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The Music of the (Neutron) Spheres

by John G. Cramer

Alternate View Column AV-92
Keywords: neutron star Rossi Explorer NASA X-ray astronomy general relativity
minimum stable orbit sonic orbit audio modulation chord
Published in the November-1998 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 04/28/98 and is copyrighted ©1998 by John G. Cramer.
All rights reserved. No part may be reproduced in any form without the explicit permission of the author.


     This AV column is about the heaviest neutron star ever observed, an object that produces the largest gravitational fields ever directly measured. This discovery by X-ray astronomers was reported at the Spring Meeting of the American Physical Society, held April-1998 in Columbus, Ohio. As will be discussed below, the very large mass of the neutron star has important implications for our understanding of the strong nuclear force at very large densities.

    A neutron star can be thought of as a gigantic spherical nucleus with more mass than our Sun, a diameter of perhaps 15 kilometers, made almost entirely of neutrons, and held together by the force of gravity. It can also be viewed as a stellar ash-heap, the leftovers from a type-II supernova, a burned-out star that has collapsed, converting the protons and electrons of its atoms into neutrons plus escaping neutrinos, an ash heal that lacked the mass to collapse all the way to a black hole. Instead its collapse was arrested by the mutual strong-force repulsion of the neutrons, so that it hovers in a delicate balance between gravity and the strong force. This balance would be overcome if the neutron star were more massive and had more gravitational attraction. Therefore, the maximum possible mass of a neutron star is of great interest to physicists because it tells us about the strong force at densities that are 2 to 5 times larger than those of normal nuclei.

    Radio astronomers first definitively observed neutron stars as pulsars, sources of periodic radio pulses that repeated a few hundred times a second. We now understand that these pulses are the result of a "lighthouse effect" of the radio beam emitted along the star's magnetic axis, a beam which scans periodically in our direction as the star rotates, producing pulses. Some neutron stars are members of a binary star system (two stars in mutual orbit), and these are often very strong sources of X-rays which occur when hydrogen gas flowing from the binary companion falls to neutron star's surface and heats it. The hot spot on the surface emits light like a light bulb filament, but in this case the temperature is about 100 million degrees K, so hot that the "incandescent light" from the hot spot is in the X-ray region. The X-ray sky is dotted with many of these sources, which mainly lie in the plane of our galaxy.

    In the Nov-92 issue of Analog in a column called "Centrifugal Forces and Black Holes", I discussed a newly- realized aspect of general relativity as applied to intense gravity fields. For circular trajectories close enough around a black hole, the centrifugal force works backwards, pointing inward instead of outward, and there are no stable orbits. This black hole situation is an extreme case, but Frederick Lamb and his co-workers at the University of Illinois have applied similar ideas to the intense gravity fields and highly curved space near neutron stars. They find that when the field is strong enough, the space curvature predicted by general relativity reduces the centrifugal force to the point where orbital stability is destroyed. For any neutron star there is a minimum-diameter stable orbit, and no stable orbits exist at smaller orbital diameters. An object with a trajectory within the minimum orbit will spiral into the gravity well and never emerge.

    A neutron star fed by gas from a close-orbiting binary companion develops a whirlpool-like accretion disk of heated gas that orbits the neutron star, undergoes repeated collisions, loses kinetic energy, and moves ever inward. When part of a clump of the gas in the accretion disk reaches supersonic speeds in the medium (at an orbit called the "sonic orbit"), it is propelled inward, spirals down to the neutron star, and crashes into its surface, creating a "hot-spot" X-ray source where it hits. The orbit at which the gas clump goes supersonic depends on how much gas flow is involved, so there is a good inverse correspondence between the sonic orbit radius and the intensity of X-ray emission, with high emission corresponding to smaller orbits having higher orbit frequencies.

    Since the clump of gas at the sonic radius is orbiting the neutron star with a particular orbital frequency (around 1 kHz), the hot spot fed by its infall should move across the surface of the star with the same frequency. This causes the X-rays to be modulated with a frequency that reflects orbital frequency of the sonic orbit. This modulation is about a factor of 10 faster than that from the rotation of the neutron star (which is around 100 Hz) and can be easily distinguished from it.

    When a very large gas flow occurs, the sonic orbit can become equal to the minimum stable orbit predicted by general relativity. At this point the relation between the intensity of X-ray emission and orbital frequency is broken because, no matter how high the gas flow, the sonic orbit can move no farther inward. Therefore, one expects the frequency to hit a maximum value with increasing X-ray luminosity, and this maximum should reflect the minimum stable orbit radius of the star, which in turn is related to its mass. Lamb and his co-workers predicted this effect in 1996, but until recently it had not been observed.

