Sodium equilibrium altitude distribution, origin, and flux calculated for earth atmospher

Ghosh, S. N.

Mitra, V.

English

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T H E  UNIVERSITY OF MICHIGAN 
C O L L E G E  OF ENGINEERING 
D e p a r t m e n t  of Aerospace  Eng inee r ing  
High  Alt i tude Engineer ing  Laboratory 
Scient  if ic R e p o r t  
ALTITUDE DISTRIBUTION, ORIGIN AND F L U X  
OF SODIUM IN THE ATMOSPHERE 
S. N. Ghosh and  Ved M i t r a  
ORA P r o j e c t  05627 
u n d e r  cont rac t  with: 
NATIONAL AERONAUTICS AND S P A C E  ADMINISTRATION 
CONTRACT NO.  NASr -54(05) 
WASHINGTON, D. C.  
a d m i n i  s t  e r  e d  through:  
O F F I C E  OF RESEARCH ADMINISTRATION ANN ARBOR 
M a r c h  1968 
https://ntrs.nasa.gov/search.jsp?R=19680012172 2020-03-12T08:29:50+00:00Z
FIJXEDiNG PAGE BLANK NOT, FILMEDc 
Table of Contents 
List of F igu res  
Abstract  
1. Introduction 
2. Equilibrium P r o c e s s e s  Involving Sodium. 
3. Equilibrium Equations. 
4. 
References 
Computation of So and Altitude Distribution of Na. 
iii 
Page  
V 
vii 
1 
2 
3 
7 
11 
List of F igures  
Page 
Figure 
1. 
+ 
Calculated diffusion coefficients of Na, Na, NaO, and N a 0 2  
as a function of height. 1 3  
2. The height distribution of N a  atoms obtained from Eqn. (9). 14 
V 
PRECEDING PAGE BLANC NOT FICME3. 
Abstract 
Taking the maximum density of Na to  be 900 a toms/cc  at  93  km, 
the equilibrium altitude distribution is calculated. It is assumed that equi- 
l ibrium is f i r s t  reached among the fast  reactions,ionization, recombination 
and oxidation of Na atoms. 
minor constituents in the atmosphere, then diffuse and a final equilibrium is 
reached under the joint effect of chemical equilibrium and diffusion. 
shown that the origin of N a  atoms is ex t r a t e r r e s t r i a l  and that approximately 
554 Na atoms a r e  deposited per sq. cm. per sec  in the atmosphere.  The 
shape of the altitude distribution curve resembles  fa i r ly  well with those 
obtained experimentally. 
Sodium a toms and their  compounds, which a r e  
It is 
vii 
1. Introduction 
Certain metallic atoms and ions have been observed in the 
atmosphere.  Rocket -borne mass  spectrometers  showed the presence of 
Na, Mg, Ca and probably F e  (Istomin, 1963; Narcisi  and Bailey, 1965). 
Again, spectroscopic studies of the night and twilight airglows have revealed 
the presence of Na, K, Li atoms and Ca ions (Sullivan and Hunten, 1964; 
Delannoy and Weill, 1958). The twilight airglow of Na, which is compara-  
tively r icher  in the atmosphere,  has been extensively studied. 
+ + +  + 
+ 
The origin of sodium in the atmosphere is a subject of contro- 
versy.  (Junge et al, 1962) These atoms may originate f rom the earthside,  
that is ,  f rom sea  salt ,  volcanic o r  continental dusts, o r  f rom outside the 
ear th  by meteors  or interplanetary or inters te l lar  dust o r  mater ia l  emanated 
f rom the sun. 
Attempts have been made to  explain the variations in the sodium 
resonance line intensity in the day and night airglows and also in the morning 
and evening twilights. 
of f r ee  sodium atoms during night, the effect of recombination of Na  ions with 
negative 0- ions is compensated and hence there  is only a slight change in  the 
intensity of twilight N a  line from the evening to  the morning. The re  can, 
however, be a large seasonal variation in  i t s  intensity. According to  Gadsden, 
the chemical reactions of N a  atoms with atmospheric  oxygen play an important 
role  in the f r ee  sodium atom abundance and height distribution. Fur thermore ,  
the height distributions of a l l  metallic a toms in the atmosphere reveal  maxima 
in  the 90-100 km altitude range (Jones,  1966),  which is also the region of maxi- 
mum concentration of atomic oxygen. 
exist some so r t  of a correlation among them. 
