Where do ion propulsion's ions go? Do they remain in the solar system or shoot out into interstellar space?

Writing this answer has got me thinking.

You can estimate the exhaust velocity of an ion engine using

$$\frac{v}{c} = \sqrt{\frac{2E}{m_0 c^2}}.$$

Choose $$E=$$ 100 keV and $$m_0 c^2=$$ 931 MeV times 50 to 200 AMU and you get between 0.2 and 0.1% of the speed of light, which at 600 to 300 km/sec is way beyond escape velocity of the Earth or the Sun.

If this happens in LEO, or on a deep space mission, where would these ions end up? Are they still trapped by the Earths' magnetic field, or the interplanetary magnetic field, or would they just go shooting straight out of the solar system and into interplanetary space? Would they instead thermalize somewhere in the solar system through collisions?

• Protons trapped in the Van Allen belts have more energy (exceeding 100 MeV) and much less mass, they are much faster than 0.2 % c. The magnetic field of Earth is therefore strong enough to trap those slow propulsion's ions. – Uwe Feb 25 '19 at 9:07
• @Uwe oh rats, I did a quick check using 100 AMU, 0.1 gauss (1E-05 T), 100 keV and got a radius of 46,957 but that was meters, and I read it as kilometers. Feel free to post the answer and correct me, at least for LEO, deep space would have to be treated separately. – uhoh Feb 25 '19 at 9:24

There are protons with energies exceeding 100 MeV in the inner Van Allen belt, source.

Choose $$E=$$ 100 MeV and $$m_0 c^2=$$ 931 MeV times 1 AMU and we get about 33 % of the speed of light.

If the Earth's magnetic field is strong enough to trap protons at 33 % c, it should be able to trap heavy and slow low energy propulsion's ions too.

The energetic (above 10 MeV) trapped proton population is confined to altitudes below 20,000 km, while lower energy protons cover a wider region, with protons below 1 MeV reaching geosynchronous altitudes. Figure 2 shows the distribution of trapped protons with energies above 10 MeV, as predicted by the NASA AP-8 MAX model [Sawyer and Vette, 1976], in invariant coordinate space. The region of space covered by higher energy protons diminishes with increasing energies and the location of the highest intensities moves inward.

AP-8 [Sawyer and Vette, 1976]

Figure 2. Invariant coordinate map of the AP-8 MAX integral proton flux >10 MeV. The semi-circle represents the surface of the Earth, distances are expressed in Earth radii.

• But if the trapped proton's energies exceed only 10 MeV like here, speed is 10 % of c. For 1 MeV 3.3 %. – Uwe Feb 25 '19 at 11:14