DOC // NL-TN-2026-005VECTOR -Z1P / 1U · MATERIALS · ORBITAL

STATUS: PUBLISHED // REV 0.1

Thermal response of a PocketQube: why a lighter spacecraft cycles harder

Time constant, single-node orbital temperature, surface finish and what 5500 eclipses a year do to solder joints

Author: nadir lab.
Published: 29 September 2026
Reading time: 11 min
Level: Academic
Tags: Thermal
Simulation
Solder fatigue
Battery
PocketQube
CubeSat

Abstract

We quantify, with two lumped models of increasing fidelity, why PocketQubes follow the orbital forcing more closely than CubeSats. Linearized radiation gives a thermal time constant of 56 minutes for a fully loaded 1P against 104 minutes for a 2 kg 1U. Integrating a single-node energy balance at 550 km gives bulk swings of 30 K for a 250 g 1P and 37 K at 200 g, against 23 K and 16 K for 1U CubeSats of 1.33 and 2.00 kg. The surface finish moves the mean temperature by tens of kelvin, while the cells that cover most faces leave little freedom. An Engelmaier screening turns the swings into median lives of about one year for germanium-based cells soldered rigidly to FR-4. We close with design rules for the bulk node, the battery and the joints.

Contents

Small bodies, short memories#

Heat capacity grows with mass, roughly with the cube of size, while a spacecraft exchanges heat with its surroundings through its surface, which grows with the square. Shrinking a spacecraft therefore reduces its heat capacity per unit of radiating area, the quantity that sets how far it cools in eclipse and how far it warms in sunlight. Under geometric similarity a 1P has half the areal heat capacity of a 1U; at the mass limits of the two standards, 75%[1]. Bouwmeester et al. singled out the high ratio of outer surface to internal heat capacity as one of the constraints of PocketQubes[2]. This note puts numbers on it.

A first estimate: the time constant#

If radiative exchange is linearized around a reference temperature $T_{0}$ , a lumped spacecraft relaxes towards equilibrium with a single time constant[3]:

τ_{th} = \frac{m c _{p}}{4 ε σ A _{tot} T _{0}^{3}}

with specific heat $c_{p}$ , emissivity $ε$ , Stefan–Boltzmann constant $σ$ and total external area $A_{tot}$ .

Lumped thermal time constant of fully loaded spacecraft (review, Eq. 2)

Form factor	Mass (kg)	Surface (cm²)	Time constant (min)	Longest eclipse ÷ time constant
1P	0.25	150	56	0.63
2P	0.50	278	61	0.59
3P	0.75	406	63	0.57
1U	2.00	654	104	0.34
3U	6.00	1,562	130	0.27

c_p = 900 J/(kg·K), ε = 0.8, T₀ = 290 K. The longest eclipse at 525 km lasts 35.7 min; the larger the ratio, the deeper the spacecraft cools in it.

A fully loaded 1P has a time constant of 56 minutes, a 2 kg 1U 104 minutes. The longest eclipse of a 525 km Sun-synchronous orbit therefore lasts 0.63 time constants for the PocketQube and only 0.34 for the CubeSat.11In the linear model a body covers the fraction $1 - e^{- t / τ}$ of the way to its new equilibrium in time $t$ : about 47% for the 1P during the eclipse and 29% for the 1U. The 2P and 3P stay close to the 1P, because mass and area grow together along the long axis.

A single-node orbital model#

The time constant describes the response, not the forcing. To see the temperature itself, the materials review integrates the energy balance of a tumbling, isothermal body of heat capacity $C$ and external area $A$ [1]:

C \frac{d T}{d t} = ν (t) [α S \frac{A}{4} + α S a F A max (cos φ, 0)] + ε q_{IR} F A + Q_{int} - ε σ A T^{4}

The terms are, in order, direct sunlight on the mean projected area $A /4$ of a tumbling convex body, sunlight reflected by the Earth, the Earth’s own infrared emission, internal dissipation and emission to space. $ν (t)$ is 1 in sunlight and 0 in eclipse, $α$ and $ε$ the absorptance and emittance, $a$ the albedo and $φ$ the angle from the sub-solar point. The orientation-averaged view factor to the Earth is

F = \frac{1}{2} 1 - 1 - (\frac{R _{E}}{R _{E} + h})^{2}

which equals 0.305 at 550 km, the value used in the thermal analysis of SMOG-1[4].

The inputs follow the review: a 550 km orbit with $β = 0°$ ( 37.2%, period 95.5 minutes), $α = ε = 0.85$ and $c_{p} = 858$ J/(kg·K) from the SMOG-1 model, $S = 1361$ W/m²[5], $a = 0.30$ and $q_{IR} = 237$ W/m²[6], and internal dissipation proportional to area, 0.25 W for a 1P and 1.0 W for a 1U. The equation is integrated with fourth-order Runge–Kutta in 2 s steps over fourteen orbits, until the temperature repeats from one orbit to the next.

