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Jumping Frog Radius

|< < Prev #3181 (December 15, 2025)

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

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 right 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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