Lecture 18 - White Dwarfs & Neutron Stars (3/23/99)

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ASTR012 Reading:
Chapter 17-2, 18-5 (ZG4)

pages 71-77

M1, the Crab Nebula, in the consellation Taurus. This supernova remnant harbors a pulsar. (Courtesy SEDS)
? Key Question: How do we know neutron stars exist?
! Key Principle: Rotational Energy and Spin-down
# Key Problem: What causes a pulsar to pulse?


  1. Neutron Stars
  2. Pulsars
  3. Classical Black Holes

Neutron Stars in Outline:

  1. Supernova Recap
  2. Neutron Stars
  3. Pulsars
  4. The Structure of Neutron Stars
  5. Accretion Disk Radiation

Black Holes:

As you compress a given mass into a smaller and smaller radius, the surface gravity grows. Eventually, it will be so strong that the escape velocity will exceed the speed of light. Such an object would appear dark, since no light could escape from its surface.

The escape velocity is the velocity at which a projectile (or particle) would have to be fired straight up so that it will eventually (infinitely far in the future) escape the gravity (come to rest at zero velocity infintely far away). The escape velocity can be calculated from the energy equation:

E = mv2/2 - GMm/R

For escape, v=0 at R=infinity, so therefore in such an orbit E=0. Therefore, at the surface (or any radius R), the escape velocity is given by:

vesc2 = 2 G M / R

Note that this velocity is higher than the (circular) orbital speed given by the centripetal velocity:

vorb2 = G M / R

by a factor SQRT(2). If you increase the speed of the Earth in its orbit by more than the factor 1.414, then it would no longer be bound in orbit about the Sun and would be free to fly about the galaxy. Q: What is the escape velocity from the surface of the Earth?

Therefore, if a mass M is compressed to a radius

Rsw = GM / c2

or smaller, then the escape velocity at the radius GM/c^2 will equal the speed of light. This radius is called the Schwarzschild Radius for the astrophysicist Karl Schwarzschild who calculated it soon after the publication of Einstein's theory in 1916.

An object with a radius equal to or less than the Schwarzschild Radius GM/c^2 is called a black hole. Light, nor anything else, can ever escape the surface of such an object, and it will appear dark. Note that this calculation uses only Newton's theory for gravity. In fact, the possibility for the existence of "dark stars" was postulated as early as 1783.

The Schwarzschild radius for 1 Msun is about 3km - if the Sun were to suddenly (and inexplicably) collapse to this radius it would become a black hole - though our orbit would remain unchanged since the gravitational force depends only on the mass and distance, not the size of the mass!

The effective radius of a black hole, the Schwarzschild radius, depends only on the mass itself, not on the actual density the the mass has (beyond the fact that it must be within its own Schwarzschild radius. As you increse the mass, the radius of the black hole increases proportionally to the mass. Furthermore, since nothing can escape, even light, the mass and size of a black hole can only increase with time.

The spherical "surface" surrounding a black hole of mass M at distance of the Schwarzschild radius GM/c^2 is called the event horizon. Once within the event horizon, matter (or radiation) is lost forever from contact with the universe outside the event horizon. The event horizon is the boundary between what we can know about and what we cannot at outside the horizon. Of course, someone unlucky to be inside the event horizon of the black hole can receive news of the outside world in a one-way information transfer.

Note that degenerate objects such as white dwarfs and neutron stars shrink as mass is added, as the increased gravity compresses the matter. The radius of a 1 Msun neutron star is around 10km, while the radius of a 3 Msun neutron star is about 9km. However, the Schwarzschild radius for 1 Msun is 3 km, and thus for 3 Msun it is 9km. Thus, a 3 Msun neutron star would be within its own Schwarzschild radius and would be a black hole (and we would never know whether it had a neutron star structure). This is the Schwarzschild limit for neutron stars. Calculations show masses greater than around 3 Msun will collpase indefinitely, as neutron degeneracy cannot withstand gravity. This is similar to the Chadrasekhar limit for white dwarfs. Barring any other sort of degeneracy or quantum effect, such a mass could collapese to infinite density, but we would never know since it is within the event horizon (unless you would wish to journey there).

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smyers@nrao.edu Steven T. Myers