Astronomy 241: Review Questions #1 Distributed: September 20, 2012 Review the questions below, and be prepared to discuss them in class on Sep. 25. Modified versions of some of these questions will be used in the midterm exam on Sep. 27. 1. State Kepler’s three laws, and give an example of how each is used. 2. You are in a rocket in a circular low Earth orbit (orbital radius aleo =6600km).Whatve- 1 locity change ∆v do you need to escape Earth’s gravity with a final velocity v =2.8kms− ? ∞ Assume your rocket delivers a brief burst of thrust, and ignore the gravity of the Moon or other solar system objects. 3. Consider a airless, rapidly-spinning planet whose axis of rotation is perpendicular to the plane of its orbit around the Sun. How will the local equilibrium surface temperature temperature T depend on latitude ! (measured, as always, in degrees from the planet’s equator)? You may neglect heat conduction and internal heat sources. 3 4. Given that the density of seawater is 1025 kg m− ,howdeepdoyouneedtodivebefore the total pressure (ocean plus atmosphere) is 2P0,wherethetypicalpressureattheEarth’s 5 1 2 surface is P =1.013 10 kg m− s− ? 0 × 5. An asteroid of mass m M approaches the Earth on a hyperbolic trajectory as shown " E in Fig. 1. This figure also defines the initial velocity v0 and “impact parameter” b.Ignore the gravity of the Moon or other solar system objects. (a) What are the energy E and angular momentum L of this orbit? (Hint: one is easily evaluated when the asteroid is far away, the other when it makes its closest approach.) (b) Find the orbital eccentricity e in terms of ME, E, L,andm (note that e>1). Then use your results for part (a) to express e in terms of ME, v0,andb. (c) What is the actual pericentric separation rp <bof the asteroid’s orbit? (d) Long after this encounter, the asteroid will again be moving in a straight line with velocity v0.Whatistheangleθ between this line and the asteroid’s initial trajectory? Figure 1: An asteroid approaches the Earth. Before being accelerated by the Earth’s gravity, its initial velocity is v0;ifitcontinuedinastraightline,itwouldpasstheEarthatminimum distance b. 1 6. The greenhouse skeptic argues “The ‘greenhouse gasses’ in the Earth’s atmosphere are cooler than the surface of the Earth. The Second Law of Thermodynamics says heat can never flow from cold to hot. Thus it’s impossible for greenhouse gasses to warm the Earth.” Your response? 7. The Space Station is in a circular orbit with a radius of aSS =6800km.Youareina spaceship in the same orbit as the Space Station, but trailing it by a distance of 100 km. (a) What happens if you try to catch up by firing your rocket engines in the direction of your orbit? (b) How would you actually fire your rockets in order to reach the Space Station? 8. Neglecting factors of order unity, the tidal torque on the Moon’s orbit due to the Earth’s tidal bulges is 2 5 MMRE τ G 6 , (1) ∼− rM where MM is the mass of the Moon, rM is the radius of the Moon’s orbit, and RE is the radius of the Earth. 2 2 (a) Verify that this expression has units of torque (angular momentum/time = kg m s− ). 6 (b) Explain why the torque is proportional to rM− . 9. Due to its rotation, the Earth is slightly flattened; the distance from the center to the North (or South) pole is Rp = 6357 km, while the distance from the center to the equator is Re =6378km. 3 (a) Assuming rock has a typical density ρ 3300 kg m− ,evaluatethemassoftheEarth’s % equatorial bulge. (b) Given that the Earth’s axis of rotation is inclined by i 24◦ with respect to its orbital % axis, estimate the torque on the Earth due to the tidal forces of the Moon and the Sun. (You can and should ignore the slight tilt between the Moon’s orbit around the Earth and the Earth’s orbit around the Sun.) (c) Finally, estimate the period of the resulting precession of the Earth’s axis of rotation. 10. We think that Mercury originally had an iron core comprising 30% of its mass (much ∼ like the Earth). After Mercury differentiated, a giant impact blasted much of the rocky mantle away without seriously disturbing themassofthecore;asaresult,Mercury’score now makes up 60% if the planet’s mass. ∼ 23 (a) Given that Mercury currently has mass MMe =3.3 10 kg, what was its mass before the impact? You may ignore the mass of the impacting× body. (b) Roughly how much energy would have been required to eject the mantle material Mercury lost? 1 (c) Assuming the impacting body had a velocity of 50 km s− ,howmassivewouldithavehad to be to deliver the necessary kinetic energy? Compare your result with Mercury’s original mass. 2 11. The four long-lived radioactive isotopes which currently heat Earth’s interior are: 238U 235U 232Th 40K 9 τ1/2 (10 yr) 4.5 0.70 14.0 1.25 12 1 εnow (10− Wkg− )2.910.1253.271.08 where τ1/2 is the half-life of each isotope and εnow is its energy release per unit mass of mantle 1 2 2 rock (note that W kg− =m s− ). (a) Which of these isotopes made the greatest contribution to Earth’s internal heating 4.5 109 yr ago? × (b) Calculate the total ε at that time. 12. If the Earth received no energy from the Sun, but was still warmed by the radioactive isotopes listed above, what surface temperature would it have? 13. Consider a homogeneous planet (i.e., one with constant internal density ρ)ofmassM and radius R. (a) At radius r R,whatisthegravitationalaccelerationg(r)? (b) Use the equation≤ of hydrostatic equilibrium, dP = ρg(r)(2) dr − to calculate the central pressure Pc. 14. Consider a homogeneous planet (i.e., one with constant internal density ρ)ofmassM and radius R.Showthatthegravitationalenergyreleasedduringitsformationis 3 GM 2 E = . (3) 5 R 15. Consider a differentiated planet of mass M and radius R.Assumethatthecorehas radius R/2andthatthecore’sdensityistwicethatofthemantle. (a) Calculate the gravitational energy released during its formation. (b) Using the equation of hydrostatic equilibrium, calculate the central pressure Pc. 1 16. Asmallasteroidofmassm impacts the surface of a planet with velocity vi =30kms− . The asteroid is completely vaporized, along with a comparable amount of the planet’s crust. 2 2 1 Assuming the vaporized material has heat capacity C 600 m s− K− ,estimatethetem- perature of the vaporized material. % 3.