A NNALES DE L ’I. H. P., SECTION A
D ANIEL B. H ENRY
The temperature of an asteroid
Annales de l’I. H. P., section A, tome 55, n
o2 (1991), p. 719-750
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719
The temperature of
anasteroid
Daniel B. HENRY Instituto de Matematica e Estatistica,
Depto. de Matematica Aplicada,
Universidade de Sao Paulo,
Caixa Postal 20570, 05508 Sao Paulo SP, Brazil
Vol. 55, n° 2, 1991, Physique theorique
ABSTRACT. - The heat
equation
in a solidbody
with radiationboundary
conditions and
periodic heating
at theboundary - modeling
arotating
asteroid - is shown to have a
unique time-periodic
solution and it isglobally attracting.
We findapproximations
to this solution for various casesrelevant to the
possible
existence of ice in the asteroid belt.RESUME. 2014 Nous montrons que
1’equation
de la chaleur dans un corps solide avec conditions aux bords radiatives et une source de chaleurperiodique
sur la frontièrepossede
uneunique
solutionperiodique qui
estun attracteur. Cette
equation
sert a modeliser un asteroide en rotation.Nous exhibons des
approximations
de cette solution que sont relevantes pour 1’existence deglace
dans la ceinture d’asteroides.INTRODUCTION
We treat the
temperature
of arotating
solidbody
with noatmosphere,
heated at the surface
by absorption
of Solar radiation and alsolosing
heat
by
radiation. We have in mind anasteroid, typically
a few kilometersAnnales de l’Institut Henri Poincaré - Physique theorique - 0246-0211
(or
few tens ofkilometers)
indiameter,
with escapevelocity
at most sometens of meters per
second,
far too small to retain anatmosphere.
Wedon’t know what asteroids are made
of,
so our caculations will considera wide range of materials. It turns out that
only
one dimensionlessparameter (~/8)
isimportant - depending
on the material near thesurface,
distance from the
Sun,
andperiod
of rotation - but we also don’t know this value.(Strictly speaking,
this is for a circularorbit;
westudy
theeffect of
eccentricity later,
and see it isnegligible
for almost anyasteroid.)
The calculations are
fairly simple
when~/8
islarge -
and it would belarge
for most
plausible
Terrestrial materials - but are far more difficult when it issmall,
and manyproblems
remain for efficientcomputation
in thiscase.
(The theory might
beapplied
toMoon,
but~/5
is small - about1 /70 - largely
due to slowrotation,
and the results are notsatisfactory.
Nevertheless,
itpredicts temperature
68 K at thepoles
ofMoon, indepen-
dent of
~/8,
and I have some confidence in thisnumber.)
We use coordinates fixed in the
body;
then theonly
effect of rotation is aperiodic
oralmost-periodic
variation of theheating
rate at agiven point
of the surface. We show there is aunique long-term
response, the limit from any initialtemperature
distribution as time goes toinfinity,
which is
equally periodic
oralmost-periodic
in time. With anyplausible
values of the
physical parameters
of anasteroid,
the thermal relaxation time is far smaller than a billion years; we may be confident the limit is attained.(Even
forMoon,
the surfacetemperature
should be determinedby
thistheory, though radioactivity
and residual "ancient" heat areimpor-
tant in the
interior.)
Thelimiting
solution varies far morerapidly
withdepth
than it doesalong
thesurface,
so a one-dimensional model -depth
below a
given point
of the surface - should be agood approximation.
The solution of this one-dimensional model
rapidly approaches
alimit, independent
oftime,
asdepth
increases.(A
few meters below the surface should beenough.)
This is used as aboundary
value to determine thetemperature, independent
oftime,
in the interior of the asteroid.We calculate in detail
only
the case of aspherical asteroid,
with constantphysical properties, though
most asteroids areprobably non-spherical
andinhomogeneous.
The calculation of the surfacetemperature - actually,
afew meters below the surface - may be used
directly
for any convexbody
with any distribution of
conductivity. (Non-convex
bodies mayproduce
troublesome
shadows,
and call for morecomplex computations, though
not different in
principle.)
Thesubsequent
calculation of the interiortemperature
willdepend
on theparticular shape
andconductivity.
Thegeneral
theoremsapply
to bodies of any(smooth) shape
with any(smooth) inhomogeneities.
Mann and Wolf
[5]
treated a one-dimensional heatequation
in(0, (0)
with a radiation
boundary condition,
constantconductivity,
constant initialAnnales de l’Institut Henri Poincaré - Physique theorique
temperature,
andasymptotically
constantheating
rate at theboundary.
