Rodolfo Amedeo Lanciani.

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mechanical equivalent which would be due to this motion of

translation alone, we must put A=lf in the factor v^ by

which ^'(Tt ia multiplied, and this gives ^'(ft. To find from
this the mean of the squares of the velocities of translation of
the compound molecules, we divide by the mass ?, and, if the
foot be the unit of length, multiplv by 64*3, whence we have
for the velocity found by taking the square root of this mean
of the squares

Determination of the curve of density for 4=1 '4 — Beginning
with a; = 1, in equations (8) and (9), we find f* = -2626 and

(— j ='8520. Developing the values of f* and (— j for «=
ll, x=L'2j &C., by means of differences we arrive at the values

jM=2145 and (^\ = 1878 when ic=4-0. Putting these values

into equations (12) and (18) we find

a/ = 5-856, A*' = 2188.
If we now allow j\d of the radius of the photosphere, or about
20,000 miles, for the height of the theoretic upper limit of the
solar atmosphere above the photosphere, and if we take the
mean specific gravity of the earth's mass at 5i, and the mean

rific gravity of the sun within the photosphere at J that of
earth, as it is known to be, these values of cc' and ^' give
us in equation (14)

^0 = 28-16,

so that the density of the sun's mass at the center would be
nearly one-third greater than that of the metal platinum.

Ourve of densiti/ for i:=lf. — For this value of k the numerical
coefl&cients in equations (8) and (9) are replaced by those in (10)



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64 J. H. Lane on Iht Theoretical Temperahure of the Sun.

and (11). Otherwise, the same process employed with the value

k = 1-4, gives, starting with cr = 1, ^ = -2875 and (-) = -8452,
and developing for a;=l-l, cc=l-2, &c., brings ns to fi=2'567

and (—1 ^1591, for x=8'0, and finally gives us

0^=8-656, iu'=2-741,
and if we now assume the same height as before for the theo-
retic upper limit of the sun's atmosphere, instead of fo=2816,
we find

^0=7-11.
The new curve of density is found in the same way as the
first, and is presented to the eye in the diagram in comparison
with it In the upper part of both curves the scale of aensity
is increased ten fold, and it is, in part only, evident to the eye
how immensely different, for the two values of i, becomes tne
density in the upper parts of the sun*s mass. It appears to the
eye only in part because the ratio of the two densities multi-
plies itself rapidly in approaching the upper limit of the at-
mosphere.

Tne above was communicated in writing as here given, to the
Academy at its late session.* The draft of the foUowing, and
a part oi the details of its substance, have been prepared sinca

Equation (20) gives in feet the square root of the mean square
of velocity of translation of molecules (8'02V}ae). At the
sun's center we find this would be 881 miles per second for the
curve of density corresponding to i=lf, and 880 miles per
second for the curve of density corresponding to ^=1*4.

In 1888 Pouillet, following the law of heat radiation given
by Dulong and Petit, estimated the temperature of the nidiat-
ing surfiwe of the sun, fix)m observations by himsdf of the
quantity of heat it emits, at firom 1461** C. to 1761"* C. Herschel,
n*om Pouillet's observations, and his own made at the Cape of
Good Hope about the same time, adopts, after allowing one-
third for the absorption of our atmosphere, forty feet as the
thickness of ice that would be melted per minute at the sun's sur-

* I desire here to state that the fannulfiB which show the relation between the
temperature, the pressore, the density, and the depth below the upper limit of the
atmosphere, so far as thej apply to the upper part of the snn*s body, were inde-
pendently pointed out by Prof. Peiroe, in a very interesting paper which that dis-
tinguished ph^dst read before the Academy at the same session, and prior to
the presentation of this paper. Also to recaD a finot whidi I first leaned firom
Prof. Peiroe's mention of it to the Academy, viz. that Prof. Henry l<mg ago threw
out the idea of the atmospheric condition to which Prof Thomson has more recently
given the term conyective equilibrium, viz., such that any portion of tiie air, on
being conveyed into any new layer above or below, would find itself reduced, by
its expansion or compression, to the temperature of the new layer.