Results#

1P, 250 g
1U, 1.33 kg
1U, 2.00 kg

Show the data as a table

Time (min)	1P, 250 g	1U, 1.33 kg	1U, 2.00 kg
0	—	—	—
20	18.6	15.7	12.6
40	9.0	9.1	8.6
60	−7.2	−3.6	−0.1
80	−2.4	−0.7	1.3
100	11.9	10.1	8.5
120	19.1	16.3	13.1
140	5.0	6.0	6.6
160	−10.0	−5.9	−1.8
180	0.7	1.5	2.7

Bulk temperature over two orbits at 550 km for a 1P PocketQube at its mass limit and 1U CubeSats of 1.33 and 2.00 kg; shaded intervals are eclipses (materials review, Fig. 6a).

Single-node temperature swings at 550 km (materials review, Eq. 12)

Case	Minimum (°C)	Maximum (°C)	Swing (K)	Fastest cooling (K/min)
1P, 200 g	−15.0	21.8	36.8	1.50
1P, 250 g	−10.8	19.5	30.3	1.16
1U, 1.33 kg	−6.6	16.8	23.4	0.83
1U, 2.00 kg	−2.2	13.6	15.8	0.52

α = ε = 0.85, c_p = 858 J/(kg·K), albedo 0.30, Earth infrared 237 W/m², internal dissipation 0.25 W (1P) and 1.0 W (1U), β = 0°.

The 250 g 1P cycles between −10.8 °C and 19.5 °C, a swing of 30.3 K, and cools at up to about 1.2 K per minute right after entering eclipse. At the 200 g of SMOG-1 the swing grows to 36.8 K. The 1U CubeSats swing by 23.4 K at 1.33 kg and 15.8 K at 2.00 kg. Three features deserve attention:

The orbit-average temperature is low. The bulk of the 1P averages about 6 °C and spends about a third of each orbit below freezing. That is a consequence of the tumbling assumption, which exposes on average only a quarter of the surface to the Sun.
The swing is not symmetric. The 1P warms at up to about 0.75 K per minute in sunlight but cools at up to 1.2 K per minute at the start of eclipse, when it is warmest and radiates most while the absorbed power drops to the Earth’s infrared and its own dissipation.
Lighter means harder. Within one form factor, every gram removed makes the cycle larger, so mass savings have a thermal price.

These are bulk values, a lower bound for anything that is poorly coupled to the core. External panels and antennas swing much more: Delfi-PQ recorded swings of up to 75 °C[7], and the finite-element model of SMOG-1 predicted extremes of about 15 °C and −26 °C, with the battery below 0 °C in eclipse in the conservative thermal-network model[4]. Delfi-PQ was accordingly qualified by thermal-vacuum cycling between −20 °C and +50 °C[8].

The surface finish sets the mean#

The ratio $α / ε$ fixes the temperature around which the spacecraft cycles; the emittance and the heat capacity fix how fast it responds. Applying one finish to the whole 1P shows the range available:

Bulk temperature of a 250 g 1P with one surface finish all over, 550 km

Surface	α	ε	α/ε	Min (°C)	Mean (°C)	Max (°C)	Swing (K)
White silicate paint	0.15	0.91	0.16	−56	−53	−51	6
Clear (sulfuric) anodize	0.35	0.80	0.44	−35	−28	−22	13
Model surface of the review (α = ε)	0.85	0.85	1.00	−11	6	19	30
Black anodize	0.88	0.88	1.00	−12	5	20	31
Triple-junction cells with cover glass	0.91	0.85	1.07	−9	9	24	32
Black polyurethane paint	0.95	0.90	1.06	−10	8	23	34

Same single-node model with representative beginning-of-life α and ε (Gilmore 2002); real spacecraft mix cells, anodized metal and boards. The ratio α/ε sets the mean temperature, the emittance how fast the body responds.

A white-painted 1P would sit near −53 °C, far too cold for its battery, and a clear-anodized one near −28 °C, whereas black anodize, black paint and solar cells all keep the mean between about 5 and 9 °C. Real PocketQubes have little freedom here: cells cover most faces, and their absorptance is fixed by the need to generate power. The remaining areas, such as the frame edges, the antenna face and the , are where a finish can trim the mean by a few kelvin, and bare aluminium there pulls the spacecraft colder.

