Orbital decay and lifetime of PocketQubes and CubeSats

Why it matters#

For a small satellite in low Earth orbit, lifetime is set by the atmosphere more than by the design. Without propulsion it is both a constraint, because the mission must finish before the spacecraft comes down, and an obligation, because the spacecraft must leave orbit within a reasonable time after the mission ends. This note derives the drag-driven decay of circular orbits with the simplest consistent model, the one used in the PocketQube review[1], compares form factors and checks the result against observed re-entries.

Drag and the ballistic parameter#

The drag deceleration of a spacecraft grows with density and with the square of its speed relative to the air:

with density $\rho$, drag coefficient $C_D$, cross-section $A$ and mass $m$. The combination $C_{D}A/m$ is the ballistic parameter; its inverse, $m/(C_{D}A)$, is the ballistic coefficient. In the free-molecular flow of LEO, $C_D\approx 2.2$ is the usual assumption[2]. An uncontrolled spacecraft presents a changing cross-section; for a convex body tumbling randomly, the mean projected area is a quarter of the total surface (Cauchy’s theorem).11A 50 mm cube has 150 cm² of surface and a mean projected area of 37.5 cm². Face-on to the flow it would present 25 cm²; along a body diagonal, 43.3 cm².

With these assumptions the ballistic parameter of a fully loaded 1P is 0.033 m²/kg and that of a fully loaded 1U 0.018 m²/kg. PocketQubes have a low mass per unit area[3], and that is the whole reason for the difference in lifetime.

The decay equation#

The specific orbital energy of a circular orbit is $\epsilon =-\mu /2a$. Drag removes energy at the rate $a_{D}v$, so

and with $v=\sqrt{\mu /a}$ the semi-major axis decays as

The rate is linear in density and in the ballistic parameter: halving $C_{D}A/m$ roughly doubles the lifetime in the same atmosphere.

A solar-driven atmosphere#

Static exponential tables, such as Vallado’s[2], ignore the main driver of thermospheric density, the Sun. The model of Kennewell[4] keeps a single scale height but ties it to solar and geomagnetic activity:

At moderate activity the scale height is about 42 km at 300 km and 47 km at 500 km; between low ($F_{10.7}=70$ sfu, $A_p=6$) and high activity (210 sfu, 18) it ranges from about 35 to 55 km, and the density at 500 km varies tenfold.

If the scale height and $\sqrt{\mu a}$ are held constant, the equation integrates analytically: the time to descend by one scale height from $a_0$ is, in the simplest form of King-Hele’s classical analysis[5],

Most of a lifetime is spent within the first few scale heights below the starting altitude, which is why this crude estimate lands within tens of per cent of the numerical result.

Integration#

We integrate the decay equation with a midpoint (second-order Runge–Kutta) scheme, limiting each step to one day and to 100 m of altitude loss, and assume re-entry at 180 km, below which the remaining life is days. These settings reproduce the lifetimes of the review to the precision quoted there[1].

Two results stand out. The ballistic parameters of 1P, 2P and 3P PocketQubes are almost identical (0.030–0.033 m²/kg), because mass and area grow together along the long axis. And CubeSats at their mass limits live much longer than PocketQubes. A lighter 1U, such as a typical 1.33 kg design, sits closer to the PocketQubes; the CubeSat rows are upper bounds.

Lifetime against altitude#

Because lifetime is roughly inversely proportional to the ballistic parameter, a PocketQube lives about half as long as a 1U from the same altitude. Both curves rise steeply with altitude: every 50 km adds a factor of two to three.

The Sun dominates the uncertainty#

The same 1P released at 500 km lives almost six years under low activity and nine months under high activity. This spread is far larger than the uncertainty in mass or drag coefficient. A lifetime forecast must follow the phase of the solar cycle at launch and over the mission, and cycle forecasts themselves move: the 2019 NOAA/NASA panel expected Cycle 25 to peak at a sunspot number of 115 in July 2025[6], while the 2023 update moved the peak to 2024 at 137–173[7]. Empirical models such as NRLMSISE-00[8] driven by such forecasts are the tools for compliance work; comprehensive studies of CubeSat decay reach the same conclusion about solar activity[9].

Checking against flight#

Transporter-3 released two well-documented PocketQubes near 525 km on 13 January 2022. Grizu-263A, a 237 g 1P, re-entered on 3 October 2023 after 628 days; Delfi-PQ, a 545 g 3P, on 6 January 2024 after 723 days[10].

For Grizu-263A, a constant mean flux of 141 sfu with the nominal ballistic parameter gives about 825 days, 31% more than observed; the review’s reconstruction with the observed monthly flux brings this to 778 days, and an effective ballistic parameter of 0.050 m²/kg matches the re-entry exactly. For Delfi-PQ, the nominal tumbling value of 0.041 m²/kg at 141 sfu gives about 790 days against 723 observed. Both residuals have the same sign and plausible sources: a single scale height, the exact release orbit, appendages such as antenna tapes that raise the drag area, and attitude that is not a uniform tumble. The TLE histories that feed such comparisons also degrade for light objects[11].

Calculation flow#

flowchart LR
  A["Form factor: dimensions, mass"] --> B["Mean area A = A_tot / 4"]
  B --> C["Ballistic parameter C_D A / m"]
  D["Kennewell density: F10.7, Ap"] --> F["da/dt = −√(μa) ρ C_D A / m"]
  C --> F
  F --> G["RK2, step ≤ 1 day and ≤ 100 m"]
  G --> H["Re-entry at 180 km: lifetime"]

Conclusion#

A first-order model with a solar-driven density captures what matters for design: PocketQubes live about half as long as CubeSats from the same altitude, and the Sun, not the spacecraft, sets most of the uncertainty. The model overestimates the observed Transporter-3 lifetimes at constant flux, in the direction expected from its simplifications. The lifetime calculator runs the same model for any spacecraft, and the five-year rule article turns it into altitude limits.