Spectral Irradiance from Stars and Planets, above theAtmosphere, from 0.1 to 100.0 Microns

R. C. Ramsey

Using published data on star irradiances in the visible region and assuming a blackbody distribution ofenergy, irradiances from stars and planets are calculated over the spectral region of 0.1 to 100 microns.Results are presented in chart form for ready application in the design of viewing systems operating inspace.

Introduction

The designer of optical systems which must operateagainst a stellar background must determine the ir-radiance from the stars and planets in the spectral re-gion to which the particular system responds. In theabsence of actual measurement data covering the entirespectral region, this determination ordinarily requirescalculations' based on published measurements ofvisible magnitude and effective temperature.

For the convenience of these designers, pertinentdata have been assembled and the necessary calculationsperformed for the complete spectral region from 0.1to 100 b'. The results are presented here in the form ofeasy reference charts, one giving the spectral irradiancevalues of the brightest stars and planets, including allthe "red stars" of importance in the infrared region (seeFig. 1) and the other giving the population of starshaving a spectral irradiance above any given value (seeFig. 2).

The data presented here pertain to the irradianceabove the atmosphere. For systems operating in theearth's atmosphere, values of spectral absorption by theatmospheric constituents can be readily applied to thechart values.

In making these calculations it was assumed that thestars behave as Planckian emitters, * and that thereforethe spectral irradiance curve for any of these sourcesmay be obtained when the visible radiance and effectivetemperature are known. In the case of the moon andplanets, the data include both self-emission and sun-

The author is in the Infrared Department of Raytheon Com-pany, Santa Barbara Operation, Santa Barbara, Calif.

Received October 1961.* The validity of this assumption is discussed in Appendix A.

reflected energy. Self-emission values lie mainly in thelonger wavelength region beyond 2.5 pu, and have beencalculated as a function of known temperature, diam-eter, and distance from the earth.

Spectral Irradiance from ParticularStars and Planets

Figure 1 represents the spectral irradiance reachingthe top of the atmosphere from the following sources:

1. The "Brightest" Stars (Fig. 1(a)). These includestars which exhibit the largest irradiance in the visibleregion as well as the more prominent red stars whichpredominate in the infrared spectral region by virtue oftheir cooler temperatures. The variable stars Betel-guex, Mira, and R. Hydrac are presented at their largestvalue of irradiance.

2. The Planets (Fig. (b)). Only the moon andbrighter planets are presented. Both the sun-reflectedenergy and that obtained from self-emission are given.The total irradiance is equal to the sum of the two kindsof curves.

Table I gives the visible magnitude and effectivecolor temperature values of the stars and planets ofFig. 1. The list contains all the stars in Schlessinger'sCatalogue of Bright Stars which are calculated to yieldan irradiance, above the atmosphere, of at least 10-12watts cm-2 in either the PbS region (1-3 /i) or thebolometer region (0.3-13.5 ,u).

It should be noted that all the curves of Fig. 1 havethe same shape on the log-log graph. This is char-acteristic of the Planck radiation function for a black-body. Figure 3 shows a well-known relationship de-rived from the Planck function, and plotted againstXT, in relative units. This function is equal to thepower radiated per unit wavelength interval X per unitarea of a blackbody at temperature T divided by the

July 1962 / Vol. 1, No. 4 / APPLIED OPTICS 465

* I

MV

-4-

-3 -

SIRUS

ACHERNARRIGALMIRACRUCISALTAIR

BETELGEUXVEGA ANTARESR. HYDRAE

f0 GRUISPOLLUX

0 CENTAURICANOPUS

2 \\ \Nt . ARCTURUSANTARESCAPELLA

0.5 1.0 2.0 50.0 100.(5.0 10.0 20.0WAVELENGTH (MICRONS)

Z

In

5:QI

5 0 10 020 0WAVELENGTH (MICRONS)

maximum power radiated per unit w'avelength interval.Since the stars are assumed to be radiating at oneeffective blackbody temperature (effective color tem-perature), the curve of Fig. 3 positioned along the ab-scissia according to appropriate values of X will give the

relative spectral irradiance for any star. The wave-length at which the peak irradianice will occur is readilyobtained from Wien's Displacement Law,

X,05 ' = 2898 ( in microns; ' il 'K).

