Radiative environment

4. Planetary thermal environment for stratospheric balloon thermal design 79

(a) (b)

Figure 4.26: (a) Original wind velocity map of 13^th June 2016 and (b) map resulting after the treatment.

convective effect of the relative velocities of the system. As explained before, they can be derived from a dynamic model using wind data presented in this paper.

(a) (b)

Figure 4.27: (a) Mean values and (b) 95 percentile values of the month of June of years 2016 and 2017.

and treated as a function of the altitude and with an hourly temporary resolution.

The region studied includes latitudes from 64^o N to 69^o N and longitudes from 20^o E to 22^o E with 1° x 1° spatial resolution. This region is large enough to analyze the potential trajectories of LDB missions during its ascent phase considering Esrange as the launch site.

Data have been retrieved from CERES [41]. The parameters to be used are the outgoing and incoming shortwave and longwave radiation for 6 different pressure levels from the surface to the TOA, defined at around 20 km. However, the float altitude for a stratospheric balloon is around 40 km. From 20 km upwards the fraction of atmosphere is small and, as is explained in [44], the inverse-square law states that the flux varies as the inverse square of the distance from the centre of the source [82]. The next sections are then focused on defining the conditions up to 20 km. Fluxes above TOA are obtained directly applying the following equation,

F(θ;d_b) =F(θ;d_a) r_E +d_a rE +db

!2

, (4.17)

wherer_E is the Earth radius,F(θ;d_b) is the flux at an altituded_b above the TOA and F(θ;da) is the flux at the reference level (TOA), for a given Solar Zenith Angle, θ.

Parameters extracted from CERES, together with the SZA, can be used in order to obtain solar irradiance (G_S), albedo coefficient (a), OLR (q_E) and sky temperature (T_sky). This way, the radiative fluxes reaching a given instrument with solar absorptance α_i, infrared emissivity ε_i, and area A_i, can be calculated as will be explained in Section 4.3.2.

The Solar Zenith Angle and its variation during the ascent are necessary to simulate the thermal behavior of the systems onboard. Moreover, thermal radiation fluxes will be derived from the instantaneous SZA. During the considered period of time (month of June), at latitudes between 64° N and 69° N, the SZA varies from 45.2^o to 89.2^o, which means that there will be a constant view of the Sun. The variation of the SZA during the month of June at Esrange is shown in Figure 4.28.

It seems quite obvious, as has been said previously, that a dependence of the four radiative parameters with the altitude due to the presence of the atmosphere exists. Moreover, analyzing the data to be used in the selection of the radiative thermal environment, the influence of the SZA can be determined, as shown in Figure 4.29. These box plots have been obtained through a discretization of the studied parameters as a function of the SZA and the altitude. The box shows the quartiles of the dataset while the whiskers extend to show the rest of the distribution, except for points that are determined to be "outliers" [83].

4. Planetary thermal environment for stratospheric balloon thermal design 81

Figure 4.28: Variation of the SZA during the month of June at Esrange.

(a) (b)

(c) (d)

Figure 4.29: Box plot representation of (a) albedo, (b) OLR, (c) solar irradiance, and (d) sky temperature as a function of the SZA and the altitude.

A dependence between these parameters and the SZA can be observed for different altitudes except for the sky temperature which mainly depends on the altitude. This dependence is explained in Refs [42, 84]. As they say, "radiative fluxes from CERES are estimated using empirical Angular Distribution Models (ADMs) that characterize the anisotropy or angular variation of the radiation field." When facing this fact, the treatment of the data in order to avoid the angular dependence

is not possible since "distinct ADMs are developed for different scene types, which are defined using a combination of variables (surface type, cloud fraction, cloud optical depth, cloud phase, aerosol optical depth, precipitable water, lapse rate)."

Since it is possible to obtain the instantaneous SZA during the ascent phase, it has been considered more appropriate to represent each parameter as a function of it. If the time range selected to compute the extreme cases is too wide, the variability of the SZA along the year could affect those values.

In order to illustrate the variability of each parameter with different SZA, they have been represented in heatmaps. Colors in the maps correspond to the parameter’s values statistically computed in order to obtain the extreme cases taking into account the probability of occurrence during the ascent phase. The details are given in the next subsections.

