The Fourth Power in the Universe

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THE FOURTH POWER IN THE UNIVERSE D. D. Jones

September 1973 WP-73-S

Working Papers are not intended for distribution outside of IIASA, and are solely for discussion and infor- mation purposes. The views expressed are those of the author, and do not necessarily reflect those of IIASA.

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September

1973

THE FOURTH POWER IN THE UNIVERSE

D.D. Jones

SIZE DISTRIBUrIONS

m

NATURE Introduction

This puper reviews a collection of non··ecological size di:3tributicma that have been obcarved in nature. The ran3~ of si~cs covcr~ 37 ordars of

C3gn!tud~. Ecologically signif:!.cant obc di:Jt~:ibul:ionsar.e r~p'J!ced eba-

~]here. Hhat ^CCUl .i'1 hope to find in this collt;)Ction'? First W~3 CCin as!< if there nrc allY seneralitie:J th1t exist, and if so, t-ihy? In th~ !?xallplea that follow on~ particular fOl~ of di3tribution is ubiquitoua.

!nlat might be the ecological significance of this? First we can ask if thene diatr:f.buticns arc 61';:P11 a result of some random stat.\~3tica1pro- cess. If not, then ~/hat are the specif:lc mechaniSm:J \>~hich counterveno

&'1d Iced' to thllse di~tributlon(J? The,l He CC1U ccmp;n-~~ th~;:;e d:f.Gtribut1on9 tdth t~108e of ,~~ological significa.nce. If the 6.:1.-,';1 type3 nppo.:~r, then

we ^l!!.':-y have n clH~~ to an m~'flLmnc!on. If Gon:e otheL" tYFe:3 .Q!,,~ fOlmd in

ani~ll co,-:mun:lt:f.co, ~1e must look fOl' tl1.) spucial n'J,~chani~m;J tlut ma.ke ccolo31ciJl size d:.!.Gtr!butio~1.3 differ'~nt.

fX£:":E

_~ ~.12!~!.;~!) ~\tion3

Size distributions arc cm!!lDonly listed in ^rnr:my diff~renl: f,.7D.ys. In

most CC8es oue r.:.,:;thod can be C~l:3i1y converted :f.nto !moeher. A~ the exarcplco

that follml W~l:C taken directly f~m.'l i:h,~ l:f.terlltUl"C, thore is no lmifornit"}.

The c\:'!ta::l.ls of the transfo;('!'.::;ticns are J.ert to the re~~d':?:rlr, iL13Jin~.tion.

Sh{~ d:f.stl'ibu'i:J.cns relate some ntt1:ib~.-,te of ^t.':J. Cl)j~ct-cJ.;lsa ( or a

phenOiY,-,~rw11) to sGr\;e chu-ractc,dci:ic dht.~:n:;iv;\ of the object:. In most of the ,~;{t-J:ll)lt:'lA, the d:1l::(:11sion 1.8 ~~:{;Pl'C.:;18'>d in 1.m:lts of l(~·ljzth. Tha attriolJto

is colt.t::0111y thi~ m~mbar p~r unit size int(!):vnl:

nCr) dr '" the numb.~r (d.th a oiza (1)

l"ithin th<;l incren.:~nV: r to r+dr.

Other attd.butea are volum,~, t11:J.S3) surfilc~ m:·~D., ete. For eX<l:ii.l'lc, the

volm;v~ dietribut1cn fer Dph';;!"CD tc

vCr)

dr ^m 4/31f

r

³

n{r)

dr, (2)

i.e., the volmnc cvut-ributcd hy the incr.I:::·~·j:\t dt" i!] t:~~~ D.1,!;i'·ilicr ulthin dr

tim~a the V(;ltm1~ of: n ::Jpher€ ,71>:1\ :-:-.:)tUU:3;:.

'I'lw _siz~ pJ).t·f.':')tcr :ts ·cit:v'}: c. l:hj;!:n: dh!':;:'f\;L"l\ (Le. COl.) or the loe:dthten of t\'lQ d:i.L·".~:.ion.

