Degenerate Pressure in Plasmas

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Questions and Answers

Match the plasma properties with their corresponding equations:

Density of plasma = $\rho = \frac{N}{V}m$ Pressure of plasma = $P = \frac{\rho kT}{m_p + m_e}$ Temperature in the core of the Sun = $T = \frac{m_p P}{2kM}\frac{4}{3}\pi R^3$ Average mass of particles = $m = \frac{m_p + m_e}{2}$

Match the constants with their usage in plasma equations:

k = Boltzmann constant m_p = Mass of a proton m_e = Mass of an electron M = Mass of the Sun

Match the variables with their corresponding units:

P = Pascals T = Kelvin m_p = Kilograms V = Cubic meters

Match the plasma components with their relative abundance in the Sun:

Protons = Most abundant Electrons = Most abundant Neutrons = Less abundant Alpha particles = Least abundant Signup and view all the answers

Match the temperature values with their corresponding descriptions:

4.4 x 10^7 K = Rough estimate of the temperature in the core of the Sun 1.5 x 10^7 K = Accurate model estimate of the temperature in the core of the Sun 300 K = Temperature at the surface of the Sun 5000 K = Temperature in the Sun's corona Signup and view all the answers

Match the mathematical symbols with their usage in plasma equations:

N = Number of particles V = Volume of the plasma R = Radius of the Sun π = Mathematical constant Signup and view all the answers

Match the following characteristics with the corresponding feature of the degenerate pressure in a plasma:

Independent of temperature = Feature of degenerate pressure Dependent on the inverse of the mass of the particle = Feature of degenerate pressure Dependent on the density as ρ2 = Not a feature of degenerate pressure Dependent on the thermal pressure = Not a feature of degenerate pressure Signup and view all the answers

Match the following expressions with their physical quantities:

𝑛𝑒 ≅ 𝑍𝑛𝑍 ≅ 𝑍/𝐴𝑚𝑝 = Density of free electrons ℎ2/8𝜋 𝑚𝑒 = Constant in the degenerate pressure equation 0.05 ℎ2 𝑍 5/3 𝜌 / 𝑚𝑒 𝐴 = Approximation in the degenerate pressure equation 𝜌5/3 = Density dependence in the degenerate pressure equation Signup and view all the answers

Match the following types of stars with their corresponding characteristics:

White dwarfs = Completely degenerate stars Neutron stars = Stars with high thermal pressure Main sequence stars = Stars with low density and high temperature Black dwarfs = Stars with zero luminosity Signup and view all the answers

Match the following conditions with their corresponding type of gas:

𝑛𝑒 ≈ 𝑛𝑄 = Degenerate gas 𝑛𝑒 << 𝑛𝑄 = Non-degenerate gas 𝑛𝑒 > 𝑛𝑄 = Partially degenerate gas 𝑛𝑒 = 0 = Ideal gas Signup and view all the answers

Match the following values with their corresponding physical quantities:

1.4 g cm-3 = Average density in the Sun 4.5 MK = Average temperature in the Sun 150 g cm-3 = Density in the core of the Sun 0.5 = Ratio of Z to A per solar nucleus Signup and view all the answers

Match the following expressions with their physical meanings:

𝑛𝑄 ≈ (4𝜋 2 𝑚𝑒 𝑘𝑇 2) / ℎ2 = Density of quantum states 𝑛𝑒 ≈ 𝑍 𝜌 / 𝐴 𝑚𝑝 = Density of free electrons 𝑃 = (1/5) (ℎ2/8𝜋) (5/3) 𝑛𝑒 5/3 / 𝑚𝑒 = Degenerate pressure equation 𝜌5/3 = Density dependence in the thermal pressure Signup and view all the answers

Match the following types of particles with their corresponding roles in the degenerate pressure:

Electrons = Main contributors to the degenerate pressure Protons = Main contributors to the thermal pressure Neutrons = Inert particles in the degenerate pressure Positrons = Antiparticles in the degenerate pressure Signup and view all the answers

Match the following conditions with their corresponding types of stars:

High density and low temperature = White dwarfs High temperature and low density = Main sequence stars Low density and high temperature = Red giants High density and high temperature = Neutron stars Signup and view all the answers

