Load libraries and set up environment


```python
import sympy
sympy.init_printing()
from sympy.physics.units import kelvin, cm, second, gram, kelvin, speed_of_light, boltzmann, kilogram, meter
import scipy.constants
erg = gram*(cm/second)**2
```

In this notebook we calculate the cooling time of the solar corona. The typical temperature is $10^7 \, \rm K$ and the typical density is $10^{15} \, \rm cm^{-3}$ (according to the [Wikipedia entry](https://en.wikipedia.org/wiki/Corona)). The energy density is given by


```python
n = sympy.Symbol('n', positive=True) # Number density
k = sympy.Symbol('k', positive=True) # Boltzmann constant
T = sympy.Symbol('T', positive=True) # Temperature
energy_density = n*k*T
energy_density
```

Assuming Bremsstrahlung is the dominant emission mechanism, the emissivity is given by


```python
alpha = sympy.Symbol('alpha', positive=True) # Fine structure constant
m_e = sympy.Symbol('m_e', positive=True) # Electron mass
c = sympy.Symbol('c', positive=True) # Speed of light
r_e = sympy.Symbol('r_e', positive=True) # Classical electron radius
bs_emissivity = alpha*(m_e*c**2/(r_e**3*(r_e/c)))*(n*r_e**3)**2*sympy.sqrt(k*T/m_e/c**2)
bs_emissivity
```

Finally, the cooling time is given by


```python
cooling_time = (energy_density/ bs_emissivity).simplify()
cooling_time
```

Substituting numbers


```python
cooling_time.subs({
    T: 1e10*kelvin,
    k: boltzmann.convert_to(erg/kelvin),
    c: speed_of_light.convert_to(cm/second),
    n: 1e15/cm**3,
    r_e: scipy.constants.physical_constants['classical electron radius'][0]*meter.convert_to(cm),
    m_e: scipy.constants.electron_mass*kilogram.convert_to(gram),
    alpha: scipy.constants.alpha
})
```

So the timescale is of the order of a minute

653fef0b375f99baccc40f62abd2dd7d5012b569
ipynb
Jupyter Notebook
solar_corona_cooling_time.ipynb
bolverk/cepheid_x_ray
3400d7ea0a3b5578e3ad43b0d32437e6a4518b09
MIT
solar_corona_cooling_time.ipynb
bolverk/cepheid_x_ray
3400d7ea0a3b5578e3ad43b0d32437e6a4518b09
MIT
2017-11-08T20:46:51.000Z
2017-11-09T16:43:20.000Z
solar_corona_cooling_time.ipynb
bolverk/cepheid_x_ray
3400d7ea0a3b5578e3ad43b0d32437e6a4518b09
MIT
Qwen/Qwen-72B
1. YES
2. YES