3 Solar Mass Black Hole Vs Neutron Star Size

Black Holes and Neutron Stars

Cole Miller
Department of Astronomy and Astrophysics, University of Chicago

The Main Point

Studying neutron stars and black holes gives us access to exotic realms that we can't explore on Earth.

Abstract

Neutron stars and black holes are among the most exotic objects in the universe. A lump of neutron star matter the size of a sugar cube would weigh as much as all humanity, and the stars have magnetic fields a trillion times Earth's. Since we can't reproduce such conditions in laboratories, we have to observe neutron stars with telescopes to figure out their properties. Recently the Rossi Explorer, a new X-ray satellite, discovered a remarkable new phenomenon of neutron stars that strip matter from their companion stars: their brightness varies almost periodically more than a thousand times per second. I will describe how this phenomenon gives us a sensitive new tool to probe the properties of neutron stars, and how it may even help us search for black holes.

Intro to Black Holes

A black hole is a region of space in which the matter is so compact that nothing can escape from it, not even light; the "surface" of a black hole, inside of which nothing can escape, is called an event horizon. The matter that forms a black hole is crushed out of existence. Just as the Cheshire Cat disappeared and left only its smile behind, a black hole represents matter that leaves only its gravity behind.

Black holes are usually formed when an extremely massive star dies in a supernova. However, some people think small black holes were formed during the Big Bang, and that the resulting "mini black holes" may be in great abundance in our galaxy.

In principle, black holes can have any mass; black holes formed by stellar death have at least twice the mass of our Sun. Unlike ordinary things (e.g., rocks), which have a size roughly proportional to the cube root of their mass, black holes have radii proportional to their mass. The event horizon of a nonrotating black hole the mass of our Sun has a radius about 3~km. Thus, large black holes aren't very dense! A black hole a billion times as massive as our Sun, such as is thought to exist in the center of some galaxies, has an average density just twenty times the density of air.

Black holes, like any gravitating objects, exert a tidal force. If you approach a black hole feet first, the gravitational force at your feet is greater than the force at your head. The tidal force at the event horizon is smaller for larger black holes: you would get torn to shreds far outside a black hole the mass of our sun, but at the event horizon of a billion solar mass black hole the tidal force would only be a millionth of an ounce!

Strange Facts About Black Holes

• Light bends so much near black holes that if you were near one and looking away from the hole, you would see multiple images of every star in the universe, and could actually see the back of your own head!
• Inside a black hole the roles of time and radius reverse: just as now you can't avoid going into the future, inside a black hole you can't avoid going in to the central singularity.
• If you stood a safe distance from a black hole and saw a friend fall in, he would appear to slow down and almost stop just outside the event horizon. His image would dim very rapidly. Unfortunately for him, from his point of view he would cross the event horizon just fine, and would meet his doom at the singularity.
• Black holes are the simplest objects in the universe. You can describe one completely by just its mass, spin rate, and electric charge. In contrast, to completely describe a dust mote you're have to specify the position and state of all of its atoms, taking at least $10^{16}$ numbers!
• As Hawking discovered, black holes can evaporate, but only very slowly. Even one the mass of a mountain will last for ten billion years, and one the mass of the Sun will only evaporate after $10^{67}$ years.

How Do We Detect Black Holes?

Black holes don't radiate light, and an object that falls inside a black hole doesn't emit light either, so detecting them can be challenging.

Just as with neutron stars, if a black hole is in a binary and it strips gas from its companion, we can detect X-rays from the resulting accretion disk (see "Observing Neutron Stars"). The light from accretion disks around black holes looks very similar to the light from disks around neutron stars, and it is not always possible to tell with certainty which object lurks at the center of the disk, although in six cases so far we're sure that the central object is a black hole.

You can also infer the presence of a black hole in the center of some galaxies. This is done by observing stars near the center of the galaxy. If the stars are moving very rapidly around some unseen object, Kepler's laws can be used to estimate the mass in the center. In some cases the mass must be at least a hundred million times our Sun's mass, in a region only a few light years across. Astronomers are virtually certain that the only explanation is a black hole, but we lack direct evidence.

The detection of black holes is very difficult and controversial, and it is being studied actively by many research groups.

Origin and Scale of Neutron Stars

A neutron star has roughly the mass of our Sun crammed in a ball ten kilometers in radius. Its density is therefore a hundred trillion times the density of water; at that density, all the people on Earth could be fit into a teaspoon! Neutron stars are born during supernova, and are held up by neutron degeneracy pressure. These stars are relatively rare: only about 10^8 in our galaxy, or one in a thousand stars, so the nearest one is probably at least 40 light years away.