Gadsden (1964) suggests that because of the oxidation 
+ 
This indicates the possibility that there  
1 
The metallic atoms, af ter  reacting with the atmospheric oxygen, 
F o r  estimating their  relative abundances and give rise to  various products. 
height distributions, various processes  which lead to equilibrium among them 
must be considered. 
In this paper, the equilibrium concentration of N a  a toms under 
the joint effect of various processes  has  been obtained. 
gives the r a t e  at  which N a  a toms are deposited in the atmosphere and their  
origin ( te r res t r ia l  or extraterrestr ia l ) .  
The analysis fur ther  
2 .  Equilibrium Processes  Involving Sodium 
F o r  determining the equilibrium processes ,  the following chemical 
reactions between sodium and oxygen a r e  considered: 
K 
I 
Na + O2 + M A ' N a 0 2  + M 
N a + O  + M-NaO+ M K2 
1 1  
N a 0 2  + 0 
NaO + 0 
In addition, the photoionization of Na atoms and the recombination 
+ 
of Na ions with 0 - i o n s  (Bates and Boyd, 1956) should be considered, namely 
Q l ,  N a  + + e N a  + hv 
1 / I  llf 
where K1, K2, K, K, Ql, Q2 are the ra te  coefficients of reaction. 
+ 
The equilibrium concentrations of Na, Na02, rJa0 and Na f o r  an  
I 1 1  1 1 1  
average atmosphere in a layer denoted by n, n, n and n'respectively,  a r e  
produced by the joint effect of reactions (1) and ( 2 )  and the diffusion of various 
sodium products. The processes  which lead to  the equilibrium of these con- 
stituents in  any layer may be represented by the following scheme. 
2 
_ _  I 
- n Q2K1n(0) n:
N i  
n' < Na -Na 
(3) K2n n n ( 0 )  
where n ( 0  ), n(O), do), and n a re  the concentrations of 02, 0, 0-  and the 
total  concentration in a layer of the atmosphere; a and a' a r e  the rates at 
which Na atoms diffuse __ out and into the adajacent layers  (that is, a is the 
number of N a  a toms which enter  the given layer  per  second per  unit a r e a  
f rom the adjacent lower layer,  and, a, those which leave the given layer to  
the adjacent upper layer).  Similarly (b  , b), (c , c) and (d , d) are the dif- 
fusion rates of Na02, NaO and N& respectively. 
2 
1 
-- 
I 1 1 
3. Equilibrium Equations 
Suppose that f rom a certain external agency (e.  g. ocean o r  
meteorites) Na atoms a r e  continuously deposited in the atmosphere and let  
S 
per  second in  a layer. 
bined products of Na should be constant, that is, 
denote the average flux o r  the number of atoms deposited per  unit a r e a  
In the steady state of diffusion, the net flux of com- 
0 
1 I 1 I 
a + b + c + d  = a + b  + C  + d  = S 
0 
o r  a + b + c + d - S o  = 0 (4) . 
Assume that the direction z pointed upward is positive, then if So  is positive, 
the flux is in  the upward direction, that i s ,  N a  atoms originate f rom the ear th-  
side. If, on the other hand, So  is negative, the origin of these atoms is 
ex t ra te r res t r ia l .  
Al l  the resulting sodium products (Na, Na02,  NaO and Na)  involve only a 
single N a  atom and this validates Eq. (4). 
While in  motion, Na atoms undergo reactions (1) and (2) .  
+ .  
3 
I I 1  I l l  
If w , w , w and be the upward vertical  diffusion velocities 
+ 
of Na, NaOZ, NaO and N a  at  an  altitude z, then 
I I  
a = n u  
b = n w  
c = n w  
I #  I I  
t i t  I l l  
- _  
d = n1 uI 
Eq. (4) then becomes 
I t  I I  I 1  r i i  I I I  - - 
n w + n w  f n  w + n ' w ' - S  = 0 
0 
The vertical diffusion velocity w of any minor constituent of con- 
centration n through a stable main atmosphere at  an altitude z is given by 
- 1 
where 
D - coefficient of diffusion 
m '  - molecular m a s s  of the diffusing constituen 
m - mean molecular m a s s  of the main gas  
H - scale height at  an altitude z 
/3 - gradient of the scale height at  z 
d - thermal diffusion factor.  
The vertical diffusion velocity w can be negative, ze ro  o r  positive. 
