this work in progress
can be found at

http://robertwolfe.com/maa-prep

Taking Earth's Temperature

Robert W. Wolfe & Mary Jane Wolfe
Rio Grande, OH

Introduction

High among today’s concerns are the potential environmental and societal impacts of global warming. The study of global warming is rooted in two major areas of research: (1) analysis of massive amounts of historical climate data and (2) development of complex, multidimensional mathematical models requiring some of the world’s largest, fastest supercomputers.

Using some simple equations, concepts, and assumptions, we can model the earth’s temperature, as well as the temperatures of other planets.

Scientific Basis

 Note: Definitions for words that appear in italic type can be found in an online glossary.

At earth’s mean distance from the sun—149.6 million km—the solar energy flux is 1.367 kW/m2. This measured quantity is called the solar constant. By making some assumptions about how earth absorbs this solar power, one can calculate the planet’s temperature. In the simplest model, the planet is considered to be a blackbody. A blackbody is one that is a perfect absorber, absorbing all radiation that falls on it. A blackbody is also a perfect radiator. The amount of solar power that would fall on a spherical body, e.g., earth, would be equal to the cross-sectional area of the planet times the solar constant, S0 (1)

All matter at a temperature above absolute zero radiates energy. The higher the temperature, the more energy is radiated. For a blackbody, according to the Stefan-Boltzman Law, the amount of energy emitted is proportional to the product of a constant—the Stefan-Boltzman constant, s, which has the value 5.67 x 10-8 Watts m-2 K-4—and the fourth power of the body’s temperature, T4.

Because spherical body would radiate over its entire surface, the total radiated energy would also be proportional to the surface area, 4pR2, resulting in the expression (2)

If the planet is in a state of equilibrium, neither heating nor cooling, the energy absorbed must be in equilibrium with the energy radiated or emitted. In mathematical terms, expression (1) must equal expression (2) (3)

This equation can be solved for the blackbody planet’s temperature, T0, which is called its effective temperature. Once you have solved equation (4) for T, check your work by placing your mouse pointer (cursor) in the chartreuse button below.

Modeling Earth’s Temperature

 To perform the calculations below, you should download the Microsoft Excel spreadsheet, PlanetData.xls. NOTE: If you do not have Excel, you can view an image of the spreadsheet here, print it, and perform the calculations with a calculator, preferrably a graphing calculator. The blue, underlined planet names are links to more data on the planets at NASA’s Goddard Space Flight Center. You will use these links later.

Your solution of equation (3) for T0 should have been (4)

Blackbody earth

Calculate the blackbody temperature of earth by entering the expression to the right of the equal sign in equation (6) as a formula in the spreadsheet cell representing earth's effective blackbody temperature (D8). This is the temperature the earth would be if it were a blackbody—which of course it is not.

some questions designed allow studentst to focus their thought and assess their understanding

Reflective earth

To refine our model, we need toconsider the fact that the planets are not blackbodies, but that they reflect some of the incident solar radiation (that’s why we can see them).

How would you expect a planet's temperature to be influenced by the fact that some of the incident solar radiation is reflected back into space? In the box below, point to the line below that correctly completes the sentence.    The fraction of incident solar radiation that is reflected by a planet is termed its albedo (A). For modeling a planet’s temperature, it is the amount of energy absorbed that is of interest. The fraction of solar radiation absorbed is (1-A), and we must modify expression (1) (5)

and the equation for temperature becomes (6)

Enter the expression to the right of the equals sign as a formula in the spreadsheet (column ‘EARTH’, row ‘Effective temperature, with albedo, K’), using the appropriate Bond albedo.

some questions designed allow studentst to focus their thought and test their understanding

Temperatures of Other Planets

Scientists develop numerical models to explain how our planet behaves, and to predict how it will behave under different conditions. The validity of these models can be tested by seeing if they will explain how earth behaved in the past, or whether the the models can predict how how the planet behaves in the very near future. But as varied as earth is, it does not match the vast array of conditions on the other planets of our solar system. So scientists turn their attention outward to learn not just how our climate works, but how climates work, not just how earth’s atmosphere works, but how atmospheres work.

The solar constant of other planets

Astronomers and planetary scientists refer a planet’s distance from the sun in terms of the astronomical unit (AU), which is the ratio between the planets mean distance from the sun and the earth’s mean distance.

In the spreadsheet low labeled Mean distance from the sun, AU, enter formulas to calculate each planet’s distance in terms of the earth’s distance, i.e., earth’s distance = 1.00.

Calculate solar constant for each planet based on earth’s solar constant and the planet’s distance from the sun using inverse square law

Sidebar on inverse square law

Calculate blackbody temperature for each planet

Plot T as a function of AU

Calculate each planet’s temperature accounting for albedo

Plot this T as a function of AU

Compare plots

Consider emissivity

Model earth's emissivity