From swing to solder fatigue#

At 15.1 orbits per day a spacecraft goes through about 5500 eclipse cycles a year. Each cycle strains the solder joints between components and boards whose expansion coefficients differ. Engelmaier’s model for rigid leadless surface-mount joints, adopted by IPC for reliability screening, estimates the median number of cycles to failure[9][10]:

N_{f} = \frac{1}{2} (\frac{Δ γ}{2 ε _{f}^{'}})^{1/ c}, Δ γ = \frac{F _{E} L _{D} Δ α Δ T}{h}, c = - 0.442 - 6 \times 1 0^{- 4} T_{sj} + 1.74 \times 1 0^{- 2} ln (1 + f)

with the distance from the neutral point $L_{D}$ , the joint height $h$ , the expansion mismatch $Δ α$ , the swing $Δ T$ , the fatigue ductility coefficient $ε_{f}^{'} = 0.325$ of eutectic Sn–Pb, the mean joint temperature $T_{sj}$ in °C and $f$ cycles per day.

Median cycles to failure of a rigid leadless solder joint (Engelmaier model)

Thermal case	Swing (K)	Δα = 11 ppm/K	Δα = 6 ppm/K	Δα = 2 ppm/K
1U, 1.33 kg	23.4	9.9 × 10³1.8 years	4.6 × 10⁴8.3 years	7.3 × 10⁵132 years
1P, 250 g	30.3	5.1 × 10³0.9 years	2.4 × 10⁴4.3 years	3.8 × 10⁵68 years
1P, 200 g	36.8	3.1 × 10³0.6 years	1.5 × 10⁴2.6 years	2.3 × 10⁵42 years

Distance from the neutral point 5 mm, joint height 0.1 mm, eutectic Sn–Pb, mean joint temperature 5 °C, 15.1 cycles per day (about 5,515 a year). 11 ppm/K corresponds to germanium-based cells on FR-4.

For the 30.3 K swing of a 1P, a germanium-based cell soldered rigidly to FR-4 ( $Δ α \approx 11$ ppm/K) reaches a median life of about 5 × 10³ cycles, less than a year in orbit. The same joint on a 1.33 kg 1U lasts about 1.8 years, and a substrate with a mismatch of 2 ppm/K moves the 1P to decades. These are median lives of idealized joints: early failures occur well before $N_{f}$ , external panels see larger swings than the bulk node, and creep and dwell are treated only implicitly. The screening still shows that, at PocketQube scale, joint life and mission duration are of the same order[1].

Mitigations follow directly from the equation: -tailored substrates for cell assemblies[11], compliant silicone attachment instead of large rigid solder areas, smaller distances from the neutral point (smaller cells or split pads), and qualification by thermal cycling over the predicted range with margin. Commercial parts with pure-tin finishes add tin-whisker risk[12], which is why high-reliability space soldering standards are built around Sn–Pb alloys and re-tinned terminations[13].

A cold interior#

The bulk temperatures above have consequences beyond the joints[1]:

Battery. Charging lithium-ion cells near or below 0 °C plates lithium on the anode. SMOG-1 restricted charging to about half of the sunlit period for this reason[4]; a heater and a temperature threshold in the charge controller are the usual answers.
Magnets and coils. The remanence of NdFeB drifts by about 12% over 100 K, against 2–3.5% for AlNiCo 5 and SmCo, and the resistance of copper traces rises by 0.39% per kelvin, so the dipole delivered at fixed voltage changes by more than 10% over an orbit.
Propellants. Water-based micropropulsion freezes unless the tank is isolated or heated.

Calculation flow#

flowchart LR
  A["Mass, area, finish α and ε"] --> B["Time constant τ = m c_p / (4 ε σ A T₀³)"]
  A --> C["Single-node balance, RK4 to a periodic state"]
  D["Orbit: altitude, β, eclipse"] --> C
  C --> E["Bulk swing ΔT, cooling rate"]
  E --> F["Engelmaier: cycles to failure"]
  G["Joint: L_D, h, Δα"] --> F
  F --> H["Compare with 5500 cycles per year"]

Design rules#

Model the bulk node first, then the periphery. The single-node swing is a lower bound; panels, antennas and anything mounted on standoffs need a thermal network or a finite-element model.
Treat mass as a thermal parameter. Every gram removed from a 1P enlarges its cycle; ballast in the right place is not always wasted mass.
Use dissipation as a resource. A heater or a transmitter burst in eclipse reduces the swing at the cost of energy that the power budget must carry.
Choose finishes for the faces you control. Cells fix most of the surface; black anodize on the frame keeps the mean close to that of the cells, bare or clear-anodized aluminium pulls it down.
Design joints for the swing you computed. Compliant attachment or CTE-tailored substrates for large rigid parts, and thermal-cycling qualification over the predicted range with the margins that test standards such as -E-ST-10-03C define[14].

The structures and thermal entry and the materials entry place these numbers in the wider design, and the materials review gives the full derivation with its parameters.