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ARHEB SERVICES TECHNICAL INfaUUHON AGENCY ARLINGTON HALL STArNN ARLOKIW 12. VIRGINIA
UNCLASSIFIED
NOTICE: When government or other drawings, speci- fications or other data are used for any purpose other than In connection with a definitely related government procurement operation, the U. S. Government thereby Incurs no responsibility, nor any obligation whatsoever; and the fact that the Govern- ment may have formulated, furnished, or In any way supplied the said drawings, specifications, or other data Is not to be regarded by Implication or other- wise as In any manner licensing the holder or any other person or corporation, or conveying any rights or pexnlsslon to manufacture, use or sell any patented Invention that may In any way be related thereto.
Heleased to ASTIA for further dlnsertlnatlon with Ottt lißitations beyond these iaposed by security regulations.
KAVWEPS REPORT 7636 NOTS TP 2648
COPY n ■
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# ̂ M
HORIZONTAL AIR-MASS TABLES
By
R. J. Stirton Test Department
and A. J. Krueger
Research Department fa/ " 3 ^3 XEROX
ABSTRACT. The illumination of artificial earth satellites dur- ing their motion into the earth's shadow is a problem of cur- rent interest. For heights above the surface greater than 20 km, the tangential solar ray incident on a satellite in a non- absorbed region of the spectrum is attenuated principally by Rayleigh scattering. The attenuation is a function of the total number of molecules, or, if the molecular weight remains constant, of the air mass along the path of the ray that is tangential to any level, h , above the surface.
Tables are given in terms of mass of air and number of molecules for a vertical path originating at h , a horizontal path ^angent to h , the horizontal quantities relative to a unit atmosphere, and the rate of change with respect to h . The tables cover an altitude range of 0 to 100 km, calculated for an atmosphere terminated at 200 km. These tables are in- tended primarily for use above a nominal altitude of 20 km, where refraction, which was not allowed for in the computations, has a negligible effect.
U.S. NAVAL ORDNANCE TEST STATION
China Lake, California 15 March 1961
r ■
■
NAVAL ORDNANCE TEST STATION
AN ACTIVITY OF THE BUREAU OF NAVAL WEAPONS
W. W. HOLLISTER, CAPT., USN WM. B. McLEAN, PH.D. Commander Technical Director
FOREWORD
This work Is part of the program for determining the ozone dis- tribution in the atmosphere by photometric observations of artificial satellites—particularly Echo I (i960 Iota one)—and was supported by Local Project 50? and Bureau of Naval Weapons Task Assignment 360FR106-2l6l-R01101001. It has been reviewed for technical accuracy by E. V. Ashburn, 0. N. Strand, and J. C. Smith.
CHAS. E. WARING IVAR E. HIGHBERG Head, Research Department Head, Test Department
Released under the authority of
WM. B. McLEAN Technical Director
KOTS Technical Publication 26kQ NAVWEPS REPORT 7636
Published by Test and Research Departments Manuscript 30/MS-I+6I Collation Cover, 12 leaves, abstract cards First printing 210 numbered copies
INITIAL DmEIBUTION
6 Chief, Bureau of Naval Weapons • DI3-31 (I) , R-12 (l) RAAV (l) RR (1) RRRE (2)
1 Chief of Naval Operations (OP 55) 2 Chief of Naval Research
Code h6l (1 Code k63 (1)
1 David W. Taylo Model Busln ■ 1 Naval Air Development Center, Johnsvllle 1 Naval Air Force, Atlantic Fleet ,* 1 Naval Air Force, Pacific Fleet 1 Naval Air Material Center, Philadelphia 1 Naval Air Station, North Island, San Diego 1 Naval Air Test Center, Patuxent River 1 Naval Ammunition Depot, Crane (Research and Development Department) 2 Naval Avionics Facility, Indianapolis (Library) 5 Naval Missile Center, Point Mugu
Member, Dat;- Reduction and Computing Working Group, IRIG (l) Member, Optical Systems Working Group, IRIG (l) Technical Library (2)
1 Naval Ordnance Laboratory, Corona 1 Naval Ordnance Laboratory, White Oak (Library) 3 Naval Ordnance Missile Test Facility, White Sands Proving Ground
Member, Data Reduction and Computing Working Group, IRIG (l) Member, .Optical Systems Working Group, IRIG (•!)