'fhe ~bovc (l:l.st',:~.~iUd.o'\:;ire) ;f.!.'!'.q~1';"·::Y t13.ztd}:,;,;;JJ);.VJ. In '::":Yj of ths c~:;':'c:iV~:J ct[:::ulGt:t.:.:~ d:btl'j\)~,;::i,nn'J :1,;:.~: ~.::;J. '!T.:; c: .. ',~l'~tiy;! 11~,';^..ber

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distribution is N(r)

Note that

a number of objects

_Jr

- n(:r') dr • o

,,71th

a

size cque.l to or ^1c:!:lg thll.'\

r.

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dN(r) 2 nCr)

""d'r-

⁽⁴⁾

The cmnulatlve distribution :l~ also expressed as I,;he numbc~r with 8 s1z~

greater than a size r.

1\;0 major forms of distribution.s uill be illustrated by _thc~ e]cwnpler;

that foHor,T. One is the lognormal di.stribution -- Hh(:re n(log(r» is normally dictributcd about some mean ^VS.lll:? of los(r). ^HOI'C \J:l.J.l he f:aid about thia for specific examples. The other type is the "hyp~rbo1.ic"

distribution:

nCr) ^H kl·-a , (5)

where k and a are constants of th£ distribution. This distribution is characterized by a otraight line on logo-log paper..

As stated, Eq. (5) extends from rc-O lJhe~e there arc an infinite number of infinitely snall particles to rr"'co where t.here B.re no particlee.

To be physi.cally meaningful the distrihution must be truncated at the cnda. ntis tnmcntion can either be chn;:D.cteri~tic.of the nntul:.:I popu- lation of particles or res~tting from the o2thod of Ero~pling.

Theory

Some theoretical intere~t has evolved to explain the r.onniotcncy

of many of the observed dietr:lbutiona. VD.riOU.'i f:!schs.nisll:s have been propo~ed

including coagulation, sedimentation, concknsation, fl·asmcnta.tic,-u, diffu- Giol\, B.nd e.bsorptioo. l)yn£lmic equ:Uibdum be t ...leen cH:t\tion end the differ··

ential rcmoyal mechanisws lead to certain "sel£-'prcservin3" or ctE.ble sir.e distributions. The exact limiting st~ble fOl~ will cle.pcnd uron the particular processes. For further reference see: l'ricdlanGcr, Fcrdan, Cadle, Gaudin ~~d Meloy, GilYarry, Bader, and Mason.

Ooe particular stable form occurs in many situationc. This is (6) that is, a graph of n(r) n ~t/dr has a clope of minus four on log-log paper. 'I1tis form if; (:qutval~nt. to the (ollr.r~vlng:

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dtUd log(r)

vs.

r [!1()F'~! ^t~ -3

N V<'^~. r r:lcr:·· ^In 3

N ve. vol

or

IMGS slop':; ^£;: ₁

dN/dv

va. v

alore

^::''1

-2

cm/dA va. ar.ea slJ~I;:;

., _-2.5

N V3. area slope:; ^"1

3.5

C1,.":2ulative cHotdbuticGiJ (N) rcay lu.we citk'.r. d,gn.

A_ i ....r, .."',~titlr) CO'l<~r:.(lu<>nC(' ('f' thi~' "<;"'·,··:'j·.t"·~on':q tl"'~" U~lf!"~ J,·.",t..)

lUI a~~I\....I-~t.... b 1- ... _ '- ._ J . • ..j '--~~ "-')"~_';o..~ .. , . . . _... .c..t:> .. / '. ' ....·r·.'~

versus 10g(r) iH II CCl1,citE'l1t. 'l'h:1t io, the v(lli.:.n~e (or Lri:.~!) c.:tnl':r·;'J"'.!Uon in any log :incrc"Nmt jn the SEiJ1H",. For fUi.:tLc~t· d:lr,cusr.d.('n sec: r'ric<11£'uder aOld Paaceri, Pc:.sceri fi.wi Fdcdlunc1m:, l.1ncl JFiize.

