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dzizzling
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- Homework Statement
- - I have to determine the density at the centre of the Sun, compare this with the mean density of the core, and to comment on the result.
- I have to determine the density of the white dwarf, Sirius B, whose mass, M = 1 M_solar_mass, and radius, r = 0.008 R_solar_radius, to compare it with the density of the Sun’s core, and to comment on the result.
- I have to determine the density at the surface of the core of the Sun. Using this density i have to calculate the rate of change of pressure with radius at that surface.
- I have to determine the mean energy generation rate per unit mass of the core 𝜀.
- Relevant Equations
- [Task a.]
- The mass distribution equation as ordinary differential equation:
$$\frac{dM(r)}{dr} = 4 \pi r^2 \rho(r)$$
and as integration:
$$m(r) = \int^r_0 4 \pi r^2 \rho dr$$
yields the mass m(r) inside a spherical shell of radius r. This equation describes the mass M(r) contained within a radius r in terms of the mass density ρ (rho).]
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[Task c.]
- The pressure within the Sun is then given by:
$$P(r) = \frac{\rho(r)}{\mu m_H} kT(r)$$
, with
- ρ = 2.45e4 kg m-3, is the density at the surface of the core at R = 0.25 Rsolar_radius
- μ = 0.5, is the mean molecular weight for the gas,
- mH = 1.67e-27 kg, is the mass of a hydrogen atom,
- k = 1.38×10−23 J K-1, is Boltzmann’s constant
- 𝑇(𝑟) = 8e6 K, is the temperature at radius 𝑟,
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[Task d.]
- The equation of stellar structure for the energy generation which is rearranged to the normal form:
$$\frac{dL(r)}{dr} = 4 \pi r^2 \rho(r) \epsilon(r)$$
to get 𝜀 the energy generation per unit mass, with
- 𝐿(𝑟) is the luminosity within the Sun at radius 𝑟,
- r = 0.25 Rsolar_radius is the radius of the solar core,
- dr is the spherical shell where the energy generation happens,
- ρ = 2.45e4 kg m-3, is the density at the surface of the core at R = 0.25 Rsolar_radius].
Assuming the Sun’s core has a mass of 0.35 Msolar_mass and taking values for other quantities from a internet background search or from the following figures
(i.e.: Radius "solar core" = 0.25 Rsolar_radius ):
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a.
Determine the density at the centre of the Sun.
Compare this with the mean density of the core.
Comment on the result.
[Hint 1: Its not about the density at R=0, which may confuse a bit as mass inside a sphere of radius R=0 is of course 0. But density is defined as mass per unit volume: You can imagine a small volume at the centre of the Sun, having mass.]
[Hint 2: You can use an derivative (i.e. an second order differential equation) of one of the "equations of stellar structure" to solve this question.
I.e.: The mass distribution equation as ordinary differential equation
$$\frac{dM(r)}{dr} = 4 \pi r^2 \rho(r)$$
and as integration:
$$m(r) = \int^r_0 4 \pi r^2 \rho dr$$
yields the mass m(r) inside a spherical shell of radius r. This equation describes the mass M(r) contained within a radius r in terms of the mass density ρ (rho).]
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b.
Determine the density of the white dwarf, Sirius B, whose mass, M = 1 Msolar_mass, and radius, r = 0.008 Rsolar_radius.
Compare it with the density of the Sun’s core.
Comment on the result.
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c.
Determine the density at the surface of the core of the Sun.
Using this density, calculate the rate of change of pressure with radius at that surface.
[Hint: The pressure within the Sun is then given by
$$P(r) = \frac{\rho(r)}{\mu m_H} kT(r)$$
, with
- ρ = 2.45e4 kg m-3, is the density at the surface of the core at R = 0.25 Rsolar_radius
- μ = 0.5, is the mean molecular weight for the gas,
- mH = 1.67e-27 kg, is the mass of a hydrogen atom,
- k = 1.38×10−23 J K-1, is Boltzmann’s constant
- 𝑇(𝑟) = 8e6 K, is the temperature at radius 𝑟,
[Hint: You will need to make use of the core density from part (a)]
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d. Determine the mean energy generation rate per unit mass of the core 𝜀.
[Hint:
Using the equation of stellar structure for the energy generation which is rearranged to the normal form
$$\frac{dL(r)}{dr} = 4 \pi r^2 \rho(r) \epsilon(r)$$
to get 𝜀 the energy generation per unit mass, with
- 𝐿(𝑟) is the luminosity within the Sun at radius 𝑟,
- r = 0.25 Rsolar_radius is the radius of the solar core,
- dr is the spherical shell where the energy generation happens,
- ρ = 2.45e4 kg m-3, is the density at the surface of the core at R = 0.25 Rsolar_radius].
(i.e.: Radius "solar core" = 0.25 Rsolar_radius ):
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------------------------------------------------------------------------
a.
Determine the density at the centre of the Sun.
Compare this with the mean density of the core.
Comment on the result.
[Hint 1: Its not about the density at R=0, which may confuse a bit as mass inside a sphere of radius R=0 is of course 0. But density is defined as mass per unit volume: You can imagine a small volume at the centre of the Sun, having mass.]
[Hint 2: You can use an derivative (i.e. an second order differential equation) of one of the "equations of stellar structure" to solve this question.
I.e.: The mass distribution equation as ordinary differential equation
$$\frac{dM(r)}{dr} = 4 \pi r^2 \rho(r)$$
and as integration:
$$m(r) = \int^r_0 4 \pi r^2 \rho dr$$
yields the mass m(r) inside a spherical shell of radius r. This equation describes the mass M(r) contained within a radius r in terms of the mass density ρ (rho).]
------------------------------------------------------------------------
b.
Determine the density of the white dwarf, Sirius B, whose mass, M = 1 Msolar_mass, and radius, r = 0.008 Rsolar_radius.
Compare it with the density of the Sun’s core.
Comment on the result.
------------------------------------------------------------------------
c.
Determine the density at the surface of the core of the Sun.
Using this density, calculate the rate of change of pressure with radius at that surface.
[Hint: The pressure within the Sun is then given by
$$P(r) = \frac{\rho(r)}{\mu m_H} kT(r)$$
, with
- ρ = 2.45e4 kg m-3, is the density at the surface of the core at R = 0.25 Rsolar_radius
- μ = 0.5, is the mean molecular weight for the gas,
- mH = 1.67e-27 kg, is the mass of a hydrogen atom,
- k = 1.38×10−23 J K-1, is Boltzmann’s constant
- 𝑇(𝑟) = 8e6 K, is the temperature at radius 𝑟,
[Hint: You will need to make use of the core density from part (a)]
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d. Determine the mean energy generation rate per unit mass of the core 𝜀.
[Hint:
Using the equation of stellar structure for the energy generation which is rearranged to the normal form
$$\frac{dL(r)}{dr} = 4 \pi r^2 \rho(r) \epsilon(r)$$
to get 𝜀 the energy generation per unit mass, with
- 𝐿(𝑟) is the luminosity within the Sun at radius 𝑟,
- r = 0.25 Rsolar_radius is the radius of the solar core,
- dr is the spherical shell where the energy generation happens,
- ρ = 2.45e4 kg m-3, is the density at the surface of the core at R = 0.25 Rsolar_radius].
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