1 Naval Postgraduate School, Monterey 2 Naval Research Laboratory (Code 2021) 2 Naval Underwater Ordnance Station, Newport 1 Naval Weapons Laboratory, Dahlgren (Technical Library) 3 Naval Weapons Plant
Code 752 (2) Code 755 (1)
i Navy Electronic a Luboratory, San Diego .1 Office of Naval Research Branch Office, Pasadena 1 Operational Test and Evaluation Force 1 Bureau of Uuvi 1 Weapons General Representative, Western District,
El Segundo 1 Bureau of Naval Weapons 3 Chief of OrcUwince
OUDTB (l) ORDTS (l) ORDTU (1)
Representative, Azusa, Calif.
2 Aberdeen Proving Ground Development and Proof Services (l) Ballistic Research Laboratories (l)
6 Army Rocket & Guided Missile Agency, Redstone "Arsenal Technical Library, ORDXR-OTL (k) ORDDW-JJDE (l) Rocket Development Laboratory, Test and Evaluation Branch (l)
1 Diamond Ordnance Fuze Laboratories * 2 Frankford Arsenal
Library (]J Pitman-Dunn Laboratory (l)
Picatinny Arsenal (Library) Rock Island Arsenal Watertown Arsenal • White Sands Proving Ground
Member, Data Redaction and Computing Working Group, IRIG (l) Member, Optical Systems V/orking Group, IRIG (l) Technical Library (3)
Headquarters, U. S. Air Force Air Force Cambridge Research Laboratories, Laurence G. Hanscom Field Air Force Misailc Development Center, Holloman Air Force Base
Member, Data Reduction and Computing Working Group, IRIG (l) Member, Optical Systems Working Group, IRIG (l)
Air Force Special V/eapons Center, Kirtland Air Force Base Air Proving Ground Center, Eglin Air Force Base Holloman Air Force Base Tactical Air Command, Langley Air Force Base (TPL-RQD-M) Wright Air De/elopment Division, Wright-Patterson Air Force Base (WWAD) Air Force Development Field Representative, Aberdeen Proving Ground
10 Armed Services Technical Information Agency (TIPCR) 1 Defense Atomic Support Agency, Sandia Base (Development Division) Ames Research Center Langley Research Center (Library) Lewis Research Center Office of Technical Services #
AeroChem Research Laboratories, Inc., Princeton, N. J. Aerojet-General Corporation, Azusa, Calif., via BuWepsRep Applied Physics Laboratory, JHU, Silver Spring
1 Armour Research Foundation, Chicago (G. A. Nothmann) 1 Arthur D. Little, Inc., Cambridge 1 AVCO Research Laboratory, Everett, Mass. (Document Control Center) , 1 General Electric Advanced Electronics Center, Ithaca (Librarian) 1 Hughes Aircraft Company, Culver City, Calif. (Research and •
Development Library) , Jet Propulsion Laboratory, CIT, Pasadena (Dr. W. H. Pickering) Rohm & Hafts Company, Redstone Arsenal Research Division (Librarian) «!,'* The Rand Corporation, Santa Monica, Calif.
1 Thiokol Chemical Corporation, Redstone Division, Redstone Arsenal, (Technical Library) ,
1 University of Maryland, Department of Physics, College Park
2 .1 1 3
2 i
1 1 1 1 1 1
1 1 1
1 1 2
1 I 1
NOTS CL 1051 (S/61 ) 210
MVWEPS REPORT 7636
!
INTOODUCTION
The orbiting of large balloon reflectors such as Echo I has provided a new tool for investigating the composition of the upper atmosphere. In addition to its primary utility as a radio communications reflector, the satellite is also particularly useful as an extra-atmospheric^reflector of light having, known reflecting properties. Although Echo I was launched in an orbit which was continuously illuminated by the sun, the orbit has a regression rate, relative to the shadow axis of the earth, of approxi- mately four degrees per day. Examination of the geometry involved shows that th^ orbital plane intersects the shadow cone of the earth about 60 percent of the time. On these 'shadow passes' the solar ray illumin- ating the satellite scans the atmosphere just before the satellite enters the shadow and just after it leaves the shadow on each rotation. At any time that the illuminating ray is traversing the atmosphere, the illumi- nance at any wavelength is a function of the distribution of any con- stituents that absorb at that particular wavelength, and is also a func- tion of the molecular density distribution, which alters the intensity by Rayleigh scattering. Since a horizontal path through any atmospheric layer is much longer t'han a vertical path, it is possible to measure the distribution of minor atmospheric constituents. •
Two methods of analysis are available. An integral technique (Ref. l) has been used for lunar eclipse mcasuremerrts of the ozone distribution, and has been modified for application to üatellite measurements (Refs. 2 and 3)• Secondly, a matrix method, which can more readily be programmed for computers, "has also been devised (Ref. h).
