AST 228 Problem Set 2 - Due Wednesday Feb. 23, 2011 1. Consider problem #3 from HW 1/Preclass #2: Calculate the gravitational force that a thin disk exerts on a point mass m at a distance z along the axis perpendicular to the center of the disk. Assume a uniform mass density ρ, total mass M, and radius R. a) Show that the area dA of a ring of mass with arbitrary radius r and thickness dr reduces to 2πrdr b) Solve the integral we developed in class to find an expression for the total force on m in terms of G, M, m, R, and z. 2. (AKA Practice Integration with Change of Variable) Consider the problem in the handout from Carroll & Ostlie (Example 2.2.1). Follow the instructions given on page 35 to set up and work out the total force. You must show work to receive credit. HINTS: (a) You will be changing variables when doing the integral over θ . When working out the new range, you will have two options for the lower bound since (R^2 + r^2 - 2rR) has two solutions. I advise you to use (r-R)^2 (b) If you’ve done everything correctly, the integral over θ should work out to something rather simple. 3. Imagine you are conducting survey for binary brown dwarfs in 2 different star forming regions: IC 348 in Perseus (d=320 pc) and the Trapezium Cluster in Orion (d=460 pc). If you are able to resolve objects with separations of 0.1” what your limiting resolution in AU for each cluster. 4. Use what you know about magnitudes and fluxes to derive the expression for Absolute Bolometric Magnitude given in Eq. 13.28 of Ryden & Peterson 5. Consider a model of the star Dschubba (Delta Scorpii, center star in the head of Scorpius). Assume Dschubba is a spherical blackbody with a surface T of 28,000K and a radius of 5.16x109m located at a distance of 123 pc from Earth. Determine the following for Dschubba: a) Luminosity b) Absolute Bolometric Magnitude c) Surface Flux d) apparent magnitude You may find the Sun’s bolometric magnitude (Mbol⊙ = 4.74), apparent magnitude (m⊙=-26.8) 26 and the Solar luminosity (L⊙ = 3.83x10 W) useful. 6. Use Wien’s law to determine the peak wavelength for the following objects, assuming they radiate as blackbodies. Please also specify the corresponding region of the EM spectrum and what we might use to detect that radiation (i.e. I can see the Sun with my eyes, optical telescope): a) Cosmic Background Radiation (T=3K) b) A Snowball (T= 273 K) c) Average Human (Skin T = 92 ºF = 306 K) d) Surface of the Sun (Teff = 5780K) e) Surface of an O star (Teff = 30,000 K) g) An Active Galactic Nucleus (Teff~1011 K).