They
showed the solutionapproaches
a constant as time tends toinfinity.
Levinson
[4]
started from a differentphysical problem,
which led to thesame mathematical
problem.
Heshowed,
for a constant initial value andperiodic heating
at theboundary,
the solutionapproaches
theunique periodic
solution with the sameperiod.
These authors workedexclusively
with the
integral equation.
We will also use anintegral equation
on theboundary
forquestions
of existence andsmoothness,
but westudy
theasymptotic
behaviorusing
the maximumprinciple
anddynamical systems theory.
Wegeneralize
the work of[4], [5]
in several directions: thespatial
dimension may be
greater
than one, theregion
may have any smoothshape
and may be bounded or not, thephysical properties (conductivity, etc.)
maydepend
onposition,
the initialtemperature
isquite arbitrary (non-negative, bounded),
and thetime-dependence
of theheating
rate isalso
general, though
theperiodic
andalmost-periodic
cases recievespecial
attention.
The interest of the mathematical
problem
is sufficientjustification,
butI believe the
problem
is alsoimportant.
Atleast,
thetemperature
is a crucial datum for animportant question:
Is there ice in the asteroids?
It may not seem the
question
isimportant;
havepatience.
Comets are active
only
a short time - some 200 revolutions a few thousand years.It is a
respectable conjecture
that "dead" comets, which have ceased visibleemission,
may very well end up in the asteroid belt.They
cannotremain very
long
in the inner SolarSystem
beforebeing swept
awayby gravitational perturbations.
Where’s"away"?
The asteroid belt may be the dust bin of the inner SolarSystem. (I
am warned there areimportant
unsolved
dynamical problems here.)
Dead comets are sometimes called
"de-volatilized",
for nogood
reason.Photos of the nucleus of
Halley’s
comet show emissiononly
frompatches,
about 10 % of the
surface,
the restbeing
coveredby
dust. A comet dieswhen its surface is
completely
covered.If the comet nucleus is not too small - at least a few kilometers across - it will still have a lot of volatiles
(ices
ofH20, HCN, NH3
andthough CH4
and CO willprobably
be lost attemperatures
above 100K)
when itarrives in the asteroid belt. If the ice can survive for billions of years under conditions encountered in the asteroid
belt,
an enormous stock of ice may have accumulated out there. This would determine the course ofdevelopment
of the SolarSystem
for centuries to come. Forexample,
it’seasy to make air from
water, cyanide
orammonia,
andsunlight. (Now
do you believe it’s
important?)
The average
temperature
at the inneredge
of the asteroid belt(2
A.U.from the
Sun)
isnearly
200K, dropping
to 150 K at the outeredge (3.5 A.U.),
and this is not verypromising
forlong
term survival of iceexposed
to a vacuum. However:
( 1 )
The averagetemperature
is not theonly temperature;
and:(2)
The ice is notexposed.
( 1 )
Even at 2A. U.,
with averagetemperature
200K,
there will be substantialregions
near thepoles
of an asteroid withtemperatures
below 150K,
and some below 120 K.(We
may use the same estimates for dead comets; see "The effect ofeccentricity" below.)
(2)
We deal withdirty ice,
which willshortly
be coveredby
dust leftbehind as the ice
vaporizes,
and the dust slows theevaporation
ofunderly- ing
ice.(This point
is treated in theAppendix.)
The
prospects
for survival of waterice,
covered withdust,
and ammonia andcyanide
ices embedded in waterice,
aregood.
axis vertical axis inclination 30°
average temperature 196 K average temperature 188 K polar temperature 0 polar temperature 101 K equatorial temperature 210 K equatorial temperature 207 K
Maximum surface temperature 280 K, "-/8= oo.
Approximate isotherms below the surface for a spherical asteroid at 2
The amount of material involved is not trivial. A
(reasonable)
rate ofone new
periodic
comet every 30 years, of(modest)
diameter 2 km anddensity (at least)
that of water, continued for 4 billion years, results in total mass - almost1021 kg - comparable
to that of the entire asteroid belt. But 2 km diameter may beexcessively modest,
and comets may have been far morefrequent
in theearly days.
We show later tha. vad comets, likeasteroids,
caneasily
retain their volatiles for billions of years, so it’sl’Institut Henri Poincaré - Physique theorique
strange
we don’t see more of them around. Butperhaps
we do seethem,
in the asteroid belt.