(a)

(b)

Fig. 1 (a), (b). Calculated spectral irradiance from planets andl)rightest stars at top of atmosphere. (b): *-calculated irra-diance from planets, at brightest, due to sun reflectance only.

# -calculated irradiance from planets, due to self-emission only.GE1-inferior planet at greatest elongation. OPP-superiorplatnet at ol)l)osition. QUAD-superior Islanet at quadrature.

Determination of Peak Spectral Irradiance

The magnitude of the peak spectral irradianee is

obtained by use of the quoted value of visual magnitudemv. The visual magnitude is a function of the ir-

radiance measured in the visible region of the spectrumfrom the particular star or planet. It is defined as

l = 2.5 logo(I/Io).

(1)

-Now if the magnitulde of irradiance at X1nax is known,

the spectral curve for the particuilar star may be deter-

miniied.

(2)

At' the top of the atmosphere, zero visible magnitude

corresponds to visible irradiance (lo) of 3.1 X 10-13

wa,.ttS (m- 2 (as shown in Appniidix B). Equation (2)

466 APPLIED OPTICS / Vol. 1, No. 4 / July 1962

-9.10

-lo10

-II10

-12

10

zZ0IL)

(\I

1)

UJ-z5

C)I

0(I) pj

i.1-1

0.2

I I

I I i 8 If I -1 I - - - - . �.. . . .. . _. _

'N

Z/

\

. . . .. .

9 o;_ -- z - C\~j

N ~~~~~~50(0

.1 0. 2.5 0 201. 0 5. 0.

-500

( 200

(0's - - - - 5_____

(0's~~~~~~1.

0. 0.2 0.5 (. 20 . (0.0 20.0 50.0 (0 0

WAVELENGTH (MICRONS)

Fig. 2. Minimum spectral irradiance of stars for various popula-tion levels, above the atmosphere.

,T CM- K)

Fig. 3. Planck function.

is plotted in Fig. 4, from which the visible irradianceI, is obtained for any star or planet.

In order to obtain the peak spectral irradiance fromthe value of 1, it is necessary to determine the blackbodyvisible response fraction. This is given by

fTo Vx( T)SeX;

sc( ) - 1VX(T)S? (3)f wx(7T)dx

where W(T) is the ordinate of the lanck blackbodyfunction at wavelength , and Sex is the fractional re-sponse of the eye at the same wavelength. The visibleresponse curve (standard observer) peaks at 0.556 ,4 and

falls to zero at approximately 0.4 and 0.7 ,. Equation(3) is plotted in 1ig. 5 as a function of temperature, overthe range of measured star temperatures, and was ob-tained by performing the numerical integration.

(Reasonably accurate values of fle( T ) may be quicklyobtained from any tables of the Planckian radiationfunction by assuming that Se is flat between 0.51 and0.61 bt, and zero everywhere else.)

The total irradiance over the complete wavelengthspectrum at the top of the atmosphere is therefore thequantity I(mv)/n7e(T).

The peak spectral irradiance is

J(1ml) W~nax(WT)(4)

The second factor on the right side of Eq. (4) is themaximum value of the Planck function (equal to 1.290X 10-" X T' watts cm-2 micron-') divided by theStefan Boltzmani function (equal to 5.679 X 10-12 X T

4

watts cm- 2 ). Figure 6 shows Eq. (4) plotted for variousvalues of the visible magnitude and effective startemperatures. The designations denoting Star Spec-tral Classifications are also indicated in the figure.These are explained below.