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J. S. Lane tm the ITieareticdl Temperature of the Sun. 65

faca The temperature of the radiating surface calculated
fix)m this datum by the formula of Dulong and Petit, and
with the co-efi&cient of radiation found by Frofi "W. Hopkins
for sandstone, the smallest co-efficient he found, is 1550® 0. or
2820** Fah. But then the solar radiation is many thousands
of times greater than the greatest in Dulong and Petit's experi-
ments, so that these calculations of the temperature of the sun^s
photosphere have little weight notwithstanding the simplicity
and accuracy with which the formula represents the experiments
Scorn which it was derived. Nothing authorizes us to accept
the formula as more than an empiric^ ona It seems desirable
that experiments similar to those of Dulong and Petit should
be made on the rate of cooling of intensely heated bodies, such
as balls of platinum not too lai^ By placing the heated ball
in the center of a hollow spherical jacket of water, either flow-
ing or in an unchanged mass, the quantities of heat radiated in
successive equal spaces of time will be determined, and the
corresponding differences of temperature in the heated baD can
at least be estimated with whatever probability we may rely
on our knowledge of the specific heat of its material At
present the best means we have of forming any judgment of
the probable temperature of the source of the sun's radiation,
is perhaps to be found in a comparison between the effects of
the hydro-oxygen blowpipe, and the recorded effects of Parker's
^reat burning lens. 1 am not aware that this method has
before been resorted to.

If the angle of aperture at the focus of a burning lens, or
combination of lenses, be called 2a, the radiation received by a
small flat surface at the focus will be sin'o, if a unit be taken
to represent the radiation the same small flat surfisu^e would
receive iust at the sun's sur&oe. Parker's lens, with the small
lens added, had, at the focus so formed, an angle of aperture of
about 47^ A small flat surfoce at its focus would therefore
receive about one-sixth the radiation that it would just at the
sun, making no allowance for absorption by the atmospheres of
the earth and sun and rays lost in transmission through the
lenses. Pouillet, fix)m the experiments already alluded to
made by himself, found his atmosphere in flne weather trans-
mitted, of the sun's heat rays, about the fraction J raised to a
power whose exponent is the secant of the sun's zenith distance.
This, of course, leaves out of view the heat rays of low inten-
sity which are totally absorbed by tiie atmosphere. He also
concluded fix>m comparison with other experiments of his
own with a moderately large burning glass, that that glass
transmitted | of the heat rays incident on it If we assume
the same fraction for each of the two lenses of Parker^s com-

AK« JOfTB. SOL^flBOOHD SbUIS, VoU L, NO. 148.-^X7LT, 1870.

5



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66 J. H. Lcme on &ic Theoretical Temperature of the Sun.

bination, and assume ftirther that the sun's zenith distance did
not exceed 48^ in the experiments made with it, we find for the
fractional multiplier expressing the ^art of the sun's heat radia-
tion which arrived at the focus unintercepted, (f)^"(J)'='55.
Hence the radiation actually received by a smaU flat siu&ce at
the focus was -09, or about one-eleventh, of what it would
receive ^ust at the sim. The heat so received by any body so
placed m the focus, must, after the body has acquired its
highest temperature, be emitted from it at the same rate. The
heat so enutted will consist : first, of heat radiated into that
part of space toward which the radiating surface of the body
looks; secondly, of heat carried of by convection of the air;
thirdly, of heat conducted away by tne body supporting the
body subjected to experiment ; fourthly, of neat rays, if any,
reflected, and not absorbed, by the boay subjected to experi-
ment Assuming it as a reasonable conjecture that fiill naif
of all this* consists of heat radiated into the single hemisphere
looking upon a flat surface, we may conclude that the body, at
its highest acquired temperature, radiated not less than ^V^h as
much heat as is radiated by an equal extent of sur&ce of the
sun's photosphere, over and above such part of that radiation
as may be intercepted by the sun's atmosphere, and sudi rays
of low intensity as are totally absorbed by our own atmosphere,
the whole of which apparently cannot be great No allowance
seems necessary for the chromatic and spnerical dispersion of
the lenses, since the diameter of the focus is stated at half an
inch, while the true diameter of the sun's image would be not
less than one-third of an inch.

