"1,000 Shares of Magnetar at 12-1/2!"

Quakes on pulsars follow the same power law
as the stock market, traffic jams

Dec. 8, 1999 - Here's a hot stock tip: the market, earthquakes, traffic jams, and magnetars follow the same power law.

This oddity of the universe won't make you rich; it certainly can't be used to predict where the market is headed. But it follows a recent theory called self-organizing criticality.

Right: "Ouch! Get the number of that fault line." A car lies crushed under what had been a townhouse in the Marina District of San Francisco after the Loma Prieta earthquake. It wasn't "the big one," although everyone knows it will happen eventually. Magnetars also have "big ones" infrequently, plus swarms and clusters of mid-size quakes. Credit: U.S. Geological Survey.

As often happens in nature, statistics can yield intriguing answers after months of individual observations, like those made by Ersin Gogus, a doctoral candidate at the University of Alabama in Huntsville, and several colleagues. Gogus earned his bachelor's degree at Middle East Technical University in Ankara, Turkey.

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"Subsystems self-organize to a critical state in which a slight disturbance can cause a chain reaction, such as an avalanche, an earthquake, or a magnetar outburst," Gogus explained. While this self-organization - a theory developed in 1988 by Per Bak - cannot predict the strength or time of the next event, "It does let us expect that the strongest events at high energies tend to occur less frequently than other events."

In the case of earthquakes or starquakes, the distribution of strength will follow a pattern called a power law.

Left: Traffic at midday in Norfolk, Virginia, is pretty manageable. In a few hours, the open spaces will be crammed with cars and with drivers and passengers who wonder how long it will take to wade through the mess. Although they are manmade, traffic jam sizes and frequencies follow the same power law as starquakes on Soft Gamma Repeaters. Credit: Virginia Department of Transportation.

"It's a good test to see how the energy distribution forms," Gogus said.

He and his colleagues analyzed the statistical properties of SGR 1900+14, one of four known soft gamma repeaters. These are neutron stars that repeatedly emit bursts of low-energy gamma rays at random intervals. The SGRs have well known locations and are within or near our galaxy (unlike true gamma-ray bursters which are once-only events that occur deep in the universe). SGR 1900+14 and other SGRs have been identified as magnetars, neutron stars with magnetic fields intense enough to drag on the star's rotation and to pump massive amounts of energy into space.

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Gogus looked at SGR 1900+14 during an extremely active period between May 1998 and January 1999. This followed an extended period with virtually no activity. For nine months, though, it let loose with more than a thousand events.

This activity was measured with the Burst and Transient Source Experiment (BATSE) aboard the Compton Gamma Ray Observatory and with the Rossi X-ray Timing Explorer.

Two interesting phenomena emerged from their data.

Left: Distribution of the waiting times between successive bursts from SGR 1900+14, as detected by the Rossi X-ray Timing Explorer. The line shows the best-fit log-normal function. The solid portion of the line indicates the data used in the fit. Links to 784x673-pixel, 108KB JPG. Credit: Ersin Gogus. Copies of graphs in this story are available in a 69KB Acrobat PDF.

"When you get a large sample set," Gogus said, "the distribution of the logarithm of recurrence times of events forms a bell curve." In other words, an ordinary graph produces a graph that few people would recognize. But putting time on a logarithmic scale, which compresses large quantities into scales that can be fit onto a sheet of paper, produces the familiar bell curve.

Gogus found that the waiting time between successive bursts ranged from 0.25 to 1,421 seconds (almost 24 minutes), with a peak at 49 seconds in the middle of the bell curve formed by the burst waiting times.

This does not translate into a way of predicting when a burst will occur, but implies some underlying mechanism within the magnetar is driving events and that they are not totally random. In addition, it doesn't seem to store up energy for "the big one" since there was no correlation between the waiting time and the intensity of the next burst.

The other interesting effect is called a power law energy distribution. That is, plotting discrete event energies against the number of events within discrete energy ranges produces a straight line.

The outbursts from SGR 1900+14 followed a power law of 1.66. Other researchers have measured similar distributions for SGR 1806-20 and 1627-41.

Power law distributions have been found for earthquakes (with power laws ranging from 1.4 to 1.8) and for solar flares (1.53 to 1.73).

Right: Scatter plot of the PCA fluence vs. duration for 281 SGR 1900+14 bursts shows a correlation. The solid line is a power law with an exponent 1.13 obtained via least squares fitting. Links to 580x502-pixel, 97KB JPG. Credit: Ersin Gogus.

Looking at similar behavior in earthquakes and even in how much sand will pile up before sliding led Bak, with C. Tang and K. Wiesenfeld, to propose self-organized criticality theory. Bak is at the Neils Bohr Institute in Copenhagen, Denmark. He proposed that large, dynamic systems will behave according to some variation of a power law.

At a top level, it seems obvious. Major earthquakes happen rarely. Moderate earthquakes come more often. Undetectable tremblers go on daily. Even a reader unfamiliar with economics with realize that the stock market has a major crash rarely, a modest adjustment more often, and small ups and downs weekly or daily.

What seems obvious, though, requires analysis and a strong mathematical footing to be accepted as a working theory. Kan did that in 1991.

Left: Differential distribution of the fluences of bursts from SGR 1900+14 as measured with Rossi X-ray Timing Explorer (diamonds) and the Burst and Transient Source Experiment (squares). The solid lines denote the interval used in the fit and the dashed lines are the extrapolations of the model. Links to 560x480-pixel, 74KB JPG. Credit: Ersin Gogus.

Scientists since then have found a range of applications for self-organizing criticality, ranging from traffic jams to magnetars.

Gogus and his colleagues have successfully applied it to SGR 1900+14, strengthening the argument that SGRs are powered by starquakes and magnetically powered flares, as the magnetar theory proposes. The mechanism is a hybrid of the two because the energy stored in the crust is mostly magnetic rather than elastic (like a fault line ready to pop).

"We can't really say that these are independent because they are part of the system," Gogus said. "They may be triggered by other bursts in the system. They are not truly independent."

The finding also helps establish limits on what kinds of emissions can be expected from magnetars, Gogus continued. This is a limit on the burst emission model, not directly on energy involved.

No matter how outrageous a star you might be, you still have to obey the laws.

Outbursts Result in Controversy -- Scientists have different ideas to explain the behavior of Soft Gamma Repeaters (SGRs). (October 20, 1999)
Happy Birthday, Magnetars -- Twenty years since SGR 0526-66 made its grand appearance to Astronomical minds. (March 5, 1999)
Crusty young star makes its presence felt: Gamma ray flash zaps satellites, illuminates Earth, and sheds light on several mysterious stellar events. (Sept. 28, 1998)
A whole lot of shakin' going on: Starquakes lead to discovery of first new Soft Gamma Repeater in 19 years (July 9, 1998)
Magnetar discovery announcement including more details, interviews and more illustrations (May 20, 1998)
Magnetars.org website

A review of Professor Per Bak's book How Nature Works: the science of self-organized criticality.

Abstract of paper The Dynamics of Money, coauthored by Per Bak.

The Rossi X-ray Timing Explorer Learning Center at Goddard Space Flight Center

Burst and Transient Source Experiment (BATSE) Home page

More Space Science Headlines - NASA research on the web

NASA's Office of Space Science press releases and other news related to NASA and astrophysics