Albedo and Outgoing Longwave Radiation

When selecting this pair of values, it is necessary to take into account not only the dependence on the altitude and the SZA but also the correlation between them [15, 14]. The process followed to select the worst cold case during the ascent phase is described in Figure 4.30.

The data retrieved from CERES correspond to Esrange and the area covered during the launch of a Polar-Summer LDB. Analyzing its relationship for each altitude, it is possible to represent the probability density function for several altitudes and a given SZA, obtaining the graphs shown in Figure 4.31.

As can be observed, as the altitude increases, data tends to be correlated. The higher the albedo, the lower the OLR and vice versa. Taking this consideration into account, it is possible to obtain the potential cold extreme cases as function of the altitude and the SZA to be found in this region during the month of June.

Three cases will be considered based in the methodology exposed in [14]. Three potential worst cold cases based on gaussian distributions of albedo and OLR are proposed by NASA [17]:

• 10 percentile albedo and its associated OLR averaged value.

• 10 percentile OLR and its associated albedo averaged value.

• 10 percentile of a filtered distribution with equal normalized variate in both albedo and OLR (X_N).

X_N = X−X_m

σ_x (4.18)

4. Planetary thermal environment for stratospheric balloon thermal design 83

Figure 4.30: Scheme of the process followed to select the worst cold case of albedo and OLR.

The potential cases obtained for the three cases respectively are shown in Figure 4.32.

Once computed the potential worst cases, it is necessary to take into account the influence of the thermo-optical properties of the considered system. As is said in [14], "the extreme hot or cold case for a particular system depends on the emissivity of the spacecraft surfaces and its absorptivity for solar radiation. Depending on the ratio between these parameters 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." It has been demonstrated that this methodology to select extreme cases is not as precise as could be the one explained in [15], but considering the complexity of this transient analysis during the ascent phase, it provides an acceptable approximation.

The pair of values of OLR and albedo that is going to minimize the temperature of the system is the one that minimizes the heat input ˙Q_a,i+ ˙Q^IR_E,i (albedo + OLR).

For altitudes above the highest represented, the albedo coefficient can be consid- ered constant since applying the inverse square law to the ratio upgoing/downgoing short-wave flux, the factor is very close to 1. However, in order to obtain the value

(a)Surface. (b) 1.5 km.

(c)5.4 km. (d)11.6 km.

(e) 18.2 km. (f) TOA.

Figure 4.31: Probability density function of albedo and OLR values for several altitudes and a SZA = 87.4°.

4. Planetary thermal environment for stratospheric balloon thermal design 85

(a)Surface.

(b) 5.4 km.

(c) 18.2 km.

Figure 4.32: Heatmap of the albedo (left) and OLR (right) for the 3 different worst cold cases.

of the OLR at a particular altitude above TOA, one could use the TOA value multiplied by the view factor to it (which in fact is applying Equation 4.17) instead of using the actual OLR at such altitude. Nevertheless, taking into account the float altitude of these kinds of flights (slightly below 40 km), this correction is negligible (< 0.7%) and values of OLR at TOA can be used.

Solar irradiance

When studying the solar irradiance reaching different layers of the atmosphere, the solar spectrum and the atmosphere opacity to some wavelengths due to its composition explain its dependence with the altitude and the SZA since "the amount of atmosphere" to pass through decreases with it.

For altitudes above the TOA, it can be considered that the solar irradiance is the solar constant, G_S, and is not affected by the atmosphere. Its value varies through the year due to the variable distance between the Earth and the Sun.

Moreover, the solar cycle modifies its value, but the variations are below 1%, so they will be considered negligible. For altitudes below the TOA, CERES data has been statistically treated obtaining the mean values for each altitude depending on the SZA, as shown in Figure 4.33.

Figure 4.33: Mean solar irradiance heatmap.

Sky temperature

As has been shown, the sky temperature, which has been derived from the incoming longwave flux, is strongly dependent on the altitude but not on the SZA. Computing the mean values for each altitude, the heatmap shown in Fig- ure 4.34 is obtained.

At the top of the atmosphere and in higher altitudes, the sky temperature could be considered as the deep space equivalent temperature since there is no significant amount of atmosphere above it. Values of 2.7 K can be used for thermal calculations.