Astronomy 170 Carl Sagan

8/13/2019 Astronomy 170 Carl Sagan

1/21

IT ASTRONOMY 70

X >

17 ebruary 965 Problem Set # 1

COMPLETED PROBLEM SETS ARE DUE AT L E C T V V R E ON MONDAY, 1 MARCH, m GENERAL NO LATE SETS WILL BE ACCEPTED.

l .a) Give a s imple der ivat ion ( i . e . , do not use spherical Bessel funct ional shoving tha t theore t ica l resolving power Q X/d, where d i a the dia- meter of the aper ture of the opt ica l instrument.

b) How i s the resolving power affec ted in pract ice when one observes a p lanet from the Ear th ' s surface?

c Howi s the resolving power affec ted in practice when one observes the Ear th ' s surface from a n ar t i f ic ia l Ear th Satel l i te? ( ( 3 pts.))

2 . ompute the genera l express ion for pressure, p , a function of alt i tude, h , in t he atmosphere of a planet of r a d i u , R i n which the temperature var ies l inear ly with height i.e., T T [ l + a ( h / f e ) ~ ] here TQ i s the su r face temperature*

0 pts

the base of t h < & Mart ian exosphere t h > number densi ty > x 10' c m *3 and the composi t ion i s p:imarilv atomic

NOTES i

3 . ssume tha t a t of par t ic les i s 5 x 1 0 ' cm"J and the composi t ion i s primarily Atomic oxygen. tomic hydrogen i s present in a bundance < 1 0 * tha t of oxygen by number (fract ional abundance } . or the temp ra ture of th e Mart ian exosphere, consider two cseas ( a ) T i s cons t a i t a t 500Kf a nd C b ? T i s constant a t 1200K. or each case, graph the tydrogen end oxygen densit ies a s functions of alt i tude. etermine t i e overa l l

atmospher ic density a t the orbits of Phobos \ n d Deimos. Be sure to a l low explici t ly for t he var i t on of gravi ty with alt i tude.

(2} Y ou should p lot two s epara te graphs for the two temperatures. n each graph wil l be two c u i r e s , one each for H a nd O.

(5pts.

There a re a number of instruct ive bounds which ca n be p i t a d on var ious physica l features of planets which a re based on t heassumpt ions of hydros ta t ic equiSbr ium and of monotonie densi ty decrease*i th radius,

Consider a non-rotat ing, spher ica l p lanet with t he f o l l o w i n g physica l parameters s R , to ta l radius; Mf t o t a l mass; p average 3ensy; p

t h e center; p_, pressure a t th e center.

c densi ty a t the center; pc , pressure

a ) Referr ing to th e equat ion of hydros ta t ic e q u J J b b r i u i i obvious condi t ion wil l

and mean radius?

u n d v r what be the smal les t for planets o given mass

Show tha t if this condi t ion obtains, than

bj Under wha t obvious ( b u t unphysical) condi t ion will p be the la rges t? how t ha t the implied inequal i ty due to this ondition om be combined wi th t ha t i n ( a ) to give

*b?6** * Vv* C C 4 r t s ) .


2/21

&STROEOMY 1 70 Prob lem Se t # 1 Page 2 .

| 5 ) . Th e gene ra l equa t ion o f mot ion o f a cu t r a l f lu id in a gravi- t a t i ona l f ie ld is

v)v -y v-f> +3 W h e r e V is veloc i ty, p is pres su ra , f > is dens i ty, t is t ime , a nd g i s a c c e l e r a t i o n d ue t o grav i ty. h i s equa t ion w a s f i r s t ob ta ined b y L. Bula r in 1755 , a nd is one o f t he fundamen ta l r e l a t i ons of f lu id dynamics .

a) S h o w t h a t in a spec i a l case , Euler ' s equa t ion wi l l r e d u c e t o t he a q u a t i o n o f h y d r o s t a t i c equi l ib r ium. h a t is t he spec i a l c a s e ?

b) If t h e r e is no ex te rna l fo rce , w h a t ca n w e sa y abou t t he pressu ra in t h e f lu id if i t is in equi l ibr ium?

c) Assume t h a t t h e r e is n o s i g n i f i c a n t compress ion o f t he f lu id under t he a c t i o n of a n ex te rna l force. n teg ra t e t h e vec to r i a l fo rm o f t he equa t ion o f h y d r o s t a t i c equ i l ib r ium f c f . p a r t a ) t o o b t a i n a n expres s ion for t he pres su re in te rms o f t he al t i tude."" Des igna te w h i c h o f your axes i s ver t i ca l ly upward .

