(a) Given that the mass of the sun is 2 ×10^33 g, estimate the number of electrons in the sun.

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(a) Given that the mass of the sun is 2 ×10^33 g, estimate...

(a) Given that the mass of the sun is $2 \times 10^{33} \mathrm{~g}$, estimate the number of electrons in the sun. Assume the sun is largely composed of atomic hydrogen. (b) In a white dwarf star of one solar mass the atoms are all ionized and contained in a sphere of radius $2 \times 10^9 \mathrm{~cm}$. Find the Fermi energy of the electrons in eV . (c) If the temperature of the white dwarf is $10^7 \mathrm{~K}$, discuss whether the electrons and/or nucleons in the star are degenerate. (d) If the above number of electrons were contained in a pulsar of one solar mass and of radius 10 km , find the order of magnitude of their Fermi energy.

(a) Given that the mass of the sun is $2 \times 10^{33} \mathrm{~g}$, estimate the number of electrons in the sun. Assume the sun is largely composed of atomic hydrogen.
(b) In a white dwarf star of one solar mass the atoms are all ionized and contained in a sphere of radius $2 \times 10^9 \mathrm{~cm}$. Find the Fermi energy of the electrons in eV .
(c) If the temperature of the white dwarf is $10^7 \mathrm{~K}$, discuss whether the electrons and/or nucleons in the star are degenerate.
(d) If the above number of electrons were contained in a pulsar of one solar mass and of radius 10 km , find the order of magnitude of their Fermi energy.

Key Concepts

Scaling Relations in Astrophysical Systems

Scaling relations help in understanding how various stellar properties such as mass, volume (or radius), and density interrelate. These relations are used to estimate key parameters like electron number density and Fermi energy in different astrophysical objects, from diffuse stars to compact objects like white dwarfs and pulsars.

Stellar Composition and Particle Number Calculation

This concept involves converting the total mass of a star into the number of particles by using typical atomic masses. For stars primarily composed of hydrogen, the mass of an individual proton is used to estimate the total number of particles, such as electrons, present in the star. This conversion is fundamental in astrophysics for relating macroscopic masses to microscopic particle counts.

Degenerate Fermion Gas

In extremely dense stellar objects like white dwarfs and pulsars, the electrons are forced into a state where quantum effects dominate and they behave as a degenerate Fermi gas. In this state, the electrons fill energy levels up to a highest level known as the Fermi energy, even at zero temperature, due to the Pauli exclusion principle, which prevents them from occupying the same quantum state.

Fermi Energy

The Fermi energy is a measure of the energy of the highest occupied quantum state in a system of fermions at absolute zero. It is crucial in understanding the pressure and thermal properties in degenerate matter. Calculation of the Fermi energy depends on the number density of fermions and provides insight into how the electrons' kinetic energy supports the star against gravitational collapse.

Stellar Degeneracy

Stellar degeneracy refers to the condition in which particles such as electrons in a star are so densely packed that quantum mechanical effects, rather than thermal pressure, dominate the pressure support. This concept explains why, in objects like white dwarfs or neutron stars, the thermal energy can be much lower than the Fermi energy, indicating that the degenerate particles are largely insensitive to temperature changes.

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