A MODEL ATMOSPHERE FOR EARTH RESOURCES …employed, and the meteorological conditions prevailing at the time of the experiment. Since the NASA Earth Resources Program remote-sensing

NASA TECHNICAL MEMORANDUM

NASA TM X- 58033 November 1969

A MODEL ATMOSPHERE FOR EARTH RESOURCES APPLICATIONS

CASE F -

COPY

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

MANNEDSPACECRAFTCENTER

HOUSTON, TEXAS

https://ntrs.nasa.gov/search.jsp?R=19700006383 2020-03-28T06:27:30+00:00Z

A MODEL ATMOSPHERE FOR EARTH RESOURCES APPLICATIONS

David E. Pitts and Kirby D. Kyle Manned Spacecraft Center

Houston, Texas

ABSTRACT

A computer subprogram set is described which permits the use of radiosonde data to provide model atmosphere data for earth resources applications.

A MODEL ATMOSPHERE FOR EARTH RESOURCES APPLiCATIONS

By avid E. Pitts and Kirby D. Kyle Manned Spacecraft Center

SUMMARY

All earth resources remote-sensing techniques a r e affected by the atmosphere lying between the target and the sensor. The computer program presented in this re- port offers a method of numerical use of radiosonde data so that atmospheric effects may be assessed and possibly removed from the signal.

INTRODUCTION

The objectives of the NASA Earth Resources Program a r e to determine the per- formance capabilities of various sensors, to discover signature criteria of resources, and to develop new sensors and systems that will evenixally enable management of earth resources. To accomplish these objectives, certain absolutes which may be used to evaluate sensing systems and techniques must be established. The laboratory usu- ally offers the best testing environment, but the type of target, the conditions of the path of the signal, and other testing parameters a r e limited. In general, the labora- tory is so restrictive that a successful laboratory test of a remote sensor is necessary but not sufficient to ensure proper operation of the sensor in an application. There- fore, much of the testing is performed in the same environment in which the instrument is expected to operate. Testing under such conditions requires that the data concern- ing the environment between the instrument platform (e. g. , an aircraft or a spacecrdt) and the target be as accurate as possible. Thus, determination of the "ground truth" and description of the state of the atmosphere in the path of the electromagnetic signal a r e necessary.

Remote-sensing techniques a r e affected by the atmosphere lying between the tar- get and the sensor. The amount of noise introduced into the signal by the interaction between the atmosphere and the signal depends upon the type of sensor, the wavelength employed, and the meteorological conditions prevailing at the time of the experiment. Since the NASA Earth Resources Program remote-sensing effort is in a developmental stage, the effects of this interaction a r e presently being determined, and hopefully, the model atmosphere for earth resources applications, presented in this paper, will facilitate analyses of such effects.

The computer subprogram set presented in this paper offers a self-consistent method for numerically calculating the state of the atmosphere based on radiosonde

data given in terms of significant levels of pressure, temperature, and temperahire- dewpoint depression. After data from the radiosonde closest to an aircraft o r space- craft remote-sensing target have been obtained and after these data have been inserted into the computer subprogram set, a programer has almost any desirable atmospheric parameter available for use in his computer programs. In particular, the subprogram se t described in this paper makes available all the necessary quantities for calculation of infrared and microwave absorption o r refraction, o r both. However, no attempt has been made in this paper to include atmospheric absorption calculations in the model at- mosphere; only the basic atmospheric data necessary for the previously mentiriraed c d - culatioas a r e provided.

The model atmosphere was written in the FORTRAN V computer language for the Univac 1168 computer. However, the program is also compatible with Control Data Corporation and IBM FQRTRAN IV compilers. Copies of the computer cards a r e avail- able upon request from David E. Pitts, TF8, Manned Spacecraft Center, Houston, Texas 77058.