What fo11o~19 is C'l. selective fH'.mpla of p.?rt1cle S:h:(l d:1.otrihuticllS for ~hYElicvJ_ objectG in the \mivcl:tJe -- st&Z'U.l"!g from.ato;;;:lc diQ.cns:i/J!LS (10- 0 0) to the unive'f.'cc. itself (10 27 11\). 1'1£,: fiEure~: ^follOi-1 tb.~ ^tC):t.

Full-sized f:t.gurc3 are Dvs.:Unble in th,;~ t-Yorkshop n.le.

ATOHIC _P.ADl~ (Figs. 1 and 2)

The Cl',mulat1.ve distd.bution for 43 ~len;2nts aho,is 3 oharp discon- tinuity l l t about 1.2 z 10-10 m. in FlC. 1. F5.cure 2 sh(j~G ^II o:imilar distribution for c~ystalline'ionicradii for 87 elements. An explanation was not found. The discont5.m.u.ty or IIknee" is indici:t1vc of a chsnge in

"process" occurring.

SlU~

PARXICLES (Figs. 3 - 10)

A lot of Dtatiat:l.cfl.l ^\10t'k has b<:'cn done on particlcD 1.n the mi.cron site range. Figure 3 sho-...a typical siz£: c'listd..bllt:f.on of ntmosph~ric

E.i:roBols. Fig\~re 4 applies to stratcspl.er.ic (\01:0;;019. B~low 0 •. 1 t::f.c.ron Farticles nre eli~inatcdby coagulatiou du~ to Dro~lniGnnation. 1110

lo.._~er limit of Datural aerosols is nbout 5 x 10-3 microns. Above 20 mict'ons particles are rapidly eliminated by gtsvi.tetionnl settling. FiG\1rC9 5

llnd 6 are the frequency and cum111ativ~ c1ietr:i.but!ons of particl.c:s in urban air.

Frequ~ntly atmosphcdc part:l.clcs c.ze plotted on loznoroa1 peper (Pi£El. 7 snd S). Lo6nOLiiJ31 ^ll.nd log-log CtlmLl1stiv~ pIou; "i.e not c~.[;:lly

related. hcn.!CVel", the frequency plots are qu"H.taU vnly r.irrJ.lar. If n lC".11oIi.::al <11£trlbUtiC·tl :1r. p!Qt ted en Fig. 3~ the rMult $.!3 2. pnrcllo1n l:M.C1¹ :l$ d;·JJ.~l:' ~.n for'L'1 to th~ natural. cJctrUmt';ton~. 1.1'J0 cote th.,t in C'~l of the plots of

ric:.

^C, ^the

a"ta

^pointG cLwjat~ frv~~ tho: ~tH..ir.ht

line in a Sj:::tc(utic l!t':~::;hGpc~d" n::aoner. '11118 form :1..[; found uhcn c. trL4"1- cntcd c1iatrtbutlon, lil:e Eq. (5), is plotted on lo[;uotQ.al fJU!Jer.

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Fir;ure 9 dw~;s the d;I.Btributioil chcu= th2 d.2:0 d:lr;tribl1tLt):t\ rumHiuG

(A'l^L "',"'-:1\'n1^~_^.••""-_:.-r,. p--",l-,'-')")nt\,Ji'~'•..._,,~ ... ,. QJ,r.<;.::.,....""""o-yi;:. to tl~-'l'"1& • • 0

ecolt)gf~::tl;l.)

of: f:1_oe Ar:1.~~onn road nUGt. Fie\\re 10 fr,.m the f:I.':tCtu4C~ of br.:1.tt:le solidr..

c~r;"-D!pl~ shculd hi! fan~.U:ln£' to all

Ill-CJ.c.lUd x·[;J.i.1. dl:'oplM:o ;:rc chom1 ~I.n

:eig.

