MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.282 March 10, 2003 Quiz 1 Name Solutions Last First (please print) 1. Be sure to attempt all problems. 2. The point values of the problems are indicated at the top of each page. 3. Closed book exam; you may use one page of notes. 4. Wherever possible, try to solve the problems using general analytic expressions. Plug in numbers only as a last step. Problem 1 (25 points) Orbital Dynamics It has long been suspected that there is a massive black hole near the center of our Galaxy. Recently, a group of astronomers determined the parameters of a star that is orbiting the suspected black hole. The orbital period is 15 years, and the orbital radius is 0.12 seconds of arc (as seen from the Earth). Take the distance to the Galactic center to be 8 kpc. Compute the mass of the black hole, starting from F=ma. Express your answer in units of the Sun's mass (M☉). (Assume that Newton's law of gravity is applicable for orbits sufficiently far from a black hole, and that the orbiting star satisfies this condition.) F = G * MBH * M / R² = M * v² v = 2πR / P G * MBH / R = 4π²R / P² G * MBH = 4π²R³ / P² Likewise: G * M☉ = 4π² * (1 AU)³ / (1 yr)² G = 4π² * (1 AU)³ / (1 yr)² * M☉ MBH = (R / 1 AU)³ * M☉ / (P / 1 yr)² R = 0.12 * (8 * 1000) * (3 * 10¹⁶ cm) = 1.4 * 10¹⁹ cm = 900 AU MBH = (900)³ * M☉ / (15)² = 3 * 10⁶ M Problem 2 (25 points) Comparison of Radio and Optical Telescopes (a) The Very Large Array (VLA) is used to make an interferometric map of the Orion Nebula at a wavelength of 10 cm. What is the best angular resolution of the radio image that can be produced? Note that the maximum separation of two antennae in the VLA is 36 km. θ = λ / D = 0.10 m / 36 * 10³ m = 2.8 * 10⁻⁶ rad. 200,000 = θ * 0.6" (b) Estimate the diameter and focal length of the primary mirror (or lens) needed in an optical telescope to achieve the same angular resolution. Take the wavelength of optical light to be 500 nm (5000 Å = 5 * 10⁻⁵ cm = 5 * 10⁻⁷ m). If you are unable to do part (a), simply adopt an angular resolution of 1". θ = 6" * 2.8 * 10⁻⁶ rad. = λ / D D = 5 * 10⁻⁵ cm / 2.8 * 10⁻⁶ = 18 cm (c) What is the size of a 10 arcminute square portion of the nebula on the detector on the optical telescope if its focal length is 200 cm? s = 0.01745 * f 10' = (1/6)° = (0.17)° width = 0.58 cm area = 3.4 cm²

2003 Quiz 1 with Solutions

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