SYMBOLS

sound computer symbol ANS(4), speed of sound, m/sec

d computer symbol DuM/D~, increment of the slant path from r to r', cm

e computer symbol ANS(19), E(X), water-vapor pressure, mbar

s conlputer symbol ANS(BO), E(X), saturation water-vapor pressure, mbar

fw computer symbol F(P, X), correction for the departure of the air and water- vapor mixture, from ideal-gas law

computer symbol ANS(5), acceleration caused by gravity, f (Z), cm/sec 2 g

surface gravity, g at Re, cm/sec 2 $0

B computer symbol H(I), geopotential altitude, m

IPa computer symbol HA, HLQW, geopotential altitude at A, where Ha < Hb, rn

computer symbol HB, geopotential altitude at B, m

H computer symbol ANS(15), pressure scale height, km P

H computer symbol ANS(16), density scale height, km P

rn

an b

"d

m 0

rn W

n '

n' '

n STP

n(z>

m.ass percentage of the ith constituent

computer symbol ANS(7), molecular weight of the atmosphere, g/g-mole

molecular weight a t H , g/g-mole b

computer symbol XMO, molecular weight sf the dry a t ~ ~ ~ o s p h e r e , g/g-mole

computer symbol XMO, molecular weight at the surface, g/g-mole

molecular weight of water, g/g- mole

computer symbol XN2, refractive index at r' + (1/2)1~2:

computer symbol XN1, refractive index at r" c (1/2)82

computer symbol ANS(17), refractive index of air at STP

computer symbol ANS(18), refractive index of air as a function of A, T, and P

computer symbol ANB(I), atmospheric pressure, mbar

Pa computer symbol PLOW, atmospheric pressure at Ha, mbar

computer symbol PHIGH, atmospheric pressure a t EIb, mbar

computer symbol ANS(1 3), specific humidity, g/kg

computer symbol ANS (1 4), specific humidity at sahrat ion, g/kg

9 computer symbol RO, universal gas constant, 8. 31432 x 10 ergs/(mole OK)

computer symbol RE, mean radius of the earth, 6371.299 km

computer symbol ANS(1 2), relative humidity, percent

computes symbol XS-XL, X-component of ,

computer symbol US-YL, U-component of

computer symbol HS-HL, Z-component of , k2-I-I

computer symbol ANS(lO), mixing ratio of the water in the atmosphere, g/kg

computer symbol S2, distance to shell Z + AZ on the refracted path, km

computer symbol $1, distance to shell Z on the refracted path, km

distance from the center of the earth to a target, km

computer symbol ANS(11), mixing ratio required for the saturation of water in the atmosphere, $/kg

distance from the center of the earth to a spacecraft, km

computer symbol S, Sutherland's constant, 110.4 K

distance

computer symbol ANS(%), kinetic atmospheric temperature, OK

computer symbol ANS(G), virtual temperature, O K

computer symbol T( ), temperature at H OK a'

computer symbol TVLOW, virtual temperature at Ha, OK

computer symbol AN$(%), temperature at Hb, OK

computer symbol TVHIGH, virtual temperature at Hb, O K

dewpoint temperature, O K

computer symbol TD( ), dewpoint temperature at Ha, OK

computer symbol ANS(9), dewpoint temperature at %, OK

molecular scale temperature, OK.

computer symbol TOS, angle between rll and rsp', rad

time

identifier of the significant-level data set of radiosonde code .

computer symbol 2, geometric altitude, km

computer symbol ZL, altitude of a target above the earth, km

computer symbol ZS, altitude of a spacecraft above the earth, %tm

computer symbol BETA, 1.458 X kp/(sec O K m)

r2tio of specific heats

computer symbol PHI, angle from the zenith down to the tangent to the path at the target, sad

computer symbol C(3), distance upward from a local station to a spacecraft, rad

computer symbol C(2), distance eastward from a local station to a space- craft, rad

computer symbol THETAL, target longitude, input card, deg (internally, rad)

computer symbol THETAS, spacecraft longitude, input card, deg (interndly, rad)

computer symbol XLAMDA, wavelength, microns

computer symbol ANS(8), coefficient of viscosity, kg/ (msec)

computer symbol SUM1 , dummy variable, rad

computer symbol C(1), distance southward from a local station to a space- craft, rad

computer symbol ANS(3), atmospheric density, g/cm 3

density of dry air, g/cm 3

3 density of water vapor, g/cm

computer symbol PHIPR, angle between r ' and the path of the ray after refraction, rad

computer symbol PHI, angle between r" and d, rad

computer symbol PHIL, target latitude, input card, deg (internally, rad)

+sp computer symbol PHIS, spacecraft latitude, input card, deg (internally, rad)

q computer symbol PSI, angle between r' and d, rad

MODEL ATMOS PWERES

Model atmospheres for earth resources applications may be described as one of three types: preflight, flight, and postflight. Preflight model atmospheres include those which have been developed from aerospace flight-support models (refs. 1 and 2) and statistical models of cloud cover over the earth (ref. 3). The last of these indicates the probability of success on spacecraft- o r aircraft-borne photographic hissions for earth resources applications.