Wishing
doesn’t make it so, andexpert opinion
seemsclearly opposed
to the notion of ice in the asteroids. In
fact,
I could find no localexpert
on
physical properties
of asteroids.My
caricature ofexpert opinion
isculled
mainly
from the last two decades of Icarus and the books:Asteroids,
T. GEHRELS
Ed.,
Univ.Arizona,
1979 and The Evolutionof
the SmallBodies
of the
,S’olar Proc. Intl. School ofPhysics
E.FERMI, 1985,
M. FULCHIGNONI and L. KRESAKEd., North-Holland,
1987.The
opposing argument
isindirect, consisting
of three claims:(i )
Unlesseverything
weknow,
or think weknow,
about the asteroids is wrong,they
arevirtually unchanged
in nature andposition
since theirformation more than 3 billion years ago.
(ii )
The conditions of theirformation,
under any usualhypothesis, preclude
volatiles like water.(iii )
The contribution of dead comets isnegligible.
Conclusion : NO ICE!
Did you notice "Unless.. "?
The
important point
is(i ).
If the claim of(i )
is wrong,(ii ) hardly
matters; the conditions of formation of Earth also seem to
preclude
water.And the
only
reason to believe in(iii )
is to be consistent with(i );
there isno
evidence,
for oragainst.
The main evidence for(i ) - the
correlation of"type" ( = color)
of asteroid with distance from the Sun - I consider inconclusive.It is assumed that
"type",
a surfaceproperty,
reflects the internal constitution of the asteroid. The fact that it’s beenroasting
in aparticular
orbit for a
long
time isirrelevant,
sincetype supposedly
does notchange
with time. It is also claimed that these orbits have
hardly changed
inseveral billions of years, which should interest those who
study
chaoticorbits. These
points
strike me asdubious,
andcertainly unproved.
If the
question
of ice in the asteroids were a theoreticalpoint
of minorinterest,
wemight
be content with anapplication
of Occam’s Razor to a state ofgeneral ignorance.
But it isimmensely important.
The
experts prefer
to call them minorplanets,
but asteroids may beradically
different from allplanets -
notprimordial rocks,
but rather young dustballs,
with buried treasure. It’s worthfinding
out.THE INITIAL BOUNDARY-VALUE PROBLEM
The
body occupies
an open set Q cRn, lying
on one side of theboundary
aS2,
and aS~ is acompact
smooth surface: classC 1 + r, (0r l)2014or C2
orC2+r,
in some later results. InD,
are definedpositive
real functions ofposition
p(density),
C(specific heat),
and thepositive-definite symmetric conductivity
matrix K = On we haveemissivity
f: andabsorptivity
a. It is very
likely
that E is close to1;
(’1, or an average value for a, is measurable as1 - (albedo).
At apoint x
ofaS2,
theintercepted
flux ofSolar energy, per unit area of aS~ and per unit
time,
is g(x, t);
we suppose is absorbed and( 1- a) g
is reflected. Then the absolutetemperature
T satisfieswhere
N (x)
is the outward unit normal to aS2 at x and 03C3 is the Stefan- Boltzmann constant,approximately a=5.67x 10 - 8 W/m2 K4.
The case of
physical
interest is when Q is a bounded set inR3,
but theargument
is the same for any dimension~ 1,
and it isimportant
for thecalculations to also treat
Q=(0, oo)
in the case of constantK,
p,C,
etc.For most of our
calculations,
Q could be bounded or not aslong
as aS2 iscompact;
but convergence results as t ~ oo are delicate in unbounded domains and the casen =1, Q=(0, CIJ), sufficiently
illustrates thepoint.
If we model the Sun as a
point
source, in the directions (x, t)
fromx E aSZ at time t
=1 ),
thenHere
,f’
is the flux of Solar energy, or Solar constant: about1,353 ±21
at the orbit ofEarth,
outside theatmosphere, decreasing
as the square of the distance from the
Sun(338 ± 5W/m2
at2 A.U.).
When the Sun is visible from x at time t, then so any
case. If Q is not convex, g will have discontinuous
jumps
as we cross ashadow
boundary.
Weusually
want g to be at leastcontinuous,
and this isphysically
reasonable since the Sun is not apoint
source,and g
will beroughly proportional
to the area of the Solar disc visible at the moment.It follows that g will be
Lipschiptzian
andprobably
better:C5/4
forgeneric
smooth surfaces 3Q.
