Help: Satellite irradiation
Table of contents
Introduction
A spacecraft in LEO will receive electromagnetic radiation from three primary
external sources. The largest source is the direct solar flux. The mean value
of this solar flux at the mean SunEarth distance is called the solar
constant. It is not really a constant but varies by about 3.4% during each
year because of the slightly elliptical orbit of the Earth about the Sun. In
addition the amount of radiation emitted by the Sun varies slightly (by about
0.1%) throughout the 11years solar cycle.
The fraction of incident sunlight that is reflected off a planet is termed
albedo. For an orbiting spacecraft the albedo value depends mainly on
the sunlit part of the Earth which it can see. Albedo radiation has
approximately the same spectral distribution as the Sun. Albedo is highly
variable across the globe and depends on surface properties and cloud cover.
It also depends on the solar zenith angle.
The third source is the Earth infrared radiation. The Earthemitted thermal
radiation has a spectrum of a black body with a characteristic average
temperature of 288 K. The Earth infrared radiation also varies across the globe
but less than the albedo. It also shows a diurnal variation which is small over
the ocean but can amount to 20% for desert areas.
The SPENVIS implementation calculates the
incident electromagnetic radiation on an oriented plate fixed to the satellite
for each source.
Direct solar flux
The solar constant is defined as the radiation that falls on a unit
area of surface normal to the line from the Sun, per unit time, outside the
atmosphere, at one astronomical unit (1 AU = average EarthSun
distance, ca. 150 10^{6} km). The solar constant has an uncertainty of
about 10 W m^{2}
[Anderson, 1994]. The value used is 1371 W m^{2}.
Direct illumination is only possible if the satellite is not situated in the
Earth shadowing cone (see figure 1). This cone has a
halfopening angle of tan(alpha) =
R_{E}/(ES^{2}R_{E}^{2})^{1/2}
with:
 R_{E}:
 Radius of the Earth;
 ES:
 SunEarth distance.

Figure 1. Geometry of the shadowing cone (not to
scale) 
If ES is the vector for the Sun position and
ET is the vector for the satellite position (origin
at the center of the Earth), then TS = ES  ET is the
vector from the satellite to the Sun. The angle beta between ES
and TS is given by:
cos(beta) = TS · ES / TS · ES
The satellite is in the Earth shadow (delta = 0) if beta <
alpha and TS >
[ES^{2}R_{E}^{2}]^{1/2},
otherwise the satellite is sunlit (delta = 1). The illumination on the
oriented plate is then given by:
I = SC · delta · max(cos(phi), 0) ·
AU^{2} · TS^{2}
with:
 I:
 illumination (W m^{2});
 SC:
 Solar constant: 1371 W m^{2};
 phi:
 angle between normal and Sun direction;
 AU:
 Astronomical Unit: 149.5988 10^{6} km.
Earth emitted infrared radiation
The average value of the Earth emitted infrared radiation is 230 W
m^{2}, but on a short time scale, this can vary between 150 and 350 W
m^{2}.
For the calculation of the Earth emitted infrared radiation
incident on an
oriented plate in a LEO, one can not make the assumption that the source
of the radiation is a point, as was done for the direct illumination. The
geometry for the contribution of an elementary surface dA is shown in figure 2.
The implementation
in SPENVIS makes use of a temperature
profile on the Earth surface, defined by two temperatures. The temperature on
the sub solar point is 306 K, on the night side it is 248 K. On the day side,
the temperature varies according to the cosine of the incident angle
(sigma in figure 2), while it is constant on the night side. The
transition at the daynight zone is continuous.