Neutron Stars and Extreme Physics

• Density at center can be several times the density of an atomic nucleus, so we can't explore this regime in laboratories. The properties of this matter are unknown, and may include such exotic things as enormous conglomerations of quarks.
• Magnetic fields are a trillion times Earth's, and more than a million times as strong as can be achieved in laboratories.
• Matter in the central parts of neutron stars is thought to be a superconductor, even at a hundred million degrees!

Neutron stars therefore have states of matter that cannot be duplicated in laboratories. Study of them helps us test our theories, and perhaps discover new physics. But how can we observe neutron stars?

Observing Neutron Stars

We see a normal star by the light it gives off during fusion. Neutron stars are very hot, more than 100,000 K for most of their lifetimes, so this sounds promising but most of the energy comes out as X-rays (not visible light). Also, neutron stars are so small that at typical distances they are ten billion times fainter than you can see with your naked eye, which is too faint for even the Hubble Space Telescope. We need some other way to see neutron stars.

One way is to see them as radio pulsars. Another way is if the neutron star is one member of a binary, in which case the gravity of the neutron star can strip gas off its companion. The gas from the companion falls onto the neutron star, and emits fantastic power in X-rays: as much as 50,000 times the luminosity the Sun produces. This is a tremendously efficient way to generate energy. Dropping a kilogram of matter onto the surface of a neutron star releases as much energy as a five megaton hydrogen bomb!

Since the neutron star is a very small target, astronomically speaking, gas can't fall onto it directly. Instead, gas spirals around the neutron star, and friction with itself releases huge amounts of energy in what is called an accretion disk. Studying the X-rays from accretion disks can give us hints about the star: for example, how does matter behave at extremely high densities?

The Equation of State

As mentioned above, we want to know the properties of the extremely dense matter in the center of neutron stars. One way to characterize the matter is by its equation of state.

The equation of state can be pictured as the relation between the density of matter and its pressure. Consider a glass of water. The shape of the water in the glass can be changed easily (e.g., by sloshing it around), but the volume, and hence the density, of the water is extremely difficult to change. Even if you apply a huge amount of pressure to the water, for example by a piston, the density changes hardly at all; this is the basis of hydraulic presses. Water may therefore be said to have a stiff equation of state. In contrast, the volume of air in an empty glass can be changed easily, with little pressure, so air may be said to have a soft equation of state. So, a knowledge of the equation of state tells us, essentially, how squeezable the matter is.

In the case of a neutron star, knowledge of both the mass and radius of a particular neutron star would tell us the equation of state. This is because gravity squeezes the star, and the more mass the star has the more gravity squeezes it. If the star has a large radius (meaning, say, 15~km!), it was relatively successful in resisting gravity and thus has a very stiff equation of state. If the star has a small radius (say, 8~km), it was not as successful in resisting gravity and it has a softer equation of state. We therefore need to estimate the mass and radius of neutron stars.

Estimating NS Masses and Radii

No easy task, this. Astronomical measurements are often challenging, because we can't go to a star and experiment on it. Neutron stars are especially tough, because they are relatively small and far away: even the closest one would appear to be the size of a bacterium on the Moon, so we have to find other ways to determine the mass or radius of a neutron star.

One way to do this is to use Kepler's laws. If we can figure out how far two stars in a binary are from each other, and the duration of their orbital period, we know something about their masses. Only for neutron stars in binaries do we have even a rough estimate of the mass, and in only a few of those cases do we know the mass accurately.

Estimating the radius is much more difficult than estimating the mass. Unlike the mass, the radius doesn't have any strong effects on what we can observe. From astronomical observations alone, neutron stars could have radii from 5~km to 30~km (although most of that range, all but about 7~km to 20~km, is ruled out by what we know of nuclear physics).

So, we need some kind of breakthrough in the evidence to allow us to further constrain the radii of neutron stars.

An Unexpected Discovery

We can only discover what our instruments can detect, so many times in astrophysics a breakthrough in our understanding has come from an improvement in instrumental capabilities.

