Atmospheric Attenuation of Solar Millimeter Wave RadiationH. H. Theissing and P. J. Caplan Citation: Journal of Applied Physics 27, 538 (1956); doi: 10.1063/1.1722418 View online: http://dx.doi.org/10.1063/1.1722418 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/27/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Coherent diffraction radiation in the millimeter and submillimeter wave region AIP Conf. Proc. 367, 482 (1996); 10.1063/1.50287 A photoacoustic joulemeter for millimeter wave radiation Rev. Sci. Instrum. 63, 166 (1992); 10.1063/1.1142950 Atmospheric attenuation of explosion waves J. Acoust. Soc. Am. 61, 39 (1977); 10.1121/1.381266 MILLIMETER WAVE RADIATION FROM INDIUM ANTIMONIDE Appl. Phys. Lett. 10, 244 (1967); 10.1063/1.1754930 Solar Radiation and Atmospheric Attenuation at 6Millimeter Wavelength J. Appl. Phys. 28, 295 (1957); 10.1063/1.1722733

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JOURNAL OF APPLIED PHYSICS VOLUME 27, NUMBER 5 MAY. 1956

Atmospheric Attenuation of Solar Millimeter Wave Radiation*

H. H. THEIS SING AND P. J. CAPLAN Signal Crrps Engineering Laboratories, Fort Monmouth, New Jersey

(Received December 3, 1955)

A lumped attenuation figure for the solar millimeter wave radiation in the water vapor of the atmosphere was measured with a thermal detection apparatus. The corresponding theoretical attenuation figure was com­puted from the solar millimeter spectrum known from previous measurements of the authors and from the Van Vleck-Weisskopf equation and turned out smaller than the first experimental figure by a factor of about 3 in terms of decibels per kilometer. With the general structure of an absorption spectrum considered well traced by theory this disagreement can only mean a failure of the computation to yield the correct absolute absorption level in the window regions to which the experiment predominantly refers. The fault may lie either in the equation itself or in inadequately known constants required in the computation.

Regardless, however, of the cause and magnitude of this discrepancy, the novel method of this paper lies in a combination of the above-mentioned experimental and computed figures to furnish a well-supported spectral attenuation function in the window region. This function, which constitutes the final result, turns out largely independent of the shortcomings in the computation part, inasmuch as theory was used to furnish merely its structure and experiment to provide its absolute level.

1. INTRODUCTION

T HE development of microwave techniques in recent years has stimulated theoretical and ex­

perimental studies of the attenuation of microwaves by atmospheric gases. As generators for various frequencies became available, they were used for absorption meas­urements either in free atmosphere or in the laboratory. To mention only a few, the H20 absorption at the 13.5-mm line was measured by Becker and Autler1 in a cavity, the atmospheric absorption (H20+02) at 8.7 mm by Marner,2 and at 4.3 mm by Tolbert and co-workers.3 In addition, measurements of O2 absorption in the millimeter range were made both in the laboratory4.6

and in the atmosphere.6

So far, no atmospheric absorption measurements as far down as the 1- to 2-mm range have been made because no sources are readily available. At that point it was of great advantage for the authors that they had recently measured the spectral intensity of solar millimeter radiation.7 This spectrum was obtained with a 60-in. diam parabolic mirror that tracked the sun, with a thermal detector and a new spectral analyzing method that employed wire mesh filters. In Fig. 1 the shaded background gives the range in which actual results lie, whereas the bold curve gives the average spectral intensity measured. It is seen that the latter rises sharply at about 0.9 mm (where the atmosphere becomes sufficiently transparent), peaks at 1.1 mm, and then tapers off toward longer waves as would be

* Presented at the URSI-Michigan Symposium (June, 1955). I G. E. Becker and S. H. Autler, Phys. Rev. 70,300 (1946). 2 G. R. Marner, Proc. lnst. Radio Engrs. 43, 257 (1955). 3 Tolbert, Britt, Tipton, and Straiton, Electrical Engineering

Research Laboratory, University of Texas, Report No. 73 (August, 1954).