(If
the horizon from agiven point,
in the direction of sunrise or sunset, has curvature different from that of the Sun’sdisc, g
will be
locally C3/2;
ifequal, g
will beordinarily C5/4,
andequality
occursonly
at isolatedpoints,
for most smooth surfacesWe will state our
results,
make some comments, and outlineproofs
later.
THEOREM 1. - Let
0fl, n~1,
and suppose 03A9 is anopen set in Rn, lying
on one sideof its boundary
compactC1+
rsurface.
Annales de l’Institut Henri Poincaré - Physique théorique
Suppose
p,C,
(’1, E arepositive
and boundedfrom
zero, or theirreciprocals
are
bounded,
andthey
areuniformly of
classCr;
K(x) _ (x)]n ~ -1
issymmetric, bounded, uniformly positive definite,
andK,
are bothuniformly of
class Cr.Finally
suppose B is apositive
constant andB(1 :
Randg : aS2 x
[to, (0) ~
R are continuous withThen the
problem (PDE)
has aunique
solution T on Q x(to, (0),
of class1 +r/2~
continuous in the closure withT (x, 0, satisfying
the differential
equation
in the classical sense in theinterior,
and theboundary
condition fort> to,
when we useThis solution has
always,
and is anincreasing ( =
non-decreasing)
function of(03C8, g),
that is: if0~03C81~03C82~B
in S2 andthe
corresponding
solutionsTj satisfy in 03A9 (t0, oo).
For any
compact
and0 8 1, t + 1] E Ce° e~2
isbounded
uniformly
fort >__ tl.
If g
uniformly,
thenTv
Tuniformly
oncompact
setsin
Q
x[to, oo).
Remark. - The local
problem
is like that in Friedman[3],
and we usea similar
argument,
with a Volterraintegral equation
on theboundary.
To extend the solution for all
time,
we use the maximumprinciple
toshow as
long
as the solution exist.We assume aS2 is
C~’
r toapply
theintegral
method. Aside from mathematicalconvenience,
we canonly plead engineering experience
inradiation
problems
tojustify
thishypothesis.
Infact,
for manyasteroids,
there is evidence from
polarimetry
that the surface isdusty.
But it isdifficult to avoid
assuming
at least aLipschitz surface,
even in very weak formulations.THEOREM 2. - Assume
Q, K,
p,C,
8, asatisfy the hypotheses of
thefirst paragraph of
Theorem1, and for convenience,
suppose 03A9 is bounded.Assume 03C8 and
g are bounded and measurable withalmost
everywhere
on 0 or aSZ x(to, (0), respectively.
For any continuous~~, g~
such that0 ~ ~1 ~ ~ ~2 ~ B,
and0 ~
g 1~ g
g2~
6B4 E/(’1,
the corre-sponding
solutionsT~
haveO _ T 1 ~ T2 ~ B,
andfor
t~ to,
fo~
a continuousfunction
B(t), independent of
the choiceof
the~~, g~.
There is a
unique
T : 0 x(to, oo)
-+ R withT 1 ~ T _ T2
a. e., for every such choice of the~r~, g~,
and it is of classC2
+ r,1 + r/2 in the interior(0), T(x, t) --~ ~ (x)
a.e. as For any081, tl > to,
andcompact
K cQ,
TI
K x[t,
t +1]
isuniformly
bounded inC8,8/2
fort >_- to.
(The boundary
condition holds in a weak sense we won’tspecify.)
Remark. - This result serves
mainly
tojustify restricting
attention tocontinuous g, but the
integral estimate,
or a similarestimate,
will be useful in another context.THEOREM 3. - In addition to the
hypotheses of
thefirst paragraph of
Theorem
l,
suppose 00 isof
classC2+r
and :g(x, t)=KN(x). t)
where G is
locally C2
+r° 1 +r~2 orequivalently:
t ~ g
(x, t)
islocally C(1 1 + r)/2’
x ~, g(x, t)
isdifferentiable,
and(x, t)
~ax
g(x, t)
islocally
Cr, r/2(where "ax"
indicatesfirst
order operatorsacting tangentially
onA sufficient condition is that g be
C1
+Y in both variables.We also suppose that
0 _- ~r (x)
B and0 ~ g (x, t) ~
6B4 £/(’1.
Then
if 03C8
iscontinuous,
the solution T isC2
+r, 1 +r/2 incompact
subsets ofQ
x(to, (0), though merely
continuous as t -+ to + .and is
compatable
with g on 00X ~ t =
that ison then the solution is
C2 +r,1
+r/2 oncompact
subsetsofQx oo).
