer day each year, the Sun penetrated to the bottom of a very deep well (and thus was directly overhead – this point is called the zenith ) in the town of Syene (now Aswan); (2) On the same day in Alexandria, the Sun at mid-day was 7 south of the zenith; (3) Alexandria was north of Syene by a distance of just under 5000 stadia, where 1 stadium is about 160 meters. Eratosthenes assumed that the Sun is sufficiently distant that its rays can be treated as parallel. Use these facts to reproduce Eratosthenes’s inference of the radius of the Earth. (You may use the value of π .) Compare this to the modern value of 6378 km. (b) Size and distance of the Moon. Aristarchus (c.310–233 B.C.) calculated these using informa- tion from a lunar eclipse. Use Timothy Ferris’s composite photograph of a lunar eclipse (on the web at http://web.mit.edu/8.901/images/moon.png ) to make a similar calculation. The dia- gram below indicates the relevant geometry. You should make use of small angle approximations where appropriate.

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1/4 o 1/4 o Umbral Shadow Earth Center of Solar Disk Center of Solar Disk Eastern Limb D Western Limb Moon i. Assume that only the darkest part of the Earth’s shadow (the umbra ) corresponds to total eclipse. Estimate the diameter of the circle roughly corresponding to the umbral shadow on the composite image, and also the diameter of one of the lunar images. Note that the center of the shadow does not lie on the line connecting the path of Moon’s center — why not?
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