• R. Beringer, Phys. Rev. 70, 53 (1946). 5 Strandberg, Meng, and Ingersoll, Phys. Rev. 75, 1524 (1949). 6 H. R. L. Lamont, Proc. Phys. Soc. (London) 61, 562 (1948). 7 Theissing, Caplan and Stelle, J. Opt. Soc. Am. 45, 405(A)

(1955); a detailed account is given in a forthcoming paper in J. Opt. Soc. Am.

expected for a temperature radiator. Black filters suppressed the very short near-infrared and visible radiation of the sun but are without noticeable influence in the millimeter range under consideration.

The use of this source of predominantly 1- to 2-mm radiation suggested itself for atmospheric absorption measurements by measuring the solar signal with the tracking parabola at various water vapor concentrations in the atmospheric path. Thus, the variation of the signal was attributed solely to changes of the product H20 vapor pressure times path, a quantity denoted by x in the following text. The O2 attenuation was added in the graphs as a constant contribution, independent of humidity. .

2. OUTLINE OF METHOD

There is, however, a difference between conventional absorption measurements and the method to be em­ployed here. The former give a direct result for the fixed frequency at which they are taken, and for the attenuation in a uniform medium, well defined, e.g., for ground level atmosphere (or in a wave guide or cavity). By contrast, in the present case one is con­fronted with an extended spectral range such as in Fig. 1 and with an atmospheric path that, under a given

~ iii z '" .. z

8 2.8 3.2 5.6 MILLIMETER WAVELENGTH

FIG. 1. Solar millimeter spectrum. Shaded background: range of spectral curves obtained from experiment. Heavy curve: average.

538

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A T M 0 S P HER I CAT TEN U A T JON 0 F MIL LIM E T ERR A D I A T ION S39

elevation angle, penetrates layers of the atmosphere in which the attenuation will differ greatly. This stratifica­tion requires that all solar signals have to be taken simultaneously with radiosonde recordings of the partial water vapor pressure Ph, temperature Th and atmospheric pressure Ph at various altitudes h. This was done at eight altitude levels up to 21000 ft in intervals of 3000 ft. The radiosonde data furnish first of all-aside from the purpose that they will serve in the following discussion-a representative value x [mm Hg partial H 20 vapor pressure times kilometers] associated with a particular solar signal reading. This x will be dealt with numerically in Sec. 3. (It may be noted in passing that the solar spectrum in Fig. 1 was also taken in conjunc­tion with a known value x= Xo in order to be useful for further analysis.) If the logarithm of the (spectrally unresolved) signal values is plotted vs the corresponding x-values (Fig. 2), one obtains the heavy drawn line

LOGe SIGNAL + I

t

o

Xx ./' fJ (THEORETICAL) -1.0 db/km ~ . -- ...........

CD .............. CD x ...........

7RY G"" "&.. ... -~~

~-X ....... ~ X ........ tl

a (EXPERIMENTAL)

• e.1 db Ikm

-I

-- X [MM Hg· KILOMETER] HUMID _e~ _________ ~10~ _________ ~20~ ________ ~30~ _______ 4~0~_

FIG. 2. Attenuation graphs for solar millimeter radiation. Experimental slope (heavy line) yields 0.= 2.7 db/km. Atmos­pheric data for bold printed crosses, lumped in a dry, medium and humid average, furnish Van Vleck attenuation for sam~ spectrum (dotted line) with theor. slope ~= 1.0 db/km. X and 0: two separate series of observations. a. and ~ refer to 1% partial Ht> pressure.

whose slope furnishes an experimental attenuation figure o{db/km], numerically evaluated in Sec. 3. But this a, which in conventional attenuation experi­ments would be a measured constant of the atmosphere, is not a final result here. In order to draw useful con­clusions from a, one has to go through the following, more elaborate procedure.