Although the methods can be applied to the determination of the atmospheric density disuribuLion, it is of immediate interest to calcu- late the Rayleigh attenuation in ordgr to separate the absorption and scattering effects. If no absorption occurs the illutainance at the satellite is given by
I'M =- I0(X.)e -l£R(X,)M
where I U) ov ' is the extraterrestrial solar intensity at wavelength A.,
k (A.) is the Rayleigh scattering cross section or coefficient, and M is the total number oL' molecules or the air mass along the ray from the sun to the satellite. An older tabulation of M-vs-altitude (Ref. r>) proved unsatisfactory because only the lower levels of the atmosphere were con- sidered. /mother tabulation (Ref. 6) gives air mass data to 150 km with rocket density data obtained at White Sauds up to 19132. Since more recent density data are available, we felt that a new calculation of M would be useful.
I:
HAVWEPS REPORT 7636
It was decided to neglect the effect of. refraction In the tables Included In this report because In the region of Interest (20 to 100 km) the errors are small. However, for completeness the data have been pre- sented from the surface to 100 km. The user should thus exercise some caution In use of the data at the lowest levels. We hope to present the data corrected for refraction at a later date.
CALCUIATI0NS
We can define 'air mass1 as the total mass-per-unit area along a specified path through the earth's atmosphere. The particular air mass calculated here we have called 'horizontal air mass*. The path in this case is a line through the atmosphere which passes the earth (assumed spherical) at some minimum distance, h , as shown in Pig. 1. .
FIG. 1. Horizontal Air-Mass Geometry.
The air mass M(h ) is then given by
M(h ) = / p(h) dx (1)
where p(h) is the mass density of the air at height h. We have assumed that th,e atmosphere is radially symmetric. Equation (l) can then be written as .
M(ho) = 2J ^(>(h) dx (2)
because of this symmetry. 0
NAVWEP3 REPORT 7^36
We now wish to Introduce h as the variable of integration In place of x. From the geometry of Fig« 1 ve have #
• ..a . /«... \2 /«„.xS x* + (lWior - (R+hr (3)
and then
^(RH*)2 - (R^o)2 >/(h-ho)(2R+h4ho)
Equation (2) now reads
M(ho). 2 r " efel w**— (5) ho [(h-ho)(2R*h4ho)]*
Although Eq. (5) is a convergent Integral for any reasonable func- tion p(h), it presents difficulties If the integration is performed numerically as is done here. The upper limit presents no problem since the earth's atmosphere can be considered to terminate, for practical calculations, at some finite h.
We can dispose of the discontinuity at h ■ ho by the following sub- stitution:
y2 » h-h^ (6) o
Then
dh • 2ydy
and Eq. (5) becomes
2 , xr„., ,2. 00 P(y +hJ[R+h +y':]äy n~ HVy fn /iRi-n ty jay M(h ) = k / 2 2 (7) 0 JO o.i 0 pi
[2(R+ho)+y2P
. MVWEPS REPORT 76^6
For practical calculations the range of y can be taken as ^
y 2 y 2 0 'max ' (8)
where
y « th -h Y' 'max * max o (9)
We shall say more about ^tea_ later.
Although the density of the atmosphere Is approximately an exponen- tial function of height, this was not considered a sufficiently accurate approximation for the present purpose. Instead, the ARDC 1959 model atmosphere (Ref. 6) was used. The properties of the atmosphere are tabulated functions of height. With this density function it is nec- essary to integrate Eq. (?) by numerical methods.
nie integration was performed with Simpson's rule. The density data of the ARDC atmosphere is tabulated at various Intervals, which are larger at higher altitudes. This fact, and the fact that Simpson's rule requires equal intervals, necessitated Interpolation for the values of
o p(y +h ). The equal Intervals must of course be in the variable of
integration y. Second order interpolation was used. The numerical work was done on the IBM 709.