The curves of Fig. were obtained from the data ofTable I, Eq. (), and Fig. 6. To facilitate the drawing

Table I. Visual Magnitudes and Effective Temperatures ofPlanets and the Brightest Visual and Red Stars

Name (planets)

1. Moon (full)2. Venus (at brightest)3. Mars (at brightest)4. Jupiter (at brightest)5. Mercury (at brightest)6. Saturn (at brightest)

Stars1. Sirius2. Canopus3. Rigel Kent (double)4. Vega5. Capella6. Arcturus7. Rigel8. Procyon9. Achernar

10. 1 Centauri11. Altair12. Betelguex (variable)13. Aldebaran14. Pollux15. Antares16. a Crucis17. Mira (variable)18. f Grruis19. R. Hydrae (variable)

Visualmagnitude

(re )

-12.2-4.28-2.25-2.25-1.8-0.93

-1.60-0.82

0.010.140.210.240.340.480.600.860.890.921.061 .211.221.611.702.243.60

Effectivetemperature

T (K)

590059005900590059005900

11 ,20062C04700

11,20047003750

13,0005450

15,00023,000

750028103130375029002810239028102250

July 1962 / Vol. 1, No. 4 / APPLIED OPTICS 467

-

I0l

0U)I-

-Io-12

0z0a:a:

IO-13

Io-15 j X I I I I I I- -4l -A ' -0 -I ) I 9 A Id 5, 6

VISUAL MAGNITUDE MV

Fig. 4. Effective irradiance in the visible region (standard ob-

server) versus visual magnitude.

of the curves, a template conforming to the curve ofFig. 3 was constructed and used. It was also found to

be very useful in performing the calculations of spectralstar populations.

Planetary Self-Emission

The spectral self-emission curves for the planets werealso calculated, and are included in Fig.lowing equation was used to determine

1. The fol-the spectral

-3TEMPERATURE (DEGREESKELVIN) (lO)

Fig. 5. Fraction of the total radiation emitted by a blackbody,at temperature, ', visible to the standard observer.

irradianee curves due to planetary self-emission.

aT Ax,,-(T'I) 7rd2Hx(Ie1Lk) . n (1 I ) ,r2(1- (5)

where

HXtpeak = peak spectral irradiance, in watts cm-2

micron-1

77 = effective surface temperature of the planet(values taken from ref. 2, Table 832),

(I-A)= assumed emissivity of the radiating body,where A is the value of the albedo in thevisible region (values taken from ref. 2,Table 837); it is assumed that the planetsradiate as grey bodies,

7rd2 = radiating surface area of the planet, d is thediameter (values taken from ref. 2,Table 831),

D = distance to the planet from the earth.This value is, of course, variable with therotation of the planets around the sun.In the case of the inferior planets, Mer-

Table II. Data Used in Calculation of Planetary Self-Emission

1)istancefrom Effective H,\(p-k)

1)iameter, (d earth, D Position of Albedo, temperature, ' (watt-cm- 2

Planet (103 km) (106 km) planet A (0K) meter-') (A)

Mercury 4.84 137.8 Greatest 0.058 690 5.84 X 10-1 4.20

Venus 12.32 103.3 J elongation 0.76 330 6.04 X 10-12 8.7

Mars 6.66 78.3 Opposition - 0.148 285 3.75 X 10-12 10.16171.8 quadrature 7.76 X 10-13

Jupiter 139.80 628.3 0.51 135 3.49 X 10-13 21.46

Saturn 115.06 1277.0 Opposition 0.50 120 3.26 X 10-14 24.14

Uranus 3.72 2720.0 0.66 90 1.96 X 10-6 32.19

Moon 0.27 0.384 0.072 400 2.51 X 10-7 7.24

Self-emission from Neptune or Pluto is insignificant.

