2.1.

Model Structure

Figure 1 shows the structure of the SXS dewar. Within the SXS dewar main-shell (DMS) are four aluminum shields, from warmest to coldest these are an outer vapor-cooled shield (OVCS), a middle vapor-cooled shield (MVCS), an inner vapor-cooled shield (IVCS), and a Joule–Thomson shield (JTS). The OVCS and IVCS are cooled by the first and second stages of the two SCs, respectively, while the JTS is cooled by the JT. The MVCS is not actively cooled, but all shields are connected to the He vent line so that they are cooled by the He vapor venting from the He tank. The He tank is suspended from the IVCS using carbon-fiber reinforced plastic straps while the IVCS is suspended from the DMS using glass-fiber reinforced plastic straps. Other shields are also supported by blocks bonded to these straps.⁷ Note that the amount of LHe was 35.6 L at the launch.

Fig. 1

(a) A cross-sectional view of the SXS Dewar and (b) the schematic structure of the thermal mathematical model of the SXS dewar. Each box represents a node except for the ApA, and solid lines show thermal connections by $K_{\mathrm{C}}$ and/or $K_{\mathrm{R}}$. Red and blue filled circles show positive (Table 1) and negative heat loads (Table 2), respectively.

Table 1

Positive heat loads to the He tank, IVCS, and OVCS nodes in units of mW. The ADR heat loads and Joule heatings are time averageda.

Table 2

Negative heat loads to the He tank, the first, and second stage of the SCs, and the JT in units of mW.

The dewar thermal structure is modeled as shown in Fig. 1. The shields, He tank, cryocooler stages, and the LHe are each represented by one node, thus ignoring nonuniformity of temperature and heat distribution. Heat capacities of the LHe and the shields are provided as a function of their temperatures. The He tank itself is not explicitly included as its heat capacity is orders of magnitude smaller than that of the LHe. Because the He vent line is connected to the He tank at a point and to the shields at nine points, we utilize 10 nodes to model the He vent line. Reflecting the structure of the support strap, four of the nine nodes are directly connected to the shield nodes, whereas the others are via nodes of pass points as shown in Fig. 1. The vapor cooling is modeled as heat transfer due to boiloff He along the vent line from the He tank to the outside, using one-way conductance among the vent line nodes. Conductive and radiative thermal conductances are defined between connected nodes, and they are denoted by $K_{\mathrm{C}}$ and $K_{\mathrm{R}}$, respectively. While $K_{\mathrm{C}}$ is determined by the size, material, and the temperature of the conductive paths, $K_{\mathrm{R}}$ depends on the area and the effective emissivity of the surface and multilayer insulations (MLIs). In the determinations of $K_{\mathrm{R}}$, the effective emissivity at IVCS–JTS and JTS–He tank is fixed at 0.03 while we allow those at DMS–OVCS, OVCS–MVCS, and MVCS–IVCS to differ from 0.03 for temperature adjustment as shown in Sec. 2.3. Note that the aperture assembly (ApA) is only weakly coupled to the DMS and the shields as shown in Fig. 1, and its contribution is not significant.

2.2.

Modeling Heat Generations and Coolings by He and Cryocoolers

Positive heat loads are input to the He tank from the ADRs (magnets, salt pills, and getter heaters) and the detector assembly (DA). Furthermore, heat conducts to the He tank through mounting structures and through off heat switches 3 and 4 (HS 3/4) from the third ADR to the He tank and JTS, respectively. Joule heatings due to magnet currents through the resistive leads dissipate at the IVCS and OVCS. The heat load from the JFET amplifiers through the harness between the IVCS and the DA is also input to the IVCS. This varies with the IVCS temperature and becomes 0.21 mW when the IVCS is at 24.5 K. ⁷ We model these positive heat loads at the He tank, IVCS, and OVCS, as shown in Fig. 1 as red circles and summarized in Table 1. Although the heat loads by the ADRs and the lead Joule heatings are actually time variable, they are represented by their time-averaged values. The ADR heat loads shown in Table 1 assume $\sim 1\text{day}$ interval between recycles. In orbit, the ADR recycle period was actually $\sim 2\text{days}$, resulting in a smaller heat load than shown in Table 1 (see Ref. 8). The cooling effects at the shields by the cryocoolers and at the He tank by the He vaporization are supplied as negative heat loads as shown in Fig. 1 as blue circles and summarized in Table 2. The nominal cooling powers of the first and second stages of the SCs, and the JT are given as functions of their temperatures, $T_{\mathrm{SC}\text{first}}$, $T_{\mathrm{SC}\text{second}}$, and $T_{\mathrm{JT}}$, respectively.^9,10 He tank cooling by He evaporation is equal to the product of the He mass flow rate ${\dot{M}}_{\mathrm{He}}$ and the LHe latent heat $L_{\mathrm{LHe}}$. In this paper, ${\dot{M}}_{\mathrm{He}}$ as a function of the He tank temperature was provided by Ezoe et al.¹¹ based on breadboard model PP tests but including a correction factor of 2/3. Refer to Ezoe et al.⁵ for detailed descriptions about the correction factor that is required to reproduce the flight-model PP mass flow rate. $L_{\mathrm{LHe}}$ depends only weakly on the LHe temperature.¹² The relation between the LHe temperature $T_{\mathrm{LHe}}$ and the heat load to the He tank $Q$ can be written as