The upper limit for h was set at either 200 km or where the density
became 10 of the initial density corresponding to h , whichever occur-
red first. The atmospheric density was taken to be zero above 200 km. The range of h was
o
0 S h S 100 km o (10)
and the air mass waa computed for every 1,000 m in this interval,
Another quantity of interest in these calculations is the 'zenith •nass '. Ti
200 km. Thus
air mass1. This is the air mass computed along the vertical from h. to
wv -X 200,000
p(h)dh (11)
Again the integration was performed by Simpson's rule. In this Integra- tion no Interpolation was used. The Interval of Integration was the interval of tabulation of h . The subroutine which performed the inte-
gration took care of the end-point automatically when there was an even number of density points at some interval of h . The basic Simpson's
rule requires an odd number of points.
The 'relative air mass1 was also computed. We define the relative air mass by
R(h0) M(ho)
(12)
that is. the zenith air mass at h * 0 is used as the reference. • o
nie final quantity which was computed was the derivative of the relative air mass with respect to height h . This was oomputed using
five-point formulas (Ref. 7).
The numerical work was checked by severed, means. First, the zenith air mass for h should be equal to the atmospheric pressure at that h
with suitable unit conversion. A small table of air pressure and accel- eration of gravity from the ARDC atmosphere, and the zenith air mass, calculated pressure, and percentage difference between the pressures are presented in Table 1. The error of about 2 in 1,000 does not seem un- reasonable, and may be ascribed to errors in Simpson's rule integration and other numerical processes. The large error at h = 100 km is proba-
bly due to the termination of the atmosphere at 200 km.
As a check on the Integration for the horizontal air mass, an ex- ponential density was assumed which passed through the ARDC density at h = 0 and h = 100 km. If we set
P(h) Ac"*11, (13)
NAVWEPS REPORT 7636
STABLE 1. Zenith Air-Mass Check
(öoters) P
(mb) " g
(m/seo )
Mz(ho) (gm/cm8)
- '- p ■ " calc
(mh)
Pressure difference
0 1.01325 x 103 9.8067 I.0356 x 103 I.OI55 x 103 ■K).23^
20,000 5.5293 x 101 9.7^52 5.685I x 101 5.5,K) x 10*1 •»0.20^
50,000 8.7858 x 10"1 9.65^2 9.1223 X.101 8.807 x 10'^ •»0.2^
70,000 6.0209 x 10"2 9.591»2 6.286U x 10"2 6.031 x 10'2 +0.1736
90,000 1.353 x lO-3 9.535 1A200 x 10'3 1.35,* x 10'3 ■fO.07^
100,000 2.138 x 10 9.505 2.238 x 10"* 2.127 x 10 -0.51?6
Equation (5) becomes
N(h0) - 2^ Ae"^ (R+h)dh
o [(h-h )(2R+h+h )]a o o
(IV
where A - 1.2250 kg/m3 and a = I.50 x 10' /m. Now if we neglect h with respect to R,
M(ho) s^/iR Aj 00 -ah dh
(15)
'o (h-ho)2
In a strict sense this approximation may not seem reasonable since we are integrating h from h to «>. However, 'infinity' here is on the order'of
200 km, as compared to the radius of the earth R at some 6,000 km. The
exponential function decays faster than the h2 term Increases. In any event, we are after an estimate to check*the order of magnitude of the numerical work. Set
x = h-h (16)
WAVWEPS REPORT 7636 • •*. »»
Then fiq« (19) read«
M(h ) - sy2RAe'ahof 0 Jo e "~ Sx. (17)
Using the same density function, the zenith air mass is
w • A/ha
e an « — e o a (18)
and the' unit air mass, M„(o), is
Mz(0) A ä (19)
nie relative air mass is therefore
M(ho) «ySitRa e
-aho (20)
A few numerical values are given in Table 2.
TABLE 2. Estimates of Relative Air Mass
h (meters)
Relative .air mass Exponential density Calculated
0 77-5 70.3
20,000 3.86 h.3h 50,000 4.28 x 10"2 6.13 x 10*2
70,000
100,000
2.13 x 10'3
2.37 x 10"5 4.95 x 10"3
1.7k x 10"5
NAVWEPS REPORT 7636 |
Exact agreement Is not expected, and this calculation was performed to check the order of magnitude.
