GEOMETRY

PROBLEM 2.

All the energy to meet needs on Earth, whether the energy is natural or synthetic, ultimately comes or has come from the Sun in the form of electromagnetic radiation. There has been much interest recently in using this radiant source of energy directly to supplement or supplant the existing power sources. Further, since our Sun is but one of many stars, it is of interest to compare its energy output with that of other celestial objects.

One measure of the total energy radiated by the Sun received at a unit area of the Earth's surface is called the "solar constant" (where radiation is summed over all wavelengths of the electromagnetic spectrum).

A radiometer flown on the Solar Maximum Mission (SMM) is able to measure accurately the intensity of solar radiation. SMM is a satellite in orbit around Earth at low altitude, and its measurements can be used to provide a good estimate of the solar constant.

The radiometer on SMM admits solar radiation through a small aperture whose area is 0.50 cm^2, and it measures the rate of entrance of this radiation accurately. The spacecraft attitude (pointing direction) is controlled so that the entrance aperture is perpendicular to the line of sight between SMM and the Sun.

a. Over one observation period, radiation entered the radiometer at the rate of 0.069 watts. What is the value of the solar constant, S, as determined by this observation? (Use an extra significant digit in the answer, since this quantity will be used in subsequent calculations.)

Solution:

S = 0.069 watts/0.50 cm^2 = 0.138 watts/cm^2

b.It is generally assumed that the Sun emits radiation uniformly in all directions. If this is true, calculate the total rate of energy radiation by the Sun.

Since the radiation energy rate measurement contains only two significant digits, we can use the Earth-Sun distance of 1.5 x 10^8 km as SMM's distance from the Sun (see Chapter 2, Problem 8). If the Sun emits uniformly in all directions, the total rate of energy radiation from the Sun is the product of the solar constant and the area of the sphere with radius 1.5 x 10^8 km, or 1.5 x 10^13 cm.

Letting

P = total rate of energy radiation from the Sun,
P = (S) (4pi x r^2 )
= (0.138 watts/cm^2) (4pi) (1.5 x 10^l3 cm)^2
= 3.9 x 10^26 watts.

c. The foregoing are typical values. Variations of approximately 0.05 percent have been observed at other times. How much do such variations affect S and P?

Solution:

Delta S = 0.05 x 10^-2 x 0.138 = 6.9 x lO^-5 watts/cm^2

Delta P = 0.05 x 10^-2 x 3.9 x 10^26 = 2.0 x 10^23 watts

(Note: These variations occur on a short time scale (day to day) and are thought to average to zero over a long time scale. A O.O5-percent systematic variation in solar radiation over a time scale of years could produce significant climate changes on Earth.)

d. In 1981, SMM lost pointing accuracy because of a component failure on the spacecraft. Suppose that the orientation of the spacecraft changed so that the line perpendicular (the normal) to the entrance aperture made an angle of 30° with respect to the Sun-SMM line, rather than being parallel to it. By how much would the radiation entering the radiometer be changed?

,b>Solution: For simplicity, let us assume the aperture is a square, ABCD (see Fig. 4.2), with side length a, where a^2 = 0.50 cm^2. Looking at this square edge-on with AD as the tilted edge, if DE is parallel to the direction of solar radiation incidence and AE is perpendicular to DE, the aperture is effectively a rectangle whose dimensions are AB and AE. We label the angles alpha, beta, gamma, and omega as shown. Since (angle alpha, angle beta) and (angle gamma, angle omega) are complementary pairs of angles, and since angle beta = angle gamma, we have angle omega = angle alpha = 30 degrees, so triangle ADE is similar to the standard 30 degrees -60 degrees -90 degrees triangle, and the ratios AD/2 , DE/1 , and AE/(3^.5) are equal, giving AE = [(3^.5)/2] AD = 0.866 a. The area of the effective aperture is therefore (0.866 a) (a) = 0.866 a^2. In other words, the radiometer will register only 87 percent of what it did before losing pointing control.

(We observe that the result holds for apertures that are not square.)

As observed in part (c) of the foregoing problem, one of the interesting outcomes of modern advances in the precision with which it is now possible to make measurements of the solar constant is that this quantity is in fact not really a constant!

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