consider a polytrope with index gamma65 the stellar radius is infinite but the mass is finite do

Name: consider a polytrope with index gamma65 the stellar radius is infinite but the mass is finite does such a star have a finite moment of inertia if yes calculate it as a function of the centra 73314
Uploaded: 2022-09-25T16:12:24-08:00
Duration: 2 min 33 s
Channel: Penny Riley
Description: consider a polytrope with index gamma65 the stellar radius is infinite but the mass is finite does such a star have a finite moment of inertia if yes calculate it as a function of the centra 73314

Step 1: Understand the properties of a polytrope with index (Gamma = frac{6}{5}). In a polyt

Consider a polytrope with index gamma65 the stellar radius...

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7. (a) It is known that for a star described by a polytropic relation P = Kρ^γ, where P is the pressure, ρ is density, K and γ are constants, the following identity holds K = N_n G M^{(n-1)/n} R^{(3-n)/n}, where n = 1/(γ − 1) is a polytropic index, G is the universal gravitational constant, N_n is a known constant depending only on the polytropic index, and M, R are stellar mass and radius respectively. Use this information to answer the following questions. (i) Consider a white dwarf star dominated by non-relativistic fully degenerate electron gas described by the equation of state P = K_1 (ρ/μ_e)^{5/3}, where μ_e is mean molecular weight per electron (assumed constant) and K_1 is a known physical constant. Use the formula above to write down the radius of such a star as a function of mass. If another white dwarf star with the same chemical composition and equation of state has a higher mass, would the radius decrease, increase or stay the same? Explain your reasoning. [4 marks] (ii) Consider a white dwarf star dominated by extremely relativistic fully degenerate electron gas described by the equation of state P = K_2 (ρ/μ_e)^{4/3}, where μ_e is mean molecular weight per electron (assumed constant) and K_2 is a known physical constant. Use the formula above to find the relationship between the mass and the radius in this case. [3 marks]

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In a polytropic process $p V^{n}=$ constant., where $n$ is called the polytropic index. For this process $$ \begin{gathered} p V^{n}=\text { constant or } T V^{n-1}=\text { constant } \\ \frac{d T}{T}+(n-1) \frac{d V}{V}=0 \end{gathered} $$ Then $$ \begin{gathered} d Q=C d T=d U+p d V=C_{V} d T+p d V \\ =\frac{i}{2} R d T+\frac{R T}{V} d V=\frac{i}{2} R d T-\frac{1}{n-1} R d T=\left(\frac{i}{2}-\frac{1}{n-1}\right) R d T \\ C=R \text { so } \frac{i}{2}-\frac{1}{n-1}=1 \end{gathered} $$ Now or, $$ \frac{1}{n-1}=\frac{i}{2}-1=\frac{i-2}{2} \text { or } n=\frac{i}{i-2} $$ Now $$ \frac{v^{\prime}}{v}=\frac{n^{\prime}}{n} \frac{\left\langle v^{\prime}>\right.}{\langle\nu>}=\frac{1}{\eta} \sqrt{\frac{T}{T}}=\frac{1}{\eta}\left(\frac{V}{V^{\prime}}\right)^{\frac{n-1}{2}} $$ $$ =\frac{1}{\eta}\left(\frac{1}{\eta}\right)^{\frac{1}{i-2}}=\left(\frac{1}{\eta}\right)^{\frac{i-1}{i-2}}=\eta^{\frac{-i-1}{i-2}} \text { times }=\frac{1}{2 \cdot 52} \text { times } $$

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Transcript

00:01 We like to solve the equations for the pressure profile.

00:07 So for part a, we know that our gravitationally force inward is gm of r times m divided by r squared.

00:17 But we can rewrite this as gm of r, row a, dr divided by r squared.

00:24 Now we can set, for a star in equilibrium, this piece plus adp is equal to zero.

00:31 So, dp d r is equal to negative g m of r row over r squared...

Question

Consider a polytrope with index Gamma=6/5 the stellar radius is infinite, but the mass is finite. Does such a star have a finite moment of inertia? If yes, calculate it as a function of the central pressure. If no, prove it.

Question

Consider a polytrope with index Gamma=6/5 the stellar radius is infinite, but the mass is finite. Does such a star have a finite moment of inertia? If yes, calculate it as a function of the central pressure. If no, ﻿prove it.

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Consider a polytrope with index Gamma=6/5 the stellar radius is infinite, but the mass is finite. Does such a star have a finite moment of inertia? If yes, calculate it as a function of the central pressure. If no, prove it.