SMITHSONIAN

CONTRIBUTIONS

to

ASTROPHYSICS

VOLUME 2

SMITHSONIAN

INSTITUTION

WASHINGTON

D.C.

Contents of Volume 2

No. 1. Notes on the Solar Corona and the Terrestrial Ionosphere,

by Sydney Chapman, with a Supplementary Note by

Harold Zirin. Pages 1-14. Published February 18,

1957.

No. 2. Chromospheric Spicules, by Sarah Lee Lippincott. Pages

15-23. Published June 14, 1957.

No. 3. Studies of Solar Granulation: I. The Statistical Interpretation of Granule Structure from One-Dimensional

Microphotometer Tracings, by Gerard Wlerick. Pages

25-34. Published June 14, 1957.

No. 4. Variations in the Thermodynamic State of the Chromosphere over the Sunspot Cycle, by R. G. Athay, D. H.

Menzel, and F. Q. Orrall. Pages 35-50. Published

June 14, 1957.

No. 5. Geomagnetism and the Emission-Line Corona, 1950-1958,

by Barbara Bell and Harold Glazer. Pages 51-107.

Published November 27, 1957.

No. 6. Ancient Novae and Meteor Showers. A New Catalog of

Ancient Novae, by Hsi Tse'-tsung; pages 109-130.

Historical Records of Meteor Showers in China, Korea,

and Japan, by Susumu Imoto and Ichiro Hasegawa;

pages 131-144. Published February 12, 1958.

No. 7. Distribution of Meteoritic Debris about the Arizona

Meteorite Crater, by John S. Rinehart. Pages 145-

160. Published January 28, 1958.

No. 8. Sunspots and Geomagnetism, by Barbara Bell and Harold

Glazer. Pages 161-179. Published March 13,1958.

No. 9. On Two Parameters Used in the Physical Theory of

Meteors, by Luigi G. Jacchia. Pages 181-187.

Published June 18, 1958.

No. 10. Orbital Data and Preliminary Analyses of Satellites 1957

Alpha and 1957 Beta, compiled by F. L. Whipple,

L. G. Boyd, J. A. Hynek, and G. F. Schilling. Pages

i-vii, 189-347. Published May 23, 1958.

No. 11. The Statistics of Meteors in the Earth's Atmosphere, by

Gerald S. Hawkins and Richard B. South worth.

Pages 349-364. Published August 5, 1958.

No. 12. Granulation and Oscillations of the Solar Atmosphere, by

Charles Whitney. Pages 365-376. Published July

29, 1958.

No. 13. Optical Properties of Saturn's Rings: I, Transmission, by

Allan F Cook, II, and Fred A. Franklin. Pages

377-383. Published November 14, 1958.

Smithsonian

Contributions to Astrophysics

VOLUME 2, NUMBER 1

NOTES ON THE SOLAR CORONA

AND THE

TERRESTRIAL IONOSPHERE

by SYDNEY CHAPMAN

with a

SUPPLEMENTARY NOTE

by HAROLD ZIRIN

SMITHSONIAN INSTITUTION

Washington, D. C.

1957
Publications of the Astrophysical Observatory

This series, Smithsonian Contributions to Astrophysics, was inaugurated

in 1056 to provide a proper communication for the results of research conducted at the Astrophysical Observatory of the Smithsonian Institution.

Its purpose is the "increase and diffusion of knowledge" in the field of

astrophysics, with particular emphasis on problems of the sun, the earth,

and the solar system. Its pages are open to a limited number of papers by

other investigators with whom we have common interests.

Another series is Annals of the Astrophysical Observatory. It was started

in 1900 by the Observatory's first director, Samuel P. Langley, and has been

published about every 10 years since that date. These quarto volumes, some

of which are still available, record the history of the Observatory's researches

and activities.

Many technical papers and volumes emanating from the Astrophysical

Observatory have appeared in the Smithsonian Miscellaneous Collections.

Among these are Smithsonian Physical Tables, Smithsonian Meteorological

Tables, and World Weather Records.

Additional information concerning these publications may be secured

from the Editorial and Publications Division, Smithsonian Institution,

Washington, D. C.

FEES L. WHIPPLE, Director,

Astrophysical Observatory,

Smithsonian Institution.

Cambridge, Mass.

February 1957.

For sale by the Superintendent of Documents, U. S. Government Printing Office

Washington 25, D. C. - Price 20 cents
Notes on the Solar Corona and the

Terrestrial Ionosphere

By Sydney Chapman

 l

Outward thermal conduction from the

solar corona

We shall consider in the simplest possible

way some properties that would characterize a

model solar corona, static and spherically

symmetrical (while recognizing that the actual

corona is dynamic and asymmetric). The

problem has already been discussed by many

writers (Woolley and Stibbs, 1953). Van de

Hulst (1953) has critically reviewed much recent

work. The problem deals partly with the

escape of coronal gas from the sun's gravitational field, e. g., as considered by Pikelner

(1948,1950) whose estimate of the rate of escape

is judged by van de Hulst to be excessive. Rosseland (1933), Biermann (1948, 1951), and

others have discussed the outward streaming of

gas from or through the corona. These matters

need further discussion, but I hope the present

simple limited treatment is worth consideration.

Let the model corona consist entirely of

protons and electrons, with equal number densities n. This and the temperature T are functions only of R, the distance from the center

of the sun. Without inquiring how this comes

to be, let us suppose that T has a maximum

(To) at a radius Ro which is taken to be 7.36

X1010 cm, namely 4X109 cm greater than RQ,

the "radius of the sun," 6.96 X1010 cm (Woolley

and Stibbs, 1953). Initially no account is taken

of magnetic fields; their influence is briefly considered in the fourth section.

The thermal conductivity K of a protonelectron gas can be calculated fairly exactly

(Chapman, 1954). It is 0.324 K,, where K,

denotes the thermal conductivity, for the same

n and T, of a gas composed of electrons alone,

half positive, half negative, supposed permanent, without recombination.