2.3.

Comparison with Ground Test in Equilibrium

Because the effective emissivities of MLI blankets are highly dependent on how they are assembled, the values of $K_{\mathrm{R}}$ could include uncertainties relatively larger than those of $K_{\mathrm{C}}$. Hence, we needed to tune $K_{\mathrm{R}}$ by comparing a simulated temperature distribution with an actual one measured in a ground test. For the comparison, we chose the latest data in the ground test derived on December 22, 2015, 04:14 (UT), in which all temperatures had achieved equilibrium. Table 3 shows the actual temperatures of the He tank, JTS, IVCS, and the OVCS, and their simulated temperatures after tuning $K_{\mathrm{R}}$. The actual temperatures were reproduced by the steady-state simulation within $\sim 15\%$ deviations.

Table 3

Temperature comparisons in equilibrium between the ground test on December 22, 2015, 04:14 (UT) and the steady-state simulation after tuning KR.

3.1.

Allowable Upper Temperature Limit Including Model Uncertainty

Although the temperatures can be reproduced to within 15% in the steady-state analysis as shown in Sec. 2.3, calculations at temperatures higher than those in the steady state are necessary in a transient analysis, which evaluates the maximum temperature after the launch. Because the LHe needs to be always in the superfluid phase in orbit, we consider that the maximum temperature must be lower than $\sim 2.1\mathrm{K}$, which is slightly margined from the $\lambda$ point. To be conservative, we decided to incorporate a 100% margin on the He tank heat loads when establishing the maximum allowable LHe temperature. At the maximum LHe temperature, Eq. (1) becomes

If we ignore the effect of the PP and assume for simplicity that the He mass flow rate is given as ${\dot{M}}_{\mathrm{He}}=G{P}_{\mathrm{sat}}(T_{\mathrm{LHe}})$, where $G$ is the He flow conductance in the He vent line and $P_{\mathrm{sat}}(T_{\mathrm{LHe}})$ is the saturated vapor pressure of the boiloff He, Eq. (2) becomes

Hence, when a 100% margin is added to the heat load corresponding to $T_{\mathrm{LHe}}=2.1\mathrm{K}$, $L_{\mathrm{LHe}}G{P}_{\mathrm{sat}}(T_{\mathrm{LHe}})$ needs to be increased by a factor of 2, and the LHe temperature becomes $\sim 1.87\mathrm{K}$. Therefore, we set the maximum allowable LHe temperature at 1.87 K.

3.2.

Transient Simulations with Various Initial LHe Temperatures

With the temperature of the DMS node fixed and initial temperatures of the other nodes set appropriately, we were able to run transient simulations of postlaunch conditions. In the nominal operation plan, the He venting starts $\sim 5\min$ after launch, and the SC operation with nominal power ($50\mathrm{W}\times 2$) starts $\sim 1$ orbit after the He venting begins (procedures 3 and 4 in Sec. 1). But given the possibility that SC startup could be delayed due to unexpected reasons, we carried out simulations by changing not only the initial LHe temperature but also the time to start SC operation. Figure 2(a) shows the LHe temperature variations with the initial temperature at 1.6, 1.65, 1.7, 1.75, and 1.8 K. It was assumed that SC operation starts half a day after the launch, with half the nominal power ($25\mathrm{W}\times 2$). The cooling power of the first and second stages of the SCs by the $25\mathrm{W}\times 2$ operations is assumed to follow $-20T_{\mathrm{SC}\text{first}}+2306\mathrm{mW}$ ($T_{\mathrm{SC}\text{first}}>113\mathrm{K}$) and $-11T_{\mathrm{SC}\text{second}}+440\mathrm{mW}$ ($T_{\mathrm{SC}\text{second}}>38.5\mathrm{K}$), respectively. In all cases, the LHe temperature did not start decreasing immediately after the start of the SC 25-W operations but reached a maximum 1 to 2 days after launch. This is because only the OVCS and IVCS are directly cooled by the SCs, and the He tank temperature responds after the shield temperatures decrease following their time constants. The maximum LHe temperature was lower than 1.87 K, when the initial LHe temperature was 1.7 K but did exceed 1.87 K when the LHe temperature at launch was higher than 1.75 K.

