Modeling Planets in a Mira Wind
Bob Cohanim
Iowa State University
REU Program
1. Introduction
1.1 Mira Stars
Mira stars are long period variable stars that are nearing the end of their lives. A Mira
star is characterized by radial pulsations of its atmosphere that extend out to distances
comparable to our own Jupiter from the Sun. These radial pulsations are also a means of
mass shedding for the star. With each pulsation, a shell of stellar material is pushed out
from the star. Dr. G.H. Bowen (Iowa State University) modeled the behavior of a Mira
wind and propagation of shocks through the system from the bottom of the atmosphere
out to great distances. If we follow a shell of material from the time of its first shock
towards the end, we will notice that it is not always traveling away from the star. The
pulsations are not strong enough for the particles to reach escape velocity immediately
(NOTE: the term particles refers to typical stellar material shed from the star and solar
dust). Particles flow out due to the initial shock, but when the shock energy dissipates the
particles begin to fall back towards the star due to the stars gravity. Before the particles
can fall all the way back onto the star though, another shock comes up behind them and
gives another “bump”. In this way particles have a net velocity outwards from the star
over a long period of time and eventually reach distances where they are able to escape
the stars gravity and leave the system.
Figure 1: Bowen’s model tracing several particles through pulsational cycles of star
This project will look at how a planet orbiting a star would interact with the wind, and
what implications it would have for observational effects.
1.2 Planet Experiencing Mira Wind
Dr. Struck (Iowa State University) and Dr. Willson (Iowa State University) have
proposed a theory based on the behavior of the wind particles as they interact with a
hypothetical planet orbiting a Mira star. For simplicity’s sake, the planet has been
modeled in a circular orbit and the winds in a periodic inflow-outflow manner in accord
with Bowen’s models for the given time it takes to run one simulation. As the planet
makes its way around the star, suppose it first experiences the outflow portion of the
wind. The outflow, along with the planets tangential velocity to the sun, act in creating a
net angular momentum of the particles in a clockwise direction around the planet
(assuming the we are looking at the model from above and are viewing it traveling in a
counter-clockwise manner). The particles would then spiral into a so-called accretion
disc around the planet. As the outflow of the particles loses energy, it will begin to fall
back towards the star and fall back onto the planet in an opposite manner. This inflow
will create an angular momentum in the counter-clockwise direction for the accreting
particles. Dr. Struck has hypothesized that the material that is slowly accreting around
(or flowing past the planet) will have to release a noticeable amount of energy to make
the particles reverse their angular momentum. This release of energy may be observable
in the light curves of Mira stars by exhibiting a bump in the curve. This bump may also
have a periodicity according to the waxing and waning of the planet around the star if we
observe it at anything except from nearly top view.
Figure 2: Sample light curve of star R Cae, showing possible bump in magnitude
Observational efforts and data mining for this effect is being done by others in the current
research group and are not in the scope of this paper. Currently, the simulation is trying
to model the correct winds and assess the affects on the particles near the planet.
1.3 Approximations
There are several key approximations that need to be addressed in the model: 1.
The planet is treated as a point mass that is not rotating. For now the planet has no
rotational effects due to atmospheric or magnetospheric coupling. 2. The planet does not
accumulate much material onto its surface and thus keeps a nearly constant mass. This
approximation comes about due to the physics of the winds and the distance the planet is
from the star. It is key to the theory because if the planet was to become noticeable more
massive it would spiral into its star due to a decrease in its tangential velocity. The star’s
mass is not decreasing at too high of a rate so that gravitational effects between the planet
and the star can be treated as a constant throughout a simulation. Current research shows
that a star loses most of its mass in a short period at the very end of its shedding stage
(Willson 2000), so there is a significant amount of time, when the star is a young Mira
(an oxymoronic statement in a way), that it is possible for planets to still be orbiting
around their star.
2. Modeling Method
2.1 General Setup of Model
The initial models are simplified and are aimed at achieving a few very particular
goals: to create well scaled particle behaviors in accord with Bowen’s models. This
entails accurate wind speeds, correct periodicity of shocks, correct periodicity of particles
between two shells, and a radial position of the planet relative to the star where the
shocks are not too strong to destroy the planet, nor to weak that effects of the winds on
the planet are hardly seen. The models are being run using a softened particle
hydrodynamics algorithm that was developed by J.J. Monaghan of Monash University,
Australia. The current model which is being tested is simplified in a several ways to
allow for efficiency in computational time and debugging purposes.
