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2. Temperature in the Sun's Interior We calculated in lecture that nuclear fusion of hydrogen can only occur at temperatures of ~10$^7$ K due to the Coulomb barrier (even after taking quantum tunneling into account), but does the temperature in the Sun's interior actually reach this high? How can we calculate this? (a) Use the virial theorem to estimate an average temperature of the Sun in Kelvin. Note that this is an average over all parcels of mass in the Sun, some of which have far higher temperatures, and some of which have far lower temperatures. Since the surface of the Sun is 5800 K (which can be calculated with Wien's displacement law using the blackbody spectrum of the Sun), what does this imply about temperatures in the Sun's core? (b) We can also estimate an average temperature of the Sun in another slightly different way: assume the Sun has a uniform pressure, density, and temperature. Although the Sun would not actually be in hydrostatic equilibrium if this were true, this simple approach neglecting all structure is called a 'one-zone model', and is a commonly-used approximation in astrophysics. Argue that in such a scenario, the $dP/dr$ equation for hydrostatic equilibrium can be reduced to: $\frac{P}{R} = \frac{GM\rho}{R^2}$ Combine this with the ideal gas law to estimate the average temperature in Kelvin.

          2. Temperature in the Sun's Interior
We calculated in lecture that nuclear fusion of hydrogen can only occur at temperatures of ~10$^7$ K
due to the Coulomb barrier (even after taking quantum tunneling into account), but does the
temperature in the Sun's interior actually reach this high? How can we calculate this?
(a) Use the virial theorem to estimate an average temperature of the Sun in Kelvin. Note that this
is an average over all parcels of mass in the Sun, some of which have far higher temperatures, and
some of which have far lower temperatures. Since the surface of the Sun is 5800 K (which can be
calculated with Wien's displacement law using the blackbody spectrum of the Sun), what does this
imply about temperatures in the Sun's core?
(b) We can also estimate an average temperature of the Sun in another slightly different way:
assume the Sun has a uniform pressure, density, and temperature. Although the Sun would not
actually be in hydrostatic equilibrium if this were true, this simple approach neglecting all structure
is called a 'one-zone model', and is a commonly-used approximation in astrophysics. Argue that in
such a scenario, the $dP/dr$ equation for hydrostatic equilibrium can be reduced to:
$\frac{P}{R} = \frac{GM\rho}{R^2}$
Combine this with the ideal gas law to estimate the average temperature in Kelvin.

2. Temperature in the Sun's Interior
We calculated in lecture that nuclear fusion of hydrogen can only occur at temperatures of 10^7 K
due to the Coulomb barrier (even after taking quantum tunneling into account), but does the
temperature in the Sun's interior actually reach this high? How can we calculate this?
(a) Use the virial theorem to estimate an average temperature of the Sun in Kelvin. Note that this
is an average over all parcels of mass in the Sun, some of which have far higher temperatures, and
some of which have far lower temperatures. Since the surface of the Sun is 5800 K (which can be
calculated with Wien's displacement law using the blackbody spectrum of the Sun), what does this
imply about temperatures in the Sun's core?
(b) We can also estimate an average temperature of the Sun in another slightly different way:
assume the Sun has a uniform pressure, density, and temperature. Although the Sun would not
actually be in hydrostatic equilibrium if this were true, this simple approach neglecting all structure
is called a 'one-zone model', and is a commonly-used approximation in astrophysics. Argue that in
such a scenario, the dP/dr equation for hydrostatic equilibrium can be reduced to:
(P)/(R) = (GMρ)/(R^2)
Combine this with the ideal gas law to estimate the average temperature in Kelvin.

Added by Jose Angel S.

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