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d. Now, the key assumption of the Eddington model is that the product $\kappa_R(r)\eta(r)$ varies much less rapidly with $r$ than either the opacity or the luminosity-to-mass ratio themselves. This is a reasonable assumption since the dominant opacity in the interior of low mass stars follows the Kramers form $\kappa_R \propto \rho T^{-3.5}$, and because the density generally decreases less rapidly than $T^{3.5}$, the opacity typically increases with increasing radius. Similarly, seeing as the luminosity reaches a fixed value $L$ outside of the nuclear-burning core, the normalized luminosity-to-mass ratio $\eta(r)$ decreases with increasing mass ($\eta(r) \propto L(r)/M(r)$) and radius outside the core. The product $\kappa_R(r)\eta(r)$ should therefore be a weak function of radius, as can be confirmed by the analysis of more realistic stellar models -- see figure 1. Make the assumption that ($\kappa_R(r)\eta(r)$) is constant throughout the star, and show that this implies that the ratio of radiation to total pressure is a constant, $1 - \beta$, at all radii, $1 - \beta = \frac{P_{rad}(r)}{P(r)}$ (14) where $\beta$ is defined to be the ratio of gas pressure to total pressure, $P_{gas}/P$. Solve for $\beta$ in terms of $L$, $M$, and ($\kappa_R(r)\eta(r)$). e. Beginning with the total pressure $P = P_{gas} + P_{rad}$, show that the constancy of the ratio of gas and radiation pressure to total pressure implies that the total pressure obeys an $n = 3$ polytropic pressure relationship, $P = K\rho^{4/3}$, where $K$ is given by $K = \left[\left(\frac{N_A k}{\mu}\right)^4 \frac{31 - \beta}{a \beta^4}\right]^{1/3}$ (15)

          d. Now, the key assumption of the Eddington model is that the product $\kappa_R(r)\eta(r)$ varies
much less rapidly with $r$ than either the opacity or the luminosity-to-mass ratio themselves.
This is a reasonable assumption since the dominant opacity in the interior of low mass
stars follows the Kramers form $\kappa_R \propto \rho T^{-3.5}$, and because the density generally decreases less
rapidly than $T^{3.5}$, the opacity typically increases with increasing radius. Similarly, seeing as
the luminosity reaches a fixed value $L$ outside of the nuclear-burning core, the normalized
luminosity-to-mass ratio $\eta(r)$ decreases with increasing mass ($\eta(r) \propto L(r)/M(r)$) and radius
outside the core. The product $\kappa_R(r)\eta(r)$ should therefore be a weak function of radius, as
can be confirmed by the analysis of more realistic stellar models -- see figure 1.
Make the assumption that ($\kappa_R(r)\eta(r)$) is constant throughout the star, and show that this
implies that the ratio of radiation to total pressure is a constant, $1 - \beta$, at all radii,
$1 - \beta = \frac{P_{rad}(r)}{P(r)}$
(14)
where $\beta$ is defined to be the ratio of gas pressure to total pressure, $P_{gas}/P$. Solve for $\beta$ in
terms of $L$, $M$, and ($\kappa_R(r)\eta(r)$).
e. Beginning with the total pressure $P = P_{gas} + P_{rad}$, show that the constancy of the ratio
of gas and radiation pressure to total pressure implies that the total pressure obeys an $n = 3$
polytropic pressure relationship, $P = K\rho^{4/3}$, where $K$ is given by
$K = \left[\left(\frac{N_A k}{\mu}\right)^4 \frac{31 - \beta}{a \beta^4}\right]^{1/3}$ (15)

d. Now, the key assumption of the Eddington model is that the product (r)η(r) varies
much less rapidly with r than either the opacity or the luminosity-to-mass ratio themselves.
This is a reasonable assumption since the dominant opacity in the interior of low mass
stars follows the Kramers form ∝ρ T^-3.5, and because the density generally decreases less
rapidly than T^3.5, the opacity typically increases with increasing radius. Similarly, seeing as
the luminosity reaches a fixed value L outside of the nuclear-burning core, the normalized
luminosity-to-mass ratio η(r) decreases with increasing mass (η(r) ∝ L(r)/M(r)) and radius
outside the core. The product (r)η(r) should therefore be a weak function of radius, as
can be confirmed by the analysis of more realistic stellar models – see figure 1.
Make the assumption that ((r)η(r)) is constant throughout the star, and show that this
implies that the ratio of radiation to total pressure is a constant, 1 - β, at all radii,
1 - β = (Prad(r))/(P(r))
(14)
where β is defined to be the ratio of gas pressure to total pressure, Pgas/P. Solve for β in
terms of L, M, and ((r)η(r)).
e. Beginning with the total pressure P = Pgas + Prad, show that the constancy of the ratio
of gas and radiation pressure to total pressure implies that the total pressure obeys an n = 3
polytropic pressure relationship, P = Kρ^4/3, where K is given by
K = [((NA k)/(μ))^4 (31 - β)/(a β^4)]^1/3 (15)

Added by Margaret A.

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