3181: Jumping Frog Radius

Jumping Frog Radius

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Title text: Earth's r_jf is approximately 1.5 light-days, leading to general relativity's successful prediction that all the frogs in the Solar System should be found collected on the surface of the Earth.

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Explanation

This is one of 55 incomplete explanations: This page was created by an A frog stuck on mars. I have added a bit about the drawing. It is important I think that the planet with the frog has exactly the rjf radius. This means the frog cannot escape but just barely. Is there a physics relation behind the fact that the jumps height seems to be very close to the radius of the planet, i.e. rjf? Also Can someone calculate the size and mass of the largest object from which a champion frog can achieve escape velocity? Are there some named asteroids that are of sow low a mass that it would be possible for frog to jump of? (Of course there are some small enough... but do any of them have real names, like the one named after Randall)? Don't remove this notice too soon. If you can fix this issue, edit the page!

The Schwarzschild radius is essentially the size of a black hole -- the maximum distance from the center where gravity is so strong that light can't escape.

It is part of a solution to Einstein's field equations. It is usually calculated as

r_s = (2*G*M) / c²

where G is the gravitational constant, M is the mass of the object, and c is the speed of light.

If M were the mass of the Earth, it would give the Schwarzschild radius for the Earth, which is about 9 mm. (If all of Earth's mass were compressed into a sphere of a bit less than 2 cm in diameter, it would become a black hole.)

The comic suggests a more useful radius: the Jumping Frog radius r_jf, which is the size of a "planet" such that its gravity keeps a champion jumping frog from being able to achieve escape velocity. Thus Randall has instead of c, the 299,792,458 m/s speed of light, used a much smaller value of 4.5 m/s, to represent the maximum speed of a jumping frog. It is possible that Randall got that value from this paper, which on page 179 puts an upper limit on the maximum velocity of adult Australian striped rocket frogs at 4.52 m/s. (The frog is shown making a "ribbit" sound, which is made by Pacific tree frogs and their relatives in North America and not by rocket frogs, but it's widely attributed to frogs all over the world.)

The drawing to the right of the formula shows a planet with exactly the radius r_jf. Thus the frog can jump really high compared to the planet's size (in this case about as high as the planet's radius), before it unavoidably falls back down, since the small planet is just massive enough to prevent the frog escaping.

The title text points out that the r_jf of the Earth is about 1.5 light days, which is about 7 times the distance to Pluto (compare to the 9 mm Schwarzschild radius). Since Earth's radius is much smaller than this, no frogs will be able to escape, so all frogs that stray into Earth's gravitational well would collect here on Earth. As far as we know, all the frogs in the Solar System are on Earth^{[citation needed]}, so the data apparently matches the theory. However, the reasoning is incorrect, as many other astronomical bodies in our solar system also have r_jf greater than their physical radius. If a frog were to be on any of those other bodies, it wouldn't be able to jump away to fall to Earth. A flawed argument neither supports nor refutes the conclusion, although it is true as far as we know that all frogs in the solar system do live on Earth.

Transcript

[The panel shows a large formula to the left and a small drawing to the right. The formula's right side is drawn above and below the division line:]

r_jf = 2GM / (4.5 ^m/_s)²

[The drawing to the left shows a very small planet with the radius indicated with a labeled dotted arrow pointing from the center straight up to the edge of the planet. A frog is shown jumping on the surface. This is indicated with a parabolic dotted line going from a frog sitting on the surface near the top of the planet, up to the frog shown soaring through the air with its limbs stretched out about as high above the surface as the planet's radius. At this point the frog is making a sound. Then the dotted line goes down to about a quarter of the way around the planet where the frog lands making a noise, with lines around the frog representing the impact.]

Arrow label: r_jf

Frog: Ribbit

Landing: Plop

[Caption below the panel:]

More practically useful than the Schwarzschild radius, the Jumping Frog Radius is the radius at which an object's gravitational pull is so strong that even a champion jumping frog can't escape.

