Estimating the Temperature of a Flat Plate in Low Earth Orbit
This
problem is a real-world problem that I first encountered in my daily
work at NASA. A flat plate is orbiting the Earth at a mean altitude
of 300 km. Its orbital velocity is 7.5 x 103 m/sec eastward.
(All spacecraft are launched eastward to take advantage of the Earth's
rotational motion.) The plate is in sunlight. Sunlight warms the plate,
and the plate radiates thermal energy back into space.
Ambient
atomic and ionic species are also present. These species are moving
at characteristic thermal velocities, which are very small compared
to the plate's orbital velocity and may, for all intents and purposes,
be ignored. The flat plate may be thought of as 'running into' populations
of stationary particles at a velocity of 7.5 x 103 m/sec.
From the plate's point of view, these particle populations appear to
be ramming it 'head-on' at 7.5 x 103 m/sec. When the particles
hit the plate, they are assumed to loose their kinetic energy to the
plate as heat. Again, as in the case for sunlight, the plate also radiates
this energy back into space.
The
balance between incoming energy and outgoing (radiated) energy causes
the plate to come to an equilibrium temperature. The problem is to calculate
this temperature.
We
begin with Stefan-Boltzmann's law for a black body in sunlight.
SSUN
=
T4 W/m2
This
law enables us to estimate the temperature T of such an object, assuming
that we know the power per unit area SSUN falling on the
plate. From direct measurement, we know that at 1 A.U,
SSUN
= 1,360 W/m2.
We
also know that
= 5.67 x 10-8 W/(m2 K4).
Thus,
the nominal temperature of an object, in space and in sunlight, is 394
°K.
Next,
we must deal with the ambient particle fluxes. In general, random directional
particle flux to the surface is calculated from
=¼(nVPARTICLE)
m-2-sec-1
where
n is the number density of the particle species per m3, and
VPARTICLE is the average velocity of the individual particles
in m/s. Here, we take VPARTICLE = 7.5 x 103 m/sec.
Since particle flux to the surface is assumed to be ram directional
(i.e., we are assuming that the plate 'sees' all the particles
coming head-on at 7.5 x 103 m/sec), the usual factor of ¼
(which arises when the directions are random) may be omitted.
From
an established knowledge of the orbital environment at 300 km, we select
the following, dominant particle species:
Oxygen
ions: O+
Atomic oxygen: O
Molecular oxygen: O2
Molecular nitrogen: N2.
Then,
for each of these selected particle species, we may define a power flux
to the surface:
SPARTICLE
= (nVPARTICLE)(EKINETIC/PARTICLE) W/m2.
where
EKINETIC/PARTICLE is the kinetic energy per particle. The
values of n and EKINETIC/PARTICLE must be determined.
If
(as we have already assumed) the particles simply deliver their kinetic
energy to the plate upon impact (totally inelastic collision), then
they may be treated as energy fluxes additional to that of the sun,
and a final temperature may be estimated from:
T4
= SSUN + SPARTICLES
Now,
we are ready to calculate. We have already calculated the temperature
contribution from the sun:
T
= 394 °K.
Next,
we calculate the kinetic energy of each of the particle species:
1.
Oxygen
ions:
O+:
mO+
= 16 x (1.67 x 10-27 kg) = 2.67 x 10-26 kg
VO+ = 7.5 x 103 m/sec
EK/O+ = ½ mO+ (VO+)2
= 7.51 x 10-19 j
2.
Atomic
oxygen:
O:
mO
= 16 x (1.67 x 10-27 kg) = 2.67 x 10-26 kg
VO = 7.5 x 103 m/sec
EK/O = ½ mO (VO)2 =
7.51 x 10-19 j
3.
Molecular
oxygen:
O2:
mO2
= 32 x (1.67 x 10-27 kg) = 5.34 x 10-26 kg
VO2 = 7.5 x 103 m/sec
EK/O2 = ½ mO2 (VO2)2
= 1.50 x 10-18 j
4.
Molecular
nitrogen:
N2:
mN2
= 28 x (1.67 x 10-27 kg) = 4.68 x 10-26 kg
VN2 = 7.5 x 103 m/sec
EK/N2 = ½ mN2 (VN2)2
= 1.32 x 10-18 j
The
number densities of each of the particle species at the altitude of
300 km are obtained from published references (cited):
nO+
= 5 x 1011 /m3 (Sp. & Planetary Env., Vol.
1, pg. 2-31)
nO = 2 x 1014 /m3 (U.S. Std. Atm.,
pg. 30)
nO2 = 1011 /m3 (U.S. Std. Atm., pg.
30)
nN2 = 5 x 1012 /m3 (U.S. Std. Atm.,
pg. 30)
Using
these quantities, we obtain the power fluxes S for each of the particle
species:
SO+
= 2.82 x 10-3 W/m2
SO = 1.13 W/m2
SO2 = 1.12 x 10-3 W/m2
SN2 = 4.96 x 10-2 W/m2
For
convenience, we sum these contributions:
SPARTICLES
= SO+ + SO + SO2 + SN2 =
1.18 W/m2.
Recall
that SSUN = 1,360 W/m2, so that the temperature
contribution from the ram particles alone must be very small in comparison.
Let us estimate this ram particle temperature contribution.
An
estimation of the change in temperature T
due to SPARTICLES
may be obtained from10
With
T = 394 °K, and SPARTICLES
= 1.18 W/m2, we find T
= 8.51 X 10-2 K. This value is too small to make a significant
difference when compared to the contribution of the sun. We therefore
ignore it, and take the plate's temperature to be 394 °K.
10 To
obtain this expression, let us re-write our working equations as follows:
Set S = SSUN, and set S + S
= SSUN + SPARTICLES
. Then, set S = T4
and S + S
= (T
+ T)4
= (T4
+ 4T3T
+ 6T2(T)2
+ 4T(T)3
+ (T)4)
(T4
+ 4T3T)
for T<<T.
Subtracting the two expressions and substitting S
= SPARTICLES
gives the expression in the text.