Fig. 2

(a) The simulated temperature variations of the He tank, calculated with various initial temperatures, 1.6 K (black), 1.65 K (magenta), 1.7 K (green), 1.75 K (cyan), and 1.8 K (orange). In the simulations, it was assumed that the SCs start working half a day after the launch with 25-W power. (b) A plot between the initial and maximum He tank temperature.

Figure 2(b) shows the relationship between the initial and maximum LHe temperatures, based on the five calculated points. The maximum allowed temperature of 1.87 K was reached when the initial (launch) temperature of the liquid was $\sim 1.704\mathrm{K}$. We also conducted a simulation by starting SC operation with nominal power one day after the launch. In this case, the liquid peaked at 1.87 K when the initial temperature was lower, $\sim 1.64\mathrm{K}$. Therefore, we adopted 1.7 K as the upper limit of the LHe temperature at launch and required SC operation to start at half the nominal power or higher within 12 h of launch. That shows that the launch would have to be aborted and the helium reconditioned if $T_{\mathrm{LHe}}$ exceeded 1.7 K. As shown by Fujimoto et al.,⁴ the LHe temperature was $\sim 1.5\mathrm{K}$ at the final prelaunch decision point, due to careful operations of the SXS team in the procedures 1 and 2 as shown in Sec. 1.

4.1.

Temperature Simulations in Orbit

We conducted a time-series analysis with initial LHe, JTS, IVCS, and OVCS temperatures set at the actual values, i.e., 1.50, 24.6, 41.0, and 158 K, respectively. By radiating in the anti-Sun direction, the DMS cooled to $\sim 260\mathrm{K}$,⁴ which was $\sim 30\mathrm{K}$ lower than that on the ground, and hence, we fixed the temperature of the DMS node at 260 K as a boundary. In the analysis, the SC 25-W, SC 50-W, and JT operations started $\sim 3\mathrm{h}$, $\sim 4\mathrm{h}$, and $\sim 1\text{day}$ after the launch, following the actual sequence.¹³ Figure 3 shows the transient simulation results. The predicted maximum LHe temperature was $\sim 1.53\mathrm{K}$, which was below the allowable upper temperature obtained in Sec. 3.2. After peaking at $\sim 1.5\text{days}$, the LHe temperature started monotonically decreasing. Similarly, the temperatures of the JTS, IVCS, and the OVCS were predicted to initially increase but then start decreasing several hours after the launch. The simulated temperatures of the LHe, JTS, IVCS, and the OVCS reached equilibrium at $\sim 1.12$, $\sim 4.3$, $\sim 22$, and $\sim 120\mathrm{K}$, respectively. All the simulated shield temperatures in equilibrium were lower than those shown in Table 3, presumably because of the lower DMS temperature. The equilibration times for the LHe, JTS, IVCS, and the OVCS temperature were $\sim 20\text{days}$, $\sim 1\text{day}$, $\sim 5\text{days}$, and $\sim 3\text{days}$, respectively. Because the shields are directly cooled down by the mechanical coolers and have relatively low heat capacity, they reach equilibrium faster than the LHe.

Fig. 3

The simulated (black) and actual (red) temperature variations of (a) the He tank, (b) the JTS, (c) the IVCS, and (d) the OVCS after the Hitomi launch. Panel (b), (c), and (d) show results of the first 12 days in the 40-days calculations and measurements. In the simulation, the initial temperatures were set at the actual values measured at the launch. The positive heat loads to the He tank, IVCS, and the OVCS from the ADRs and current Joule heatings in Table 1 were implemented after being time averaged in the calculation.

4.2.

