# How much mass do the volcanoes of Io erupt past escape velocity?

While trying to do some calculations to answer this quesion, I got stuck missing a crucial piece of data: I have no clue how much mass the volcanoes of Io manage to throw out of the gravity well of the Jupiter moon.

It is clear that most of the erupted material falls back to the surface, and the various velocity estimates I have found are quite a bit lower than the escape velocity (~2.56 km/s). For instance "up to 1 km/s".

One number that is possible to find is how much mass the magnetosphere of Jupiter strips of Io. Most are listing that as 1000kg/s (example). That is about as much as the wheat production of Canada.

Are the volcanoes directly throwing away a lot more than that? A lot less? Not at all?

• Might want to start w/ escape velocity from Io -- some portion of that mass, depending on direction, will orbit alongside Io, some will be ejected "forwards" and thus at the least go to a higher orbit, etc. – Carl Witthoft Oct 4 '16 at 12:25
• My comments on the inspiring question may be worth echoing here: the primary method of mass loss expected for Io is through Jupiter's magnetic field pulling off charged particles from the upper reaches of Io's Hill sphere. This is estimated to occur at the rate of 1 tonne/second. I otherwise echo polyphant's answer: the mass loss expected from ejecta hitting escape velocity is 0. – zibadawa timmy Oct 8 '16 at 20:54
• @zibadawatimmy your comment got me thinking, if the ejecta are released when the volcano is facing Jupiter would there be a greater kick from it's gravity that would noticeably accelerate any erupted mass toward the planet? I doubt it would be much. A fun calculation though. – christopherlovell Oct 19 '16 at 16:37

## 1 Answer

The velocity required to escape the gravitational attraction of a massive body is given by the following equation:

$$v_{\mathrm{escape}} = \sqrt{\frac{2GM}{R}}$$

where $G$ is the gravitational constant ($G = 6.67 \times 10^{-11} \; \mathrm{Nm^{2} {kg}^{-2}}$), $M$ is the mass of the body from which you are escaping, and $R$ is its radius.

Inputting the values for Io's mass and mean radius, $M = 0.015 \, M_{\oplus}$ and $R = 0.286 \, R_{\oplus}$,1 we get an escape velocity of

$$v_{\mathrm{escape}} = 2560 \; \mathrm{m/s}$$

However, the explosive ejecta are ejected from the top of volcanoes, so we should strictly add this to our radius. The tallest volcano on Io is approximately $2.5 \;\mathrm{km}$ above the surface; including this we get a marginally lower velocity

$$v_{\mathrm{escape}} = 2559 \; \mathrm{m/s}$$

which is higher than the $\sim 1000 \; \mathrm{m/s}$ maximum velocity of ejecta calculated in McEwen & Soderblom (1983). Therefore, no mass is ejected from the surface of Io through volcanic eruptions.

For comparison, the earth's escape velocity is much higher, $11.2 \; \mathrm{km/s}$. As discussed in the linked paper, the most extreme ejecta can reach heights of $500 \; \mathrm{km}$ before falling back to the surface.

1where $M_{\oplus} = 5.972 \times 10^{24} \; \mathrm{kg}$ and $R_{\oplus} = 6371 \;\mathrm{km}$

• Yes, since none of the mass reaches escape velocity. Answer updated. – christopherlovell Oct 19 '16 at 14:27
• There is more to it than that, Io ejects a lot of material and contributes significantly to Jupiter's magnetosphere, "Io is a strong source of plasma in its own right, and loads Jupiter's magnetosphere with as much as 1,000 kg of new material every second." – Cody Oct 19 '16 at 17:26
• Check the link, the charged particles come from the volcanoes. It may not be the force of the volcano directly, but through outside help the volcanoes are ejecting a lot of mass from Io – Cody Oct 19 '16 at 18:43
• @Cody I stand corrected. That's pretty neat. It certainly complicates the analysis I present above. Now you need some ionisation factor, and an atmospheric escape fraction. – christopherlovell Oct 20 '16 at 15:04
• Io doesn't really have an atmosphere, so any ejecta are almost immediately exposed to a plasma and the ionizing radiation from the sun. Thus, some fraction of the ejecta is ionized when expelled producing what is called the Io torus -- a ring of partially ionized gas following Io's orbital path -- in the Jovian magnetosphere. Once ionized, electromagnetic forces can easily overcome gravity and the escape velocity is much less of a constraint. – honeste_vivere Jan 17 '17 at 16:17