A. Thicknesses and Densities

Transcription

1 10 Lab0 The Eath s Shells A. Thcknesses and Denstes Any theoy of the nteo of the Eath must be consstent wth the fact that ts aggegate densty s 5.5 g/cm (ecall we calculated ths densty last tme). In othe wods, 5.5 g/cm s a constant on theoes of the ntenal consttuton of the Eath. Quanttatve Concepts and Sklls Weghted aveage The natue of a constant olume of sphecal Concept that an ntegal s a sum At the begnnng of next lab, tun n the answes to the lab questons, a had copy and electonc copy of the speadsheet ceated n slde You have pobably leaned aleady that the Eath conssts of concentc composton layes (): an outemost cust, a thck shell called the mantle, and an nteo coe. You pobably also leaned that the oute coe s lqud and the nne coe s sold. The Eath s Shells Cust Mantle Oute coe Inne coe Knowledge that the Eath s composed of these comes fom the ntepetaton of tavel tmes of eathquake geneated sesmc waves. Eathquakes occu nea the suface of the Eath (up to depths of ~700 km), and sesmc waves can tavel fom one sde of the Eath to the othe, whee the avals ae ecoded by sesmogaphs. As we wll dscuss late n the semeste, the tavel tmes gve an ndcaton of the densty of the mateal that the sesmc waves pass though. Ths module exploes the combnaton of denstes and shell thcknesses that poduce an aggegate densty of the Eath of 5.5 g/cm. 1

2 PREIEW The coe mathematcs of ths lab s the weghted aveage. We want the oveall aveage densty of the Eath. We cannot calculate the oveall aveage densty as the smple aveage of the shell denstes. The mantle s much lage than the cust, fo example, and so the densty of the mantle contbutes much moe to the densty of the eath than the densty of the cust does. Gong nto the calculaton, we do not know ethe the thckness of the fou o the denstes. To stat, Sldes and 4 use a weghted aveage whee the denstes ae mee guesses, and the thcknesses ae equal (1/4 of the adus of the Eath). The calculaton n Slde 4 uses thckness as the weghtng vaable n the weghted aveage, and Sldes 5 and 6 dscuss why that s an ncoect appoach. Sldes 7 and 8 gets us on the ght tack: the weghtng paamete must be the volume of the, and slde 10 pefoms the calculaton fo the same combnaton of thcknesses and denstes as n Slde 4. The esult s not 5.5 g/cm, howeve, and so the est of the module does follow-up calculatons adjustng the shell thcknesses and denstes untl the bottom lne of the speadsheet poduces the coect 5.5 g/cm fo the oveall aveage. PROBLEM Gven that the densty of the eath s 5.5 g / cm, what ae the denstes of the cust, mantle, nne coe, and oute Coe? Don t look ths up n a book (not yet anyway). Fo now, just make some guesses. Remembe that the adus of the Eath s 6,70 km, so the shell thcknesses have to sum to that value. The smplest guess, to set up the speadsheet, s to make all the the same thckness. Hee ae some guesses Shell Cust Mantle Oute Coe Inne Coe Densty (g/cm ) Thckness (km) Make a speadsheet that calculates the aveage densty fom these guesses fo the shell thcknesses and denstes. 4

3 Hee s one appoach: Calculate the aggegate densty as the weghted aveage of the denstes usng thckness as the weghtng facto. B C D Shell Thckness (km) Densty (g/cm ) cust mantle oute coe nne coe Sum (km) 670 Sumpoduct (km-g/cm ) weghted aveage (g/cm ) 5.5 Ths speadsheet uses the SUMPRODUCT functon (Cell D). Altenatvely, you can add Column E fo the poduct of Columns C and D n Rows -6, and then sum all those poducts. The esult would be the same as n Cell D. Note, the esult (5.5 g/cm ) does not confom to the constant (5.5 g/cm ). Moe mpotantly, the appoach s not coect. Why s ths an ncoect appoach? 5 The weghtng facto should be volume, not thckness. Explanaton: Densty s mass ove volume, M Eath Eath Eath (1) The mass of the Eath s the sum of the masses of all of the, M Eath The volume of the Eath s the sum of the volumes of all of the, Eath Combnng the thee equatons poduces the weghted aveage Eath () () (4) 6