11; th~ form p:rf'cC'n~ t.o be eXl'onent1.al rat.hl"!:r tb:m h;tperl:cJJ.c (:l.t ir: a linear 81:;;:"", Gcnlp..). Figure lIn is a fr,,"qw~ncy d:l.sti"ibution. Fi(:urc 12 appHcG to falling dropn.

A sCiJilp1e for suspended silt in a D.C. streen 1.G choml on ~ lognormal plot in Fig. 13. Sand and gr~""el sizes nrc Gho,m in F:lg. 14. (l'loto courtesy of Art Tautz)

LlJNAR SURFliCE l[ATED.Il~LS

l1eloy and 0 'Kcefe discus£! the early Snrveyor I photoeTc:.phs '-'here the visible pC!'I:icles v.cre governed by the equutioil

N(r) '" Nor-m

,,11th III "" 1.71.

HETEO~(Fig. 15)

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The size distr:f.bution of extraterreatinl influx (at a rete of 100-1000 tooa/day) is Sh01ID in Figure 15. The Dlope of the lllrr;C p8T.ticlno is about -1.0. Hhen thia :f.a converted from a cumulative mans distributic.n to

&l/dr_t the slope becomes -4.0 again.

ASTEROIDS

(Fig.

16)

The size dif;tdbution of astero1.dG appears to be in D. tr~nn!tion from

tID or:1.sina1 gaussian to a h}~re.rbCllic ^fO]:i'l1 (F1C. 16).

T1H~ {U-str1huUon of pl,,~J~t,c; fmd r,3teJJ.itCf.: in the r.:o1.:'r cystr.:<l do r:ot fit a hypc>:l,ol:l.c fcnm. Both group::; combine into a un:J.fotlJ co] It·ct1cm,

however.

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-5-

co~mTS

(Fig. 18)

The size distribution of comets was not easy to find. Instoad, the distr5.hut:Jon of orbit sizes is chotm in Fig. 18. The discontinu:i.ty at 1 n.v. (earth-sun _di6t~ncc) is nttributed to n reduced probahility of observation for large. orbits.

The visual magnitude of sttrra in the solar neighborhood 1.s 8ho';:\"1 in Fig.

19.

The magnitude (a line-at' scale) is related to the log of the radius.

PLANETAr'YJ~nULAE(Fig. 20)

Planetary nebulae are shelle of ejected 8as expanding about c(';rtain extremely hot sters. ?ne data corrcapond to our galaxy only. The size measure is the angular di;meter, \)hich io a function of tt'uc diai"\,<;tcr and

distance

from

earth.

GALAXIEE.-(171g. 1.9)

The visual mSBnitude of galaxies is related to the cl.-c:\ulativc frequency by

log N ^A O.5m-7.2

Th.1.s

equation

is plotted on Fig. 19.

THE COSNOS (Fig. 21)

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This Figure is redrawn fl.·om an intriguins paper by Al.bert l·lih:(\n on the hierarchical structure of the Universe. The regiol1~ of £lizes E::nd L~aS3es

of various ltpa.l."ticles'l are shown. The gaps in the 5:1ze raTlzes axe uppa-

r~ntly very real and numerically related to variol~ fundlli~entalatcruc.

electromagnetic, and gravitational parameters.

II. _geographical _Featt\~es LAKES (F'ig. 22)

The a:r~as of natural fr£':zh--water l~kes are ~hO'.m in FiC. 22. l~or

lakes in the U. S. the dis tribo..\ Uon ic hypt:rb oUe if tt~ Gte at I,::',~':.e0 lU:-C

excluded. The lCtkC8 of the \<mrlcl have E'.I1 upper size cut-off bCe<::llicc of the limitation imposed by contiu~nt size.

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RIVERS (FJg. 23)

The l.J:::tjor r:!.vct"s of the Hor}d ShOH a fdJdlnr limitation at the upper end.

The cm::";uJ.<'ltive size d:tstribut:ton of oaria n:oon craters is $hO~1.l :i.n Figure 24. The distr:Lbutions of r,'OOll Cf;:;ters end I::..eteori.tcw are BUpCl:-

in:posed in Hg. 25. HM:S craters cf;timated fn,m early l'J.3r.).n(',f flights are in Fig. 26.