    Since X-rays cannot penetrate the Earth's atmosphere, the X-rays from neutron star binary systems have only been mapped in the last 30 years using a succession of X-ray telescopes launched into orbit. One of the most recent of these telescopes is NASA's Rossi X- Ray Timing Explorer (RXTE) satellite, which was launched from the Kennedy Space Center on December 30, 1995. It has the distinction of being the first X-ray telescope that can determine not only the spatial direction of the emitted X-rays but also their time structure at frequencies up to the megahertz region. Rossi has an all- sky monitor that can detect rising X-ray activity and very-large-area detectors that can be rapidly pointed at a source to detect X-rays with energies between 2 and 200 keV.

    The emphasis on timing in the Rossi satellite was something of a gamble, since there was no guarantee that any interesting time structure would be found in astronomical X-ray sources. The Rossi design was based on the astronomy rule of thumb that the most rapid possible time change of a stellar source is given by the size of the source divided by the speed of light. Since the speed of light is 0.3 million kilometers per second, an object with a 1 MHz time structure would have to be less than 0.3 kilometers across, and an object with a 1 kHz time structure would be less than 300 kilometers across. Neutron stars, predicted to be around 15 kilometers in diameter, fall nicely between these two values.

    William Zhang and his coworkers at NASA's Goddard Space Flight Center in Greenbelt, MD have used the Rossi telescope to observe an X-ray source with the catchy name of 4U 1820-30, which has been identified as a neutron star binary. They observed a remarkable effect: the X-rays from the source change at audio frequencies, showing a double modulation at frequencies about 270 Hz apart. When the luminosity of the X-ray source increases, these frequencies increase together until they reach maximum values of 1.050 kHz and 0.780 kHz. When these audio frequencies are converted to sound waves, they make quite a pleasing musical tone which Zhang has characterized as a "Cosmic Chord", perhaps the first modern observation of what the ancients called the music of the spheres (in this case, neutron spheres).

    From these data it was deduced that 270 Hz is the rotational speed of the neutron star and 1.050 kHz is the frequency of the minimum stable orbit. The 0.780 kHz frequency is interpreted as the minimum orbit frequency and star's rotation frequency combining to produce a beat frequency which is the difference of the two. There is also some evidence of a modulation at 1.320 kHz, the sum of the rotation and orbit frequencies, but this component is much weaker.

    The big surprise in this analysis is the very low value of the minimum stable orbit frequency. This frequency implies the most intense gravitational field that has ever been measured directly, and it also corresponds to a very large mass for the neutron star, about 2.3 times the mass of the Sun. This value comes as a shock to many in the physics community. Before this observation, it was widely assumed that neutron stars could have masses larger than about 1.4 to 1.8 times the mass of the Sun. Beyond this mass it was believed that gravity would overwhelm the strong-force repulsion between the neutrons and the object would collapse to a black hole.

    Apparently there is a flaw in the chain of logic that underlies this conclusion. It depends on the relation between pressure and density in the neutron star, which is called "the equation of state". A "hard" equation of state means that the pressure rises steeply as the density is increased due to gravitational compression, while a "soft" equation of state means that the pressure rises more gently with increased density. During the past decade, nuclear theorists have been speculating on a plethora of speculations that would soften the equation of state: 6-quark bags, kaon condensates, strange matter, and so on.

    However, it seems that these theorists have been pushing in the wrong direction. A neutron star of 2.3 solar masses is only possible if the equation of state is extremely hard, requiring very strong repulsion between adjacent neutrons. This repulsion is so large that it exceeds that which can be produced when the strong interaction acts between only two particles. The suggested mechanism is to get the additional repulsion from "three-body forces", that is, forces that only act when three neutrons are simultaneously in close proximity.

    The information from nuclear collision experiments on such three-body forces is very limited, so the neutron stars are providing information about nuclear forces from an entirely new direction. We are learning new lessons about the forces within ordinary nuclei by observing the musical tones from 4U 1820-30, a "nucleus" 15 kilometers in diameter.

John G. Cramer's 2016 nonfiction book (Amazon gives it 5 stars) describing his transactional interpretation of quantum mechanics, The Quantum Handshake - Entanglement, Nonlocality, and Transactions, (Springer, January-2016) is available online as a hardcover or eBook at: or

SF Novels by John Cramer: Printed editions of John's hard SF novels Twistor and Einstein's Bridge are available from Amazon at and His new novel, Fermi's Question may be coming soon.

Alternate View Columns Online: Electronic reprints of 212 or more "The Alternate View" columns by John G. Cramer published in Analog between 1984 and the present are currently available online at: .


Rossi X-Ray Timing Explorer (RXTE):
Information available fron NASA at the following URL:
Minimum stable orbit:
M. C. Miller, F. K. Lamb, and D. Psaltis, Astrophysics Journal, (in press, 1998); (LANL Preprint astro- ph/9609157).
Observation of "Cosmic Chords":
W. Zhang, A. P. Smale, T. E. Strohmayer, and J. H. Swank, W, Bull. Astrophysics Journal Letters, (accepted for publication, 1998); (LANL Preprint astro-ph/9804228).

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