The power emitted by a surface at temperature T (K) is given by the
StefanBoltzman equation:
P = 5.6703 10^{8} T^{4}

Figure 2. Geometry for the locally emitted IR (resp.
local reflection) on the Earth surface (resp. TOA) (not to scale) 
As the Earth infrared radiation is uniformaly distributed over a hemisphere
(solid angle 2 pi), the contribution to a solid angle can be written as:
I_{E} = EIR · dA · max(cos(tau), 0) ·
(2 pi sin(theta) LT^{2})^{1}
with:
 EIR:
 the average infrared radiation emitted by the Earth: 230 W m^{2};
 tau:
 angle between normal and zenith direction;
 theta angle between the local normal n and direction of the
radiation.
Integrating this over all elementary surfaces dA where cos(theta)
> 0 gives the total Earthinfrared radiation incident on the oriented plate.
Earth reflected radiation
Albedo values are only applicable when a portion of the Earth that is seen by
the satellite is sunlit. Albedo is also dependent on solar zenith angle.
For the albedo calculation, a number of assumptions are made:
 The sunlight is reflected at the top of the atmosphere (TOA), i.e. at
R_{a} = R_{E} + d(atm), with d(atm) the thickness of the
atmosphere.
 The Earth atmosphere behaves as a Lambertian reflector, i.e. a
perfectly diffusing surface with the property that the amount of light
reflected from a small area toward the viewer is directly proportional to the
cosine of the angle between the direction to the viewer and the normal to the
surface.
 Albedo is supposed to be constant across the globe and independent of
cloud cover.
The radiation power incident on a infinitesimal surface dA on the
reflecting surface (i.e. top of atmosphere surface) is given by:
P_{i} = SC · max(cos(sigma),0) ·
(1AU)^{2} · LS^{2} · dA
with:
 sigma:
 angle between the sun direction LS and the local normal n.
The reflected radiation is proportional to this value by a factor
albedo. As the surface is supposed to be Lambertian, and reflection
occurs only into one hemisphere the reflected
radiance per unit solid angle is:
P_{r} = 
P_{i} / (2 · pi · sin(theta)) 
theta < pi / 2 
0 
theta > pi / 2 
The target plate and orientation determine the solid angle seen from the point
L as:
Psi = A · max(cos(tau),0) ·
LT^{2}
Finally, integrating over the Earth surface gives the total albedo as seen
by the oriented plate.
Segment and mission integration
The incident energy is the integration over time of the incident power.
It can also be defined as the product of the average power over a certain time,
and this time. As the power is computed for a discretised number of times
(corresponding to the orbital positions), this definition is used in the
illumination tool. Energy for each orbit as well
as for the whole mission are calculated.
Segment energy
A recurrent trapezoid method is used. The energy phi_{i} at
orbital position i is given by:
phi_{i} = phi_{i1} + 0.5
(f_{i}+f_{i1})
(t_{i}t_{i1}) ,
where f_{i} is the power at orbital position i and
t_{i} the corresponding Julian date, and
phi_{0}=0.
The average power is then the total energy divided by the segment duration.
Mission energy
A similar method is used for the cumulated energy from the beginning of the
mission. The energy F_{j} after orbit j is given by:
F_{1} = Phi_{1}
(T_{1}t_{1})
and
F_{j} = F_{j1} + 0.5
(Phi_{j}+Phi_{j1})
(t_{j}T_{j1}) +
Phi_{j} (T_{j}t_{j}) ,
where Phi_{j} is the average power over orbit j,
T_{j} is the Julian date at the end of orbit j, and
t_{j} is the Julian date at the start of orbit j. The
mission total energy is the energy after the last orbital arc.
The same integrations apply for the calculation of the exposure time and the
effective exposure time, with adapted integrands (0 or 1 for the the exposure
time and; cos(incident angle) for the effective exposure time).
References
ECSSE1004, European
Cooperation On Space Standardization, Draft 03, 2000.
Anderson, B. J., editor, and R. E. Smith, compiler, Natural Orbital
Environment Guidelines for Use in Aerospace Vehicle Development, NASA TM 4527,
chapters 6 and 9, June 1994.
This description is based on
ECSSE1004 clause 6.
Last update: Mon, 12 Mar 2018