Remark. - Theorem 3 follows from the Schauder
estimates,
once we have asufficiently
smooth solution[C3, 3/2]
for the case of smoothdata;
and this is
proved by
examination of theintegral equation
used forTheorem 1. f
Outline of
proof
of Theorem 1The
argument
is similar to that of Friedman[3],
Section 2 ofChapter 5,
so many details will be omitted.
We first extend p,
C,
K to the whole space,preserving
smoothness andpositivity properties,
and construct the fundamental solutionT (~,
t; y,s),
for t>s, and for any continuous bounded cp and
any x~Rn,
as t ~ s+,Annales de l’Instiut Henri Poincaré - Physique theorique
In case p,
C,
K are constants andand in
general,
this is theleading (most singular) part
of r when p,C,
Kare evaluated at the
point
y. Since the coefficients don’tdepend
ontime,
The solution is
represented
aswhere
for
t> to,
is a continuous extension to all Rn with0~B[/
(~)~B and
p is a continuous function onoo)
which is not toosingular
as tto:
uniformly
for x E aSZ as t-~ to + .
In this case, T is a solution of the differential
equation
of classC2 +r,1
+r/2in Q x
(to, (0)
which extendscontinuously
to the closureQ
x[to, (0)
withT
(x, t) ~ BjJ (x)
as t ~ to,uniformly
for x incompact
sets inQ.
We willchoose p
so theboundary
conditionholds,
and we find that thisimplies
the "not too
singular"
conditionabove,
so this is indeed a solution ofproblem (PDE).
According
to Friedman[3],
Thm. 1 of Sec.2, Chap. 5,
theboundary
condition holds at
(x, t)
E aSZ X(to, oo)
if andonly
ifThis is a Volterra
integral equation
with amildly singular (absolutely integrable) kernel,
and a local solution may be foundby
iteration.
It is easy to show
and
uniformly
incompact
sets of x, as t -~ s+,
soby continuity
of~, BJ/,
also2014BJ/(;c, t) = o (t - to) -1 ~2
whent ~ to + , uniformly
incompacts.
It folows 3~t) = o (t - to) -1~2,
as desired.It is also
easily proved that,
ifBfív ~ Bfí uniformly
on Rn g,uniformly
on SQ x[to,
thecorresponding Tv T, uniformly
on com-pact
sets inQ
x[to, t 1 ] .
Suppose
theintegral equation (* *)
has asolution p
on(to, t*),
tot*oo,
such thatf
is bounded onSQ x
(to, t*).
It follows that thesolution p (x, t)
converges as t~ t*-, uniformly
onaS2,
and the solution may be extendedbeyond
t*. Thus ana bound for T will ensure
global existence,
and such a bound follows from the maximumprinciple.
First assume the solution exists on
{ to _ t _- tl ~
and there is a constant Ö such that08B,
,5~B[/~B-8 - ’Y -
onRn
, and -~~~(B-5)~
08
(x) _( )
on Then we show
0 T (x, t) B
on Infact,
5~B[/~B-8 always,
andT(;c,/)-B[/(x,~)-~0 as x ( ~ oo, (so
dist(x, oo), uniformly
into __ t _- tl. Thus,
if the resultfails,
there exists(finite)
x inQ
and t in(to,
such that 0 T B in03A9 [to, t),
but T = 0or T = B at
(x, t). Perhaps
x~03A9 - but then we violate Thm.1,
Sec.1, Chap. 2,
of[3].
Therefore and[or ~0]
where T = 0[or, T = B]
at(x, t),
so~~0 [or g > 6 B4 E/a],
both of which are false.By
continuousdependence, 0 ~ B[/ ~
Band0 ~ ~ a B4 E/ (’1 imply 0 ~ T ~
Bon the interval of
existence,
so the solution exists for all Therepresentation
of the solution and Friedman’s[3],
Thm.3,
Sec.4, Chap. 5, give
uniform bounds inOutline of
proof
of Theorem 2We
only
prove theintegral
estimate forv = T2 - T1.
