Problem Set #1Ge/Ay 133Due Thursday, 6 October 20111. Consider a planet of mass Mpthat orbits a star of mass M∗at orbital distance a, or,more precisely, the star and the planet go around their common center of mass. For astar some R parsecs distant, use Kepler’s l aws to derive the velocity of the planet and themaximum radial velocity of the star. How large i s the velocity for the Sun and Jupiter(the sun is 2 × 1033g, Jupiter is 1000 times less ma ssive, and Jupiter orbits at 5.2 AU).Now imagine looking at t he sun from 10 parsecs away (the distance of typical nearby stars,where 1 pc is 3 × 1016m). By what angle do es the Sun move due to Jupiter? What arethe velocity and angular movement of the Sun due to the earth (1 AU, 6 × 1027g)?2. If the Jupiter-mass planet around the sun-like star 10pc away orbits precisely edge onto us, it wil l, once an orbit, pass directly in front of the star, block out a small amount oflight from the star, and cause a perceptible dimming of the star.a. Transit depth. How does the transit depth scale with orbital distance from the star?The radius of the Sun is easil y remembered as 2 light seconds. The radius of Jupiteris 10 times smaller. What is the magnitude of the dimming of t he Sun? How does themagnitude of the dimming change if the Jupiter-sized planet is at 1 AU instead of 5 AU?What if the sta r is moved to 100pc instead of 10pc?b. Transit timing. Show, for an equato rial transit of the star, thatτ =RpaPπwhere a, P are the planet semi-major axis and period, and Rpis the radius of the planet;and thatT =R∗aPπwhere T is the full-width-half-maximum of the transit and R∗is the radius of the star.Here, the following additional times are defined: T he full transit from first to last contact(tT). The transit time over which the planet fully occults the star (tF). The ingress/egresstime (τ = (tT− tF)/2).For Jupiter at 5 AU, what are the values of tT, tF, and τ ? The last tells you aboutthe cadence you’d like to take dat a with. Can the relative va lues of tTand tFtell you atwhat “latitude” the planet crosses t he sta r?Finally, usng scaling relations (that is, do not worry about all the constants, justworry about the functional forms of various relationships), show thatgP∼K∗PaRp2∼K∗Pτ2ρ∗∼aR∗3P−2∼PT3where K∗is the speed of the star a s it moves in orbit around the center-of-mass of thestar-planet system (from problem #1). To do this, you’ll need Kepler’s Third Law in thelimit that mp≪ M∗Interestingly, this shows that from timing alone the model independent parametersthat can be derived are the surface gravity of the planet and the (bulk) stellar
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