468 APPLIED OPTICS / Vol. 1, No. 4 / July 1962

v _ - ,1 -. ,

6 8

z00

U)

Li

0

0

CL

lo-13

M=-5

F ~~~~~~~~~-3

-42

2 - -7

(I -. -~~~~~~~

(I ~~~ -. -2~~~2I

- I Y fl W1 s I 0 1 iŽZu(0 IL0' o~

umEn 0co

2 4 6 8 10 12 14 16 18 20 22 24 26TEMPERATURE, T, (DEG-K) x 10-3

Fig. 6. Graph used to obtain peak spectral irradiance forvarious values of visual magnitude versus effective temperatureor spectral class.

cury and Venus, the values of D werechosen corresponding to their greatestelongation. For the superior planets,values of D used correspond to theiropposition. In the case of Mars, theclosest superior planet, the value of D,when the planet is at quadrature, is alsoused. For a definition of these terms,see ref. 3, Chapter IX.

Knowing HX(peak) and the wavelength value at whichthis peak occurs

(/max = 2898)T(°K))

the spectral irradiance curves can be readily con-structed.

Table II gives all of the values used in the calcula-tions.

Spectral Star Populations

Figure 1 yields data for the brightest stars andplanets. It is also of interest to obtain an idea of thespectral irradiance from the total star population.

Li

zC

-

Fig. 7. Population of

NUMBER OF STARS

stars brighter thanmagnitude.

104 v l5

a particular visual

As reported in ref. 3, page 604, the stars are dividedinto classes designated by letters, with decimal classifi-cation denoting intermediate types (Draper Classifica-tion). On page 705 it is stated that 99% of all thestars are included in six principal spectral classes, B, A,F, G, K, and M. Reference 3 also includes a table of thepercentage of stars of the various spectral classes for dif-ferent regions of visual magnitude and, on page 622, alist of the number of stars brighter than a given magni-tude. Krasno 4 has extended this table to the smallvalues of visual magnitude. Figure 7 shows this plot.The combination of the table of star percentages versusspectral classes and the data in Fig. 7 yields the actualnumber of stars brighter than a given visual magnitudefor each of the six principal spectral classes. Thesevalues are listed in Table III.

The data of Table III are plotted in Fig. 8, assumingthat in each spectral class, the spectral irradiance curvesof the stars peak at the same wavelength value. Theplot was readily obtained by use of the templatecorresponding to Fig. 3. For example, the data in row1 of Table III correspond to the curves in Fig. 8 thatcross the visual magnitude line at a value of 2.24, each

Table IlIl. Number of Stars in the Principal Spectral Classesabove a Given Magnitude

B A F G K M(BO (B8 (A5 (F5 (G5 (K5

Magnitude to to to to to tolimit B5) B5) F2) GO) K2) M8)

2.24 15 15 4 6 8 73.24 52 43 18 23 40 234.24 135 157 56 85 222 655.24 270 562 236 265 672 2156.24 484 2189 792 693 1871 4717.24 976 5879 2268 2415 5930 13328.24 1684 15,083 6162 8079 19,030 4164

July 1962 / Vol. 1, No. 4 / APPLIED OPTICS 469

X5 .1 . 05 10 20 50 1. 00 5.

In.

Z (01

Lr

WAVELENGTH MICRONS)

Fig. 8. Minimum spectral irradiance of stars for various popula-tion levels and spectral classes, above the atmosphere.

curve peaking at the wavelength of the correspondingspectral class. The visual magnitude line is taken fromthe data of Fig. 2. For example, in Fig. 4, a value ofmv equal to zero corresponds to a visible irradiance of3.1 X 10-13 watt cm-'. Now assuming that the visualspectral region peaks at 0.55 .and extends over a 0.1 region, the spectral peak value of 0.55 fo equals 3.1 X10-cr watt cm-a micron-l.

The information in Fig. 8 is not immediately usefulin this form. Therefore, Fig. 2 ias constructed. Itessentially represents the drawing of contour lines ofequal population values through the curves of Fig. 8.Appendix C explains the procedure used in obtainingthe data of Fig. 2.