CO pts.n


3/21

A S T R O N O M Y 170

24 February 1965

roblem Set

C O M P L E T E D B G B L . B 4 ETS ARE U E AT L E C T U R E N E D N E S D AY, 10 A R C H

1, An deal as onsista of molecules ach of mass at a emperature and contained a olume T 0

Oka) alculate he mean peed* , f he nlscuies 0

I b) Calculate he oofc^mean^aquare epeed 8 fmB f he moleculeSo

I c) alculate he mo3t probable velocity, ^9 f he molecules. m fllA) alculate he most probable peed, f he-molecules mc

I e) how hat f v is any artesian component of , hen 'ttv^ p>)' value or he paed of ound n a gas

" . his s Nowton

2 f) how hat he raction of mol seules hat ave peeds reater han ome arbitrary valuv is

v/v

i

where m HINTS* (1) V -fvvUv

C 3) ntegrate by parts. (3)

(4) T he rror unctioiTis ( efined y

f


4/21


5/21

^^ Zt^oT e* ^fonleeo) erodynamic 5*., s a orce par- unit mass. 8 the *tno PhePlo density,.,s fee apianen above crreet ^ + . clearly oin-, u iv iu no I?J S ""* QaXt * If ot, ~ ou way T ; s not aad what ^^^ argunent houU b ^ 6

(tS pts. )

At . 0 he vMoeity distribution unction is iren by

*i M ^* t Tu L? tS 6n 2aas,litude * peed. otice hat his produce. owever? a ssu t ^ ll i 2nj am> distribution, 30 hat t s hard o i. heh hl I * *T l0iaSh0W beett ^ duoed at . 0. he normalisation

tYt a w many molecules re to hs right 0 f he plane x 0 at 0

(kb) hat is he otal inetic .ergy of he gas

V ^^ if * ** - *C -N rovided he walls constant, b ot nKct LusiSsf " ** *"* ^ a -*cati ve

4 ... * .

4 pts.))

ullZ ?l: V? :n f,V* of planeir * --. - u pr cut,ofT o he atmosfera. n parts and h below o o for he Earth. ASSUMPTIONS OE *ARTS and b (1) Magnetopause is at-14 Earth radii.

/ti ? e "* x sphere is t ~600 ms. I Sl ^ a He f * a) Be-intgerate he efinition of he h. ^-,- cut-off occurs a t he nagetonause FiST T ^f? 5 ^ . ssuming hat he atmosphericat he base of he eSC ?nSuS Li gression or he umber enstv

Estimate he rror which s oS?Si f^? /^* 0 11 of Cavity with ipneo.Vc) 1 he fr pW VSESgf

* deflnitio * * he exosphere by astlni*

Vc) nder what onditions for ome otheret *** an atmospheric cuWf e mport,5 pts.))


6/21

ASTROHOMY 170 10 March 1965

roblem Set # 3

COMPLETED PROBLEM SETS ARE DUE AT LECTURE OH MQHDAY, 2 2 MARCH,

1 . (4 p o i n t e ] ) C a Calculate th e equilibrium pressure of xenon in thelunar atmosphere. ssume that

( 1 ) the surface temperature ^400K.

(2) thermal escape i s th e only loss mechanism.

(3) xenon i s produced by th e bombardment of th e lunar surface by cosmic rays, with a cosmic ray flux of 2 c m *2 s e c * * .

C 4 ) each cosmic ray magically produces one xenon atom, on th e average.

(5) the cosmic ray intensity in space i s isotropic.

Why does your answer i n (& } disagree with th e bound given in lecture for t he lunar atmosphere? n seeking the explanation of thisdiscrepancy, compare th e equilibrium time - i.e., the time required for the xenon to accumulate to its equilibrium density - with th e age of the solar s y s t s s a .

2 . (6 pointsD Calculate fo r t he Martian atmospheres

( a ) B , C c y c . .

assume that 1 S th e lower atmosphere can b e considered for this

problem to b e isothermal, and it consists mainly of molecular nitrogen

m ^surf ~230K, P8urf ~* 1/20 atm.

from the altitude at Ts h i c h diffusive equilibrium obtains to the base of th e exosphere, the atmosphere i s isothermal at ~/ 900K.

th e exosphere i s isothermal at ^^900^.

(continued next page)


7/21

ASTROSOM 1 7 0 Problem f c e t # 3 ^continued?