Now we are not without the means of forming a probable
approximate estimate of this temperature at which the radiation
becomes ^V^h, more or less, of that of the sun's photosphere.
We are told that in the focus of Parker's comipound lens 10
grains of very p\ire lime (" white rhomboidal spar ) were melted
m 60 seconds. We may presume that in that length of time
the temperature of the mne, aft^er parting with its carbonic
acid, made a near approximation to the maximum at which
it would be stationary, a presumption confirmed by the period
of 75 seconds said to have been occupied in the fusion of 10
CTains of camelian, and by the considerable period of 46 seconds
for the fusion of a topaz of only 8 grains, and 25 seconds for
an oriental emerald of but 2 grains, and in &ct sufi&ciently

* As to the heat carried off by oonyeotion of the air, if itB quantity be calcu-
lated by the fOTmula glTen by Dnloqg and Petit ibr that pomae, it ocmea out
utterly indgniflcant In comparison with the heat received htm m buniiDg gjaaa.
The CQi^eotural allowanoe of fths in aU, of thiS) ia likely, therefore, to be mudi
too la^ge. Not mudi reliance, indeed, can be placed upon the fcvmula here men-
tioned, at such a temperature aa 4000® Fah., yet, as by it the convection ia taken
proportional to the 1*233 power of the difference of temperature, it aeems nnlikttlj
that it givea a quantity very many fold leas tiian the truth.



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J. n. Lome on the fheoretical Temperature of the Sun. 67

proved, it would seem, by observing that the heat we have
estimated to fidl at the focus, upon a flat surface, would suffice,
if retained, to raise the temperature of a quarter of an inch
thick of lime 4000^ Fah. in 6 seconda I^ then, we may take
the temperature maintained at the focus of Parker's lens to
have been at the melting point of lime, we may conclude that
it is also not &i fix)m the temperature given by the hydro-oxygen
blowpipe. Dr. Hare, who was the fiist inventor of this instru-
ment, and the discoverer of its great power, melted down, by
its means, in partial fusion, a very small stick of lime cut on a
lump of that material, which we understand to have been a
very pure specimen. Burning glass and blowpipe seem each
to nave been near the limit of its power in this apparently
conmion effect But Deville found the temperature produced
by the combination of hydrogen and oxygen under the atmos-
pheric pressure to be 2600^ dent As tne lime in the heated
blast would radiate rapidly, its temperature must have been
lower than that of comoin^ hydrogen and oxygen, and I have
called it 2220*" Cent or 4000'* Fah.

The formula of Dulong and Petit, with the co-efficient found
by Hopkins, as abready mentioned, gives for the quantity of
heat radiated in one minute by a square foot of surface of a
body whose temperature is 0+t centigrade, into a chamber
whose temperature is centigrade, when expr^sed with ike
unit employed by Hopkins,

8-877 (1-0077/ [(l-0077)'-l].

It will be convenient, and, in the discussion of the high tem-
peratures with which we are concerned, will involve no sensible
error, to use the hypothesis that the space around the radiating
body is at the temperature of 0** 0. and the formula for the
radiation then becomes,

8-877 [(1 •0077)^-1]. (21)

The unit used by Hopkins, in the formula here given, is the
quantity of heat that will raise the temperature of 1000 grains
of water 1^ centigrada Expressed by the same unit, the
quantity adopted by Sir J. Herschel as the amount of the sun's
radiation, viz. that which would melt 40 feet thick of ice in a
minute (at the sun's surface), is 1,280,000. The ^Vth of this,
or 64,000, expresses, therefore, the quantity which we have
estimated the lime under Parker's lens to nave radiated, per
square foot of its surface, at its estimated temperature of 4000^
¥six. If now we calculate its temperature by the above formula,
firom the estimated radiation, the result is 1166^ Cent or 2180^
FaL This is manifestly much below the real temperature,
and so fEur below that there can be no doubt the formula of
Dulong and Petit has fsdled at the melting point of lima If



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((8 J. H. Lane en the Theoretical Tevhperature of the Sun.

