A Type Ia supernova is powered by a thermonuclear runaway in electron-degenerate white dwarf mat

A Type Ia supernova is powered by a thermonuclear runaway in...

Key Concepts

Thermonuclear Runaway

Thermonuclear runaway refers to a process in which nuclear reactions accelerate uncontrollably as the increased temperature from energy release further enhances reaction rates. In the context of stellar explosions, a subtle perturbation in temperature can lead to a self-amplifying cycle where nuclear burning becomes explosive rather than self-regulated, driving phenomena such as supernovae.

Electron Degeneracy

Electron degeneracy is a quantum mechanical state of matter that occurs under extremely high densities when electrons are squeezed into a limited volume, and their pressure is governed by the Pauli exclusion principle instead of thermal motion. In white dwarfs, this degeneracy pressure supports the star against gravitational collapse, yet it decouples pressure from temperature, thereby playing a crucial role in the onset of thermal instabilities during nuclear burning.

Polytropic Equation of State

A polytropic equation of state is a simplified relationship that connects the pressure and density of a stellar material through a fixed polytropic index, encapsulating the balance between gravitational forces and internal pressure. This model is instrumental in describing various astrophysical objects, including white dwarfs, where it provides insights into their structural properties under conditions dominated by degeneracy pressure.

Lane-Emden Equation

The Lane-Emden equation arises in astrophysics as a dimensionless form of the equations of hydrostatic equilibrium for a polytropic fluid sphere. It enables the determination of the density and pressure profiles within stars and is critical for analyzing central conditions where instabilities might trigger runaway phenomena, thereby linking stellar structure to dynamical behavior.

Stability Analysis in Stellar Interiors

Stability analysis involves assessing how small perturbations in physical parameters like temperature and density evolve over time within a stellar interior. By studying the coupled variations and ensuring that changes in density and temperature propagate with the same sign, one can determine whether the system will return to equilibrium or diverge into a runaway process, which is essential for understanding explosive events such as supernovae.

Generic Equation of State with Temperature and Density Sensitivities

The generic equation of state that relates fractional changes in pressure to those in density and temperature, characterized by non-negative parameters, illustrates how pressure responds to thermodynamic variations. This framework is crucial in assessing whether a system stabilizes against perturbations or experiences runaway instability, especially in contexts where nuclear burning and electron degeneracy interact.

Transcript

00:01 Hi there, so for this problem, we are told that a start that is too heavy to stabilize as a white dwarf can collapse further to form a neutron star.

00:10 That is a start made internally of neutrons supported against gravitational collapse by degenerate neutrons pressure.

00:19 So we need to repeat the steps of the previous problem for the neutron star to determine the following.

00:25 The mass radius relation.

00:28 The radius, density, firm it, energy, and fermin temperature of a one solar mass neutron star and the critical mass above which a neutron star becomes relativistic and hence unstable to further collapse.

00:46 So in a neutron star, the kinetic energy comes from the neutrons, and the number of this is simply the mass of the entire star divided by the number the mass of neutrons in that star, where m, of course, is the total mass, and mn is the mass of a neutron.

01:11 So we can write the kinetic energy as to follow.

01:15 The kinetic energy, in this case, then is equal to 3 over 5 times n, the number n of neutrons, times the fermi energy.

01:30 And now we can write this as 3 over 5, and we write the expression for the fermi energy that we know is plans constant to the square divided by n 8 times the mass of a neutron and this times three times the number and divided by pi times the volume and this elevated to 2 over 3 now we can simplify this expression as follows we can simplify this as 3 times plumps constant to the square divided by 40 times the mass of a neutron, and this times the mass of the star, the total mass, divided by the mass of a neutron, that elevated to 5 over 3, and this times 9, divided by 4 times pi to the square times the radius of the star to the 3, and this elevated to 3 to 2 over 3.

02:30 So that's the expression that we have in here for the kinetic energy.

02:41 Now, so adding the negative gravitational potential energy, we have for the total energy.

02:52 So the total energy is equal to the potential energy plus the kinetic energy.

02:57 So in here we will find that this is minus.

03:02 And we're going to call this alpha divided by the radius, plus beta divided by the radius square, where we are going to define what is gamma and what is beta.

03:15 So alpha, sorry, alpha is equal to 3 over 5 times the gravitational constant g times the mass of the star is square.

03:26 Meanwhile, beta, and from the previous expression, we can deduce that this is equal to 0 .028 times the, plums constant to the square times the mass of the star elevated to 5 over 3, and these divided by the mass of a neutron elevated to 8 over 3.

03:50 So as with a white dwarf star, the equilibrium radius is the one that minimizes the total energy.

04:00 So we're going to set the derivative of the total energy with respect to the radius is equal to 0, and we are going to solve for the radius to find the maximum.

04:12 So it's quite easy to obtain from there that when we derive by this expression and we set that equal to zero, we obtained that the radius is equal to 2 times beta divided by alpha.

04:26 So if we put all of this expression in here, we will have that this is 0 .093, and this times 0 .093.

04:36 Three, and these times plums constant to the square divided by the gravitational constant times the mass of an electron elevated to 8 over 3 times the mass of the start elevated to 1 over 3.

04:53 And here again, the equilibrium radius increases with increasing mass due to the greater gravitational attraction.

05:02 For a one solar mass neutron, this model predicts the following motor produced.

05:07 So we just simply substitute the mass of the sun.

05:15 So we will have.

05:16 In the denominator we have the gravitational constant that we know is 6 .67 times 10 to the minus 11, newton's time meters squared divided by kilograms square...

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