For this purpose it is first necessary to introduce a theoretical expression which gives the attenuation of gases 'Y (v) as a function of frequency v. This so-called Van Vleck-Weisskopf equationS is shown in its typical form in Eq. (1) and in a form arranged for numerical

8 J. H. Van Vleck and V. F. Weisskopf, Revs. Modern Phys. 17,227 (1945); J. H. Van Vleck, Phys. Rev. 71,413 (1947) and 71,425 (1947); it would exceed the scope of this predominantly experimental paper to go into such topics as derivation of the equation, comparison with another approach (statistical theory), comparison with experiments, etc. For a good brief review of these questions see footnote 13.

- WAVELENGTH M M

1000 4 1!5 12 I

000 0.15

1

100 I 00

..

r 0 I \ II

II ~ ~ I

BV ~ ~ SINe

I ~ MEDIU !

1

°lMTNr~ '-..!!i '='- ",0 VAPOR PR~ I

10 I~ • 0 • F"1QUENCYt-1

.I o 100 400 KMC

• • • \'6.k Ill? ell-I

0-...

o

o

FIG. 3. Attenuation r(ji)[db], integrated over total slant atmospheric path for three conditions dry, medium, and humid specified in Fig. 2.

computation in conjunction with the use of Tables I and II in Eq. (Sa) and (Sb). 'Y(v) contains a number of constants for the molecule in question and the variable meteorological data p, P, and T. Obviously, the experimentally obtained quantity a contains in lumped form the frequency dependent 'Y(v) function integrated over altitudes h (because of changing Ph,

Ph, T h) and integrated over frequencies v, whereby it is weighted at each v with the solar millimeter wave intensity at that particular v. By duplicating this process by which a must have originated one may build up a theoretical substitute fJ for it. The usefulness of this procedure will be shown in the discussion of the results.

The theoretical attenuation figure fJ is arrived at by the following reasoning. For three points in the a plot (Fig. 2) labeled "dry", "medium," and "humid," one averages the radiosonde data Ph, Th and Ph from the nearest measured points (heavy crosses in Fig. 2) for each altitude ·h. Inserting these in the Van Vleck­Weisskopf equation (Sec. 4) gives the H 20 component of frequency-dependent attenuation 'Y(v) [db/km] for each of the eight altitudes and for each humidity condi­tion dry, medium, and humid, which makes 24 curves in all. Each condition is then graphically integrated over altitudes, a constant altitude-integrated O2 com­ponent is added for each condition, and elevation angles are taken into account. This yields (Fig. 3) three spectral attenuation graphs r(V)dry, r(V)med. and r(vhum.[dbJ. These rev) graphs represent the com­puted attenuation for the actual slant atmospheric paths at precisely the meteorological conditions under which the three points dry, medium, and humid on the plot (Fig. 2) had been originally observed.

It is now important to note that the dry condition was chosen such that its abscissa xo[mm HgXkm] was the same as the value of the bracket product prevailing when the spectral intensity distribution Fig. 1 was measured. If, in this way, spectrum Fig. 1 and r(V)dry correspond to each other, one can readily plot the

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540 H. H. THE ISS I N G AND P. J. C A PL A N

,. !: .,

o.~

~ 0.20 !Ii

10•10

CALCULATED

\

---'-"-'~-~

1.6 2.0 2.4 2.B 3.2 __ WAVELENGTH M M

FIG. 4. Upper­most curve: ex­perimental milli­meter spectrum taken under specified dry con­dition. Lower curves: medium and humid spectra, com-

3.& puted by means of Fig. 3.

spectra (Fig. 4) that correspond to r(V)med. and r(vhum. in Fig. 3. The areas sub tended by the three spectra (Fig. 4) represent the total solar signals that would be measured if the Van Vleck-Weisskopf equa­tion were quantitatively correct. The logarithms of their areas were consequently plotted for the same abscissas x that belong to the three humidity conditions chosen before. One obtains the dotted line in Fig. 2 whose slope yields the desired theoretical attenuation figure {3. This {3, however, turns out smaller than cx. The significance of this result will be discussed in Sec. 5.