3 . (6 points) onsider a planet whose atmosphere and exosphere

originally contains 9 5 9 S molecular hydrogen, 49 6 water vapor and 1s i lane (SiH 4 D b y number. onsider th e plarut and atmosphere other-

wise to have propert ies s imi lar to t hose o the Earth at the present

t ima i .e. , same g, same exosphere temperature same radius, etc.

Heglect th e pho tod i s soc ia t ion and ionisation of molecular hydrogen

and the ionisat ion of s i lane and water. onsider the photodissocia t ion

times of water and silane* and their cheiical react ion t imes, to b e shor t compared to the escape t ime of s a o Jocular hydrogen.

( a ) Find th e t ime when there will e an equal number of water and hydrogen molecules .

Find th e subsequen t t ime when water will l o se th e dominant ro le in the composi t ion which it has gained in part f a ) i .e. . when there will b e l i t t le water comp ared with other substances in the atmosphere. ou may define "littlo" in any convenient way.

C c D s o w soon will th e surface b e coated with sand and th e ataospher relat ively depleted of s i l .me? ndicate the chemical reactions r e spons ib le fo r these evento.

) A t the t ime found in part ( c ) , what is the compos i t ion of th e atmosphere? o w long will th e planet retain an appreciable amount of this atmosphere?


8/21

ASTRONOME 1 70 22 M a r c h 1965

rob lem Se t #4

COMPLETED PROBLEM SETS A R EDU E A T LECTURE ON WEDNESDAY 31 MARCH.

1 . ( 6 poin ts ) ( a ) Findoe , i^ , a nd kg for a pe r fec t gas . ( b j Find t h e pres su re - t empra tu re r e l a t i o n in s c o n -

vec t ivo f lu id for each o f t h e fo l l owing t wo equa t ions of s t a t e s

m const.) [}f f ? 3 - /o 5 J p0 y- jo \r J 0 -i

which i s c a l l e d t he M u r n a g h a n - B i r c h equa t ion of s t a t e ; a nd

(ii)

/3 p - (const . ) (o , * >, w h i c h is t he equa t ion o f s t a t s for

s comple te ly degene ra t e e l ec t ron ga s obey ing Permi -Di rac s t a t i s t i c s .

2 . ( 3 poin ts ) ( a ) Compute t o f i r s t orde r t he e ff ec t o f t he v a r i a t i o n of g w i t h a l t i t u d e o n a dry a d i a b a t i c l apse ra te .

( b ) Is t he a b o v e e ff ec t r e a l i s t i c a l l y s ign i f i can t a t h igh a l t i t udes? xp la in .

3 . ( 6 poin ts ( a ) Compute t h e number dens i ty, n(h) , in t h e Ear th ' s a tmosphe r s a t a l t i t udes o f 5 a nd 1 0 l a s , as suming a d i a b a t i c equi l ib r ium.

Der ive t he a d i a b a t i c s c a l e he igh t , de f ined in t e rms o f t he l oga r i t hm^der iva t iv s o f t he a d i a b a t i c dens i ty prof i le .

(c) C a l c u l a t e t h e v a l u e of t he a d i a b a t i c s c a l e he igh t for a r ep re sen ta t ive a v e r a g e t empera tu re o f t he t r oposphe re say, 2 5 0 K .

f d ) See h o w good a n a p p r o x i m a t i o n t o a n a d i a b a t i c dens i ty pro f i l e i s provided b y t he expres s ion

n(h) nQ exp|-h/H ad y

w h e r e n s t he number dens i ty a t t he Ear th ' s su r f ace , b y us ing ths express ion to c a l c u l a t e t he dens i t i e s a t 5 a nd 1 0 km.

(cont inued next page}


9/21

ASTKONOM* 1 70 Prob lem Set #4 ^cont inued

4. ( 5 po in t s ) ( a ) In c l a s s w e ob ta ined a d i ff e ren t i a l equa t ion fo r t he v a r i a t i o n o f p w i t h d e p t h in t h e i n t e r io r o f a convec t iva Ear th . Al lowing for t he v a r i a t i o n o f g w i t h dep th , i n t eg ra t e t he above equa< t i on unde r t h e assumpt ions t

< t )

( i i )

cons t .

( a + bp)

y a nd -1

K n o w i n g t h a t 3 g m c m "3 , w h a t va lues o f kg a nd a a nd b, r e spec t ive ly, a ra requi red t o g ive ? 5 .5 p c a T3 for t h e Ear th fo r t he two cases?

