Hydrodynamic Escape

Simply because of their buoyancy, smaller particles will tend to rise up through the atmosphere. As they do, they run into larger particles, and some of those larger particles are swept up along with them. When the escape parameter, l , discussed in the Jean’s escape page, is greater than two, the flow of the particles associated with it is supersonic.

A way to visualize this process is to consider a room that is about half filled with balloons full of air bouncing around. If the ceiling is covered with pins then hitting the ceiling takes a balloon out of the system. Just with this system that will occasionally happen and will be analogous to Jean’s escape. To model hydrodynamic escape we add a layer of helium filled balloons along the ground and then release them. As the helium balloons rise through the air filled ones they will occasionally bump into them and can raise them to the ceiling just due to collisions.

Below is an applet that demonstrates this effect. This simulation begins with 15 large particles in a box like that used in the Jean’s applets. But very frequently a particle is introduced at the base of the box with a large upward velocity. These new particles have 1/16^th the mass of the large ones (something like hydrogen vs. atomic oxygen) and hence even at their higher velocities, a single collision can’t do all that much to the larger particles. However, because the flux of small particles is large the repeated impacts of upward moving smaller particles with the larger ones drives the larger ones slowly to the ceiling. To view this simulation simply click the "Start" button.

Analytically the relation between the flux of smaller particles and the flux of larger particles that it drives can be expressed as , where the two subscript denotes the heavier particles and the one subscript is for the lighter particles. The symbol X is the mole fraction of that population and m is the mass per particle. The value m_c is a crossover mass for which hydrodynamic escape works and has a value of , where b is related to the diffusion constant and is a function of temperature.

Let’s look first at what this crossover mass means in our mental analogy using balloons. Consider using balloons filled with CO₂ instead of regular air. The helium balloons would probably still manage to boost some of them to the ceiling, but not as many. However, if they were filled with water there is no way that collisions with the helium balloons would ever be able to boost one to the ceiling. So somewhere between CO₂ and water there is a critical mass (in this case it is actually a density) where all lighter materials can be boosted to the ceiling and heavier ones can’t.

This expression of F₂ will take on negative values if m₂>m_c. This negative flux doesn’t actually occur in nature, instead the flux goes to zero in this case. One important fact to notice about this formula is that it is linear in m₂. This is compared to the exponential relationship for Jean’s escape. Because of this, hydrodynamic escape can be much more efficient at removing larger particles, and fractionating isotopes.

In the model this analytic expression is implemented taking the calculated Jean’s escape rates to be the fluxes of the lighter particles. Technically this formula is only accurate for a system with only two populations, but it is plausible to extend it to more as long as the smaller particles dominate. This is not necessarily the case in the model, but the inaccuracies it induces are not out of order for the purposes of this simulation.