Calculating albedo and eccentricity

In summary, the Earth would need an eccentricity of 1.39*10^11 in order to maintain a temperature equilibrium with irradiation from the Sun.
  • #1
Chief17
2
0
I have a three part question:

Background: For a planet on an orbit with semi-major axis a and eccentricity e, the distance of closest approach to the Sun is r = a(1 − e) and the farthest approach is r = a(1 + e).

(1) Assuming an albedo A = 0.2, estimate the temperature on Earth in equilibrium with irradiation from
the Sun. Estimate the correction factor necessary due to the greenhouse effect to bring us up to a balmy 300 K.

I don't know what to do.

(2) Assuming that this correction factor does not change, how large an eccentricity could the Earth have before the temperature extremes reach the point where the Earth reaches either boiling or freezing point?

Freezing:

Based off the equation T=T⊙((1-A)/4)^0.25(R⊙/r)^0.5,

273.15=5778((1-0.2)/4)^0.25(6.96*10^8/r)^0.5

0.005=(6.96/r)

r=1.39*10^11

Boiling:

373.15=5778((1-0.2)/4)^0.25(6.96*10^8/r)^0.5

r=7.46*10^10

These both seem reasonable to me except for one thing. The actual distance from the Sun to the Earth is 1.5*10^11. This means that the Earth is actually farther than the distance I calculated for the boiling part, which doesn't make sense.

Is this the equation I should use, and is the work (and answer) correct? Or did I do something wrong?

(3) If we define habitability as having a level of irradiation between these two extremes, consider the
habitable zone around a lower mass star. Assuming circular orbits again, and the same greenhouse correction factor as above, where is the habitable zone around a 0.5M⊙ star, which has radius 0.5R⊙ and effective temperature 3700 K?


Not sure where exactly to get started here.

Do I use the equations

L=4*pi*R⊙^2*stefan-boltzmann constant*T⊙^4
Labs=((R⊙^2*stefan-boltzmann constant*T⊙^4*pi*R^2)/r^2)(1-A)?

Not sure where the mass of the star fits in here.
 
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  • #2
(1): You have the formula in (2), you just have to plug in numbers for the first part. And then find out which prefactor you need in the equation to get the actual average temperature.

Chief17 said:
The actual distance from the Sun to the Earth is 1.5*10^11. This means that the Earth is actually farther than the distance I calculated for the boiling part, which doesn't make sense.
That comes from correction factor you have to include.

(3): right
Chief17 said:
Not sure where the mass of the star fits in here.
You don't need it.
 
  • #3
Thanks for the reply.

mfb said:
And then find out which prefactor you need in the equation to get the actual average temperature.

Not sure what you mean by prefactor. Would I just multiply the albedo times a value (i.e. x) and then solve for x? So:

300=5778((1-0.2x)/4)^0.25(6.96*10^8/(1.5*10^11))^0.5

----> x=-1.75
 
  • #4
I'm sure you have a definition of the correction factor somewhere.
 

1. What is albedo and why is it important?

Albedo is the measure of a surface's reflectivity, or how much light it reflects compared to how much it absorbs. It is important because it affects the Earth's energy balance and has an impact on climate and temperature.

2. How is albedo calculated?

Albedo is calculated by measuring the amount of incoming solar radiation and the amount of reflected radiation. The albedo is then calculated by dividing the reflected radiation by the incoming radiation.

3. What factors influence albedo?

The albedo of a surface is influenced by factors such as color, texture, and composition. Darker surfaces tend to have a lower albedo, while lighter surfaces have a higher albedo. Smooth surfaces also tend to have a higher albedo than rough surfaces.

4. What is eccentricity and how does it affect the Earth's orbit?

Eccentricity is a measure of how elliptical an orbit is. A perfectly circular orbit has an eccentricity of 0, while a highly elliptical orbit has an eccentricity close to 1. The Earth's orbit around the sun is slightly elliptical, with an eccentricity of 0.0167. This affects the Earth's distance from the sun, which in turn affects the amount of solar radiation it receives.

5. How is eccentricity calculated?

Eccentricity is calculated using Kepler's First Law, which states that the ratio of the distance between the foci of an orbit to the length of the semi-major axis is equal to the eccentricity. It can also be calculated using the Earth's aphelion (farthest point from the sun) and perihelion (closest point to the sun) distances.

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