If the constituent concentration decreases  slowly with altitude, the second t e r m  
in the above equation outweighs the f i r s t  and the t ransport  is in the downward 
direction. If, on the otherhand, the constituent concentration falls off rapidly 
with altitude, the first t e r m  predominates and the t ransport  is upward. 
tuting the valuesof w', w ' l ,  w l l l  and into Eqn. (5) we have 
Substi- 
4 
I I  I 1 1 1  I I I  I I  I I  1 1  
I m '  I I  D n  + o ( + - ) +  - 
m H m m 
+ 2(/3 p H 
+ In the expression for  2 ,  as N a  - electron plasma diffuses through 
a neutral  gas, ambipolar diffcsion coefficient (which is double of the diffusion 
+ 
coefficient of Na) is used (Ratcliffe and Weeks, 1960). Also, for  the mass  of 
Na, the mean molecular m a s s  of ion-electron i. e. G'/2 is used. Since below + 
the F layer  of the atmosphere, an ion suffers  collisions mainly with the neutral 
particles,  the diffusion coefficient of N a  can be calculated using the same + 
formula as fo r  the neutral  particles. 
The continuity equation during the steady s ta te  of diffusion for  all 
sodium products is given by 
Thus, in  the steady state of diffusion, while the combined concentration of Na, 
N a 0 2 ,  NaO and N a  remains constant with t ime in  a layer ,  the concentration of 
an  individual constituent may change. Therefore, besides the overall  steady 
+ 
state diffusion established f rom layer to layer  af ter  a long period of time 
by the diffusion process  (diffusion is slow in  the 80-120 km region of the 
present  study), local equilibrium among the various products in  a layer  can be 
established independently by rapid processes .  The equilibrium concentration 
of the various sodium products can be obtained in any layer  by considering 
Ki .  n! n. n (02)  + Kb . nl. n. n(0)  + Qlnl = K"'n1I1n(O) + Q 2 n! n(a )  
and KI1. nl. n. do2) = K". n". n (0 )  
5 
+ 
Equating N a  ion production and loss r a t e s  we have 
- 
Qlnl = Q2n1 n(O) 
The concentrations of the various sodium products in  any layer  
can, therefore,  be written in t e r m s  of the concentration of N a  a toms in that 
layer ,  thus 
nil = n '  
n l l l  = nI 
I 
K1. n. n(0,) 
K t l .  n (0)  
I 
+ -  
K1. n. n(0,) 
K"'. n (0)  K 
Under the joint effect of chemical equilibrium and steady diffusion, the 
+ constituents Na, Na02, NaO and N a  satisfy Eq. (8) and independently 
Eq. (6 ) .  Substituting the values of nl' and n'" and in t e r m s  of nr and 
neglecting thermal diffusion, we have f rom Eq. (6 )  
6 
To eliminate the uncertainty involved in the 0- distribution, 
it is assumed that 
is obtained f rom the above equation. 
= n. The maximum concentration of sodium, nlmaX, 
4. Computation of So  and Altitude Distribution of Na 
F r o m  the observed maximum concentration of sodium and i t s  
altitude, S can be determined from Eq. (10) provided other quantities are 
known. So, thus obtained, can be  used in the differential Eq. (9) for  com- 
puting the height distribution of Na atoms in the atmosphere: 
distributions can then be calculated f rom Na atom concentrations by using 
Eqns. (8a) and (8b). 
0 
NaO and NaOZ 
3 
Values 
At Saskatoon, the average value of n' 
and occurs  at  an  altitude of about 93  km (Sullivan and Hunten, 1964). 
of n, n(02) ,  n(O), m, T, H, a n d p  at  different altitudes are obtained at  an  
interval  of 1 km f rom the Cospar International Reference Atmosphere (Champion, 
1965). 
value of n' 
latitude. 
is about 900 a toms /cm max 
A l l  the atmospheric data belong to an average atmosphere.  Since the 
belongs to Saskatoon (52ON), a l l  other data should belong to this 
max 
In the absence of such latitudewise data, the average atmospheric 
data a r e  used. 
7 
In the absence of known values of r a t e  coefficients of reactions 
between sodium and oxygen, values corresponding to reactions between 
hydrogen and sodium a r e  adopted a s  was done by Blamont and Donahue (1964) .  