Match the given equations with their corresponding physical concepts:

dFG = −G M(r)ρ(r)dr dA / r2 = Gravity force on the small cylinder dFP = −dPdA = Force due to pressure unbalance dP/dR = −G M(r)ρ(r) / r2 = Equation of hydrostatic equilibrium PV = NkT = Ideal gas law Signup and view all the answers

Match the variables with their corresponding physical quantities:

M(r) = Mass of the small cylinder ρ(r) = Density of the fluid dA = Cross-sectional area of the small cylinder P(r) = Pressure of the fluid at radius r Signup and view all the answers

Match the expressions with their corresponding physical concepts:

dFG + dFP = 0 = Equilibrium condition dP/dR = −G M(r)ρ(r) / r2 = Hydrostatic equilibrium PV = NkT = Ideal gas law dP = −G M(r)ρ(r) / r2 = Pressure gradient Signup and view all the answers

Match the given physical quantities with their corresponding units:

Pressure = Pa Density = kg/m³ Mass = kg Gravity = N/kg Signup and view all the answers

Match the expressions with their corresponding physical concepts in the context of the Sun:

dP/PC ≈ M / R³ = Pressure in the core of the Sun PC ≈ G M² / R⁴ = Pressure in the core of the Sun dP/dr = −G M(r)ρ(r) / r2 = Hydrostatic equilibrium PV = NkT = Ideal gas law Signup and view all the answers

Match the variables with their corresponding physical quantities in the context of the Sun:

R = Radius of the Sun M = Mass of the Sun PC = Pressure in the core of the Sun <ρ> = Average density of the Sun Signup and view all the answers

Match the equations with their corresponding applications:

dFG = −G M(r)ρ(r)dr dA / r2 = Gravity force on a small cylinder dP/dr = −G M(r)ρ(r) / r2 = Pressure in the core of the Sun PV = NkT = Ideal gas law for a non-relativistic plasma dFP = −dPdA = Force due to pressure unbalance Signup and view all the answers

Match the physical quantities with their corresponding effects on the pressure gradient:

Density = Increases the pressure gradient Gravity = Increases the pressure gradient Mass = Decreases the pressure gradient Radius = Decreases the pressure gradient Signup and view all the answers

Study Notes

Degenerate Pressure

For a plasma, which is a highly ionized gas, the degenerate pressure is given by the formula: P = (ℎ² / (5 * 8π * me)) * (ne^(5/3) / (Am*p))
The degenerate pressure does not depend on the temperature.
It depends on the inverse of the mass of the particle, specifically 1/me, which means that electrons are the main contributors to degenerate pressure.
The pressure depends on the density as ρ^(5/3), leading to completely degenerate stars, like white dwarfs, at high density and low temperature.

Degeneracy Condition

Electrons in a plasma are degenerate if the density of free electrons (ne) is greater than or equal to the density of quantum states (nQ).
The condition for degeneracy is: ne ≥ nQ = (4π / (3 * ℎ²)) * (me * kT)^(3/2)

Example: Sun's Core

In the Sun's core, the density (ρ) is approximately 150 g/cm³, and the temperature (T) is around 15 MK.
The density of free electrons (ne) is approximately 4.6 x 10²⁵ cm⁻³, which is less than the density of quantum states (nQ), making the gas non-degenerate.

Equation of State

The equation of state for an ideal gas is: PV = NkT, where N is the total number of particles, k is the Boltzmann constant, and T is the temperature.
This approximation is generally good for non-relativistic plasmas.

Hydrostatic Equilibrium

The gravity force acting on a small cylinder in a plasma is given by: dFG = -G * M(r) * dm / r²
The force due to the pressure unbalance is given by: dFP = -dP * dA
At equilibrium, the sum of these forces is zero, leading to the equation of hydrostatic equilibrium: dP / dr = -G * M(r) * ρ(r) / r²

Core Pressure

The approximate pressure in the core of the Sun can be calculated using the equation: PC ≈ (G * M²) / (4 * R)
This gives a pressure of approximately 10¹⁵ Pa in the Sun's core.