Let k denote Boltzmann's constant, e the

electrostatic unit of charge, m the electron

mass. Then, allowing for a factor 4/3 to correct

the first approximation to the formula for K

(Chapman, 1954, p. 155), we obtain

25

At(2) =2 -2*7(1+z»);

_AkT( kT V'2

where

and

by use of the Debye cutoff distance (Spitzer,

1956) instead of the cutoff distance that I

proposed in 1922 (Chapman, 1922); the adoption of a fixed cutoff distance is a weakness in

the calculation, but the resulting value of

K is probably correct within 10 percent. With

values of T and n appropriate to the corona,

x is large, and approximately

A2(2) =35.5+13.8 log (T/n1'").

Thus, if T=106 and n=10», ^1,(2) =76.9 (the

use of my cutoff distance would give 55.8).

If 2 = 109 and n=109, 4,(2)=90.7. If T=

2X108 and 71=10*, 42(2)=94.9. The dependence of Aa(2) on T is not great, and on n is

decidedly small. Here A3(2) will be taken to

have the value 85, independent of T and n.

> Geophysical Institute, College, Alaska, and High Altitude Observatory, Boulder, Colo.
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Then for proton-electron gas, at temperature

T,

K=A T5* erg/cm sec°K (1)

where ^4=5.2 X10~7; Woolley and Allen (1950)

adopted 5 X 10~7 as the value of A (Woolley

and Stibbs, 1953, p. 234), taking ii,(2) as 50,

and not using the factors 0.324 and 4/3, not

then published (Chapman, 1954). The present

estimate of A may be uncertain to 10 percent.

It is convenient to rewrite equation (1) as

K=Ko(T/T0); (2)

so that KQ denotes the thermal conductivity

at To, and s=5/2.

The model corona will extend to infinity,

where we suppose that T=0. The conditions

being steady, there will be a constant outward

flow of heat by thermal conduction; the flux

F over the sphere of radius R will be independent of R, and is given by the equation,

F=-4irRiKdTldR. (3)

The gas being supposed everywhere fully

ionized, so that equation (2) is valid at all

distances, equation (3) may be rewritten as

where

(4)

(5)

As T— To at R=R0, the integration of equation

(4) gives

2V+ (6)

1—T*1

=C/Ro— C/R.

As T=0 at infinity,

and therefore

Hence also

T=T0(Ro/R)l'

it+v

,

(7)

(8)

which gives T at any radius R. As s=| , T

varies as 1/R2/7, decreasing only slowly from

the sun.

In table 1 are given the meagi distances x of

the planets from the sun in astronomical units,

and the corresponding values of lO'T/To according to equation (8); if To is 106

, column c

gives the inferred values of the temperature

of the coronal gas at the distance of each

planet. If To is greater or less than 106, the

values in column c must be increased or decreased proportionately.

Owing to the eccentricities of the orbits,

especially of Mercury, Mars, and Pluto, the

temperature of the gas encountered in their

orbits will vary; the variations of 106 T/To are

as follows for these three planets:

Mercury 270, 000° to 308, 000°

Mars 185,000° to 195,000°

Pluto 72, 000° to 83,000°

The variation of T near the sun, and at

greater distances up to the earth (RB/RQ=215)

and beyond, are illustrated in table 2 by the

ratio T/TQ. (The distance 1.06 RQ represents

i?0; more exactly the chosen value of Ro is

1.058 RQ.) Table 2 also illustrates the distribution of the (negative) radial temperature

gradient, in degrees per km; it refers to T0=108,

and for other values of To the values in table

2 need to be multiplied by T0/106, because from

equation (8) it follows that

dR

2 T_2 To

7 R~7 Ro R

The table well shows how small is the temperature gradient, as compared with that in the

earth's own atmosphere, especially for distances

greater than 3 i?0.

TABLE 1.—Properties of corona model in neighborhood

of planets.

a

Mercury

Venus

Earth and

Moon

Mars

Jupiter

Saturn

Uranus

Neptune

Pluto

b

X

0.387

0. 723

1

1. 524

5. 503

9.539

19. 191

30.07

39.6

c

10« T/To

288, 000

240, 000

219, 000

190, 000

135, 000

115, 000

94,000

83,000

77, 000

d

E(ev)

37.2

31.0

28.2

24.5

17.4

14.8

12. 1

10.7

9.9

e

v+

106

77

74

69

58

53

48

45

44

/

10-* vo

4.56

3.30

3. 16

2.94

2.48

2.29

2.07

1.94

1. 87

g

vp

48

32

30

24

13

10

7

5

5
SOLAR CORONA AND TERRESTRIAL IONOSPHERE

TABLE 2.—Coronal temperature gradients.

R/R©

1.06

1. 1

1. 2

1.3

1.5

1.7

2.0

2.6

3.0

4.0

T/To

1.000

.989

.964

.943

.905

.873

.833

.773

.742

.733

-dT/dRin

degrees/km

forT0=10«

.388

.369

.330

.298

.248

.211

. 171

. 122

. 102

.072

R/R©

5

6

10

15

50

100

150

200

215

250

T/To

. 642

. 609

.526

.469

.332

.273

.243

.224

.219

.210

-dT/dRin

degrees/km

for T0=10«

.0527

.0417

.0216

.0128

. 00273

. 00112

. 00066

. 00046

.00042

.00034

The thermal flux inward, from the corona to

the chromosphere, has been estimated by Alfv6n

(1941), Giovanelli (1949), and Woolley and

Allen (1950) (see also Woolley and Stibbs, 1953,

p. 234) with different results, depending on

the different estimates of K and dT/dR. The

value given by Woolley and Allen is 1.8X10*

ergs/cm2 sec, or, over the whole corona,

inward heat flux=l.lX1027 ergs/sec.

From equation (7), for T0=108, it is calculated

that

outward flux=1.38X1026 ergs/sec.

This is naturally decidedly less than the inward

flux, because the estimated inward temperature

gradient (12.5°ykm) so much exceeds the outward gradient.

If To differs from 108, this result must be

multiplied by (r0/108)

7/2

, a factor here given

for a few other values of To:

Tt 500,000 1,500,000 2X10» 3X10«

(7yiO«)V» 0. 088 4. 13 11. 3 46.8

This shows how considerably the outward conductive loss of heat from the corona to outer

space would alter if To should change from a

half million to 1.5 million degrees.