Use of Figures in System Design

It is felt that Figs. 1 and 2 yield infrared backgroundinformation on the planets and stars that is immediatelyuseful in system design. For example, assume a hypo-thetical infrared system operating in the wavelengthregion around 4 A u, with an equivalent bandwidth of 2,.

A value of NEPD (noise equivalent power density) ofthe system equal to 10-2. watt cm-' corresponds to anaverage spectral irradiance of 5 X 10-14 watt cm-2

micron- at 4 . From Fig. 2, one observes that thereare about 12 stars that would be "seen" by the system.Most of these 12 stars can be immediately identifiedfrom Fig. 1, and the signal to noise from each calculated,as well as the signal to noise obtained from the plants.

The above assumes that the system is operating

above the atmosphere. However, if below the at-mosphere, absorption information may be readily ap-plied to any data obtained from Figs. 1 or 2.

Appendix A. Blackbody Radiation from Stars

Figure 9 (taken from ref. 5) shows the measuredirradiance from the sun as extrapolated to the top of theatmosphere, and compared with 57000 K and 6000'Kblackbody curves. It shows that a Planckian curvegives a reasonable fit and gives some measure of thedeviation or error obtained when using the blackbodyassumption.

The work of Stebbins and Whitford6 provides a goodbasis for assuming the star radiation to be that of ablackbody. 238 stars of all spectral types from 0 to Mwere observed in six spectral regions from 0.353 to1.03 y. The observed colors of the stars agree closelywith the colors of a blackbody at suitable temperatures.

Measurements performed by Hall7 cast doubt on thevalidity of the use of the color temperature in calcu-lating irradiance from the cool stars (classes K andM). The calculations of Larmore (which are identicalto those in this report) predict irradiance values fromthese stars in the PbS region that are a factor of 2 to 4below Hall's measured values.

Appendix B. Calculation of the EffectiveIrradiance 10, in watts/cm2 , in the Visible,Corresponding to a Visual MagnitudeValue of Zero

In ref. 2, Table 827, page 730, it is stated that theapparent magnitude of a standard candle, at a distanceof 1 meter, is -14.2.

z00

'I'

0U)

0z'5

a:-j

IL)wa.0,

0.2

0.1

0.2 0.4 0.5 0.7 1.0 1.5WAVELENGTH (MICRONS)

Fig. 9. Measured spectral irradiance above the atmosphere fromthe sun compared with blackbody curves (from ref. 5).

470 APPLIED OPTICS / Vol. 1, No. 4 / July 1962

The relation between the visible light received fromtwo stars and their magnitudes is expressed by theformula (see Eq. (2)):

11log - = 0.4(m2 - n).12

Since a unit candle source emits one lumen per stera-dian, when m2 equals - 14.2, 12 equals 10-4 lumen/Cm 2.

For

in, = 0,

I, = (10-4 lumen/cm 2) X 10-5.68

= 2.09 X l0-l0lumen/cm2.*

In ref. 2, Table 73, page 94, the value for the me-chanical equivalent of light is given as 0.00147 watt/lumen. Therefore, for zero magnitude the effectiveirradiance is

Jo = 0.00147 X 2.09 X 10-10

- 3.1 X 10-l3watt/cm2.

Appendix C. Method of Obtaining Fig. 8The contour data of Fig. 2 were obtained by use of

the template corresponding to Fig. 3. The methodused is as follows:

1. For example, a wavelength of 1.0 IL and a spectralirradiance value of 4 X 10-14 watt cm- 2 micron-'

* Allen' states that this quantity is 2.43 X 10-0 lumen/cm2 .Thus Io would equal 3.6 X 10-13 watt/cm

2 . This is larger thanthe value calculated above by 16%. If this new value were usedin the calculations, all the curves of Figs. 1, 2, and 8 relating tothe stars and the planets (sun-reflected irradiance only) wouldmove upwards by 0.16.

determines a point on the graph paper.2. Place the template on the graph so that the peak

lies along the wavelength (0.155 IL) corresponding to theB spectral class.

