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Simulating Giant Waves in Newborn Neutron Stars

Supernovas and Neutron Stars
Simulating Newborn Neutron Stars
Building a Computer Model of a Neutron Star

ost stars are very stable, their rates of rotation changing only slowly. But not newborn, rapidly-rotating neutron stars. In these dense stars astrophysicists have predicted bizarre behavior in which huge oscillations are excited by the emission of gravity waves--those elusive ripples in the fabric of space-time predicted by Einstein’s general theory of relativity. Now Caltech astrophysicist Lee Lindblom and colleagues have simulated the life cycle of these oscillations using NPACI supercomputers. In addition to surprising new insights into neutron stars, these simulations are yielding a clear "signature" to search for as the new Laser Interferometer Gravitational Wave Observatory (LIGO) tries to detect gravity waves for the first time in history.

Figure 1. Crab Nebula and Neutron Star
At the center of the Crab Nebula, the glowing remnants of a supernova explosion, is a dense neutron star. Simulations (Figure 2) show that if rotating fast enough, a newborn neutron star exhibits rapidly growing r-mode oscillations, and that in only about a minute the star spins down, radiating away 40% of its angular momentum and 50% of its rotational energy in surprisingly monochromatic gravity waves that provide a clear signature astronomers hope to detect for the first time ever with LIGO. Image courtesy of Palomar Observatory.

Supernovas and Neutron Stars

On a clear night in a.d. 1054 as Chinese astronomers gazed at the heavens, they observed a brilliant new star, clearly visible to the naked eye and so bright that for a time it could even be seen during the day. The light came from an exploding star or supernova, whose glowing remnants can be seen today as the Crab Nebula (Figure 1). As one of the most cataclysmic events in the universe, supernova explosions involve a range of exotic phenomena, from radiating so much light that they can briefly outshine their entire galaxy to being a principal mechanism for creating the elements heavier than iron.

But the whole star doesn’t disappear--formed at the center of the collapsing star as it detonates into a supernova is a tiny core called a neutron star a mere 12 miles (20 km) or so in diameter, but with a mass comparable to the Sun’s. The density of neutron stars is so high--some one hundred trillion times that of water--that it is similar to that of the atomic nucleus.

Like a twirling dancer who pulls in her arms, the collapse of matter into this dense core can leave the newborn neutron star spinning extremely rapidly, some 60,000 revolutions per minute. Now, Lee Lindblom of the Theoretical Astrophysics group at Caltech and colleagues Joel Tohline of the Department of Physics and Astronomy at Louisiana State University and Michele Vallisneri of Caltech have carried out the most detailed computational modeling ever done of the stability of these stars.

"We’ve simulated the dynamics of newly formed neutron stars and found that, if the star is spinning fast enough, gravity waves will feed growing oscillations on the star. Some of these oscillations grow very rapidly, overcoming internal dissipation and becoming extremely large within just a few minutes as enormous amounts of energy are radiated away from the star in gravity waves," said Lindblom. The largest of these oscillations are called r-modes, after the Rossby or planetary waves resulting from the rotational Coriolis force, which are familiar to terrestrial meteorologists and oceanographers.

In addition to wanting to learn more about neutron stars, scientists are also interested in the gravity waves these stars may emit. Just as electromagnetic waves are radiated by accelerating electric charges and currents, gravity waves are radiated by accelerating masses and currents. Since gravity is so much weaker than electromagnetic forces, however, gravity waves are only detectable from very large and rapidly accelerating masses, as in neutron stars and black holes.

Long predicted, gravity waves have yet to be detected, and scientists are anticipating that a new generation of instrumentation in the soon-to-be-completed gravity wave observatory, LIGO, will finally be capable of teasing out the whisper of remaining energy in gravity waves that reach the Earth. When found, these waves are expected to carry new information about the violent events of their origins as well as the nature of gravity itself. Measuring them will provide important tests of one of the most fundamental theories of modern physics, Einstein’s general theory of relativity, checking its predictions of such things as the speed and polarization of gravity waves in the extreme conditions of massive black holes and neutron stars.

Another important consequence of being able to detect gravity waves is that this will give astronomers an entirely new window, beyond light and other electromagnetic radiation, through which to view and learn about the universe.

