Wolfram Computation Meets Knowledge

Wolfram Summer School


Bram Boroson

Summer School

Class of 2003


Bram is an astrophysicist who has done research at the Harvard-Smithsonian Center for Astrophysics, MIT, and the NASA Goddard Space Flight Center. He has served on committees determining who uses the Hubble Space Telescope and Chandra X-ray Telescope. He has taught at the Claremont colleges and the College of Wooster in Ohio. In high school, Bram was one of 40 winners of the Westinghouse Science Talent Search for a paper on “the Boroson Rectangle”, an array similar to Pascal’s Triangle. At Oberlin College, his undergraduate majors were math and physics; he won an award for the highest score at Oberlin for the Putnam mathematics exam. His thesis at the University of Colorado at Boulder examined X-ray binaries, star systems that contain neutron stars or black holes in orbit with normal stars. These star systems have been the main subject of Dr. Boroson’s astrophysics research, with focus on accretion disks and stellar winds, although he has also worked on the interstellar medium and RS CVn systems.

Project: Stellar Wind Cellular Automaton

The hottest stars, “types O and B” (and the stars they evolve into, Luminous Blue Variables and Wolf-Rayet stars), emit strong stellar winds. These winds are not analogous to the Sun’s “solar wind”. The key to these winds is that they are “radiation-driven”, that these bright stars are in fact so brilliant that their outgoing radiation pushes out considerable mass (up to 10^-5 solar masses a year). In this way, radiatively driven winds are similar to the tails of comets, although comets have several tails driven not only by solar radiation but by ions from the Sun.

The mechanism of radiative driving is thought to lead to special complications, and many numeric simulations have found chaos down to such scales that the model is not fully applicable. (See the discussion in “Introduction to Stellar Winds” by Lamers and Cassinelli, page 250.) One of the reasons for this chaos in stellar winds is that the force is mainly the effect of starlight being absorbed (scattered in a random direction, actually) by atomic transitions with exact wavelengths. If a patch of stellar wind has relatively constant velocity, the radiative force will not be great. The parts of the wind closer to the star will absorb the starlight, but the further parts within the parcel will then see less radiation, as there is no relative motion for a Doppler shift. The further parts of an accelerating parcel of wind will feel the force of stellar radition absorbed by line transitions, thanks to the Doppler effect.

Below I show a very idealized continuum-valued cellular automaton, essentially a discretization of the differential equation of a stellar wind. The analytic solution to a simple 1-d stellar wind is v=vmax (1-1/r)^beta, where r is the distance from the star’s center, measured in units of the star’s radius, and beta=0.5. More complete analyses in 3-d suggest that beta=0.8. In a Hubble Space Telescope study of a stellar wind affected by an embedded neutron star (which also acts as a probe), I found beta=1.4 (Boroson, Kallman, McCray, Vrtilek, and Raymond. Published in the July 1, 1999 Astrophysical Journal (519, 191)). In the model below, I perturb an analytic beta=0.5 wind with random variations.

I allow a time step small enough so that only a fraction of the wind in each cell moves into the neighboring cell. Naively, I ignore all effects of wind density. A cell “donates” velocity to a neighboring cell, which simply takes on a velocity value given by the sum of old and new velocities.

The core of the concept of radiatively driven wind chaos, however, is kept in the model. That is, the acceleration (which is added to the velocity each time step) depends on the difference in velocity between neighboring cells. The gravity of the normal star causes a counter-acceleration, pulling the gas inward (although it is partially balanced by the radiation force of the stellar continuum radiation.

Although the primitive model below shows spikelike superpositions on the standard analytic radius vs. velocity curve, these spikes go away when the initial intrinsic randomness is reduced to zero.

My proposal is to extend the model to two dimensions (cylindrical coordinates) where (1) the results can be visualized more intuitively, and (2) effects of stellar rotation, including initial motion tangent to the star, and centrifugal acceleration, can be included. These additional effects may provide structure within the stellar wind from which intrinsic randomness can grow.

A second, unrelated project will involve analytic, combinatorial proofs calculating the distribution of sides of polygons in trivalent networks.

Favorite Three-Color Cellular Automaton

Rule Chosen: 3225087886188

Reason: Originally, I planned to perform “exhaustive searches” to find interesting automata. My plan was to calculate something like the “two-point correlation function” used to measure the distribution of galaxies in the sky. Interesting automata may tend to have two colors especially close together (on the other hand, close together colors would force others to be further away.) Being a novice at Mathematica, I decided instead to look directly at the bits (actually base 3 “trits” or “trinary bits”) that defined the automaton. This strategy was similar to that used by John Horton Conway when he designed the 2-d automaton, “Life”–that rule was chosen to avoid either uncontrolled growth or ubiquitous extinction.

A more rigorous approach toward bit-design of automata could surely be made, and I could compare over several trials whether bit-designing or exhaustive searches discovered more interesting (class 3 or 4) automata given the same time period.

Once I found an automaton that seemed interesting, I often fiddled with the individual bits of the automaton number–sometimes by logic, sometimes by intuition, and sometimes randomly. The principles were as follows: at first, I wanted a balance between the digits 0, 1, and 2. I wanted to make sure that the digits 1 and 2 were not treated identically (i.e. 101->1 and 202->2, etc.) or the 3-color automaton would be too obviously an emulation of a 2-color automaton. (The numbering of 3-color automata compared with the 256 2-color automata assures us strict emulations are rare, but a human bit-designing an automaton should be aware of this pitfall. Also, one color could die out quickly, leaving a 2-color automaton.) I tried to avoid too much symmetry, either left/right, or between the colors.