3. Move the template up or down so that the curvepasses through the above-mentioned point.- 4. Read off the value of visual magnitude deter-

mined by the position of the template and the m, line.(For this case, mv is +2.6.)

5. From the data in Table III, column 1, corre-sponding to B spectral class, which is plotted as a con-tinuous function on a separate graph, a total of 24 starscorresponds to a value of mv equal to 2.6.

6. This same operation is performed for all six of thespectral classes, and the total number of stars obtainedfor this point. (Total equals 890 stars.)

7. Steps 1 to 6 are repeated for sufficient points onthe graph paper so that accurate contours of starpopulations may be plotted.

References

1. L. Larmore, "Infrared Radiation from Celestial Bodies,"U.S. Air Force Project, Rand Research Memorandum, 293-1(17 March 1952).

2. Smithsonian Physical Tables (Smithsonian Institution,Washington, D.C., 954), 9th revised ed.

3. H. N. Russell, R. S. Dugan, and J. Q. Stewart, Astronomy(Ginn, New York, 1945), Vols. I and II.

4. M. R. Krasno, private communication.5. Handbuch der Physik, S. FlUgge, ed. (Springer-Verlag,'Berlin,

1957), Vol. 48.6. J. Stebbins and A. E. Whitford, Astrophys. J. 102, 318

(1945).7. F. F. Hall, Jr., and C. V. Stanley, Appl. Opt. 1, 97 (1962).8. C. W. Allen, Astrophysical Quantities (Athlone Press, Univ.

of London, 1955).

Books for review should be submitted to the Managing Editor,APPLIED OPTICS, 115516th St., N.W., Washington 6, D.C.

Fundamentals of Photographic Theory. Second edition. ByTHOMAS H. JAMES AND GEORGE C. HIGGINS. Morgan and Morgan,New York, 1960. 3 45 pp. $7.50.

It is a pleasure to see a new edition of Fundamentals of Photo-graphic Theory published. The earlier text provided an excellentsource of information useful both for classroom and reference pur-poses. The new edition has incorporated much recent work inphotography and related fields which should prove of inspira-tional and practical value to the serious student of photography.However, in providing some of this additional information, theauthors have been forced to use concepts and mathematics whichmany students who have "a basic knowledge of physics andphysical chemistry" may find unusually difficult.

Throughout the 14 chapters of the book, about 20% is eithernew or rewritten. The book has been completely re-set, and onlya few typographical errors are noticeable. For example, on page165 a complete line in the text has been omitted, and for those of

you who may wonder what it should be, the first edition gives"color change is known as fading. It is the result of a reactionbe-."

Chapter 1 has been left essentially unchanged, but Chapter 2,on emulsion, includes considerably more information on emulsionpreparation and the sensitizing process than was given in theformer edition. Also, the added comments on special emulsionsfor ultraviolet, cosmic rays, and nuclear particles enhances thepractical aspects of the text.

During the past ten years advances made in solid state physicshave given a new impetus to an understanding of the latent image.Chapter 3 deals with these problems and presents informationwhich should whet the appetite of physicists as well as photo-graphic students. Even though some of the concepts may provedifficult for beginning students, sufficient background material isgiven to impart much useful information.

Chapter 4 remains very much the same as in the first edition butsome new material on reciprocity failure draws on the solid stateconcepts developed in the previous chapter.

The changes in Chapter 5 stress the physical mechanisms in thedevelopment process with much of the material rewritten. Inparticular, the material on direct development, physical develop-ment, filament formation, and the charge effect seems to bring the

continued on next page

July 1962 / Vol. 1, No. 4 / APPLIED OPTICS 471