Figure 2. Giant Waves on Neutron Star
Movie frames from simulation of rapidly spinning newborn neutron star showing final stages of growing oscillations driven by gravitational radiation. These
r-mode or Rossby waves, involving the rotational Coriolis force, grow exponentially until the waves begin to break on the surface and strong shockwaves quickly damp them. Entire cycle of wave growth and damping takes only a few minutes. From star’s center outward, orange, yellow, red, and blue translucent areas indicate surfaces of constant density (0.8, .0.1, 0.005, and 0.0001, respectively, of star’s central density). A movie is available online at the CACR Web site.

Quick Time Movie of Simulation

Simulating Newborn Neutron Stars

Supernovae and the formation of neutron stars are spectacular physical events in the universe, and produce equally spectacular simulations. "Among other things, in just a couple of minutes these r-modes grow from microscopic ripples into enormous waves," said Lindblom.

When the researchers started, their main questions were how big the waves would grow before something limited their growth and how long this would take. "We knew that something had to stop them, but we had no idea what or how soon. So one of the surprises was how large the r-mode waves grow before being damped--to a dimensionless amplitude greater than three--and that it’s strong shockwaves, which form as the r-mode waves break like ocean waves, that damp the energy. Both of these were new results that were unexpected," said Lindblom.

As the rapidly rotating star emits gravitational radiation, in order to conserve energy and angular momentum there is a back reaction on the fluid--a gravitational reaction force--that drives the swift growth of the r-mode waves. Although they are primarily a circulation mode, as the r-mode waves grow larger changes in density also grow, until finally cresting waves appear on the surface in which strong shockwaves dissipate the energy and damp the waves.

In their simulations the researchers found that the star spins down dramatically as it emits gravitational waves, losing around 40 percent of its initial angular momentum and 50 percent of its rotational kinetic energy before the r-mode waves are damped (Figure2). The nonlinear evolution also causes the fluid to develop strong differential rotation, concentrated near the surface and poles of the star.

Another unexpected result of the simulations has to do with the frequency of the gravitational waves. "We thought that as the star spins down by something like 40 percent, that the frequency of the gravity waves would also decrease by a similar amount. But to our surprise, the frequency of the gravity waves remains almost constant at about 950 Hz, it’s very monochromatic," said Lindblom. "Besides what this can tell us about neutron stars, it’s important because this stable frequency provides an unusually clear ‘signature’ or pattern that can be looked for, making this a good source of gravity waves for LIGO to try to detect."

Lindblom also notes that astronomers have not observed any young, fast-spinning neutron stars, which is consistent with the prediction of the simulations that such stars are quickly spun down by their gravity wave radiation, and so are rotating rapidly for only a brief time.

Building a Computer Model of a Neutron Star

In putting together an accurate computational model to simulate neutron stars, the researchers began with previous predictions and what is already known. Even in the exotic conditions of massive newborn neutron stars, researchers believe these stars behave like a fluid that can be reasonably well described by the same classical equations used to describe the motion of air and ocean currents on Earth, but with the added importance of gravity.

The researchers use a 3-D Newtonian hydrodynamics code to which they have added the effects of Einstein’s equations with the back reaction of the gravitational radiation. "That’s the extra force you wouldn’t find on the Earth or the Sun. And since gravity is important, at every time step we have to solve the gravitational potential equation, which is very computationally intensive, requiring supercomputers," said Lindblom.

The simulation code solves the full nonlinear gravitational hydrodynamics equations in a rotating referenceusing a computational algorithm developed by astrophysicist Joel Tohline to study a variety of astrophysical problems. The model uses a cylindrical grid, 64 steps in the radial direction, 128 steps in the axial or z direction, and 128 angular zones on the star, for a total of about one million grid points.

In brief, the code performs an explicit time integration of the equations, using a finite-difference technique that is accurate to second order in both space and time. The code was written in Fortran 90 using the MPI parallel library and run on NPACI’s Hewlett-Packard V2500 at the Center for Advanced Computing Research at Caltech. From the simulation results the researchers have produced a movie that shows the final stages of the nonlinear evolution of the stellar model (Figure 2).

Looking ahead, the researchers are planning to extend the model by including more physics such as buoyancy forces and eventually magnetism, in order to simulate neutron stars even more realistically.

Project Leader
Lee Lindblom

Joel Tohline
Louisiana State University
Michele Vallisneri

Lee Lindblom, Joel E. Tohline, and Michele Vallisneri (2001), Phys. Rev.
Letters 86, 1152-1155.
Lee Lindblom (2001), to appear in Gravitational Waves: A Challenge to
Theoretical Astrophysics, edited by V. Ferrari, J. C. Miller, and L.
Rezzolla (ICTP, Lecture Notes Series).