What's the theoretical dV of the SNAP-10A satellite?

Question

SNAP-10A, launched in 1965, was the first nuclear reactor tested in orbit. It was intended to produce 500 watts of electricity for a year, but it only lasted 43 days; a voltage regulator in the satellite's control systems failed and took the reactor with it.

Interestingly, it was also the first test of electric propulsion in orbit; the satellite included an 8.5mN cesium ion thruster which ran off batteries charged by the nuclear reactor, in an hour-on, fifteen-hour-off cycle. This, too, died early, not even completing its inaugural burn before the EMI it produced forced the operators to command it off.

It strikes me that this combination of reactor and electric thruster would make SNAP-10A the first spacecraft to use nuclear propulsion... assuming everything worked as intended for the full year, then, what was SNAP-10A's theoretical maximum delta-V? Hell, spacecraft usually outlast their design lifespans; what would the satellite's delta-V be if it ran until it depleted its uranium fuel?†

_{^{† Presumably the delta-V would be limited by available thruster fuel long before it was limited by available nuclear fuel, but I couldn't find a source for how much was onboard, so I'll ask you to fill in whatever quantity you think would make the answer plausible/interesting.}}

Answer 1

TL;DR: I'd estimate that SNAP10A had about 3200 m/s of $\Delta{v}$, limited by the amount of ion-engine fuel aboard, which would have run out in about a month.

Let's do some calculations:

Specific Impulse of the engine

We have the information:

The ion-beam power supply was operated at 4500 V and 80 mA to produce a thrust of about 8.5 mN

From here, we can roughly calculate the power of the ion beam at $P = VI$ so $360 \text{ W}$. Then, if we use the formula for kinetic energy in the exhaust velocity:

$$P=\frac{1}{2}\dot{m}v_e^2$$

Where $v_e$ is the exhaust velocity and $\dot{m}$ is the change in mass along with the formula for thrust:

$$T=\dot{m}v_e$$

we can rearrange these formulas into:

$$\dot{m}=\frac{T}{\sqrt{2P}}$$

Plugging in our values, we get $\dot{m} \approx 3.17 \times10^{-4} \text{ kg/s}$ for the engine, and then, using the Specific Impulse formula:

$$I_{sp}=\frac{T}{\dot{m}\times{}g_0}$$

We get a resulting $I_{sp} = 2732 \text{ s}$ which is reasonably in line with what we expect from electric propulsion, especially considering this took place in the 1960s.

Estimating fuel content for ion thruster

We have this information:

Launch mass: 440 kg

The SNAP-10A reactor was designed for a thermal power output of 30 kW and unshielded weighs 290 kg

This means that we have about 150 kg of unaccounted mass, which is used for the radiators, thermal-electric converter, all the spacecraft systems, and the ion thruster fuel. I couldn't find anything about the mass breakdown, so for simplicity's sake, let's just allocate 50 kg to radiators and electric generator, 50kg for spacecraft systems and structure, and a final 50kg for fuel.

I think this is a rather optimistic number, because from what I've read, the "vibe" of SNAP10A was definitely more about the reactor and the ion thruster was more of a tech demo or experimental payload. I wouldn't be surprised if it actually only had single-digit kilos of fuel for the ion thruster aboard.

Mission as planned

Now that we have calculated an $I_{sp}$ and have an estimated fuel mass, we can use the rocket equation:

$$\Delta{v} = I_{sp} \times{} g_0 \times \ln{\frac{m_0}{m_f}}$$

If we plug in all our values, we get $\Delta{v} = 3227 \text{ m/s}$

How long would this take though?

If we assume that the thruster consumes $3.17 \times10^{-4} \text{ kg/s}$ of fuel while active and we have $50 \text{ kg}$ of fuel, we can easily calculate that we have enough fuel for about 44 hours of continuous thruster firing.

So, taking the "one hour on, fifteen hours off" usage pattern, we would need about 700 hours to empty the fuel tank, or about 30 days of operation. Since the reactor worked for 43 days before a component failure, and was designed to work much longer, the limiting factor in this case was definitely the amount of fuel.