5. ( 5 poin ts ) Read pages 82 a nd 83 o f B r a n d t a nd Hodge . s s u m e t h a t s

f l ) Convec t iva f l ux r a d i a t i v e input ; ( 2 ) F l u x o f r a d i a t i o n a t t h e t o p o f t h e Ear th"s a tmosphe re

2 oa l c m "2 min** 3 ) A b o u t 40 % of t he r a d i a t i o n inc ident o n t h e Ear th is r e f l ec t ed

b a c k i n to space .

C a ) Using mix ing l eng th theory, f ind h o w super a d i a b a t i c t h e Ear th0sa tmosphe re c a n be .

( b ) Find t h e Vrm o f convec t ivo mot ion .

{{c} Dete rmine t h e m e a n l i f e t ime of a convec t iva bubb le .


10/21

29 M a r c h 1965 ASTRONOMY 1 70

Hour Examina t ion

Bol tzmann0 s c o n s t a n t k - 1 . 3 8 x 1 0 "16 erg ( K ) 1 . N e w t o n i a ngrav i t a t i on c o n s t a n t O 6.67 x 1 0 "8 dyne c m2 g m ~2

M a s s of hydrogen a t o m m - 1.67 x ICT24 gm.

1 . C30%| i amonds a re found in b o t h i ron a nd s tone meteor i tes , n t he l abo ra to ry, pressure o f bout 3 5 , 0 0 0 a tmosphe res a re requi red t o produce d iamonds f rom graphi te . s s u m e t h a t t h e d iamonds in meteo r i t ew e r e formed unde r h igh pressures f rom graph i t e .

If t h e d iamonds w e r e produced in t he in te r iors of meteor i t e pa ren t bodies , w h a t wa s t h e cha rac t e r i s t i c d imens ion o f t h e s e pa ren t bod ies? h a t ob jec t s in t h e so la r s y s t e m t o d a y h a v e such d imens ions?

Al te rna t ive ly, w e c a n assume t h a t t h e d iamonds w e r e produced w h e n t he meteo r i t e s impac t ed the Ear th . r e t h e pres su res exe r t ed even for a b r i e f per iod of t ime dur ing such a n impac t su ff i c i en t t o produce t he d iamonds?

2. 30XJ h e h y p o t h e t i c a l plane t Y Velorua VIII h a s a mass 1 0 0 t imes t h a t o f t he Ear th a nd a radiun to t h e t o p of i ts c louds 1 0 t imes t h a t o f t he Ear th . t s o m e re ference l eve l a b o v e t h e c louds o f th i s p lane t t he a tmosphe re is composed l toa t exc lus ive ly of molecu la r hydrogen . Th e t empera tu re a t t h i s l eve l is s o l ow t h a t v ib ra t lona l a nd ro t a t iona ldegrees o f f reedom a re not exci ted. h e a c t u a l t empera tu re grad ien t a t t h i s l eve l is 2 K / k r a a t the moment o f observa t ion . h e t e m p e r a t u ra t t h e t op of t h e c louds is 200K.

M If no condensa t ion occurs nea r t h e r e f e rence level , is t h e a tmosphe re s t ab l e a g a i n s t convect ion?

If very ex tens ive condensa t ion occu r s a t t he r e f e r e n c e level , is t he a tmosphe re s t ab l e aga i imt convec t ion?

C c | No w as sume t h a t t he re is no condensa t ion a b o v e t he c louds , w h a t is t h e m a x i m u m d i s t a n c e be tween t h e t o p o f t he c l ouds a nd t he t r opopause?

(cont inued nex t page)


11/21

f t S T B O H C V S * 70 Hour Examination 2 .

3 . ^ t i f c t ) ccording to th e planetary cosmogony of Kuiper, the Earth formeras a mass ive protoplanet of cosmic composit ion. os t of the initial-mass subsequen t ly escaped into space.

