SE! F - C ON SISTEN T CA! C!! ATION O F T! E ! ISSOCIATION O F O! Y G EN IN T! E PART II! T! REE - BO! Y REC O M BIN ATIO N S. b y ! ARRY E.

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2 NEW Y O R!! NI! ERSITY W AS! ING TO N SQ! ARE CO!! EG E O F ARTS AN! SCIENCE S M AT! EM ATICS RE SEARC! G RO! P RESEARC! RE PO RT N o. C! -8 fi b c cc x x * SE! F - C ON SISTEN T CA! C!! ATION O F T! E! ISSOCIATION O F O! Y G EN IN T! E! PPER ATM OSP! ERE PART II! T! REE - BO! Y REC O M BIN ATIO N S b y! ARRY E. M O SES and TA - Y O! CO NTRACT NO. AF - l9 ( l22) - 463

4 New York! nivers i ty Washington S q uare C o l l e g e of Arts a nd Science Mathemat ics Resea rch Gr Res earch Repo rt no. A Self- Cons isten t Ca lcul a t ion of the! is socia t ion o f gen in the! pper Atmosphere Part I I! Three-Body Recomb in a tions by! arry E. Moses ins Proj ec t! irect or research in this document been made poss ible through The reported has spons orship by Geo hys rch ivis ion support and extended the p ics Resea! o A ir rce Cambr idge Res rch Cent under Contract f the Fo ea er. No. AF - 19 (1 2 2) 1-4 is published or techni l info t ion only does 63 It f ca rma.. and no necess ly represen ti ons conclus ions the spon t ari t recommenda o r of April. 1953

6 Iabl e of Conten t s I. Introduct ion The diss oci at i on reg ion of oxygen molecules is trea ted in the same "self- cons is tent m anner des cr ibed in a prev ious p a p erl z. ex cept that now the dens i t ies of atomic a nd molecular oxygen have been calcul ate d a s functions of the alt i tude. on the assumpt ion that the ma in recombination proce ss between ox y gen a t oms is a three- body non ra d iat ive process. The result of the present work ind icates that diss oc iat ion occurs a t a n al t itude about 5! m higher than t h e corre s pond ing alt i tude of r e ference 1. Furthermore, the region of d issoc i at ion is broad er than in the model of reference 1. Th e maximum a tomic dens ity 1 is l x cm 3.

8 I. Introduction In two previous p a p e rs l ' 2 the dens i ties of a tomic a nd mo lecular ox y gen in the upper a tmosphere were ca lculated in terms o f two different models. Both of these models w ere carried out in a "self- cons ist ent m anner, that is, no a pr iori assumpt i ons were made a s to the distribution of mo lecular or at om ic ox y gen. ( S uch assumptions have been made by previous authors in the fiel d, a s wa s pointed out in references 1 a nd In the firs t model it wa s p oss ible to calculate the distribution o f temp era ture a s wel l a s the dis tributi on of the atomic a nd the molecular ox y gen dens ities. In the second model, a temperature dis tribut ion wa s assumed in order to avo id the necess ity of imposing a severe assumption, used in the fi rs t model, with respect to energy balance. In. both of these m odels the pr inc ipal recombinati on process wa s assumed to b e a two- b ody ra diation process. Th e prese nt paper is to be regarded a s a direc t ex tens ion of refere nce 2 ; the ass umption of two- body recomb inat ions is replaced by the assump tion tha t the ox y gen a toms recomb ine in a thre e body process. Therefore ins tead o f repea ting the var ious details of the mo del of reference 2 a nd of the metho d of solut i on of the e q uat ions for the dens i ties, we shal l mere ly indicate the mo dif ications. Bo II. The Three- d y R ecomb tion Process ina Th princ ipal icat ion of reference which we intro duce in the e modif 2 wo rk is to replace body recomb in t ion process there present t h e two- a (1 1 ) the bo t ion process rate of r ecomb t ion b y d three- y recomb ina The ina of t h e ox y gen a toms is t aken to be Cn M n g where C is the three- bod y re comb ination coe ffic ient a nd n M is the number dens i ty of the third body M.! arious add i t ional assumptions a re now necess ary, s ince the recomb in a tion coefficient C a nd the lumber dens i t y n M a re not known. F irs t of a ll

10 2 we s hall assume tha t a l l atoms or molecules a re e h u a l l y e ff icient a s third b odi es. Once one makes this assumption i t is clear tha t it is suffic ient to take n M a s the number dens i ty o f n i trogen mo lecules, s ince n i trogen predominates s o gre atly in the reg ion of interes t. W e further assume tha t it obeys its own ba rome tric e q uation of the usual e x ponential type when the t emperature is taken a s cons tant. W e have prescribed that j us t below the al titude a t which dissoc i a ti on bec omes important, the number dens ity o f nitrogen molecules will be four t imes that of ox y gen, that is, the oxygen n it ro g en ra t io is the same a s a t s ea - leve l. have taken two values namely F 32 o r C we, C If)" a x - nd C These a re typ ica l values given by che mis ts fo r three- body processes a nd used by previ ous workers on the present problem (see e. g. re ference Th e fo rmulas and me thod of proce dure a re subs tantia lly the s ame a s those of re ference 2. We need mere ly replace the two- b od y recomb ina tion c oefficient B ' by B! whe re n M is a func tion of the a ltitude a nd is given by the barometric e qu a t ion a s described ab ove. (Inc i den tally there is a mispr int in reference 2. The let te r B in e q uat ions (15 ) throug h (1 7 ) shoul d be replaced by III. Results The results of the calcul a t ion a re summari ze d in Figure l. The temper ture taken a wa s a s 30 0 tude!. The a lt i x. o wa s chosen to correspond to the value n2 2 x l O l B / cm. 3. Only above this a l t itude does d issoc iat i on be come im p ortant. If we use the pressure vs. a lt i tude results of reference b, we f ind that x 0 is about 90! n; The graphs of n 2 a nd u p a re given in Fi g ure 1. W e ha v e als o superposed the graph of n 2 a s it would be if no d issoc i a tion took place the values a re ob ta ined from the ordinary baro me tric e q uation.

12 opposed to l ess than 15! m. of referenc e 2. sect ions is ava ilable ; then it w ill be c lear which model should be used. which a re available a t present m us t be accepte d only with great reserve. - Q! It is seen that in the presen t mode l dissoc ia tion occurs a t higher alt i tudes (about 5! m. higher ) than in the model of reference 2. This result is a conse q uenc e of the us e in this paper of a greater va lue for the effec ive bo dy recomb in ation section ence in t Cn two- cross-.! the M present mo del the ox gen toms recomb more ly than in the y a ca n ine rea di model of referenc on ly higher alt itudes issoc iat ions e 2 a nd a t a re d numerous eno gh to ensure number Another str difference u iki balance. n g between the present results those o reference is the broadness a nd f 2 of the region in which dissociation takes p lace about 25! m., a s These differences between the results o f the present paper a nd thos e of reference 2 ( a nd reference 1 ) indicate the high degree of sens itiv it y of the results to the model used. W e mus t conc lude that the problem of obta ining rea lly useful numeric al results for the dissociation of oxygen cannot be solved unt i l much more reliab le data on the var ious cross This result, tho u gh no t surpris i ng, indicates that a ny numerical results

16 References!. E. Moses a nd T. Y. W u, Phys. Rev. Q!, !. E. Moses a nd T. Y. W u, Phys. Rev R. Pe nndorf,! our. Ge op h y s. Res (19h9), Th e Rocker Panel, Phys. R e v. 88,

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A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N