Flight model atmospheres a r e calculated from sounding-type remote-sensing de- vices aboard spacecraft or aircraft. Flight model atmospheres a r e not presently well developed, but when they a r e well developed, they will represent the ultimate in knowl- edge of the "air truth'' until special-purpose instruments that will perform atmospheric noise extraction in real time a r e developed.

Postflight model atmospheres a r e based upon standard meteorological soundings and a r e used to assist in the development of flight model atmospheres. These post- flight model atmospheres may be described as predictive and nonpredictive.

Predictive postflight model atmospheres use equations of motion, thermodynam- ics, and continuity and standard meteorological soundings to predict (in time and space) the state of the atmosphere near the target for a remote sensor mounted on an instru- ment platform. This type of model atmosphere is not presently well developed. Non- predictive postflight model atmospheres offer a self-consistent method of calculating a model atmosphere at the position of a radiosonde which may be located near the experi- ment platform. The subprogram model atmosphere set discussed in this paper has the capability of performing either as a nonpredictive postflight model atmosphere o r a s a preflight model atmosphere, depending on the form of the input data.

EQUATIONS FOR THE MODEL ATMOSPHERE

The model atmosphere may generally be considered to be in a state of quasi- static equilibrium. That is, when the equations of motion, thermodynamics, and con- tinuity a r e scaled and when closed sets a r e found, the large-scale (i. e. , the first order) vertical-component solution will show that, except near clouds with high-velocity updrafts, the hydrostatic equation

applies well. In equation (I), P is atnnosyheric pressure, Z is geemetsic a'ltibde, p is atmospheric density, and g is the acceleration caused by gravitg. At pressures anad temperatures experienced in the atmosphere of the earth, the ideal-gas law is usu- a l l y accurate to withthin 1 percent. The equation of state

is a form of the ideal-gas law, where m is the molecular weight of the atmosphere, R is the universal gas constant, and T is the kinetic atmospheric temperahre.

With certain reasonable and valid assumptions, the proper combination of the hydrostatic equation (eq. (1)) and the ideal.-gas law (eq. (2)) results in eq~~at ions (3) and (4)) which a r e derived in detail in reference 4. If ~ T * / ~ I I [ 0, where T* is virtual temperature and PI. is the geopotential altitude, then

and if a ~ * / a H = 0, then

b a P = P exp

In equations (3) and (4), Pb is the atmospheric pressure a t Hb, Pa is the at-

mospheric pressure at Ha, Tb* is the virtual temperature at Hb, Ta* is the virtual

temperature at Ha, go is the surface gravity, m is the molecular weight of the dry d atmosphere, Ha is the geopotential altitude at A, and Hb i s the geopoteniial altitude

a t B. In the upper atmosphere, a fictitious temperature designated as molecular scale temperahre Tm is defined in order to include variations in molecular weight

(caused by molecular dissociation) and temperature in one variable.

where ma is the molecular weight at the surface. Similarly, in the lower atmosphere, 0

a quantity designated a s virtual temperature T* is defined in order to include varia- tions in molecular weight (caused by water vapor) and temperature in one variable.

Therefore, T* and Tm may be used interchangeably in equations (3) and (4); this

fact eaables the use of equations (3) and (4), which were derived for planetary atmos- pheres in reference 4.

As shown in appendix A, the proper combination of the equation of the state of dry air, the equation of the state of moist air, and equation (6) gives the exact expres- sion of %* as a function of temperature, pressure, and water-vapor pressure.

where fw is the correction factor for the departure of the air and water-vapor mix-,

ture (from the ideal-gas law) and e is water-vapor pressure. Equateotms (3) and (4), whidl a r e the fundamental equations of subroutine RlODATM calculations, a r e used in different forms to find the altitude of the significant levels and to find the pressure a t a level between significant levels.

Subroutine MODATM

When atmospheric data at a particular altitude a r e desired, either geometric altitude is used a s the calling variable, or pressure is used a s the calling variable and a corresponding geometric altitude is calculated by using equations (3) and (4). Geo- potential altitude H is calculated by

where Re is the mean radius of the earth. Geopotential altitude is then used to cal-

culate temperature, virtual temperature, and molecular weight.