Woolley and Allen (1950) estimated the radiative loss of energy from the corona to outer

space to be 2.7X1028 ergs/sec, or about twice

the outward conductive flux here given. This

radiative loss of energy proceeds mainly from

the layers of the corona near the sun, where

metal ions are most abundant and the density

is greatest. Beyond a few solar radii the radiative loss of energy must be a small fraction of

the conductive loss.1 Their estimate of the

total power needed to maintain the corona at

108 degrees is little altered by the inclusion of

the outward conductive flux: at 1.5 X1027 ergs/

sec it is about 4X10~7 of the total solar output.

Column d of table 1 gives the energy E per

proton or electron, in electron volts, for a gas

whose T equals the number shown in column c;

columns e and / give the corresponding root

mean square speeds v+, ve of the protons and

electrons, in km/sec. Column g gives the mean

orbital speed vP of the planet, in km/sec. The

speed v+ exceeds the speed of sound V in the

ratio (3/7)1/2, where 7 denotes the ratio of the

specific heats: for a proton-electron gas, 7=5/3,

hence (3/7)

 1/2

= 1.34. Comparison of columns

e and g shows that if T0=106 the planetary

motion is decidedly subsonic in the coronal gas,

and the Mach ratio vP/V decreases from about

1/3 for Mercury to about 1/12 for Pluto.

Column d shows that if T0=108 the mean

particle energy exceeds 13.6 ev, which is the

ionization potential of atomic hydrogen, up to

and beyond the orbit of Saturn. As E decreases, the proportion of neutral hydrogen

atoms will slowly increase. This will modify

the radial variation of the thermal conductivity

K, and of the temperature gradient and density.

Statical equilibrium of the model corona

The equation of statical equilibrium of the

model corona is

dp——pg dR

or

d]np=-(dR)!H, (9)

where

H=kTIMg. (10)

Here p and p denote pressure and density, H

the scale height, M the mean molecular mass;

for a proton-electron gas, M is 8.37 XI0"*8,

half the mass of a hydrogen atom. Also

g=g,(R0/Ry, (11)

where g and g0 denote the value of solar gravity at

1

 See the accompanying Supplementary Note by H. Zlrln dealing with

this question.
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

R and 2?0; as the surface gravity on the sun, at

5o=6.96X101*, is 2.74X104, the value of g0

is 2.45X104. If TWO6, the value (Ho) of H

at J?o is 6.73 X10* cm. Let

y=R}R,, u=7i2o/5J7o,

for To= 10*, u=15.31.

Substitution in equation (10) for T from

equation (8), and for g from equation (11),

gives

7

, (12)

(13)

Hence p decreases steadily outwards, not to

zero but to the finite limit p»e~

m

; if To=lO6, the

factor «"• is 2.24 X10"7.

For the number density n(=pfkT),

£ = S JW' 7 exp{-u(l-y->")}; (14)

the factor TQ/T increases outwards from the

sun, the factor p/p0 decreases outwards; n has

a minimum value at Rm, where

and so equation (9) leads to the solution

The minimum number density nm is given by

no

For To= 106 the minimum n is at 1.21 X1013 cm

or 0.809 astronomical unit; and nm/n® is

1.435X10"6. The corresponding value of

at the earth's distance is only slightly higher,

by less than .5 percent; at ten astronomical

units n/no is 2.11 X10"4.

The distance Rm and ratio n/no is rather

sensitive to a change in To. For a few values

of To table 3 gives values of Rm (expressed in

astronomical units as xm), nm/no, and also of

na/»o, where na signifies the number density

at the earth's distance.

Table 3 indicates a considerable variation

of the distance of minimum n as T ranges

from 8X10* to 1.5X108; for the lowest of these

temperatures, the minimum is attained just

beyond the earth's orbit; for the highest of

these temperatures, it is attained about half

way between the sun and the earth. Over this

considerable range of T and of xmi however,

the variation of n is very slight, near and beyond the distance of minimum n, and at the

earth's distance n has almost the same value

as at the minimum.

Table 3 shows also that at the earth's

distance, n/no increases by a factor 620 as

To increases from 8X105 to 1.5 X106. Van de

Hulst (1950) finds that n0 decreases from sunspot maximum to sunspot minimum; for

i?-1.06 solar radii (approximately the same

as Ro) his sunspot maximum value of no is

2.35X108, the same in all solar latitudes; at

sunspot minimum his polar value of no, 8.7 X 10r,

is notably less than the equatorial value,

1.3X10*. If To also is greater at sunspot

maximum than at sunspot minimum, the

disparity of no at the two phases will make the

number density n at the earth vary by an extra

factor of about two as compared with what

table 2 would indicate, if no were assumed

constant. !

TABLE 3.—Minimum density and density at 1 A. U. for coronal models.

(In this table, numbers of the form a" signify oX 10")

io-«r.

n»/iH

n,/n,(*=l)

0.8

1.011

&35-T

a43"7

0.9

0.899

2.73-7

2. 74"»

1

0.809

1. 44"»

1. 44-«

1. 1

0.735

5.57-*

5. 64-«

1.2

0. 674

1. 71-*

1. 75"«

1.3

0.622

4. 45"5

4 57"5

1.4

0.578

9. 98"8

1. 04-*

1.5

0.539

2. 01"*

2. 12-«
SOLAR CORONA AND TERRESTRIAL IONOSPHERE

Behr and Siedentopf (1953) have estimated

that near the earth n (=71*) is about 600. If

n0 is taken to be 2X108, their value gives

3X10~* for nB/iio at the earth's distance.

According to table 2, this would correspond

to a value of To between 108 and 1.1 X106. If

the actual corona varies with To at all like the

model corona here discussed, it would suggest

that the average value of To over the sun

undergoes only moderate changes.

The changing radial distribution of the

proton-electron gas (if To changes) involves an

outward movement of gas if To increases; if

To decreases, the gas must draw back towards

the sun. Changes propagated with the speed

of sound would take about two months to

reach the earth's orbit. The actual corona

seems to show a nonuniform changing distribution of To. The irregularities of To over

the corona will influence the proton-electron

gas at a distance from the sun, but in a

smoothed-out manner.

