Neutron Stars and Supernova Remnants. The Crab Nebula is a...

Neutron Stars and Supernova Remnants. The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light-years from the earth (Fig. P9.98). It is the remnant of a star that underwent a supernova explosion, seen on earth in 1054 A.D. Energy is released by the Crab Nebula at a released by the Crab Nebula at a rate of about $5 \times 10^{31} \mathrm{W},$ about $10^{5}$ times the rate at which the sun radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning neutron star at its center. This object rotates once every $0.0331 \mathrm{s},$ and this period is increasing by $4.22 \times 10^{-13} \mathrm{s}$ for each second of time that elapses. (a) If the rate at which energy is lost by the neutron star is equal to the rate at which energy is released by the nebula, find the moment of inertia of the neutron star. (b) Theories of supernovae predict that the neutron star in the Crab Nebula has a mass about 1.4 times that of the sun. Modeling the neutron star as a solid uniform sphere, calculate its radius in kilometers. (c) What is the linear speed of a point on the equator of the neutron star? Compare to the speed of light. (d) Assume that the neutron star is uniform and calculate its density. Compare to the density of ordinary rock $\left(3000 \mathrm{kg} / \mathrm{m}^{3}\right)$ and to the density of an atomic nucleus (about $10^{17} \mathrm{kg} / \mathrm{m}^{3} ) .$ Justify the statement that a neutron star is essentially a large atomic nucleus.

Neutron Stars and Supernova Remnants. The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light-years from the earth (Fig. P9.98). It is the remnant of a star that underwent a supernova explosion, seen on earth in 1054 A.D. Energy is released by the Crab Nebula at a released by the Crab Nebula at a rate of about $5 \times 10^{31} \mathrm{W},$ about $10^{5}$ times the rate at which the sun radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning neutron star at its center. This object rotates once every $0.0331 \mathrm{s},$ and this period is increasing by $4.22 \times 10^{-13} \mathrm{s}$ for each second of time that elapses. (a) If the rate at which energy is lost by the neutron star is equal to the rate at which energy is released by the nebula, find the moment of inertia of the neutron star. (b) Theories of supernovae predict that the neutron star in the Crab Nebula has a mass about 1.4 times that of the sun. Modeling the neutron star as a solid uniform sphere, calculate its radius in kilometers. (c) What is the linear speed of a point on the equator of the neutron star? Compare to the speed of
light. (d) Assume that the neutron star is uniform and calculate its density. Compare to the density of ordinary rock $\left(3000 \mathrm{kg} / \mathrm{m}^{3}\right)$ and to the density of an atomic nucleus (about $10^{17} \mathrm{kg} / \mathrm{m}^{3} ) .$ Justify the statement that a neutron star is essentially a large atomic nucleus.

Key Concepts

Spin-down Rate

The spin-down rate is the rate at which a rotating object, such as a neutron star, loses its angular velocity over time. This gradual slowing of the rotation is due to the loss of rotational kinetic energy, which may be emitted in the form of radiation or particle winds. The spin-down rate provides important insights into the energy loss mechanisms and the evolution of the star's rotation.

Relativistic Speed Considerations

Relativistic speed considerations involve comparing the linear speeds (such as those at the equator of a rapidly rotating neutron star) to the speed of light. Ensuring that these speeds do not exceed the speed of light is important for maintaining consistency with special relativity. This concept is crucial when modeling astrophysical objects to ensure that the derived physical properties are realistic and conform with the known laws of physics.

Density Calculation

Density calculation involves determining the average density of an object by dividing its mass by its volume. For astrophysical objects like neutron stars, calculating the density is crucial because it illustrates just how extreme the conditions are, often comparable to or exceeding that found in atomic nuclei. This comparison helps in understanding the state of matter under such high pressures and densities.

Uniform Sphere Model

The uniform sphere model is a simplification used in astrophysics to approximate the properties of spherical bodies by assuming that mass is evenly distributed. This model allows for straightforward calculations of key physical properties such as moment of inertia, radius, and density, and is particularly useful when detailed internal structure is either unknown or too complex for basic estimates.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion. In the context of astrophysical objects like neutron stars, it relates the mass distribution of the star to its angular velocity and kinetic energy. Calculating the moment of inertia helps in estimating how rotational energy is stored and lost as the star spins down over time.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation. For objects like neutron stars, this energy is significant and can be converted into other forms, powering phenomena such as the radiation observed from supernova remnants. The conservation and loss of rotational kinetic energy are key to understanding the evolution of spinning astrophysical objects.

Supernova Remnants

Supernova remnants are the expanding clouds of gas and dust left behind after a star has exploded in a supernova. These remnants, such as nebulae, continue to emit energy over extended periods and provide clues about the explosion dynamics and the properties of the progenitor star. They also interact with the surrounding interstellar medium, influencing subsequent star formation.

Neutron Stars

Neutron stars are the extremely dense remnants of massive stars that have undergone supernova explosions. They are primarily composed of neutrons and have masses comparable to that of the Sun but packed into a sphere with a radius of only about 10 kilometers. Their extreme density results in intense gravitational fields and unique physical properties that challenge our understanding of matter under extreme conditions.

Transcript

00:01 Hi, everybody.

00:01 So for a is if the rate of the rich energy is lost by neutron star is equal to the rate at which the energy is released by the nebula, final inertia of the neutron star.

00:11 So for this, we have a equals k of connect energy of nebula equals one -half inertia times the rotation velocity equals one -half inertia, two pi over two.

00:30 T squared and this is an equal i 4 pi squared to two two square equals two inertia pi squared t squared which is gonna equal to negative inertia four pi squared t squared t cubed d t over d t and if we have the nebula k and the d t can be four pi i t cubed d t and if we have the inertia it's going to equal to d k over b t and that's nebula t cube over four pi squared um times one over d d bt over bt and let me just get off and if you plug in the numbers that we are given so we have five times 10 to 31 watts equals 0 .031 seconds cubed divided by four pi squared times times 1 times 4 .2 squared times 1 times 4 .22 times 10 to 12 and this one get 1 .09 times 10 to 38 kilograms meters squared.

02:07 Okay.

02:09 And now for p, we're asked theories of supernova, modernly neutron stars star square, kind of like its radius in quantum meters.

02:21 So we have mass equals 1 .4 times an s, which is, 1 .4 times 1 .99 times 10 at 230 kilograms.

02:36 So we're going to keep it like that.

02:38 So inertia equals 2 5mr squared.

02:44 And we solve for r.

02:46 And we get 5l2m .m.

02:52 And this equals 5 times 1 .09 times 10 to 38.

03:05 We need this 38 out of the way.

03:11 Okay, and it's going to be two times 1 .4 times 1 .99 to count for 30 square roots.

03:21 And here we get 98 90 meters.

03:25 Okay, and for c, we need to find the velocity...