The method outlined in the foregoing was originated by the authors and has been disclosed before on the following occasions. In a paper presented in April, 1955,9 it was shown that the experimental spectrum (Fig. 1) agreed well with the Rayleigh-Jeans v2 law attenuated by r(")dry[db] (and modified as discussed in Sec. 5 of this paper). A brief description of the method was published in April, 1955.10 A complete presentation of the graphs and results shown in the present paper was given at the URSI-Michigan Symposium, June, 1955.11 A variation of this treatment appeared in November, 1955 in a paper by Sinton.12

3. EXPERIMENTAL RESULTS

The fact that the sun had to be tracked with the parabolic mirror restricted the observations to clear days. Figure 2 shows the experimental graph log signal 'Vs partial water vapor pressure times kilometer path. The circles and crosses refer to two series of observa­tions with different detector cells. Each point was taken in coincidence with daily radiosonde ascensions at Evans Signal Laboratory. Therefore the variation in abscissa values was due to actual humidity changes much more than to varying elevation angle. In order to cover a sufficient range of abscissa values, the ob­servations were extended through all seasons of 1954. A standard reference source was provided' in the parabolic mirror against which all solar readings were

• Presented at the meeting of the Optical Society of America in New York, April 8, 1955.

10 Bulletin of the American Physical Society, Vol. 30, No.3 containing the abstracts of the Washington, D. C. meeting, April 28-30, 1955. The presentation of the paper was, however, postponed to the URSI-Michigan Symposium.

11 Inst. Radio Engrs. Transactions on Antennas and Propaga­tion Vol. AP-3 Suppl. No.1, 1955 (see App. A2-9).

12 W. M. Sinton,]. Opt. Soc. Am. 45, 975 (1955).

compared in order to eliminate the influence of possible variations of sensitivity.

The partial water vapor pressure Ph at each altitude level h [kilometer] was weighted in its influence on attenuation by writing the abscissa

0.914 Ph(TO)4 x=-- L Ph- - [mmHgXkm].

sin8 h Po Th Here 0.914 km=3 000 ft is the altitude interval, (J the angle of elevation, and the summation is carried out over the eight altitudes mentioned before. The factor with Ph is built in accordance with FI in Eq. (6) and expresses x in terms of [mm HgXkm] at Po= 760 mm Hg, To = 291 oK. The graph Fig. 2 has to be visualized as a superposition of a number of attenuation curves for incremental wavelength ranges and has, therefore, a slight curvature. Neglecting the lower curved end sec­tion one finds a representative value for the slope 0.0795 per (mm HgXkm). If this is multiplied by Po= 7.6 mm Hg corresponding to 1% partial pressure,

cx= 10 logloe-o.079H6= 2.7 db/km at 1% H 20

is obtained as the result for the experimental attenua­tion cx of the solar millimeter radiation at standard ground level conditions.

Other absorbers such as oxygen, ozone, dust, etc. may have increased the scattering of observed points but cannot have influenced the slope cx which was plotted with respect to partial H20 vapor pressure Xpath.

4. COMPUTATIONS

The formalism by which the attenuation 'Y(v) for the H 20 component was computed for the meteorological data obtained by radiosonde recordings will now be briefly outlined on the basis of the treatment by T. F. Rogers.13•14

The Van Vleck expression for the attenuation 'Y(v,vp )

db/km at frequency v, if only the absorption line Vp

were present is

pv2106Iogloe·81r2. N· I J.l.p12 exp( - W p/O.695T) 'Y(v,vp)=----------------­

P·3·c·k·T·Gf:.v

For standard ground level condition (denoted by sub­script 0) Po= 7.6 mm Hg, Po= 760 mm Hg, po/Po = 1% partial water vapor pressure, To= 291 OK (consistent with 7.5 g/m3 H 20) and with N=9.66·1018 Po/To = total number of molecules per cm3, f:.iio = f:.vo/ c

