Calculation of the Earth's Effective Temperature.
The effective temperature of a planet is the temperature it would have if
it acted like a black body, absorbing all the incoming radiation received
at its surface and reradiating it all back to space.
With the following facts you can easily calculate the effective
temperature of the Earth and, if you know the distance to the other planets,
their effective temperatures as well.
Distance from Earth to Sun = 150x10^{6}
kilometers.
Radius of the Sun = 6.96x10^{8} meters.
Radius of the Earth = 6.4x10^{6} meters.
Surface area of a sphere = 4pr^{2}.
Area of a circle = pr^{2}.
p = 3.1416.
Maximum output from the Sun occurs at a wavelength of 0.5 microns.
Lets walk through the steps together:
- Since we know the wavelength of the Sun's maximum output, we can use Wein's
law to calculate the Sun's temperature. Go ahead and do it. You should get
a number near 5796 K. For convenience you can round this off to 5800 K.
- If we know the Sun's temperature, we can calculate the intensity of the solar output (I, flux per square
meter), by using the Stephan Boltzman equation. Go ahead and calculate it.
(To calculate the fourth power of a number with your hand calculator square
it twice.)
- To calculate the Sun's total output (total flux or luminosity, L)
we must multiply the intensity (I, flux per square meter) by the total
number of square meters on the Sun's surface. Use the radius of the sun and
the equation for the surface area of a sphere, given above, to do this.
- Now, since energy is conserved, any sphere of any radius with the Sun at
its center will receive the same total radiation from the Sun as is emitted
from the Sun's surface. Therefore the intensity of solar radiation at the
Sun-Earth distance is related to that distance by the intensity to distance relationship you learned
about earlier.
The Earth intercepts a small part of the total radiation that arrives
in its neighborhood of the solar system (its distance from the Sun). If
we could hold an enormous sheet of paper on the other side of the Earth
from the Sun and orient it so the plane of the paper is perpendicular to
the rays of the Sun, then the earth would cast a shadow on the paper and
the area of that shadow would be the amount of solar radiation the Earth
intercepts. This circular shadow will have a radius equal to the radius
of the Earth. If you calculate the area of the shadow and multiply it by
the intensity of the solar beam at the Earth's distance from the sun, you
will have the total energy the Earth intercepts. Now the Earth rotates like
a chicken on a rotisserie and is heated on all sides. Therefore, the area
that receives the radiant energy from the Sun is not that of a disk, like
the shadow of the Earth on the paper, but that of a sphere which has a different
surface area. Calculate the average radiation (Watts/square meter) the Earth's
surface receives from the Sun.
- Now you have arrived at the last step. Since you know the
average intensity of the radiation the Earth's surface receives from the Sun
(and if the Earth is a black body it will reradiate the same amount to space),
you can use the Stephan-Boltzman equation to calculate the effective temperature
of the Earth.