Temperahre is calculated by

where Tb is the temperature at Hb, and Ta is the temperature at 3,. Virtual tem-

perature is calculated by

Molecular weight is calculated by

where m is the molecular weight at Hb. b

When P and T* a re known, a form of the equation of state (eq. (2))

Pmd p=- RT*

is used to calculate density. Then, additional quantities related to altitude, pressure, density, molecular weight, temperature, and virtual temperature a r e calculated. The equations for the speed of sound Csound, acceleration of gravity g, coefficient of vis-

cosity p, saturation mixing ratio r S' saturation specific humidity qs, pressure scale

height H and density scale height H a re as follows: P' P

RT* H =- P mdg

where y is the ratio of specific heats, /3 is 1.458 X lom6, S is Sutherland's constant, and es is the saturation water-vapor pressure. Equations (13), (15), (18), and (19)

a r e derived in reference 1, equation (14) is derived in reference 4, and equations (16) and (17) a r e derived in reference 5. The fw-factor is calculated by a function subpro-

gram simulating tables 89 and 90 given in reference 6.

For calculations of variables describing the amount of water vapor in the atmos- phere, dewpoint temperature Td is calculated as follows:

where T is the dewpoint temperature a t Hb, and T is the dewpoint tempera- d, b d, a

ture a t Ha. The equilibrium vapor pressure over a plane surface of water (ref. 6) is

then calculated.

The formula for the vapor pressure over ice (ref. 6) may also be used.

The choice of the temperature ranges during which each of the previously mentioned equations for e is used is determined by the programer (function E(X)). As presently se t up, only equation (21) is used. E q u a ~ o n s (21) m d (22) a r e used for cdcuIating eS by using T in place of Td.

With the previously discussed basic quantities available, the remaining atmospheric qunt i t ies may be cak- culated. The equations for the mixing ratio r, relative humidity Rel, specific humidity q, refractive index n STP

(in wavelength), and refractive index n ( ~ ) (in P, T, and waveleneh) a r e as follows (ref. 5) :

For the infrared region (ref. 7)

and

where X is wavelength. If the wavelength is in the microwave region (A > 12 5Q0 mi- crons, i. e. , X > 1. 25 centimeters), then

as shown in reference 8.

The input variables of MODATM a re included in the calling argument, and all output variables (i. e . , the variables calculated by equations (3) to (28)) a r e stored in a "common block" in the array ANS. Detailed instructions on the use of subroutine MODATM a r e included in comment cards. For data-card information, see the discus- sion on subroutine INPUT in this report.

Subroutine l NPUT

The purpose of subroutine INPUT is to read the input data cards necessary to set up the significant levels of various atmospheric parameters (i. e. , altitude, pressure, temperature, and dewpoint temperature) for subroutine MODATM. Subroutine INPUT is initiated by MODATM whenever pressure (i. e . , A ~ s ( 1 ) ) is set equal to a number which is less than zero, and because of this fact, many sets of radiosonde data may be used successively, but not concurrently.

The input data may be of the form given in the significant levels (i. e . , VV) of pressure, temperature, and temperature-dewpoint: depression for a radiosonde. Table I shows an example of radiosonde data and the key to the radiosonde code. Table 11 gives the input data cards for the example shown in table I.

Subroutine INPUT is also constructed to accept input data other than radiosonde code VV. If the first data card encountered is blank, then each of the next data cards

will. be read in uncoded form (i. e. , as al.tihde, temperahre, and relalive hur-kridily). AM example of the inwt dab cards necessary to set up the 15' N m n u d model (ref. 2) i s inchded in table 111.

Levels of possible condensation a r e indicated by the word "condensation'; in the print-out of the significant levels. This occurreiace is detersnined by T - Td c 2O K at 1500 meters and T - Td : 8" K at: 9000 meters, which i s expressed

by the approximate expression

'P - T~ < 1.0 + 0.000777B (meters) (29 >

Subroutine REFRAC

Subroutine REFRAC is included to assist in making refracted path cdculations throughout the atmosphere. The basic equations a re developed (ref. 9) from Snell's law

n' sin @' = n" sin * 630)

and from the law of sines

as shown in figure 1. In equations (30) and (3f) , n' is the refractive index at r' + ( f / 2 ) ~ % , @' is the angle between r' and the path of the ray after refraction, n" is the refractive index at r" + (1/2)h%, Q is the angle between r' and d, @" is the angle between r" and d, r' is the distance to shell Z + AZ on the refracted path, and r" is the distance to shell % on the refracted path.