instead of the co-efficient 8*877 we had used tie larger co-effi-
cient 12*808 "which Hopkins gives for unpolished limestone, the
formula would have been reduced only 68** Cent It best suits
the direction of our inqidry to use the smallest co-efficient which
Hopkins' experiments gave, since we aie seeking the highe$t
temperature which can be plausibly deduced from the sun's
radiation. For ease of expression, the curve which we will
imagine for representing the actual relation of radiation to
temperature, the horizontal ordinate standing for the tempera-
ture and the vertical ordinate for the radiation corresponding
thereto, may be called the curve of radiation. The course of
this curve from the freezing point of water to a point somewhat
below the boiling point of mercury is correctly marked out to
us by the formula. Beyond tliat we have but the rough
approximation which we can get by means of the above com-
parison, to the single point of the curve where the radiation is
,Vth that of the sun's photosphere. The attempt, from these
data, to extend the curve till it reaches the fiill radiation
of that photosphere, must be mainly conjecturaL As a
basis for the most plausible conjecture I am able to make
let us assume : first, that the upward concavi^ of the curve of
radiation, which increases very rapidly with the temperature as
&r as the curve follows the formula of Dulon^ and Petit, is at
no temperature greater than that formula woiud five it at the
same temperature; secondly, that the curve of radiation is
nowhere convex upward. If, then, we set out from diese two
coniectural assumptions— of the degree of probability of which
eacn one must form his own impression — ^tne greatest tempera-
ture the sun's photosphere could have consistently with the
radiation of 64,000 at the temperature of 4000** Fah., is foimd by
drawing through the point representing that radiation and that
temperature a straight line tangent to the curve of the formula.
The line so drawn would cross the real curve of radiation in a
greater or less angle at the radiation of 64,000 and t^npera-
ture of 4000^ Fah., and at higher temperatures would fidl more
or less helow that curve, and its intersection with the sun's
radiation of 1,280,000 would be at a temperature greater than
that of the curve, that is to say, greater than the tempera-
ture of the sun's photosphere. This greater temp«*atare is
55,450** Fah.

A different train of coniecture led me at first to assume a
temprature of 54,000° Fan., and this last number I will here
retain since it has been already used as the basis of some of
the calculations we now proceed to give. It must be here
recollected that we are discussing the question of chuda of
soUd or at least jluid particles floating in non-radiant gas, and
oonstituting the su&'a photosphere. K the amount of radiaii&n



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J. H. Lane on the Theoretical Temperattire of the Sun. 69



I

I



6. fc^

ExplanaHon.—A'ni., Assumed tlieoretic upper limit of atmosphere ; Phot.. Pho-
tosphere; C.T.K=lf, Arbitrary Curve of temperature for A:=l|^; C.T.K.=1 4,
Arbitzaiy Curve of temperature forX;=l'4; C.D.K.=1'4, Absolute Curve of density
for ksmiA; aD.K.=sl|, Absolute density for k=ll.



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70 J. H. Lane on the Theoretieal Thnperahare of 6ie Sun.

would lead us to limit the temperature of such clouds of solids
or fluids, so also it seems diflicult to credit the eoctstence in the
solid or fluid form, at a higher temperature than 54,000® Fah.
of any substance that we know of

K then we suppose a temperature of 54,000** FaL, what would
he the density of that layer of the hypothetic gaseous body
which has that temperature, and what length of tune would lie
required, at the observed rate of solar radiation, for the emis-
sion ol all the heat that a foot thick of that layer would give
out in cooling down under pressure to absolute zero? The
latter question depends on the mechanical equivalent of this
heat for a cubic foot of the layer of gas, and the two questions,
together with that of the depth at which the layer would be
situated below the theoretic upper limit of the atmosphere, are
answered by equations (17), (18), and (19), provided we knew
the value of k and the value of <r in the body of gaa The less
the atomic weight of the gas the greater the value of a, and
the greater the density of the layer of 54000® Fah. and the
greater the quantity of heat which a cubic foot of it would
give out in cooling down. I therefore base the first calculation
on hydrogen as it is known to us. The value of o^ is in that
case about 800 feet, and the value of k about 1*4, nearly the
same as in common air. These values would give for the
layer of 54000® Fah. a specific gravity about -00000095 that
of water, or about one 90th that of hydrogen gas at common
temperature and pressure, and the mechanical equivalent of the
heat that a cubic foot of the layer would give out in cooling
down under pressure to absolute zero would be only about
9000 foot pounds, whereas the mechanical equivalent of the
heat radiated by one square foot of the sun's surface in one
minute is about 254,000,000 foot pounds. The heat emitted
each minute would, therefore; be fully half of all that a layer
ten miles thick would give out in cooling down to zero, and a
circulation that would dispose of volumes of cooled atmosphere
at such a rate seems inconceivable.

It may possibly appear to some minds that the difficulty
presented by this aspect of the case will vanish if we suppose
the photosphere, or its cloudy particles, to be maintained by
radiation at a temperature to almost any extent lower than that
of convective equilibrium. This would enable us to place the
theater of operations in a lower and denser layer of atmos-
phere, but the supposition seems to me difficult to realize
unless, as the hot gases rise fix)m beneath, precipitation could
commence at a temperature many times higher than the 54000®
Fah. which we have estimated mr the upper visible surface of
the clouds, and this, as before intimateo, seems to me itself
extremely improbable.