We havein Qx
(to, oo), always,
and on ~03A9(to, oo),
with
Annales de 4 l’Institut Henri Poincare - Physique ’ theorique ’
and then
For
any ~>0
there existsCE
so thatfor every
(Q),
hencewith ~=1,
we find for allt~ to,
Ontline of
proof
of Theorem 3As noted
earlier,
weonly
need to show T is of classC3,3/2
when thedata are smooth. Assume that
~03A9, K, p C,
f;(’1 are Coo andalso 03C8 and
gare We will not
yet
assumecompatability
at 00.X ~ t =
(1)
Examination of the construction of the fundamental solution r in[3], Chap. 1,
shows -where
r’ (x,
t; y, s;z)
is for t>s with~t
-S)~a +a’~~~ (i + I z )N ~ ~~t
+as)°~,, a~ ~’ ~~x
+r ~x’ ~;
y’ s;z)
uniformly
bounded for bounded and all choices of x, y, zeR"and all
non-negative N, oca’,
..., y.[In
our case,T (x,
t;y, s) = T
~x, t - s; y, ©)-~
(2)
Definefor
x~~03A9,
t > o.LEMMA. - For any t* > o and
integer
there is a numbersuch that is a bounded operator on to itself with norm
andj= l, 2.
The kernel of the
integral operator Fj
may beexpressed
t-n/2 I> (x,
t, y,0; (x - y)/, Jt- s) where
satiafies estimates similar tothose above for
f,
inparticular,
for anyN~,
there is a constantCN
sothen
is bounded on
0~~*
for ;ce3Q. If r issmall,
we may"straighten"
theboundary
near x soThis
gives
the result for m =1.Let a be a smooth
tangent
vector field on If cp iscontinuously
differentiable on aSZ and t>
0,
where
1>cr
is obtainedby
differentiation of a and1>,
and satisfies the same sort of estimates orf,
so the sameargument gives
the result form =1,
andby induction,
for all m.(3) Choosing to
=0,
theintegral equation
for p may be writtenwhere For any
~0, B)/
(.,
are we may solve theintegral equation
for a continu-ous /?:
[0, (0)
~ C"" such thatso the
compatability
condition saysprecisely that p
’(0)
= 0.Annales de l’Institut Henri Poincare - Physique ’ theorique ’
Suppose
and for small ö>o define~(~)=-(~(~+8)-~(~)).
Ö Then
Since p (0)
=0,
qõ(t)
-~p (t)
as 8 -~0,
wherep
is the solution ofThus
/?:[0,
iscontinuously
differentiable. Since weonly
assumed zero-order
compatability [~(0)==0],
ingeneral ~(0)~0,
but asimilar
argument shows/?(/)
isC1
for with as~ 0
+,
so p E([0.
Cm(~Q)).
(4)
In the interiorQ,
the solution is so it isenough
to provesmoothness near the
boundary. Since p
isquite smooth,
it is easy to showtangential
derivatives extendcontinuously
to theboundary.
We find thesame holds for the time-derivatives
using
From the
boundary condition,
KN.VT is smooth on theboundary.
Introducing appropriate
coordinates near aSZ andchanging
variables inthe differential
equation,
we see the normal derivatives are also smooth to theboundary,
so T islocally C3° 3/2,
as claimed.Asymptotic
behavior of solutionsExamination of the calculations for Theorem 2 shows we could find much better estimates if we had a
positive
lower bound for thetemperature
on We obtain such an estimate from the maximum
principle
on anycompact
set inQ,
such asprovided
aS2 isC2.
With mild"compactness"
and
"positivity" hypotheses
on g, we obtain apositive
lowerbound,
uniformly
fott ~
const., and thus obtain convergence as t ---~ oo . If theheating
rate isperiodic
oralmost-periodic
intime,
thelimiting "steady-
state" solution is
equally periodic
oralmost-periodic.
LEMMA. 2014 Assume ~03A9 is
C2 (ar
has an interiorsphere at
eachpaint)
and the
hypotheses of
Theorem 1 hold.7/) for
andQ,
we have
T(x, t)=0,
thenQ
xt]. Equally, if 03C8~0 on 03A9 or g~0
on ~03A9
[to, t],
thenT (x, t)>0.
Proo.f : -
the result is immediate from Thm.1,
Sec.1, Chap.
2of
[3].
If inQ
xt]
then T>O inQ x (0, t)
and inQ x {~},
whileT (x, t)=0,
so KN . V T 0 at(x, t)
andg (x, t)0,
which is false.(Fried- man’s,
Thm.14,
Sec.5, Chap.
2 of[3],
does notapply,
but a similarargument - using
aparabolic "cup"
instead of anellipsoid - gives
theresult. For the one-dimensional heat
equation,
theargument
is in Cannon[2].)