The combination of equations (30) and (31) gives

@' = sin -1 n"r" sin 0" ( nt, )

and

Q = sin

Thus, by using known values for r", r', A, and $" and by initiating MODATM to obtain values for n" and n', the angles $' and 51/ a re calculated. If a continuous path is desired, $" should be set equal to $', and r" and r' should be incre- mented. Then, subroutine REFMC should be called again.

Slant-path calculations a r e also made available by using the law of sines to cal- culate the increment d of the slant path from r to r' as follows:

d = r" sin($" - $) ,

sin I&

Since subroutine MODATM is called by subroutine REFRAC and since subroutine MODATM is called last for the altitude corresponding to the middle of d, the array AN$ may be used externally to calculate the amount of water vapor or the total atmos=- pheric mass that was traversed over distance d. For the initial calculation at the target point, the angle % (i. e . , $") is needed; therefore, subroutine PATH is pro- vided to calculate % for the programer.

subroutine PATH

The principal purpose of subroutine PATH is to calculate the angle c; however, while calculating c , it is also convenient to calculate the columnar mass and the pre- cipitable water vapor along this path. These three quantities a r e stored in the array .ANS. If subroutine PATH is called prior to the calling of subroutine MODATM, ANS(1) will be se t equal to -1.0, and subroutine MODATM will be called such that subroutine INPUT is activated, eliminating the future need to call subroutine INPUT externally. Subroutine PATH is thus programed to be called only once for each radiosonde sounding.

The initial guess at is calculated by finding (-1, the vector from the SP

target (1) to the spacecraft (sp), as shown in figure 2 and as developed in reference 10.

) are The components of (r SP

and

RZ = Re + Z ) sin $ - ( SP SP

(Re + Z sin GL 1)

where B is the longihde of the spacecrsuft, e is the jlongihde of the hrget , G3 SP -

is the lat ihde of the target, @SP

is the latitude of the spacecrdt , Z is $tile altihde SP

of a spacecraft above the earth, and Zl is the altitude of the target a.bove the earth.

The components ( Ry, and R ) a r e found by coordimte transformation in Z liEhe c o o r d i m ~ , ~ ays&ern of the target to be [", qvv, and {" , which are 4!1e respective disbnces southward, eastward, and upward from a local sht ion to the target,

'a' r

e l sin sin e -cos Q

cos e 0 438)

c o ~ $ ~ s i n e ~ s in$ l

The unrefracted zenith angle

can then be found. Next, the angle TOS between rll and r ' is calculated by using the definition of the dot product SP

$. * TOS = cos 1 ( 1 r p )

lrspll . 15'1

so that Lee best refracted path from the target to the spacecraft (fig. I) may be found by iteration.

Iteratiort of paths from the equations developed in the description of subroutine REFIUC is used to find @' and @ for each level, and since

integration proceeds until

Then, 23 86 is compared to TOS for the purpose of iterating on < as follows

[ (t c At) = [ (t) - (Z A[ - 110s) 2

until 1.Z At - TB( a 0.0001 radian (0.0057'). This procedure yields an accuracy on

of apprmtimately 3 X 10-' radian (0.17'). The quantities eolurnlmr mass and pre- cipi%;s%bl@ centimeters of water along this refracted path a r e calculated, respectively, in the f oBEov~ing equations.

and

The increments on AZ are made to be nlultiples of 10 smaller than Z - Z , such that sP 1

where i is 10, 100, 1000, et cetera and AZ r 0.2 kilometer.