ATMOSPHERIC Ttm.r:tJLENCE (Fig. 27)

AtmlJsphel'ic turbulence can be resolved f.nto distributions of variou!3 sized "eddies ". Tt\"o exc-,1,ples 1o,: elccr (lil' tttrbulence are shmm in Pir.. 27.

ATO~'fIC ABUNJ)t~HCES (Fig. 28)

The abundl.tnce of the elements in the earth' c crus t: if: 6ho~m in 1"1e. 20.

The "sizc" in this ClWC is the extent of concentration in parts per ltd,llion.

• Social Phenomena

CITY SIZES (Fig. 29)

The cumulntivc distribution of tT.S. cities in 1910 s1¹d 1960 is sho'im in Fig. 29.

COLO~~ES

(Figs. 30 and 31)

The distribution of mealy-bugs in coloni-en 1.9 Shcrwl1 in Fiy,. 3D. The relationship is not clear from the gl·aph. ~JH1i~6 conch,fled thrlt the colonies were distributed normally on log3 1ntel~·als. Toc popul~tion aixc of pods on cocoa tre.es is 6ho\!,tl1 :tn Fig. 31 -- the Ghnpe :to eiIil.l1ar to that of lo'ig. 30.

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.

A'OI-liIC 'RADII ("13 r-'\ttl'.u.:"£O CL.!"1£AlT~)

FI(j i ^Fllj

2

101,\ Ie C RY5TAL RAol1 .

r (p.)

..

10-1 r ^IO-i

(p.)

N (2'{')

\0 100

10'S-I-,- - - r - - - r - -

1 100

10 - \ - - - , . - - -.S' ..,

, + - - - j - - - 1 o~

~ ~

-

_- !'.

..

_::: _!: _!: _~

"I"I.~,~^p

r--/;;.J~-- ...--...--.

: U--+--+--+---i---t---

I ^r ^I

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'0!---=-.+-'''',----.->..,,_--t-

"

, ^,

".

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...

-

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99.9')

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blJ:·nc'lm.,1 pilI' ort~l'ic;.rl di11riuullIllU indt.lnall~tp"tllltl"llair. {t"I'IIl;

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"rd·7.'3'·"

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spheriC'3croli.lls. (-IH\"l.' 1,\i"r"Hthc !lnler,,:r.lto~

5ph~r(and IJ)(II(:rlIHlld.:.;Jbo ..~ ~t)I...m Cu,\'C'i

1and 3 OJrc C'ltirn.Jtcul.'lllllld~'IH':CIlllll!('" (C()lfrrt"~y

Dj''''.A."U',inur,\I~·lf'nrl;/')_r./..rJlS()('h'ty) [£PCOt:x."l'O ~.=.., ('Ilt>~£.

FlO. 6.JnlcZrJ.ltdSil~~rrcq'JCncydiHri1>ul:O.:l0('.(;:jJS'I;pe",)

fCOo-\ ~U'''', Ct,<>.l.

~izC' tfi~lriluJli"n n( ronl' ,\'U'll,", rc.:'lol rllI ..l. Tlli ...1I1\1.~frt1'-' 1h"l tilt"rOUlll;'l~{ r~.;(~:1o ~~"":lll,.-nl'::, l:lIh.li,,:.;l.~· Ir,IUi,', :I1:d llln,l :lrtioll '':Inl,ro.."I"~:Ill id,·;tl1,>r·_rl....':Ir:

dl~,nt.u•• on,

FCDO'\ el\oce.

" ft ..n

Flq 8

-"~-'--'---rT--r--r-r---.

tintitln",ti.O~hO

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, • ..L--L_l._J_l 1-1.1.LL_I_J_L

I I: • l' ~ • • '" " n M " " .. "

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X., "

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""011.'tlnt Il,,~I

(O<:O(v:(f)n:"""" Lr;,,-

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