TABLE 4.—Empirical and model densities in corona.

(In this table, numbers expressed as o» signify oX10y)

R/Ro

1.06

1. 1

1.2

1.3

1.5

1. 7

2.0

2.6

3.0

4 . 0

5.0

6.0

10.0

u n 1O. \f

50.0

100.0

150.0

200.0

215.0

250.0

n

(van de

Hulst:

Sunspot

max.)

2.35"

1. 608

7. O87

3. 76*

1. 48*

7. 11«

2.81*

6.65s

3.13s

9. 0«

4.4*

2.9*

r

T0=10«

(2. 35»)

1. 548

6. 597

3. 08*

&97»

3. 33«

1. 04«

2. 16s

1.02s

2.9*

1.3*

7.2*

2.2*

1 1*

X. x

4.2*

3.5*

3.4*

3.4*

3.4*

&4*

i (calculated)

1.1X10*

(2. 35«)

1. 61*

7. 3F

3. 70'

1. 207

4 91«

1.70*

4. 18s

2.11s

a 7*

3.3*

2.0«

&6»

i it

o. /•

1.5*

1.3*

1.3*

1.3*

1.3*

1.3*

1.2X108

(2. 35»)

1. 67«

a ir

4.34*

1. 557

6.89*

2. 66«

7.26s

3.89s

1.4s

7.1*

4 4*

1.7*

Q 99

y. or

46*

4. 1*

4 0 *

4 1*

4 1*

42*'

Baumbach (1937) and van de Hulst (1950),

using eclipse coronal data, have given estimated

distributions of n as a function of R up to a few

solar radii. The extension of the solar corona

has also been studied by observing how the radio

waves from the radio star in Taurus vary in

intensity when they traverse the outer corona

(Hewish, 1955). The radial distribution of n

estimated by van de Hulst (1950) for sunspot

maximum is compared in table 4 with the distributions given by equation (14) by taking

three different values of To and adopting van

de Hulst's value of no, namely, 2.35 X108.

None of the three distributions exactly fits van

de Hulst's series of estimates, but the "fit" of

the calculated values, depending on only one

chosen parameter, To (no having been adopted

to agree with the eclipse estimate at R Q), seems

satisfactory, for To= 1-1X10* or for some value

between this and To= 1.2 X108. This value has

the order of magnitude indicated by coronal

temperature measurements; it gives a value of

n at the earth's distance of the same order as

that indicated by the zodiacal light observations; and at the intermediate distances, 5 to

15 times RQ, to which the Taurus data refer

(Hewish, 1955), the calculated values of n seem

to be consistent with the radio star observations,

though these only set lower limits to n.

In some recent discussions of the far extension

of the corona it has been supposed that n varies

as some inverse power of R; there seems to have

been no suggestion that n might have a minimum value. The formula in equation (14)

suggests a different kind of distribution. Table

3 shows that near the earth's distance the calculated radial variation of n is very slight for

the three values of To there considered; this is

also implied by the low values of dTJdR given

in table 2. In this respect the present calculations differ from the results given by Behr and

Siedentopf (1953); it is difficult to see how a

rapid outward decline of n near the earth could

be produced.

The mean free path, collision frequency

and collision interval for the electrons

The thermal conductivity in the protonelectron gas is mainly due to the electrons

(Chapman, 1954); it is reduced by the presence
SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.J

of the protons. The collision frequency v for

an electron (of mass TO) is given by

p=2n(TkT/m)»(2oa+2*<ri

3

), (17)

where <r, a\ are the mean collision distances for

electron-electron and electron-proton collisions

respectively. These may be evaluated as

follows. Let D and K, denote the first approximations to the coefficient of diffusion in a

proton-electron gas, and to the coefficient of

thermal conductivity in an all-electron gas

(see first section). Formulae for D and Kt

(Chapman and Cowling, 1952)1 are calculated

(a) by regarding the particles as rigid elastic

spheres, and (b) by considering them to be

centers of Coulomb forces; D depends only on

au and K, only on <r. By comparing the two

forms for D and the two forms for K,, we

obtain expressions for <r and <ri. They are

found to be almost the same (Chapman and

Cowling, 1952, pp. 218, 235):

(18)

and (Chapman and Cowling, 1952, p. 245),

(19)

where (Chapman and Cowling, 1952, p. 179)

in the notation in the first section,

(20)

Thus Ai(2) differs from % A2{2), when x is

large (as here), only by 1. For A2(2)=85, it

is sufficiently accurate to take

Hence

»= 14.14 n/TH,

•(21)

(22)

and the mean collision interval t is 1/v.

At #0, taking 7io=2.35XlO8, 2T

0=106

, gives

vo=3.32/sec, <o=O.3O sec,

and at RB, the distance of the earth,

y*=4.7X10-5/sec, ^=2.1X10* sec=5.9 hours.

The mean free path L is given by the expression,

Hence at Ro, taking To and w<> as before,

Zo=1900 km,

and at RB,

Z*=6.4X107km.

Thus the mean free path of an electron at the

distance of the earth is given as rather less than

half the distance to the sun. As table 4 indicates, over this distance the number density of

the coronal gas varies only very slightly; also

the temperature gradient is small, this being

ensured by the long electron paths. The proton

mean free paths are of similar magnitude.

The question may be raised: Is the thermal

conductive flux rightly given by equation (3)

when the number density is so small as 103,

or even 108? A question of this kind has been

considered by Cowling and myself (Chapman

and Cowling, 1952, p. 260) for a simple gas.

The value r of the conductive term in $(1),

where Z10* and/( 0 ) (1+$(1)) are the first and

second approximations to the velocity distribution function /, is there given as of order

45 /2\H _j _ dT

32 W na

2

TdR'

1 Second edition. Pagination cited herein al»o applies to first edition'

1MB.

wherein use has been made of formula (1) on

page 218 of Chapman and Cowling (1952).

Substituting for a

2

 and dT/dR gives the formula

r=105 T*lnR.