4 The calculaton that used shell thckness as the weghtng facto s ncoect because t assumes that the aeas of all of the ae the same. Explanaton: Statng wth the basc equaton, Eath you can get to the weghted aveage usng thckness (b ) as the weghtng facto aveage f each of the volumes has the same aea (.e., A A ). Thus b A A b b kates layes b b b A A b b Ths wll wok fo a stack of ectangula layes, but t won t wok fo concentc. Why? Layes vs. 7 To use Eath you need to calculate the volume of sphecal. Poblem: Fnd the volume of a sphecal shell wth nne adus 1, and oute adus. 1 mcoshell thckness d Stategy: Dvde the shell nto a bazllon mco each wth ncedbly small thckness d. Fnd the volume of each of the mco and add them up. Let be the ntenal adus of any gven mcoshell. Recall the suface aea of a sphecal shell wth adus : A 4p Then, wte down equaton fo volume of each mcoshell 4 And, add them up: 4p d 4 d ( p p - 1 ) Ú mco 1 The above s smple calculus. Don t woy all you need s ths fomula fo volume 4p d An ntegal s a sum. Thckness, d, s nfntesmal. 8 4

5 Now that we have the fomula fo the volume of a sphecal shell, we can lay out a speadsheet that coectly calculates the aveage densty (usng the same guessed values fo the thckness and densty of the ). depth to 1 thckness volume densty base (km) (km) (km) (km ) (g/cm ) 4 suface cust mantle oute coe nne coe Sum Sumpoduct (km -g/cm ) 1 weghted aveage (g/cm ) You ae to put fomulas nto these dak yellow cells. depth to 1 thckness volume densty base (km) (km) (km) (km ) (g/cm ) 4 suface cust E mantle E oute coe E nne coe E Sum E+1 11 Sumpoduct (km -g/cm ) 4.4E+1 1 weghted aveage (g/cm ) 4.01 What a dffeence. The po esult was 5.5 g/cm. Note, the esult (4.01 g/cm ) stll does not confom to the constant (5.5 g/cm ), but at least the esult s coect fo the assumptons. What we need now ae bette values fo the depth to the shell boundaes and the densty of the. Tme to consult a geology textbook. 10 5

6 Hee ae values fom Fowle, C.M.R., 10, The Sold Eath: An Intoducton to Global Geophyscs, Cambdge Unvesty Pess, p. 11. Shell Densty (g/cm ) Depth to base (km) Cust Mantle Oute Coe Inne Coe Redo the speadsheet usng these textbook values. Assume the densty of the cust and nne coe ae.8 and 1 g/cm, espectvely. depth to 1 thckness volume densty base (km) (km) (km) (km ) (g/cm ) 4 suface cust mantle 81 7 oute coe nne coe Sum 11 Sumpoduct (km -g/cm ) 1 weghted aveage (g/cm ) What do these values have to be fo ths value to come out to 5.54 g/cm? Tal and Eo 11 Usng tal and eo, you pobably had seveal wong answes befoe you came up wth one that confoms to the constant (Cell G1). depth to 1 thckness volume densty base (km) (km) (km) (km ) (g/cm ) 4 suface cust E mantle E oute coe E nne coe E Sum E+1 11 Sumpoduct (km -g/cm ) 6.00E+1 1 weghted aveage (g/cm ) 5.54 Does you answe match ths o dd you fnd othe values that wok? What ae some othe values that confom to the constant? Columns B, C, and G compse a model fo the densty stuctue of the Eath. The constant allowed you to elmnate many possble models. Stll, thee ae many othe possbltes that have not been elmnated. 1 6

7 Lab Questons 1. Answe the queston n Slde 7. Expand you answe to nclude an explanaton of why the esult n Slde 10 (4.01 g/cm ) ) s smalle than the esult n Slde 5 (5.5 g/cm ) Skp #. Accodng to Dante s Infeno, Hell s at the cente of the Eath. Assume that Hell has a adus of 1000 km, and that the est of the Eath s the densty of nomal ocks (say.8 g/cm ). What would the oveall densty of the Eath be? What would g be?. Suppose that a planet conssts of two of equal thckness. Deve a fomula fo the densty of the planet as a functon of thee vaables: R, the adus of the planet, and 1 and, the denstes of the two. Test you fomula by usng a modfcaton of the Skp speadsheet # n Slde 1. Does the ode of the denstes matte? In othe wods, f you exchange the value of 1 wth that of, does the oveall aveage change? Explan. 4. Fnd thee possble answes fo the denstes of the mantle and oute coe that esult n the coect oveall aveage (Slde 1). 5. Slde 1 s an example of fowad modelng: guessng values of the paametes that poduce a known end esult. Wte a paagaph on fowad modelng, and nclude the wods constant and unque soluton. Also tun n an electonc copy of the speadsheet ceated n slde