GEOS 32060 / GEOS 22060 – Spring 2018 – Homework 2 DueinclassTuesday24April.Nocreditwillbegivenforanswerswithoutworking.ItisOKtousee.g.Mathematica,butifyoudo,pleaseprintoutthework.Q1.DustingbyPhobos.Phobos, a darkmoon, is accelerating towardsMars andwill disintegrate in 20-70Myrduetotidalforces1.

Black & Mittal, Nature Geoscience 2015 (k2 and Q refer to tidal dissipationparametersforMars).Thisquestionisabouttheclimaticconsequences.

(a) Calculate the global-equivalent depth of Phobos dust (Phobos diameter~10km)followingdisintegrationofPhobosandreentryofthefragments.Assume allmaterial arrives as dust. (The duration overwhich fragmentsreentercouldbeMyr,orevenlonger).

(b) AssumePhobosdust (albedo=0.1) fallson thenorthpolarwater icecapand Mars obliquity (tilt of spin axis relative to incoming sunlight) = 45degrees, Mars semimajor axis a is unchanged (1.52 AU) 2 and Marseccentricity e is zero. What is the theoretical maximum polar melt rate(kg/m2/hr)?(Assumeallabsorbedsunlightgoesintomeltingwaterice).

1Thedetails,whicharenotnecessaryforthishomework,areinBlack&Mittal,NatureGeoscience,2015.Theaccelerationhasbeenconfirmedby(amongothermethods)Phobos-2Semimajoraxischangeisnegligiblesince>3Gya.

(c) Correctyouranswerforupwellinglongwaveradiationatthemeltingpoint(Stefan-Boltzmann law) – taking into account energy lost to upwellinglongwave radiation, what is the corrected melt rate? Assume dusty iceradiates in the thermal infrared as a blackbody, and neglect the Marsgreenhouseeffect(whichisweak).

(d) Correctyouranswerto(c)forevaporiticcoolingusingthefollowingtable(from Ingersoll, Science, 1970), assuming an atmospheric pressure of 25mb3–whatisthemeltrateincludingthissecondcorrection?

(e) AsthemeltwaterreactswithatmosphericCO2andwith thePhobosdust,carbonates will form, reducing CO2 pressure. Assuming Henry’s lawsolubilityofCO2inthemeltwater-3.4x10-2mol/(literxbar)–andthatthepolar cap covers 10% of the planet, what is the maximum (dissolution-limited) rate of CO2 consumption? How long would it take for theatmospheretodisappearatthisrate?

(f) In reality, CO2 consumption will stop when either when the carbonate-formingpotentialofPhobosdust isuseduporwhenincreasedevaporiticcooling (due to the lower total atmospheric pressure) prevents furthermeltwaterproduction–whichevercomesfirst.Show,byquantitativeuseofTable 1, whether reactant-mass or water availability will limit the CO2consumption.AssumePhobos dust has density 2 g/cc and is 10wt%Mg(noCa).ApproximateinterpolationsareOK.

The one-sided negative feedback you have just worked through (minus thedisintegratingmoons,althoughitisquitepossiblethatPhobosismerelythelatestina chain of inspiralling moons) is one hypothesis for what regulates atmosphericpressureontherealMars.Kahn(Icarus,1985)iscreditedwithsuggestingthisone-sidednegativefeedback.Q2:Venussurfacetemperatureis~750K,Marssurfacetemperatureis~210K.

3ThisassumesthatburiedCO2iceandadsorbedCO2isreleasedathighobliquity:probablebutunproven.

(a) DrawthebrightnesstemperatureversuswavelengthforVenusandMarsiftheybothradiatedtospaceasablack-bodyattheirobservedsurfacetemperatures(noatmosphereoneitherplanet).(1/5ofcredit)

(b) BelowisbrightnesstemperatureversuswavelengthforVenusandMars.WhyistheaveragebrightnesstemperatureforVenusandMarssimilar?Explainindetail.(3/5ofcredit)

(c) SupposethatthebrightnesstemperatureofVenuswasslightlylessatallwavelengthsthanthatofMars.Howmightthisbetrue,giventhatVenusisclosertotheSun?(1/5ofcredit)

fromdePater&Lissauer2001