Astronomy - Greenhouse Effect (integral troubles mainly)

In summary, the conversation discusses the use of equations to find the temperature at the photosphere using the optical depth measured at ground level. The first equation relates the temperature gradient to the flux of radiation, while the second equation defines the optical depth. The equation for radiation flux is also provided, and it is used to solve for the temperature at the photosphere. The conversation ends with a request for further clarification on how to proceed with the given information.
  • #1
NIQ
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Given equations:
[tex]\frac{d T^4}{dr} = - \frac{3 \kappa \rho}{a c} F_{rad}[/tex]
where [tex]a = 4 \sigma_{B} / c[/tex] with [tex]\sigma_{B}[/tex] being the Boltzmann constant.
Also, define the optical depth [tex]\tau = \int \kappa \rho dr[/tex]
Optical depth measured at ground level is [tex]\tau_{g}[/tex]
Where [tex]\tau_{g} = \int^\infty_{ground} \kappa \rho dr[/tex] and the optical depth at the photosphere equals 2/3
Also [tex]F_{rad} = \frac {L_{r}}{4 \pi r^2} = \frac {A \sigma_{B} T_{g}^{4}}{4 \pi r^2} = \sigma_{B} T_{g}^{4}[/tex]
Now, I have to use the first equation given to find:
[tex]T^4_{g} = T^4_{p} [1 + \frac{3}{4} (\tau_{g} - \frac {2}{3})][/tex]
So what I did was take the integral of the first equation and try to work from there but I am having difficulty understanding what to do with the left hand side of the integral but here's what I have so far:
1) [tex]\frac{d T^4}{dr} = - \frac{3 \kappa \rho}{a c} F_{rad}[/tex]
2) [tex]\int^{photosphere}_{ground} d T^4 = - \frac{3}{a c} F_{rad} \int^{photosphere}_{ground} \kappa \rho dr[/tex] (because F_rad is constant)
3) [tex](T^4_{g} - T^4_{p}) = - \frac{3}{a c} F_{rad} ( \int^\infty_{ground} \kappa \rho dr - \int^\infty_{photosphere} \kappa \rho dr)[/tex]
4) [tex](T^4_{g} - T^4_{p}) = - \frac{3}{a c} F_{rad} (\tau_g - \frac{2}{3})[/tex]
5) [tex](T^4_{g} - T^4_{p}) = - \frac{3}{4} T_{g}^{4} (\tau_g - \frac{2}{3})[/tex]
I know I'm close and it's not complete and I'm stuck... Did I make a mistake anywhere or what? If anyone could help me out that'd be great.
Thanks!
 
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  • #2
Could you be more explicit about what you're trying to show?
 

1. What is the greenhouse effect?

The greenhouse effect is a natural process that occurs in Earth's atmosphere, where certain gases (such as carbon dioxide, methane, and water vapor) trap heat and warm the planet's surface. This helps regulate the Earth's temperature and keep it within a habitable range for life.

2. How does the greenhouse effect contribute to climate change?

The increase in greenhouse gases, mainly carbon dioxide, in the atmosphere due to human activities, such as burning fossil fuels, has enhanced the natural greenhouse effect. This results in more heat being trapped, causing the Earth's temperature to rise, leading to global warming and climate change.

3. What are the main sources of greenhouse gases?

The main sources of greenhouse gases are human activities, such as burning fossil fuels for energy, deforestation, and industrial processes. These activities release large amounts of carbon dioxide and other greenhouse gases into the atmosphere, contributing to the greenhouse effect.

4. What are the potential consequences of the enhanced greenhouse effect?

The enhanced greenhouse effect can lead to various consequences, including rising sea levels, more frequent and severe weather events, changes in precipitation patterns, and disruptions in ecosystems. It can also have a significant impact on human health, agriculture, and economies.

5. How can we reduce the effects of the greenhouse effect?

To reduce the effects of the greenhouse effect, we can take actions to decrease our greenhouse gas emissions. These include using renewable energy sources, such as solar and wind power, improving energy efficiency, and promoting sustainable practices. It is also essential to protect and restore natural ecosystems, which act as carbon sinks and can help mitigate the effects of the greenhouse effect.

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