An initial particle grid is set up on portion of the orbital trajectory between two
spherical shells. The boundaries are set up around the particles to keep them from
expanding and floating away from the test section. The lower boundary oscillates
periodically, providing an outward push, creating a shock wave of particles streaming
through the test section which is to interact with the planet. Currently the upper boundary
is fixed for simplicity, but in essence it should oscillate as well, slightly out of phase from
the lower boundary. Particles in the grid are given initial velocities based on a function
that makes particles with smaller radius initially falling in the most, to those with the
greatest radius expanding initially. This is concurrent with Dr. Bowen’s models of
periodicity in the wind. Noise is added to the model to keep in a bit of randomness which
is concurrent with nature in general.
Figure 3: Initial Grid of particles, all coordinates in AUs and the planet at 5 AU
2.2 Future of the Method
Currently the planet being used in the models is a 50-Jupiter mass planet, which is
quite large. Since the particle grid has particles spaced out at quite a distance from each
other, it is more difficult to see the effects it has on a smaller planet, so the larger planet
allows us to see the effects more greatly. Better resolution in the particle grid would
allow for the use of any size planet, but would greatly increase computational time, so for
now a 50-Jupiter mass planet it is. Thermal physics is currently modeled as an adiabatic
gas with simple, approximate radiative cooling.. The current particle interaction with the
planet forms a small accretion disc with spiral arms stemming out from it. In the future
the hope is to resolve the accretion disc better. This will come along with particle
resolution, stronger shocks, better thermal and magneto-hydrodynamics.
3. Results
Many simulations have been run, each improving on an aspect from a previous
run. The simulation process began with a planet traveling in a line through a stream of
particles that were flowing in a perpendicular direction to the planets trajectory with a
constant velocity. These initial tests were run to determine if material would accrete
around the planet, if a bow shock would form in front of the planet, and to view the wake
pattern the planet created. The evolution of the model has led to a polar coordinate grid
circling around an origin (wherein lies the star), phased initial particle velocities as a
function of radius, stiff boundary conditions, oscillating lower boundary, periodic inflow
and outflow of particles across the planet, developed shocks propagating through system,
a bow shock in front of the planet, accreting material around the planet, and a wake
structure trailing the planet that is influenced by the inflow and outflow of particles.
A numerical instability in the code caused by particles getting too close together
haunted the model for quite some time. The instability caused enormous, dense shocks to
propagate through the entire grid and increase particle velocities and energies to
tremendously high (non-physical) values, which in turn cut down the time step of the
integrator, effectively stalling the simulation. After much hunting and testing, the cause
of the instability was narrowed down and protection was added to the computer code to
prevent the runaway instability from stalling the simulations. Now the focus of the
simulations are to develop shocks that are strong enough to accrete and reverse accreted
material, but not completely strip the planet of all material.
Shown here are several frames of a simulation where a bow shock, accreting
material, a propagating shock, and a developed wake is well defined. The figures
inserted below demonstrate the current models. Frames included show a large scale view
as well as a zoomed in view near the planet. All distances are in AUs.
Figure 4: Large scale view of simulation, showing a long wake and propagating shock
Figure 5: Zoomed in view on planet with well defined bow shock leading the planet and accreting material
streaming around the planet
4. Discussion
The main goal of the simulations are to create an accretion disc around the planet
and to see that accretion disc reverse its angular momentum. Analysis will be done on
particles very near the planet to determine how much energy was released in the reversal
process to see if theoretical models of light release coincide with observational bumps in
the light curves of Mira stars. The process of hard analysis is just starting, as the
simulations are becoming more and more refined. With stronger shock development
should come well defined accretion disc reversals. The simulations already demonstrate
a well defined wake trailing the planet, whose properties are also interesting and may be
studied in the future. A large encompassing bow shock forms early on and accreting
material forms a small disc with arms inside the planets bow shock when particles are
flowing back towards the star. Accretion reversal will require stronger propagating
shocks to break through the planets bow shock and affect the particles near the planet.
With the new protection that has recently been added to the code, stronger shocks are
being used. One may wonder, what is the limiting shock strength? For one, initial shock
speeds need to be in accord with Dr. Bowen’s models. Secondly, it has been
demonstrated in a few errant runs that if the shock is too strong, it can completely strip
the planet of its bow shock and all accreted material. Development of the models are
creating better conditions with every run and will soon hopefully yield interesting results.
Secondary analysis codes, which will take the raw data of particle numbers,
velocities, and energies in the near vicinity of the planet, are already beginning to be
developed and will yield the information about how viable such a process is. If energies
cannot be obtained from this simulation which correlate to enough light output from such
a process, then the physics of the simulations may be enhanced to include more
complicated thermal and magneto-hydrodynamics.