Comparing the Earth with a basketball?

Question

(Please excuse my English)

The picture above is an image of the Earth without water. We know that the picture above is a very exaggerated one. If the Earth were to shrink to the size of the picture above, it would look like a very smooth and round sphere.

I wonder what it would look like if the Earth were reduced to the size of a basketball. I think the basketball-sized Earth will have a smoother surface than actual basketball. But I don't know how smooth the surface of the basketball-sized Earth will be.

What would the basketball-sized Earth look like? Will it look like a marble ball with shinning smooth surface?

Answer 1

The picture above is an image of the Earth without water. We know that the picture above is a very exaggerated one.

About the image

The image shown in the question is very widely claimed to show an image of the Earth without water. A google image search for that image comes up with the nice logo "earth form without water". This image pops up on the internet again and again. This image does not show an Earth without water, even greatly exaggerated.

Note that the image in the question has a fuzzed out title and legend. All of the images on the internet that claim this is an image of the Earth without water have a fuzzed out title and legend. A non-spinning version of that image with the correct title and legend is below. (I do not like reading around animated gifs.)

^{Source: Bezděk, Aleš, and Josef Sebera. "Matlab script for 3D visualizing geodata on a rotating globe." Computers & Geosciences 56 (2013): 127-130.}

The above image is by the author of the image. The title says what the image shows, which is the geoid height. The legend shows that the variation from the highest to lowest geoid height is less than 200 meters. Compare that with the 19.777 km altitude difference between the highest mountain above sea level and the deepest trench below sea level. The image shows the Tibetan plateau as below the reference ellipsoid and shows the Java trench (not visible in the above) as above the reference ellipsoid. This is not a world without oceans.

So what does the image show?

What the image does show is the geoid height, greatly exaggerated. A geoid is the surface of constant gravitational plus centrifugal potential that is the best fit in a least squares sense to mean sea level. In other words, the geoid portrays the Earth's gravitational field. A simpler model of mean sea level is an oblate ellipsoid. The geoid height at some point is the altitude difference between corresponding points on the geoid and on the reference ellipsoid. Red areas in the image (positive geoid height) show where the geoid is above the reference ellipsoid while blue areas (negative geoid height) show where the geoid is below the reference ellipsoid.

Look at the Tibet Plateau in the rotating image. It is blue, which means the geoid there is below the reference ellipsoid. Look at the Java Trench. It is the very deepest red, which means the geoid there is above the reference ellipsoid. The Rocky Mountains are white, no different than the rest of the northern North America. The Alps are reddish, but not as red as is the eastern North Atlantic.

The Tibet Plateau shows up as a depression in this image instead of as a large elevated land mass because it is are very large blob of rock that has less than average density. The rock near the Java Trench is very dense basalt.

The image does not represent anything close to what many say it represents. The image does represent what the author intended it to represent. The image is interesting to those who know what the image portrays. The problem is that the image is so often described incorrectly.

An image of the Earth without water

An image of the Earth without water, slightly exaggerated, is shown below.

^{Source: : Jack Cook, Woods Hole Oceanographic Institution, Howard Perlman, USGS, Astronomy Picture of the Day 2012 May 15}

The Earth is round rather than lumpy. For portrayal purposes, a ball of water about 1365 km across hovers over the western US. That ball represents all of the Earth's surface waters.

Comparison to a basketball

The difference between the Earth's equatorial and polar radii is about 21.385 km. The difference between the height of the highest mountain on the Earth above sea level and the Earth's deepest trench below sea level is about 19.777 km.

Shrinking these down to the size of a basketball with a circumference of 29.5 inches (I used US units because that is what NBA regulations specify) shrinks the 21.385 km difference between the equatorial and polar radii to a mere 0.4 millimeters, shrinks the height of Mount Everest above sea level to 0.165 millimeters, and shrinks the depth Challenger Deep below sea level to 0.204 millimeters.

A basketball is supposed to be round(ish). That the Earth's polar radius is 99.66% of its equatorial radius makes the Earth fairly "roundish". The roundness of the Earth is in line with the roundness of a billiard ball, and probably for a basketball. (While I did find roundness requirements for a billiard ball, I was unsuccessful in finding roundness requirements for a basketball.)

What about smoothness? Basketballs are not smooth. They are intentionally designed with stipples (pebbles) and channels so as to make basketballs easier to handle. There are patents galore regarding the size, shape, and placement of those pebbles and channels. The pebbles are a bit higher than 0.17 millimeters and the channels are significantly deeper than 0.2 millimeters. The Earth is smoother than a basketball because basketballs are unsmooth by design.

Answer 2

The deviations of Earth from a sphere are of order 20 km. This approximates both the difference between equatorial and polar radii (defining an ellipsoid) and the difference between the highest and lowest points (Mount Everest and Mariana Trench) relative to the ellipsoid. A basketball is about 50 million times smaller than Earth, and the corresponding deviations on a basketball would be 0.4 mm.

The depth of the main grooves or "ribs" on a real basketball is about 2 mm. I am unable to find a source for the thickness of the numerous bumps or "pebbles" on a real basketball, but it is plausibly close to 0.4 mm. Thus, the bumps that cover the ball are somewhat to scale with the largest 20 km features of Earth, though of course Earth does not have such density of 20 km features.

Answer 3

A basketball is a sphere while Earth is an oblate spheroid, meaning Earth's equatorial radius is larger than its polar radius (as mentioned already in David Hammen's answer and nanoman's answer, I have just introduced the name for this phenomenon).

What exactly is an oblate spheroid?

If you take an ellipse and rotate it around its major (longer) axis you get a prolate spheroid, much like an Australian football:

If you rotate the ellipse around its minor (shorter) axis, you get an oblate spheroid, more like a lentil, M&M chocolate, or mandarin fruit:

Here are the oblate and prolate spheroids side-by-side:

This is a sphere:

.

Earth is an oblate spheroid, so a basketball is not the best model. A mandarin fruit is better.