13 T. F. Rogers, Cambridge Research Center Rept. E5078 (1951).

14 R. G. Newton and T. F. Rogers, Cambridge Research Center Rept. E5111 (1953).

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AT M 0 S P HER I CAT TEN U A T ION 0 F MIL LIM E T ERR A D I A T ION 541

TABLE 1. Basic molecular data used in the numerical evaluation of Eq. (2). Peak attenuation 1'01' of eleven H 20 resonance lines at standard ground level condition (Po= 760 mm Hg, To=291°K, Po= 7.6 mm Hg, 1% partial H 20 pressure).

Term Term values v. notation [em-I]

[em-I] dimensionless W.

0.74 5_1 6_5 446.50 447.24 6.12 22 3_2 136.15 142.27

10.86 40 5 __

315.73 326.59 12.67 31 4_3 212.16 224.83 14.98 33 4_1 285.40 300.38 18.58 LI 11 23.79 42.37 25.11 2_2 20 70.06 95.17 32.93 I. 2_2 37.13 70.06 36.59 3_3 3_1 136.77 173.36 37.13 O~ 10 0 37.13 38.80 3_1 31 173.36 212.16

=0.10 cm-l = half-intensity half-width of absorption line, I ).tpI2=Sp· gp.).t2 [e.s.u.]; ).t= 1.84.10-18 cm6/2 g1/2

sec1=dipole moment of H 20 molecule, sp=line strengthl5 of line lip (dimensionless), gp=weighting factor16 of line lip (dimensionless, either 1 or 3), G=par­tition function (dimensionless) = 170 for To= 291 oK; (G=170=0.034·T03/2), Wp=lower term of line lip in em-I, k=Boltzmann's constant= 1.38.10-16 erg/degree, c=3 .1010 cm/sec, i{cm-1]= II/c=wave number per cm, Eq. (1) takes the following form for the peak of the absorption line vp, i.e., for v= vp, where the bracket is very nearly unity:

With the foregoing listed numerical values, Co, which is constant for all lines, is found equal to 14.5. The absorption peak values of all lines that contribute most effectively to the A range under investigation were computed from Eq. (2) and are listed, along with the factors that make up the expression for 'YOp, in Table I. They are in good agreement with the peak values of absorption graphs presented by T. F. RogersY

For any arbitrary v, but still for standard ground

Line strength exp( -0.6~;To) Weight factor

s. ==EOp dimensionless dimensionless

K. dimensionless "0. (db/km)

0.0549 0.110 3 0.14 0.1015 0.510 1 27.6 0.0891 0.210 1 31.5 0.1224 0.350 3 297 0.1316 0.244 3 308 1.5000 0.889 3 19600 2.0739 0.708 1 13200 0.7557 0.833 1 9710 2.1809 0.509 3 63500 1.0000 1 1 19650 2.5434 0.424 3 69400

level condition, as above, 'Yo(v,vp) is

with 'Yo(v,v p) = (n+ +n_)ov2'Yop/vp2 (3)

The 'Yo(v,vp) are listed in Table II. The sum of all entries (neglecting the small numbers in parentheses) in a horizontal row is listed in the last column at right. This sum

'YO(v) = ji2 E 'Yop(n+ +n_)o/vi[db/kmJ (4) p

gives the actual attenuation at frequency v due to all H 20 absorption lines under standard ground level condition. These 'Yo(v) also check with Rogers' graphsP

For other than standard conditions, p, P, T, one has .1. v = .1.voPTOI/2(1 +kp/ PO)/POTI/2(1 +k) where k=:,0.012 makes a small correction,18 and, off resonance, n+ +n_ = (n++n_)o.1.v2/.1.vo2• With the abbreviation EOp=exp (-W p/O.695 To) which is a function, defined for standard condition, of each individual line vp , one obtains Eq. (Sa) for v at a window and (5b) for v at