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Discussion

firstQwertyuiopfromdefly (talk) 05:17, 16 December 2025 (UTC)

Question: Would a correct interpretation be "if a champion jumping frog were to be located just under 1.5 light-days from earth, and if there we're no other gravitational bodies nearby, and if said frog then performed its mightiest jump directly away from earth, then the frog would eventually be overcome by Earth's gravitational field and would eventually land on Earth's surface"? Pgn674 (talk) 06:26, 16 December 2025 (UTC)

I guess that is exactly how it should be interpreted. Or more interesting if it was just outside this radius and somehow could gain exactly 4,5 m/s extra speed then it would escape Earth (if there was anything to push of against that was heavy enough to move basically only the frog forward, then that would change the mass behind the frog so... That was why I wrote gain exactly rather than jump). --Kynde (talk) 07:36, 16 December 2025 (UTC)

or its mightiest jump in any direction (that doesn't cause it to crash through the Earth) since the escape speed is the same in all directions (relevant xkcd:https://what-if.xkcd.com/68/ ) --178.197.223.163 09:21, 16 December 2025 (UTC)

The only two variables are rjf and M, so plotting a 2 axis graph plotting the relationship between M and rjf should be possible. Zabadoh (talk) 08:20, 16 December 2025 (UTC) ^{[You sign after your contribution]}

As frogs usually collect on the surface of worlds ^{[citation needed]}, the *surface* escape velocity is most important. The crossover point for a planet with earth-like density (5515 kg/m³) is 2.6km, above that, the rjf falls below the surface, and the planet can accumulate frogs. Smaller bodies are, however, usually less dense; an interesting borderline candidate is Chicxulub, which had an rjf of 3-4km, and a radius of 5-6km so could have just about held onto its frogs, for a while at least. JeffUK (talk) 10:04, 16 December 2025 (UTC)

It would be interesting to look at the R_jf values of a frog, to consider where new limits are put upon the frog for M-masses that aren't totally dominating the scenario of "frog leaves mass"... 82.132.237.93 11:03, 16 December 2025 (UTC)

I interpreted it as a reference to the Mark Twain short story The Celebrated Jumping Frog of Calaveras County. Gustaveeiffel314 (talk) 12:25, 16 December

I also suspected an allusion to Twain's short story, but then I read it at archive.org/details/celebratedjumpin00twai and found no parallels. The earth's radius wasn't the problem, it was 5 pounds of quail shot. That frog didn't land with a "plop" but "as solid as a gob of mud." There is no mention of "champion" in the story. The 1865 population of Calaveras County (post Gold Rush) was down below 15,000. That is, the frog shown in #3181 probably came from somewhere else that really knows how to breed frogs with muscular legs, maybe France. Before I risk overthinking this, I'm going to conclude that #3181 is not a Twain reference. Bismuthfoot (talk) 14:37, 16 December 2025 (UTC)

What's with all that text in the incomplete explanation warning box? It seems like it belongs in the discussion. Barmar (talk) 15:05, 16 December 2025 (UTC)

Erm, the current text has a statement that rjf < 4.5m/s for other planetary bodies. Seems like it is mixing measurements, a radius would be a distance, not a velocity. It might be trying to say that other planetary bodies have an ESCAPE VELOCITY of more than 4.5 m/s, so jumping frogs on the surface of those planetary bodies couldn't get out of that planet's gravity well. ~~ 57.140.32.36 (talk) 15:53, 16 December 2025 (please sign your comments with ~~~~)

Don't recognise your statement (until I check the current state of the main explanation), but a radius can be defined as a vector, as can a velocity. Pretty sure that's not what it says (or should be saying), but there is a possible interchangability if analysed in the 'right' way. 82.132.237.93 17:00, 16 December 2025 (UTC)

(ETA: Nope, can't see where "the current text has a statement that rjf < 4.5m/s for other planetary bodies" - Unless I'm missing some obscure reference to it that you're not!) 82.132.237.93 17:04, 16 December 2025 (UTC)

It might be worth pointing out that frogs found on the surfaces of other planets in our solar system will have other reasons for not being able to jump to escape velocity (eg., they are no longer alive) 2A09:BAC2:6188:123C:0:0:1D1:CF 01:20, 17 December 2025 (UTC)

A frog does not have to be alive to jump, it could be a mechanical one. SDSpivey (talk) 02:44, 17 December 2025 (UTC)
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