At i?0 this has the value 0.006, showing that in

the hottest part of the corona the usual kinetic

theory of thermal conduction is adequate. At

the earth's distance RB, taking n0 and To as

before, r is slightly less than 1. Even under

these conditions of very low density, the disturbance of the Maxwellian velocity distribution function /0, though substantial, is moderate; but in view of this value of r it is desirable

to consider what contribution is made to the

flux F by the next term in/. The third approximation to/is discussed in Chapter 15 of Chap-
SOLAR CORONA AND TERRESTRIAL IONOSPHERE

man and Cowling (1952). It is there shown

(p. 265) that the flux of conducted heat is

unaffected by the further approximation, if the

gas is at rest; no new terms are introduced into

the heat flux "depending on the second or

higher space-derivatives of T, nor on squares

or products of derivatives of T." The third

approximation to /, however, can supply a

thermal flux even when T is uniform, if the gas

is in motion. In the present discussion the

gas is supposed to be everywhere at rest,

although this is clearly an idealization of the

actual changing corona. Hence it is concluded

that the above calculations of T and n for our

model corona are unlikely to be seriously in

error even up to and somewhat beyond the

earth's orbit.

Possible influence of the solar magnetic

field

It is known (Chapman and Cowling, 1952, pp.

325-329) that in a magnetic field the thermal

conductivity is reduced for heat flow perpendicular to the field. It is still uncertain whether

the sun has a general magnetic field extending

into space like the field of a dipole in vacuo.

If it has, and even if the strength at its poles is

no more than 1 gauss, the outward heat flow

here calculated must probably be much reduced,

except in high heliomagnetic latitudes. Thus

the temperature would decrease outwards more

rapidly, and the extension of the coronal gas

would be more limited. There is order-ofmagnitude agreement between the density near

the earth as here calculated (if we assume only

Wo, To and no solar magnetic field for R^>Ro)

and that inferred from zodiacal light observations. This may be taken as an indication that

the sun's general magnetic field (if any) does

not pervade the corona. The present calculations would be much less affected by local

sunspot fields, or by local magnetic fields convected by solar clouds or streams issuing from

the sun. This question of the possible influence

of magnetic fields on the heat flow in the corona,

and on the far extension of the corona, seems

worthy of further consideration, but this will

not be attempted here.

 !

How does the coronal gas affect a nonmagnetic planet?

The mean free path of the protons and electrons

of the coronal gas much exceeds the radius of

the planets that are immersed in the gas. The

mean random speeds of the protons exceed the

planetary speed, and the excess is still greater

for the electrons. Hence, as in the case of

meteors (Herlofson, 1948) traversing the higher

layers of our atmosphere, we may infer that no

cap of compressed gas will form in front of the

planet. Planets without magnetic fields will be

subject to the direct impact of the coronal

protons and electrons. If the planet has an

atmosphere, the coronal particles will penetrate

its outermost layers; if there is no atmosphere,

as, for example, is supposed to be the case for

the moon, the coronal gas will be in direct

contact with the planetary surface. It is of

interest to consider the impact of the coronal

gas, supposing there is no magnetic field.

The flux of particles of number density n and

mean random speed v, upon a fixed surface in a

gas in equilibrium, is }i nv (Chapman and Cowling, 1952, p. 100). Now n is the same for the

protons and electrons, whereas v4 far exceeds v+.

Hence the flux of electrons upon a planet would

much exceed that of the protons. The impinging flux may be balanced for each kind of

particle by an equal reflected or scatter flux.

Suppose, instead, that the particles are absorbed

by the planet or its atmosphere. Then the

excess electron flux will rapidly charge the

planet negatively, so as to equalize the two

fluxes. Let the planet, for the present, be

supposed at rest relative to the gas.

The electric field will retard the radial component speed of the electrons, and increase that

of the protons. As the protons and electrons

must keep together, until they enter the atmosphere or body of a planet, the electrostatic

energy gained by the protons equals that lost by

the electrons; hence their combined energy is

constant. If Wb, w+ are the radial speeds at a

distance, and w their common speed at the earth,

(23)
8 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

where mH denotes the mass of a hydrogen atom.

As the mean value of \m,w? and Jm+w+2 is \kT,

it follows that

where the bar signifies that the mean value of

the function below it is to be taken. For

T= 219,000, w=42.5 km/sec. The radial proton

speed is increased in the ratio ^/2, and the radial

electron speed is reduced to this value.

The energy supplied to the planet will be

greater for particles of speed higher than the

average; a simple calculation suggests that on

this account the average energy brought to the

planet per proton-electron pair is not 3kT but

AkT.

If ap is the radius (from the center of the

planet) at which the coronal protons and electrons are absorbed into the planet or its atmosphere, the total flux of energy to the atmosphere

(still neglecting the orbital motion of the

planet) would be

nw (4kT)

or

If we take T= 219,000 (corresponding to

7

7

o=lO8), this is

1.6X10-*naP

2

.

The orbital motion of the planet will affect

this calculation. The order of magnitude can

be estimated, very crudely, as follows. Consider the flux of energy per cm2

 sec on to the

planet at its foremost and rearmost points,

where the collisions are respectively head-on and

overtaking. The fluxes are

£nmB (wtvp) <u2+tP+(io±vP)

2

li

where the mean value indicated by the overbar

is to be taken over all molecules with a positive

impact speed; u and v denote the lateral component speeds. By making crude approximations, the energy flux averaged over the fore and

rear areas is of order (per cm* sec)

where the mean values are now supposed to be

taken as if vP were zero. The ratio of this to the

flux for a stationary planet is of order

l+3vP

2

/ufi. (24)

If vP=30 km/sec and w is 42.5 km/sec, this

number is 2.5. For impacts between the fore

and rear points along the orbit the ratio will be

less. Until the ideas in this paper have been

subjected to critical consideration by those

learned in this field, it does not seem worthwhile

to go more elaborately into this question. But

it seems reasonable to take 1.5 as an order-ofmagnitude estimate of the over-all factor by

which the total coronal energy flux into the

atmosphere on the moving planet exceeds what

it would be on a stationary planet for which vP

and Thave the supposed values. The motion of

the planet will cause the energy flux to be greater

over the forward than over the rear hemisphere;

this must produce a daily variation of temperature in the high layer of the atmosphere that

absorbs the coronal particles.