TABLE II. Results of the numerical evaluation of Eqs. (3) and (4): attenuation 1'o(v,vp) (db/km) due to a single H20 line vp as a function of v and vp; attenuation 1'o(v) due to all eleven lines as a function of v. All values are for standard ground level condition.

jip[cm-1]_ 0.74 6.12 10.86 12.67 14.98 18.58 25.11 32.93 36.59 37.13 38.80 ii[cm-I ]

"o(ii.;;.) (db/km)

2 0.0084 0.00218 (0.000137) (0.00001) (0.00052) 0.0136 0.0027 0.00069 0.00285 0.00083 0.00246 5 0.148 0.0031 0.00934 0.0043 0.1025 0.0187 0.0059 0.0187 0.00694 0.0161 6.12 27.6 (0.0048) 0.0180 (0.0077) 0.1718 0.0298 (0.0071) 0.0288 (0.0082) 0.0247 7 0.453 0.0092 0.0305 0.0119 0.250 0.0413 0.0096 0.0387 0.0109 0.0331 8 0.136 0.0214 0.0569 0.0196 0.377 0.0581 0.0130 0.0524 0.0146 0.0445 9 0.0746 0.0632 0.1143 0.0330 0.561 0.0800 0.0173 0.0686 0.0192 0.0582

10 0.0519 0.355 0.262 0.0574 0.845 0.1090 0.0225 0.0888 0.0250 0.0749 10.86 (0.0416) 31.50 0.670 0.0976 1.205 0.1410 (O.0280) 0.109 (0.0311) 0.0918 11.5 0.0367 0.602 1.76 0.1520 1.580 0.1705 0.0328 0.1265 0.0367 0.1059 13.5 (0.0278) 0.071 4.80 1.14 4.11 0.308 (0.05) 0.195 (0.056) 0.161

15 Cross, Hainer, and King, J. Chern. Phys. 12,210 (1944). 16 Randall, Dennison, Ginsburg, and Weber, Phys. Rev. 52, 160 (1937). 17 Navy Millimeter Wave Conference in Washington, D. C. (1953) (unpUblished, COllrtesy private cOIIlIIlunica.tion). 18 Compa.re reference 14, p. 40.

"om (db/km)

0.0344 0.334 27.90 0.888 0.7914 0.9894 1.898 33.92 4.603

10.92

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542 H. H. THEISSING AND P. ]. CAPLAN

TABLE III. Meteorological data at eight altitudes for the three humidity conditions entering in the computation of the theoretical attenuation figure of Sec. 4.

Dry

Height in P. T. ph P. 1000 ft mmHg OK mmHg mmHg

0 770 278 2.81 761 3 683 269 1.68 683 6 609 270 0.55 609 9 544 266 0.46 544

12 482 264 0.55 482 15 428 258 0.27 428 18 378 252 0.16 378 21 333 245 0.08 333

a peak

'Y(ii) = F 1 L 'Yo(ii,iip)Eop(To/T)-l (Sa) p

'Y(ii) = F 2'YOpEOp(To/T)-1+ F 1 L' 'Yo(ii,iip)EOp(To/T)-l (5b) p

whereby the second term in Eq. (5b) is small compared to the first. The factors F 1 and F 2

F1=pPTo4(1+kp/po)/poPoP(1+k) = 1.23·106pP(1+0.0016p)/P

F2=F1(T/To) (PO/P)2 (6)

are dimensionless quantities depending only on mete­orological data but independent of lines. All 'Yo(ii,iip) are listed numerically in Table II and all EOp and 'Yop

in Table 1. In Eq. (Sa) Lp indicates summation over all peaks listed in Table II, while in the summation Lp' of Eq. (5b) the peak at which ii occurs is excluded.