We will use the
following hypotheses:
The compactness
assumption :
For anycompact
interval J c R and any sequence there is asubsequence
such that{g(., .
+|~03A9
xJ }
is aCauchy
sequence in C xJ, R).
The
positivity assumption :
For somepositive
constants ’to, co, we haveTHEOREM 4. -
Suppose
a~2 isC2
or has interiorspheres
and thehypo-
theses
of
Theorem 1 hold. We also suppose that either Q is bounded or thatn
=1,
0 =(0, (0),
and p,C, K,
E, a are constants. We assumegiven
asolution
T of
theproblem (PDE)
on S2 x(to, CIJ).
(a) I. f ’
the compactnessassumption holds,
any sequencetn
~ oo has asubseguence tn.
such thatexist
uniformly
on compact setsof
aS2 xR, Q
xR, respectively,
and T * is asolution
of (PDE) -
withg*
inplace of
g - on all S2 xR, satisfying 0 ~ T* _
Beverywhere.
(b) If
the compactness andpositivity assumptions hold,
thenfor
any compact K cQ, T (x, t) ~
const. > 0 on K x[to
+ io,(0).
Given any sequencetn
~ oo, such that lim g( . , .
+tn)
=g*
existsuniformly
on everycampact
set in aSZ x
R,
itfollows
thatAnnales de Henri Poincaré - Physique theorique
exists
uniformly
on compacts inn x R, the
limit sotutionT* depends only
an
g* (independent of
theinitial value
and the sequencetn)
andsatisfies T* ~const.> 0, uniformly
intime,
oneach
compact subset ofÕ.
( c) If
t ~ g( ., t)
isuniformly
almostperiodic [or periodic with period
or
constant], 0~g~B
asusual, and g~0, then
the compactnessand positivity assumptions
hold.There
is
aunique non-negative bounded solution
of on all n xR, and
it is almostperiodic in
t,uniformly for x~03A9,
withfrequency
module
contained in that of g[or
isperiodic
withperiod
~; or is constant intime, respectively],
and it is bounded from zero on every set K x~,
I~compact
iriQ. Any
solutionT
of{ta, oo)
satisfies
uniformly
oncompact
setsof x,
and the convergenceis exponential
whenQ is
bounded.
(~x) ( . , t),
is bounded inCo (K),
for anycompact
K and 061,
it is in acompact
subset. We maychoose; by
thediagonal
process, a such
that g (., .
+ ~g*
and alsoexists
uniformly
oncompact
setsin 03A9
for1,2,3,
... It followseasily
that T* =lim (.,.
+tn,)
existsuniformly
oncompacts
in R andn’
is the limit solution claimed in
(a).
is
compact
and liminf There existso
’~’ (xn, tn) --~ o.
We may assume,by (a)>
thatT (.,.
+tn)
~T*, uniformly
oncompacts, and xn
-4x*,
so T*(x*, 0)
= 0.By
the
lemma,
T* = 0 on 03A9(-
oo,0]
sog*=0
on(2014
oo,0], contracting
the
positivity hypothesis.
Thus~
const. > 0 on K x[ta
+ to,(0).
Ifg ( . , . ~- t~) -~-~ g*, by (a),
we have convergence of translates ofT by
som~subsequence of {tn} .
But we prove the limit solution T* isunique, depend- ing only
ong~,
so infact, T (.,.
+t,~) --~
T*uniformly
oncompacts.
Indeed,
T*(x, t) ~
c > 0 on(recall
on iscompact)
so for anybounded, non-neatie
solutiont
of on all Q x R --- withg~‘
inplace of g -
the difference ~ ==T* -1 satisfies
and
|v|~const.~
on so0c1~Q
for some constantc~. We show r=0.
If Q is
bounded,
there is aC3
> 0 so thatThus,
so
when ~ ~.
Let s -~ 2014 oo toIf Q is not
bounded, Q=(0, (0)
and p,C, K,
a, E are constant.From the maximum
principle, ~)~Mw(~, t - s)
fort >__ s,
whereM =
max I v (., s) I and w
is the solution ofand
with w =1 for
/=0,
It suffices to show as too,uniformly
for x incompact
sets.We rescale t and x to
get
atx = 0,
with w = 1 atBy Laplace
transformation in t, we findwhich is (9
(t -1 ~2)
as t ~ oo with x bounded. Infact,
and,
from anotherrepresentation, 01-w(x,t)exp-x2/y,
for:c > 0,
It is tather weak
"convergence
to 0" sincew(x, t)
-~ 1 as x oo for(c) By almost-periodicity,
we maychoose tn
-> oo so thatuniformly;
for any solutionT, lim T (. ,. + tn) = T
existsuniformly
oncompacts,
and T is theunique positive
bounded solution of(PDE)
onQ x R. The
argument
above shows T(x, t + tn) -
T(x, t)
-+- 0uniformly
oncompact
sets ofQ
xR,
but we needuniformity
on K x R for everycompact Kc=Q.