When we insert the orbital-motion factor 1.5

in the expression for the energy flux into the

atmosphere, this becomes

2.4 X10~3naP

2

. (25)

g nmB 4

This energy is kinetic energy; each electronproton pair also brings its energy of ionization,

13.6 ev. This increases the energy supply by

nearly 20 percent, if this additional energy is

"absorbed" by the atmosphere. The kinetic

energy will be absorbed, and will be increased

by the gravitational pull of the planet, but the

increase per proton-electron pair, for a planet

like the earth, is less than 0.6 ev, and may be

left out of account.

Applying this calculation to the moon, supposed to be devoid of an atmosphere and any

hindering magnetic field, aP must be identified

with the actual radius of the moon. Hence the

flux of coronal kinetic energy to the surface is

2.4X10-3n/4ir or l.QXlO"4TM ergs/cm2 sec. If

n= 103, this is 0.19 erg/cm2 sec. At the subsolar point of the moon, the flux of incident

solar radiation is 2 cal/cm2 minute or 1.4 X 109

erg/cm2 sec. Thus the heat flux from the

corona is about 10~7 of the maximum solar
SOLAR CORONA AND TERRESTRIAL IONOSPHERE 9

radiation flux on the sunlit half of the moon.

Hence the coronal gas seems unlikely to have

any appreciable influence on the temperature of

the dark side of the moon. This temperature

is now inferred, from the study of microwave

thermal radiation (Piddington and Minnett,

1949; Jaeger, 1953), to be of order 150° K,

superseding the much lower values still sometimes quoted (Duncan, 1946).

If the planet has an atmosphere, the coronal

particles will penetrate only its outermost layer,

even on the forward hemisphere. The kinetic

energy of the coronal particles will be shared

with the particles of the atmosphere* if these

likewise are mainly protons and electrons, the

energy can be shared only as kinetic or heat

energy; if they include other kinds of ions and

neutral particles, a fraction of the coronal

energy may go into excitation and ionization,

but most of it will be converted into local heat

energy. In a steady state this must be removed at the same rate as it is supplied; the

removal will be by downward conduction. The

energy will descend as heat to lower layers

where it can escape, probably by long-wave

radiation from polyatomic molecules. This

downward heat flux will be superposed on the

other processes of supply and dispersal of

energy, by solar wave or corpuscular radiation.

It implies an upward increase of atmospheric

temperature towards a high value, perhaps of

the same order as that of the coronal gas.

This rise of temperature will facilitate the

escape of lighter gases of the atmosphere, beyond the rate that would obtain in the absence

of the coronal gas.

Zodiacal dust

The zodiacal light (Allen, 1946; van de Hulst,

1947) indicates the presence round the earth of

multimolecular particles ranging in radius up

to 0.03 cm (van de Hulst, 1947), or on another

view, 0.001 cm (Allen, 1946); such particles are

supposed to be similar in size to telescopic

meteors. Van de Hulst considers that only

a small proportion of the zodiacal light particles

have speed sufficient to render them visible

when they enter our atmosphere. Watson

(1937) estimated that about 5 tons of meteoric

matter falls daily upon the entire earth (Whippie, 1955); van de Hulst estimated the total

accretion to be 10* times as great, mainly by

zodiacal particles. If the bulk of this matter

enters the earth's atmosphere with the speed

simply of free fall (about 10 km/sec), the energy

flux over the earth will be of order 3X1017

erg/sec. Some of this energy may be absorbed

below 100 km, but the main contribution, by

the smaller particles, would be made at higher

levels.

Influence of the coronal gas upon the outermost atmosphere of the earth

The geomagnetic field complicates the consideration of the terrestrial influence of the protonelectron gas. This question has already been

discussed by Dungey (1954), in the light of the

theoretical discussions of magnetic storm phenomena by Ferraro and myself (Chapman and

Ferraro, 1931; 1932; 1933; 1940). As Dungey

says, the problem is difficult, and no attempt is

made to solve it here. Dungey concludes that,

as in the theory of magnetic storms, the geomagnetic field will exclude the coronal gas from

a space around the earth. He estimates the

radius of the cavity in the plane of the equator

to be 9a, where a denotes the earth's radius.

When a solar cloud or stream of fast-moving

gas is first projected towards the earth, a cavity

will be formed in it; this is a passing event.

But if the coronal gas surrounds the earth for

long periods, it seems to me probable that the

cavity will slowly fill up. Suppose the coronal

temperature were to fall to a low value, causing

the equilibrium density n near the earth to

Become inappreciable. If, then, To rose again

to a steady value of 10* degrees or more, the

corona would expand, and on reaching the

earth a cavity would be formed in it. The

particles, protons and electrons, would be magnetically deflected back into the gas at the

surface of the cavity. Collisions there would

be rare, but when they occurred they would

scatter the colliding particles, which could

remain in the vicinity or spiral along the lines

of force towards the earth. They could also

slowly diffuse laterally into the cavity, by the

action of further collisions, with spiralling motion. If the density of the coronal gas remains

fairly uniform over long periods, the cavity

might thus fill up with gas, forming an extension

of the terrestrial atmosphere. At first the
10 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.2

surface of the cavity would act like an almost

perfect reflector, and no energy would be lost

to the coronal gas. If the cavity filled up, the

extended atmosphere would trap some of the

energy of the impinging coronal particles, and

this energy could be conducted downwards to

lower levels in the atmosphere. The heat conductivity along the lines of force is unimpeded

by the magnetic field, but the conduction

transverse to them is reduced by the field.

The expression (25) for the kinetic energy

brought to the radius of penetration gives an

upper limit to the downward flux of energy.

In the terrestrial case, suppose that the

radius aP of the penetration level is 5a, and n

is 103. If the air temperature rises to be of

order 105 at this level, the mean temperature

gradient between this level and the F2 peak

would be about 4°/km. The maximum coronal

heat available to the whole of the earth's

atmosphere at this level would be 2.4 X1019

erg/sec.

Whether any significant fraction of this

energy is actually trapped by the earth must

remain uncertain until the trapping process

above suggested, or some alternative process,

is established by theory or until air temperature

measurements above the F layer show that

there, also, heat energy is flowing downwards.