Equations (5) were used to evaluate numerically the attenuation 'Y(ii) in the layers of the atmosphere for those meteorological conditions that actually prevailed when the points denoted dry, medium, and humid on the graph Fig. 2 were taken. These meteorological data which were averaged from the heavy crosses in Fig. 2 surrounding the points dry, medium, and humid are listed in Table III. They were inserted in Eqs. (5) and (6), and the resulting 'Y(ii) was plotted at fixed fre­quencies as a function of altitude. The values in the ii column of Table II were taken as those fixed frequencies. Graphical integration of these plots over altitudes furnished the H 20 component of total attenuation r(ii)dry, r(ii)med. and r(iihum. shown in Fig. 3 with the angle of elevation for the dry, medium, and humid observations worked in. Figure 3, therefore, gives the Van Vleck attenuation in the total slant atmospheric path with the meteorological data actually prevailing at the time when the representative points in Fig. 2 were measured.

Figure 3 would not be complete as a graph presenting atmospheric attenuation if it did not contain, in ad­dition, the O2 component, although the latter does not enter the computation directly. The altitude-integrated O2 attenuation was superimposed which manifests

Medium Humid

T. pA PA TA pA OK mmHg mmHg OK mmHg

289 8.28 758 295 12.65 285 5.83 683 290 11.28 282 2.26 609 285 7.56 277 4.43 544 281 3.22 270 2.86 482 275 2.81 264 1.40 428 270 1.89 259 0.98 378 264 0.64 252 0.37 333 257 0.35

itself predominantly in the region x> 2 mm where the spectral intensity of Fig. 1 has already tapered off con­siderably. For this reason a constant oxygen contribu­tion seemed adequate. This was integrated over alti­tudes from an approximate expression that had been found to trace the O2 behavior sufficiently well for this case where O2 plays a minor role, and which is valid for 1'>50000 Mc (X<6 mm): ('Y(I') in db/km)

'Y(I') = v2[0.392 . 10-20 (n++n_) a+ 1.20· 10--22 (n+ + n-hJ

(n±)a= {1 + ( ii± iia )2}-1 0.0746K

{ ( ii±iib )2}-1

(n±h= 1+ -- . 0.020K

(7)

The line width,t greater than the theoretical value, compensates for the neglected O2 continuum in this approximation. Va = 60 kMc and I'b=117 kMc are the two O2 peak frequencies. The altitude influence (altitude h in kilometers) comes into the formula through K con­tained in the n± expressions:

K=PhTol/2/PoThl/2; Ph/PO=exp( -h/7.6); T h/To=1-h/55.!4

The integration had to be extended to much higher altitudes than was necessary for H 20 vapor.

It was remarked in Sec. 2 that the r(ii)dry graph in Fig. 3 refers to the same value xo [partial pressure XpathJ as that for which the solar spectrum Fig. 1 had been measured. If one therefore multiplies this original spectrum (uppermost in Fig. 4) by factors corresponding to the decibel difference r(ii)med.-r(ii)dry (solid arrows in Fig. 3), one obtains the medium spectrum in Fig. 4. Similarly, the humid spectrum in Fig. 4 is obtained by factors corresponding to the dotted arrows in Fig. 3. However, since the experimental spectrum did not show absorption bands because of lack of resolution, they were likewise omitted in the computed spectra, and the peak regions were interpolated in dotted lines. The areas under the three spectral graphs represent the

t i.e., effective line width of envelope of many individual com­ponents.

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ATMOSPHERIC ATTENUATION OF MILLIMETER RADIATION 543

three solar signals for the dry, medium, and humid points as they would be obtained if the Van Vleck relation were quantitively correct. The logarithms of the three areas were therefore inserted back into Fig. 2 at the abscissas of the three points. Thus, the dotted line was obtained whose slope is seen to give a consider­ably smaller attenuation figure (3= 1.0 db/km for the solar spectrum.