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J. n. Lane on the Theoretical Tem'peroOyre of the Sun, 71

I may mention here that my friend Dr. Craig, in an unpub-
lished paper, following the hint thrown out by Frankland, is
disposed to &yor the idea that the sun's radiation may be the
ramation of hot gases instead of clouds. At present I shall
offer no opinion on that point one way or the other, but will
only state it as my impression that if the theory of precipitated
clouds, as above presented, is the true one, something quite
unlike our present experimental knowledge, or at least much
beyond it, is needed to make it intelligible.

The first hypothesis which offers itself in an attempt to
make the theory rational is suggested by one point in Clausius'
theory of the constitution or the gases, already alluded ta
In forming his theory Clausius found that the mown specific
heats of the gases are all much too great for firee simple atoms
impin^in^ on one another, and he therefore introduced the
hypomesis of compound molecules, each compound molecule
bemg a system of atoms oscillating among each other under
forces of mutual attraction. Now if this were accepted as the
actual constitution of the gases it is of course easy enough to
conceive that in the fierce collisions of these compound mole-
cules with each other at the temperatures supposed to exist in
the sun's body, Iheir component atoms might be torn asunder,
and might thenceforth move as free simple molecules. In this
case, stall retaining the hypothesis of Clausius' theory, that the
average length of the patn described by each between collisions
is lai^ compared witn the diameter of the sphere of effective
attraction or repulsion of atom for atom, the value of k would
reach its maximum of If. Experiment has not shown us any

Sits in this condition, and for the present it is hypothetical
ven in hydrogen the value of k does not materially, if any,
exceed the value of 14 which it has in air. But if it were
found that the hydrogen molecule is compound, and that in
the body of the sun the heat splits this molecule into two equal
simple atoms, and in &ct that all the matter in the sun's body
is split into simple fi*ee atoms equally as small, then, while the
value of k would be If, the value of a would be about 1600
feet If with these values we repeat the calculation of the
density of the layer of 54000® Fan. we find its specific gravity
to be 0-000363 of that of water, or 4-35 times that of hydrogen
ffas at common temperature and pressure and in its known con-
dition, or S"? times that which the hydrogen in the hypothetic
condition would have if it retained that condition at common
temperature and pressure. We find also that the mechanical
ec[uivalent of all the heat that a cubic foot of the layer wo\ild
give out in cooling down, under pressure, to zero, would be no
less than 13,500,0W) foot pounda Instead, therefore, of a layer
ten miles thick, it would now require only a thickness of 38 leet



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72 J. H. Lane on the TheoretuxH Temperature of the Suru

to give out, in cooling down to zero, twioe the heat emitted by
the sun in one minuta It will be seen^ (equations (17) and (19)),
that this thickness, retaining the constant value i=lf, would
diminish with the 2 J power of the masses of the atoms into
which the sun's body is hypothetically resolved (the reciprocal
of the value of a), and I leave each to form his own impression
how far this view leads towards verisimilitude.

It is important to add that the depth of the layer of 54000®
Fah. below the theoretic upper limit of atmosphere, when cal-
culated with value 4=1*4, <^=800 feet, comes out only 1107
miles, and with the values A=lf and a=1600 feet only 1681
miles. This calculation of the depth, unlike the other results
above, may be said to be independent of the question of the
constitution of the sxm's interior mass. It is alike difficult, on
any plausible hvpothesis, to reconcile a tOTiperature no higher
than 54000° FaL with any perceptible atmosphere extending
manv thousand miles above, and yet no less an authority than
Prof Peirce has assigned a hundred thousand miles as the
height of the solar atmosphere above the photosphere, at the
same time, however, pointing out the enormous temperature
which, under convective equflibrium, this would imply at the
level of the photosphera But all are not yet agreed that the
appearances seen at such distances &om tne sun are proof of
the existence of a true atmosphere thera It will be seen that
the numbers I give above were obtained fix)m a first hypothesis
of an atmospheric limit 20,000 miles above the photosphere,
but for the purpose of this paper it is of no consequence to
repeat the calculation from a oinerent limit

it is, I believe, recognized on theoretical grounds that in an
atmosphere containing a mixture of gases of unequal density
the lighter gases might be expected to diffuse in greater propor-
tion into the higher parts of the atmosphere and the neavier
gases into the lower parts. But perhaps the supposed circula-



Online LibraryRodolfo Amedeo LancianiThe American journal of science and arts → online text (page 60 of 109)