A third check was made, using an atmosphere of constant density. The integration with a finite thickness can easily be performed since the density is constant. Here
M(ho) - 2 r pdx - 2 XjP
From the geometry in Fig. 2,
FIG. 2. Geometry for Check Calcu- lation With Constant Density.
it is easily seen that
In this case
A s (R+hi)2 " (R+h0)2 " (h1-h0)(2R+h1+ho)
M(h ) = 2p[(h -h )(2R+hn+h )]= x o r x 1 o'% 1 o
(21)
(22)
(23)
The unit zenith air mass M (O) is
1
M (0) = / z Jo päh ~ ph {2k)
a
NAVWEPS REPORT 7636
M(h ) •lit 5rT§T«E^[(h1.ho)(2R+h1+ho-)]
z 1. , (25)
Table 3 shows values of the relative air mass computed by the machine pro- gram and by hand from Eq. (25).
TABLE 3. Relative Air Mass for Constant Density
h Relative air mass .^„difference (meters) Machine Ed. 25
0 I6.O77 16.08^ O.Ohk 20,000 15.261 15.271 0.066 50,000 13.9^5 13.956 0.079 70,000 12.982 13.003 0.162
100,000 li.»t09 11.1H7 . O.1U5
The last check was made on the derivative of the relative horizontal air mass. This was done with the same formulas that were used in the machine. Agreement was noted to the last figure In the machine tabula- tion of the derivative for several values of h. .
Strictly speaking, the air mass, M, can be used for calculating Rayleigh scattering only if the molecular weight is constant. If the latter is not constant the number density is the proper density to use. Calculations of the total number of molecules per square centimeter (integrated number density) for horizontal and Zenith paths were also made, using Equations (7), (ll), and (12), and substituting number density for mass density.
The horizontal air mass and associated quantities are presented in Table k. The two-digit number following the. entries indicates the power of ten by which the number -should be multiplied. Thus, .12250-02 = .12250x10" The air-mass entries may be considered accurate to about 0.2^6 on the average if a single figure may be quotfecl for this quantity. The derivative error is estimated to be less than 2^.
The integrated number density and associated quantities are given in Table 5. The relative horizontal air mass from Table k and the relative integrated number density from Table 5 depart by (at most) 0.3^ at the highest altitudes. For most practical work either table may be used.
NAVWEPS REPORT 7636
-
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NAVWEPS REFORT 76^6
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11
MAVWEPS REPORT 76^6
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MAVWEPS REPORT 7636
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NAVWEPS REPORT 76^6
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NAVWEPS REPORT 76^6
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NAVWEPS REPORT 7fo6
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NAVWEP3 REPORT 76^6
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17
KAVWEPS REPORT 7636
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18
HAVWEPS REPORT 7636
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19
MAVMBP3 REPORT 7636
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NAVWEPS REPOBT 7636
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NAVWEFS REPORT 7636
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22
NAVWEPS REPORT 7636
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NAVWEPS REPORT 7636
REFERENCES
1. Vigroux, E. "Spectrophotometrie de 1*Eclipse de Lune du 29-30 Janvier 1953/' AM D ASI^OPffifSIQUE, Vol. 1? (195^), p. 399-
• 2. U. S. Naval Ordnande Test Station. A Note on a Certain Abel-Like
Integral Equation With Application to the Atmospheric Ozone Distri- bution, by R. J. Stirton. China Lake, CalTLf., HOTS, December i960. (IDP1220).
3. , Determination of the Ozone Distribution by a Satellite Eclipse Method, by A. J. Krueger. China Lake, Calif., NOTS. (un- published) .
k. Venkateswaran, S. V., and others. "Determination of the Vertical Distribution of Ozone by Satellite Photometry," (to be published in J GEOPHYS RESEARCH).
5. Link, F. and Z. Sekera. "Dioptrlsche Tafeln der Erdatomsphare," Publikace Prazske Hvezdarny, Praha, 19^0.
6. 'Air Research and Development Command. The ARDC Model Atmosphere, 1959, by R. A. Mlnzner and others. Bedford, Maas., ARDC, August 19'?9. (Air Force Surveys in Geophysics, No. 11^; AFCRC-TR-39-267).
7- Milne, W. E. Numerical Calculus. Princeton, N. J., Princeton University Press, 19^9-
ACKNOWLEDGMENT
The authors wish to acknowledge the help«ot" Robert Dancey in pro- grammlng the computer for the computations of the tables.
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