Recently Hunten (1967) has applied corrections to  the ra te  coefficients of 
sodium-hydrogen reactions to give best guesses  f o r  the corresponding sodium- 
oxygen reactions. H i s  values a r e  
K1 = 2x10 cm sec 
K2  = 7x10 cm sec 
K" >/ 1 0 - l l c m  sec 
K = 4x10 cm sec 
I - 3 3  6 - 1  
I - 3 3  6 - 1  
3 -1 
I l l  -11 3 -1 
Treating molecules a s  rigid elastic spheres ,  their  diffusion 
coefficients can be calculated by the following formula 
8 n y 2  L  IT m1m2 J 
where 
6 =  -1+ 62 
12 n 
L 
and 
cl, 7 - molecular diameters  of the minor and main gas  respectively 
K - Boltzmann constant 
T - absolute temperature  
ml ,  m2-  m a s s e s  of the two kinds of molecules 
n - molecular density of the main gas.  
8 
It has  been shown (Golomb and Mac Leod, 1966) that the values 
of the diffusion coefficients estimated by using the above formula a r e  in 
good agreement with those measured f rom the radial  growth of chemilu- 
minous t r a i l s  deposited in the atmosphere. 
of the collision diameters  of atoms, molecules and ions and their  s t ruc tures  
and s izes  are obtained from the knowledge of their  chemical bonds and val-  
encies. The diameter of the sphere which circumscribes  their  chemically 
determined s t ructure  has  been tentatively taken to be equal to their  collision 
diameter.  
In the absence of known values 
t The radii  of Na and Na a r e  respectively 1 .57A and .95A.  The 
0 -0 bond distance in the N a 0 2  molecule has  been reported to be 1 . 2 8  f . 01A 
(Paul ing,  1960). 2p ) has two unpaired p-electrons 
which are orientated at right angles and therefore one may expect the Na -0-0 
angle to  be nearly 90° (Slater,  1931) and hence the approximate shape of N a 0 2  
molecule can be obtained by drawing two spheres  ( 0 molecules) each of radi i  
1 . 4 A  with a distance 1. 28A between their  centers,  one of which is touched in a 
perpendicular direction by another sphere (Na molecule) of radius 0.95A. 
a molecule wi l l  be circumscribed by a sphere of radius 2. 54A. 
collision diameter of N a 0 2  molecule may be taken equal to 5 .08A.  
A s  all the alkali metals form some linkage, N a - 0  bond distance 
2 4  The oxygen atom (1s2 2 s  
- -  
+ Such 
Therefore,  the 
+ 
in  NaO molecule may be taken equal to  the sum of the radii  of N a  and 0- ions 
which are respectively 0.95A and 0 .81A.  
scribing such a molecule which may be taken as i t s  collisional diameter, is 
The daimeter of the sphere c i rcum-  
therefore  equal to  3. 52A. 
Mean collisional diameters  of the air molecules at  different 
altitudes between 85 and 120 km a r e  obtained by taking collisional diameters  
of N2,  O2 and 0 equal to  3. 15A, 2 .98A and 1 .49A respectively. 
9 
+ 
The diffusion coefficients of Na,  NaO, N a 0 2  and Na calculated 
with the help of the above values between the altitude range 85-120 Jsm are  
plotted in  Figure 1. 
-2 -1 s e c  
Thus it is concluded that the origin of N a  is ex t ra te r res t r ia l .  The source  
may be meteorites,  inters te l lar  dust o r  so la r  mater ia l  and approximately 
554 Na atoms a r e  deposited per  sq. cm. per  sec  in the atmosphere.  The 
value of S 
values of the atmospheric data and other constants used in  the above corn.- 
putation. 
changed by the revised values because the positive t e r m s  in Eq. (10)  are  
severa l  o rde r s  smaller  than the negative t e rms .  
F r o m  Eq. ( l o ) ,  the value of So  comes out to be -554 c m  . 
is however, subjected to a revision by more  accurate  and exact 
0 
It may, however, be noted that the sign of So  is not likely to be 
Using the above value of So,  the altitude distribution of N a  a toms 
is obtained with the help of Eq. (9 )  by using an IBM 7090 computer. 
tribution curve (Fig.  2)  resembles  fairly well those obtained experimentally 
( Sullivan and Hunt en, 19 6 4). 
The d is -  
10 
-. 
References 
Bates, D. R . ,  and H. S. W. Massey, The Basic Reactions in the Upper 
Atmosphere I, P roc .  Roy. SOC., A187, 261-296, ( i 946) .  
