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(Please excuse my English)

Earth without water

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?

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    $\begingroup$ I am sure I used less spherical than the Earth basketballs to play basketball some 30 years ago. Some of them were brand new. I am not sure the balls are much better today. $\endgroup$ – fraxinus Aug 17 at 7:08
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    $\begingroup$ The portrayed image, and related ones, can be found at many sites on the internet. The typical description of what that image portrays is not even wrong. $\endgroup$ – David Hammen Aug 17 at 9:35
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    $\begingroup$ @a_donda - It shows the geoid height variations, the difference between the geoid (a surface of constant gravitational+ centrifugal potential) and the best fit oblate spheroid that contain the same volume. The deviations between the two are small, ± 150 meters IIRC. $\endgroup$ – David Hammen Aug 17 at 12:52
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    $\begingroup$ Relevant xkcd What If? $\endgroup$ – agweber Aug 17 at 15:51
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    $\begingroup$ Bowling balls and billiard balls are supposed to be very smooth (ignoring the three holes on a bowling ball). Basketballs are intentionally unsmooth. There are patents galore on how to manufacture a large (29.5 inch circumference) ball so that it is "generally round", bounces well and predictably, but can be held with one sufficiently large hand. $\endgroup$ – David Hammen Aug 17 at 17:06
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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.)

Exaggerated image of the Earth's geoid height, with legend 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.

An image of the Earth with all of it waters gathered into a ball about 1365 km across hovering over the western US 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.

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    $\begingroup$ These are very interesting points, but in it's current form this doesn't answer the.question that was asked- $\endgroup$ – Emil Aug 17 at 7:30
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    $\begingroup$ @pdh0710 - It is not your fault. It is the fault of the people who created that image, and even more so, it is fault of the many people who have incorrectly promulgated that image across the internet. No matter what exaggerations one applies, this is not how the Earth looks, with or without water. $\endgroup$ – David Hammen Aug 17 at 9:16
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    $\begingroup$ So interestingly, the Himalaya would have about the height of one of the basketball pebbles! $\endgroup$ – Peter - Reinstate Monica Aug 17 at 10:23
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    $\begingroup$ @jpa - IIIRC, Phil Plait ("the Bad Astronomer") compared the Earth to a billiard ball and wrote that the roundness of the Earth is well within spec, but that the smoothness was not. Those who have played pool in a bar or a pool hall know that when racking the balls, one has to rotate the balls on the corners of the rack just right so that they roll towards the other balls in order to result in a nice tight rack. If one doesn't do tha, there will be a few balls that move a lot the break and a cluster of balls that don't move much at all. The table isn't flat and the balls aren't quite round. $\endgroup$ – David Hammen Aug 18 at 18:15
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    $\begingroup$ @Spencer Himalaya is 8km/12700km = 6.3E-4 , and the basketball pebble is 0.17mm/238.8mm = 6.7E-4. $\endgroup$ – Peter - Reinstate Monica Aug 18 at 22:11
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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.

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  • $\begingroup$ I have found patents galore regarding basketballs. Patents on basketballs go back almost 100 years. The first one I found dates back to 1928. Yet very few recent basketball patents cite any prior art (what???). Not a single one of those patents I read specified how deep the channels are or how large the pebbles are. I'm used to reading patents that give exquisite detail that would let me recreate the object in question if I had the requisite skills and equipment. That's not possible with the basketball patents I have read. $\endgroup$ – David Hammen Aug 17 at 9:09
  • $\begingroup$ BTW, welcome aboard the Astronomy Stack Exchange. Nice answer. $\endgroup$ – David Hammen Aug 17 at 9:28
  • $\begingroup$ And with the migration, welcome aboard the Earth Science StackExcange! $\endgroup$ – David Hammen Aug 17 at 13:01
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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.

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    $\begingroup$ Yes, the Earth is an oblate spheroid, but barely. It's much closer to a perfect sphere than to a mandarin, which is visibly oblate. $\endgroup$ – CJ Dennis Aug 18 at 5:22
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    $\begingroup$ I can see without measuring it that the mandarin is oblate. I can't see without measuring that the Earth is oblate. To the naked eye, the Earth looks perfectly spherical. $\endgroup$ – CJ Dennis Aug 18 at 9:02
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    $\begingroup$ The mandarin in your picture is visibly oblate. The Earth is not visibly oblate. A visibly oblate basketball would not pass referee inspection. The Earth shrunken to the size of a basketball most likely would pass inspection on the basis of roundness. $\endgroup$ – David Hammen Aug 18 at 9:37
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    $\begingroup$ I would think the basketball on the ground would be an oblate spheroid as well. The Earth's oblateness is about 1/1000th it's diameter. Basketballs are inflated to 8 psi and are made of somewhat stiff material, but the weight would tend to squish it just a little. With a radius of 4.5 inches looking down the circular area would be about 64 square inches. Very roughly 512 pounds of pressure tending to overcome gravity and keep it spherical. However the weight of 22 ounces would tend to flatten it, making it oblate. The ratio is 1/372, about 3 times the Earth's oblateness. $\endgroup$ – Jason Goemaat Aug 18 at 21:08
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    $\begingroup$ @JasonGoemaat And here I thought to recall that the Earth's oblateness was 1/298, so surprisingly close to your back-of-the-envelope oblateness of a basketball $\endgroup$ – Hagen von Eitzen Aug 19 at 6:04

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