The temperature of the outer ionosphere

The temperature at the Fa peak itself is still

uncertain, but several types of evidence indicate a value of about 1000° K; this is supported

by Gallet's studies (unpublished) of the absorption of radio waves in the F layer.

The thermal economy of the upper atmosphere has been discussed by Spitzer (1952),

Bates (1951), Johnson (1956), and others.

Some of these discussions (Spitzer, 1952;

Bates, 1951) mention the possibility that a

significant energy contribution may come from

particles in the space around the earth, but

mainly they proceed on the supposition that

there is no such contribution. Spitzer drew

attention to the importance of thermal conduction. He estimated K (using a rigid-elasticsphere formula) to be 4.1 X 10s

 erg/°cm sec.

He determined the temperature gradient required to enable conduction to dispose of the

energy absorbed in the F layer from solar

radiation. This energy has not yet been

measured, and it was assumed to be X times

the energy in black body solar radiation at

6000° K for wavelengths shorter than 910 A.

The conclusions drawn were: If X<5 , T will

be constant above the O2 dissociation level,

probably about 150 km; if 5<X<10, Twill be

determined jointly by the absorbed radiation

and by conduction, and an average value of

1500° K, reached some distance above 300 km,

would be consistent with all known facts; if

X>100, there will be radiative equilibrium,

with 27>2000° from about 300 km upwards

(according to Spitzer (1952, p. 239), "this value

is much too great for an average temperature,

but might reasonably be expected during a

burst of intense ultraviolet radiation from the

sun").

Bates (1951) gave estimates of K using a

formula showing a greater dependence on T

than the formula used by Spitzer. For

T=1000 he found 2T=5.9X103 erg/°cm sec.

He estimated the supply of solar radiation

energy needed to ionize the F layer as observed.

He found the chief loss process to be conduction,

but with the estimated energy supply this

would imply a temperature gradient of only

0.4°/km (at sunspot minimum); this would not

accord with the supposed high temperature at

the F2 peak. Hence he sought other possibilities of energy supply, and suggested tentatively that this is mainly associated with

unobserved F layer ionization.

The total heat flux descending below the F2

peak may be estimated as 9.5X1017 ergs/sec,

taking i£=5X103 and dT/dh as 3.57km.

This is about 4 percent of the maximum supply

here estimated as possibly coming from the

coronal gas; it is thrice as much as the estimated

amount that zodiacal dust would supply. All

three estimates, of course, are uncertain,

particularly those of the heat possibly coming

from particles outside. The zodiacal dust,

though it may be photoionized, is not so subject

to the influence of the geomagnetic field as the

coronal gas, and is likely to make some contribution to the heat balance of the atmosphere.

The estimation of the supply from the coronal

gas involves more difficult considerations. In

view of the other unsolved difficulties in the

problem of the heat balance of the upper
SOLAR CORONA AND TERRESTRIAL IONOSPHERE 11

atmosphere, it seems desirable to bear in mind,

and to examine, the possibility that the observed

heat flux below the F layer has come through

the F layer from above, and is not merely a loss

process of the F layer itself. If this be the

case, our picture of the outermost atmosphere

must be much modified. I hope elsewhere to

deal with these ionospheric questions more in

detail.

In conclusion, I wish to thank Dr. Eugene

N. Parker and Dr. E. W. Dennison for helpful

discussions on the subject of this paper.

This research, in part, was supported by the

Air Force Cambridge Research Center, Geophysics Research Directorate, and by the

Boulder Laboratories of the National Bureau of

Standards.

References

ALFVEN, H.

1941. Ark. Mat. Astron. Fys., vol. 27, No. 25A.

ALLEN, C. W.

1946. Monthly Notices Roy. Astron. Soc, vol.

106, p. 137.

BATES, D. R.

1951. Proc. Phys. Soc, ser. B, vol. 64, p. 805.

BAUMBACH, S.

1937. Astron. Nachr., vol. 263, p. 122.

BEHR, A., AND SIEDENTOPF, H.

1953. Zeitschr. Astrophys., vol. 32, p. 19.

BIERMANN, L.

1948. Zeitschr. Astrophys., vol. 25, p. 161.

1951. Zeitschr. Astrophys., vol. 29, p. 274.

CHAPMAN, S.

1922. Monthly Notices Roy. Astron. Soc, vol. 82,

p. 294.

1954. Astrophys. Journ., vol. 120, p. 151.

CHAPMAN, S., AND COWLING, T. G.

1952. Mathematical theory of non-uniform gases,

ed. 2. Cambridge University Press.

(1st ed., 1939.)

CHAPMAN, S., AND FERRARO, V. C. A.

1931. Journ. Terr. Mag., vol. 36, pp. 77, 171, 186.

1932. Journ. Terr. Mag., vol. 37, pp. 147, 421.

1933. Journ. Terr. Mag., vol. 38, p. 79.

1940. Journ. Terr. Mag., vol. 45, p. 245.

DUNCAN, J. C.

1946. Astronomy, p. 141. Harper and Bros.,

New York.

DUNGEY, J. W.

1954. Physics of the ionosphere. Rep. Phys.

Soc. London, p. 229.

FERRARO, V. C. A.

1952. Journ. Geophys. Res., vol. 57, p. 15.

GALLET, R.

. Unpublished.

GIOVANELLI, R. G.

1949. Monthly Notices Roy. Astron. Soc, vol.

109, p. 372.

HERLOFSON, N.

1948. Phys. Soc. London, Reports, vol. 11, p. 444.

HEWISH, A.

1955. Proc. Roy. Soc. London, Ser. A, vol. 228,

p. 238.

HULST, H. C. VAN DE

1947. Astrophys. Journ., vol. 105, p. 471.

1950. Bull. Astron. Inst. Netherlands, vol. 11,

p. 135.

1953. In Kuiper, ed., The sun, p. 207. University of Chicago Press.

JAEGER, J. C.

1953. Australian Journ. Sci. Res., vol. 6, p. 10.

JOHNSON, F. S.

1956. Journ. Geophys. Res., vol. 61, p. 71.

PlDDINGTON, J. H., AND MlNNETT, H. G.

1949. Australian Journ. Sci. Res., ser. A, vol. 2.,

p. 63.