Such was the case when the Rossi X-ray Timing Explorer was launched on December 30, 1995. Its many outstanding properties include an unprecedented sensitivity to very rapid variations of the X-ray intensity of accreting neutron stars, i.e., neutron stars stripping mass from their stellar companions. This led to the discovery of a completely unexpected phenomenon: fast intensity oscillations, sometimes more than a thousand times per second!

Kilohertz Intensity Oscillations

Figure 1 shows the X-ray brightness from one neutron star system, as a function of time. The intensity goes up and down nearly 1000 times per second. There are at least 10 known neutron stars that show this, and we have discovered that:

• The intensity variations are fast, up to 1200 times per second
• For a given neutron star, the frequency of the variations goes up and down with time: in one case, the variation can be anywhere from 500 per second to 1100 per second

The dramatic change in frequency means that it can't be something simple like the spin frequency of the neutron star, since the star can't easily be spun up or down. However, the common occurrence of this phenomenon and its other properties mean that it is telling us something fundamental about the flow of matter onto neutron stars.

Click to see my proposed explanation of this phenomenon.

Implications

• The frequency with which the clump goes around the star is calculated by Kepler's laws: the higher the frequency, the closer the clump has to be to the star. This limits the radius of the star.
• When the effects of Einstein's general relativity are included, it turns out that we also get an upper limit to the mass of the star.
• The result is that the neutron stars in these systems must have masses less than 2.2 times our Sun's mass, and radii less than 17~km. This is the first convincing observational limit to the radius of neutron stars.

Key Points

• The matter in the center of neutron stars is incredibly dense, and we can't reproduce it on Earth.
• So, the study of neutron stars can tell us things about the universe that would otherwise remain forever undiscovered.
• One way to find out about the dense matter of neutron stars is to determine the equation of state of neutron stars, which is the relation between their pressure and density.
• The equation of state is known if we know both the mass and radius of a neutron star.
• But, astronomical observations are indirect: we can't experiment on stars.
• Neutron stars are particularly difficult, since they are relatively small and distant.
• Luckily for us, the recently-discovered phenomenon of rapid X-ray intensity oscillations may allow us, for the first time, to estimate both the mass and radius of some neutron stars and thus know their equation of state.
• As it turns out, if we knew the equation of state of neutron stars, we would also know their maximum mass. This ends up helping in the search for black holes in our galaxy.
• This phenomenon is still new, and we continue to learn things about it at a rapid pace, both observationally and theoretically; these are exciting times!

Glossary

Accretion disk:

the pattern of flow of matter from a normal star to a neutron star or black hole, which is flattened and thus disk-like.

Degeneracy pressure:

a quantum-mechanical phenomenon; fermions, such as electrons or neutrons, obey Pauli's exclusion principle, so that no two fermions can occupy the same state. Thus, if fermions are squeezed together they resist even if there is no temperature and no energy generation. This resistance to squeezing is degeneracy pressure.

Equation of state:

the relation between the pressure and density of a given type of matter, which is an indication of how the matter resists squeezing. If the matter resists squeezing strongly (e.g., water), the equation of state is stiff; if it resists squeezing only weakly (e.g., air), the equation of state is soft.

Event horizon:

in a black hole, the point beyond which events cannot be detected. This is the point of no return; an object that falls inside the event horizon can't get out.

Kepler's laws:

rules for the orbital motion of planets or anything else bound by gravity. The law of most interest here is that the square of the orbital period is proportional to the cube of the orbital separation, and inversely proportional to the mass. Thus, if we see an orbital period, we can estimate the mass or orbital separation and therefore constrain the mass and radius of a neutron star.

Singularity:

in a black hole, the "center point", at which densities, tidal forces, and other physical quantities become infinite. Our current physical theories break down at this point.

Tidal force:

the force an object feels because of the differential pull of gravity at different distances.

Related Web Resources

http://jovian.physics.uoguelph.ca/~droz/inside/

Black Holes:The inside story. Includes lots of diagrams and suggestions for further reading. The explanations are thorough and very readable.

http://www.astro.umd.edu/~miller/nstar.html

Moderately technical guide to various aspects of neutron star physics

http://antwrp.gsfc.nasa.gov/htmltest/rjn_bht.html

Robert Nemiroff's page on virtual trips to black holes and neutron stars. Neat animation showing gravitational light bending!

http://physics7.berkeley.edu/BHfaq.html

Black hole FAQ at Berkeley. Accurate, yet accessible

3 Solar Mass Black Hole Vs Neutron Star Size

Source: https://www.astro.umd.edu/~miller/poster1.html

Share this post