10.3 Stellar Evolution, Very Briefly
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star is formed. A neutron star is even smaller and denser than a white dwarf, of order
10 km in radius, and with roughly nuclear density, 10
14 times water. Many neutron
stars have been observed as pulsars and all have masses greater than or about the
Chandrasekhar limit. Pulsars are neutron stars that may spin at rates of up to about
10
3 Hz and emit electromagnetic radiation in regularly spaced pulses. The quantity
2m/r may be of order 1/10 for a neutron star, indicating a much stronger gravitational
field than occurs in the solar system. The Hulse-Taylor binary pulsar system PSR
1913 + 16 mentioned in Chap. 9 is a pulsar in orbit with a companion neutron star;
the companion does not emit radiation in our direction (Hulse 1975).
A very heavy star will emit radiation for a still shorter time, and may also undergo
a supernova explosion. If the core remnant of the explosion is sufficiently massive
however it cannot form a neutron star. There is an upper limit to the mass of a
neutron star, called the Tolman-Oppenheimer-Volkoff or TOV limit, analogous to
the Chandrasekhar limit for a white dwarf (Tolman 1939). The numerical value of
the TOV limit is not as precisely known as the Chandrasekhar limit, but it is thought
to be roughly two or three solar masses. The uncertainty is due mostly to lack of
knowledge of the equation of state of the neutron fluid and the effect of rotation
in the star. For a stellar remnant heavier than the TOV limit the internal pressure
cannot support it against gravity and it shrinks to smaller and smaller size, until it
finally approaches the Schwarzschild radius. The collapse of the remnant towards the
Schwarzschild radius is thus somewhat like the fall of a particle onto the surface of a
black hole that we studied in a previous section; it continues forever as viewed by an
outside observer and the surface approaches the Schwarzschild radius asymptotically.
In the final stage of the collapse the light from the surface of the star is redshifted
to longer and longer wavelengths, and finally, according to theory, the star vanishes
as an invisible black hole (Wiki NS). We will pursue black hole formation further in
the next section.
10.4 Collapse of a Dust Star
For a sufficiently heavy stellar remnant internal pressure cannot halt the collapse to a
black hole. In order to understand the process qualitatively we will make the drastic
approximation of ignoring pressure entirely. The stellar model with no pressure is
often called a dust ball or dust star. It will give us a rough qualitative picture of what
happens in the collapse of a real star, and is an easy theoretical exercise. In fact we
have already done all the mathematics needed and only some additional words are
required.
Specifically, our model is a spherically symmetric ball of dust or gas with negligible internal pressure, which therefore collapses under the influence of gravity. This
is illustrated in Fig. 10.5.
It is important to emphasize that we do not need to know the metric in the interior
to understand the exterior, only that the exterior metric is Schwarzschild.
149
star is formed. A neutron star is even smaller and denser than a white dwarf, of order
10 km in radius, and with roughly nuclear density, 10
14 times water. Many neutron
stars have been observed as pulsars and all have masses greater than or about the
Chandrasekhar limit. Pulsars are neutron stars that may spin at rates of up to about
10
3 Hz and emit electromagnetic radiation in regularly spaced pulses. The quantity
2m/r may be of order 1/10 for a neutron star, indicating a much stronger gravitational
field than occurs in the solar system. The Hulse-Taylor binary pulsar system PSR
1913 + 16 mentioned in Chap. 9 is a pulsar in orbit with a companion neutron star;
the companion does not emit radiation in our direction (Hulse 1975).
A very heavy star will emit radiation for a still shorter time, and may also undergo
a supernova explosion. If the core remnant of the explosion is sufficiently massive
however it cannot form a neutron star. There is an upper limit to the mass of a
neutron star, called the Tolman-Oppenheimer-Volkoff or TOV limit, analogous to
the Chandrasekhar limit for a white dwarf (Tolman 1939). The numerical value of
the TOV limit is not as precisely known as the Chandrasekhar limit, but it is thought
to be roughly two or three solar masses. The uncertainty is due mostly to lack of
knowledge of the equation of state of the neutron fluid and the effect of rotation
in the star. For a stellar remnant heavier than the TOV limit the internal pressure
cannot support it against gravity and it shrinks to smaller and smaller size, until it
finally approaches the Schwarzschild radius. The collapse of the remnant towards the
Schwarzschild radius is thus somewhat like the fall of a particle onto the surface of a
black hole that we studied in a previous section; it continues forever as viewed by an
outside observer and the surface approaches the Schwarzschild radius asymptotically.
In the final stage of the collapse the light from the surface of the star is redshifted
to longer and longer wavelengths, and finally, according to theory, the star vanishes
as an invisible black hole (Wiki NS). We will pursue black hole formation further in
the next section.
10.4 Collapse of a Dust Star
For a sufficiently heavy stellar remnant internal pressure cannot halt the collapse to a
black hole. In order to understand the process qualitatively we will make the drastic
approximation of ignoring pressure entirely. The stellar model with no pressure is
often called a dust ball or dust star. It will give us a rough qualitative picture of what
happens in the collapse of a real star, and is an easy theoretical exercise. In fact we
have already done all the mathematics needed and only some additional words are
required.
Specifically, our model is a spherically symmetric ball of dust or gas with negligible internal pressure, which therefore collapses under the influence of gravity. This
is illustrated in Fig. 10.5.
It is important to emphasize that we do not need to know the metric in the interior
to understand the exterior, only that the exterior metric is Schwarzschild.