Example Mission Scenario: Jovian Gravity Assist with Kinetic Energy Capture

1. Mission Setup (Illustrative Parameters)

Assume:

• Spacecraft mass (wet): $$ M \approx 2\,000 \,\text{kg} $$
• Baseline heliocentric cruise speed before Jupiter fly‑by: $$ v_{1} \approx 10\,\text{km/s} $$
• Target post‑assist cruise speed: $$ v_{2} \approx 15\,\text{km/s} $$ (net gain $$\Delta v_{\text{GA}} \approx 5\,\text{km/s}$$)
• Main kinetic rotor system: three rotors, each with:
• radius $$ r \approx 1\,\text{m} $$
• mass $$ m \approx 500\,\text{kg} $$
• maximum operating speed $$ \omega_{\max} \approx 300\,\text{rad/s} $$ (≈ 3 000 rpm; illustrative).

The moment of inertia of each rotor, approximated as a thick ring:

$$
I \approx m r^{2} \approx 500 \,\text{kg·m}^{2}
$$

Stored rotational energy per rotor:

$$
E_{\text{rotor}} = \tfrac{1}{2} I \omega^{2}
\approx 0.5 \times 500 \times (300)^{2}
\approx 22.5 \,\text{MJ}
$$

For three rotors:

$$
E_{\text{rot,total}} \approx 3 \times 22.5 \,\text{MJ} \approx 67.5 \,\text{MJ} \ (\approx 18.7 \,\text{kWh})
$$

2. Gravity Assist Energy Scale vs Rotor Scale

The orbital energy change associated with the Jovian gravity assist (very roughly) is:

$$
\Delta \epsilon \approx \tfrac{1}{2} (v_{2}^{2} – v_{1}^{2})
= \tfrac{1}{2} (15^{2} – 10^{2}) \,\text{km}^{2}\text{/s}^{2}
= \tfrac{1}{2} (225 – 100)
\approx 62.5 \,\text{km}^{2}\text{/s}^{2}
$$

In Joules, multiplying by spacecraft mass:

$$
\Delta E_{\text{orb}} \approx M \times \Delta \epsilon
\approx 2\,000 \,\text{kg} \times 62.5 \times 10^{6} \,\text{J/kg}
\approx 1.25 \times 10^{11} \,\text{J}
$$

So:

• The orbital energy gain from the gravity assist is of order $$10^{11} \,\text{J}$$.
• The fully charged rotor system stores of order $$7 \times 10^{7} \,\text{J}$$.

This illustrates that:

• The energy scale of the gravity assist is thousands of times larger than the rotor storage capacity.
• Consequently, diverting even a very small fraction of the dynamical “budget” into internal kinetic storage is, in principle, sufficient to fully charge the rotors multiple times.

3. Kinetic Capture During the Fly‑by

3.1 Pre‑Fly‑by

Before the close approach:

• Rotors are spun up to a moderate baseline speed, say $$ \omega_{0} \approx 150 \,\text{rad/s} $$.

Stored energy per rotor:

$$
E_{0} = \tfrac{1}{2} I \omega_{0}^{2}
\approx 0.5 \times 500 \times (150)^{2}
\approx 5.6 \,\text{MJ}
$$

For three rotors:

$$
E_{0,\text{total}} \approx 16.8 \,\text{MJ}
$$

3.2 During Gravity Assist

The fly‑by includes a critical high‑dynamics phase lasting on the order of a few hours. For illustration, take an effective “high‑load interaction window” of:

• $$ t_{\text{GA}} \approx 2 \,\text{hours} $$ (7 200 s).

During this window:

• Attitude corrections and structural loads are at a maximum.
• The control system uses the rotors both for attitude control and as controllable sinks for mechanical work, accelerating them (“motor mode”) when advantageous.

Assume that, averaged over this window, a mechanical/electrical power of order:

• $$ P_{\text{into,rotors}} \approx 20 \,\text{kW} $$

can be directed into the rotor system (via controlled internal actuation and electromechanical coupling). Then:

$$
E_{\text{into,rotors}} \approx P \times t
\approx 20\,000 \times 7\,200
\approx 1.44 \times 10^{8} \,\text{J}
$$

Since the rotor system’s maximum capacity is $$\sim 6.75 \times 10^{7} \,\text{J}$$, in practice:

• The rotors cannot absorb the full 1.44×10⁸ J;
• Instead, the control algorithm ramps rotor speed from $$\omega_{0} \approx 150 \,\text{rad/s}$$ towards $$\omega \approx 300 \,\text{rad/s}$$, approaching but not exceeding safe limits and effectively adding on the order of $$ \sim 50 \,\text{MJ} $$ of stored energy.

3.3 Post‑Fly‑by

After the gravity assist:

• Rotors are near their maximum operational speed, $$ \omega \approx 300 \,\text{rad/s} $$.
• Total stored kinetic energy is $$ E_{\text{rot,total}} \approx 67.5 \,\text{MJ} $$.

This energy is in addition to the translational orbital energy gained from the gravity assist itself and is entirely available for later conversion to electrical power.

4. Post‑Assist Electrical Recovery

Suppose we choose to use 70% of the stored rotor energy (leaving a safety margin):

$$
E_{\text{usable}} \approx 0.7 \times 67.5 \,\text{MJ} \approx 47 \,\text{MJ}
$$

With an overall mechanical‑to‑electrical conversion and storage efficiency of:

• $$ \eta_{\text{mech-elec}} \approx 0.8 $$,

the net electrical energy delivered to the super‑capacitor banks is:

$$
E_{\text{elec}} \approx \eta_{\text{mech-elec}} \times E_{\text{usable}}
\approx 0.8 \times 47 \,\text{MJ}
\approx 37.6 \,\text{MJ} \ (\approx 10.4 \,\text{kWh})
$$

This 10.4 kWh can, for example:

• operate a 5 kW ion thruster continuously for roughly 2 hours,
• power life‑support and avionics at 1 kW for about 10 hours,
• or sustain low‑power modes (e.g. 200 W) for several days.

5. Approximate Δv Contribution

Using the earlier ion thruster example:

• Electrical input power: $$ P_{\text{in}} = 5\,\text{kW} $$
• Thrust: $$ T \approx 0.2 \,\text{N} $$
• Spacecraft mass: $$ M \approx 2\,000 \,\text{kg} $$

Burn duration with $$ E_{\text{elec}} \approx 37.6 \,\text{MJ} $$:

$$
t_{\text{burn}} \approx \frac{E_{\text{elec}}}{P_{\text{in}}}
\approx \frac{37.6 \times 10^{6}}{5\,000}
\approx 7\,520 \,\text{s} \ (\approx 2.1 \,\text{hours})
$$

Resulting acceleration:

$$
a = \frac{T}{M} \approx \frac{0.2}{2\,000} = 1 \times 10^{-4} \,\text{m/s}^{2}
$$

Approximate Δv:

$$
\Delta v \approx a t
\approx 1 \times 10^{-4} \times 7\,520
\approx 0.75 \,\text{m/s}
$$

In absolute terms, this Δv is small compared to the $$\sim 5\,\text{km/s}$$ gained from the gravity assist itself. However:

• it is achieved without consuming chemical propellant,
• it can be repeated after multiple assists and long cruise phases,
• and it is valuable for fine trajectory trimming, formation flying and precision rendezvous in deep space.

6. Illustrative Disclaimer

Illustrative Note
All numerical values in this scenario are order‑of‑magnitude estimates intended solely to illustrate scaling relationships between gravity‑assist energy, internal kinetic storage and electric propulsion. They do not represent a validated design and would require detailed structural, thermal and mission analysis for any practical implementation.

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