The subroutine ATMM3 reproduces the U. S. Standard Atmosphere, 9962 (ref. 1). Subroutine ATMQS3 is called with geometric altitude from which geopotential altitude is calculated. The equations which a r e subsequently used for ATMOSS are many of those developed for subroutine MODATM. Equations (3) to (5) and (8) to (15) a r e common to both subroutines. The main difference between subroutines ATMW3 and MODATM is that in subroutine A T M a 3 , all the significant levels a r e included in a data statement so t%at no data cards a r e necessary, and the output variables a re more limited; that is, only the Iisst eight variables in array ANS a re available. These variables a re pressure, temperature, density, speed of sound, acceleration of gravity, molecular scale temperature, molecular weight, and coefficient of

viscosity. The main purpose for including subroutine ATMW3 i s that if atmospheric data above the maximum-altihde radiosonde data a r e required of subrsrrtine >4OBATM, &en ATMm3 is aukomaticdly called. The main impact subroutine ATMWS has on m a y s s s is that if the mmimum usable radiosonde altitude is 4 0 kilc~saeters, sigxazi- cant water v q o r will be imored since the subroutine ATMm3 includes no water vapor* Ia~ tmct ions on the use of subroutine ATM-3 a re included in comment cards in the subp~ogagam. The comjuter print-out, including A1 subroutines, is s h o ~ ~ n %D

appendk B.

CONCLUD I NG REMARKS

It is hoped that this nonpredictive model atmosphere for earth resources applica- tions will fill the need for atmospheric data until predictive postflight or flight models can be developed.

Manned Spacecraft Center NationaS Aeronautics and Space Administration

Houston, Texas, November 15, 1969 160- 75-03-00- 72

REFERENCES

1. U. S, Committee on =tension to the Standard Atmosphere (COESA): U. S. Standard Atmosphere, 1962. U. S. Government Printing Office, Dec. 1962.

2. U. S. Committee on Extension to the Standard Atmosphere (COESA): U. S. Standard Atmosphere Supplements, 1966. U. S. Government Printing Office.

3. Sherr, PmE E. ; Glaser, Arnold E. ; and Barnes, James C. : World-Wide Cloud Cover Distributions for Use in Computer Simulations. NASA CR-61226, 19638.

4. Pitts, David E. : A Computer Program for Calculating Model Planetary Atmos- pheres. NASA TN D-4292, 1968.

5. Saucier, Walter J. : Principles of Meteorological Analysis. University of Chicago Press, 1955.

6. List, Robert J. : Smithsonian Meteorological Tables. Sixth ed., Publication No. 4014, Smithsonian Institution, Washington, D. C., 1966.

7. Anding, David: Band- Model Methods for Computing Atmospheric Slant- Path Molecular Absorption, report 9142-21-T, Willow Run Laboratories, Univ. of Michigan, Feb. 1967. (Also available as NAVSO P-2499- 1. )

8. Valley, Shea E., ed. : Handbook of Geophysics and Space Environments, McGraw- Hill Book Co., 1965.

9. Smart, William M. : 'I'ext- Book on Spherical Astronomy, Cambridge Univ. Press , 1960.

10, Pitts, David E. : A Mathematical Technique for Programming Automatic Picture Transn~ission Tracking Angles, Appl. Meteor. , vol. 7, no. 6, Dec. 1968, pp. 9036- 1038.

May 10 1963 80802

2nd Trans

a ~ h e significant level code is VV. Far VV, the code is iippp TTTdd where

ii = identifier of a set of data; the two characters are identical (e. g., 00, 11,22,33).

ppp = pressure in mbar except the 4th character from the right is suppressed (e. g. , 970 = 970 mbar, and 016 = 1016 mbar).

TTT = temperature, + if last digit is event, and - if last digit is odd.

dd = dewpoint temperature. If 00-49, multiply by 0. 1 for "C; 50 = 5.0" C; 51-55, not used; 56-99, subtract 50 for "C.

That is, 02 = 0.2, 56 = 6.0, 60 = 10.) Slashes indicate no data.

TABLE a. - INPUT DATA CARDS FOR LAKE CBBTCEES, LOIbLSUNA,

RADIOSBNDE DATA

k t - Earth -b-Atmosphere L-

Figure 1. - Refraction-path geometry through a spherically symmetric atmosphere.

. Figure 2. - Resultant vector f rom the target to the spacecraft in fked-ear th center coordinates.

APPEND I X A

DERIVATION OF VIRTUAL TEMPERATURE T *

The equations of the state of dry a i r

af water vapor

and of wet air

can be used with the mass percentage formula for molecular weight

to give a formula for the relationship of temperature, molecular weight, pressure, and water -vapor pressure

Equation (As), when simplified, becomes

By employing the definition of T*

and by using equation (A6), the exact expression for T* may be found in terms of T, e, and P

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