Wiiuhowed in c lass tha t th e thermal escape f lux from the base ofa planetery exosphere, L can b e given b y

1 -, &n T JL- j L2lrH j ] 1/2

Re R/Hj

(a) Show t h a t t h e above equa t ion impl ies t h a t b e c a u s e o f t he rma l e s c a p e , , th t func t iona l dependence of t h e t o t a l pro top lane ta ry mass ont ime fol lows a n exponen t i a l d e c a y law. ormula t e t h e prob lem in such a w a y t h a t t he number dens i ty a t t h e b a s e of t he exosphe re0 n j does no t ap |?ar exp l i c i t l y in t he r esu l t .

lj

how t h a t t he mass escape r a t e is a m a x i m u m i f R / B 1. s ingt h e equa t ion der ived in ( a ) for Tfl lt) a nd for t h e c a s e RAi ~^1 0 c o m - pu te a rough c h a r a c t e r i a t i e t ime for t he Ear th t o h a v e los t 1 = ( l /)of i t s i n i t i a l , nans. s s u m e t h a t t he m e a n exosphe re t empera tu re

V 1 0 O K during this period,


12/21

1 J LXV /I /.A

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13/21

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H f t txf- /Kjt^vvv.

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14/21

w .

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S^^M-n^A

allots .* .?S < 3 = > rg


15/21

C O Tk^= "(TL) - ~* UJUJL ^^


16/21

ASTROKQW 170 12 pril 96-

roblem Set 5

C O M P L I E D mOh. fET ARS Dg E C T U R E 71 WEDNESDAY, 21 PR X L.

1. (5 o.ntsD et ^^ u enote he adiation ressure nd adiant energy e\sity es 4 nvely* of adiation ncident n erfectly absorbing surface. H % ec ^ he mitted aclant luas.

(a) in Seeking K eiationship between nd 0 recisely or what volume s a" o e ovalu ? Explain.

Show chat o:r he su3 f n form hemispherical adiant luxand otal scrptioji , p u/.

(G) eriva e at ion etween and or he ase hat eam f radiation e orma ; ay_ ncident n erfectly bsorbing urface.

2. J4 oints) 3 .w 884 Soltzmann duced heoretically hat he otal rate f missicn f adiant nergy by n deal adiator CbiackbodyS i 8proportional < he th ovrer f \e Kelvin emperature. It an be shewn rom his hat he adiant \ergy ensity, 0 within n nclosure whose walls re at uniform emp rature 0 is lso roportional o T.

Since laecbody adiation aribe escribed y he oordinates p V, and it eay e reated s . hemical ystem,, and hermodynamics may be pplied .o it.

flaj how rci hermodynaraic re Lai; ions erived n lass hat if s the nternal ntrgy f hemical yitem hen

T/fet - /p\ Wv The a l x > v e e q u a t i o n , , known a ; i th e enerjy equation enables us to draw conclusions abo a t fo r any sys tem whore equation of s ta te is known.

f b ) B y applying the e n e r t j y equation to the case of b laekbody radiat ion encii>sed in an evacuated container whose walls are perfectly* reflecting t h e : . * m a l insulators show that

u b T4

whore b s onstant. From his xpJ8ion, erive he blackbody emission lux.

CO&TItUED EJ2F PAGE


17/21

ASTAiomr 1 7 0e Problem Set #5 ^continued):

3 . (5 j ointsI ( a ) Show that th saturat ion vapor pressu re over i c e ^ , PS 9 Ci l b e given approximate ly b y

Ps =c i

What asumpt ions are required in th e derivation of the above equation?What a r th e values of c. and c , ?

C b J 1 the clouds of Venus are composed of ice h o w much water vapor s h e , i d we expect above the c louds? ompare your resul t with the rang* of spec t roscop ic results fo r the water vapor above the clouds of'/enuss

1 0 '1 g c m *2 CDollfusi 3 2

clouds of Jupiter are composed of ammonia, h o w much ammonia vapcj should we expect above them?

HIHTS: 1 . Lant heat of vapor iza t ion for water 2.83 x 1 0 ergs g ra2. Th iriple point of water is at f ? 0 C9 6 mbK 3 * Infsired bolometr ic temperature of th clouds of Venus 234. Infr.red bolometr ic temperature o f the c louds of Jupi ter

~190K. 5 , Laten, heat of vapor iza t ion fo r ammonia = 1.37 x 1 0 ergs

4. [ 7 points } f a Consider the sa tura t ion adi&batic process of an air parcel lifted pas\ the level at which it becomes saturated. how that the First Law of Thermodynamics for this process may b e written as

* L dw. c dT + p d c x v * where L i s th l a t e n ' ; heat of vapor iz a t io n e w s the mass of water condensed per unit r c i s s of air and #* , _

Combine the relat ion in f a | with the equation of hydros ta t ic equi l ibr ium to obta in an express ion fo r th saturat ion adiabatic lapse rat P .

i j c j Compare the relat ion derived in C b | with the relat ion that wa s derived in class . alculate a rough va lue of Y) fo r the earth 's atmosphere.


