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Satellite irradiation

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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 Sun-Earth 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 11-years 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 Earth-emitted 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 Earth-Sun distance, ca. 150 106 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 half-opening angle of tan(alpha) = RE/(|ES|2-RE2)1/2 with:

RE:
Radius of the Earth;
|ES|:
Sun-Earth distance.
Solar shadowing cone
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-RE2]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) · AU2 · |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 106 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 day-night zone is continuous.

The power emitted by a surface at temperature T (K) is given by the Stefan-Boltzman equation:

P = 5.6703 10-8 T4

Local geometry
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:

IE = 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 Earth-infrared 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 radiation power incident on a infinitesimal surface dA on the reflecting surface (i.e. top of atmosphere surface) is given by:

Pi = 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:

Pr = Pi / (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 phii at orbital position i is given by:

phii = phii-1 + 0.5 (fi+fi-1) (ti-ti-1) ,

where fi is the power at orbital position i and ti the corresponding Julian date, and phi0=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 Fj after orbit j is given by:

F1 = Phi1 (T1-t1)
and

Fj = Fj-1 + 0.5 (Phij+Phij-1) (tj-Tj-1) + Phij (Tj-tj) ,

where Phij is the average power over orbit j, Tj is the Julian date at the end of orbit j, and tj 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

ECSS-E-10-04, 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 ECSS-E-10-04 clause 6.


Last update: Mon, 12 Mar 2018