Bates, D. R. and M. J. Seaton, Theoretical Considerations Regarding 
the Formation of the Ionized Layers,  Proc .  Phys. SOC. , B 3 ,  129-140,  
(1950).  
-
Bates,  D. R., and T.J.M. Boyd, Inelastic Heavy Part ic le  Collisions Involving 
the Crossing of Potential Energy Curves IV. Ionic Recombination, 
P roc .  Phys.  SOC., A69, - 910-916,  (1956).  
Balmont, J. E . ,  and T. M. Donahue, Sodium Dayglow: Observation and 
Interpretation of a Large Diurnal Variation, J. Geophys. Res . ,  69, 
4093-4126,  (1964) .  
-
Champion, K. S. W . ,  A i r  Fo rce  Cambridge Laboratories Report AFCRL-65-443 
(Mean Atmospheric Propert ies  in the Range 30-300 Km), 1965. 
Champion, K. S. W . ,  A i r  Fo rce  Cambridge Laboratories Report AFCRL-66 -564 
(U.  S. Supplementary Atmosphere), 1966. 
Chapman, S. , and T. G. Cowling, The Mathematical Theory of Non-Uniform 
Gases,  Cambridge University Press , p. 245, (1952).  
/ 
Delannoy, J . ,  and G. Wei l l ,  Observation d'une nouvelle raie d'emission 
crepusculaire atmosphe'rique, Compt. Repd. , 2 47, 806-807, (1958). - -  
Gadsden, M., On the Twilight Sodium Emission - 2. A Theoretical  Model of 
Sodium Abundance, Ann. de Geophys. 20, 383 -396, (1964) .  
Golomb, D.,  and M. A.  MacLeod,, Diffusion Coefficients in the Upper Atmosphere 
f rom Chemiluminous Tra i l s ,  J. Geophys. Res . ,  - 71, 2299-2304, (1966) .  
Hunten, D. M., Spectroscopic Studies of the Twilight Airglow, Space. Sci. Rev. , 
- 6, 493-573, (1967).  
Istomin, V. G.,  Ions of Ext ra -Ter res t r ia l  Origin in the Ea r th ' s  Ionosphere, 
Planet, Space Sci. ,  11, 173-181,  (1963).  
Jones,  V. A , ,  Abundance of Metallic Atoms in the Atmosphere, Ann. de Geophys., 
- 22, 189 -197, (1966).  
Junge, C. E. ,  0. Oldenberg and J. T. Wasson, On the Origin of the Sodium 
Presen t  in the Upper Atmosphere, J. Geophys. Res. , - 67, 1027-1039, 
(1962).  
Mange, P., Diffusion P rocesses  in the Thermosphere, Ann. de Geophys. , - 11,
1 5 3  -168, (1955).  
11 
I 
Narcisi ,  R. S . ,  and A. D. Bailey, Mass Spectrometric Measurements 
of Positive Ions at Altitudes f r o m  64 to  1 1 2  Km, J. Geophys. Res . ,  
- 70, 3687 -3700 (1965).  
Pauling, L. , The nature of the Chemical Bond, Cornel1 University Press, 
Ithaca, p. 352 (1960).  
Ratcliffe, J. A . ,  and K. Weekes, Physics of the Upper Atmosphere, Ed. by 
J. A. Ratcliffe, Academic Press, New York and London, p. 391, (1960) .  
Slater, J. C.,  Directed Valence in Polyatomic Molecules, Phys.  Rev.,  - 37,
481-489, (1931).  
Sullivan, H. M . ,  and D. M. Hunten, Lithium, Sodium, and Potassium in 
the Twilight Airglow, Can. J. Phys . ,  - 42, 937 -956, (1964) .  
1 2  
DIFFUSION COEFFICIENT (xIO-~CM* SEC-’ ) 
Figure Captions 
Fig. 1 Calculated diffusion coefficients of Na, Ni ,  NaO, and NaOa 
as a function of height. 
125 
120 
I I5 
I IO 
105 
I oc 
95 
9c 
85 
8C 
75 
7c 
Sodium Number Density 
I I I 1 I I I I I 
100 200 300 400 500 600 700 800 900 I 
C 0 N C E N T RAT I ON ( PART IC L ES C M-3 
Fig. 2 The height distribution of Na a toms obtained f rom Eqn. (9). 


Altitude distribution, origin and flux of sodium in the atmosphere

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