5. DISCUSSION

If a is found larger than its computed equivalent fJ and if-as there will be little doubt-theory traces the general structure of the absorption spectrum correctly, then a/fJ=2.7 should be a scale factor by which the over-all level of the the9retical H 20 component of the absorption curve must be raised. This was checked by raising the H 20 component of the three r(ii) curves in Fig. 3 by a factor 2.7 in the decibel scale, finding the corresponding three new spectra as in Fig. 4 before and plotting the logarithms of their areas corresponding to the dotted line in Fig. 2. The new (3' obtained in this fashion was still slightly smaller than a, and it turned out that the raising factor '1 for the r(ii) curves had to be 3.3 (instead of a/{3=2.7) in order to make a=fJ'. A similar evaluation was made on the basis of a previous result7•9 of the authors. It was noted that the experi­mental solar spectrum (Fig. 1), if allowance is made for poor resolution and experimental error, came close to the product of the Rayleigh-Jeans v2 law reduced by 'lXr(ii)dry[db]. Consequently, v2 was multiplied by the three r(ii) curves in Fig. 3 whereby three spectral dis­tributions resulted that now replace the old ones in Fig. 4. Again, the three areas were logarithmically plotted as before in Fig. 2 furnishing fJ= 1.2 db/km and hence a/fJ= 2.2. By repeating the process as before it was found that a raising factor for the r(ii) curves '2=2.6 (instead of a/{3=2.2) was necessary to make a=fJ'.

The fact that the factors, turned out not too different from the corresponding a/fJ values may be considered as a support for the idealized picture of separation in structure and scale factor. The cause for such a factor could be inherent in the Van Vleck-Weisskopf equation particularly since the measurements in this paper refer mostly to the wings where too low theoretical values had been suspected earlier,19 but it could partly come from inadequately known constants used in the computation like, e.g., .1iio=O.lO cm-I • Errors in .1iio would generally affect peaks and wings in opposite direction and therefore alter the structure which in this idealized picture was considered traced correctly. But since the peaks and their immediate vicinity played a negligible part for the solar signal at all H 20 concentra­tions, a change of .1iio will here practically reduce to the

lj Compare reference 13, p. 23ff.

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FIG. 5. Attenuation for 1 % partial water vapor pressure under standard ground level conditions. Heavy curve: computed from Van Vleck's formula. Dotted curves: region of modified attenua­tion according to results of this paper.

role of another factor. This would not be strictly true if .1iio were not the same for each resonance line. There may very well be such a line-to-line variation which would account for the fact that the desired raising factor, was not immediately given by the ratio a/fJ.

Regardless, however, of what causes the inadequacy of the theoretical computation, the experimental result remains. That is, the actual window attenuation 'Y'(v) is given by , times the computed 'Y(v). This result is indeed independent of the absolute level of 'Y(v). If, for example, a still lower value of .1iio had been used in the Van Vleck equation, 'Y(v) would have been lowered in the window regions, the fJ slope (Fig. 2) would have turned out still shallower, the required value of , would have been larger, but the resulting 'Y'(v)=r·'Y(v) would have remained essentially the same. Thus, in Fig. 5, the atmospheric attenuation for standard ground level condition has been plotted in a solid curve. Its water vapor component 'Yo(p) [compare Eq. (4)] was then raised by '1 to give the upper dotted curve and by '2 to give the lower dotted curve, '1=3.3 and '2=2.6 being the two raising factors derived before from the two spectral distributions. These two dotted curves are then the experimental results which are largely inde­pendent of the absolute level of the solid theoretical curve. The Van Vleck-Weisskopf equation has therefore been used for the shape of the spectral attenuation curve, while the experimental data have provided the absolute level of the attenuation function in the wings of the low millimeter region.

ACKNOWLEDGMENTS

The authors acknowledge the continued support of this project by Dr. H. J. Merrill and the valuable assistance of Mr. H. Blake, Mr. M. J. Lowenthal, and Mr. P. Stelle.

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