Annales de l’Institut Poincaré - Physique theorique
Let vn (x, t)
= T +tn) -
T(x, t) ;
thenwhere
gn = g ( . , . +f’n)
and *0
1 for all n, t, x.Define w
(x, t)
as the solution ofwith w =1 when ~=0. Then for any
real s,
where M = sup T
(Q
xR) and En
= gI Allowing s
-~ - oo , we find9QxR
on QxR. When n -~ oo, and
uniformly, proving almost-periodicity:
see Ameiro and Prouse[ 1 ] .
If g
isperiodic
withperiod p,
and if T is a "limit"solution,
boundedand defined for all
time,
thenT (x, t + p)
is another limitsolution;
there isonly
one limitsolution,
so it isperiodic
in time withperiod
p. If g isconstant in
time,
it isp-periodic
for every p >0,
so the same is true of thelimit
solution,
which is constant in time.The convergence results are
proved
asabove, by comparison
with w.Consider a
homogeneous, isotropic
ball of radius R inR3 ;
the solutions of
tend to the surface
temperature Ts
likeexp ( -
whereis the thermal relaxation time. For water-ice and various
types
ofrocks,
the
diffusivity K/p C
is about 0.005m2/s,
so for R =1 km[10 km, 100 km],
the relaxation time is about
2,300
years[230,000
yr, 23 x106 yr].
Thus wecan
expect
thermalequilibrium
in any asteroid.APPROXIMATE SOLUTIONS
An asteroid’s
"day" - with
fewexceptions,
from 3 to 50 hours - is many hundreds of times smaller than its"year" -
3 to 6 of our years. We will firstneglect
theyearly variation,
considerperiodic (daily) heating,
andfind the
temperature
a short distance below the surface(perhaps
ameter).
This is
independent
of thedaily
rotation but still variesduring
the year.A second
( yearly)
averagegives
atemperature independent
oftime,
stillnear the
surface,
which we use as aboundary
value for anequilibrium problem
in the interior.We rescale time so the
daily
variation hasperiod
2 ~. We also rescaledepth
in units §(the
skindepth, specified later-probably
less than 10cm).
The
daily variation
is confined to adepth
of order8,
far smaller than the diameter of theasteroid,
and the variation oftemperature
withdepth
isfar more
rapid
than the variationalong
the surface. Thus the heatequation
becomes
approximately
at
depth
ös below agiven point
of thesurface,
203C0/(period
ofrotation).
We choose S=/2014~20142014 . 2014 ("skin depth")
and also define"conduction
depth" 03BB
=To N . f
and "maximum surfacetemper-
ature"To=(03B1/~03C3)1/4. (Applied
to Moon andMercury,
we findTo
= 390and 700 K
respectively,
which are in fact close to the observed maximum surfacetemperatures.
Atemperature gradient T o/À produces
heat fluxequal
to the maximum absorbed heat fluxa ~‘’.)
Normalizing temperature
and theheating
rateg/a .,~’= y,
asfunctions of normalized time t and normalized
depth
s, we obtainwhere y is
203C0-periodic
in(angle
from zenith toSun)
when theSun is
visible, y=0
when it is notvisible,
Thisequation
has aunique non-negative
bounded solution M on{
2014oo/oo,~0},
whichis in fact
strictly positive (0M1)
and203C0-periodic
in time. We write00
~) = ~ where ~ == (:I:
so~ = ~
2014 00
and the
boundary
condition becomesThe
important quantity
is Mo= lim the limit isessentially
attai-s -+ +00
ned when so the
temperature
at adepth
5.8, independent
ofdaily
variation.Before
studying (* *),
wecompute À
and S for various materials for thecase
To = 280 K, f = 300 kcal/hr.m2, a =8=1,
which wemight expect
at2 A.U.,
and forperiod
6 hours.(Recall 8
isproportional
to/period,
sode Henri Poincaré - Physique theorique