If the Terrestrial Poles Were to Melt...

Problem:
If the terrestrial poles were to melt, then estimate how much the earth's oceans would rise.

Solution:
If the poles melted, then

First, let us estimate the total volume of the south polar ice. We begin by noting that, on the Earth, 1 degree of latitude corresponds to a distance of 69.2 mi. Now, using a well marked map, Antarctica may be estimated as having an angular radius of 18 degrees of latitude, or a radius in miles of r_A = 69.2 mi/deg x 18 deg = 1.25 x 10³ mi. Its estimated surface area is then r_A² = 4.91 x 10⁶ mi². If we assume the average thickness of the Antarctic ice to be 1 mi, then the total ice volume is

If this ice melts, the resulting volume of liquid water is 4/5 of this value or

Now, let this volume of water be spread in a uniform, thin shell over a uniform Earth (for the moment, we will ignore the continents). The total surface area of the Earth is A_E = 4r_E² = 1.97 x 10⁸ mi², where r_E = 3.96 x 10³ mi. Since the thickness of this shell, r_E, may be assumed to be << the Earth's radius, r_E , we may write

3.93 x 10⁶ mi³ A_Er_E¹

or, r_E = 1.99 x 10^-2 mi = 105 ft.

If we now return the continents to their proper place, it is easy to see that the value of r_E must become larger. Since only 2/3 of the earth's surface area is ocean, the thickness of the water shell must be increased by a factor of about 3/2:

r_E = (3/2) x 105 ft = 158 ft. ²

Thus, we have estimated that if the polar ice were to melt, the Earth's oceans would rise by about 150 ft. Reference to a map showing land elevations allows us to determine that coastal areas and major river valleys (such as the Mississippi or the Amazon river valleys) would be flooded. The land distribution would appear somewhat differently than it does on the globe today. But the majority of land would still be above sea level.

Actually the volume of the water shell is V_Water = (4/3)(r₂³ - r₁³), where r₁ is the radius of the Earth prior to flooding (3.96 x 10³ mi), and r₂ is the radius of the Earth after flooding (r₂ = r₁ + r_E). We may simplify the above expression for V_Water to read V_Water = (4/3)(r_E³), where (r_E³) = (r₁ + r_E)³ - r₁³= 3r²r_E + 3r₁(r_E)² + (r_E)³ 3r²r_E, since r_E << r₁. Thus V_Water 4r²r_E = A_Er_E. This last expression is the one given in the text.

² The actual volume of Antarctic ice is 7 x 10⁶ mi³. Using this value rather than the value 4.91 x 10⁶ mi³ previously calculated, we find r_E = 225 ft., close to the values given in text books.