(a) Estimate the temperature of the sun's surface given that...

(a) Estimate the temperature of the sun's surface given that the sun subtends an angle $\theta$ as seen from the earth and the earth's surface temperat ure is $T_0$. (Assume the earth's surface temperature is uniform, and that the earth reflects a fraction, $\varepsilon$, of the solar radiation incident upon it). Use your result to obtain a rough estimate of the sun's surface temperature by putting in "reasonable" values for all parameters. (b) Within an unheated glass house on the earth's surface the temperature is generally greater than $T_0$. Why? What can you say about the maximum possible interior temperature in principle?

(a) Estimate the temperature of the sun's surface given that the sun subtends an angle $\theta$ as seen from the earth and the earth's surface temperat ure is $T_0$. (Assume the earth's surface temperature is uniform, and that the earth reflects a fraction, $\varepsilon$, of the solar radiation incident upon it). Use your result to obtain a rough estimate of the sun's surface temperature by putting in "reasonable" values for all parameters.
(b) Within an unheated glass house on the earth's surface the temperature is generally greater than $T_0$. Why? What can you say about the maximum possible interior temperature in principle?

Key Concepts

Stefan-Boltzmann Law

This law relates the power radiated per unit area of a blackbody to the fourth power of its temperature. In contexts like estimating the temperature of the Sun or any radiating body, it provides the fundamental relationship between energy flux and temperature, forming the backbone of the energy balance calculation used in these problems.

Blackbody Radiation

The assumption that objects such as the Sun and the Earth (or its surface) behave like blackbodies simplifies the problem by allowing the use of idealized emission laws. This concept is central to determining how much radiation is emitted based on temperature and, conversely, estimating temperature from observed radiative flux.

Radiative Equilibrium and Energy Balance

These concepts refer to the condition where the energy absorbed by a system equals the energy it emits. In the context of the problem, balancing the incident solar energy (modified by geometric factors and reflection) with the thermal emission from the Earth's surface or the interior of a glasshouse is crucial for estimating equilibrium temperatures.

Geometric Dilution and Angular Size

When radiation travels from a distant source like the Sun, the flux decreases with distance, which can be understood by the subtended angular size of the source. The angle determines the fraction of the total radiated energy intercepted, making geometric factors essential in relating the solar surface temperature to the energy received on Earth.

Albedo and Reflectivity

Albedo is a measure of how much incoming radiation is reflected by a surface rather than absorbed. In the problem, the fraction ? representing the reflection of solar radiation by the Earth is directly tied to albedo, impacting the energy balance and, consequently, the surface temperature estimation.

Greenhouse Effect

The greenhouse effect explains how certain materials (like the glass in a greenhouse) allow the passage of shortwave solar radiation while trapping longwave thermal radiation emitted from the Earth’s surface. This trapping of heat results in interior temperatures that exceed the ambient external temperature, defining the maximum possible interior temperature under passive conditions.

Transcript

00:01 In this question, we are trying to model the climate of earth based on just a thermal equilibrium that would exist between the sun and the earth respectively.

00:18 So the equilibrium arises because we have the sun providing energy based from its radiation and the earth itself emitting radiation on the earth, in all directions as well, right? because it is a black body.

00:42 We're considering it to be a black body in this case, right, where it emits radiation based on its temperature, given in the stapha -bosman's equation.

00:59 Right? so the first part of this question is to find how much energy the earth is receiving from the sun.

01:10 So this amount of energy.

01:13 And we can do that by multiplying the intensity of the radiation coming from the sun with the area intercepted by the earth.

01:23 And this area, you can see because the sun rays are parallel would be the amount of area that is intercepted by earth is simply a circle.

01:39 A circle from if you are to view from the sun's perspective.

01:48 It's just a circle with radius equals to the radius of our earth.

01:59 This is area intercepted is pi r square.

02:05 And so the area that must be multiplied by our intensity would be just pi r square to give us the total power absorbed by earth.

02:24 It's just s times piar square.

02:34 Now assuming that the earth reflects 30 % of the solar energy, we're going to find the rate at which earth absorbs energy.

02:45 So rates of absorption would be s times pi r square, but only times 70%, because the 30 % is reflected.

03:02 So basically just one minus the albedo.

03:09 Alright, so this is the rates of absorption.

03:19 For the final part, we'll find the temperature at which earth radiates the energy at the same rate.

03:26 We equate this to the rate of emission by our earth, because at thermal equilibrium, there shouldn't be any change in the heat, amount of heat.

03:43 And therefore, the incoming energy must be equals to the outgoing energy.

03:50 The outgoing would be biol radiation in all direction, right, from the earth.

04:00 Assuming that the emissitivity is 1 for earth, then this is just sigma times, the surface area of our earth, which will be 4 pi square, times the temperature to the power 4...

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