Comparison between Simulations and Measurements

In Fig. 3, we superimpose the actual temperature curves of the LHe, JTS, IVCS, and the OVCS measured in the orbit from February 17, 2016, to March 25, 2016,^4,13 and simulation results. The LHe temperature variation was well reproduced by the simulation, except for a few % underestimations at $\sim 0$ to 1 days and $\sim 10$ to 20 days, and periodic temperature changes that corresponded to the ADR recycles. Note that the positive heat loads of the ADRs were implemented only as time-averaged values, which were updated by orbital measurements. ⁸ The IVCS and OVCS temperatures were also well reproduced, although deviations of up to 10% were seen. The measured IVCS and OVCS temperatures also showed periodic temperature changes due to the ADR recycles, because the Joule heating by the current leads was larger during the recycles while only the time-averaged heat loads, which were also updated by orbital values,⁸ were implemented in the calculations. Note that for days 1 to 4, the actual JTS temperature deviated from the simulation by 1 to 3 K. This is because the JT was turned off autonomously and restarted $\sim 1.1\text{days}$ and $\sim 1.9\text{days}$ after the launch, respectively, and then its power was adjusted to fine-tune its operation.¹³

The JTS temperature measured in the orbit decreased monotonically just after He venting started, which is different from the simulation, and the simulation overestimated it by as much as $\sim 20\%$. The deviation is larger than the uncertainty obtained in Sec. 2.3 (Table 3). This may mean that the simulation cannot accurately reproduce the JTS temperature when it is much higher than its nominal operating temperature ($\sim 4.5\mathrm{K}$). Possible cause is inaccuracy of the JTS heat capacity in the thermal mathematical model.

4.3.

He Lifetime Estimation

As described in Sec. 2.2, the He mass flow rate via the PP is calculated as a function of the He tank temperature. Therefore, the model can predict the He mass flow rate and, by integrating, a running total of the vented mass. The results are shown in Fig. 4. The He mass flow rate increased at first, then started decreasing at $\sim 1\text{day}$, and reached equilibrium at $\sim 34\mu \mathrm{g}/\mathrm{s}$ in a few tens days after the launch. The trend is similar to the curve of the simulated He tank temperature as shown in Fig. 3. The amount of He vented in the first few days was consistent with that needed to cool the liquid, and the loss rate eventually stabilized $\sim 20\text{days}$ after the launch. Figure 4(a) also shows the He mass flow rates estimated from the actual temperatures of the PP and the He tank in the orbit, using the flight-model PP mass flow rate model and two-thirds of that, in order to take the hydrostatic head effect into account.⁵ Our prediction is close to the latter in the first $\sim 2\text{days}$, but then, coincides with the former very well. Refer to Ezoe et al.⁵ for detailed descriptions about the PP mass flow rate model and comparisons with our simulations.

Fig. 4

(a) Black shows the simulated variation of the He mass flow rate. Red and green show those measured by utilizing the temperatures of the PP and the He tank in orbit, with assumptions of the flight-model PP mass flow rate and two-thirds of that, respectively.⁵ (b) The consumed He amount calculated by integrating the simulated He mass flow rate in panel (a). The initial amount of the LHe in the He tank was $\sim 5162\mathrm{g}$.

The initial LHe amount was $\sim 35.6\mathrm{L}$, which corresponded to $\sim 5162\mathrm{g}$. By extrapolating the curve of the consumed LHe in Fig. 4(b) to $\sim 5162\mathrm{g}$, the lifetime of the LHe with the beginning-of-life cryocooler performance was estimated at $\sim 4.7\text{years}$, which is $\sim 0.2\text{years}$ longer than that predicted by Yoshida et al.⁷ before the launch. This is because the initial LHe amount was $\sim 6\%$ larger than they assumed (33.6 L). Note that cryocooler performance is expected to gradually degrade, making the IVCS and OVCS temperatures increase. The higher IVCS temperature provides the higher positive heat load from the DA to the He tank, making the He mass flow rate increase. This makes the LHe lifetime shorter, and hence, we also carried out the calculation with the end-of-life (EOL) performance of the cryocoolers, to obtain a lower lifetime limit. We replaced the negative heat loads at the first and second stages of the SCs in Table 2 by the EOL performances, $-28T_{\mathrm{SC}\text{first}}+2885\mathrm{mW}$ ($T_{\mathrm{SC}\text{first}}>103\mathrm{K}$) and $-17T_{\mathrm{SC}\text{second}}+260\mathrm{mW}$ ($T_{\mathrm{SC}\text{second}}>15\mathrm{K}$), respectively, and simulated the temperature distribution in equilibrium. In this case, the He mass flow rate was $\sim 42\mu \mathrm{g}/\mathrm{s}$, giving a helium lifetime of $\sim 3.9\text{years}$ in the EOL case, which still satisfies the requirement of over 3 years. As a conclusion, the lifetime of the He vaporization cooling operation would be at least $\sim 3.9$ to 4.7 years, and the SXS cooling system was working well as designed in the orbit.