PlKELNER, S. B.

1948. Izv. Krymskaya Astrophys. Obs., vol. 3,

p. 51.

1950. Izv. Krymskaya Astrophys. Obs., vol. 5,

p. 34.

ROSSELAND, S.

1933. Publ. Univ. Obs. Oslo, No. 5.

SPITZER, L., JR.

1952. In Kuiper, ed., The atmospheres of the

earth and planets, p. 211. (1949 ed., p.

213.) University of Chicago Press.

1956. Physics of fully ionized gases. Interscience,

New York.

WATSON, F. G.

1937. Ann. Astron. Obs. Harvard, vol. 105, p. 623.

1941. Between the planets. Blakiston, Philadelphia.

WHIPPLE, F. L.

1955. Astrophys. Journ., vol. 121, p. 750.

WOOLLET, R. V. D. R., AND ALLEN, C. W.

1950. Monthly Notices Roy. Astron. Soc, vol.

110, p. 358.

WOOLEY, R. V. D. R., AND STIBBS, D . W. N .

1953. The outer layers of a star, p. 234. Clarendon Press. Oxford.
1 2 SMITHSONIAN CONTBIBUTIONS TO ASTROPHYSICS VOL. a

Abstract

For a model solar corona, static and spherically symmetrical, the radial distribution of temperature T and number

density n is considered. For coronal gas composed solely of protons and electrons, the thermal conductivity varies

as T"8, and depends little on n. A steady conductive heat flow to infinity (T=0) implies that T varies as l/r*/7, and

that if the coronal temperature To is of order 10*, the interplanetary gas is hot (about 200,000° near the earth). If

in the hottest level of the corona the number density (no) is about 2 x 10* cc, near the earth n is of order 10*, depending

rather sensitively on 7V This accords with zodiacal light indications, but differs from them as regards their suggestion

of a rapid decrease of n near the earth. It accords also with inferences drawn from radio studies of "whistlers."

It is shown that the coronal gas will convey little heat to the dark side of the moon.

It is suggested that the hot coronal gas surrounding the earth may be the cause of the downward flow of heat

below the Fj peak in the terrestrial ionosphere; but difficult questions related to the influence of the geomagnetic

field upon the fully ionized coronal gas render this suggestion speculative. It might be settled by observing the

density or temperature above the Ff peak.

The satellites to be launched during the International Geophysical Year will move in regions where such observations will be of special value in this connection.
Supplementary Note1

By Harold Zirin2

The preceding remarks on the corona by

S. Chapman raise the following important question : Is the flux, assumed constant in equation

(3), indeed independent of R? To answer this

it is necessary to consider the nonconductive

heat loss of the outer corona.

Since the mean molecular weight of the metals

in the corona is around two or three compared

to n=\ for hydrogen, the concentration of

metals will fall off sharply with height. Furthermore, whatever processes of loss of coronal

energy take place at lower levels (say less than

0.2 solar radii above the photosphere) are inextricably bound up with the unknown processes

of energy input. We are therefore limited in

our consideration of the losses of heat by the

outer corona to emission by hydrogen and possibly helium (n=%).

Hydrogen can emit radiation by three processes, free-free, free-bound, and bound-bound

transitions. For the first two it is possible to

treat the hydrogen as completely ionized. The

rates of free-free and free-bound emission, respectively, are (Allen, 1955):

(1A) Free-free 1.435 X 10~27 T1'

2

 n*

ergs/cm3/sec

(2A) Free-bound 5.45 X 10"22 71"

1

'

2

 n,2

ergs/cm3/sec,

where the number of protons has been set equal

to the number of electrons.

1

 This work was supported by the Office of Naval Research under Contract Nonr 393(04).

> High Altitude Observatory, Boulder, Colo.

The energy emitted in the line spectrum of

hydrogen is equal to the number of H I atoms

times the rate of coUisional excitation, times

the energy per transition. But the number of

H I atoms is determined by the ionizationrecombination equilibrium, viz.:

(3A)

N(H I) recombinations

N(H II) ionizations

So the hydrogen line emission is given by:

,. A v N(H II) X recombinations . .

(4A) ta-a=— ;—: : X

ionizations

excitation X x

where x is the ionization energy of hydrogen.

But Bethe (1933) shows that, at the electron

energies under consideration, the rates of collisions! ionization and of coUisional ionization

of hydrogen are roughly equal. Thus the line

emission is

(5A) *B-B=N(H II) X recombinations

= 3 .

the latter formula being from Cilli6 (1932). At

T=106 this can be very well represented by

(6A) «B-B= 1.64 X IO-19T-W ergs/cm'/sec.

We can now integrate up the loss of energy

by radiation, starting at the surface of the sun

18
14 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

and extending out to the earth's orbit.

radiation to all three processes is

(7A)

The

+1.64X10-19J-1]<&\

It is necessary to integrate each contribution

separately, if we assume the temperature and

density distribution given by equations (8) and

(14). If «B is small compared to the total outward flux, then the temperature-density model

presented, which neglects this loss of energy,

will stand unaltered.

Making the substitutions referred to and

setting y=R/R0, we get

(8A) x [1.435X

dy.

Since u= 15.31, the integrand falls off extremely

rapidly, but increases with great distance because of the y

18'7

 factor. However, evaluation

of (8A), if we leave out the exponential in the

integrand (as is appropriate for large y), shows

that even at y=1000 the contribution is extremely small. Evaluation of the integral close

in shows that only the free-free emission is

important; this gives a value of 0.237X1026

ergs/sec, roughly one-eighth of the outward

flux. Furthermore, most of this energy is

radiated at y<1.10, where it is inextricably

bound up with the unknown input of energy to

the corona.

It should be noted that observational data on

the inner corona do not show a fall-off with

height of the heavy atoms relative to hydrogen.

This possibility would introduce an additional

large sink for heat.

References

ALLEN, C. W.

1955. Astrophysical quantities. University of

London, The Athlone Press.

BETHE, H. A.

1933. Handbuch der Physik, 2d ed., vol. 24.1, p.

442.

, G.

1932. Monthly Notices Roy. Astron. Soc, vol.

92, p. 820.