Contents
2.1 STEM
2.1.1 Methodology
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Contents
remove this dependence, ERBE albedo observations,aθ, are converted into equivalent TOA values at SZA equal to zero, aθ=0, where,
aθ=0 =aθ−cθ, (2.1)
The empirical correction term cθ given by Ref. [14] is,
cθ =c1θ+c2θ2+c3θ3 +c4θ4, (2.2) where c1 = 1.3798×10−3, c2 = 2.1793× 10−5, c3 = 6.0362× 10−8 and c4 = 4.9115× 10−9 for 0 ≤ θ ≤ 90◦. However, STEM only considers data where 0≤θ ≤65◦ due to the increasing uncertainty for large θ. In addition, if the aim of STEM is to provide constant albedo values considering they do not vary with SZA, an orbital average correction term must be used. As SZA is limited by SBA (θmin =β), and this can be considered to be constant throughout the orbit period, it is possible to obtain a correction term as a function ofβ as shown in Figure 2.1.
Figure 2.1: Orbital-average albedo correction term as a function of the orbital beta angle and correction at minimum SZA [17].
NASA data processing [17] requires to consider two more parameters in order to develop an appropriate methodology:
• The sample time, ∆tsample, which is the period of time between one sample of thermal environmental values and the next one in the satellite orbit.
• The average time, ∆tavg, which is the period of time the thermal environmental parameters are averaged centred in each sample.
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The optimum sample time depends on the characteristics of the satellite and the amount of independent data required for performing the statistical analy- sis. Nevertheless, if the aim is obtaining independent data about the thermal environment, the sample time must be related to the maximum average time through the following relationship:
∆tsample ≥∆tavg →∆tavg,max = ∆tsample (2.3) By doing so, no duplicated values would be used to obtain the average cor- responding to each sample. In addition, the average time must be related to the characteristics of the satellite or at least to one critical part of it. The temperature variations with time are smoothed with higher values of time constants, providing a smaller range between extreme values. Besides, there is less variability in temperature extremes from one orbit to other. The appropriate data average time for a system with a characteristic time, tc, should be between tc/4 and tc. High values of tc make the environmental changes during an orbit period negligible in terms of temperature response of the system. In contrast, low tc values make the system sensitive to short-period variations of the thermal environment. STEM uses 7 average times ranging from 16 seconds (corresponding to the sample time of ERBE) and 24 hours. As mentioned above, non-overlapping time averages have been used in the data processing.
When selecting the worst-case values to analyse the thermal behaviour of the system, the risk of occurrence during the mission lifetime should be considered.
Anderson and Smith presented a statistical study considering different percentiles.
These percentiles represent the probability that a given value will not be exceeded.
However, the limited data available did not allow them to established such criteria.
The required data set should cover a much longer period of time compared to the mission life. Thus, these percentiles are to be associated with the fraction of time the values are expected to be exceeded; not the probability that they will be exceeded over a mission lifetime.
In order to simplify the selection criteria regarding the thermal risk, STEM provides just two sets of statistics. Mission critical values, which are shown in Figure 2.3, correspond to 0.04 and 99.96 percentiles (corresponding to the ± 3.3σ for a Gaussian distribution). Mission non-critical values for 5 and 95 percentiles are also provided. The use of these values do not ensure that they will be exceeded 5 % of the time since it is only an approximation.
A deep study is also carried out to determine the influence of orbital beta angle, latitude and inclination. The results led to the conclusion that both albedo
(referenced at SZA 0) and OLR strongly depend on latitude and therefore, inclination should be explicitly considered in the statistical study since it determines the range of covered latitudes. In addition, once removed the influence of SZA from the albedo, the orbital beta angle does not contribute significantly. It was not considered in the statistical study but was its contribution through the albedo correction term as well as the Earth’s shadow. A similar behavior was found for the time-of-day since its contribution is mainly due to the latitude excursions.
Time-of-year effects on albedo and OLR distributions are relatively small and they were not considered in the analysis.
Figure 2.2: Extreme hat and cold cases (albedo, OLR and combined) [17].
From the spacecraft perspective, however, what proves to be the extreme hot or cold case for a particular system depends on its thermo-optical properties.
Depending on the ratio between solar absorptance and infrared emissivity, the extreme spacecraft temperatures may be associated with extreme OLR cases, extreme albedo cases, or some intermediate "combined" case where both OLR and albedo run high (or low) together but neither is near its individual extreme.
• For extreme OLR cases, the highest or lowest OLR value bins were selected until at least 0.04 percent of the data set is accumulated, then the paired albedo values are averaged.
• Extreme albedo cases are obtained in the same way as OLR.
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• The combined case consider both distribution of albedo and OLR and selects the subset with equal normalized variate, xn, in both albedo and OLR,
xn= x−xm
σx (2.4)
where xm is the mean of the distribution and σx the standard deviation.
For a medium inclination orbit with 128 seconds of time average, these 6 cases (hottest and coldest albedo, OLR and combined) are shown in Figure 2.2.
Figure 2.3: Mission critical hot and cold extreme environments at low, medium and high inclinations, and averaging times from 16 seconds to 24 hours [17].
The result of the NASA statistical study was the development of parametric tables for critical and no-critical application that are shown in Figure 2.3 and 2.4 respectively. In these tables, 3 ranges of orbit inclinations, 7 average times and 3 extreme types for the hot and cold case are considered.
The program developed by NASA allows to define a generic system in a defined orbit with established thermo-optical properties and a considered characteristic time. The thermal behaviour of this system is evaluated in order to provide the extreme cases. However, depending on the complexity of the system and the variety of components with different time constants, it may be convenient in some cases to use non-constant profiles of albedo and OLR. STEM considers the use of pulses modeled as step functions, on top of the appropriate base value with a duration just higher than the considered time constant. The hot case albedo and OLR
step functions are specified to encompass orbital noon so the maximum short-term values are encountered at the time of greatest heating. Likewise, the cold case step functions are applied at the opposite side of the orbit, orbital noon plus 180°, when the vehicle is in a shadow or at least the albedo radiation is at a minimum.
Figure 2.4: Non-critical hot and cold extreme environments at low, medium and high inclinations, and averaging times from 16 seconds to 24 hours [17].
As an example, a 400-seconds time constant system is considered in Figure 2.5.
In this case, STEM provides as a base value for a multiple pulses of various duration for hot and cold cases. This implementation allows to consider a system with multiple time constant components.