Chapter 25 — Key Takeaways (Power Systems)

A one-page reference. Reread this before an exam, or before you size any spacecraft's power.

The equations that size a power system

Quantity Equation Use it to find
Solar flux at distance $S(d) = \dfrac{1361}{d^2}\ \text{W/m}^2$ sunlight available at $d$ AU from the Sun
Array power (BOL) $P = S\,\eta\,A\cos\theta$ electricity from area $A$ at sun angle $\theta$
Required array output $P_{sa} = \dfrac{P_d T_d/X_d + P_e T_e/X_e}{T_d}$ array output to run loads and recharge for eclipse
Array area (EOL-sized) $A = \dfrac{P_{sa}}{S\,\eta\cos\theta\,L_d}$ panel area that still meets load after degradation
Eclipse fraction $f_e = \dfrac{1}{\pi}\arcsin\!\left(\dfrac{R}{r}\right)$ fraction of a circular orbit spent in shadow
Degradation factor $L_d = (1-D)^{y}$ surviving array fraction after $y$ years at loss $D$/yr
Battery capacity $C = \dfrac{P_e T_e}{\text{DoD}\times\eta}$ installed W·h to deliver one eclipse
RTG decay (thermal) $P(t) = P_0\left(\tfrac12\right)^{t/t_{1/2}}$ heat remaining after $t$ years ($t_{1/2}=87.7$ yr)
Harness current $I = P/V$ why higher bus voltage → lighter harness ($I^2R$)

What every symbol means (and its units)

Symbol Name Units Notes
$S$ solar flux / solar constant W/m² 1361 at 1 AU; owned by Ch. 24
$\eta$ cell efficiency — space multi-junction ≈ 0.30
$\theta$ sun angle from array normal ° cosine loss $\cos\theta$; 0 is best
$X_d, X_e$ daylight / eclipse path efficiency — ≈ 0.85 / 0.65 (eclipse pays battery losses)
$L_d$ life degradation factor — $(1-D)^y$, $\le 1$
DoD depth of discharge — fraction of battery used per cycle
$C$ battery capacity W·h install ≫ delivered energy
$t_{1/2}$ half-life of Pu-238 87.7 yr governs RTG fade
$V$ bus voltage V 28 legacy → 100–160 high power

The inverse-square law decides the source

Destination AU Flux (W/m²) Practical source
Earth / LEO / GEO 1.0 1361 solar + battery
Mars 1.52 586 solar (bigger, dust-derated) or RTG
Jupiter 5.2 50 solar (huge, e.g. Juno) or RTG
Saturn and beyond ≥ 9.5 ≤ 15 RTG / fission only

Sizing method (do it in this order)

  1. Power budget first (§25.6): loads by mode + harness (+5%) + margin (+25%) → budgeted load, in the worst mode (often eclipse, because of heaters).
  2. Required array output $P_{sa}$ — about 2× the average load (eclipse duty + storage losses).
  3. Array area at end of life — apply $\cos\theta$ and $L_d$; the array grows ~20% for degradation alone.
  4. Battery — from the worst eclipse energy $P_e T_e$ and a DoD set by cycle count.
  5. Bus voltage — raise it to keep the harness light.

Battery: DoD is set by how often you cycle

Orbit Eclipses/yr Cycles (life) Usable DoD Battery for same eclipse energy
LEO ~5,500 ~40,000 ~25% (shallow) heavy
GEO ~90 ~1,350 ~70% (deep) ~⅓ as heavy

Rule: more cycles → shallower DoD → bigger, heavier battery. Li-ion ≈ 150 W·h/kg.

Power sources compared (specific power, W/kg)

Source Specific power Power range Best where
Solar + battery (1 AU) ~10–50 W–100s kW near the Sun, near planets
RTG (Pu-238 + thermocouples, ~6–7%) ~3–5 100–300 W deep space, shadow, decades
Fission (U-235 + Stirling, ~30%) ~2–6, rising with size kW–MW high power in the dark; EP; bases

Common pitfalls

Pitfall Reality
Sizing the array to beginning-of-life, square-on. Size to end of life, worst sun angle, worst mode — or it browns out later.
Battery mass = eclipse energy ÷ energy density. Divide also by DoD × efficiency; a LEO battery is ~4× its deliverable energy.
"Just fly a bigger RTG for kilowatts." Pu-238 is bred at ~1.5 kg/yr; kW-electric needs ~30 kg. Use fission.
"Solar works everywhere." Flux $\propto 1/d^2$: ~1/27 at Jupiter, hopeless past Saturn.
A battery is a power source. It only stores; with no array charging it in the dark, it just runs down. Deep dark needs an RTG/reactor.
Keeping a low bus voltage as power grows. High current → $I^2R$ loss and a copper-brick harness. Raise the voltage.

Numbers worth memorizing

  • Solar constant 1361 W/m² at 1 AU; array ~300 W/m² at the panel after losses.
  • Space cells ~30%; Li-ion ~150 W·h/kg; Pu-238 0.54 W/g, half-life 88 yr, RTG ~6–7%.
  • LEO eclipse ~35 min of a ~95 min orbit (~38%); GEO eclipse only near equinoxes, ≤ ~72 min.
  • Required array output ≈ 2× average load. Lunar/Mars night breaks store-through-the-dark.

Mission / astrotools additions this chapter

  • MDR: built your mission's power budget (loads by mode → source selection → array/battery sizing → bus voltage). Major increment.
  • power.py: solar_flux(au), required_array_power(...), array_area(...), battery_capacity_wh(...).