18/21

AST

5

COMPILED PROBLEM Sl^S A R E B U S IK R ( OBSERVATORY, O H FRIDAY, 1 4 MAY.

1 . ( 6 paints) ( a ) Consider the case 3 the surface temperature o f a plane paz

T for O t<

T = T fo r fL t < P o 2

Solve the one-dimensional equation of beat c o i temperature at all depths and t imes.

Approximate th e s tep driving-function in { : .; few terms perhaps 4 or 5 of a Fourier series , h o w well the s tep function i s approximated, i th the lation of the s tep driving-fusction, solve the one-diia conduction equation again and compare the result with t i l obtained in ( a ) .

2. ( 4 points \ Radioactive potass ium, thorium, and uranium centrations found in ehondri t ic meteori tes release ~10~ :

sec 1 of heat . onsider a largo, initially "cold" objec t ehondri t ic composi t ion in interstellar space. s s u m e it s conductive propert ies to b e those of any rocky material.

neglecting heating hy star l ight , eompute it s equi l ibr ium temperature.

3 . S 3 poin tsD onsider the Planck distr ibution fo r the two special cases: ( i ) long wavelengths and high temperatures y and C i i J s h < wavelengths and low temperatures. t surface temperatures of terrestrial planets and at microwave frequencies, which of t^ese approximations is valid? h at are th e bounds on th e vaveleagth which this approximat ion is valid fo r those temperatures?


19/21

5 M ay 1 9 6 5 ASTRONOMY 1 7 0

Problem S et #8

COMPLETE PROBLEM SETS AR E DU E IN ROOM A-103, HARVARD COLLEGE O B S E RVA ^ RY, O N FRIDAY, 14 MAY.

a. ( 7 po ia ( a ) Murray, Wildey, and Westphal (Astrophys. J. 139: 986, 1964) have obtained brightness temperature maps of Jupi ter in the 8-14 mivron region. s ing the approximat ion to the Planck distribution, in their wavelength and temperature regimes, of B oC T , whtre n 8 . i , determine from their Pigure 1 an aoproxi- mate l imb-dastening lau of the form L^oC/u", where arc c o s i c - the angle between the loca? planetary normal and the l ine of sight, and CK i s a constant.

( b ) Assume th*t we i r e seeing an a tmospher ic region in the 8-14

micron regime on Jupiter characterized b y connective equi l ibr ium

and pure absorpt ion. etermine the values of t , >1 , and s required to expla in the obaerve-.ions and discuss whether the eonvective-pure- absorpt ion model can p'-ausibly expla in th e observations.

2. C 5 points) ( a ) Compsie th e temperature gradient for radiat ive equil ibrium using the El3ington approximat ion with the temperature gradient for convective aquilibriura, under th e boundary condit ion

> pft as < lo

Write own he chwarzscfcild nstability criterion or his case analogous o he criterion

n^fls ) Y-

>4

derived in c lass .


20/21


21/21

ASTRONOMY 1 70 Problem Set #6 (Cont inued)

3. ( 3 poin ts ) Murray, wildey. a n d Westpha l (Astrophvs. j . 1 39: 9 86. 1964) h a v e determined a m e a n br igh tnes s t empera tu re of t he sa t e l l i t e J u p i t e r IV in t he 8- t o 1 4 - / 6 r eg ion a s 168 .5K . o m p a r e th i s br igh tnes s t empera tu re w i t h t h e equ i l i b r ium t empera tu re which you expec t for J u p i t a r IV a nd exp la in quan t i t a t i ve ly w h y Murray, Wildey, a nd We s t p h a l found t he i r resu l t t o f e e perp lex ing .

4. ( 4 poin ts ) ( a ) Der iva t he Roche In s t ab i l i t y c r i t e r ion for two t es t pa r t i c l e s in synchronous r o t a t i o n abou t t he i r pr imary.

C b ) Imag ina t h e S un su r rounded b y a nebula of h a l f t h i cknes s equa l t o t he r ad ius o f Jup i t e r a n d w h i c h eve rywhere exceeds t he Roche dens i ty, so i t is grav i t t i ona l ly s t ab l e aga ins t t i da l d i s rup t ion b y t h e Sun.

W h a t is t h e mass o f t he nebu la? t o t a l present mass o f t h e p lane t s?

H ow does i t compare wi th t he

Astronomy 170 Carl Sagan

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