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# Berkeley ASTRON C162 - Problem Set 3

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Astro 162 – Planetary Astrophysics – Problem Set 3Due Thursday in class.Readings: Landstreet section 5.9; also pages 139-141 on radioactive dating; skim section5.6Problem 1. Disk HeavenConsider once again the minimum-mass s olar nebula, a circumstellar disk of gas anddust of solar composition orbiting the young sun.X-ray satellite observations of young stars reveal that they are much more lumin ous inX-rays than the present-day sun. Some of these X-rays will strike the circumstellar diskand ionize its outermost layers of gas.Assume this ionized outer layer of hydrogen is everywhere at a tem perature of T = 104K.(One could compute its temperatur e f rom first principles by equating the heating ratedue to photoionization to the cooling rate due to radiative de-excitation of collisionallyexcited ions.)(a) What is the thermal speed of ions in this hydrogen plasma? Express in [km/s].(b) At w hat critical disk radius, rcrit, does the vertical scale height, h, of this T = 104Kplasma equal the disk radius? In other words, at what critical d isk radius does theaspect ratio, h/r, of the plasma disk equal 1? Answer symbolically in terms of G, M,k, T , and µ (the usual symbols; µ is the mass of the hydrogen ion), and numerically in[AU]. Remember that h is determined by hydrostatic balance in the vertical direction;we derived an expression for the vertical scale height in class, which you may either useor derive yourself.(c) Imagine that you are a parcel of plasma at disk radius r = rcritand hovering abovethe disk midplane at the full scale height z = h. Calculate the escape speed from thesun at your position, and compare to your thermal speed. Would you expect to remaingravitationally bound to the sun?(d) Repeat (c) but at r  rcritto explain quantitatively why plasma might or mightnot remain bound to the sun at those great distances. Repeat also for r  rcrit.(e) Comment on the relevance of your answers above to the metallicity gradient withhelio centric distance in the giant planets. In other word s , can you u nderstand whyJupiter at 5 AU is closest to solar composition but Neptune at 30 AU is heavily enrichedwith metals?1Problem 2. Terrestrial Rain Fall(a) Estimate the mean annual rain fall [in cm] that would result if all the solar radiationintercepted by the earth went into evaporating water.(b) List 2 factors that would reduce your estimate in (a).Problem 3. Forming TerraIs gravitational focussin g required to form the Earth?Give a yes or no answer and justify it quantitatively. Credit awarded for care andprecision in estimation.Problem 4. Live Aluminum-26 in the Ancient Solar SystemIn 1976, Lee et al. (1976, Geophys. Rev. Letters) published the results of isotope analysesof Aluminum (Al) and Magnesium (Mg) in a calcium-aluminum inclusion (CAI) in theAllende chondritic meteorite. Different minerals in the CAI (fassaite, m elilite, anorthite-B, anorthite-G) gave different isotope ratios. But together these ratios formed a trend:higher abundances of26Mg were found with higher abundances of27Al. The data lookedsomething like this:MINERAL27Al/24Mg26Mg/24MgFassaite 0 0.1400Melilite 10 0.1405Anorthite-B 133 0.1467Anorthite-G 240 0.1520Here27Al/24Mg is the number abundance of27Al atoms relative to the number abu n-dance of24Mg atoms. The same notation applies to26Mg/24Mg. You can verify thatthese 4 points make a pretty good line.Now it is known that26Al is an unstable isotope of Aluminu m that beta-decays with ahalf-life of a mere 0.7 × 106yr into26Mg. Thus, some of the26Mg found in the CAI maybe radiogenic (it came from26Al). By contrast,24Mg is non-radiogenic and is used hereas a reference. Furthermore,27Al is completely stable.Use arguments similar to those used in U-Pb dating (as discussed in class), and the dataabove, to estimate the26Al/27Al abundance ratio at the time the CAI solidified. (Thisproblem does not demand any fanciness; please don’t think too complicatedly.)We may see later that live26Al was a major heat sour ce in ancient planetesimals, ableto melt their

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